Recent Trends in Naval Engineering Research (STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health) 3030641503, 9783030641504

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STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health

Anthony A. Ruffa Bourama Toni  Editors

Recent Trends in Naval Engineering Research

STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health

STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health Series Editor Bourama Toni Department of Mathematics Howard University Washington, DC, USA

This interdisciplinary series highlights the wealth of recent advances in the pure and applied sciences made by researchers collaborating between fields where mathematics is a core focus. As we continue to make fundamental advances in various scientific disciplines, the most powerful applications will increasingly be revealed by an interdisciplinary approach. This series serves as a catalyst for these researchers to develop novel applications of, and approaches to, the mathematical sciences. As such, we expect this series to become a national and international reference in STEAM-H education and research. Interdisciplinary by design, the series focuses largely on scientists and mathematicians developing novel methodologies and research techniques that have benefits beyond a single community. This approach seeks to connect researchers from across the globe, united in the common language of the mathematical sciences. Thus, volumes in this series are suitable for both students and researchers in a variety of interdisciplinary fields, such as: mathematics as it applies to engineering; physical chemistry and material sciences; environmental, health, behavioral and life sciences; nanotechnology and robotics; computational and data sciences; signal/image processing and machine learning; finance, economics, operations research, and game theory. The series originated from the weekly yearlong STEAM-H Lecture series at Virginia State University featuring world-class experts in a dynamic forum. Contributions reflected the most recent advances in scientific knowledge and were delivered in a standardized, self-contained and pedagogically-oriented manner to a multidisciplinary audience of faculty and students with the objective of fostering student interest and participation in the STEAM-H disciplines as well as fostering interdisciplinary collaborative research. The series strongly advocates multidisciplinary collaboration with the goal to generate new interdisciplinary holistic approaches, instruments and models, including new knowledge, and to transcend scientific boundaries. More information about this series at http://www.springer.com/series/15560

Anthony A. Ruffa • Bourama Toni Editors

Recent Trends in Naval Engineering Research

Editors Anthony A. Ruffa Naval Undersea Warfare Center Newport, RI, USA

Bourama Toni Department of Mathematics Howard University Washington, DC, USA

ISSN 2520-193X ISSN 2520-1948 (electronic) STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health ISBN 978-3-030-64150-4 ISBN 978-3-030-64151-1 (eBook) https://doi.org/10.1007/978-3-030-64151-1 Mathematics Subject Classification: 93-XX, 44-XX, 65-XX, 15-XX, 76-XX, 33-XX © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

This is a book about overcoming mathematical challenges, challenges representative of a variety of important Navy problems. Some years ago, I met Professor Toni at a technical review of research that I sponsor. One presentation in particular captured his keen interest. It focused upon a mathematical theorem with important implications for stability and optimality of sensor systems in the face of unavoidably limited objective field data. I was oblivious to the fact that for a couple of hours I was speaking with the Head of the Mathematics Department at Howard University – but not surprised later when I learned that he is. This is a book about broadly important things, not just Navy problems. Extracting information, at the highest level of fidelity, given the reality of limited, noisy data is the theme of this book that resonates with me. Improving the precision of balance between expectation and observation can begin with better models that reliably define higher fidelity expectations. Economic game theory can rely upon a mix of data and models to identify stable solutions for design and employment of complex systems, solutions with certifiable claims of quasi-optimality. Better materials produce higher fidelity measurements of real phenomena to produce better data – with more information about the things we care most about and less about noise, and ideally at lower cost. In a statistical sense, real data are of course not Gaussian. It is important to know when it is a Gaussian assumption that is controlling the fidelity of performance measurement, and vital to have a way to deal with that reality to know objectively both the limit of achievable performance and hierarchically the causes of limited performance. Washington, DC, USA July 2020

Michael Traweek

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Preface

This book, Recent Trends in Naval Engineering Research, is the second to feature the applications of mathematical and statistical sciences and their applications in the multidisciplinary STEAM-H (Science, Technology, Engineering, Agriculture, Mathematics and Health) series; the series brings together leading researchers to present their work in the perspective to advance their specific fields and in a way to generate a genuine interdisciplinary interaction transcending disciplinary boundaries. All chapters therein were carefully edited and peer-reviewed; they are reasonably self-contained and pedagogically exposed for a multidisciplinary readership. Contributions are invited only and reflect the most recent advances delivered in a high-standard, self-contained way in line with the goals of the series, that is: 1. To enhance multidisciplinary understanding between the disciplines by showing how some new advances in a particular discipline can be of interest to the other discipline, or how different disciplines contribute to a better understanding of a relevant issue at the interface of mathematics and the sciences 2. To promote the spirit of inquiry so characteristic of mathematics for the advances of the natural, physical, and behavioral sciences, and here naval engineering, by featuring leading experts 3. To encourage diversity in the readers’ background and expertise while at the same time structurally fostering genuine interdisciplinary interactions and networking Current disciplinary boundaries do not encourage effective interactions between scientists; researchers from different fields usually occupy different buildings, publish in journals specific to their field, and attend different scientific meetings. Existing scientific meetings usually fall into either small gatherings specializing on specific questions, targeting specific and small group of scientists already aware of each other’s work and potentially collaborating, or large meetings covering a wide field and targeting a diverse group of scientists but usually not allowing specific interactions to develop due to their large size and a crowded program. Here contributors focus on how to make their work intelligible, accessible to a diverse audience, which in the process enforces mastery of their own field of expertise. vii

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Preface

This volume, as the previous one on naval engineering, strongly advocates multidisciplinarity with the goal of generating new interdisciplinary approaches, instruments, and models including new knowledge, transcending scientific boundaries to adopt a more holistic approach. For instance, it should be acknowledged, following Nobel laureate and president of the UK’s Royal Society of Chemistry, Professor Sir Harry Kroto, “that the traditional chemistry, physics, biology departmentalised university infrastructures–which are now clearly out-of-date and a serious hindrance to progress–must be replaced by new ones which actively foster the synergy inherent in multidisciplinarity.” The National Institutes of Health and the Howard Hughes Medical Institute have strongly recommended that undergraduate biology education should incorporate mathematics, physics, chemistry, computer science, and engineering until “interdisciplinary thinking and work become second nature.” Young physicists and chemists are encouraged to think about the opportunities waiting for them at the interface with the life sciences. Mathematics is playing an ever more important role in the physical and life sciences, engineering, and technology, blurring the boundaries between scientific disciplines. The series, through contributed volumes such as the current one, is to be a reference of choice for established interdisciplinary scientists and mathematicians and a source of inspiration for a broad spectrum of researchers and research students and graduate and postdoctoral fellows; the shared emphases of these carefully selected and refereed contributed chapters are on important methods, research directions, and applications of analysis including within and beyond mathematics. As such, the volume implicitly promotes mathematical sciences, physical and life sciences, engineering, and technology education, as well as interdisciplinary, industrial, and academic genuine cooperation. The current book entitled Recent Trends in Naval Engineering Research as a whole certainly enhances the overall objective of the series, that is, to foster the readership interest and enthusiasm in the STEAM-H disciplines, stimulate graduate and undergraduate research, and generate collaboration among researchers on a genuine interdisciplinary basis. The STEAM-H series is hosted at Howard University, Washington DC, USA, an area that is socially, economically, intellectually very dynamic and home to some of the most important research centers in the USA. This series, by now well established and published by Springer, a world-renowned publisher, is expected to become a national and international reference in interdisciplinary education and research. Washington, DC, USA July 28, 2020

Bourama Toni

Acknowledgments

We would like to express our sincere appreciation to all the contributors and to all the anonymous referees for their professionalism. They all made this volume a reality for the greater benefits of the community of Science, Technology, Engineering, Agriculture, Mathematics, and Health.

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Contents

Introduction: Mathematical Sciences and Naval Engineering . . . . . . . . . . . . . . Anthony A. Ruffa and Bourama Toni

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Microlattice Materials and Their Potential Application in Structural Dynamics and Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lisa M. Dangora

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An Alternative Convolution Approach to the Cagniard Method for Transient Ocean Acoustic Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roy L. Deavenport and Matthew J. Gilchrest

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Acoustic Transmission in a Low Mach Number Liquid Flow . . . . . . . . . . . . . . . Scott E. Hassan Lift Production Using Differential Cavity Ventilation on a Symmetric Hydrofoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aren M. Hellum and David E. Yamartino An Exact Solution for a Class of Kalman Smoothers . . . . . . . . . . . . . . . . . . . . . . . . Anthony A. Ruffa and Tod E. Luginbuhl

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Metallic Foam Metamaterials for Vibration Damping and Isolation . . . . . . 123 Mark J. Cops, J. Gregory McDaniel, Elizabeth A. Magliula, and David J. Bamford The Other Navy Seals: Seal Whiskers as a Bio-inspired Model for the Reduction of Vortex-Induced Vibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Christin T. Murphy, William N. Martin, Jennifer A. Franck, and Joy M. Lapseritis A Series of Multidimensional Integral Identities with Applications to Multivariate Weighted Generalized Gaussian Distributions . . . . . . . . . . . . . 163 Anthony A. Ruffa and Bourama Toni

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Minimum Uniform Search Track Placement for Rectangular Regions . . . . 203 Richard D. Tatum, John C. Hyland, and Jeremy Hatcher Antennas in the Maritime Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 David A. Tonn The Destabilizing Impact of Non-performers in Multi-agent Groups. . . . . . 257 Thomas A. Wettergren Improving Inertial Navigation Accuracy with Bias Modeling . . . . . . . . . . . . . . 277 Tod E. Luginbuhl, Ahmed Zaki, and Eugene Chabot Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Contributors

David J. Bamford Naval Undersea Warfare Center, Division Newport, Newport, RI, USA Eugene Chabot Naval Undersea Warfare Center, Newport, RI, USA Mark J. Cops Department of Mechanical Engineering, Boston University, Boston, MA, USA Lisa M. Dangora Naval Undersea Warfare Center, Division Newport, Newport, RI, USA Roy L. Deavenport Naval Undersea Warfare Center, Newport, RI, USA Jennifer A. Franck Department of Engineering Physics, University of WisconsinMadison, Madison, WI, USA Matthew J. Gilchrest Naval Undersea Warfare Center, Newport, RI, USA Scott E. Hassan NAVSEA Newport, Newport, RI, USA Jeremy Hatcher Naval Surface Warfare Center Panama City Division, Upper Grand Lagoon, FL, USA Aren M. Hellum Naval Undersea Warfare Center, Newport, RI, USA John C. Hyland Naval Surface Warfare Center Panama City Division, Upper Grand Lagoon, FL, USA Joy M. Lapseritis Bio-Inspired Research and Development Laboratory, Naval Undersea Warfare Center Division, Newport, RI, USA Tod E. Luginbuhl Naval Undersea Warfare Center, Newport, RI, USA Elizabeth A. Magliula Naval Undersea Warfare Center, Division Newport, Newport, RI, USA William N. Martin Bio-Inspired Research and Development Laboratory, Naval Undersea Warfare Center Division, Newport, RI, USA xiii

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Contributors

J. Gregory McDaniel Department of Mechanical Engineering, Boston University, Boston, MA, USA Christin T. Murphy Bio-Inspired Research and Development Laboratory, Naval Undersea Warfare Center Division, Newport, RI, USA Anthony A. Ruffa Naval Undersea Warfare Center, Newport, RI, USA Richard D. Tatum Naval Surface Warfare Center Panama City Division, Upper Grand Lagoon, FL, USA Bourama Toni Department of Mathematics, Howard University, Washington, DC, USA David A. Tonn Submarine Electromagnetic Systems Department, Naval Undersea Warfare Center, Division Newport, Newport, RI, USA Thomas A. Wettergren Naval Undersea Warfare Center, Newport, RI, USA David E. Yamartino Naval Undersea Warfare Center, Newport, RI, USA Ahmed Zaki Naval Undersea Warfare Center, Newport, RI, USA

Introduction: Mathematical Sciences and Naval Engineering Anthony A. Ruffa and Bourama Toni

Mathematical modeling and analysis of system behavior are at the core of the development of high-tech system. Modern engineering sciences, to include naval engineering, call for more sophisticated mathematical and statistical methods in order to efficiently understand and resolve the increasing complexity of the tasks at hand. There is a need of a deeper understanding of the mathematical tools most needed to proper support naval research. Indeed, invariably, in the press to “get on with it,” highly complex issues are addressed by engineers using existing tools and knowledge to produce approximate methods and results that have not been rigorously substantiated and are thus subject to unintended consequences, most importantly, breakage/failure in unpredictable circumstances. Nowhere is this situation more chaotic than in the field of information extraction, signal and image processing, acoustics, and nonlinear vibration analysis where a “witches’ brew” of inadequately posed concepts, questionable/unrecognized assumptions, approximations of convenience, and heuristically defined metrics combined to yield what could be reasonably described as bordering on “technological sorcery.” Most mathematical models are multivariate/multidimensional: e.g., in science, finding a wave function of some electron cloud requires solving a Schrodinger’s equation in 276 dimensions. Multivariate problems, when handled by engineers and computational scientists, suffer oftentimes from the curse of dimensionality, mostly due to their limited training in sophisticated mathematics. Many engineers have even limited or hollow exposure to something as fundamental to their research as

A. A. Ruffa Naval Undersea Warfare Center, Newport, RI, USA B. Toni () Department of Mathematics, Howard University, Washington, DC, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_1

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advanced linear algebra and matrix analysis. Many data scientists have shockingly no expertise in topological data analysis, which requires training in topology and real analysis. Most applied sciences rely heavily, for their own intellectual comfort, on inadequate old age mathematical methods sometimes dating back to Newton, with the more and more inadequate real number systems. For example, still many scientists do not realize that ultrametricity is a natural property of high-dimensional spaces and emerges as a consequence of randomness and the law of large numbers and the p-adic number system, unknown to a lot of researchers, is closely associated with ultrametricity topology; a very interesting feature of such spaces and number systems is that they allow error-free computation; when carrying out data coding and analysis in very-high-dimensional spaces, ultrametric techniques allow to bypass Bellman’s curse of dimensionality, using different perspectives on nearest neighbor searching or best matching searching. That is, for instance, random data are increasingly ultrametric in proportion to the increase in spatial dimensionality and sparsity. For example, dendrogram is a convenient data structure for points in an ultrametric space. Search dimensionality is given by the ratio of mean to variance of given metric space distances, as a large and/or small variance of distances implies exponential increase in nearest neighbor searching, typical of high-dimensional spaces. When analyzing/investigating natural, physical, or social phenomena, the most common formal models are the mathematical model for real-world problems and the model of computation for computer simulation. Both require rigor and high degree of accuracy and should have advanced mathematics at their core. This book, Recent Trends in Naval Engineering Research, implicitly serves as a model of an efficient way to maintain a good balance between utility in the sense of being able to obtain necessary statistical parameters, mathematical tractability, and physical reality in the sense of adequately describing and addressing real-world conditions, in particular, related to naval applications. Of important note is the following with respect to the undersea naval research: the undersea domain is wholly the concern and responsibility of the Naval Service. This domain possesses two salient characteristics: It provides an operating environment immensely more complex than above-water conditions; to wit, acceptable characterizations of radio signals propagated through the atmosphere are different from the acoustic signals through the ocean. Also observe that the speed of sound is so much smaller than the speed of light and Doppler effects become much less negligible at the kind of velocities likely to be encountered in practice. In addition, the undersea domain is particularly “information starved”: compared to the above-water environment, almost everything propagates with high loss, i.e., high-frequency signals are strongly attenuated, which massively limits the potential sources for information content. Therefore, these characteristics put an absolute premium on the precise extraction of information in the undersea environment, as there is so little information to be had in the first place and actually correctly extracting it without waste so utterly complicated.

Introduction: Mathematical Sciences and Naval Engineering

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This book features in a single volume a variety of problems and approaches relevant to naval engineering. The contributors are seasoned experts in their field with a greater sense of the need of an interdisciplinary approach to resolving issues important to the navy. The book contains the following chapters and topics: The chapter on “Microlattice Materials and Their Potential Application in Structural Dynamics and Acoustics,” by Lisa Dangora, discusses some theoretical and practical topics pertinent to digital design of microlattice materials, including metamaterials, fabrication techniques, and test methods; such materials could be analyzed using the mathematical and computational method of finite element, a numerical technique of solving partial differential equations, by subdividing the larger system into simpler parts. In the chapter on “Acoustic Transmission in a Low Mach Number Liquid Flow,” Scott Hassan uses the finite element method to analyze the steady-state twodimensional acoustic velocity due to a piston source on a body in a low Mach number liquid flow, assuming inviscid and irrotational acoustic and hydrodynamic fields. The steady-state hydrodynamic field is computed using the boundary element method applied to the Laplace equation for incompressible flow. Steady-state pressures, obtained from the hydrodynamic solution, are used along with an equation of state to compute the local density. The local speed of sound is computed as a function of the steady-state pressures using an empirical expression along with the assumptions of constant temperature and salinity. Far-field acoustic beam patterns are obtained using the finite element solution to the acoustic velocity potential on the computational domain boundary and a special form of the Helmholtz integral applicable to acoustic propagation in a moving homogeneous medium. The case of a piston source located symmetrically about the hydrodynamic stagnation point on a rigid body is investigated as a function of Mach number, transmit frequency, and transmit angle. The far-field beam patterns were found to exhibit a strong Mach number dependence, with increases in both main lobe beam width and side lobe levels over the M = 0 case. Additionally, at the high Mach numbers, the beam width does not exhibit the classical monotonic decrease with frequency and was found to reach a minimum value. Further increases in frequency resulted in a monotonic increase in beam width. It was also found that for transmission off the main response axis, the far-field main lobe angle is greater than the specified transmit angle. These effects are due to an effectively inhomogeneous and anisotropic medium as a result of the sound speed and flow velocity gradients within the fluid. Deavenport and Gilchrest’s chapter proposes an “Alternative Approach to Cagniard Method for Transient Ocean Acoustic Modeling”; Cagniard method, based on convolution, was initially used by seismologists to treat transients in the earth; the authors adopt an alternate approach, also based on convolution and time-domain procedure, but without requiring homogeneous layers and allowing attenuation, to simulate a transient signal propagating in a shallow-water waveguide; attenuation is here necessary as the ocean is a low-pass filter with a nonlinear absorption. In both methods, one convolves an arbitrary source with the impulse response of the medium to determine the transient field. The significant difference between the two methods is the manner for obtaining the medium (system) impulse

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response. Cagniard derived his impulse function using integral transforms in the complex plane using a Heaviside step function source. In the present chapter, an alternative method is used to determine a bandlimited impulse function by calculating the inverse Fourier transform of a general Green’s function of the medium. The transient field is then found by convolving the bandlimited impulse with an arbitrary transient source. Example of a bandlimited impulse is given using the image solution transfer function for a Pekeris shallow-water environment. The chapter on “Lift Production Using Differential Cavity Ventilation on a Symmetric Airfoil” by Aren Hellum and David Yamartino considers the well-known phenomenon of cavitation on a lifting surface; the authors show that cavitation can be harnessed to produce predictable and useful control forces, by introducing noncondensable gas on one side of a submerged hydrofoil to produce an artificial cavity. Their data lead to the conclusion that a differentially vented foil at 0 degree angle of attack can produce a significant amount of lift equivalent to 4 degree angle of attack for a fully wetted version of the same foil. Luginbuhl, Tod, and Ruffa, Anthony, in the chapter “An Exact Solution for a Kalman Smoother” develop an exact solution for a Kalman smoother. The solution for a single nonzero right-hand side (RHS) term involves both exponentially growing and exponentially decaying solutions that meet at the update number corresponding to the nonzero RHS term. The envelope defined by these two solutions provides precise guidance for the optimum interval length for a given choice of the tracking index. This result is valid everywhere because of a translational symmetry with respect to the update number of the nonzero RHS term, provided that it is not close to the batch edges. A solution corresponding to an arbitrary RHS vector can then be constructed with an appropriate superposition of such solutions. The chapter by Elizabeth Magliula is on “Metallic Foam Metamaterials for Vibration Damping and Isolation.” Metallic foam metamaterials consist of an open-cell metallic microstructure completely saturated by a viscous material. The dynamic responses of metamaterials containing foams with two distinct pore sizes as well as six different saturating materials are measured in controlled laboratory settings. The vibration damping and isolation characteristics of the metamaterials are then analyzed using a lumped element approach. For all metamaterials tested, the saturating material increased the damping ratio due to increased viscous dissipation arising from metal foam ligaments interacting with the viscous fluid. For the bestperforming saturated foam subject to a transient excitation, an order of magnitude increase in damping ratio is observed, compared to the same off-the-shelf foam with no saturation. These results warrant further research into using metallic foam metamaterials for vibration damping and isolation applications, including optimization of the metal foam microstructure/pore size, and optimal selection for saturating material including other viscous fluids as well as rubbers or elastomers. “The Other Navy Seals: Seal Whiskers as a Bio-inspired Model for the Resolution of Vortex-Induced Vibrations” is a chapter by Christin Murphy, William Martin, Jennifer Franck, and Joy Lapseritis. Bio-inspired engineering looks to nature for inspiration for new technology or solutions to existing challenges. The Science and Technology research community for the US Navy has looked to bio-

Introduction: Mathematical Sciences and Naval Engineering

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inspired research to aid in a variety of design challenges for operation in the ocean environment. One particular model system is the whiskers of seals which have unique bumpy surfaces that aid in vibration reduction as these structures move through the water. Vortex-induced vibration (VIV) can occur when water flows over a bluff body. This is ubiquitous in operations in the marine environment and can create many engineering challenges. The unique morphology of the seal whisker may reduce VIV by disruption of the shed vortices. While vibrations are reduced, a small vibrational signal remains, which provides important sensory input to the seal. This study examines the spectral characteristics of the elicited vibrations from real seal whiskers under various flow conditions and explores the mechanism of their VIV reduction in relation to their geometry utilizing whisker geometry models. Anthony Ruffa and Bourama Toni develop in the chapter “A Series of Multidimensional Integral Identities with Applications to Multivariate Weighted Generalized Gaussian Distributions” new multidimensional integral identities using a novel substitution approach that involves at least two nested integrals, each with integration limits from 0 to 1. The identities are expressed in compact formulas in terms of common special functions such as gamma, Bessel, and hypergeometric functions. Applications include multivariate weighted generalized Gaussian distributions. Involved in the study are most special functions such as gamma function, Bessel function, hypergeometric function, and complementary error function. In the chapter “Minimum Uniform Track Placement for Rectangular Regions,” Richard Tatum, John Hyland, and Jeremy Hatcher study the problem of search track placement over a rectangular region that considers a priori target information to maximize coverage. Our approach allocates uniformly spaced parallel tracks over sections of homogeneous sensor performance. We solve the search planning problem as a MILP optimization problem that is subject to a priori target distribution constraints, as well as minimum sectional and total effective coverage constraints. Additionally, we provide a derivation of a general method to evaluate the effective coverage that remains agnostic to the assumption of dependence or independence between tracks, as well as account for the inevitable overlap of parallel tracks. We include simulation results that demonstrate the efficacy of our approach. “Antenna Behavior in the Maritime Environment” is a chapter by David Tonn that considers the basic principles of antennas and what effects the air-sea interface has on the behavior of the antenna and the electromagnetic field that it produces. Included are some applications in the areas of antenna measurement, and highlighted are some recent developments in antenna engineering involving the use of advanced materials. The world of wireless communications could not exist without antennas. Antennas form the basic interface that converts voltages and currents into radiated electromagnetic waves and vice versa and are essential components in all forms of radio communication, including systems that operate at or near the air-sea interface. The presence of the surface wave must be taken into account provided that the frequency of operation is below roughly the middle of the VHF band. Tonn also presents how recent developments in the field of anisotropic materials can be

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employed to improve the bandwidths of several basic antenna types, all of which have potential application to communications in the maritime environment. The chapter “The Destabilizing Impact of Non-performers in Multi-agent Groups,” by Thomas Wettergren, examines the question of how a small number of non-performing individuals can impact the performance of a large multi-agent group. In particular, Wettergren looks at models from the mathematical ecology community that describe the behavior of simple group aggregation by individuals that interact only through observation of their proximal neighbors. By taking this approach, the author limits his focus to the interaction and grouping effects of the aggregation phenomena, rather than looking at specific detailed behaviors of individuals. He considers non-performing individuals to be agents that do not follow the stated rules of interaction of the rest of the group, but are otherwise identical to the others. Numerical simulations are performed to demonstrate the resulting effects of the non-performing individuals on these groups. These types of effects must be considered when designing engineered systems that model emergent natural behaviors, as reliability limitations on physical systems will create some small numbers of non-performing agents. Ahmed Zaki, Tod Luginbuhl, and Eugene Chabot in the chapter “Improving Inertial Navigation Accuracy with Bias Modeling” study navigation-grade gyroscopes and accelerometers, well known to have bias in their output. This bias contributes significantly to the error in the time-evolving position solution of the inertial navigation systems (INS). The innovation in this work is the approach of modeling the bias at the solution level as linear function and estimating the bias using unbiased velocity measurement which is available in the sensor suite for underwater vehicles. Moreover, a model switching scheme is used to more accurately correct the position estimates when the vehicle maneuvers. Motion models for nearly straight line motion and turning models are used.

Microlattice Materials and Their Potential Application in Structural Dynamics and Acoustics Lisa M. Dangora

1 Outline This chapter will discuss theoretical and practical topics pertinent to digital design of microlattice materials, like those shown in Fig. 1. The work is not intended to be all-encompassing, but will provide a high-level overview of the general concepts and ideas. It opens by introducing the emergent class of metamaterial and identifying influential factors governing the response characteristics before moving onto a discussion of some potential naval application areas. The chapter then covers some common fabrication techniques and test methods for property evaluation. A section on computational techniques follows to provide details and instructions for analyzing these materials using the finite element method.

2 Nomenclature Asterisk Overbar Subscript c Subscript eff Subscript k Subscript s

Indicates either (1) property of a cellular solid or (2) a complex value (as specified) Average property Property of the core material in a sandwich composite Indicates an effective property Indicates dependence on wavenumber Property of the bulk solid material (continued)

L. M. Dangora () Naval Undersea Warfare Center, Division Newport, Newport, RI, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_2

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L. M. Dangora Superscript  Superscript  Superscript Re superscript Im a c f k t u v w B Cijkl E G G11 G12 H M N P R T TK V VRVE Z a b k r x x¨ ∼ x C F K M MAC S α γ ij δ δk εij

Storage modulus Loss modulus Real component of complex number Imaginary component of complex number Periodic distance Sound speed Frequency Angular wavenumber Time Displacement degree of freedom in the 1-direction Displacement degree of freedom in the 2-direction Displacement degree of freedom in the 3-direction Beam flexural rigidity Components of material stiffness in index notation Young’s modulus Shear modulus Auto-power spectrum Cross-power spectrum Transfer function Maxwell degree of determinacy Total number of elements Sound pressure Acoustic reflection coefficient Wave period Temperature Volume Total volume of the representative volume element Degree of connectivity Primitive translation vector of the direct lattice Basis vector of the reciprocal lattice Wavevector Position vector Displacement vector Acceleration vector Eigenvector, mode shape Material stiffness matrix Force vector Global stiffness matrix Global mass matrix Modal assurance criterion matrix Material compliance matrix Acoustic absorption coefficient Engineering shear strain Phase lag Attenuation constant Strain components in tensor notation

[m] [m/s] [Hz] [rad/m] [s] [m] [m] [m] [N·m2 ] [Pa] [Pa] [Pa] [Pa2 ] [Pa2 ]

[Pa] [s] [K] [m3 ] [m3 ] [m] [rad/m] [rad/m] [m] [m] [m/s2 ] [Pa] [N] [N/m] [kg] [Pa−1 ]

[rad] [rad/m] (continued)

Microlattice Materials and Their Potential Application in Structural Dynamics. . . Δ  ζ η λ λj ν ν ij ρ ρ

Used to indicate a change in value Phase constant Damping ratio, percent of critical damping Loss factor, loss tangent, loss coefficient, damping factor Wavelength jth eigenvalue Linear wavenumber Poisson’s ratio Density Relative density

σ ij ζj φ φj ω

Stress components in tensor notation Coordinate value in the j-direction Porosity Phase in j-direction Angular frequency, circular frequency, radial frequency

Octahedron

Hyper-Kagome

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[rad/m]

[m] [(rad/s)2 ] [m−1 ] [kg/m3 ] [Pa] [m] [rad] [rad/s]

Octet-Truss

Tetrahedron

Fig. 1 Lattice materials formed from differing base units (highlighted in orange)

3 Overview Through deliberate material arrangement – used to affect efficient manipulation of the load path – digitally-designed, truss-based cellular solids can be made simultaneously lightweight, stiff, and strong. By engineering the microstructure, lattice materials – like those shown in Fig. 1 – can therefore significantly outperform conventional stochastic foams in structural applications. Topologies of these advanced materials are created by spatial tessellation of a repeatable base unit to from a network of interconnected struts comprising the periodic, open-cell solid. Unlike lattice “structures” seen in architectural design – e.g., truss bridges and roof frames – the term “material” applies to microlattice because their base features exist on a scale much smaller than the macroscopic part lengths [1, 2]. The development of such highly-ordered, engineered materials has been facilitated by advances in additive manufacturing (AM). Nanolattice and microlattice – referenced here to denote characteristic strut diameters on the order of 10−9 –10−7 m and 10−6 –10−4 m, respectively – have transitioned from theoretical notions to tangible realities. Although production scalability is presently limited, the proof of concept has alluded to promising futures for these cellular solids. Accompanying

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400

search term: 'microlattice' search term: 'microlattice','stretch-dominated'

Number of Publications

350 300 250 200 150 100 50 0 1990

1995

2000

2005 2010 Publication Year

2015

2020

Fig. 2 Open literature publications in the field of microlattice materials (results compiled from Google Scholar)

the technological progress in fabrication techniques has been a correlated increasing interest in the materials themselves. As evident by the recent surge of publications (Fig. 2), the field is becoming more widely studied and is producing an expanding volume of literature. With AM offering precision and control over the microstructure, digitallydesigned solids are expanding the property space of available materials, and the limits of their potential remain to be seen. As such, numerous research efforts are underway to explore their behaviors and understand how to push the bounds of their performance. Some specific areas of focus have included fabrication methods and parent materials [3–5]; manufacturing repeatability, reproducibility, and sensitivity to tolerancing and imperfections [6, 7]; elastic characterization, collapse mechanisms, and computational methods for predicting such behaviors [8– 12]; vibrational [13] and acoustical properties [14–16]; topology optimization and reverse homogenization [7, 17, 18]; and applications (e.g., heat exchangers [19] and impact survivability [13, 20, 21]).

3.1 Influential Factors Because the microstructure is tailored, there are a large number of design parameters that can be controlled in constructing the engineered solid. The mechanical properties, however, are said to be most influenced by three critical factors: phase material, relative density, and connectivity [1].

Microlattice Materials and Their Potential Application in Structural Dynamics. . .

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Phase Material Cellular materials consist of two phases: (1) a solid constituting the frame struts or cell walls and (2) a fluid – gas or liquid – occupying the volume in between. For quasi-static applications of open-cell materials, the fluid’s contribution to stiffness and strength may be considered negligible; therefore, its mechanical properties are often directly derived from, and defined in relation to, those of the bulk material. Regarding rate-dependent behaviors, however, the porous solid will inherit viscoelastic properties (if any) of the parent material; but, because viscous damping will occur as deformation forces fluid from one cell to the next, viscosity of the pore fluid can be a significant contributor as well [22].

Relative Density Relative density, ρ, is frequently defined in open literature as the ratio of the cellular solid’s density, ρ * , to that of the bulk material from which it is made, ρ s ; however, the calculation of ρ * in such texts neglects the mass of the fluid phase. It is, therefore, more accurate to define relative density as the solid-phase volume fraction – i.e., the ratio of volume occupied by the solid phase to the total volume of the cellular material. Porosity, or void fraction, is thus given by φ = 1 − ρ. As can be seen through this relation, when relative density decreases, the pore volume expands, and the cell walls thin out. Therefore, slenderness ratio – i.e., strut length to diameter – is also implicitly defined by relative density. Because members of the frame are more susceptible to buckle under load as slenderness ratio increases, the absolute strength of cellular solids degrades with decreasing relative density [1].

Connectivity Deformation mechanics of lattice materials are governed by the topology of their microstructure. Despite having joints that are neither hinged nor rigid, the mechanics of a lattice material can be closely related to that of a pin-jointed framework [1], for which Maxwell’s stability criterion identifies the degree of determinacy. Determinacy is quantified through the relation M = b – 3j + n where b and j are the number of struts and joints, respectively, and n = 3 for 2D systems or n = 6 for 3D systems [23]. If M < 0, the pin-jointed frame is classified as a “mechanism” rather than a “structure.” To relate this metric to lattice materials, imagine locking a four-bar frame at its joints; if M < 0, the stability criterion is not satisfied, and the struts will bend under application of external load (Fig. 3a). If instead M = 0, such as from the addition of a crossbar (Fig. 3b), the determinate condition is met, and, although the struts will still see some bending, the frame is a “structure” that deforms primarily by extensional strain of its members; note M > 0 characterizes a redundant structure. Cellular solids can, therefore, be broadly classified by their

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determinacy as either bend-dominated or stretch-dominated [25]. The reader is referred to a review paper by Fleck et al. [2] for a comprehensive overview of these deformation mechanisms and taxonomy.

Classification: Stretch-Dominated Versus Bend-Dominated Standard foaming processes are used to manufacture conventional foams, like that shown in Fig. 3c. In addition to producing random pore sizes and void distributions, the process tends to result in a low degree of frame connectivity – with, on average, only three or four cell walls merging at a single node (i.e., Z = 3 or Z = 4) – to minimize surface energy during bubble expansion of the injected gas phase [8]. Consequently, stochastic foams are typically compliant materials that deform primarily through bending of cell walls. The bend-dominated deformation mechanisms result in a parabolic relationship, causing mechanical properties to degrade rapidly as density decreases (refer to the equation in Fig. 3c relating relative stiffness and density). While compliance of bend-dominated microstructures generally results in high capacity for energy absorption, their low stiffness and strength restrict suitability in structural applications [1]. Conversely – because the microstructure is controlled in digitally-designed solids – lattice materials (e.g., Fig. 3d) can be engineered to behave more rigidly by facilitating extensional and compressive straining of the struts. When the microstructure is designed with truss-based networks for efficient load-carrying capabilities, engineered cellular solids can exhibit a nearly constant specific stiffness. On a pound-for-pound basis, this produces a material with exceptional stiffness and strength as compared to most other media. Although the study of stretchFig. 3 As illustrated by the simplified frame schematics, the primary deformation mechanisms in cellular solids are described by (a) bending or (b) tension of the cell walls; (c) stochastic open-cell foams [24] are bend-dominated, whereas (d) lattice materials can be designed to be stretch-dominated

Microlattice Materials and Their Potential Application in Structural Dynamics. . .

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STRETCH-DOMINATED

Hyper-Kagome Octahedron Z=6 Z=8

Tetrahedron Z = 12

Octet-Truss Z = 12

BEND-DOMINATED

Tetrakaidecahedron (Kelvin Foam) Z=4

Fig. 4 Stiffness trends for unit cells of several lattice topologies constructed from the same base material; discrete point values obtained via finite element analysis

dominated microlattice is still an emerging research field, as indicated by the low publication numbers in Fig. 2, the materials being produced are beginning to occupy formerly uncharted regions of Ashby property diagrams [2]. Figure 4 provides a visual aid for the comparison of such stretch- and benddominated lattice materials. The line’s slope on the log-log plot defines a power law describing the proportionality relationship between relative stiffness, E* /Es , and relative density,ρ. Five unit cell architectures are shown: one bend-dominated and four stretch-dominated. The bend-dominated Kelvin foam – a space-filling tetrakaidecahedron commonly used to approximate the microstructure of stochastic foams – has a low degree of connectivity that allows “soft-mode” deformation mechanics to drive the compliant response; a slope value of two confirms its classification and indicates a quadratic decay in stiffness with decreasing density. The four stretch-dominated topologies instead exhibit a more linear relationship between specific stiffness and density, with the trigonal lattices being most efficient; cubic lattices, i.e., octahedron and octet-truss, are slightly more compliant. Hyper-kagome (Z = 6) and tetrahedron (Z = 12) microstructures offer the stiffest responses per unit mass; although they perform similarly, the tetrahedral configuration provides a more redundant structure that is less sensitive to imperfections and eccentricities in truss alignment. The property curves begin to converge not far above 10% relative density, as the sections exhibit more of a bulk response and loading becomes more complex. That is to say, as the frame members thicken, the lattice struts no longer behave as slender beams, which support only axial and bending stress; shear and torsional deformations are not negligible.

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4 Naval Interests The Department of the Navy (DON) has identified Advanced Materials and Manufacturing Technologies as a key strategic investment area in its “30 Year Research and Development Plan” [26]. Metamaterials, such as microlattice, fall within this overarching category; these are broadly described as materials with microstructures engineered to augment properties of the bulk solid – from which they are made – to produce behaviors not otherwise naturally occurring. Investments in this research field can help to maintain maritime superiority by developing tools and cultivating a knowledge base that offer a competitive advantage over other naval powers. Metamaterials are therefore recognized as critical technologies for sustaining and advancing capabilities of the US military power. Metamaterials that offer suitable functionality in several application areas are particularly desirable and can be employed to integrate multiple subsystems of larger assemblies. Microlattice, and cellular solids in general, possess significant potential for such multifunctionality; some practical uses include combinations of load bearing, impact energy absorption, acoustic and noise control, vibration damping, electromagnetic shielding, and thermal insulation [1, 27]. Microlattice show promise to be a solution across multiple platforms, but of primary interest is their combined suitability in structural, buoyancy, vibration, and acoustic applications.

4.1 Structural and Buoyancy Maintaining vehicle buoyancy can be a challenge for subsea applications. When operational requirements demand survivability at great depths, deep-diving marine vessels must withstand significant external pressures. Monocoque designs are often untenable for such purposes; to prevent implosion under substantial hydrostatic loads, they require thick shell walls, adding parasitic mass that decreases net buoyancy. Sandwich composites – which are commonly substituted in structural applications for meeting stringent size, weight, and stiffness requirements – are a suitable alternative to single-shell constructions. Figure 5 illustrates the benefit by comparing monolithic and composite beam sections. For little added mass, a lightweight core material can separate high-stiffness face sheets from the neutral axis – where bending stress is null – thereby increasing flexural rigidity. The improved bending stiffness offered by dual-wall shells increases the critical buckling pressure to enhance structural stability of the vessel and help the design comply with operational requirements. Digitally-designed materials can produce ultra-low-density cores that efficiently transfer load between the face sheets. Therefore, microlattice could become prime candidates as core materials in sandwich constructions. Stretch-dominated deformation mechanics of certain lattice configurations allow these materials to surpass

Microlattice Materials and Their Potential Application in Structural Dynamics. . .

(a)

(b)

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(c)

Fig. 5 Replacing (a) monolithic constructions by (b) sandwich structures can (c) increase bending rigidity

(a)

(b)

Fig. 6 (a) Importance of various submarine noise sources as a function of frequency and speed at periscope depth (adapted from [28]) and (b) schematic illustrating radiated noise and self-noise resulting from UUV shipboard machinery

the performance of conventional composite cores by offering greater stiffness and higher collapse strengths per unit weight [9]. Microlattice cores can also serve to improve survivability during impacts and underwater explosions (UNDEX). The energy from these events can be expended through collapse mechanisms, plastic work, and fracture energy to maintain structural health of the overall vehicle. Microlattice have high promise for absorbing such substantial dynamic loads through permanent deformation, but they are additionally of interest for vibration suppression, which concerns only small strains and recoverable deformations.

4.2 Vibrations and Acoustics Vibration-induced sound from shipboard machinery is a primary source of selfnoise and radiated noise in underwater vehicles (refer to Fig. 6a) [28]; some contributors in this area can include the main propulsion plant, exhaust system, coolant pump, propeller shaft, maneuvering systems (e.g., steering, bow and stern planes), reduction gears, and other auxiliaries. Figure 6b illustrates the concept of noise contamination using an aft-situated rotating component, such as a motor or

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Fig. 7 Microlattice have potential for effective acoustic quieting by absorbing sound waves to prevent (a) strong surface reflections; performance may be further enhanced by designing them as (b) functionally graded materials or (c) waveguides

propulsor. Vibrational energy from such mechanisms can transfer – via a direct transmission path through the hull – to the Guidance and Control Section where it degrades signal excess of the onboard sonar. Alternatively, the energy can radiate from the hull to the surrounding marine environment and – via a fluid path – reach the hydrophones or transducers to produce the same effect. Furthermore, disturbances that reach the fluid-structure interface generate pressure waves that can propagate in the water column and disclose the vehicle’s location to listening adversaries. Microlattice may be employed in such vibroacoustic applications as an isolation material – for example, in machine mounts – to decouple vibrating components from the structural hull. If these materials can be used for vibration suppression, they would serve to reduce the vehicle noise signature, thereby improving stealth and effective sensor range. Furthermore, as their porous nature facilitates absorption of incident waves, digitally-designed lattice materials have potential to reduce, or even redirect, reflected signals from external sonar sources that actively probe the marine environment. For example, hierarchical design architecture may be employed to create functionally graded microlattice with a fluid-matched acoustic impedance on the surface that gradually increases as the wave entrains deeper into the material (Fig. 7b); a broadband absorption may be achieved by funneling acoustic waves in this manner. Alternatively, a microstructure could be designed to create preferential paths for energy transport in the material, thus producing an effective acoustic waveguide (Fig. 7c); therefore, when pressure waves from an active source impinge on the submerged object, the angle of reflection differs noticeably from the angle of incidence. To physically realize such constructions, control and precision in the fabrication technique is essential.

5 Fabrication Methods 5.1 Additive Manufacturing As has been previously mentioned, advances in additive manufacturing are enabling development and fabrication of these emergent metamaterials. The process initiates by drafting the microlattice architecture in a digital framework with computer-aided

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design (CAD) software. Surfaces bounding the continuous geometry are discretized into a collection of triangles and normal vectors that may be exported and read for printing. The meshed surface is then sliced – in discrete steps of constant thickness – along the build direction, and the AM production equipment deposits material one layer at a time to produce the desired 3D shape; printer hardware dictates limits on layer thickness and resolution. There are numerous AM techniques that can be used to fabricate digitally-designed materials, but two of the most common and scalable methods are selective laser sintering (SLS) and stereolithography (SLA). For a comprehensive list and discussion of AM processes, the reader is referred to review papers by Vaezi et al. [29], Huang et al. [30], and Wong and Hernandez [31]. Selective laser sintering is a powder-based method that uses a laser to melt and fuse particles together. SLS can therefore be used to fabricate digitallydesigned materials from certain thermoplastics, metals, and combinations thereof. The production occurs in a temperature-controlled chamber to keep the raw material near its melting point. A thin layer of powder is deposited on a build platform, and a laser beam scans a pattern to produce a cross-sectional slice of the 3D geometry. The platform is then lowered by the layer thickness, and the procedure is repeated – i.e., a fresh coat of powder is deposited, the next cross-sectional slice is traced by the laser, and the build plate is lowered – until the full print has been erected. This process is self-supporting, which makes it amenable for delicate constructions; the unfused powder supports the weight of free-standing sections until the build is complete and the cooled geometry is removed from this excess. Stereolithography is similar to SLS except it is a liquid-based method rather than a powder-based one. The process uses a liquid photopolymer, for which crosslinking of the polymer chains is catalyzed by an ultraviolet (UV) light source. In scanning SLA, a laser is used to cure a solid cross-sectional profile from liquid along the beam’s track; in projection SLA, the entire cross-section is projected and cured at once. The build platform moves by the thickness of one layer, and the liquid is replenished to repeat the cycle until the complete geometry has formed. The microlattice can then be post-cured after removal from the printer.

5.2 Other Methods Concurrent Curing via Self-Propagating Photopolymerization While useful for precision and control of the microstructure, AM techniques can be very slow and difficult to scale [30]. Furthermore, tolerancing errors in the layer-bylayer deposition process can introduce eccentricities that disrupt the load path and weaken the microstructure. Interstitial bonds and interlaminar strengths of the strata system – which develops from the sequential curing of each layer – may also limit performance of additively manufactured microlattice [32]. A “Self-Propagating Photopolymer Waveguide” technique – not to be confused with “acoustic waveguides” – was developed at HRL Laboratory to overcome

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Fig. 8 Self-propagating photopolymer waveguide process for fabrication of microlattice materials (adapted from [3])

these standard AM deficiencies [3]. The fabrication process – depicted in Fig. 8 – begins with a reservoir of liquid monomer that, as in SLA, can cure through photopolymerization. A mask, patterned with holes, covers the reservoir, and collimated UV light is incident on the mask at angles consistent with producing the desired truss alignment. UV beams are able to transmit through holes in the mask, and liquid in the light’s path solidifies. The solid microlattice frame can then be removed from the excess liquid monomer and post-cured in an oven. This process can be used to rapidly cure near-net-shape three-dimensional objects with truss-based microstructures [33]. Its production scale and speed can make manufacture of engineered solids a more practical venture. Moreover, since curing occurs concurrently in the beam paths, this method eliminates eccentricities along the truss axis; it also produces no interlaminar regions, which could otherwise introduce catalyst sites for failure. It should be noted, however, that this concurrent curing technique is presently limited in compatibility with certain cell architectures; for example, the hyper-kagome and Kelvin foam in Fig. 4 could not be fabricated by these means. Conventional AM methods therefore offer more flexibility, especially for constructing hierarchical designs and for creating acoustic waveguides.

Hollow Microlattice Both AM and concurrent curing techniques can be used to produce sacrificial scaffolds for construction of hollow microlattice. This is accomplished by coating the polymer truss network with a ceramic or metal, e.g., via atomic layer deposition [34] or electroless plating [4]. The polymer can then be chemically removed or etched out to leave only a hollow ceramic or metal frame. Such a method has been used to produce some of the lightest man-made materials, with reported densities as

Microlattice Materials and Their Potential Application in Structural Dynamics. . .

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low as 0.9 kg/m3 (assuming the mass of the fluid phase is ignored) and solid-phase volume fractions ranging from 0.001% to 0.3% [35].

6 Property Evaluation For engineering purposes, it is generally considered sufficient to characterize cellular solids from a macroscopic perspective; therefore, standard test methods that have been developed for homogenous materials (e.g., ASTM D5592 [36], MIL-HDBK755 [37]) are often employed in experimental programs to evaluate “apparent” properties of porous materials. However, deviation from the customary specimen sizes may be necessary due to the relative scale of heterogeneity [38]. While there are many properties to describe material behaviors, the brief discussion here is limited to those most relevant for the principal application areas identified in Sect. 4 (i.e., buoyancy, structural, vibration, and acoustic). For additional information, the reader is referred to the writings of Banhart [27], Jaouen et al. [39], and the references cited therein.

6.1 Physical Properties Physical quantities defining mass and volume are used in the design of buoyancy solutions. Because microlattice are digitally-designed materials, physical properties – such as density, mass distribution, porosity, specific surface area, and general tomography – may be easily evaluated computationally without the need for testing. Imaging techniques, e.g., optical microscopy and computed tomography, can be used for quality assurance of the printed parts.

6.2 Mechanical Properties Mechanical properties are used to determine suitability for structural applications. Unlike conventional foams – where variability across sample populations may produce significant experimental scatter [40] – lattice materials display statistically repeatable properties; this uniformity can be attributed to the controlled, periodic microstructure offering a consistent distribution of mass and stiffness. The material may be tested in tension [41, 42], compression [43, 44], bending [45, 46], and shear [47, 48] and evaluated quasi-statically, at higher strain rates (e.g., via split Hopkinson pressure bar), at lower strain rates (i.e., creep under constant load), or for fatigue. Quasi-static uniaxial compression tests are, by far, the most frequent employed in experimental programs [49]; such test configurations offer the most

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simplicity. Furthermore, compressive modes are generally the most common seen in-service [50]. Additional standard core-testing methods [51–53] are appropriate in characterizing microlattice for application in sandwich composites.

6.3 Damping To evaluate materials for application in vibration isolation, their damping properties must be assessed. Viscoelastic properties may be characterized using a dynamic mechanical analyzer (DMA), which is an instrument capable of measuring material stiffness as a function of time, temperature, and frequency [54]. Different mounting fixtures are available to test in compressive [55], tensile [56], bending [57, 58], and shear modes [54]; a typical configuration used for compression testing is shown in Fig. 9a for reference. A sinusoidal forcing function is applied during testing to induce strain. The amplitudes of force and displacement are used, in conjunction with the phase shift between them (Fig. 9b) and the specimen geometry, to calculate a complex modulus, E*, for the material; the real component is termed the storage modulus, E , and the imaginary component is the loss modulus, E, such that E* = E + iE [59]. The derived dynamic modulus is dependent on the test mode; for example, when the shear test configuration is used, complex shear modulus, G*, is evaluated. The damping factor can be determined from the phase lag, δ, which is the product of harmonic frequency, ω, and the time delay between maxima of the applied and measured signals, Δt (refer to Fig. 9b). The tangent of this phase lag – i.e., tan δ – is the loss factor, η, which is interchangeable as the loss tangent, loss coefficient, or damping factor. Damping can alternatively be specified as a damping ratio or percent of critical damping, ζ , which is roughly half of the loss factor at resonance [60].

(a)

(b)

Fig. 9 (a) Compression test configuration for viscoelastic characterization on a dynamic mechanical analyzer with (b) applied and measured signals shown

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Fig. 10 Time-temperature superposition principle for estimating viscoelastic response at frequencies beyond the direct measurement range of the dynamic mechanical analyzer

DMA testing is a frequently used characterization method because it is relatively quick, simple, and inexpensive [[59]; it is not without shortcomings, however. These dynamic analyzers are limited in capacity for force transmittal. They also have a very limited dynamic range due to mount resonances and instrument inertia, which generally restrict the upper drive frequency to well below 200 Hz. The principle of time-temperature superposition (TTS) may be applied in some cases to extrapolate the response to frequencies outside of the machine limits. TTS is accomplished by cycling the specimen in an isothermal frequency sweep over a range of temperatures. The frequency-dependent response curves for each isothermal condition are then shifted temporally about data at a reference temperature [61]. The TTS process is depicted in Fig. 10 for visualization; the curve at the reference temperature remains stationary, while higher-temperature curves shift lower in frequency, and lower-temperature curves shift higher in frequency until overlaid. Note, however, the time-temperature equivalence principle is considered applicable over small deformations for homogenous, isotropic, amorphous materials [62]; therefore, the behavior profiles generated via TTS may not always be accurate representations for cellular solids. Furthermore, for DMA testing to be appropriate, the scale of heterogeneity must be sufficiently small – relative to the sample size – to display a representative response [54]. Similar experimental setups can be configured with an electromagnetic shaker, force transducer, and accelerometer to (1) test larger specimen sizes, (2) characterize stiffer materials, and (3) obtain direct-frequency measurements over a broader range; details for such alternate configurations can be found in [27, 39].

6.4 Acoustic Absorption An impedance tube may be used to evaluate the acoustic performance of cellular solids. Using a broadband white noise excitation and the transfer function between two sensors, the frequency-dependent normal-incidence absorption coefficient, α, may be evaluated quickly and precisely [63]. The simple experimental configuration is illustrated in Fig. 11.

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Fig. 11 Impedance tube test configuration for measurement of normal-incidence acoustic absorption coefficient

An impedance tube is instrumented with two microphones connected to a data acquisition system (DAQ). The tube itself must be thick-walled and rigid enough to prevent external noise from contaminating the measurement. The material sample is placed at one end of the duct and backed by a rigid plunger; note the specimen should fit snugly in the tube’s hollow cross-section. The openness of lattice materials facilitates a volume-dominated absorption, which produces a spectral behavior that is relatively insensitive to orientation; therefore, it is not critical to test both sides (i.e., front and back) [27]. A speaker at the opposite end is fed by signal generator and used to produce broadband noise; the resulting spherical pressure wave travels along the length of the impedance tube. The microphones and test specimen must be a sufficient distance away from the sound source such that the signal may fully develop as plane waves; then, with negligible multidirectional components to the noise, the system physics are simplified and may be analyzed in a one-dimensional context. Sound pressure is measured at the two discrete microphone locations and recorded by the DAQ as a function of time. Post-processing algorithms decompose the digitized pressure signal into its incident and reflected components so that the absorption can be quantified. This process entails (1) removing sensor drift and bias; (2) calibrating for atmospheric conditions and test configuration parameters; (3) performing linear block averaging; (4) computing the fast Fourier transform (FFT) of the sound pressure time series data; (5) calculating the auto- and crosspower spectra, G11 and G12 ; (6) computing the acoustic transfer function, H; (7) correcting for the phase and amplitude mismatch between the two microphones; and

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(8) calculating the complex reflection coefficient, R, and the absorption coefficient (as in Fig. 11) as a function of frequency. Note the usable frequency range of the measured spectral response is governed by the spacing between microphones as well as the tube length and diameter [63].

7 Computational Methods When comparing candidate materials for a particular application, it can often be more efficient and cost-effective to explore the property space using computational techniques rather than to develop and execute a full experimental program. Numerical methods – e.g., finite difference (FD), finite volume (FV), and finite element (FE) – are used to analyze complex geometries that cannot be adequately defined by closed-form analytical solutions. Such approximate methods accommodate geometry and material behaviors that are completely general. The finite element method (FEM), in particular, is most widely employed. As such, the FEM will be the basis of the content discussed here. Numerical homogenization techniques of computational micromechanics can be implemented to derive effective mechanical properties of periodic materials at a macroscopic level [64]. This can be useful in evaluating microstructures for various applications or tailoring the cell architecture to meet a desired functionality. It is also beneficial – and often necessary – to model heterogeneous materials as homogenous continuums when performing structural analyses at the “part” or “assembly” level. The mesh resolution required to discretely consider each mesoscopic feature at a macroscopic scale would present an arduous problem that either (1) exhausts memory and storage capacity of the computing hardware or (2) requires impractical solution time; therefore, discretization of the microstructure on such a scale is computationally prohibitive. Although the mechanical response is local at fine spatial resolutions – e.g., mesoscopic, microscopic, and atomic levels – the global behaviors concerning general engineering applications can effectively be considered spatially invariant. Homogenization is therefore accomplished by analyzing a single representative volume element (RVE). The RVE is a small volume of the heterogeneous material for which macroscopic constitutive relations are sufficiently accurate to describe a mean physical behavior that is statistically repeatable over a broader volume [65]. For periodic microstructures, it is convenient to select the unit cell as this RVE. The homogenization methods discussed herein for effective properties consider volume averages of the field variables – viewed as measurable macroscopic physical quantities – and assume compliance with the correspondence principle [66].

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7.1 Static The computational homogenization approach described in [64] is commonly employed for deriving static continuum properties of composites; the method is summarized here for completeness. Equations in this section are written with reference to the Cartesian material coordinate system shown in Fig. 12a and have adopted the index notation convention of the general-purpose finite element solver, Abaqus/Standard [67]. The constitutive relation for an orthotropic elastic medium is given by Eq. (1), which uses the material stiffness matrix, C, to relate the strain field and Cauchy stress state. The first three components in the left-hand-side array denote direct stress values along the principal material axes, as indicated by the subscript; the last three components of the column matrix define shear stresses in the respective subscripted orientation. Strain variables specified in the right-hand-side array result from a product of the Reuter matrix and tensor strains; as such, the components of shear are reported as engineering strain, γ , which relate to tensor strains as γ ij = εij + εji (for i = j). ⎧ ⎪ σ11 ⎪ ⎪ ⎪ ⎪ σ22 ⎪ ⎪ ⎨ σ33 ⎪ σ12 ⎪ ⎪ ⎪ ⎪ σ13 ⎪ ⎪ ⎩ σ23

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎡

C1111 C1122 C1133 0 0 ⎢ 0 0 C2222 C2233 ⎢ ⎢ 0 0 C3333 ⎢ =⎢ ⎢ ⎪ 0 C 1212 ⎪ ⎢ ⎪ ⎪ ⎪ ⎣ SY M C1313 ⎪ ⎪ ⎭

⎤⎧ 0 ⎪ ⎪ ⎪ ⎪ 0⎥ ⎪ ⎥⎪ ⎨ ⎥⎪ 0⎥ ⎥ 0⎥⎪ ⎪ ⎥⎪ ⎪ 0⎦⎪ ⎪ ⎪ ⎩

C2323

ε11 ε22 ε33 γ12 γ13 γ23

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(1)

Each column of the material’s stiffness matrix can be independently populated through computation of the RVE’s volume-averaged stress – as in Eq. (2a) – after

Fig. 12 (a) RVE of octet-truss and six strain cases – (b) three normal and (c) three shear – applied for homogenization routine

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applying an average directional strain (i.e., pure normal or shear); all other far-field strains are constrained to zero. Numerical evaluation of this integral equation takes the form shown in (2b). The average stress, σ ij , is calculated by summing – over all N elements – the volume-weighted stress, at the centroid of each element, n. In Eq. (2b), Vn is the volume of element n, whereas VRVE is the total volume of the unit cell.  1 σ ij = σij dV (2a) V V σ ij =

1 VRV E

N  

σij V

 n

(2b)

n=1

Consider, for example, enforcing a strain exclusively in the 1-direction of the principal material coordinate system (i.e., ε11 = ε0 11 and ε22 = ε33 = γ 12 = γ 13 = γ 23 = 0). Averaging the stress over the RVE in that same direction (i.e., 0 . For this same load σ 11 ) will yield the first cell in the matrix: C1111 = σ 11 /ε11 case, averaging stress in the 2-direction (i.e., σ 22 ) gives the next term in the matrix, 0 , and so on until the column values are all determined. C2211 = σ 22 /ε11 Through independent application of all six pure strain cases – as defined in Fig. 12b, c – the stiffness matrix in Eq. (1) can be fully populated, one column at a time. Once all components have been identified, the matrix inverse can be taken to compute the compliance. The compliance may then be related to the 12 engineering constants in Eq. (3); note, however, only 9 of the constants are independent due to symmetry of the matrix. ⎡

S = C−1

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

⎤ 1/E1 −ν21 /E2 −ν31 /E3 0 0 0 1/E2 −ν32 /E3 0 0 0⎥ − ν12 /E1 ⎥ ⎥ 1/E3 0 0 0⎥ − ν13 /E1 −ν23 /E2 ⎥ 0 0⎥ 0 0 0 1/G12 ⎥ 0⎦ 0 0 0 0 1/G13 0 0 0 0 0 1/G23

(3)

To properly enforce the far-field applied strain on the RVE, periodic constraints must be used. That way, while a complex state will develop locally within the unit cell, the volume-averaged strain over the RVE will be equivalent to the applied value. Note that reflective-symmetry boundary conditions are not appropriate for constraining the unit cell as they (1) do not permit application of shear strains and (2) require planes of symmetry to remain planar under deformation. Displacementbased periodic boundary conditions (PBCs) tie the motion of parallel faces bounding the RVE domain to simulate an infinite medium. Therefore, even in a deformed state, the unit cell can be tessellated in space. Although some commercial finite element programs, e.g., COMSOL Multiphysics, have built-in capabilities to accommodate periodic boundary conditions, many do not yet have this functionality integrated in

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their base coding. As such, the proceeding content is provided in a general context to be relevant across a variety of platforms. Referring to the 1-2-3 principal material coordinate system for the RVE in Fig. 12a – and denoting displacement degrees of freedom (DOFs) as u, v, and w for translations in the respective 1-, 2-, and 3-directions – the periodic constraint equations (CEs) are given by Eqs. (4a), (4b), and (4c). The variable ςi denotes the i-coordinate value in the domain bound by the volume of the unit cell – i.e., ς1 ∈[−x,+x], ς2 ∈[−y, +y], and ς3 ∈[−z, +z]. The right-hand side of Eqs. (4a), (4b), and (4c) defines the displacement necessary to enforce the applied strain, εij , over the ςj domain. For example, the separation of node pairs at the same (ς2 , ς3 ) position but on opposite ς1 periodic faces (i.e., ς1 = ±x) is increased by 2xε11 along the 1direction, in accordance with the first equation in (4a). The nine unit tensor strains associated with the loading cases of Fig. 12b, c are explicitly defined in Table 1. For node pairs on ± x faces : u (+x, ς2 , ς3 ) − u (−x, ς2 , ς3 ) = 2xε11 v (+x, ς2 , ς3 ) − v (−x, ς2 , ς3 ) = 2xε21 w (+x, ς2 , ς3 ) − w (−x, ς2 , ς3 ) = 2xε31

(4a)

For node pairs on ± y faces : u (ς1 , +y, ς3 ) − u (ς1 , −y, ς3 ) = 2yε12 v (ς1 , +y, ς3 ) − v (ς1 , −y, ς3 ) = 2yε22 w (ς1 , +y, ς3 ) − w (ς1 , −y, ς3 ) = 2yε32.

(4b)

For node pairs on ± z faces : u (ς1 , ς2 , +z) − u (ς1 , ς2 , −z) = 2zε13 v (ς1 , ς2 , +z) − v (ς1 , ς2 , −z) = 2zε23 w (ς1 , ς2 , +z) − w (ς1 , ς2 , −z) = 2zε33

(4c)

Multi-point constraints (MPCs) are used to enforce these conditions on periodic pairs in commercial software (e.g., Abaqus/Standard, ANSYS, LS-DYNA, MSC Nastran, etc.). For implementation in Abaqus/Standard specifically, the periodic conditions are applied as linear constraint equations following the format defined in the Abaqus Analysis User’s Guide [68]; Wu et al. also provide detailed instruction for this execution [69]. To prevent the FE model from being over-constrained, the first degree of freedom specified in any CE is eliminated. As such, the eliminated DOF cannot be used in subsequent constraints or boundary conditions. Therefore, nine free-space “dummy” nodes – not associated with the RVE mesh – are introduced: one for each of the nine tensor strains specified in the heading row of Table 1. These detached nodes should be allocated as secondary terms when used in the linear constraint equations, such that their DOFs are not eliminated by the CE and may be reserved for application of displacement boundary conditions. Because degrees of freedom are eliminated in MPCs, the equations of (4a), (4b), and (4c) must be reformulated for special treatment on the 12 edges. By combining the constraint equations for diagonally paired edges, the corresponding periodic boundary conditions are explicitly stated in Eqs. (5a), (5b), (5c), (5d), (5e), and (5f).

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Table 1 Average unit tensor strains applied in six loading cases for homogenization routine Strain case 1 Strain case 2 Strain case 3 Strain case 4 Strain case 5 Strain case 6

ε11 1 0 0 0 0 0

ε21 0 0 0 ½ 0 0

ε31 0 0 0 0 ½ 0

ε12 0 0 0 ½ 0 0

ε22 0 1 0 0 0 0

ε32 0 0 0 0 0 ½

ε13 0 0 0 0 ½ 0

ε23 0 0 0 0 0 ½

ε33 0 0 1 0 0 0

Likewise, the constraint equations on the eight vertices must be uniquely defined using the relations in Eqs. (6a), (6b), (6c), and (6d). For node pairs on ± xy edges : u (+x, +y, ς3 ) −u (−x, −y, ς3 ) =2xε11 +2yε12 v (+x, +y, ς3 ) −v (−x, −y, ς3 ) =2xε21 +2yε22 w (+x, +y, ς3 ) −w (−x, −y, ς3 ) =2xε31 +2yε32 (5a) For node pairs on ± yz edges : u (ς1 , +y, +z) − u (ς1 , −y, −z) = 2yε12 + 2zε13 v (ς1 , +y, +z) − v (ς1 , −y, −z) = 2yε22 + 2zε23 w (ς1 , +y, +z) − w (ς1 , −y, −z) = 2yε32 + 2zε33 (5b) For node pairs on ± zx edges : u (+x, ς2 , +z) − u (−x, ς2 , −z) = 2zε13 + 2xε11 v (+x, ς2 , +z) − v (−x, ς2 , −z) = 2zε23 + 2xε21 w (+x, ς2 , +z) − w (−x, ς2 , −z) = 2zε33 + 2xε31 (5c) For node pairs on ∓ xy edges : u (+x, −y, ς3 ) −u (−x, +y, ς3 ) =2xε11 −2yε12 v (+x, −y, ς3 ) −v (−x, +y, ς3 ) =2xε21 −2yε22 w (+x, −y, ς3 ) −w (−x, +y, ς3 ) =2xε31 −2yε32 (5d) For node pairs on ∓ yz edges : u (ς1 , +y, −z) − u (ς1 , −y, +z) = 2yε12 − 2zε13 v (ς1 , +y, −z) − v (ς1 , −y, +z) = 2yε22 − 2zε23 w (ς1 , +y, −z) − w (ς1 , −y, +z) = 2yε32 − 2zε33 (5e)

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For node pairs on ∓ zx edges : u (−x, ς2 , +z) − u (+x, ς2 , −z) = 2zε13 − 2xε11 v (−x, ς2 , +z) − v (+x, ς2 , −z) = 2zε23 − 2xε21 w (−x, ς2 , +z) − w (+x, ς2 , −z) = 2zε33 − 2xε31 (5f) For corner pair (+x, +y, +z) | (−x, −y, −z) : u (+x, +y, +z) − u (−x, −y, −z) = 2xε11 + 2yε12 + 2zε13 v (+x, +y, +z) − v (−x, −y, −z) = 2xε21 + 2yε22 + 2zε23 w (+x, +y, +z) − w (−x, −y, −z) = 2xε31 + 2yε32 + 2zε33 (6a) For corner pair (+x, −y, +z) | (−x, +y, −z) : u (+x, −y, +z) − u (−x, +y, −z) = 2xε11 − 2yε12 + 2zε13 v (+x, −y, +z) − v (−x, +y, −z) = 2xε21 − 2yε22 + 2zε23 w (+x, −y, +z) − w (−x, +y, −z) = 2xε31 − 2yε32 + 2zε33 (6b) For corner pair (+x, +y, −z) | (−x, −y, +z) : u (+x, +y, −z) − u (−x, −y, +z) = 2xε11 + 2yε12 − 2zε13 v (+x, +y, −z) − v (−x, −y, +z) = 2xε21 + 2yε22 − 2zε23 w (+x, +y, −z) − w (−x, −y, +z) = 2xε31 + 2yε32 − 2zε33 (6c) sFor corner pair (+x, −y, −z) | (−x, +y, +z) : u (+x, −y, −z) − u (−x, +y, +z) = 2xε11 − 2yε12 − 2zε13 v (+x, −y, −z) − v (−x, +y, +z) = 2xε21 − 2yε22 − 2zε23 w (+x, −y, −z) − w (−x, +y, +z) = 2xε31 − 2yε32 − 2zε33 (6d) If appropriate, symmetry of the RVE may be exploited to ensure perfectly paired nodes for the periodic constraint equations. For topologies with threefold symmetry, one-eighth of the unit cell can be modeled and meshed; the mesh can then be mirrored over the three planes of symmetry and coincident nodes merged, as illustrated in Fig. 13. By doing so, the nodes on opposite faces are guaranteed to have the same (ςi , ςj ) position. When the finite element program does not have builtin capabilities to define PBCs, it can be most efficient to forgo exclusive use of the

Microlattice Materials and Their Potential Application in Structural Dynamics. . .

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Fig. 13 Method for generating perfectly paired nodes

commercial pre-processor in generating the model for analysis. Instead, an objectoriented scripting language – such as MATLAB or Python – may be used to build the full analysis input deck (assuming the analyst has familiarity with the required solver input data structure). A general outline of the procedure is provided in Fig. 14. Post-processing is also conveniently accomplished external to the software’s graphical user interface; with a coded script, Eq. (2b) can be numerically evaluated for integrating the field variables over the RVE.

7.2 Dynamic Static homogenization acknowledges that material properties are local in space, but serves to relate these properties to nonlocal macroscopic behaviors. Dynamic homogenization goes a step further by considering the properties are local in time as well. It is sometimes difficult to digest content in multi-variable domains; so, to better understand wave propagation concepts in this section, consider the spatialtemporal analog illustrated in Fig. 15. Angular wavenumber, k – i.e., the magnitude of wavevector, k – is the spatial equivalent to angular frequency, ω, in a temporal domain; therefore, the wavenumber can be thought of as a spatial frequency defining the number of waves that can occupy a unit of space. Likewise, the wavelength, λ, is analogous to wave period, T. The domain where dynamic homogenization techniques can be used – while still considering the dispersive nature of wave propagation – is identified in [70] as the dynamically homogenizable regime (Fig. 16a). The upper bound of this region is characterized by long wavelengths (Fig. 16b) where wave behavior is nondispersive

30

Fig. 14 Pseudocode for generating FE models for static homogenization

Fig. 15 Spatial-temporal analog for wave propagation

L. M. Dangora

Microlattice Materials and Their Potential Application in Structural Dynamics. . .

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(b) (a) (c)

Fig. 16 (a) The dynamically homogenizable regime bounded by the (b) long-λ limit and (c) shortλ limit

and propagation can be defined in relation to time-independent averages of the material constants; in other words, at low frequencies – where the wavelength is much larger than the scale of the periodicity – the response will be quasi-static. A lower limit is imposed at very short wavelengths (Fig. 16c) where propagation characteristics are dominated by interfacial scattering and separation of scales cannot be respected; generally speaking, for dynamic homogenization techniques to be appropriate, the wavelength should be greater than twice the scale of periodicity [70]. Dynamic homogenization methods – although less precise and well-established than their static counterparts [71] – are necessary for wave propagation and vibration analyses when the material is expected to exhibit sensitivity to operational frequencies. While there are a number of techniques available for such homogenization – e.g., multiple scattering models [72], micromechanical approaches [73], mixed variational methods [74], etc. – the content that follows is limited to a periodic eigenstrain approach. Again, micromechanics of the RVE are considered, and volume averages of the field variables are taken to derive quantities that describe macroscopic behaviors. Therefore, this method can be thought of as the dynamic complement to the static homogenization theory previously discussed. For more general and extensive information pertaining to dynamic homogenization techniques, the reader is referred to a review paper by Srivastava [70] and the references cited therein.

Finite Element Implementation for Free-Wave Propagation The finite element formulation for the equation of motion is given by (7) for an undamped mechanical system, where M and K denote the global mass and stiffness matrices and x denotes the nodal displacement vector, x¨ its second time derivative, and F the nodal force vector.

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M¨x + Kx = F

(7)

Classical solutions to this governing differential equation describe simple motion as a function of time, t. Where propagating waves are of interest, the analogous wave equation introduces a solution that is dependent on position, r, as well. Through application of Bloch’s theorem – which, for a periodic system, presents solutions to the wave equation as the product of a plane wave, eik•r , and a function having the same periodicity as the material [75] – the assumed displacement for time-harmonic wave propagation is given by ∼

x (r, t) = x k (r) eik·r eiωt

(8)

where k is the wavevector and ω the angular frequency. Construction of an arbitrary Bloch waveform for a one-dimensional periodic material is depicted in Fig. 17 for ∼ visualization. The first term, x (r), defines vibration of the unit cell, which may vary within the internal subdomain but retains the same periodicity as the material (i.e., invariant under lattice translations). The second term, eikx , defines spatial periodicity for the plane wave, and the third term, eiωt , dictates the temporal periodicity. Admitting free-wave, time-harmonic solutions of Bloch form according to (8), the equation of motion can be rewritten as the eigenfunctions in (9). Note all variables – as implied by the subscript – become dependent on the wavevector, which may be complex-valued [76]. For the admissible solution, eigenvalues (λ1 , λ2 , . . . , λn ) can be solved for over discrete vectors of k. These eigenvalues are used to identify frequencies (i.e., λj = ωj 2 ) permitting wave motion through ∼

∼

∼

the material at the specified wavevector. Likewise, eigenvectors ( x 1 , x 2 , . . . , x n ) from the analysis give the unit cell normal mode of vibration for each frequencywavevector pair. 

∼ −ω2k Mk + Kk x k = 0

(9)

Fig. 17 Illustrative example of Bloch waveform construction at an instantaneous point in time

Microlattice Materials and Their Potential Application in Structural Dynamics. . .

33

Floquet’s principle in one dimension [77] – and Bloch’s extension for higherorder domains [78] – therefore allows wave motion through the lattice material to be predicted by considering only the representative volume element. Again, for computing results that correspond to full-field solutions, it is critical to use proper boundary conditions when reducing the problem to a single periodic base unit. The static constraints in Eqs. (4a), (4b), (4c), (5a), (5b), (5c), (5d), (5e), (5f), (6a), (6b), (6c), and (6d), however, are not sufficient to describe the dynamic response characteristics. To allow each mode that can be excited in the lattice material to travel uninhibited through the periodic representation, the boundary conditions should facilitate phase and amplitude gradients across the medium. Deriving from point solutions of Eq. (8) at periodically paired boundary nodes, constraint equations on the translational degrees of freedom are given by u (r + Δr) = u (r) eik·Δr v (r + Δr) = v (r) eik·Δr w (r + Δr) = w (r) eik·Δr

(10)

where the same conventions from Section 7.1 apply. The finite element solver inherently enforces similar traction boundary conditions in the eigenanalysis. It should be noted that the wavevector k = k1 ςˆ 1 +k2 ςˆ 2 +k3 ςˆ 3 is, in general, complexvalued; each wavenumber can be expressed as k = δ k + i. The real component δ k – known as the attenuation constant – defines spatial attenuation of the wave as it moves from one cell to the next; the imaginary component or phase constant, , describes the change in phase over single unit cell [79]. For the content presented here, effects of spatial decay are ignored; therefore, the attenuation constant is assumed null, and the wavevector reduces to only its imaginary components. This also ensures that the mass and stiffness matrices of Eq. (9) are Hermitian [76]. The periodic boundary conditions of (10) are applied to a three-dimensional RVE as demonstrated in Fig. 18 for the first translational degree of freedom. The assumption of free-wave propagation eliminates the need for dummy nodes, since no external forces are applied; however, a set of reference nodes on the RVE mesh are reserved for use across multiple constraint equations. These reference nodes should be allocated as secondary terms when used in the linear constraint equations, such that a single CE does not remove their DOFs. For the example unit cell of Fig. 18, constraint equations on the corners are written with respect to the node at c1; likewise, edge equations are expressed in relation to e1, e4, and e9, and face CEs are defined with respect to f1, f3, and f5. As many commercial FE codes do not readily handle complex-valued functions, the components of the Hermitian mass and stiffness matrices in Eq. (9) can be separated and augmented – as detailed in [80] – to produce real symmetric matrices for numerical implementation. Complex expressions for mass (M = MRe + iMIm ), ∼

∼Re

∼Im

stiffness (K = KRe + iKIm ), and displacement ( x = x + i x ) are used to expand the n-order equations of (9); the expanded form is then separated by the complex classifier into two sets of n-order linear equations and recombined in matrix form

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Fig. 18 Constraint equations for enforcing Floquet-Bloch periodicity on the translational degree of freedom in the 1-direction

(of order 2n) as ⎫ ⎧   Re   Re  ⎨ ∼Re ⎬ Im Im x M −M K −K −ω2 + =0 Im ⎩∼ MIm MRe KIm KRe x ⎭

(11)

Although the subscript k has been dropped, wavevector dependence is implicit for all variables. Two identical meshes of the unit cell are used to represent each complex component – i.e., one for the real domain and one for the imaginary domain [81]. Therefore, the complex Hermitian Bloch eigenvalue problem may be replaced by an equivalent real-valued function at the expense of increased computational cost (since the matrix dimensions have doubled). The eigensolutions to (11) result in double eigenvalues (λ1 , λ1 , λ2 , λ2 , . . . , λn , λn ) and yield a similar pair of orthogonal ∼Re

∼Im

∼Im

∼Re

eigenvectors, { x j + i x j } and {− x j + i x j }, for each repeated root. Likewise, the complex constraint equations enforcing periodicity must also be decoupled. Consider, for example, two points separated by a unit vector of the direct lattice: one point being located at (ς1 , ς2 , ς3 ) and the other at (ς1 + ς1 , ς2 + ς2 , ς3 + ς3 ). Denoting the displacement of each point along the ς1 -direction as uA and uB , the motions of these two nodes are related by the periodic condition as uB = uA ei(φ1 +φ2 +φ3 )

(12a)

where the phase, φ j , is the product of the wavenumber’s phase constant,  j , and node separation distance, Δςj , along the j-axis (i.e., φ j =  j Δςj ); recall the attenuation constant is assumed null. The equation can be rewritten explicitly in terms of real and imaginary components such that they may be separated. To do so, Euler’s identity is used for expressing the complex exponential as trigonometric functions. Equation (12a) then becomes

Microlattice Materials and Their Potential Application in Structural Dynamics. . .

35

  Im Re Im uRe + iu = u + iu [cos (φ1 + φ2 + φ3 ) + i sin (φ1 + φ2 + φ3 ) ] B B A A (12b) and separating the real and imaginary terms yields Re Im uRe B = uA cos (φ1 + φ2 + φ3 ) − uA sin (φ1 + φ2 + φ3 ) Im uIBm = uRe A sin (φ1 + φ2 + φ3 ) + uA cos (φ1 + φ2 + φ3 )

(12c)

Equation (12c) is therefore used to couple the two identical RVE meshes to enforce the proper periodic condition. A separate eigenanalysis is executed for each discrete wavevector considered, since the trigonometric coefficients of the (12c) constraint equation must be updated for each case. An example pseudocode flowchart for generating dynamic homogenization models is shown in Fig. 19. Once the FE solver has computed solutions to the eigenanalysis, the volumeaveraged eigenstress and eigenstrain may be calculated in similar fashion to Eq. (2b). Note that the average is taken separately for each of the two meshes to obtain the “real” and “imaginary” components. The averaged values are then combined as a complex number and used in Eqs. (1) and (3) to determine effective stiffness properties for a given wavevector-frequency pair. Unlike the static homogenization process, no external strains are enforced in this dynamic homogenization routine. Therefore, if the “strain” of the voided volume is not considered, Eq. (3) will regurgitate the engineering constants of the frame’s bulk material. To obtain a proper solution for the voided material, empty regions of the RVE should be discretized with elements (Fig. 20) and assigned negligible mass and stiffness. The “dummy” material properties for the void should be small enough

Fig. 19 Pseudocode for generating FE models for dynamic homogenization

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(a)

(b)

(c)

Fig. 20 FE representation of the (a) voided and (b) solid regions forming the (c) octet-truss RVE for dynamic homogenization

so as not to influence the results – i.e., the voided elements should just be “along for the ride” with the frame motion; the stiffness-to-mass ratio, however, should be large to discourage low-frequency eigenvalues – corresponding to local void motion (i.e., void resonance with stationary frame) – from presenting.

Dispersion Surfaces The dispersion curve, or band diagram, can also be generated from this set of eigenanalyses to provide more insight into the material characteristics. Secant and tangent slopes of a dispersion curve yield respective phase and group velocities for propagation of that mode through the medium [82]. Band diagrams are also used to distinguish frequencies of propagation for a given wavevector. Likewise, they can identify stopbands, where frequencies over a certain range do not support any Bloch waves at all; the energy states in these regions instead cause spatial attenuation of the waveform [75]. Band diagrams are created by plotting frequency as a function of wavenumber; see, for example, Fig. 21. Recall that the wavevector has multidirectional components – k = k1 ςˆ 1 + k2 ςˆ 2 + k3 ςˆ 3 – and the attenuation constant is assumed null (i.e., kj is imaginary). If only one component is varied, a 1D band diagram can be obtained (Fig. 21a). If two components are varied, another dimension is added, and dispersion surfaces are plotted (Fig. 21b). The dispersion curves form by considering the lattice material in reciprocal space. Whereas the direct lattice defines periodicity of the medium in physical space, the reciprocal lattice defines periodicity of the waves propagating through the medium. Basis vectors {b1 , b2 , b3 } of the reciprocal lattice are derived from the primitive translation vectors {a1 , a2 , a3 } of the direct lattice as b1 = 2π

a2 × a3 a3 × a1 a1 × a2 , b2 = 2π , b3 = 2π a1 · a2 × a3 a1 · a2 × a3 a1 · a2 × a3

(13)

Microlattice Materials and Their Potential Application in Structural Dynamics. . .

(a)

37

(b)

Fig. 21 (a) 1D and (b) 2D band diagram showing several propagation modes; the real component of frequency is plotted

For the Euclidean geometry shown in Fig. 12a, the direct and reciprocal lattice vectors are given by Eqs. (14a) and (14b), respectively. a1 = 2x ςˆ 1 , a2 = 2y ςˆ 2 , a3 = 2zςˆ 3

(14a)

π π π ςˆ 1 , b2 = ςˆ 2 , b3 = ςˆ 3 x y z

(14b)

b1 =

When observed in reciprocal space, it becomes evident that angular frequency is a periodic function of the wavevector. Because of this periodicity, only the first Brillouin zone (FBZ) requires consideration to cover all independent solutions; the FBZ is bound by the normal planes bisecting basis vectors [83]. Therefore, the range of phase, φ j , that has physical significance is bound by ±π; phase outside of this range will repeat lattice motions obtained within the range [75]. Note that although the most interesting phenomena typically occur along the boundaries defining an irreducible Brillouin zone (IBZ), it is not always sufficient to traverse only these boarders; to ensure band extrema are located accurately, the entire first Brillouin zone should be sampled [82, 84]. As can be seen in the examples of Fig. 21, dispersion surfaces for different modes may cross and can be difficult to distinguish in high-density areas of the chart. The modal assurance criterion (MAC) can be a useful tool to quickly identify

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Fig. 22 Modal assurance criterion used for identifying mode swapping

such mode swapping [85]. The MAC quantifies the degree of correlation between ∼ two mode shapes (or eigenvectors, x) and is calculated as in Fig. 22. Each mode shape for wavevector ka is compared to each mode shape for wavevector kb to yield a scalar value between zero and one. When all n modes are compared for the two wavevectors, an n × n MACab matrix is populated. Similar eigenvectors with a high degree of correlation produce a value near unity, while dissimilar modes yield a number near null. If no modes cross, the matrix diagonal is populated with the highest values.

8 Locally Resonant Lattice Materials The aforementioned computer-numerical techniques may be used to design microstructures tailored to a specific application and optimized for performance. As such, the computational framework provides a digital space to also explore more advanced topics in lattice materials research, like hierarchical architectures (Fig. 7b) and waveguide topologies (Fig. 7c). Another forward-looking concept is the design of locally resonant sonic materials [86]. It is postulated that a dispersion of plate-like protrusions in the microlattice can enhance dissipation of acoustic and vibrational energy by using incident pressure waves or structure-borne vibrations to excite damped oscillations of cantilevered micro-resonators, as illustrated in Fig. 23. Additional losses may be introduced by factors such as sound scattering off the high aspect ratio protrusions or heat dissipation via material hysteresis. Such a medium will be referred to herein as a locally resonant lattice material (LRLM).

Microlattice Materials and Their Potential Application in Structural Dynamics. . .

(a)

39

(b)

Fig. 23 (a) Microlattice embedded with high aspect ratio protrusions that (b) dissipate energy from incident pressure waves or structure-borne vibrations through damped oscillation of cantilevered platelets Fig. 24 Scattering and reflections of incident sound wave in open-celled material (reprinted from [87])

8.1 Acoustics Open-cell porous materials are often employed for applications requiring sound absorption. Their high specific surface areas facilitate efficient dissipation mechanisms introduced by interactions between a fluid medium and an elastic frame. For fluid-saturated poroelastic solids, energy can transfer from the acoustic pressure wave in the fluid and manifest as strain energy in the elastic structure (e.g., flexure of the cell walls); it can then dissipate in the solid due to relaxation and viscoelastic effects of the restoring force. Within the flow itself, the influence of surface tension at the interstices induces shear in the viscous boundary layer; as a result, a portion of the fluid’s kinetic energy is converted to heat. Additional heat generation from frictional forces causes energy to be dissipated at the fluid-structure interface. Furthermore, with a large number of cavities in porous media, the characteristic impedance mismatch (i.e., difference in sound speed and density of the fluid and solid) can cause many internal reflections of the sound wave (Fig. 24); this scattering results in effective attenuation of the acoustic energy as it is dissipated by frictional, shear, viscoelastic, and relaxation forces [66, 87].

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Based on these principal phenomena, it is expected that addition of cantilevered ancillary features protruding into the cavity volume (Fig. 23) will enhance microlattice performance in acoustic absorption. Since the cantilevered platelets would be more compliant than the primary cell walls, these local resonators could deform more readily and transfer energy from the fluid into the structure. Once in the solid body, a portion of the strain energy may be lost to relaxation and viscoelastic effects in the restoring forces. The micro-resonators in LRLMs also introduce more exposed surface area to (1) increase scattering and diffusion in the cavities, (2) establish more sites to develop the viscous boundary layer for shear dissipation, and (3) create more interstitial regions for frictional dissipation.

8.2 Vibrations In addition to improving acoustic performance, LRLMs may also increase the damping capacity in structural-dynamic applications by directing energy into motion of ancillary features rather than creating net global movements. Consider, for example, the configuration illustrated in Fig. 23, where ancillary bodies (i.e., plate-like protrusions) are arranged within the primary structure (i.e., lattice frame) to damp its dynamic response. Disturbing vibrations in the main body can cause excitation of the auxiliary features, which thereby may exercise an out-of-phase reaction force back onto the primary structure; in this manner, global vibrations of the primary structure will be diminished by oscillations of the local resonators. The response may further be attenuated by frictional, hysteretic, and viscoelastic effects in the relaxation time necessary for the perturbed system to evolve toward a new state. A LRLM would therefore serve to be both a vibration absorber, opposing excitation forces, and a vibration damper, dissipating kinetic energy. To absorb vibrational energy by exploiting local substructural resonances, one can apply concepts used in tuned absorber design. In its most basic form, a tuned absorber can be described as a mass-spring system that is mounted to the primary structure and designed to limit motion of that main body for a specific excitation frequency [88]. This is accomplished as the tuned absorber resonates out-of-phase to exert an equal and opposite force to that which is supplied by the disturbance [89]. To illustrate the concept, refer to the single degree of freedom (SDOF) system in Fig. 25a consisting of a spring, K, and mass, M. At a specific harmonic excitation frequency, f, the mass oscillates with a certain amplitude about the neutral position. To attenuate the response of the system at this excitation frequency, a tuned absorber can be added. For this illustration, the absorber has a stiffness, k, and mass, m. If perfectly tuned, the vibration absorber will resonate at the excitation frequency, and the reaction force imposed by the absorber will annul vibration of the primary M-K system (Fig. 25b). To contemplate the applicability over a broader frequency spectrum, consider a vibration absorber that is not perfectly tuned to the frequency of excitation (Fig. 25c); although movement of the primary structure is not completely inhibited, an

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41

Fig. 25 Vibration in (a) a dynamic system oscillating under harmonic excitation is (b) annulled by addition of a tuned absorber designed for that specific excitation frequency or (c) attenuated by out-of-phase motion of an absorber having a resonance not associated with the frequency of excitation. (d) Comparison of (a), (b), and (c)

attenuating effect can still be achieved when the bodies move out-of-phase. With proper design to facilitate destructive interference, it is anticipated that a LRLM can exhibit enhanced damping performance over that of the primary structure alone. Although this “local resonator” concept has yet to be explored in the field of lattice materials, the principles and underlying theory of vibration suggest it should be effective.

9 Summary This chapter has provided a cursory introduction to the subject of lattice materials. Such digitally-designed, truss-based cellular solids are pushing the bounds of attainable material properties, which is spurring interest in the research field. In addition to their taxonomy and governing characteristics, naval interests were discussed with particular attention to multifunctional applications involving structures, buoyancy, vibrations, and acoustics. Brief coverage of some production methods was included, and test methods – following standard procedures developed for homogenous materials – were discussed for evaluation of apparent properties. A more extensive section was provided on computational micromechanics. Periodicity of the lattice microstructure allows full-field behaviors to be derived from only the base unit cell. Numerical homogenization can therefore be performed using a representative volume element; volume averages of the field variables can be taken to derive quantities describing macroscopic properties. Implementation in a finite element framework was detailed for both static and dynamic behaviors. The chapter also touched briefly on more intricate microstructures – e.g., to produce hierarchical and waveguide topologies – before covering locally resonant lattice materials for enhancement of acoustic and vibration properties.

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An Alternative Convolution Approach to the Cagniard Method for Transient Ocean Acoustic Modelling Roy L. Deavenport and Matthew J. Gilchrest

1 Introduction A desirable requirement for underwater acoustic models is the ability to simulate a transient signal propagating in an ocean waveguide. Relevant broadband signals are generated, for example, by airguns used in seismic exploration, transducers used for underwater communication systems, and vocalizations produced by marine mammals. Single-frequency acoustic models based on rays, modes, wavenumber integration, or parabolic equation representations are often used for this purpose by employing a Fourier synthesis method. In this case, the multiple-frequency medium transfer function is filtered by the spectrum of the source and then inverse Fourier transformed to yield the time-domain waveform. In contrast, direct convolution of a source signal with the medium’s impulse response can be carried out in the time domain. As a result, transient signals that do not have Fourier transforms required by the synthesis method can be accommodated. Propagation models that are based on standard representations for the acoustic field have been developed into reliable and efficient computer codes, e.g., ray-based [30], wavenumber integration [26], normal mode [25], and parabolic equation [14]. For each of these representations of the acoustic field (treated in [21]), a timeharmonic, (e−iωt ), single-frequency point source is considered. This continuouswave (CW) source derives from the historical motivation that underlies each of these representations, namely, to provide single-frequency estimates of transmission loss (TL) for use in Navy sonar performance prediction applications.

R. L. Deavenport () · M. J. Gilchrest Naval Undersea Warfare Center, Newport, RI, USA e-mail: [email protected]; [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_3

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Any of the single-frequency acoustic model representations given above can be used to generate transient signals in a range-independent ocean waveguide by using a Fourier synthesis approach. Since the effective spectral bandwidth is determined by the frequency content of the source waveform, a number of single-frequency acoustic predictions must be carried out that span this bandwidth using a frequencysampling interval determined by the time spread of the multipath arrivals at the range of interest. The resulting medium transfer function is then multiplied by the spectral content of the source, and the product is inverse Fourier transformed to yield the transient waveform. In the time domain, the Fourier synthesis method is equivalent to evaluating the convolution of the source waveform with the medium’s impulse response. An alternative convolution approach for simulating a transient signal was developed in [15] and is based on a ray-based model of underwater sound propagation. An important definition to recall here is that of an eigenray (i.e., a single ray trajectory that connects the source to a receiver). In this previous work, the impulse response of the medium was convolved with a nonlinear explosive source waveform whose peak pressure level is given semi-empirically in terms of a scaled range (geometric similitude). An important assumption that underlies the use of the polar form of the ray theory model used in [16] is that each eigenray amplitude is assumed to be independent of frequency. As a result, the phase variation of the nth eigenray is readily handled solely via ω in the factor eiωTn , where Tn is its delay time between source and receiver. This phase variation is handled prior to summing across the individual eigenray contributions (multipaths) in the calculation of the medium’s transfer function at a given range. After inverse transforming to the time domain (via an FFT), the resulting impulse response is convolved with the source waveform to generate the simulated transient signal in the waveguide. The convolution method developed in [15] was first used to propagate a shockwave pulse waveform in both deep and shallow-water configurations to exhibit the distortion effects due to bandlimited signals. Subsequently this procedure was used to demonstrate multipath propagation in a shallow-water waveguide [16] similar to the one analyzed previously by Jensen et al. [20]. Jensen et al.’s downwardrefracting, range-dependent waveguide was replaced with one of Pekeris type and used to propagate a low-frequency Ricker wavelet signal. This range-independent environment enabled the use of the wavenumber integration code SAFARI [26] to provide time-domain simulations computed using the Fourier synthesis method to use for comparison. In addition, it was convenient to model the travelling wave representation in terms of image theory where each lossy bottom-interacting image is modified by the plane-wave reflection coefficient at the appropriate grazing angle. Numerical justification for using image theory in this context is provided in the section on numerical results. In the following sections, we review the theory underlying two convolution methods for computing the time response in a shallow-water Pekeris waveguide due to a transient source: Cagniard’s delta-function impulse versus bandlimited impulse methods. The concept of expressing a bandlimited transfer function in polar form is originally due to J.M. Tattersall (Private communication).

Alternative Convolution Approach to Cagniard

49

2 Cagniard Method Cagniard [7] used integral transforms from Carson [8] to develop a model convolving a two-layer homogenous impulse response with a general transient signal. His method is used primarily in seismology and has been generalized by de Hoop [19] and Gilbert and Helmberger [18]. A modern online treatment has been given by Dobrushkin [17]. Chapman [9, 10, 12] and Kennett [23] provide an excellent treatment for the Cagniard method and generalized ray theory. Aki and Richards [1] also provide a readable account of the Cagniard method. In this chapter, an alternative convolution-based approach is presented that is based on a travelling wave or polar-form representation for the propagation model [6, 11]. The two methods are subsequently used to simulate a transient signal propagating in a shallow-water, lossy-bottom Pekeris waveguide. In the following, we review some basic concepts of the classical Cagniard method and how it differs from the new convolution method. Both methods start with the time-dependent wave equation. Noted here, portions of Eqs. (1)–(10) will be recalled in subsequent sections. This will only be described for the acoustic case, since the seismic case is much more complicated. The time-dependent equation for the pressure field in the ocean in cylindrical coordinates (r, z) is given by (assuming rotational symmetry)   δ(r)δ(z − z0 ) ∂2 1 2 ∇ − 2 P (r, z, t) = −S(t) , (1) 2 2π r c (r, z) ∂t where the Laplacian in cylindrical coordinates is given by   ∂ ∂2 1 ∂ 2 ∇ = r + 2. r ∂r ∂r ∂z

(2)

Here c(r, z) is the sound speed as a function of range r and depth z, S(t) is a real, linear point-radiated signal at the channel (filter) input, and P (r, z, t) gives the instantaneous pressure at the output. A point source is located at (r, z) = (0, z0 ). Most treatments of the Cagniard method use the Heaviside step function H (τ ) to define the source for the impulse response. The function H (τ ) is defined by ⎧ ⎨ 0 for τ < 0 H (τ ) = , ⎩ 1 for τ > 0 where τ = t − t  and t  denotes the initial response time. In the following development, use will be made of the sifting property of the Dirac delta function, namely,  ∞ f (t)δ(t − t  )dt = f (t  ). (3) −∞

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Application of the Fourier transform, defined by  p(r, z, ω) =

∞ −∞

P (r, z, t)e−iωt dt

(4)

to Eq. (1), yields the following equation satisfied by p(r, z, ω):   δ(r)δ(z − z0 ) ∇ 2 + k 2 (r, z) p(r, z, ω) = −S(ω) , 2π r

(5)

where k(r, z) = ω/c(r, z) denotes the total wavenumber. The spatial Green’s function G(r, z, ω) is defined as the solution of the Helmholtz equation in Eq. (5) for a single-frequency e−iωt , unit point source, namely [3]   δ(r)δ(z − z0 ) ∇ 2 + k 2 (r, z) G(r, z, ω) = − . 2π r

(6)

Several methods of solving Eq. (6) include wavenumber integration [26], parabolic equation [14], and normal modes [25]. In this chapter, we are focused on ray-type solutions (e.g., Gaussian bundle [30] or images [16]) that involve individual propagation paths and which can be expressed in terms of travelling waves having the polar form described in [6, 11], i.e., G(r, z, ω) =

N 

An eiωTn +iφn ,

(7)

n=1

where An is the amplitude, Tn is the delay travel time, and φn is the phase shift associated with boundary and/or caustic interactions of the nth eigenray path. Finally, we will require the use of the impulse response h(r, z, t) defined as the inverse Fourier transform of the Green’s function G(r, z, ω), i.e., h(r, z, t − t  ) =

1 2π



∞ −∞



G(r, z, ω)eiω(t−t ) dω.

(8)

It is worth pointing out that for a single frequency ω0 , Eq. (8) becomes for the travelling wave representation in Eq. (7) 

h(r, z, t − t ) =

∞  n=1

=

∞ 

An e

iω0 Tn +iφn



1 2π



∞

−∞

An eiω0 Tn +iφn δ(t − t  ),

e

−iω(t−t  )

 dω ,

(9)

n=1

where the last result follows from the definition of a delta function. Each singlefrequency arrival at t  = Tn in Eq. (9) is associated with an infinite bandwidth.

Alternative Convolution Approach to Cagniard

51

In the following analysis we develop the integral equation method of Cagniard for the specific case of a Pekeris shallow-water waveguide. Following Towne [28] (see also Britt [5]), we first perform a Laplace transform, i.e.,  ∞ P (r, z, s) = P (r, z, t)e−st dt, (10) 0

on Eq. (1) for the specific uniform-fluid, spherical-wave point source F (t−R/c)R −1 to obtain ∂ 2P 1 ∂P s2 ∂ 2P + + − P = −F (s)R −1 e−sR/c r ∂r ∂r 2 ∂z2 c2  ∞ = −sF (s) J0 (sur)e−s|z−z0 |μ (u/μ)du,

(11)

0

where μ = (u2 + c−2 )1/2 . In deriving the Hankel transform in Eq. (11), use was made of an integral identity that is related to an expansion of the point source into its plane-wave spectral components (see [28] for details). Here c is the sound speed of the fluid and R = [r 2 + (z − z0 )2 ]1/2 is the slant range between the source at (0, z) and the receiver at (r, z0 ). Next, define a step-function response, A, analogous to a Green’s function, for a Pekeris shallow-water waveguide as the propagation medium, ∂ 2 A 1 ∂A ∂ 2 A s 2 δ(r)δ(z − z0 ) + 2 − 2 A = −H (s) . + 2 r ∂r 2π r ∂r ∂z c

(12)

On comparing Eqs. (11) and (12), we find P = sF (s)A(r, z, s).

(13)

Now by invoking the convolution theorem for Laplace transforms (e.g., see [27]), we can write the final Cagniard solution in terms of a general transient source F and step-function response (Green’s function) A in the form  t P (t) = F  (t − ν)A(r, z, ν)dν. (14) 0

From the derivative of a convolution (e.g., see Bracewell [3]), Eq. (14) can be written as  t F (t − ν)A (r, z, ν)dν. (15) P (t) = 0

The step-function response A is now defined by convolving the Green’s function h(r, z, t − t  ) with the Heaviside step function H (t) to yield 

∞

A(r, z, t) = 0

h(r, z, t − t  )H (t  )dt  .

(16)

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R. L. Deavenport and M. J. Gilchrest

Substituting Eq. (9) into Eq. (16) yields 

∞

A(r, z, t) =

N 

0

 An e

iω0 Tn +iφn



δ(t − t ) H (t  )dt  .

(17)

n=1

Carrying out the convolution of the step function H with the multiple image function, we determine the multipath step function and its derivative, namely, A (r, z, t) =

 0

 = 0

N ∞

An eiω0 Tn +iφn H  (t − t  )dt 

n=1 N ∞

An eiω0 Tn +iφn δ(t − t  )dt  .

(18)

n=1

Finally, we apply in Eq. (15) in convolution with the Arons [2] shock waveform with the derivative of the multipath step-function response to find the Cagniard solution for the transient field P (r, z, t) = P01 + P02 +

4 ∞  

Pj ,

(19)

=1 j =1

where P01 and P02 represent the direct and surface-reflected plane waves, respectively, and the double summation represents those paths that have interacted with the bottom boundary of the waveguide. This classical Cagniard result for the Pekeris shallow-water case is the same as the proposed alternative convolution method except that the impulse response for the alternative method is bandlimited. Using the image Green’s function to determine the step-function response avoids the use of Sommerfeld contour integrals. As demonstrated elsewhere [16], the approximate image solution provides excellent agreement with the full-wave wavenumber integration model SAFARI.

3 Alternate Methods to Cagniard A good review of the acoustic wave equation and transient sources can be found in [21]. In the following we first describe the Fourier synthesis method and then the alternate convolution method for treating a general bandlimited impulse response. In underwater acoustics, Fourier synthesis is traditionally used to determine timedependent solutions for broadband transient signals. The reason for this is that any of the usual numerical approaches developed for a single-frequency source (ray, wavenumber integral, normal mode, or parabolic equation) can be run (without

Alternative Convolution Approach to Cagniard

53

modification) for a number of frequencies that span the bandwidth of the transient source. From the convolution theorem, the resulting transfer function can be multiplied by the spectrum of the source and inverse Fourier transformed to yield the time-domain signal. From Eqs. (6) to (11), the time-domain solution for the pressure is given by P (r, z, t) =

1 2π



∞

−∞

G(r, z, ω)S(ω)e−iωt dω,

(20)

which provides the desired Fourier synthesis in terms of the product of the source spectrum and the medium transfer function. A notable numerical model based on the Fourier synthesis method is the SAFARI code developed by Henrik Schmidt [26]. There exist situations where the Fourier synthesis approach can be computationally difficult, such as when the number of frequencies required is large or when the source signal does not possess an analytic Fourier transform. The usual way that convolution is derived is to first to perform a Fourier synthesis of harmonic solutions as a function of frequency and then use properties of Fourier transforms to convert the synthesis into the time domain. However that method involves taking the Fourier transform of the signal. For example, in the case of an initial value problem, the Fourier transform of the source waveform may not exist. In these cases, alternative methods for modelling convolution need to be considered, such as the Cagniard method treated above. Direct convolution in the time domain does not involve Fourier transforming the signal. Our convolution procedure is different than the Cagniard method in that it does not require homogeneous layers and allows for attenuation. Nevertheless, in both methods one convolves an arbitrary source with the impulse response of the medium. The primary dissimilarity between the two methods is the method used to obtain the medium (system) impulse response. Cagniard developed his solutions using integral transform methods to develop the impulse response. In the present convolution method Eq. (1) can be written symbolically in the form L · P (r, z, t) = −S(t)

δ(r)δ(z − z0 ) , 2π r

(21)

where the linear differential operator L is defined by L = ∇2 −

∂2 1 . c2 (r, z) ∂t 2

(22)

From Eq. (10), the retarded time-domain Green’s function impulse, h(r, z, t −t  ), for a point source in space and time is defined by the partial differential equation L · h(r, z, t − t  ) =

δ(r)δ(z − z0 )δ(t − t  ) , 2π r

(23)

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R. L. Deavenport and M. J. Gilchrest

where t  is the initial response time. Note that the impulse function h(r, z, t − t  ) depends only on relative time (t − t  ) and that all terms in equation Eq. (23) must be dimensionally homogeneous. Multiplying both sides of Eq. (23) by S(t  ) and integrating with respect to t  yield 

∞

L · h(r, z, t − t  )S(t  )dt  = −S(t)

0

δ(r)δ(z − z0 ) . 2π r

(24)

If all initial conditions are assumed to be zero, then the order of differentiation and integration in Eq. (24) can be interchanged since L was defined to be linear. Consequently, Eq. (24) can be put in the equivalent form 

∞

L

h(r, z, t − t  )S(t  )dt  = −S(t)

0

δ(r)δ(z − z0 ) . 2π r

(25)

Comparison of Eq. (25) with Eq. (20) reveals that the convolution integral for the instantaneous pressure P (r, z, t) is given by  P (r, z, t) =

∞

h(r, z, t − t  )S(t  )dt  .

(26)

0

Equation (26) gives the pressure solution to Eq. (1) in terms of a convolution of the bandlimited linear impulse response h(r, z, t − t  ) with a real point-radiated signal S(t). This is an alternative convolution approach to that developed by Cagniard. When a transient signal is transmitted in the ocean, it is spread in time resulting in time dispersion. Dispersion in time refers to environmental conditions that cause a finite pulse to be spread. There are basically two types of time spread: multipath and distortion dispersion [22]. Multipath dispersion occurs when individual paths arrive at a receiver via different geometric travel times. These paths satisfy Snell’s law: each individual path itself is not spread . Distortion dispersion is a wave effect and depends on frequency. Distortion dispersion is when the individual arrivals are spread in time, which can happen in ray-theoretic models where the amplitudes and phases of the eigenrays depend on frequency. Distortion dispersion also occurs in wave models because of diffraction and scattering. Individual multipaths are resolved when the transmitted signal bandwidth B is greater than 1/Δt, where Δt is the multipath separation. If, however, there is distortion dispersion, then the individual multipaths are spread and the entire received signal may appear as a single spread packet. The effects of distortion dispersion are treated by the bandlimited impulse as it is believed that the primary cause of dispersion is the ocean and its boundaries. The approach taken in this study is to keep the frequency dependence in the phases but to evaluate the amplitudes at the carrier frequency. The term bandlimited is understood to mean the source bandwidth since the coherent bandwidth of the medium, without scattering, is nearly infinite. For long-range propagation, the ocean acts as a low-pass filter [24]. As the source bandwidth is finite, a bandwidth

Alternative Convolution Approach to Cagniard

55

restriction can be applied to the output of the convolution. The received signal, therefore, is bandlimited by physical sources such as the source itself and physical oceanography. This method is similar to that used by Jensen et al. [21, p. 613] in the section on frequency windowing. This procedure is defined by sampling the Green’s function G(r, z, ω) over a finite range of frequencies, therefore creating a bandlimited transfer function. Similarly, a series of adjacent bandlimited transfer functions can be created to account for frequency-dependent characteristics such as volume absorption of the ocean or multilayered bottom properties. Now the Green’s function G(r, z, ω) is determined from the eigenray amplitudes and phases for a finite range of frequencies. With this representation, a significant simplification occurs if the amplitude of each eigenray is independent of frequency. In this case, the only frequency-dependent part of Eq. (7) occurs in the eiωTn factor. As a result, only a single-frequency calculation is needed by the eigenray model, and the bandlimited transfer function can be assembled very efficiently. It remains to transform G(r, z, ω) for a receiver at each range and depth via an inverse FFT to obtain the bandlimited impulse response that is then convolved directly with the time-domain source signature. This provides the instantaneous pressure time series as given by Eq. (16). Up until now, an example of a specific source waveform F (t) has not been considered. Small explosive charges have been traditionally used as sound sources in underwater acoustics experiments. In particular SUS (signal underwater sound) charges provide an easily deployed, broadband transient signal for carrying out transmission loss experiments at sea. The initial portion of the shockwave pressure history has been found empirically to be well described by a scaled exponential decay law of the form [13] F (t) = Pm exp(−t/tc ),

(27)

where F is a modified form of the F used in Eq. (12) that is specific √ for the explosive charge waveform. The peak pressure is given by Pm = 50.4×1012 ( 3 W /R)1.13 μPa at 1 m, W is the charge √ weight in kg, and R is the slant range in m. The geometric similitude factor ( 3 W /R)1.13 accounts for attenuation and nonlinear effects that are not predicted by a linear theory. In addition, a slower decrease in fall-off from Pm has been√accounted for experimentally by a decay constant of the form tc = √ 81.2 × 10−6 3 W ( 3 W /R)−0.14 s. In Fig. 1 the shock waveform of Eq. (27) is shown for a SUS (W = 0.82 kg) charge at a range r = 1000 m. The Cagniard method does not include the effects of absorption, so the impulse for the direct path is the same as the direct path defined above (see [5]). The convolution of h(r, z, t − t  ) in Eq. (9) with F (t) in Eq. (27) gives P (r, z, t) = Pm e−t/tc

 N

n=1

An eiω0 t+iφn δ(t − Tn ).

(28)

56

R. L. Deavenport and M. J. Gilchrest 0.025

Pressure [106 Pa]

0.02

0.015

0.01

0.005

0

-0.005 0.6795

0.68

0.6805

0.681

0.6815

Time [s]

Fig. 1 Pressure history for the direct path pulse due to a 0.82-kg SUS charge at source depth z0 = 100 m, receiver depth z = 20 m, and range r = 1000 m for a sound speed c = 1475 m/s Fig. 2 Environmental parameters for the Pekeris waveguide

1475 0 100

α=0.0 dB/λ ρ=1.0 g/cm3

F=500 Hz 1600

200 α=0.5 dB/λ ρ=2.0 g/cm3

z [m]

c [m/s]

4 Image Theory for the Pekeris Waveguide The lossy Pekeris waveguide used in the numerical calculations is shown in Fig. 2. It consists of a uniform ocean of depth h = 200 m, water sound speed c = 1475 m/s, density ρ = 1 g/cm3 , and absorption α = 0 dB/λ overlying a fluid half-space bottom of sound speed cb = 1600 m/s, density ρb = 2 g/cm3 , and absorption αb = 0.5 dB/λ. Here λ is the acoustic wavelength in the relevant medium. For this configuration, image theory can be used to provide the eigenray information in Eq. (16) that is needed to determine the bandlimited transfer function [4, 16]. Each eigenray follows a sequence of straight-line segments that reflect from the surface and bottom of the waveguide (i.e., there is no refraction). The surface- and bottom-reflected eigenrays are modified by the plane-wave reflection coefficients

Alternative Convolution Approach to Cagniard

57

Vs and Vb , respectively. Although this representation is an approximation when Vb = Vb (θ ), it can be quite accurate in certain propagation situations. For cb > c, the representation can be improved by taking ray displacements at the sea-bottom interface into account for those grazing angles θ smaller than the critical ray angle, θc = cos−1 (c/cb ). As shown in [16], however, this ray displacement correction is not needed for the shallow-water Pekeris waveguide used in the numerical calculations below. The single-frequency image solution for the Pekeris waveguide can be cast in the form [16] (adapted from the treatment in [4, p. 346]): ∞

p(r, z) =

exp(ikRj ) exp(ikR01 ) exp(ikR02 )   + Vs + Vj (θj ) , R01 R02 Rj 4

(29)

=1 j =1

  where R01 = r 2 + (z − z0 )2 and R02 = r 2 + (z + z0 )2 are the slant ranges for the  direct path (first term) and surface-image path (second term), respectively.

2 , j = 1, . . . 4, are the slant ranges for the th set of four bottomRj = r 2 + zj bounce image paths with reflection-coefficient products Vj given by

V1 V2 V3 V4

= Vs−1 Vb , = Vs Vb , = Vs Vb , = Vs+1 Vb ,

θ1 θ2 θ3 θ4

= tan−1 (z1 /r), z1 = tan−1 (z2 /r), z2 = tan−1 (z3 /r), z3 = tan−1 (z4 /r), z4

= 2h − z − z0 , = 2h + z − z0 , = 2h − z + z0 , = 2h + z + z0 .

(30)

Here k = 2πf/c is the wavenumber in 0 < z < h. For the pressure-release sea surface used here, Vs = −1. The slant ranges of the eigenrays Rj have grazing angles θj at the receiver. The frequency-independent fluid–fluid bottom losses Vb are given by

Vb (θj ) =

m sin θj − m sin θj +

 

n2 − cos2 θj n2 − cos2 θj

,

(31)

where m = ρb /ρ and n = c/(cb + iβ) with β = α/(40π log10 e) Nepers/m. Since kRj = ωRj /c, each eigenray term of Eq. (29) can be put in the polar form of Eq. (7) in terms of its travel time Tj = Rj /c, namely, pj = |Vj /Rj | exp(iωTj + i arg Vj ).

(32)

The phase Tj is the same as the delay time that Chapman [11], [12, p. 19] uses in his WKBJ formulation of seismic wave propagation. Moreover, it is analogous to Hamilton’s principal function of classical mechanics as described by Whittaker [31] in his development of a quantum mechanical principal function involving noncommuting operators.

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R. L. Deavenport and M. J. Gilchrest

5 Numerical Example for the Pekeris Model Although the image representation described in the previous section is not exact for a shallow-water layer over a lossy half-space, it can be sufficiently accurate for certain waveguide parameters [16]. To provide some justification for its use in the Pekeris model of Fig. 2, we show in Fig. 3 a coherent transmission loss TL = −10 log10 |p/p0 |2 dB re 1 m versus range comparison between the image model and the well-known wavenumber integration model SAFARI [26] at a center frequency of 500 Hz for the range interval 0.5–1.5 km. Here p0 is a reference pressure at a range of 1 m, of unit magnitude for the image representation in Eq. (3). The source and receiver depths are at z0 = 100 m and z = 20 m, respectively. It is clear that the interference patterns obtained by both propagation models are in close agreement. The waveguide parameters considered here were adapted from the ones used by Jensen et al. [20] in their broadband signal propagation study with the exception of the sound speed profile near the surface of the water column. To evaluate the bandlimited delta-function responses in this waveguide, we compare the signal waveforms for the shockwave source of Fig. 1 (with R = 1 m) for the receiver at a range of r = 1 km. Figure 4 shows the transient propagation results at a range of r = 1 km obtained using the bandlimited convolution method and the image-based propagation model of Eq. (29). Only the first six arrivals are included in this figure. In addition, in Table 1 the full arrival information is tabulated. These calculations are based on 40 SAFARI Images

Freq: 500.0 Hz SDep: 100.0 m RDep: 20.0 m

TL [dB Re 1 m]

50

60

70

80

90 0.6

0.8

1

1.2

1.4

Range [km]

Fig. 3 Coherent transmission loss (TL) comparison at 500 Hz between the wavenumber integration model SAFARI and the image model for the Pekeris waveguide of Fig. 2

Alternative Convolution Approach to Cagniard

59

1e+10

Amplitude [Pa]

5e+09

0

-5e+09

-1e+10 0.66

0.68

0.7

0.72 0.74 Time [sec]

0.76

0.78

0.8

Fig. 4 Bandlimited convolution signals for a receiver at (r, z) = (1000, 20) m in the shallowwater Pekeris waveguide due to an explosive charge at (r, z) = (0, 100) m. Only the direct arrival, surface-image arrival, and the four first bottom-bounce arrivals are shown Table 1 Image theory arrival information at a range of 1 km for the first 6 arrivals Arrival type [j ] 01 02 11 12 13 14

Arrival angle [Deg] −4.57 6.84 −15.64 17.74 −25.64 27.47

Transmission loss [dB] 60.03 60.06 61.03 61.22 64.91 65.90

Phase angle [Deg] 0.00 180.00 −54.95 136.66 174.47 −4.14

Delay time [Sec] 0.680132 0.682830 0.704041 0.711832 0.752023 0.764149

Bottom loss [dB] 0.00 0.00 0.71 0.79 4.00 4.86

single-frequency images computed at f = 500 Hz. Since the complex planewave reflection coefficient is independent of frequency for a uniform half-space bottom (with absorption that varies linearly with frequency), only a single-frequency calculation within the band is required to determine the amplitude and phase of each arrival. As a result, the waveguide transfer function is readily filled in by simply populating the frequency-dependent term eiωTn in the polar form in Eq. (7) across the full ω-bandwidth for each image, summing over the images at each frequency, and then taking the inverse Fourier transform to the time domain. The resulting impulse response is then convolved with the source waveform to produce the signal that is propagated in the waveguide. For the bandlimited shockwave pulses shown in Fig. 4, a transfer function bandwidth of 1 kHz was used.

60

R. L. Deavenport and M. J. Gilchrest

The first two arrivals in Fig. 4 comprise the direct path and its surface image, respectively. The next four arrivals comprise the single bottom-bounce paths. The critical angle at the sea bottom for the acoustic parameters in Fig. 2 is given by θc = cos−1 (cw /cb ) = 22.8◦ . As seen in Table 1, the two early bottom-bounce arrivals have grazing angles |θ | less than θc , while the latter two bottom reflections have grazing angles |θ | that are greater than θc . As a result, they undergo approximately 4 dB more loss and exhibit smaller amplitudes. The relatively small bandwidth of 1 kHz is sufficient to resolve the six arrivals at this range. For comparison to the above results, Fig. 5 shows the transient waveforms for the delta-function impulse responses (identical to those obtained using the Cagniard method) that are convolved with the shockwave source function. The horizontal scale is the same as the one displayed in Fig. 4, while the vertical scale is larger to accommodate the higher amplitudes. It is evident that the delta-function pulse arrivals occur at the same arrival times as for the bandlimited ones, but the time resolution is much sharper. In essence, each delta-function arrival exhibits the nature of the explosive source signature in Fig. 1. This is more clearly seen in Fig. 6 that shows the first two arrivals of Fig. 5 on an expanded time scale. Here it is clearly seen that the arrivals show the characteristic exponential decay of the similitude shock waveform. 6e+10

Amplitude [μPa]

4e+10

2e+10

0

-2e+10

-4e+10

-6e+10 0.66

0.68

0.7

0.72 0.74 Time [sec]

0.76

0.78

0.8

Fig. 5 Delta-function response corresponding to the same configuration presented in Fig. 4. Note the change in the amplitude scale from Fig. 4

Alternative Convolution Approach to Cagniard

61

6e+10

Amplitude [μPa]

4e+10

2e+10

0

-2e+10

-4e+10

-6e+10 0.676

0.678

0.68

0.682 0.684 Time [sec]

0.686

0.688

Fig. 6 Expanded time scale of Fig. 5 showing just the direct path and surface-image arrivals

6 Conclusions In this paper, two convolution-based, time-domain procedures were described for simulating a transient signal propagating in a shallow-water waveguide. The deltafunction response Cagniard method was discussed and compared with a bandlimited alternative method. This alternative convolution method provides another technique to solve the transient problem in the time domain. The classical Cagniard method is applicable only for homogenous layers and only allows for multipath diffraction since the impulse is for the Heaviside step function. Further, this Cagniard method is limited by not including absorption or scattering effects that cause the ocean to behave as a low-pass filter. In contrast, the alternate convolution approach allows for nonuniform layers, which can accommodate more realistic sound speed profiles in the water column as well as allowing absorption within the ocean and the sea-bottom layers. In the frequency domain, the signal was taken to have the form of a sum of eigenray arrivals whose amplitudes could be taken to be constant over the effective frequency band of the source. In this case, only a single-frequency propagation run is required at the range of the receiver: the frequency-dependent part of the total signal is contained in the time phasor eiωTn and is readily incorporated into the medium’s impulse response before inverting via an inverse Fourier transform. This is in contrast to the usual Fourier synthesis method of generating bandlimited signals in the time domain: by running a single-frequency model many times at a set of

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frequencies across the band, filtering by the source spectrum, and then taking an inverse Fourier transform. It is worthwhile pointing out that there are situations, e.g., for a multilayered sea bottom, where the assumption of frequency-independent eigenray amplitudes is invalid. In this instance, the alternate convolution method would also require making multiple single-frequency calculations across the band in order to construct the transfer function. Moreover, it is not clear that simply modifying each bottom-reflected image by the appropriate plane-wave reflection coefficient at each frequency would be a sufficiently accurate approximation. A similar kind of situation involving a Gaussian modulated signal propagating in dispersive media has been discussed by Wait [29] for electromagnetic applications. Acknowledgments The authors gratefully acknowledge the support of Dr. Elizabeth A. Magliula and Dr. Anthony A. Ruffa for funding this research as part of ONR 2019 ILIR funding and the Chief Technology Office of the Naval Undersea Warfare Center, Division Newport (CTO NUWCNPT). Thanks to Dr. David J. Thomson for his help in understanding the Cagniard method and for using his previously published image method. Also, special thanks to John M. Tattersall (formerly of NUWC Division Newport) for introducing the bandlimited transfer function method.

References 1. K. Aki, P. Richards, Quantitative Seismology: Theory and Methods, vol. 1 (W.H. Freeman, New York, 1980) 2. A. Arons, Underwater explosions shock wave parameters at large distances from the charge. J. Acoust. Soc. Am. 26, 343–346 (1954) 3. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 2000), pp. 126–127 4. L. Brekhovskikh, Waves in Layered Media (Academic Press, New York, 1960) 5. J. Britt, Linear theory of bottom reflections. Rep. NOLTR 69–44, Naval Ordance Laboratory, White Oak (1969) 6. M. Brown, A Maslov-Chapman wavefield representation for wide-angle one-way propagation. Geophys. J. Int. 116, 513–526 (1994) 7. L. Cagniard, Reflection and Refraction of Progressive Seismic Waves (McGraw-Hill, New York, 1962) 8. J. Carson, Electric Circuit Theory and the Operational Calculus (Chelsea, New York, 1953) 9. C. Chapman, Generalized ray theory for an inhomogeneous medium. Geophys. J. R. Astron. Soc. 36, 673–704 (1974) 10. C. Chapman, Exact and approximate generalized ray theory in vertically inhomogeneous media. Geophys. J. R. Astron. Soc. 46, 201–233 (1976) 11. C. Chapman, Ray theory and its extensions: WKBJ and Maslov seismograms. J. Geophys. 58, 27–43 (1985) 12. C. Chapman, Fundamentals of Seismic Wave Propagation (Cambridge University Press, Cambridge, 2004) 13. N. Chapman, Measurement of the waveform parameters of shallow explosive charges. J. Acoust. Soc. Am. 78, 672–681 (1985) 14. M. Collins, The split-step Padé solution for the parabolic equation method. J. Acoust. Soc. Am. 93, 1736–1742 (1993)

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15. R. Deavenport, M. Gilchrest, Time-dependent modeling of underwater explosions by convolving similitude source with bandlimited impulse from the CASS/GRAB model. Rep. 12,176, Naval Underwater Warfare Center, Newport (2015) 16. R. Deavenport, M. Gilchrist, D. Thomson, Acoustic modelling of a transient source in shallow water. Appl. Acoust. 150, 227–235 (2019) 17. V. Dobrushkin, Mathematical tutorial for the Second Course, Part VI: The Cagniard Method. http://www.cfm.brown.edu/people/dobrush/am34/Mathematica/ch6/cagniard.html (2020) 18. F. Gilbert, D. Helmberger, Generalized ray theory for a layered sphere. Geophys. J. R. Astron. Soc. 27, 57–80 (1972) 19. A.D. Hoop, A modification of Cagniard’s method for solving seismic pulse problems. Appl. Sci. Res. B 8, 349–356 (1960) 20. F. Jensen, C. Ferla, P. Nielsen, G. Martinelli, Broadband signal simulation in shallow water. J. Comput. Acoust. 11, 577–591 (2003) 21. F. Jensen, W. Kuperman, M. Porter, H. Schmidt, Computational Ocean Acoustics, 2nd edn. (Springer, New York, 2011) 22. F. Kelly, Multimode and dispersive distortion in the very-low-frequency channel. Radio Sci. 5, 569–573 (1970) 23. B. Kennett, Seismic Wave Propagation in a Stratified Medium (Cambridge University Press, Cambridge, 1983) 24. P. LeBland, L. Mysak, Waves in the Ocean (Elsevier, New York, 1981) 25. M. Porter, The KRAKEN Normal Mode Program. Memo. SM–245, SACLANT Undersea Research Centre, San Bartolomeo, Italy (1991) 26. H. Schmidt, SAFARI Seismo-Acoustic Fast field Algorithm for Range-Independent environments. User’s Guide. Rep. SR–113, SACLANT Undersea Research Centre, San Bartolomeo, Italy (1988) 27. I. Sneddon, Fourier Transforms (McGraw-Hill, New York, 1951), pp. 31–32 28. D. Towne, Pulse shapes of spherical waves reflected and refracted at a plane interface separating two homogeneous fluids. J. Acoust. Soc. Am. 44, 65–76 (1968) 29. J. Wait, Distortion of pulsed signals when the group delay is a nonlinear function of frequency. Proc. IEEE (Lett.) 58, 1292–1294 (1970) 30. H. Weinberg, R. Keenan, Gaussian ray bundles for modeling high-frequency loss under shallow-water conditions. J. Acoust. Soc. Am. 100, 1421–1431 (1996) 31. E. Whittaker, On Hamilton’s principal function in quantum mechanics. Proc. R. Soc. Edin. A 61, 1–19 (1941)

Acoustic Transmission in a Low Mach Number Liquid Flow Scott E. Hassan

1 Introduction The theoretical and numerical treatment of acoustic waves in a moving medium has primarily been addressed by researchers in the atmospheric acoustics and aeroacoustics communities. Texts by Morse and Ingard [1], Ostashev [2], and Goldstein [3] collectively provide a broad theoretical background along with special cases that provide insight into the many salient features associate with the influence of a moving medium on sound propagation. Technical papers by Pierce [4], Blokhintzev [5], and Godin [6] provide a detailed theoretical foundation with regard to the governing differential equations and underlying assumptions. There are many special forms of the governing differential equations with associated derivations dependent on the underlying thermodynamic and hydrodynamic assumptions. Furthermore, acoustic assumptions such as small variations in properties over a wavelength are commonly used to further simplify the governing equations. Closed form solutions to the acoustic field in a moving medium are very limited. Typically, these solutions are restricted to cases of simple sources in a uniform steady flow field. The numerical solution to acoustic radiation in a moving medium has been previously addressed using both finite element and boundary element methods. Astley [7] reviews the development of computational methods applicable to acoustic propagation in subsonic flows with emphasis on radiation from turbofan engines. Resolution requirements and dispersion are also discussed in this paper. Boundary conditions applicable to acoustic perturbations in a mean flow have been addressed

S. E. Hassan () NAVSEA Newport, Newport, RI, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_4

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S. E. Hassan

by Eversman and Okunbor [8], and Eversman [9]. The use infinite wave envelope elements for radiation boundary termination was introduced by Astley [10] and extended for use in a uniformly moving medium by Eversman [11] and Parrett and Eversman [12]. A unique boundary element method was developed by Wu and Lee [13] that is applicable to sound propagation in a uniform mean flow. Their work provides an exact theoretical approach to predicting the far-field acoustic quantities under the restriction of a homogeneous mean flow. It is noted that acoustic propagation through a moving inhomogeneous liquid medium, to the far field, has not been previously addressed. This chapter focuses on acoustic radiation, from near field to far field, due to a piston source on a body in a low Mach number liquid flow. As a result of the fluid flow over the body, this problem gives rise to an inherently inhomogeneous and anisotropic medium through which the acoustic waves propagate. The theoretical and numerical development is presented in Sect. 2. Numerical results are presented and discussed for two cases in Sect. 3. These cases include a point source in uniform homogeneous flow, and a piston source located symmetrically about the stagnation point on a rigid body.

2 Theory The problem of interest consists of a body with a piston source generating active harmonic acoustic transmissions while in a low Mach number inviscid and irrotational flow field as shown in Fig. 1. Flow is incident on a rigid body defied by surfaces Sa , Sb , and Sc . The piston source is identified as surface Sa with uniform motion normal to the surface. The mid-body, Sb , is defined as an ellipse extending from the piston surface Sa , to the rigid termination, Sc . The resulting hydrodynamic flow field includes two regions: an inhomogeneous near field and homogeneous far field separated by the bounding surface, S∞ . The precise location of S∞ is dependent on the flow perturbations due to the body. The equations governing the hydrodynamic mean flow field are presented in Sect. 2.1. The equations describing the near-field acoustic velocity potential and pressures are introduced in Sect. 2.2. The integral expression for the far-field velocity potential is introduced in Sect. 2.3. Boundary conditions on the body and S∞ are derived in Sect. 2.4, followed by the finite element formulation in Sect. 2.5. In the following development, uppercase and lowercase variables are associated with the incompressible hydrodynamic mean flow and acoustic perturbations to the mean flow, respectively.

Acoustic Transmission in a Low Mach Number Liquid Flow

67

Fig. 1 High-speed flow incident on a body with an active acoustic surface Sa ( ), mid-body surface defined by an ellipse Sb ( ), and body termination surface Sc ( ). The external flow field includes two regions: an inhomogeneous near field and homogeneous far field separated by the bounding surface S∞ (—)

2.1 Hydrodynamic Mean Flow The steady-state hydrodynamic mean flow about the body is assumed to be inviscid and irrotational. It follows that the velocity field can be expressed as the gradient of a scalar potential field, , where − → V = ∇Φ,

(1)

and explicit dependence on position (x, y) is omitted for convenience. The total potential field in Eq. (1) is the summation of the perturbation potential and incident flow potential. For the problem of interest, this can be expressed as Φ = Φ  − U∞ x.

(2)

where  is the perturbation potential due to the presence of the body. Under the additional assumption of a low Mach number incompressible flow (M < 0.3), the perturbation velocity potential satisfies the Laplace equation ∇ 2 Φ  = 0.

(3)

For the problem of interest, Neumann boundary conditions are specified on the body surface. It follows from Eq. (2) that: ∇Φ  · nˆ = nˆ x U∞ ,

− → x ∈ Sa ∪ Sb ∪ Sc

(4)

For this chapter, the solution to Eqs. (3) and (4) is obtained using a standard boundary element method [14]. The implemented approach assumes constant

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S. E. Hassan

elements on the body and results in a linear system of equations for the unknown source strength densities. The perturbation potentials in the field, and associated gradients, are then evaluated using a weighted summation of the source strength densities. The resulting steady-state pressures can be obtained by utilizing Eq. (1) and Bernoulli’s equation. It follows that the steady-state static pressures can be expressed as P = P0 + ρCp U2∞ /2,

(5)

where the pressure coefficient Cp is expressed as − →2 Cp = 1 − V /U2∞ .

(6)

The P0 in Eq. (5) accounts for atmospheric and hydrostatic components of pressure. For this chapter, this term is assumed constant with a nominal value of 2 atmospheres (i.e., P0 = 202.650 kPa). The local density is computed using the pressure from Eq. (5) and a standard equation of state for seawater [15] with the assumption of constant temperature (20 ◦ C) and salinity (30 ppt). Local speed of sound is computed using the Del Grosso equation [16].

2.2 Acoustic Velocity Potential and Pressure in the Inhomogeneous Flow Region The inhomogeneous mean flow conditions described in Sect. 2.1 give rise to a spatially varying density, speed of sound, and flow velocity. This results in an inhomogeneous and anisotropic medium through which acoustic propagation must occur. Of particular relevance to this chapter is the linearized small amplitude wave equation in a moving steady-state inhomogeneous flow. Under these assumptions, the developments found in Pierce [4] and Blokhintzev [5] are most applicable and are utilized in the subsequent development. These formulations express the irrotational perturbation velocity associated with the acoustic field using the velocity potential, φ. It follows that: − → v = ∇φ,

(7)

where the ejωt time dependence is assumed for the acoustic quantities and suppressed for convenience. The acoustic pressure in the moving medium can be obtained from the velocity potential as [1]: p = −ρDω φ,

(8)

Acoustic Transmission in a Low Mach Number Liquid Flow

69

where ρ is the local density and the operator Dω in Eq. (8) is expressed as − → Dω = jω + V · ∇.

(9)

− → The spatially varying steady mean flow velocity vector, V , is obtained from the solution to the inviscid and irrotational steady flow problem developed in Sect. 2.1. The applicable linear wave equation for the acoustic velocity potential in an inhomogeneous medium with spatially varying mean flow is expressed as [4, 5] ∇ 2 φ + k2 φ +

 1 − → − → − → ∇ρ· ∇φ − M · ∇ M · ∇φ − 2jk M · ∇φ ρ

− →  2 M · ∇c  − → jkφ + M · ∇φ = 0, + c

(10)

where the acoustic wavenumber, k = ω/c, and c is the local speed of sound in the − → − → liquid. The Mach number vector is defined as M = V /c using local values of the velocity and speed of sound. The speed of sound in the above expression is noted to be a function of the salinity, temperature, and pressure. At a sufficiently large distance from the body, outside S∞ , the incident homogeneous flow will be unperturbed by the body and Eq. (10) can be simplified to a form suitable for a homogeneous medium with uniform flow in the x-direction. For this special case, it follows that: ∇ 2 φ + k2 φ − M2x

∂ 2φ ∂φ = 0, − 2jkMx ∂x ∂x2

(11)

where Mx is the x-component of the Mach number vector.

2.3 Far-Field Pressures The far-field pressures are obtained using a boundary element approach that has been developed for homogeneous flow [13]. Fundamental  to this approach is the → − existence of a free space Green’s function g − x |→ x 0 satisfying the following inhomogeneous form of the convected wave equation, Eq. (11), → − → −  →  2 −   − → − ∂g − x |→ x0 → − → → 2 − 2∂ g x | x 0 ∇ g x | x 0 + k g x | x 0 − Mx − 2jkMx ∂x ∂x2 −  → − → = −δ x − x 0 , (12) 2

where δ (·) is the Dirac delta function. Solution to Eq. (12) is obtained using the method described by Ostashev [17] that utilizes a Prandtl-Glauert transformation to obtain the free space Green’s function applicable to a moving homogeneous flow:

70

S. E. Hassan 

 Mx k0 (x−x0 ) 2 1−Mx

→ −  g − x |→ x 0 = −je √ j

1−M2x

4

 (2) H0

k0 1−M2x



   2 2 (x − x0 ) + 1 − Mx (y − y0 ) , 2

(13) where H(2) 0 (· ) is the Hankel function of the second kind, order zero. It is noted that the Green’s function does not satisfy reciprocity. This is due to the hydrodynamic flow, and is intuitively evident by noting the differences with acoustic propagation in the upwind and downwind directions. The resulting acoustic velocity potential → due to a point source of strength q0 located at the origin, − x 0 = (0, 0) can therefore be expressed as 

 k0 x

   −  jq0 e   k0 (2) → 2 2 2  φ x = H0 x + 1 − Mx y . 1 − M2x 4 1 − M2x j

Mx 1−M2x

(14)

The velocity potential and associated gradient on surface S∞ are used to compute the far-field pressures in the fluid. The basic approach utilizes an extension to the Helmholtz integral equation applicable to acoustic propagation in a homogeneous flow [13]. The derivation utilizes the convected wave equation, Eq. (11), and it is adjoint to derive a boundary integral expression for the velocity potential due to an arbitrary source distribution and bounding surface motion in homogeneous flow. Using this method, the far-field velocity potential in a homogeneous medium, with Mach number Mx in the x direction, can be expressed as → → → !  − →  ∂φ (− →  ∂g(− →  x 0) x |− x 0) → φ − x = g → x |− x0 −φ − x0 − 2jk0 Mx φ − x 0 nˆ x0 ∂ nˆ 0 ∂ nˆ 0 S∞     → → →  ∂φ (− →  ∂g(− x 0) x |− x 0) → → x |− x0 nˆ x0 dS. −M2x g − −φ − x0 ∂x0 ∂x0

(15) The velocity potential and normal derivative on S∞ are obtained as part of the finite element solution developed in Sect. 2.5. The acoustic pressures in the homogeneous flow, with velocity in x-direction, can be obtained from Eq. (8). It follows that:   ∂φ . p = −ρ jωφ + U∞ ∂x

(16)

where the velocity potential is obtained from Eq. (15). Far-field beam patterns are obtained as a function of the angle θ from the pressures in Eq. (16). The standard expression for a beam pattern is of the following form:

Acoustic Transmission in a Low Mach Number Liquid Flow

H (θ ) = 20log10 where R =



p (R, θ ) max {p (R, θ )}

71

,

(17)

R→∞

x2 + y2 and θ = tan−1 (y/x).

2.4 Boundary Conditions The governing differential equation, Eq. (10), is subjected to boundary conditions on the mid-body and body termination surfaces (Sb ∪ Sc ), active acoustic surface (Sa ), and the fluid surface (S∞ ) as indicated in Fig. 1. The mid-body and body termination surfaces are both assumed to be rigid and subjected to the Neuman boundary condition: ∇φ· nˆ = 0,

− → x ∈ Sb ∪ Sc

(18)

The boundary conditions on the active acoustic surface are assumed as a velocity distribution, normal to Sa . This imposed boundary condition is the source of acoustic waves propagating through the fluid medium and is specified as ∇φ· nˆ a = v0 (y) e−jk0 ycos(θt )

x = 0, −L/2 ≤ y ≤ L/2.

(19)

The v0 and θt in Eq. (19) are the specified complex amplitude of the velocity on Sa , and acoustic transmit beam steering angle. For this chapter, v0 (y) is assumed constant resulting in a piston source transmitting at angle θt . The computational domain for the unknown acoustic velocity potential, φ, is truncated at the surface S∞ as indicated in Fig. 1. The boundary conditions require the specification of φ, ∇φ· nˆ ∞ or the ratio of these quantities on S∞ . An approximation for homogeneous flow, asymptotically exact in the acoustic far field, can be obtained from the solutions for a point source given in Eq. (14). This ratio can be defined as Zφ =

φ , ∇φ· nˆ ∞

− → x ∈ S∞ .

(20)

Substituting Eq. (14) for the terms in Eq. (20) and assuming that S∞ is a circular cylindrical boundary results in the following expression for the boundary condition on S∞ :

72

S. E. Hassan − → 1− M ∞

Zφ =

k

2

(2)

H0 (ψ) "

,    2  − → → − → − − → 2 (2) (2) j M ∞ M ∞ ·ˆn∞ H0 (ψ)−H1 (ψ) 1− M ∞ 1− M ∞ ·ˆn∞

− → x ∈ S∞ . (21)

where: kR ψ= − → 1 − M∞

# 2

− → 2 R2 − M ∞ cos2 (θ ),

(22)

(2)

and H1 (· ) is the Hankel function of the second kind, order 1. In addition to the above boundary conditions, the acoustic impedance on S∞ will be useful in the subsequent development. The impedance is defined as the ratio of acoustic pressure to normal velocity. It follows from Eqs. (7) to (8) that the impedance can be expressed as

Z=

  − → −ρ jωφ + V · ∇φ ,

∇φ· nˆ ∞

− → x ∈ S∞ .

(23)

For the case of a uniform homogeneous flow in the −x direction, Eq. (23) can be written as ⎡ ⎤ (2) (2) H jH0 (ψ) − √ Mx cos(θ) (ψ) 1−M2x sin2 (θ) 1 ⎢ ⎥ − →  Z = ρc ⎣ x ∈ S∞ , (24) ⎦, (2) 2 2 −jMx cos (θ ) + H1 (ψ) 1 − Mx sin (θ ) where the x-component of the Mach number is Mx = − U∞ /c. For the limiting case where Mx → 0 the classical acoustic impedance of 2-dimensional circular ring is obtained: (2)

Z ∼ jρc

H0 (kR) (2)

H1 (kR)

,

Mx → 0.

(25)

2.5 Finite Element Formulation The method of weighted residuals [18] is implemented in this chapter to solve the governing partial differential equation and boundary conditions. The weighted residual method requires the inner product of a position-dependent weight function with the residual associated with the approximate solution to vanish. For this chapter, the residual of Eq. (10) is multiplied by a weighting function, w, and

Acoustic Transmission in a Low Mach Number Liquid Flow

73

integrated over the domain. The resulting equation, referred to as the weak form, is obtained after integration by parts and results in the following: −  −  !!!  → → ∇w · ∇ϕ − k2 wϕ − w ∇ρ·∇ϕ − M · ∇w M · ∇ϕ dV+ ρ V  −  %   − → !!!  $ − → → − → jk w M · ∇ϕ − M · ∇w ϕ − w 2 Mc·∇c jkϕ + M · ∇ϕ dV = V    !! − → − → − →  w ∇ϕ − M M · ∇ϕ − jk M ϕ · nˆ dS S∞ +Sa +Sb +Sc

(26) The surface integrals on the right-hand side of Eq. (26) can be simplified by substituting the boundary conditions from Eq. (19) into the integral over Sa with the following final result:    $  − → − → − → % w ∇ϕ − M M · ∇ϕ − jk M ϕ · nˆ dS = wv0 (y) e−jk0 ycos(θt ) dS. Sa

Sa

(27) Substituting Eq. (18) into the integral over the mid-body and body termination surfaces (Sb ∪ Sc ) results in:   $  − → − → − → % w ∇ϕ − M M · ∇ϕ − jk M ϕ · nˆ dS = 0. (28) Sb ∪Sc

The radiation impedance defined in Eq. (23), along with Eq. (20), can be used to simplify the integral over S∞ . It follows that: ⎧ ⎫ − →   $  ⎬  M 1 ⎨ − → − → − → % Zˆnx dS. w ∇ϕ − M M · ∇ϕ − jk M ϕ · nˆ dS = w ϕ 1+ ⎭ Zϕ ⎩ ρc S∞

S∞

(29) With the finite element method, the domain of integration in Eq. (26) is divided into discrete elements containing a specified number of nodes. Associated with each element are local interpolation functions, which operate on the nodal point unknown velocity potentials. These interpolation functions satisfy continuity of the velocity potential at the nodal points. In this chapter, Lagrange polynomials are used as interpolation functions and quadrilateral isoparametric elements with linear interpolation functions are utilized. The Galerkin finite element method requires the weight functions, w, to be made equal to the interpolation functions. This choice

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S. E. Hassan

of functions has the distinct feature of generating symmetric matrices. This overall approach has been implemented using Matlab.

3 Numerical Results and Discussion Numerical results are presented in this section for two special cases. The first case consists of an acoustic point source in a homogeneous flow and is used for model verification purposes. The second case is that of a piston source located symmetrically about the hydrodynamic stagnation point on a rigid body as shown in Fig. 1. For this case, far-field beam patterns are investigated as a function of Mach number, transmit frequency, and transmit angle.

3.1 Point Acoustic Source in a Uniform Homogeneous Flow The analytical solution for the velocity potential due to a uniform point source in a homogeneous moving medium provides a means to verify the finite element model solution. The analytical solution, for a point source of strength q0 and Mach number M, is given in Eq. (14). The finite element mesh is shown in Fig. 2 with an acoustic source located at the origin and uniform homogeneous flow in the

Fig. 2 Finite element mesh for a point acoustic source in uniform flow

Acoustic Transmission in a Low Mach Number Liquid Flow Table 1 Physical properties of the fluid

(a)

75 Property ρ∞ c∞

Value 1000 kg/m3 1500 m/s

(b)

Fig. 3 Point source solution to the velocity potential at k0 = 4π (lines-analytical, symbolsnumerical); (a) real and imaginary components of φ along x-axis at M = 0.1, (b) φ in the far-field normalized by the solution at M = 0

−x direction. The fluid properties for this case are listed in Table 1. The resulting velocity potential, due to the unit source in the homogeneous flow at M = 0.1 and a wavenumber of k0 = 4π is shown in Fig. 3a. The real and imaginary components of the finite element solution are noted to be in excellent agreement with the analytical solution. These results exhibit the well-known differences in upwind and downwind propagation. In particular, the wavenumber in the upstream direction (+x axis) is increased with k = k0 /(1 + M), corresponding to a reduction in wavelength. Downwind propagation (−x axis) wavenumbers are reduced with k = k0 /(1 + M), corresponding to an increase in wavelength. Far-field velocity potential results at M ≤ 0.3, normalized by the M = 0 solution, are shown in Fig. 3b. These results were obtained using the finite element solution on S∞ and the boundary integral expression for the velocity potential using Eq. (15). These numerical results are in agreement with the analytical solution and exhibit the theoretical dependence of velocity potential on the observation angle (θ) relative to the flow.

3.2 Piston Source Located Symmetrically About the Hydrodynamic Stagnation Point on a Rigid Body The problem of interest is depicted in Fig. 1 where uniform flow is incident on a body that consists of three surfaces Sa , Sb , and Sc . The active acoustic surface has

76 Table 2 Normalized geometric parameters

S. E. Hassan Parameter Active acoustic surface length, L Body termination length, Lc Mid-body ellipse (major axis, a) Mid-body ellipse (minor axis, b)

Value 1 1 2.5 0.35

a specified acoustic velocity, normal to Sa , given in Eq. (19). The rigid mid-body surface Sb is defined by an ellipse. The boundary conditions for the mid-body and body termination are given in Eq. (18) and correspond to a zero acoustic velocity normal to the surface. Normalized dimensions for the mid-body ellipse parameters, active acoustic surface, and body termination are listed in Table 2. Inside the dashed line, indicated as S∞ in Fig. 1, is the computational region where inhomogeneous flow is assumed. In this chapter, this boundary is represented as a circular cylindrical surface and the finite element computational domain is bounded by S∞ and Sa ∪ Sb ∪ Sc . The hydrodynamic velocity field in the region local to Sa is shown in Fig. 4a as a vector plot. The velocity field was computed using the boundary element method described in Sect. 2.1. The velocity vectors are normalized by the free stream velocity magnitude and, assuming incompressible flow, are independent of the incident Mach number. The local flow velocity introduces an anisotropic component to the quiescent acoustic medium. The computed velocity field is used to compute the scalar pressure coefficient using Eq. (6). It follows that the pressure coefficient is also independent of incident Mach number. Spatial variation of pressure coefficient, over the computational domain used for the inhomogeneous flow, is shown in Fig. 4b. The pressure coefficient clearly shows the expected stagnation points on Sa and Sc located on the plane of symmetry (y = 0). Large static pressure increases are noted near the stagnation points, with reductions along the mid body Sb particularly in the region where the body radius of curvature is smallest. In general, the speed of sound is a function of temperature, salinity, and static pressure. This chapter assumes constant temperature (20 ◦ C) and constant salinity (30 ppt) resulting in a speed of sound that is dependent only on static pressure. The spatially varying static pressure is the result of atmospheric, hydrostatic, and hydrodynamic components as given in Eq. (5). This study assumes a constant atmospheric and hydrostatic component of 2 atmospheres. The spatially varying static pressures are therefore due to the steady-state hydrodynamic flow field. The resulting speed of sound is therefore only a function of the local flow velocity magnitude and is computed using the Del Grosso equation [16]. The spatial variation in the speed of sound, normalized by c∞ from Table 1, is shown in Fig. 5 for M = 0.1, M = 0.2, and M = 0.3. It is clearly evident that the hydrodynamic flow significantly influences the speed of sound. The influence is greatest at higher Mach numbers where a 1.2× increase is noted in the stagnation region at M = 0.3. Furthermore, the speed of sound is significantly reduced in the area where Sa transitions to Sb due to the high local flow velocities and resulting decreases in

Acoustic Transmission in a Low Mach Number Liquid Flow

(a)

77

(b)

Fig. 4 Hydrodynamic solution to the velocity field, (a) normalized velocity vectors near Sa , and (b) pressure coefficient

(a)

(b)

(c)

Fig. 5 Spatial variation of the normalized speed of sound at (a) M = 0.1, (b) M = 0.2, and (c) M = 0.3

static pressure. These variations in the speed of sound give rise to an inhomogeneous medium through which sound must propagate. The normalized magnitude of the field pressures at frequency L/λ = 4.0 and transmission angle θt = 0◦ are shown in Fig. 6a and b for M = 0.0 and M = 0.3, respectively. The solution for the field pressures is obtained from Eq. (8) and the acoustic velocity potential is obtained using the finite element method developed in Sect. 2.5. The boundary condition on the active acoustic surface, Sa , is applied using the expression in Eq. (19) with a constant v0 (y). The boundary conditions on S∞ are applied using Eq. (21). Results are normalized by the maximum values of pressure magnitude for each case. These results provide a qualitative understanding of the hydrodynamic flow influence on acoustic field pressure levels. Significant differences between the M = 0 and M = 0.3 results are noted in the computational domain local to Sa . The M = 0 results in Fig. 6a exhibit a single

78

S. E. Hassan

(a)

(b)

Fig. 6 Normalized magnitude of field pressures in the finite element computational domain due to transmission from Sa at L/λ = 4.0 and θt = 0◦ ; (a) M = 0.0, and (b) M = 0.3

beam at the transmission angle with side lobes evident on S∞ . The M = 0.3 results in Fig. 6b differ significantly from the M = 0 results with the near-field beam split symmetrically about the transmission axis and larger side lobe levels qualitatively evident on S∞ . The splitting of the beam is due to the anisotropic and inhomogeneous medium induced by the flow as indicated in Figs. 4 and 5. The increase in sound speed and flow velocity gradients near the stagnation region causes refraction of the acoustic rays away from x-axis. This gives rise to the beam splitting and a broadened main lobe that are qualitatively observed in Fig. 6b. Far-field beam patterns are an important characteristic on acoustic transmit and receive system. The effect of Mach number on beam patterns for a transmission angle, θt = 0◦ , are shown in Fig. 7a and b for frequencies of L/λ = 2.0 and L/λ = 4.0, respectively. The influence of mean flow is more pronounced at the higher frequencies and is attributed to increased refraction with increases in L/λ. Additionally, it is evident that the main lobe beam width increases with corresponding increases in Mach number. These features are most evident in Fig. 7b where both beam width and side lobe levels are significantly increased due to the mean flow. The case of M = 0.3 in Fig. 7b is most significant with the broadened main lobe and lack of distinct side lobes. These features were also evident, in the qualitative sense, in Fig. 6b. This is due to the refraction that occurs in the near field of the surface Sa . At higher Mach numbers (M > 0.3), it is likely that the main lobe will become a local null, and the main beam will split into two beams symmetric about θ = 0. It is recommended that additional studies are pursued at higher Mach numbers to verify this assertion. The presented results have illustrated that far-field beam patterns are significantly influenced by the steady-state hydrodynamic flow field. The flow field directly introduces an anisotropy due to the velocity vector perturbations caused by the presence of the body. Additionally, the flow field indirectly introduces an inhomogeneity into the acoustic medium as a result of the sound speed dependence on static pressure.

Acoustic Transmission in a Low Mach Number Liquid Flow

(a)

79

(b)

Fig. 7 Influence of the Mach number on beam patterns at θt = 0◦ ; (a) L/λ = 2.0, (b) L/λ = 4.0

(a)

(b)

Fig. 8 Influence of the flow field on beam patterns at L/λ = 4.0 and θt = 0◦ , (a) M = 0.1, (b) M = 0.3

The relative influence of these two effects on the beam patterns at M = 0.1 and M = 0.3 are shown in are shown in Fig. 8a and b for a frequency of L/λ = 4.0. Both cases are plotted along with the M = 0 case. The condition referred to as neglecting flow in Fig. 8 refers to neglecting the Mach number terms in Eq. (10) while variations in the speed of sound are retained. It is clear that both speed of sound variations and local Mach number variations significantly influence the beam patterns. The influence of local Mach number variations increases with Mach number as shown in Fig. 8b. Arrays for generation and reception of acoustic signals will typically have the capability to transmit at selected angles. The influence of the flow field on beam patterns at a transmit angle θt = 20◦ is shown in Fig. 9a and b for frequencies of L/λ = 2.0 and L/λ = 4.0, respectively. A shifting of the maximum response axis to the transmit angle, θt , occurs at M = 0 for both frequencies. As Mach number is increased, the main response axis is shifted to angles greater than θt . This is due to refraction away from the x-axis resulting from the sound speed and flow velocity

80

S. E. Hassan

(a)

(b)

Fig. 9 Influence of the flow field on beam patterns at θt = 20◦ , (a) L/λ = 2.0, (b) L/λ = 4.0

gradients in the near field of Sa as indicated in Fig. 4a and b. Compensation for this Mach number, frequency, and transmit angle-dependent shifting was not addressed in this chapter. The main lobe beam width of the transmitted acoustic signal is a fundamental acoustic systems design parameter. A standard measure of beam width is the angle (degrees) where the response is reduced from the main response axis by −3 db. The classical beam pattern from a system at M = 0 is characterized by a beam width that is inversely proportional to frequency. The main lobe beam width as a function normalized frequency (L/λ) at θt = 0◦ and θt = 20◦ is shown in Fig. 10a and b, respectively. Both cases exhibit a beam width that increases significantly as a function of Mach number. There is also a monotonic decrease in beam width as a function of frequency for all cases, except M = 0.3. The M = 0.3 case exhibits a minimum achievable angle, due to near-field refraction. This is an important finding as it can significantly impact acoustic systems design. It is unknown whether this effect occurs at the lower Mach numbers and higher frequencies. Finite element mesh limitations limited the frequency range of this chapter.

4 Summary A unique finite element and boundary element method was developed and implemented to gain physical insight into the problem of acoustic transmission in a low Mach number liquid flow. The special case of a piston source located symmetrically about the hydrodynamic stagnation point on a rigid body was investigated as a function of Mach number, transmit frequency, and transmit angle. The results exhibited a strong Mach number dependence on the far-field beam patterns. In particular, increases in beam width were noted along with a minimum achievable beam width, at the highest Mach number investigated. It was also found that main beam axis shifted as a function of Mach number for transmission off the

Acoustic Transmission in a Low Mach Number Liquid Flow

(a)

81

(b)

Fig. 10 Main lobe beam width as a function of L/λ; (a) θt = 0◦ , (b) θt = 20◦

main response axis. These effects are due to the inhomogeneous and anisotropic components of the fluid that are introduced by a moving medium. Additional studies to investigate the beam width minimum, at higher transmit frequencies, are recommended. Acknowledgments This work was supported by funds from the In-house Laboratory Independent Research Program at the Naval Undersea Warfare Center Division, Newport, Rhode Island.

References 1. P.M. Morse, K.U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968), pp. 698–737 2. V.E. Ostashev, Acoustics in Moving Inhomogeneous Media (Spon, London, 1997) 3. M. Goldstein, Aeroacoustics (McGraw-Hill, New York, 1976), pp. 1–54 4. A.D. Pierce, Wave equation for sound in fluids with unsteady inhomogeneous flow. J. Acoust. Soc. Am. 87, 2292–2299 (1990) 5. D. Blokhintzev, The propagation of sound in an inhomogeneous and moving medium I. J. Acoust. Soc. Am. 18, 322–334 (1945) 6. O.A. Godin, An exact wave equation for sound in inhomogeneous, moving, and non-stationary fluids, in IEEE OCEANS 2011 (2011), pp. 1–5 7. R.J. Astley, Numerical methods for noise propagation in moving flows, with application to turbofan engines. Acoust. Sci. Technol. 30(4), 227–239 (2009) 8. W. Eversman, D. Okunbor, Aft fan duct acoustic radiation. J. Sound Vib. 213, 235–257 (1998) 9. W. Eversman, The boundary condition at an impedance wall in a non-uniform duct with potential mean flow. J. Sound Vib. 246, 63–69 (2001) 10. R.J. Astley, G.J. Macaulay, J.P. Coyette, L. Cremers, Three-dimensional wave envelope elements of variable order for acoustic radiation and scattering. Part I. Formulation in the frequency domain. J. Acoust. Soc. Am. 103, 49–63 (1998) 11. W. Eversman, Mapped infinite wave envelope elements for acoustic radiation in a moving medium. J. Sound Vib. 223, 665–687 (1999) 12. A.V. Parrett, W. Eversman, Wave envelope and finite element approximations for turbofan noise radiation in flight. AIAA J. 24, 753–759 (1994)

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13. T.W. Wu, L. Lee, A direct boundary integral formulation for acoustic radiation in a subsonic uniform flow. J. Sound Vib. 175, 51–63 (1994) 14. C.A. Brebbia, J. Dominguez, Boundary Elements: An Introductory Course (McGraw-Hill, New York, 1989), pp. 45–134 15. N.P. Fofonoff, R.C. Millard, Algorithms for the computation of fundamental properties of seawater. UNESCO Tech. Papers Mar. Sci. 44, 1–53 (1983) 16. V.A. Del Grosso, New equation for the speed of sound in natural waters (with comparisons to other equations). J. Acoust. Soc. Am. 56, 1084–1091 (1974) 17. V.E. Ostashev et al., Equations for finite-difference, time-domain simulation of sound propagation in moving inhomogeneous media and numerical implementation. J. Acoust. Soc. Am. 117, 503–517 (2005) 18. O.C. Zienkiewicz, The Finite Element Method in Engineering Science (McGraw-Hill, London, 1971), pp. 698–737

Lift Production Using Differential Cavity Ventilation on a Symmetric Hydrofoil Aren M. Hellum and David E. Yamartino

1 Introduction Background When the local pressure in a fluid is reduced below the vapor pressure, the liquid is said to have “cavitated” at that location. Cavitation near lifting surfaces is a subject of significant naval interest because highly loaded propeller blades can cavitate locally, both reducing the lift [1] and potentially inviting damage to the surface at the point of cavity collapse [2]. The former was first observed by Parsons in the trials of the pioneering craft Turbinia [3]; for that craft, additional shafts were employed in order to reduce blade loading. Supercavitating propellers, which are intended to be run with a cavity over a significant part of their chord, have since been designed (e.g., [4]) as a way of mitigating the problem for certain applications. A different thread of work has investigated cavitation as a means of producing drag reduction underwater. By fully enveloping a submerged body in a volume of gas, significant drag reduction can be achieved [5]. The apotheosis of this is underwater ballistics, in which a well-designed bullet running within such a cavity can approach [6] or exceed [7] the speed of sound in water. As the size of the cavityrunning body increases, artificial ventilation can be employed [8] in order to produce a gas bubble at speeds below those required to produce a natural (vapor) cavity. Theory: Creating Lift with Gas The research on cavitating hydrofoils has largely treated the subject as a problem to be mitigated and furthermore created as a side effect of the angle of attack of the foil. However, the fact that loss of lift due to

A. M. Hellum () · D. E. Yamartino Naval Undersea Warfare Center, Newport, RI, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_5

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cavitation is a significant problem also indicates that controlling the cavity near a foil could create a controllable lift force. A conceptual model indicating the direction and approximate magnitude of this force follows. Figure 1a is a depiction of the coordinate system and important flow features of a hydrofoil which is designed to use this method of lift production. A photograph of the vented fin in operation is depicted in Fig. 1b. The free-stream speed and pressure are U∞ and p∞ , respectively. The magnitude of the gas flow rate Q is not used in this analysis and is indicated only by its presence or absence. On the wetted side (Q = 0), the streamline originating at the cavitator reattaches at xr , and the outward pointing normal is nˆ w . On the cavity side (Q = 0), the contact location of the cavity is given by xc , and the outward pointing normal is nˆ c . Since the foil is symmetric, nˆ c = −nˆ w . For this simple analysis, it is assumed that the pressure in the dead region on the wetted side, 0 ≤ x ≤ xr , is equal to the reference pressure and that CP (xr < x ≤ c) is the same as found on the fully submerged foil. On the cavity side, we likewise

Fig. 1 (a) Cross-sectional view of the coordinate system and flow features of a differentially vented symmetric hydrofoil. The foil is pictured in a vent-down orientation. The streamline from the cavitator to xr captures a liquid volume, while the streamline from the cavitator to xc captures the cavity. (b) Still image taken of vented fin in operation. Flow is from left to right

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assume that CP (xc < x ≤ c) matches that found on the fully submerged foil and that the cavity pressure pc is uniform. The pressure coefficient CP and the cavitation number σ c are: CP (x) =

p(x) − p∞ p ∞ − pc , σc = 2 2 (1/2) ρU∞ (1/2) ρU∞

Applying the foregoing assumptions (after some manipulation) yields an estimate for the force in the y-direction (FLift ) produced by the geometry above: FLif t

1 2 = − ρU∞ S 2



xc xr

   Cp (x) + σc nˆ w · jˆ dx +



xr

   ˆ σc nˆ w · j dx

0

(1) This quantity will generally be positive, indicating a force toward the wetted side of the foil; for a standard foil, CP (x) < 0 over most of the chord, and |CP (x)| > σ c if a stable cavity is being produced. The second integral is small for small xr , because of both the integration range and minimal y-projection of the surface normal. This force can be converted to a lift coefficient CLift for easier comparison to standard fully wetted hydrofoils. A proof of concept of this theory is presented in the following section.

2 Results Apparatus The data in the present work were acquired at the Naval Undersea Warfare Center in Newport, Rhode Island (NUWC). The water tunnel used has a 12-by-12 test section. The apparatus employed is shown in Fig. 2. The 8” chord foil has a NACA0012 profile. The foil span is nominally equal to that of the tunnel, but care was taken to avoid mechanically grounding the foil on the tunnel edges so that force data can be acquired. The fences shown in Fig.2a are used to keep the ventilation gas from leaking from the vented to the unvented side of the foil. These fences are transparent to enable better imaging of the cavity. The isometric view depicted in Fig. 2b shows the position of the cavitator relative to the location where the ventilation gas is expelled. The stainless steel disk shown in the top center of Fig. 2a and the top right of Fig. 2b is mechanically coupled to an ATI Delta 6-DOF load cell which is outside the tunnel. Basics of Lift Production The relationship between the volume flow rate of ventilated gas Q and the lift production is depicted in Fig. 3a. All data in this figure were acquired under the following conditions: U∞ = 3.35 m/s, ½” cavitator, ventilated side down, and with zero angle of attack. The “Q steady” data were acquired by quickly ( 0 for typical airfoils [9]; the introduction of gas in only in that region therefore adds lift to the ventilated half-foil by reducing the magnitude of the positive CP in this forward region. An improved apparatus has been built to permit measurement of the profile CP (x) using a series of pressure taps; measurements using this setup are ongoing. Scaling The results presented in Fig. 3 have also been observed at other values of U∞ . These results are presented in dimensional form in Fig. 4a. The same trend is obvious for each of these curves—negative lift at low Q, followed by a step change to positive lift and subsequent insensitivity to additional gas. As expected, increasing U∞ increases both the magnitude of the lift produced and the amount of ventilation gas required to produce the positive step. The results shown in Fig. 4b have been scaled to remove the dependence on U∞ . These relationships are as follows: CQ =

Q U∞ d S ,

CLif t =

Fr =

U∞ √ gd

FLif t 2 cS (1/2) ρU∞

In the above, ρ is the density of the water, g is acceleration due to gravity, c is the chord of the foil, d is the cavitator size in the y direction, and S is the foil’s span. These coefficients are used commonly in the literature (e.g., [9, 10]), but some discussion of the length scales chosen is warranted. The characteristic area chosen for CLift is cS, which is the planform area of the foil. This area has been chosen so that the lift coefficient of a differentially ventilated foil can be compared to a

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Fig. 4 Force produced at ten values of U∞ . Colors are shared between velocities. (a) Unscaled relationships. (b) Scaled relationships, including a reference curve for a fully wetted foil at nonzero angle of attack

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standard foil lacking a cavitator; as a reference point, the CLift of a fully wetted ◦ NACA 0012 foil at an angle of attack of 4 has been included in the figure. In contrast, we have notionally taken the characteristic area used in the ventilation coefficient CQ to be the cavitator area. This is because the cavity dimensions √ are dependent upon the drag produced by the cavitator. An additional factor of 1/ CD0 is likely to be appropriate in the CQ scaling for a general forebody [11]. The length scale used in the Froude number Fr has been chosen because it characterizes the distance between the streamlines separating from the upper and lower termini of the cavitator in the gravity-normal direction. This number is most interesting in describing deviations from the main trend in Fig. 4b. The two lowest values of Fr display the transition to positive flow at notably lower values of CQ than at other speeds. This makes sense, given the orientation of the foil in these tests; because vents are beneath the foil, the buoyancy of the gas causes it to be trapped, forming a continuous cavity at lower speeds. The Froude √ number captures this effect because it uses a gravity-based characteristic velocity gd. We expect that this trend will be reversed in a vent-up configuration, such that low Fr runs will require more gas than the main trend. The deviation from the trend at the highest measured Fr is more likely to be the result of mechanical deflection at those speeds; following the Fr = 16.3 run, a higher speed was attempted which damaged the apparatus. The improved apparatus, built primarily to permit the measurement of Cp (x), employs a metal strongback as reinforcement. Hysteresis Preliminary investigation into the hysteretic behavior of the cavity is presented in Fig. 5. It has been observed that less gas is required to maintain a

Fig. 5 Hysteretic behavior, in which the cavity is maintained using significantly less gas than required to establish it

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cavity than to establish it. Likewise, a rapid decrease in ventilation gas does not instantly decrease the lift force provided. Quantification of this phenomenon for a differentially ventilated system is in progress and is necessary for developing useful control systems based on the differential ventilation phenomenon.

3 Concluding Remarks ◦

The present data show that a differentially vented foil at 0 angle of attack can ◦ produce a significant amount of lift, equivalent to 4 angle of attack for a fully wetted version of the same foil. This indicates that it should be possible to produce useful levels of vehicle control forces without changing the angle of attack of the control surfaces. These forces are delivered as a “step” when a minimum level of ventilation is provided. A working theory for the production of lift has also been developed and is consistent with the measurements obtained to date. Work to more fully confirm the theory is ongoing, in concert with efforts to more fully understand the hysteretic behavior. Acknowledgments The apparatus presented in this work was designed by Charles Henoch and Dana Hrubes, informed by proof-of-concept testing they performed on an earlier implementation of the idea. The support of the In-House Laboratory Independent Research program at NUWC-Newport is gratefully acknowledged.

References 1. M. Potter, J. Foss, Fluid Mechanics (Great Lakes Press, Okemos, 1982) 2. A. Philipp, W. Lauterborn, Cavitation erosion by single laser-produced bubbles. J. Fluid Mech. 361, 75–116 (1998) 3. C. Parsons, The application of the compound steam turbine to the purpose of marine propulsion. Transactions. Inst. Nav. Archit 38, 232 (1897) 4. M. Tulin, Supercavitating Propellers: History, Operating Characteristics, Mechanism of Operation (Hydronautics Inc., Laurel, 1964) 5. I. Kirschner et al., Supercavitation research and development, in Undersea Defense Technologies, (Undersea Defense Technologies, Waikiki, 2001) 6. T. Truscott, B. Epps, J. Belden, Water entry of projectiles. Annu. Rev. Fluid Mech. 46, 355–378 (2014) 7. J. Hrubes, High-speed imaging of supercavitating underwater projectiles. Exp. Fluids 30(1), 57–64 (2001) 8. J.P. Franc, J.M. Michel, Fundamentals of Cavitation (Springer Science & Business Media, 2006) 9. I. Abbott, A. von Doenhoff, Theory of Wing Sections, Including a Summary of Airfoil Data (Courier Corporation, 1959) 10. A. May, Water entry and the cavity-running behavior of missiles, SEAHAC TR 75-2., 1975 11. V. Semenenko, Artificial Supercavitation: Physics and Calculation (von Karman Institute, Brussels, 2001)

An Exact Solution for a Class of Kalman Smoothers Anthony A. Ruffa and Tod E. Luginbuhl

1 Introduction Ideally, one would like to know how long the interval should be when using a fixed interval smoother. For linear, Gaussian assumptions, it is reasonable to think that the effective interval length is determined by the state transition, output, measurement covariance, and process noise covariance matrices. Numerical solutions have been known since the 1960s [1, 2] and can be used to determine when increasing the interval (or batch) length ceases to improve the solution. In particular, defining the optimum interval length as the one where increasing the length by one or more updates no longer decreases the covariance matrix corresponding to the smoothed state estimate for the middle of the batch. To find the solution, it is necessary to run the covariance recursion for the fixed interval smoother for each interval length until the covariance no longer decreases. While this is straightforward to do and is no longer computationally demanding, it is a brute force approach. Solving the difference equations corresponding the fixed interval smoother exactly and using the solutions to determine the appropriate smoothing interval length is a more elegant and insightful approach. To this end, exact analytical solutions for linear, Gaussian discretized continuous-time kinematic models with one dimensional measurements and one- and two-dimensional state vectors are derived in this chapter. When these solutions are parameterized using the track index [3], they provide clear insight into the smoothing interval length. Kalata and Chmielewski [4] used the track index to find expressions for the smoothed state covariance matrices for the α − β and α − β − γ filters. They

A. A. Ruffa · T. E. Luginbuhl () Naval Undersea Warfare Center, Newport, RI, USA e-mail: [email protected]; [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_6

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defined the smoothing improvement ratios as the smoothed variance of the position, velocity, and acceleration estimates divided by the corresponding (forward) filter variances and plotted these ratios as a function of the steady-state feedback gain for position, α. Kalata and Chmielewski did not discuss optimum interval length However, since the track index is proportional to (1 − α)−1 , for highly maneuvering objects, their plots show that smoothing provides no improvement as α approaches one. In a closely related paper, Ogle and Blair [5] examined the performance of the fixed-lag smoother as a function of the track index for α − β filters. They describe a straightforward algorithm for computing the smoothed state covariance matrix at the chosen fixed lag. Ogle and Blair define the smoothing gain as the filtered state covariance matrix minus the fixed-lag smoothed covariance. The authors plot the smoothing gain as a function of the tracking index for a variety of fixed-lag intervals. These plots clearly show that there are optimal fixed-lag intervals for different values of the track index. Like the results for [4], Olge and Blair’s results show that as the track index becomes large, fixed-lag smoothing provides no improvement over filtering. Unlike [4] and [5], the filter or smoother here is not in steady state, but the linear, Gaussian models describing an object’s motion are time invariant. When these kinematic models converge to the steady state, they are equivalent to the α and the α − β filters [6]. However, [4] and [5] use the discrete-time kinematic models, whereas discretized continuous-time kinematic models are used here because the process noise covariance matrices are invertible [6]. This makes the derivation of the Kalman smoother as a block tridiagonal system of difference equations more straightforward. The block tridiagonal system of equations corresponding to the Kalman smoother is parameterized by the track index. Then the methods of solving the difference equations from [7] and [8] are used to solve the block tridiagonal system of equations. Analytic solutions are found for the block tridiagonal systems that correspond to discretized continuous-time kinematic models with one-dimensional measurements and one- and two-dimensional state vectors.

2 Kalman Smoother Derivation The fixed interval Kalman smoother is the solution to the following estimation problem. Suppose one needs to estimate a sequence of state vectors, X = {xn }, for 0 ≤ n ≤ N, from a sequence of measurement vectors, Z = {zm },

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for 1 ≤ m ≤ N. Each state vector, xn , is real with dimension Dx , and each measurement vector, zn , is a real with dimension Dz . The state vector sequence, {xn }, is a linear Gauss-Markov process defined by xn = F xn−1 + ωn where ωn is a zero mean Gaussian random vector of dimension Dx with covariance matrix Q and A is a real Dx × Dx matrix. ωn is statistically independent of ωm for ¯ all n = m. The prior state vector, x0 , is Gaussian with mean x¯ and covariance P. Each measurement vector, zn , equals zn = H xn + ηn where ηn is a zero mean Gaussian random vector of dimension Dz with covariance matrix R and H is a Dz × Dx matrix. The Gaussian random vectors ηn are statistically independent of one another and are statistically independent of ωm for all m. ˆ is The maximum a posteriori (MAP) estimate of the state vector sequence, X, obtained by solving a block triadiagonal system of linear equations [9]. Because all the component distribution are Gaussian, the MAP solution is identical to the posterior mean square error estimate as well [10]. This system of block tridiagonal equations is given by ⎡

A0 ⎢ Bt ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢0 ⎢ . ⎢ . ⎢ . ⎢ ⎢ ⎢ ⎢ ⎣

B 0 0 A1 B 0 Bt A1 B . . 0 .. .. .. .

··· ··· ..

.

.. .. .. . . . · · · 0 Bt A1 · · · 0 0 Bt

⎤

⎡ ⎤ ⎡ ⎤ x0 z˜ 0 ⎥ ⎥ ⎢ x ⎥ ⎢ z˜ ⎥ ⎥⎢ 1 ⎥ ⎢ 1 ⎥ ⎥ ⎢ x ⎥ ⎢ z˜ ⎥ ⎥⎢ 2 ⎥ ⎢ 2 ⎥ ⎥⎢ . ⎥ ⎢ . ⎥ ⎥⎢ . ⎥ ⎢ . ⎥ ⎢ . ⎥=⎢ . ⎥ .. ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ . ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ 0 ⎥⎢ ⎥ ⎣xN −1 ⎦ ⎣z˜ N −1 ⎦ ⎦ B z˜ N xN A2

where the matrices in the block tridiagonal matrix equal Bt = −Q−1 F, A0 = P¯ −1 + Ft Q−1 F, A1 = Q−1 + Ht R−1 H + Ft Q−1 F, A2 = Q−1 + Ht R−1 H, and the vectors on the right side equal

(2.1)

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z˜ 0 = P¯ −1 x¯ and z˜ n = Ht R−1 zn for 1 ≤ n ≤ N. This system of difference equations is too general to obtain analytic solutions that have any advantage over conventional solutions due to the complexity of the analytic solutions. Consequently, only the discretized continuous-time kinematic models with one-dimensional measurements and one- and two-dimensional state vectors are considered here.

3 Two Special Cases These two special cases are commonly used in the tracking literature. In these two cases, the measurement at time nT , zn , and the measurement noise ηn are scalars. The variance of the measurement noise is σr2 . For these two cases, it is useful to move the measurement variance from the right side of (2.1) to the left side. This allows the filters to be parameterized in terms of the tracking index. The tracking index is a dimensionless parameter introduced by Kalata in [3]. The tracking index is the ratio of an object’s position uncertainty due to maneuverability and the measurement uncertainty. Following [6], the track index is defined here as λ2 =

T 3 σq2 σr2

(3.1)

where T is the sampling period (time between updates), σq2 is the power spectral density of the object’s process noise, and σr2 is the measurement variance. When the tracking index is less than one, the measurement noise variance is larger than the process noise variance. When the tracking index is greater than one, the opposite is true. Reasonable values for the track index, λ2 , range from 10−3 to 103 . Intuitively, large values of the tracking index mean the object is maneuvering a lot and the measurement noise is small. When the tracking index is small, the problem is dominated by the measurement noise and the object is not maneuvering very much. To parameterize the tridiagonal system of equations in terms of the tracking index, the matrices on the left side of (2.1) and the measurement vector on the right side need to be redefined. Multiplying both sides of (2.1) by the measurement variance, σr2 , yields Bt = −σr2 Q−1 F, A0 = σr2 P¯ −1 + σr2 Ft Q−1 F,

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A1 = σr2 Q−1 + σr2 Ht R−1 H + σr2 Ft Q−1 F, A2 = σr2 Q−1 + σr2 Ht R−1 H for the left side. The vectors on the right side are now z˜ 0 = σr2 P¯ −1 x¯ and z˜ n = Ht zn for 1 ≤ n ≤ N. This allows the matrices A0 , A1 , A2 , and B to be parameterized by the tracking index.

3.1 One-Dimensional State Vector Since the filter is a first-order filter, Dx = 1, and each unknown state vector is simply scalar: xn = x(nT ) where T is the sampling period. As stated previously, the measurement is also scalar: z˜ n = z(nT ), so Dz = 1 as well. Since all of the terms in (2.1) become scalars, they are denoted here without the bold typeface. In this case, F is just the scalar, 1, and the variance of the process noise, ωn , equals σq2 T 3 /3. The output, H, also equals 1, and the variance of the measurement noise equals σr2 as mentioned previously. Using these definitions, the matrices on the right side of (2.1) scaled by the measurement variance equal B t = −3λ−2 , A0 =

σr2 + 3λ−2 , P¯

A1 = 6λ−2 + 1, and

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A2 = 3λ−2 + 1 where P¯ is the variance of the initial state, x. ¯ On the left side, the first entry is z˜ 0 = σr2 P¯ −1 x, ¯ and all the rest are z˜ n = zn . These equations simplify further when P¯ equals some scaler times the process noise covariance, Q. If P¯ = cQ, then A0 = 3(1 + 1/c)λ−2

(3.2)

2 ). While this choice for the initial covariance may seem contrived, ¯ and z˜ 0 = 3x/(cλ there is some physical intuition behind it. If the state is propagated forward without any measurements, then the uncertainty in the state is proportional to Q because F = 1 in this case (set P¯ = Q and propagate the state). Consequently, cQ for some c > 0 is a reasonable choice for the initial covariance in the absence of other information.

3.2 Two-Dimensional State Vector Each xn in the vector of unknowns is a 2 × 1 real vector (i.e., Dx = 2) such that  xn =

x(nT ) x(nT ˙ )



where T is in seconds and represents the update interval. The measurement vector on the right side of the linear equation is made up of 2 × 1 vectors: z˜ n =

  z(nT ) . 0

For a constant velocity model with random accelerations, the system matrices are given by   1T F= , 0 1 & ' H= 10 ,  3  2 2 T /3 T /2 , Q = σq T 2 /2 T and the measurement covariance matrix, R, is simply the scalar measurement noise variance σr2 . The process noise power spectral density equals σq2 .

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Multiplying the left and right side of (2.1) by the scalar measurement noise variance, σr2 , gives the following definition of the matrices in the block tridiagonal system. The matrix A1 equals A1 =

  1 24 + λ2 0 . 0 8T2 λ2

The final 2 × 2 matrix on the diagonal equals   1 12 + λ2 −6 T A2 = . 5 λ2 −6 T 4 T 2 The matrix on the upper diagonal equals   1 −12 −6 T B= 2 . λ 6T 2T2 Bt appears on the lower diagonal. After multiplying the right side of the tridiagonal system of equation by σr2 , the first entry equals z˜ 0 = σr2 P¯ −1 x¯ . When P¯ = c Q, z˜ 0 = c−1 σr2 Q−1 x¯ . The rest of the vectors on the right-hand side are 

 z(nT ) z˜ = . 0 The 2 × 2 matrix A0 is given by   1 12 6 T + σr2 P¯ −1 . A0 = 2 λ 6T 4T2 If the prior covariance P¯ = c Q, then   1 12(1 + 1/c) 6 T (1 − 1/c) A0 = 2 . λ 6 T (1 − 1/c) 4 T 2 (1 + 1/c) Setting P¯ equal to c Q may seem even more contrived than in the one-dimensional case. However, a similar reasoning applies in two dimensions. Assume P¯ = Q, and propagate the state vector forward with no measurements. The resulting covariance

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for the state is a linear function of F and Q. This is true no matter how many times in row the state is propagated forward with no measurements. A simple approximation to the resulting covariance is a positive, nonzero, constant times Q.

4 Solution Approach 4.1 One-Dimensional State Vector Equation (2.1) is typically solved using either of the methods from [1] or [2]. If the covariance matrices corresponding to the state estimates are of no interest, then a conventional linear equation solver (e.g., MATLAB “backslash”) can be used. Here we instead employ an approach that leads to an analytical solution, which allows us to understand the mechanisms governing the behavior of the onedimensional filter. The solution approach is specifically tailored to the structure of (2.1), i.e., a symmetric tridiagonal Toeplitz system. While this analytic approach like a conventional linear equation solver does not yield the covariance matrices of the state estimates, the analytic solution does provide insight into the required smoothing interval. If for all n the measurements z˜ n equal zero except for some n0 , then the analytic solution involves both exponentially growing and exponentially decaying solutions that meet at the update number n0 . The envelope defined by these two solutions provides precise guidance for the optimum interval length for a given choice of the tracking index, i.e., λ2 . This result is valid for the entire solution because of a translational symmetry with respect to the update number, n0 , provided that it is not close to the batch edges. A solution corresponding to an arbitrary RHS vector can then be constructed with an appropriate superposition of all N individual solutions. The resulting solution for the state vectors is identical to solutions obtained using a conventional solver. Ruffa et al. [7] and Ruffa and Toni [8] show that the solution of a symmetric tridiagonal Toeplitz system with a single nonzero right-hand side (RHS) term has the functional form xj = ej φ , where j is the update number. Substituting this into a tridiagonal system with row structure [B, A1 , B] leads to B + A1 eφ + Be2φ = 0.

(4.1)

After substituting for B and A1 in (4.1), the result can be solved for φ to obtain φ1,2

    1 2 2 6 + λ ± λ 12 + λ . = ln 6

(4.2)

Figure 1 shows |φ1 | and |φ2 | as a function of λ2 . Since φ1 = −φ2 with an error of O(10−13 ) or less for 10−3 ≤ λ2 ≤ 103 , |φ1 | and |φ2 | essentially overlay each other.

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6

5

|f1|,|f2|

4

3

2

1

0 -3

-2

-1

0

1

2

3

log10(l2)

Fig. 1 |φ1 | and |φ2 | shown as a function of the track index log10 (λ2 )

The solution to (2.1) for a single RHS term having unit value at j = p exhibits an exponential growth and decay functional form [7, 8] as follows: xj = κeφ1 (j −p) ; 1 ≤ j ≤ p; xj = κeφ2 (j −p) ; p ≤ j ≤ N;

(4.3)

where κ=

Be−φ1

1 1 = . φ 2 + A1 + Be A1 + 2Beφ2

(4.4)

Solution (4.3) has a translational symmetry with respect to p (i.e., the equation number associated with the single nonzero RHS term), except when p ≈ 1 or p ≈ N . When the interval length, N, is very large relative to the decay rate, the solution for xp and xp is the same provided that p and p are well away from the edges of the batch. The solutions in (4.3) meet at j = p. The forward solution satisfies the rows in (2.1) up to j = p − 1, and the backward solution satisfies the rows back to j = p + 1, leaving the row corresponding to j = p, i.e., Bxp−1 + A1 xp + Bxp+1 = z˜ p .

(4.5)

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Position xj

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 500

600

700

800

900 1000 1100 Update number j

1200

1300

1400

1500

Fig. 2 The normalized position xj as a function of the update number j for N = 2000, p = 1000, and λ2 = 10−3

Setting z˜ p = 1 and substituting the solutions (4.3) for xp−1 , xp , and xp+1 into (4.5) leads to (4.4). As an example, when λ2 = 10−3 , p = 1000, and N = 2000, Fig. 2 shows the forward solution, which grows exponentially for 1 ≤ j ≤ p, and the backward solution, which decays exponentially for p ≤ j ≤ N. The solution envelope defines an interval length for λ2 = 10−3 that is on the order of 500 updates. Figure 3 shows how the solution envelope varies as λ2 ranges from 10−3 to 103 in powers of 10. In this figure, as λ increases, the solution envelope converges to a function that is one at p = 1000 and zero everywhere else. Consequently, as the track index, λ, increases, the optimal smoothing interval length shrinks and eventually converges to 1. The interval length required by a particular choice of the tracking index, λ2 , is defined as the number updates that the solutions is above a threshold. Choosing a threshold equal to 0.01 (i.e., 1%) yields the plot shown in Fig. 4, which provides a visualization envelope for 10−3 ≤ λ2 ≤ 103 . The influence of the RHS term at j = p is limited to approximately 250 update number terms on either side for λ2 = 10−3 . As λ increases, the number of update terms influenced decreases (Fig. 4). Because of the translational symmetry, this result is valid for the entire solution, except at the batch edges. Since (4.3) only solves the equations represented by the row structure [B, A1 , B], the first and last equations in (2.1) are not satisfied. In this example, the error

An Exact Solution for a Class of Kalman Smoothers

105

1 0.9 0.8

Position xj

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 500

600

700

800

900

1000

1100

1200

1300

1400

1500

Update number j

Fig. 3 The normalized position xj as a function of the update number j for 10−3 ≤ λ2 ≤ 103 for N = 2000 and a single nonzero RHS term at p = 1000

Number of Update Number Terms Influenced

300

250

200

150

100

50

0 -3

-2

-1

0

1

2

3

log10(λ2)

Fig. 4 The approximate number of update number terms influenced by a single RHS term for 10−3 ≤ λ2 ≤ 103 for N = 2000 and a single nonzero RHS term at p = 1000

106

A. A. Ruffa and T. E. Luginbuhl

corresponding to both of those equations is O(10−21 ). However, when p ≈ 1 or p ≈ N , those errors are not negligible, but the solution can be modified to eliminate them. The first and last equations are as follows: A0 x1 + Bx2 = 0; BxN −1 + A2 xN = 0.

(4.6)

A solution xˆ e can be constructed making use of x from (4.3) that removes the “edge errors” (i.e., the errors occurring when p ≈ 1 or p ≈ N ) having the form xe = x + xˆ ; z˜ e = z˜ ; xˆ = δ1 xˆ F + δ2 xˆ B ; zˆ = δ1 zˆ F + δ2 zˆ B ; (4.7) where xˆjF = eφ1 (j −N ) ; xˆjB = eφ2 j ; zˆ 1F = A0 xˆ1F + B xˆ2F =

3eφ1 (1−N ) (1 + c − ceφ1 ) ; cλ2

zˆ 1B = A0 xˆ1B + B xˆ2B =

3eφ2 (1 + c(1 − ceφ2 )) ; cλ2

F F F zˆ N = B xˆN −1 + A2 xˆ N =

3 − 3e−φ1 + λ2 ; λ2

B B B zˆ N = B xˆN −1 + A2 xˆ N =

eφ2 (1−N ) (eφ2 (3 + λ2 ) − 3) . cλ2

(4.8)

Using (4.7), δ1 and δ2 can be found by solving 

zˆ 1F zˆ 1B F zˆ B zˆ N N

( ) ( ) δ1 −e1 = , δ2 −eN

where   3eφ1 (1−p) 1 + c − ceφ1 e1 = A0 x1 + Bx2 = ; cλ2

(4.9)

An Exact Solution for a Class of Kalman Smoothers

    eφ2 (N −p−1) eφ2 λ2 + 3 − 3 eN = BxN −1 + A2 xN = . λ2

107

(4.10)

This process essentially utilizes an exponentially decaying solution to satisfy the first equation and an exponentially growing solution to satisfy the last equation. The resulting solution is identical to that obtained with a conventional solver. The solution for an arbitrary RHS vector is then the superposition of N individual solutions, each with a single nonzero RHS term weighted by the corresponding value of the RHS vector for that term. Again, this solution is identical to that obtained with a conventional solver with a fully populated RHS vector; however, this approach offers further insights into the mechanisms governing the behavior of the Kalman smoother, especially for the two-dimensional state vectors (discretized continuous-time kinematic models with one-dimensional measurements and twodimensional state vectors).

4.2 Two-Dimensional State Vector The discretized continuous-time kinematic models with one-dimensional measurements and two-dimensional state vectors lead to a symmetric block tridiagonal Toeplitz system with the block row structure [Bt , A1 , B]. Here we develop an exact procedure via an extension of the approach used in the previous section. For two dimensional state vectors, there are two exponentially growing solutions and two exponentially decaying solutions in response to a single nonzero RHS term because each state vector contains two variables (e.g., position and velocity). The block row structure supports a solution having the functional form xj = e2j φ . Otherwise, the approach for solving this system is similar to that used previously for onedimensional state vectors. The solution for a single nonzero RHS term having index j = p (where p  1, p  2N +2, and N is the number of 2×2 matrices or blocks) is primarily governed by the structure of A1 and B, since these matrices comprise the symmetric block tridiagonal Toeplitz system. A0 and A2 mainly affect the solution when p ≈ 1 and p ≈ 2N + 2, which will be discussed later. We will mainly focus on solutions that are free of edge effects in order to study the nature of the solutions as a function of the track index λ2 . If   a11 a12 , (4.11) A1 = a21 a22 and   b11 b12 B= , b21 b22

(4.12)

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A. A. Ruffa and T. E. Luginbuhl

and   x2j +1 , xj = x2j +2

(4.13)

 z˜ 2j +1 z˜ j = , z˜ 2j +2

(4.14)

and 

then the solution has the following functional form: x2j +1 = e2j φ x1 ; x2j +2 = e2j φ x2 .

(4.15)

Consider the first block tridiagonal row in (2.1), representing the third and fourth equations, i.e., b11 x1 + b21 x2 + a11 x3 + a12 x4 + b11 x5 + b12 x6 = 0; b12 x1 + b22 x2 + a21 x3 + a22 x4 + b21 x5 + b22 x6 = 0.

(4.16)

Substituting (4.15) into (4.16) leads to b11 x1 + b21 x2 + a11 e2φ x1 + a12 e2φ x2 + b11 e4φ x1 + b12 e4φ x2 = 0; b12 x1 + b22 x2 + a21 e2φ x1 + a22 e2φ x2 + b21 e4φ x1 + b22 e4φ x2 = 0,

(4.17)

or ( ) ( )  0 (b11 + a11 e2φ + b11 e4φ ) (b21 + a12 e2φ + b12 e4φ ) x1 = . (b12 + a21 e2φ + b21 e4φ ) (b22 + a22 e2φ + b22 e4φ ) x2 0

(4.18)

Because of the Toeplitz nature of (2.1), every row having the block structure [Bt , A1 , B] will lead to (4.18). The only exceptions are the first and last rows, which will be addressed later. Since the RHS vector in (4.18) is zero, the determinant of the coefficient matrix must also be set to zero. Substituting the components of A1 and B into the result leads to an equation for φ, i.e., 6 + 6e8φ + (e2φ + e6φ )(λ2 − 24) + 4e4φ (λ2 + 9) = 0.

(4.19)

Note that T cancels in (4.19), so that φ is independent of T . There are four solutions for φ. When λ2 < 24,

An Exact Solution for a Class of Kalman Smoothers

φ1,3

1 = ln 2

φ2,4 =

1 ln 2

  λ2 λ 2 1− + λ − 144 ± 24 24    1 √ λ4 − 96λ2 − (λ3 − 24λ) λ2 − 144 ; 12 2   λ 2 λ2 − λ − 144 ± 1− 24 24    1 4 2 3 2 √ λ − 96λ + (λ − 24λ) λ − 144 . 12 2

109

(4.20)

(4.21)

For λ2 < 24, φ1,3 represents the two exponentially growing solutions, and φ2,4 represents the two exponentially decaying solutions. For λ2 ≥ 24, however, (4.21) represents the exponentially growing solutions, and (4.20) represents the exponentially decaying solutions. To keep the notation consistent (i.e., to retain φ1,3 as the two exponentially growing solutions and φ2,4 as the two exponentially decaying solutions for all values of λ), we denote the following for λ2 ≥ 24:

φ1,3

1 = ln 2

φ2,4 =

1 ln 2

  λ2 λ 2 1− − λ − 144 ± 24 24    1 4 2 3 2 √ λ − 96λ + (λ − 24λ) λ − 144 ; 12 2   λ2 λ 2 1− + λ − 144 ± 24 24    1 √ λ4 − 96λ2 − (λ3 − 24λ) λ2 − 144 . 12 2

(4.22)

(4.23)

All four values of |φ| are plotted in Figs. 5 and 6. When λ < 12, all four values overlay each other. However, when λ ≥ 12, | Im(2φ)| = π (so that oscillatory nature of the solution changes to an exponential growth/decay nature), and two pairs of two values of |φ| overlay each other. λ = 12 means the object’s process noise standard deviation is 12 times larger than the measurement noise standard deviation; so the object is maneuvering a lot, and these maneuvers are measurable because the measurement noise is small relative to the size of the maneuvers. Specifically, for λ ≥ 12, the solutions show that the interval length is effectively one update. This makes sense because the object is highly maneuverable in low noise, and there is no benefit to averaging over multiple updates.

110

A. A. Ruffa and T. E. Luginbuhl 6

5

|2f|

4

3

2

1

0 -3

-2

-1

0

1

2

3

log10(λ2)

Fig. 5 The four values of |2φ| as a function of the track index log10 (λ2 ). Note that φ should be doubled to be consistent with the one-dimensional state vector in the previous section 3.2 3

|Re(2f)|, |Im(2f)|

2.8 2.6 2.4 2.2 2 1.8 100

110

120

130

140

150

160

170

180

λ2

Fig. 6 The four values of | Re(2φ)| (lower lines) and the four values of | Im(2φ)| (upper line) as a function of the track index λ2 in the region where the splitting occurs

An Exact Solution for a Class of Kalman Smoothers

111

Table 1 The four values of x2(k) as a function of the track index λ2 λ2 10−3 10−2 10−1 100 101 102 103

x2(1) −0.0314 − 0.0314i −0.0559 − 0.0559i −0.0995 − 0.0995i −0.1778 − 0.1776i −0.3350 − 0.3241i −0.9542 − 0.4325i −12.0035

x2(2) 0.0314 − 0.0314i 0.0559 − 0.0559i 0.0995 − 0.0995i 0.1778 − 0.1776i 0.3350 − 0.3241i 0.9542 − 0.4325i 12.0035

x2(3) −0.0314 + 0.0314i −0.0559 + 0.0559i −0.0995 + 0.0995i −0.1778 + 0.1776i −0.3350 + 0.3241i −0.9542 + 0.4325i −0.7694

x2(4) 0.0314 + 0.0314i 0.0559 + 0.0559i 0.0995 + 0.0995i 0.1778 + 0.1776i 0.3350 + 0.3241i 0.9542 + 0.4325i 0.7694

For each of the four values of φk , the state variables are as follows: (k)

x1 = 1; (k)

x2 = − =

b11 + a11 e2φk + b11 e4φk b21 + a12 e2φk + b12 e4φk

(4.24)

λ2 − 48 sinh2 φk . 12T sinh(2φk ) (k)

(4.25) (k)

Although φk is independent of T , x2 is not. The state vectors x2 for selected values of λ2 are listed in Table 1. This solution also has a translational symmetry with respect to p when it is not close to the batch edges (i.e., when p  1 and p  N). The four values of φk support four solutions to (2.1), which we call x F , x f , B x , and x b , i.e., two forward solutions, which grow exponentially as a function of the update number, and two backward solutions, which decay exponentially as a function of the update number. The equations for the first forward solution are F 2φ1 (j −p) x2j ; −1 = x1 e

1 ≤ j ≤ p;

F = x2 e2φ1 (j −p) ; x2j

1 ≤ j ≤ p;

(1) (1)

F F F F F z˜ 2p−1 = b11 x2p−3 + b21 x2p−2 + a11 x2p−1 + a12 x2p

=

(tanh φ1 + 1)(48 + λ2 + λ2 coth φ1 ) ; 4λ2

F F F F F z˜ 2p = b12 x2p−3 + b22 x2p−2 + a21 x2p−1 + a22 x2p

=−

6T e−2φ1 2T (λ2 − 48 sinh2 φ1 ) e−2φ1 T (λ2 − 48 sinh2 φ1 ) + ; + λ2 3λ2 sinh(2φ1 ) 6λ2 sinh(2φ1 )

F F F = b11 x2p−1 + b21 x2p =− z˜ 2p+1

12 (λ2 − 48 sinh2 φ1 ) ; + λ2 2λ2 sinh(2φ1 )

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A. A. Ruffa and T. E. Luginbuhl

F F F z˜ 2p+2 = b12 x2p−1 + b22 x2p =−

6T T (λ2 − 48 sinh2 φ1 ) . + λ2 6λ2 sinh(2φ1 ) (4.26)

The equation for the second forward solution are x2j −1 = x1 e2φ3 (j −p) ;

1 ≤ j ≤ p;

x2j = x2 e2φ3 (j −p) ;

1 ≤ j ≤ p;

f

(3)

f

(3)

f

f

f

f

f

z˜ 2p−1 = b11 x2p−3 + b21 x2p−2 + a11 x2p−1 + a12 x2p =

(tanh φ3 + 1)(48 + λ2 + λ2 coth φ3 ) ; 4λ2

f

f

f

f

f

z˜ 2p = b12 x2p−3 + b22 x2p−2 + a21 x2p−1 + a22 x2p =−

6T e−2φ3 2T (λ2 − 48 sinh2 φ3 ) e−2φ3 T (λ2 − 48 sinh2 φ3 ) + ; + λ2 3λ2 sinh(2φ3 ) 6λ2 sinh(2φ3 )

f

f

f

f

f

f

z˜ 2p+1 = b11 x2p−1 + b21 x2p = − z˜ 2p+2 = b12 x2p−1 + b22 x2p = −

12 (λ2 − 48 sinh2 φ3 ) ; + λ2 2λ2 sinh(2φ3 ) 6T T (λ2 − 48 sinh2 φ3 ) . + λ2 6λ2 sinh(2φ3 ) (4.27)

The equations for the first backward solution are B 2φ2 (j −p) x2j ; −1 = x1 e

1 ≤ j ≤ p;

B = x2 e2φ2 (j −p) ; x2j

1 ≤ j ≤ p;

(2) (2)

B B B B B z˜ 2p+2 = a21 x2p+1 + a22 x2p+2 + b21 x2p+3 + b22 x2p+4

=

6T e4φ2 2T e2φ2 (λ2 − 48 sinh2 φ2 ) e4φ2 T (λ2 − 48 sinh2 φ2 ) + ; + 2 λ 3λ2 sinh(2φ2 ) 6λ2 sinh(2φ2 )

B B B B B z˜ 2p+1 = a11 x2p+1 + a12 x2p+2 + b11 x2p+3 + b12 x2p+4

=

48 − λ2 coth φ2 + (48 + λ2 ) tanh φ2 ; 4λ2

B B B z˜ 2p = b21 x2p+1 + b22 x2p+2 =

6e2φ2 T e2φ2 T (λ2 − 48 sinh2 φ2 ) ; + λ2 6λ2 sinh(2φ2 )

An Exact Solution for a Class of Kalman Smoothers

B B B z˜ 2p−1 = b11 x2p+1 + b12 x2p+2 =−

113

12e2φ2 e2φ2 (λ2 − 48 sinh2 φ2 ) . − λ2 2λ2 sinh(2φ2 ) (4.28)

Lastly, the equations for the second backward solution are (4) 2φ4 (j −p) b x2j ; −1 = x1 e

1 ≤ j ≤ p;

b x2j = x2 e2φ4 (j −p) ;

1 ≤ j ≤ p;

(4)

b b b b b z˜ 2p+2 = a21 x2p+1 + a22 x2p+2 + b21 x2p+3 + b22 x2p+4

=

6T e4φ4 2T e2φ4 (λ2 − 48 sinh2 φ4 ) e4φ4 T (λ2 − 48 sinh2 φ4 ) + ; + 2 λ 3λ2 sinh(2φ4 ) 6λ2 sinh(2φ4 )

b b b b b z˜ 2p+1 = a11 x2p+1 + a12 x2p+2 + b11 x2p+3 + b12 x2p+4

=

48 − λ2 coth φ4 + (48 + λ2 ) tanh φ4 ; 4λ2

b b b z˜ 2p = b21 x2p+1 + b22 x2p+2 =

6e2φ4 T e2φ2 T (λ2 − 48 sinh2 φ4 ) ; + λ2 6λ2 sinh(2φ4 )

b b b z˜ 2p−1 = b11 x2p+1 + b12 x2p+2 =−

12e2φ4 e2φ2 (λ2 − 48 sinh2 φ4 ) . − λ2 2λ2 sinh(2φ4 ) (4.29)

The solution at any update away from the edges of the smoothing interval is a linear combination of the four individual solutions: x = κ1 xF + κ2 xf + κ3 xB + κ4 xb ;

(4.30)

z˜ = κ1 z˜ F + κ2 z˜ f + κ3 z˜ B + κ4 z˜ b .

(4.31)

Note that all of these solutions have nonzero RHS terms for j = 2p − 1 ≤ j ≤ 2p +2. The linear combination of these four solutions defines a set of four equations to find κ1 , κ2 , κ3 , and κ4 for a specific nonzero RHS term in (2.1). For example, if z˜ 2p−1 = 1, then ⎤ f ⎧ ⎫ ⎧ ⎫ b F B z˜ 2p−1 z˜ 2p−1 z˜ 2p−1 z˜ 2p−1 ⎪1 ⎪ ⎪κ1 ⎪ ⎥ ⎢ F f ⎨ ⎪ ⎬ ⎨ ⎪ ⎬ ⎪ b ⎥⎪ B ⎢ z˜ 2p z˜ 2p z ˜ z ˜ 0 κ 2 2p 2p ⎥ ⎢ = f ⎥ κ3 ⎪ ⎪0 ⎪ . ⎢z˜ F b B ⎪ ⎣ 2p+1 z˜ 2p+1 z˜ 2p+1 z˜ 2p+1 ⎦ ⎪ ⎩ ⎪ ⎭ ⎩ ⎪ ⎭ ⎪ f 0 κ4 z˜ F z˜ B z˜ z˜ b ⎡

2p+2

2p+2

2p+2

2p+2

(4.32)

114

A. A. Ruffa and T. E. Luginbuhl 1

Normalized Position

0.8 0.6 0.4 0.2 0 -0.2 900

920

940

960

980

1000

1020

1040

1060

1080

1100

Update Number

Fig. 7 The normalized position for λ2 = 10−3 for N = 2000 and a single nonzero RHS term at p = 1000 1 0.8

Normalized Velocity

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 900

920

940

960

980

1000

1020

1040

1060

1080

1100

Update Number

Fig. 8 The normalized velocity for λ2 = 10−3 for N = 2000 and a single nonzero RHS term at p = 1000

When λ2 = 10−3 , p = 1000, and N = 2000, Figs. 7 and 8 show the solutions for the position and the velocity, respectively. This example shows that when the measurement noise standard deviation is much larger than the object’s process noise standard deviation, significant information is gleaned about the object’s position and velocity from a large number of updates. Figures 7 and 8 both show that approximately 40 to 45 updates are used to estimate position.

An Exact Solution for a Class of Kalman Smoothers

115

The response to a single nonzero RHS term will vary depending on whether it corresponds to an unknown representing position or velocity. For example, when p = 1999, the position plot is symmetric, and the velocity plot is antisymmetric. On the other hand, when p = 2000, the velocity plot is symmetric and the position plot is antisymmetric. This is a consequence of the structure of A1 . The influence plots shown here are based on p = 1999. Because this solution only satisfies the equations represented by the row structure [Bt , A1 , B], the first two and last two equations are not satisfied. In this example, the error for those equations is O(10−52 ). However, when p ≈ 1 or p ≈ N, the error is not negligible. The solution can be modified to eliminate those errors. Here we develop a solution xe that is valid even when p ≈ 1 or p ≈ N. It is an extension of the approach used in for one-dimensional state vectors in that we use the exponentially decaying solutions to satisfy the first two equations, and we use the exponentially growing solutions to satisfy the last two equations. More specifically, the approach involves creating a new solution xˆ so that the error vector e from the original solution forms the basis for the RHS vector for the new solution, i.e., zˆ = −e. We then remove the RHS vector from the new solution xˆ , which generates a new error vector that matches the original error vector. The original solution x is valid far away from the edges of the batch. The solution xe that is valid everywhere is constructed as follows: xe = x + xˆ ; z˜ e = z˜ . If  0 0 a a12 A0 = 11 0 a0 a21 22 and  2 a2 a11 12 A2 = 2 2 , a21 a22 

then we can express the four solutions as follows. The equations for the first forward solution are F 2φ1 (j −N ) xˆ2j ; −1 = x1 e

1 ≤ j ≤ N;

F = x2 e2φ1 (j −N ) ; xˆ2j

1 ≤ j ≤ N;

(1) (1)

0 F 0 F zˆ 1F = a11 xˆ1 + a12 xˆ2 + b11 xˆ3F + b12 xˆ4F   12 1 + 1c e2(1−N )φ1 12e2(2−N )φ1 e2(2−N )φ1 (λ2 − 48 sinh2 φ1 ) = − − λ2 λ2 2λ2 sinh(2φ1 )   1 − 1c e2(1−N )φ1 (λ2 − 48 sinh2 φ1 ) ; + 2λ2 sinh(2φ1 )

116

A. A. Ruffa and T. E. Luginbuhl 0 F 0 F zˆ 2F = a21 xˆ1 + a22 xˆ2 + b21 xˆ3F + b22 xˆ4F   6T 1 − 1c e2(1−N )φ1 6T e2(2−N )φ1 e2(2−N )φ1 T (λ2 − 48 sinh2 φ1 ) = + + 2 2 λ λ 6λ2 sinh(2φ1 )   1 + 1c e2(1−N )φ1 T (λ2 − 48 sinh2 φ1 ) ; + 3λ2 sinh(2φ1 )

F F F 2 F 2 F zˆ 2N −1 = b11 xˆ 2N −3 + b21 xˆ 2N −2 + a11 xˆ 2N −1 + a12 xˆ 2N

=

−3(32 + λ2 ) + 2λ2 coth φ1 + 3(48 + λ2 ) tanh φ1 ; 10λ2

F F F 2 F 2 F = b12 xˆ2N zˆ 2N −3 + b22 xˆ 2N −2 + a21 xˆ 2N −1 + a22 xˆ 2N

=−

6T 6T e−2φ1 T (λ2 − 48 sinh2 φ1 ) T e−2φ1 (λ2 − 48 sinh2 φ1 ) + . − + 5λ2 λ2 15λ2 sinh(2φ1 ) 6λ2 sinh(2φ1 ) (4.33)

The equations for the second forward solution are xˆ2j −1 = x1 e2φ3 (j −N ) ;

1 ≤ j ≤ N;

xˆ2j = x2 e2φ3 (j −N ) ;

1 ≤ j ≤ N;

f

(3)

f

(3)

f

f

f

f

f

f

f

f

f

f

0 0 xˆ1 + a12 xˆ2 + b11 xˆ3 + b12 xˆ4 zˆ 1 = a11   12 1 + 1c e2(1−N )φ3 12e2(2−N )φ3 e2(2−N )φ3 (λ2 − 48 sinh2 φ3 ) = − − 2 2 λ λ 2λ2 sinh(2φ3 )   1 − 1c e2(1−N )φ3 (λ2 − 48 sinh2 φ3 ) ; + 2λ2 sinh(2φ3 ) 0 0 zˆ 2 = a21 xˆ1 + a22 xˆ2 + b21 xˆ3 + b22 xˆ4   6T 1 − 1c e2(1−N )φ3 6T e2(2−N )φ3 e2(2−N )φ3 T (λ2 − 48 sinh2 φ3 ) = + + 2 2 λ λ 6λ2 sinh(2φ3 )   1 + 1c e2(1−N )φ3 T (λ2 − 48 sinh2 φ3 ) ; + 3λ2 sinh(2φ3 ) f

f

f

f

f

2 2 zˆ 2N −1 = b11 xˆ2N −3 + b21 xˆ2N −2 + a11 xˆ2N −1 + a12 xˆ2N

=

−3(32 + λ2 ) + 2λ2 coth φ3 + 3(48 + λ2 ) tanh φ3 ; 10λ2

An Exact Solution for a Class of Kalman Smoothers f

f

f

f

117 f

2 2 zˆ 2N = b12 xˆ2N −3 + b22 xˆ2N −2 + a21 xˆ2N −1 + a22 xˆ2N

=−

6T 6T e−2φ3 T (λ2 − 48 sinh2 φ3 ) T e−2φ3 (λ2 − 48 sinh2 φ3 ) + . − + 2 2 5λ λ 15λ2 sinh(2φ3 ) 6λ2 sinh(2φ3 ) (4.34)

The equations for the first backward solution are (2)

1 ≤ j ≤ N;

(2)

1 ≤ j ≤ N;

B 2φ2 j xˆ2j ; −1 = x1 e B = x2 e2φ2 j ; xˆ2j

0 B 0 B zˆ 1B = a11 xˆ1 + a12 xˆ2 + b11 xˆ3B + b12 xˆ4B

=

e4φ2 (−24 + 24e2φ2 − λ2 − 2c sinh φ2 (λ2 eφ2 + 48 sinh φ2 )) ; cλ2 (e4φ2 − 1)

0 B 0 B zˆ 2B = a21 xˆ1 + a22 xˆ2 + b21 xˆ3B + b22 xˆ4B

=

1 2φ2 e2φ2 T (1 + c) e T (1 + coth(2φ2 )) + 6 6c sinh φ2 cosh φ2

+

e2φ2 T (12(−3 + 5c + c cosh(2φ2 ) + c sinh(2φ2 ) − 2(2 + c) tanh φ2 )) ; 6cλ2

B B B 2 B 2 B zˆ 2N −1 = b11 xˆ 2N −3 + b21 xˆ 2N −2 + a11 xˆ 2N −1 + a12 xˆ 2N

=− −

12e2(N −1)φ2 e2N φ2 (12 + λ2 ) e2(N −1)φ2 T (λ2 − 48 sinh2 φ2 ) + + 2 λ 5λ2 2λ2 sinh(2φ2 )

e2N φ2 T (λ2 − 48 sinh2 φ2 ) ; 10λ2 sinh(2φ2 )

B B B 2 B 2 B zˆ 2N = b12 xˆ2N −3 + b22 xˆ 2N −2 + a21 xˆ 2N −1 + a22 xˆ 2N

=− +

6T e2(N −1)φ2 6T e2N φ2 e2(N −1)φ2 T (λ2 − 48 sinh2 φ2 ) − + λ2 5λ2 6λ2 sinh(2φ2 )

e2N φ2 T (λ2 − 48 sinh2 φ2 ) . 15λ2 sinh(2φ2 ) (4.35)

Finally, the equations for the second backward solution are (4)

1 ≤ j ≤ N;

(4)

1 ≤ j ≤ N;

b 2φ4 j xˆ2j ; −1 = x1 e b xˆ2j = x2 e2φ4 j ;

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A. A. Ruffa and T. E. Luginbuhl 0 b 0 b zˆ 1b = a11 xˆ1 + a12 xˆ2 + b11 xˆ3b + b12 xˆ4b

=

e4φ4 (−24 + 24e2φ4 − λ2 − 2c sinh φ4 (λ2 eφ4 + 48 sinh φ4 )) ; cλ2 (e4φ4 − 1)

0 b 0 b zˆ 2b = a21 xˆ1 + a22 xˆ2 + b21 xˆ3b + b22 xˆ4b

=

1 2φ4 e2φ4 T (1 + c) e T (1 + coth(2φ4 )) + 6 6c sinh φ4 cosh φ4

+

e2φ4 T (12(−3 + 5c + c cosh(2φ4 ) + c sinh(2φ4 ) − 2(2 + c) tanh φ4 )) ; 6cλ2

b b b 2 b 2 b zˆ 2N −1 = b11 xˆ 2N −3 + b21 xˆ 2N −2 + a11 xˆ 2N −1 + a12 xˆ 2N

=− −

12e2(N −1)φ4 e2N φ4 (12 + λ2 ) e2(N −1)φ4 T (λ2 − 48 sinh2 φ4 ) + + λ2 5λ2 2λ2 sinh(2φ4 )

e2N φ4 T (λ2 − 48 sinh2 φ4 ) ; 10λ2 sinh(2φ4 )

b b b 2 b 2 b zˆ 2N = b12 xˆ2N −3 + b22 xˆ 2N −2 + a21 xˆ 2N −1 + a22 xˆ 2N

=− +

6T e2(N −1)φ4 6T e2N φ4 e2(N −1)φ4 T (λ2 − 48 sinh2 φ4 ) − + λ2 5λ2 6λ2 sinh(2φ4 )

e2N φ4 T (λ2 − 48 sinh2 φ4 ) . 15λ2 sinh(2φ4 ) (4.36)

The solution for j = 1 and j = N is a linear combination of these for solutions: xˆ = γ1 xˆ F + γ2 xˆ f + γ3 xˆ B + γ4 xˆ b ;

(4.37)

zˆ = γ1 zˆ F + γ2 zˆ f + γ3 zˆ B + γ4 zˆ b .

(4.38)

This leads to the following set of linear equations for γ1 , γ2 , γ3 , and γ4 : ⎡

⎤⎧ ⎫ ⎧ ⎫ f zˆ 1F zˆ 1B zˆ 1 zˆ 1b −e1 ⎪ γ1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎢ F ⎥⎨ ⎬ ⎨ f zˆ 2B zˆ 2 zˆ 2b ⎥ γ2 −e2 ⎢ zˆ 2 = , ⎢ F ⎥ f b B ⎪ −e2N −1 ⎪ ⎣zˆ 2N −1 zˆ 2N ⎦ ⎪γ3 ⎪ ⎪ ⎪ ⎪ −1 zˆ 2N −1 zˆ 2N −1 ⎪ ⎩ ⎩ ⎭ ⎭ f b F B γ4 −e2N zˆ 2N zˆ 2N zˆ 2N zˆ 2N where 0 0 e1 = a11 x1 + a12 x2 + b11 x3 + b12 x4

(4.39)

An Exact Solution for a Class of Kalman Smoothers

119

= κ1 zˆ 1F e2φ1 (N −p) + κ3 zˆ 1 e2φ3 (N −p) ; f

0 0 e2 = a21 x1 + a22 x2 + b21 x3 + b22 x4

= κ1 zˆ 2F e2φ1 (N −p) + κ3 zˆ 2 e2φ3 (N −p) ; f

2 2 e2N −1 = b11 x2N −3 + b21 x2N −2 + a11 x2N −1 + a12 x2N B −2φ2 p b −2φ4 p = κ2 zˆ 2N + κ4 zˆ 2N ; −1 e −1 e 2 2 e2N = b12 x2N −3 + b22 x2N −2 + a21 x2N −1 + a22 x2N B −2φ2 p b = κ2 zˆ 2N e + κ4 zˆ 2N e−2φ4 p .

(4.40)

The solution xe that is valid everywhere is given by xe = x + xˆ and z˜ e = z˜ . Note that zˆ is not added to z˜ in order to cancel the original error. Figures 9 and 10 show the normalized position and the velocity, respectively. The curves are continuous for the lower values of λ2 (i.e., λ2 = 10−3 and λ2 = 10−2 ), but not for the higher values of λ2 (i.e., λ2 = 102 and λ2 = 103 ). This is not a numerical artifact since the exact solution to (2.1) is plotted for all values of λ2 . The number of influenced terms is significantly fewer relative to the onedimensional state vector case (Fig. 4). The curve in Fig. 11 also becomes discontinuous as the number of influenced terms decreases from four to three and from three to two. Because of the translational symmetry, this result is also valid for the entire solution, except at the batch edges.

1

Normalized position

0.8 0.6 0.4 0.2 0 -0.2 980

985

990

995

1000 1005 Update number

1010

1015

1020

Fig. 9 The normalized position for 10−3 ≤ λ2 ≤ 103 for N = 2000 and a single nonzero RHS term at p = 1000

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Normalized velocity

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 950

960

970

980

990

1000

1010

1020

1030

1040

1050

Update number

Fig. 10 The normalized velocity for 10−3 ≤ λ2 ≤ 103 for N = 2000 and a single nonzero RHS term at p = 1000

Number of Update Number Terms Influenced

45 40 35 30 25 20 15 10 5 0 -3

-2

-1

0

1

2

3

2

log10(l )

Fig. 11 The approximate number of position (lower line) and velocity (upper line) update number terms influenced by a single RHS term at p = 1000 for 10−3 ≤ λ2 ≤ 103 for N = 2000

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121

5 Conclusion The analytic solutions to the fixed interval Kalman smoother difference equations derived in this chapter for discretized continuous-time kinematic models with onedimensional measurements and one- and two-dimensional state vectors clearly show that there is an effective interval length for a given choice of the tracking index, λ2 . This enables one to determine what interval length should be used for a particular choice of the tracking index. The analytic solutions come in pairs where one solution is in the forward solution in time (update) and the other is the backward solution in time. Both of these solution decay exponentially, and the effective interval length is the width of this envelope determined by the magnitude of the forward and backward solutions being greater than some threshold. The one-dimensional state vector has a pair of solutions for position only, whereas the two-dimensional state vector has a pair of solutions for both position and velocity (four solutions in all). The effective interval length for one-dimensional state vectors and two-dimensional state vectors is shown in Figs. 4 and 11, respectively. Figure 11 shows the effective interval length for both position and velocity and shows that the effective interval length for velocity is always larger than the effective interval length for position. Intuitively, this makes sense because position is measured, and velocity must be inferred from at least two updates. The fact that the analytic solutions come in pairs where one is the forward solution and the other is the backward solution is consistent with [2]. Fraser and Potter’s solution to the Kalman smoother is obtained by combining the estimates from forward and backward filters [2]. The analytic solutions derived here for twodimensional state vectors also are broadly consistent with the results from [5] even though Ogle and Blair studied the fixed-lag smoothing problem with steady-state models (α − β filters) rather than the fixed interval smoothing problem. Ogle and Blair found that there was no benefit in terms of variance reduction to smoothing once the tracking index was larger than 1 or 3 for position and velocity, respectively. The results for the analytic solutions derived here and shown in Fig. 11 indicate the effective interval lengths are longer, but the problems are slightly different (fixed lag with steady-state models versus fixed interval smoothing and non-steady-state models). The behavior of φ as a function of the tracking index, λ2 , for two-dimensional state vectors given by Equations (4.22) and (4.23) is very interesting and shown in Figs. 5 and 6. Figure 6 shows that when the tracking index, λ2 , equals 144, the magnitude of the imaginary parts of all four solutions for φ equals π/2 (the figure plots 2φ). This shows that when the tracking index is greater than 144, the solution no longer oscillates away from a particular smoothing point far from the interval edges because the exponent of the exponential is always an even number times φ; so the imaginary part is zero. Consequently, the estimate for a particular state, xp , away from the edges is a weighted average where all the weights are positive. The weighted average rapidly becomes the Kronecker delta function δ[j − p] as λ2 increases above 144. Thus, the interval length rapidly goes to one for λ2 > 144.

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The behavior can be seen in Figs. 9 and 10. Physically, this means for highly maneuvering object in low noise, the estimate of the object’s state xp is only obtained from measurements near and at time p. This rapidly converges to the estimate of xp equaling the measurement at time p. When tracking index, λ2 , is less than 144, the solution is not simply a weighted average where all the weights are positive. The solution is more interesting because the magnitude of the imaginary part of 2φ is less than π. While the estimate of xp is a weighted average of positive values for values of j near p, the weights eventually become negative and eventually decay to nearly zero at the edges of smoothing interval. This is clearly visible in Figs. 7, 8, 9, and 10. In this situation, (144 > λ2 → 0), the measurement noise is dominant, and the object is not maneuvering much. This means measurements over a wide block of time are needed to estimate an object’s state. Thus, the smoothing interval length needs to increase as the tracking index goes to zero. Presumably, the negative weights in the average correct influence from measurements far away from the current object state, and eventually, the negative weights correct for edge effects of the smoothing interval.

References 1. H.E. Rauch, F. Tung, C.T. Striebel, AIAA J. 3, 1445 (1965) 2. D.C. Fraser, J.E. Potter, IEEE Trans. Autom. Control 7(4), 387 (1969) 3. P.R. Kalata, IEEE Trans. Aerosp. Electron. Syst. AES-20(2), 174 (1984) 4. P.R. Kalata, T.A. Chmielewski Jr., in 1992 American Control Conference (1992) 5. T.L. Ogle, W.D. Blair, IEEE Trans. Aerosp. Electron. Syst. 40(4), 1417 (2004) 6. Y. Bar-Shalom, X.R. Li, T. Kirubarajan, Estimation with Applications to Tracking and Navigation: Theory, Algorithms, and Software (Wiley, New York, 2001) 7. A.A. Ruffa, M. Jandron, B. Toni, Parallelized solution of banded linear systems with an introduction to p-adic computations, in Mathematical Sciences with Multidisciplinary Applications (Springer International Publishing, Cham, 2016), pp. 431–464 8. A.A. Ruffa, B. Toni, Exact solutions to the spline equations, in Advanced Research in Naval Engineering (Springer International Publishing, Cham, 2018), pp. 105–124 9. R.L. Streit, T.E. Luginbuhl, Probabilistic multi-hypothesis tracking. NUWC-NPT Technical Report 10,482, Naval Undersea Warfare Center, Newport (1995) 10. H.L.V. Trees, K.L. Bell, Z. Tian, Detection, Estimation, and Modulation Theory Part I: Detection, Estimation, and Filtering Theory, 2nd edn. (Wiley, Oxford, 2013)

Metallic Foam Metamaterials for Vibration Damping and Isolation Mark J. Cops, J. Gregory McDaniel, Elizabeth A. Magliula, and David J. Bamford

1 Introduction This chapter presents metallic foam metamaterials developed for vibration damping and vibration isolation. Metamaterials can be composite materials in which two or more homogeneous materials have been combined at small scales to create an inhomogeneous material. The properties of a metamaterial are determined by the way in which the homogeneous materials have been combined [1]. Metallic foam metamaterials of the present work are produced by saturating metallic foams with fluids and solids to achieve vibration isolation and vibration damping. In the present work, metallic foams are the scaffolds on which the metamaterials are built. For background information on metallic foams, see [2–4]. Reference [2] describes nine different ways to make metallic foam, five of which are used in commercial production. Reference [5] is a more recent review of manufacturing methods and how they affect foam properties. A photograph of a typical metallic foam used in this work is shown in Fig. 1. In this photograph, the foam exhibits a characteristic topology of voids and thin metallic structures. This is an open-cell foam, which means that the cells are inter-connected, and as a result fluid may flow through and saturate the foam, filling the pores with fluid. The geometry of a metallic foam is characterized by several parameters. The void fraction φ is the ratio of the void volume Vv to the total volume Vt (1) where the

M. J. Cops · J. Gregory McDaniel Department of Mechanical Engineering, Boston University, Boston, MA, USA e-mail: [email protected]; [email protected] E. A. Magliula · D. J. Bamford () Naval Undersea Warfare Center Division Newport, Newport, RI, USA e-mail: [email protected]; [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_7

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Fig. 1 An open-cell aluminum foam 20 PPI and 90% porosity, manufactured by ERG Aerospace Corp

subscript v denotes void material, s denotes solid material, and t denotes total. The effective density of the foam ρ ∗ is given by ρ ∗ = ρv φ + ρs (1 − φ).

(1)

The number of pores per inch (PPI) is used to measure the average inverse size of pores. The effective elastic properties of open-cell metallic foams can be commonly approximated by using equations from [3]. The effective elastic modulus E ∗ is given by E∗ ≈ Es



ρ∗ ρs

2 (2)

.

The effective shear modulus G∗ 3 ≈ Es 8



ρ∗ ρs

2 .

(3)

Most foams have a Poisson’s ratio given by ν∗ ≈

1 . 3

(4)

Figure 2 shows a comparison of Eq. 2 plotted alongside measurements of foam elastic modulus for some aluminum open-cell foams from the literature [6–12]. The scaling law predicts overall correct order of magnitude; however there is large variation in reported values, especially near 8% relative density. The variation could be due to a number of assumptions. First, the scaling laws are derived based solely

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Fig. 2 The effective elastic modulus (E ∗ in Eq. 2) versus relative density for open-cell aluminum foams. The experimental data is taken from measurements reported in the literature [6–12], and Es is taken to be 69 GPa

on beam bending arguments. Depending on the microstructure and relative density, it is possible that beam stretching may also contribute. Additionally, the analysis assumes that stiffness is independent of pore size. Finally there may be variation in the experiments due to orientation of the sample as some directions may be stiffer than others. This can result from the effects of gravity during the manufacturing process. Designers are often interested in the stiffness-to-weight ratio, otherwise known as specific stiffness. This ratio is important in situations where one desires high stiffness and low weight, as in the aerospace industry. For materials in tension, this ratio is given by f =

E . ρ

(5)

The ratio of the specific stiffness of metallic foam to the specific stiffness of the solid material is f∗ =

ρ∗ fs ρs

(6)

Since ρ ∗ < ρs for most foams, metallic foams do not provide a tensile stiffnessto-weight ratio that is greater than the solid material from which they are made. However, a significant advantage of open-cell metallic foams is their porous nature, which allows saturation by liquids and solids to achieve vibration damping and vibration isolation. The remainder of this chapter is organized as follows. A literature review of previous work on foam vibration is given in Sect. 2. Section 3 presents an experimental investigation into the steady-state vibration of a metallic foam saturated with various fluids. Section 4 presents an experimental investigation of transient vibrations of a metallic foam saturated with a petroleum jelly in its semi-solid state. Finally, Sect. 5 presents conclusions and future work.

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2 Literature Review The concept of controlling vibration by saturating foams is not new, going back at least to the 1939 paper by Kosten and Zwikker [13]. They explored rubber foams for damping vibration and shock and investigated the role played by the saturating air. This work was followed by the 1966 paper by Gent and Rusch [14] who studied the effects of foam dimensions, fluid viscosity, and frequency but did not include the effects of fluid inertia. Fluid inertia was included in the analysis presented by Hilyard and Kanakkanatt in 1970 [15]. These works all investigated foams made of materials soft relative to metal, such as polyurethane and rubber latex. In 2006, Goransson [16] presented an investigation into acoustic and vibrational models of saturated foam based on Biot theory. This was followed by a 2009 paper from Rayess that investigated the vibration damping of a metal foam polymer composite, with attention to the interface between the metal and polymer materials. In 2013, Bianchi and Scarpa [17] presented an investigation into vibration damping of auxetic and conventional foams, concluding that the auxetic foam provided enhanced viscous damping. Finally, Yin and Rayess [18] investigated polymer-metal foam composites in which the metal foam served as the “skeleton” and the polymer filled the pores. They created this composite by pouring liquid rubber into aluminum foam and waiting for the rubber to cure into a solid. The loss factor and dynamic stiffness were increased; however it should be noted that the study only included results at specific frequencies of 1 and 2 Hz.

3 Steady-State Vibration When a structure is subject to base excitation, some of the energy is transmitted through the structure, and some of the energy is lost to heat due to damping. To determine the transmitted vibration, a common experimental method is to excite the base of a structure over a desired frequency range and then measure acceleration at the top and at the base of a structure. Transmissibility, the ratio of the accelerations, can be calculated using these two acceleration readings. The mass, stiffness, and damping properties of the structure all determine how much energy is transmitted and how much energy is dissipated. This section concerns steady-state vibration, in which it is assumed that the structure is excited at each frequency of interest for a sufficiently long period of time that the transient portion of the response has vanished.

3.1 Experimental Setup The experiments described in this section were designed to enable measurement of steady-state vibration transmissibility through fluid-saturated aluminum foams

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Fig. 3 20 PPI (left) and 40 PPI (right) aluminum foam samples bonded to 3-inch diameter aluminum discs. The discs are tapped with 1/4–20 threads allowing the foam to be mounted Fig. 4 Schematic of experimental setup for steady-state vibration experiments. The metallic foam is supported from the bottom, attached to a shaker, and placed in a tub allowing for complete fluid saturation

subjected to a frequency-dependent base excitation. Two-inch cube foam samples were specially prepared for these tests. To enable mounting and attaching of masses, aluminum discs (3 inch outside diameter and 1/4 inch thick) were bonded to the top and bottom of foam samples with a two-part, slow-setting structural epoxy. Additionally, the aluminum discs were tapped with 1/4–20 holes at the center. This allowed for threaded connections to masses and support structures. The prepared foam samples are shown in Fig. 3. The experimental setup is shown in Fig. 4. The foam was seated in a small tub, approximately 2 liters in volume. The tub had a small through hole and seal at the bottom, which allowed the foam to be fastened to the shaker stinger underneath the tub. Modular steel masses (3 inches outside diameter and 2 inches tall) were also

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machined with threaded ports on the top and bottom, allowing for interconnecting arrangements of masses to be added on top of the foam. The entire setup was attached to the shaker by a 4-inch-long stinger, approximately 1/8 inch outside diameter. The shaker used in this experiment was a permanent magnetic shaker, model LDS V408, with a specified useful frequency range of 5 Hz–9 kHz. The shaker was not designed to support any static load. As a result, the entire static weight was loaded on a support structure, which was comprised of an 80/20 frame and rubber padding. This support structure protected the shaker from the high static load while still enabling the shaker to exert a dynamic load. The accelerometers used in this experiment were Brüel & Kjær type 4534-B. They were mounted with beeswax to the top dead center of the mass and bottom dead center underneath the foam tub. The shaker was connected to an amplifier type PA500L. Additionally both the amplifier and accelerometer signals were connected to the data acquisition box, module LAN-XI. This acquisition hardware was connected to a desktop computer and had three inputs and one output, which simultaneously controlled the shaker and recorded acceleration levels from both accelerometers. For the tests described in this section, a payload of approximately 2.7 kg was attached to the foam. The excitation to the shaker was a white noise signal in the range of 5–800 Hz. In this frequency range, the amplitude of base displacement did not exceed 2×10−6 m. The Brüel & Kjær software PULSE Reflex was used to control the shaker and record the acceleration measurements. Post processing of the data was done in MATLAB. This experimental setup allowed for many different experiments by varying the saturating fluid. First, a control test was run with air as the saturation fluid. Subsequent tests were done with other saturating fluids by filling and emptying the tub and drying out the foam. Descriptions of the fluids and resulting vibration characteristics are presented in the next section.

3.2 Experimental and Analytical Results An image of the test setup with motor oil as the saturating fluid is shown in Fig. 5. Five separate tests were run with five different fluids: air, motor oil, water, glycerol, and petroleum jelly. For all fluids except the petroleum jelly, it was straightforward to saturate the foam by pouring the liquid and filling the tub. The flow resistivity of the foam was sufficiently low enough to allow these fluids to completely fill the pores. Since the petroleum jelly is a semi-solid, it was first melted down, poured in the foam pores, and then allowed to solidify within the foam pores. In a similar way for removal, the foam was heated, changing the petroleum jelly to a liquid and allowing it to drip out. The acceleration transmissibility magnitude for the five different saturating materials is shown in Fig. 6. At low frequency it continues to approach one, indicating a rigid body motion where the top and bottom of the structure move in unison. There is one resonance, occurring near 340 Hz for all materials except the

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Fig. 5 Shaker setup with a 20 PPI aluminum foam saturated with motor oil. The mass load is approximately 2.7 kg

petroleum jelly which is near 395 Hz. This resonance is similar in characteristics to the resonance of a single degree of freedom harmonic oscillator, when the base is providing small displacement and the energy is amplified by the structure, resulting in high transmissibility. The frequency of this resonance is determined by the payload mass on the foam as well as the apparent stiffness of the foam. Since the petroleum jelly is a semi-solid, it likely contributes greater to the bulk stiffness of the foam, thereby pushing the resonance to higher frequency. The magnitude of transmissibility at resonance is determined by the damping. When the foam is vibrating, it is globally compressing and expanding which causes some of the fluid to move relative to the foam. Depending on the viscosity of the fluid, more or less energy may be dissipated to the fluid. Therefore, the entire system may be approximately modelled and understood as a mass-spring-dashpot system excited at its base. In this model, the top weights are the mass, the metallic foam is the stiffness, and the saturating fluid interacting with the metallic foam is the dashpot. To quantify damping for a transfer function in the frequency domain, the halfpower bandwidth method can be used. Considering light damping, the damping ratio, ζ , can be computed by 1 ζ = 2



ω2 − ω1 ωn

 ,

(7)

where ωn is the natural frequency and ω1 and ω2 are frequencies above and below √ resonance where the amplitude drops by a factor of (1/ 2) (3 dB). The computed damping ratios as well as approximate fluid properties are given in Table 1.

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Fig. 6 Acceleration transmissibility versus frequency for a 20 PPI aluminum foam saturated with various fluids for (a) the entire test frequency range and (b) zoomed in near resonance

Table 1 Damping properties for a 20 PPI aluminum foam saturated with various fluids Fluid Air Motor oil Petroleum jelly Glycerol Water

Dynamic viscosity (Pas) 1.98e-5 8.00e-2 6.40e1 1.00e0 8.90e-4

Kinematic viscosity (m2 /s) 1.65e-5 9.33e-5 7.11e-2 7.94e-5 8.90e-7

Damping ratio (non.dim) 0.0068 0.0237 0.0352 0.0137 0.0101

Frequency (Hz) 343.2 343.1 394.1 346.4 340.3

From the table, it is clear that the metallic foam saturated with petroleum jelly has the highest damping with a damping ratio approximately 5 times greater than the control sample. Figure 7 shows the non-dimensional damping ratio plotted versus the dynamic viscosity of the saturating fluid for all of the experiments (dynamic viscosity of the

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Fig. 7 Damping ratio versus dynamic viscosity for five different saturating materials

Fig. 8 Comparison between experimentally measured transmissibility for metallic foam saturated with petroleum jelly and an analytical model given in (8). Parameters in the model are fn = 395 Hz and ζ = 0.033

fluids is plotted on a logarithmic scale because the values span nearly 6 orders of magnitude). The calculations of damping ratio implicitly assume a lumped parameter model of the experiment in which the saturated foam is modeled as a spring in parallel with a dashpot between two rigid masses. An analytic model of this system gives the transmissibility T as T =

ωn2

ωn2 + i2ζ ωωn + i2ζ ωωn − ω2

(8)

This model was fit to the experimentally measured transmissibility for petroleum jelly. The fit was performed by estimating ωn and ζ in the model and computing the root-mean-square error between the measured transmissibility and the model in (8). The values of ωn and ζ were automatically varied by a simplex search algorithm to minimize the error. The resulting modal properties are listed in the caption of Fig. 8.

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4 Transient Vibration Transient vibration occurs when a structure is subject to an arbitrary loading condition which causes some time-dependent response. The initial condition may be a force, displacement, or velocity that is applied somewhere on the structure.

4.1 Experimental Setup The experiments described in this section were designed to enable measurement of transient vibration through saturated metallic foams subject to an impact force. The same metallic foam specimens with bonded aluminum discs (described in Sect. 3.1) were used again. For this test, first a 0.9 kg mass was attached to each end of the control sample (air saturated metallic foam). The assembly was then suspended by two elastic supports. This was meant to test the assembly in a configuration with free-free boundary conditions for longitudinal vibration. A single accelerometer was placed dead center on one of the masses. The other mass was struck dead center by an instrumented impact hammer (Brüel & Kjær type 8206-002). The test setup is shown in Fig. 9. In addition to the control test, the metallic foam was saturated with petroleum jelly as described in Sect. 3.2. Since the petroleum jelly is a semi-solid, it was selfcontained within the foam and did not require an additional housing to support the saturating material. The metallic foam saturated with petroleum jelly is shown in Fig. 10. In the next section, the measured acceleration versus time data is presented for both the control sample and the sample saturated with petroleum jelly. Fig. 9 Schematic of experimental setup for transient impact experiments. The metallic foam is mass loaded and suspended by elastic supports. The foam is struck at one end by a hammer, and the acceleration is measured at the opposite end

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Fig. 10 A 40 PPI, 92% porosity aluminum foam saturated with petroleum jelly that is mass loaded at each end and suspended by elastic supports

4.2 Experimental and Analytical Results The results for the impact experiments are shown in Fig. 11. The acceleration versus time plots indicate that the assembly is ringing down in a single mode. In terms of lumped elements, the assembly can be thought of as two masses that are connected with a spring and dashpot. It can be seen that the petroleum jelly greatly reduces the number of cycles and therefore the amount of time to reach resting position (steady state). For each experiment, a modal fit was performed to determine modal properties. The modal fit is based on a model of the experiment in which a mass is connected to ground by a spring and a dashpot. The mass element is the steel mass attached to the foam, and the spring and dasphot represent the saturated metallic foam. For such a model, the transient acceleration is given by a(t) = [A cos(ωd t) + B sin(ωd t)] exp(−ζ ωn t),

(9)

where the undamped and damped frequencies are related by  ωd = ωn 1 − ζ 2

(10)

The fit was performed by estimating ωn and ζ and performing a least-squares fit to data to determine A and B. Next, a root-mean-square error was computed between the modal fit and the data. The values of ωn and ζ were automatically varied by a simplex search algorithm to minimize the error. The resulting modal properties are listed in the captions of Figs. 12 and 13.

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(a)

(b) Fig. 11 (a) Comparison of transient ring-downs for a 40 PPI, 92% porosity aluminum metallic foam, saturated first by air and secondly by petroleum jelly. The decay time to nearly zero acceleration is reduced from approximately 0.18 to 0.015 s by saturating the foam with petroleum jelly. (b) Fast Fourier Transform of the ring-downs in the top plot

Fig. 12 Transient ring-down for a 40 PPI, 92% porosity aluminum metallic foam saturated by air. Parameters in the model are fn = 783 Hz, ζ = 0.006, and amax = 139 m/s2

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Fig. 13 Transient ring-down for a 40 PPI, 92% porosity aluminum metallic foam saturated by petroleum jelly. Parameters in the model are fn = 874 Hz, ζ = 0.070, and amax = 108 m/s2

For the petroleum jelly-saturated foam in Fig. 13, it can be seen that the damping ratio is over an order of magnitude greater than the control sample. Also the natural frequency is higher for the petroleum-saturated foam, consistent with the same trend observed in Sect. 3.2. Because of the different mass loading in this experiment compared to the shaker experiment, the natural frequencies were different, and the damping ratios, although both higher than the control samples, were also different. These results suggest that the petroleum jelly-filled metallic foam has a frequencydependent damping ratio. The model described above is now used to estimate the natural frequency of 783 Hz that was observed in Fig. 12 for the aluminum foam saturated by air. The mode shape associated with the fundamental frequency is symmetric about the midpoint of the foam. The fundamental frequency is thus approximated by 1 fn ≈ 2π

#

k m

(11)

where the stiffness of the foam is approximately [19] k≈

EA , L

(12)

where L is half the length of the foam due to symmetry, the measured mass is m = 0.9 kg, and E denotes the Young’s modulus. There are two means of estimating the Young’s modulus of the foam. The first of these is to use the approximate equation given in (2). Use of this equation for an

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aluminum foam with 92% porosity results in a Young’s modulus of E = 435 MPa and a natural frequency of fn = 1,200 Hz. This natural frequency should be compared to the measured natural frequency of fn = 783 Hz, resulting in a difference of approximately 53%. A second means of estimating the foam modulus is to use measured values published in the open literature. Figure 2 shows a wide range of measured values for an aluminum foam with 92% porosity. Using a value of E = 200 MPa, near the lower end of the range, gives a natural frequency of fn = 802 Hz, which yields a 2.4% difference between the estimated and measured natural frequencies.

5 Conclusions and Future Work This chapter has presented experimental results and associated analyses that quantify the vibration damping and vibration isolation of saturated metallic foams. In the case of steady-state vibration, the correlation between the viscosity and the damping ratio was investigated. For four out of five fluids, a higher viscosity resulted in a higher damping ratio. The highest damping ratio resulted from petroleum jelly in its semi-solid state. This suggests that saturating the foam with a damped solid would significantly improve the damping ratio. Specifically, the composites described by Yin and Rayess [18] but only tested at 1 and 2 Hz might hold tremendous damping potential at higher frequencies. The results presented for transient vibration, in which the foam played the role of a damped spring between two steel masses, also showed significant damping from petroleum jelly in its semi-solid state. Specifically, petroleum jelly reduced the ringdown time by an order of magnitude and increased the damping ratio an order of magnitude. These results also suggest further investigations into semi-solids and solids that saturate metallic foams in their liquid states. For example, the present authors have recently fabricated composite foams for acoustic absorption [20] that might be useful for vibration damping and vibration isolation.

References 1. M. Kadic, G.W. Milton, M. van Hecke, M. Wegener, 3D metamaterials. Nat. Rev. Phys. 1(3), 198–210 (2019) 2. M.F. Ashby, T. Evans, N.A. Fleck, J.W. Hutchinson, H.N.G. Wadley, L.J. Gibson, Metal Foams: A Design Guide (Elsevier, 2000) 3. L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties. Cambridge Solid State Science Series, 2nd edn. (Cambridge University Press, 1997) 4. L.J. Gibson, Mechanical behavior of metallic foams. Ann. Rev. Mater. Sci. 30(1), 191–227 (2000) 5. U.M. Mahadev, C.G. Sreenivasa, and K Shivakumar, A review on production of aluminium metal foams. IOP Conf. Ser.: Mater. Sci. Eng. 376, 012081 (2018)

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6. E. Andrews, W. Sanders, L.J. Gibson, Compressive and tensile behaviour of aluminum foams. Mater. Sci. Eng.: A 270(2), 113–124 (1999) 7. E.W. Andrews, G. Gioux, P. Onck, L.J. Gibson, Size effects in ductile cellular solids. Part II: experimental results. Int. J. Mech. Sci. 43(3), 701–713 (2001) 8. P. Fanelli, A. Evangelisti, P. Salvini, F. Vivio, Modelling and characterization of structural behaviour of al open-cell foams. Mater. Des. 114, 167–175 (2017) 9. W.-Y. Jang, S. Kyriakides, On the crushing of aluminum open-cell foams: part I. Experiments. Int. J. Solids Struct. 46(3), 617–634 (2009) 10. T.G. Nieh, K. Higashi, J. Wadsworth, Effect of cell morphology on the compressive properties of open-cell aluminum foams. Mater. Sci. Eng.: A 283(1), 105–110 (2000) 11. J. Zhou, P. Shrotriya, W.O. Soboyejo, Mechanisms and mechanics of compressive deformation in open-cell al foams. Mech. Mater. 36(8), 781–797 (2004). Mechanics of Cellular and Porous Materials 12. ERG Aerospace Corporation, Duocel aluminum foam (2019). http://ergaerospace.com/ materials/duocel-aluminum-foam/ 13. C.W. Kosten, C. Zwikker, Properties of sponge rubber as a material for damping vibration and shock. Rubber Chem. Technol. 12(1), 105–111 (1939) 14. A.N. Gent, K.C. Rusch, Viscoelastic behavior of open cell foams. Rubber Chem. Technol. 39, 389–396 (1966) 15. N.C. Hilyard, S.V. Kanakkanatt, The dynamic mechanical behaviour of liquid-filled foams. J. Appl. Phys. D: Appl. Phys. 3, 906–916 (1970) 16. P. Göransson, Acoustic and vibrational damping in porous solids. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 364(1838), 89–108 (2006) 17. F. Scarpa, M. Bianchi, Vibration transmissibility and damping behaviour for auxetic and conventional foams under linear and nonlinear regimes. Smart Mater. Struct. 22, 084010 (2013) 18. S. Yin, N. Rayess, Characterization of polymer-metal foam hybrids for use in vibration dampening and isolation. Proc. Mater. Sci. 4, 311–316 (2014). 8th International Conference on Porous Metals and Metallic Foams 19. S.S. Rao, Mechanical Vibrations (Addison-Wesley Longman, Incorporated, 1986) 20. M.J. Cops, J. Gregory McDaniel, E.A. Magliula, D.J. Bamford, J. Bliefnick. Measurement and analysis of sound absorption by a composite foam. Appl. Acoust. 160, 107138 (2020)

The Other Navy Seals: Seal Whiskers as a Bio-inspired Model for the Reduction of Vortex-Induced Vibrations Christin T. Murphy, William N. Martin, Jennifer A. Franck, and Joy M. Lapseritis

1 Bio-inspired Design in Naval Applications The US Navy is interested in bio-inspired solutions to the technological challenges of operating in and on the ocean. The marine environment presents a suite of obstacles for the people and systems tasked with maritime security. Temperature, pressure, salinity, light, and sound create extreme operating conditions, requiring exhaustive efforts to maintain thermal windows, prevent implosion, avoid corrosion or short-circuiting, see, hear, or remain hidden. Marine organisms have solved these and other challenges, by evolving strategies to survive and thrive in the oceans. Borrowing from these strategies promises more robust and effective design solutions, making biologically inspired (bio-inspired) technology an attractive research topic for the US Navy. In the last two decades, the Naval Undersea Warfare Centers have invested in supporting bio-inspired technology basic and applied research. The term “bioinspired” has emerged in strategic guidance documents only recently, generating focused interest in biomimetic systems and processes. Areas of interest include structures for propulsion and manipulation, sensory systems, signal processing and control, exploitable behaviors, and novel materials and chemical processes. Funding for this research has been supplied by the Office of Naval Research as well as

C. T. Murphy () · W. N. Martin · J. M. Lapseritis Bio-Inspired Research and Development Laboratory, Naval Undersea Warfare Center Division Newport, Newport, RI, USA e-mail: [email protected]; [email protected]; [email protected] J. A. Franck Department of Engineering Physics, University of Wisconsin-Madison, Madison, WI, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_8

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In-House Laboratory Independent Research (ILIR) and Naval Innovation Science and Engineering (NISE) Section 219 internal investment instruments. The reduction of flow-induced vibrations is one area for which Navy Science and Technology research has looked to nature for solutions. Vibrations can be catastrophic in many structures due to fatigue stress or buckling; thus, vibration mitigation in both air and water are integral to the design, maintenance, and performance of a wide range of technologies. In the most undesirable conditions, the coupling of the unsteady flow phenomena over a bluff body can resonate with the natural frequency of the structure in a positive feedback loop in a process called vortex-induced vibration (VIV). Mitigation of VIV and subsequent unsteady hydrodynamic loads has precedent in the biological world in the whiskers of seals, which can offer insight into bio-inspired technology.

2 Physics of Vortex-Induced Vibrations (VIV) A streamlined structure is that in which the surrounding fluid flow closely follows the body, creating smooth, steady velocity and pressure profiles in a uniform flow (Fig. 1a). However, in many engineering applications, a cylindrical cross section or similar bluff-body geometry is required (Fig. 1b), as would be the case with an underwater mooring line or pylon support structures for a ship dock. With a blunter shape, or cross section, the flow is not able to overcome the adverse pressure gradient as it moves around the body and the streamlines separate from the surface in an alternating manner rolling up into coherent vortices. The alternating vortex structures form a pattern (Fig. 2) known as the von Kármán vortex street [33, 42]. These vortex streets occur in all types of fluids and can vary dramatically in size, as they scale with the structure creating the disturbance. For example, dock pilings in a water current can create vortices of around a meter in diameter, while a repulsive laser beam moving through a superfluid gas can create microscale vortices of 50 μm Fig. 1 Water flow over (a) a streamlined body and (b) a bluff body. Vortex shedding occurs downstream of the bluff body for sufficiently large Reynolds numbers

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Fig. 2 Vortices over Alejandro Selkirk Island in the southern Pacific Ocean. (Photo credit: NASA Earth Observatory)

diameter [26]. On the other extreme, very large vortices, kilometers in diameter, are formed in the clouds by the wakes of high islands (Fig. 2). The von Kármán vortex street is very predictable in terms of the shedding frequency at which the alternating vortices form. The non-dimensional frequency, or Strouhal number, St = fD/U, is defined as the shedding frequency, f, times the bluff-body width, D, divided by the freestream velocity, U, and is commonly found to be St ~ 0.2 for a wide variety of flow regimes [5]. However, this aesthetically pleasing and predictable flow pattern can also wreak havoc on many engineering designs. As the vortex sheds, the body experiences a reaction force from the lowpressure region formed by the vortex, and thus the body is exposed to a steady occurrence of fluctuating forces perpendicular to the flow direction. Depending on the structural stiffness, these unsteady forces can cause a side-to-side vibration at the same frequency of the vortex shedding. This phenomenon is typically called vortex-induced vibration and is usually abbreviated as VIV [5, 34].

3 VIV and Its Impact on Engineering One of the most prominent examples of VIV is that of the Tacoma Narrows Bridge in 1940. When wind was blowing perpendicular to the bridge length, vortices were shed off of the deck. This vortex shedding caused cyclic pressure loading that excited the torsional flutter (twisting) mode of the bridge. Within a short amount of time, the flutter mode became phased-locked with the vortex shedding frequency, which subsequently caused the catastrophic failure of the bridge [14]. Since this tragedy, extensive research has been conducted on VIV and mode excitation [2,

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5, 19, 41, 44]. Consequently, this was the last major suspension bridge failure as engineers now better understand and design for aerodynamic forces. Although an airplane wing is designed to be streamlined, even a streamlined body can produce flow separation during certain maneuvers that require high angles of attack. This could initiate a similarly complex flutter mechanism as the Tacoma Narrows Bridge and be equally catastrophic. Extensive work has gone into predicting, sensing, and alleviating aerodynamic flutter. This includes developing multi-physics modeling tools to capture the dynamics, aerodynamics, and structural changes of a wing in flight and is one of the most challenging aspects of modern aircraft design. Ironically, this undesirable and passive vibrational motion of a fixedwing can be beneficial in the flapping flight of insects, birds, and bats that have been shown to capitalize on the unsteady vortices shed from their wings [39]. Some engineering applications also seek to capitalize on the motion produced by VIV. Recently, VIV has been utilized as a water turbine in which the oscillatory motion of the vortex shedding drives a generator to produce power. Research at the Marine Renewable Energy Lab developed the VIVACE (Vortex Induced Vibration for Aquatic Clean Energy) prototype, which has multiple large bluffbodied cylinders whose purpose is to maximize VIV by carefully selecting the array configuration and desired flow mode for optimal power generation [3, 4]. In a similar fashion, the Leading Edge team from Brown University developed an ocean-renewable energy device whose energy production relies heavily on the strong vortex shedding of flapping foils, a motion inspired from the flapping flight of animals [6, 23, 35, 40]. VIV is still a major concern in many industries, including architecture, oil and gas exploration, renewable energy, and design of support structures [7, 25]. As wind turbines become taller, they reach more wind velocity resources; however, the support structures, on land and in water, will almost always suffer from some mode of VIV and must be structurally designed around these unsteady loads [21]. Likewise, the oil industry has moorings, risers, and pipelines, which are all cylindrical in structure. Two typical vibration mitigation techniques used in the drilling industry are helical strakes and fairing tails [45]. These can be manufactured in lengths relative to the pipe or cable and are easy to deploy. However, small strake heights can be affected by marine growth, and added water mass and unsteady flows can cause problems for tail fairings.

4 VIV Mitigation in Navy Applications The Navy, like many other industries, must contend with VIV complications in both air and water environments. Poles, masts, and cables in air or underwater are subject to enhanced drag, vibration, and potential failure. In many scenarios, this is complicated by the accompanying acoustic noise that emanates from the vibrations, which subsequently interferes with instrumentation and sensing.

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Fig. 3 Example of ribbon fairing on a section of cable

One simple mitigation solution has been the introduction of ribbon fairing, which is widely used in underwater cables. Shown in Fig. 3, wrapping flexible and piecewise “ribbons” is effective in breaking up the coherent vortex shedding by shifting the cross-sectional body shape from a bluff-body cylinder (Fig. 1b) to a more streamlined body such as that shown in Fig. 1a. This shift in streamlines around the structure reduces the mean drag and reduces the intermittent and unsteady forces due to vortex shedding. The implementation of ribbon fairing allowed for faster towing without interference from cable strumming. While ribbon fairing is currently utilized broadly, improvements and emerging solutions for VIV reduction are continuously sought. As recently as 2019, the Navy has patented a type of ribbon fairing with ribbons that rest closer to a towed deployment angle of the cable (rather than the typical ribbon fairing that comes off of the cable at a 90 degree angle). This reduces the added drag caused by the fairing [8]. Other types of modifications to bluff bodies have been successfully deployed, with the goal of reducing the magnitude and severity of the coherent vortices. Adding helical strakes, dimples, or protrusions onto an otherwise smooth structure will add turbulence and decrease the coherence, and thus the pressure force of any shed vortices.

5 VIV Reduction in Nature Nature has been solving this problem long before the Navy has taken it up. While many shapes in nature are naturally streamlined – such as the body shapes of fish, penguins, dolphins, and seals – vortices are still shed from these bodies. Nature has evolved strategies to either reduce or utilize these vortices and VIV. While overall flexibility allows plants to withstand strong winds, the leaves of some species have an often overlooked method of VIV mitigation to facilitate this. Wild ginger (Hexastylis arifolia) and wild violet (Viola papilionacea) have developed leaf structures that curl up into cone shapes with the vertex pointing into

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the flow, when put into high wind situations. A stable recirculation zone forms inside of the cone, extending into the downstream air flow. This creates a low-pressure zone that assists in tightening the cone as flow speed increases while reducing vortex shedding [28]. The pectoral flippers of the humpback whale (Megaptera novaeangliae) represent an iconic bio-inspired model for reducing drag and lift in large bodies. These massive marine mammals mitigate flow separation by channeling fast-moving water across their long bladelike flippers using upstream bumps or tubercles. Without these bumps, a smooth flipper would stall the whale in mid-turn due to turbulent vortices [10, 29]. Using a different mechanism in air, owls have evolved a specialized wing/feather structure that enables near-silent flight by eliminating VIV and associated noise. The leading edges of the wing feathers are covered in small projections that break up air flow along the wing into smaller, more stable flows. Interestingly, these features seem to work best at steep wing angles when owls are in the final phase of an attack, similar to the humpback whale. At the posterior of the wing, the narrow air flow is further broken down by a fringed trailing edge that ultimately eliminates VIV and dampens wing sound [24]. On the other extreme, swimmers and flyers in nature have also been shown to purposefully manipulate the unsteady vortices from flow separation to provide greater propulsion and/or lift forces. Flapping flight on all scales, from insects, bats, and birds, have shown that unsteady vortex shedding, or dynamic stall, is critical to enhance lift beyond the steady-state equivalent [22]. In aquatic propulsion, the sinusoidal kinematic motion of the fins operate in a specific frequency regime to generate a vortex pattern optimal for thrust generation, called a reverse von Kármán vortex street [1, 37]. Whether they are mitigating or manipulating shed vortices, each of these organisms has developed mechanisms to control the unsteady vortex shedding targeted toward their local environment and their specific functions. Learning the physics and impact of these mechanisms can lead to more innovative and effective solutions for how to control VIV for the next generation of engineering designs.

6 Seal Whiskers as a Bio-inspired Model One particular model for the reduction of VIV in nature that has been studied by Navy researchers is the seal whisker. Seals use their highly sensitive whiskers (Fig. 4) to hunt, by feeling the hydrodynamic disturbances generated by their swimming prey. This ability is well known in the harbor seal (Phoca vitulina) [15, 38, 43] and presumed for most other true seal (Phocidae) species. The harbor seal is often used as a bio-inspired model due to its common nature and sample accessibility, as well as the wealth of information on its sensory ecology. Harbor seals and most of the other true seal species have undulations on the surface of their whiskers (Fig. 5). This repeated series of crests and troughs along the

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Fig. 4 The whisker array of a harbor seal with the whiskers held in an active position

Fig. 5 The smooth whisker of a sea lion (top), compared to the undulated whisker of a harbor seal

surface of the whisker is unique to seals. The whisker surface in all other mammals is smooth. Even the closely related sea lions, fur seals, and walrus have smooth whiskers. Detailed description of the bumpy morphology can be found in Ginter et al. [11–13] as well as Hanke et al. [17]. The seal whisker undulations have been shown to change the nature of the fluid-structure interaction when compared to a smooth vibrissa [17, 18]. Due to the three-dimensionality of the whisker, the strong vortices that drive vibrations are modified creating a smaller footprint and shifting the natural frequency of oscillation. In summary, the strong VIV response is diminished through the breakup of coherent vortical structures that drive the vibration. However, there is much more to the explanation that is still unexplored, in terms of the specific fluid dynamics mechanisms that drive the breakup, what is the response of the whisker at varying

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angles or flow conditions, and how multiple whiskers may work together to relay sensory information back to the seal. Here we focus on understanding the vibrational responses of the seal whiskers in various flow conditions and explore the fluid dynamics associated with VIV reduction. To study this system comprehensively, experimental data was collected on real seal whiskers in a water tunnel and compared to modeled data from an idealized and modified whisker geometry. Three dimensional (3D) prints of the modeled whiskers were used for preliminary visualization studies to serve as a guide for future experimentation.

6.1 Water Tunnel Experiments with Real Seal Whiskers For the experiments, whiskers were dissected from a harbor seal carcass recovered by the California Marine Mammal Stranding Network. The subcutaneous capsule (follicle) was removed, but the base of the whisker was maintained intact. Six individual whiskers were selected for testing, and each whisker was tested individually. Whisker lengths ranged from 8.3 cm to 10.2 cm. The whisker tapers along its length. The diameter for each whisker was measured with calipers at the point on the shaft where recordings were made (12.5% up the whisker length). For each whisker, the diameter of the major axis was measured at four peaks and four troughs. For all of the measurements taken, the peak values ranged from 0.7 to 1.2 mm (mean = 0.96 mm), and the trough values ranged from 0.67 to 1.10 mm (mean = 0.88 mm). Whiskers are made of keratin material and may be stored dry without the need for preservative. Samples were rehydrated for at least 1 hour prior to testing to allow the keratin to rehydrate and retain its pliability. Testing was conducted at the Naval Undersea Warfare Center in Newport, Rhode Island. During testing, the bottom 1 cm of the whisker was clamped in a metal holder. The holder was inset in the tunnel floor (see Fig. 6) and equipped with a rotary stage that allowed the sample to be fixed at different angels of attack. A laser vibrometer (Polytec OFV-5000) was utilized to directly measure the vibration of the whisker. Measurements were taken in the cross-stream direction at 12.5% up the whisker length. This position, close to the base of the whisker, provided good signal return to the vibrometer. A small silver Sharpie dot was marked on the whisker surface at 12.5% up the whisker length to allow better reflectivity for the laser and ensure accurate positioning. Vibrations were measured in the cross-stream direction (as in [30]). Recordings were limited to cross-stream measurements as it was not feasible to record with the laser from downstream or upstream of the whisker. At this time, we are not aiming to determine structural or hydrodynamic coupling effects of vibrissa vibration between the streamwise and transverse (lateral) directions, although these effects may exist. The water tunnel provided a controlled flow environment in which the water speed could be precisely adjusted and the flow-induced vibrational response of the

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Fig. 6 View of the laser water tunnel experimental setup with the laser vibrometer focused on a whisker in the test section. The recording location can be seen as a red dot reflecting off of the whisker’s surface

whisker could be measured. Whiskers were exposed to the full range of biologically relevant flow speeds (0.5–2.5 m/s). Testing was conducted in .25 m/s increments, for a total of nine test speeds. This corresponds to a Reynolds number range of 370–2800. Reynolds number was calculated for each whisker based on the average diameter of four peaks and four troughs, at the point on the whisker where laser vibrometer recordings were taken. Water was pumped through a recirculating system and smoothed by upstream flow conditioning structures, so that flow in the test section was laminar. Water speed was controlled by rpm of the pump, which was calibrated by laser Doppler velocimeter to give accurate flow speed. Data recordings were 1 minute in length at a sample rate of 65,536 Hz (216 samples per second). This sample rate was used to ensure that no high-frequency content was overlooked and proved to be much higher than necessary for the frequency content of the signals. Signals were lowpass filtered during sampling with a 5000 Hz cutoff so that no aliasing from higher frequencies would be created. Two recordings were collected for each sample for quality assurance purposes. For calculations of peak frequency for the dimensional and non-dimensional plots, signals were high-pass filtered at 20 Hz in order to filter out low-amplitude, low-frequency responses of the whisker holding system. For creating frequency spectra such as those in Fig. 9, the signal was processed using Welch’s power spectral density estimate with segment length of 215 , with 214 overlapped samples, and with 215 DFT points. In addition to being undulated, the whiskers of the harbor seal have an elliptical cross section. For comparison, the smooth whiskers of the California sea lion are somewhat elliptical [30], while those of terrestrial mammals are circular in cross section. This compressed cross section creates a distinct major and minor axis of the whisker, and therefore angle of attack must be considered. Whiskers were tested at both a 0◦ and 90◦ angle of attack (Fig. 7). The results showed that whiskers vibrate when exposed to water flow and the spectra of those vibrations are affected by water speed and angle of attack. This

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Fig. 7 In the 0◦ angle of attack condition (top), the narrow face of the whisker cross section points into the flow. At the 90◦ condition, the broad face points into the flow

confirms and expands upon the prior findings from Murphy et al. [31], which completed similar recordings at low speed only. When tested across the range of biologically relevant swim speeds in laminar flow, the peak frequencies of the whisker vibrations ranged from 54 to 937 Hz. This is within the vibrotactile sensitivity range for this species [32] of 10 to 1000 Hz. The prior published harbor seal sensitivity data was obtained in a behavioral study, in which a trained seal touched its whisker array to a vibrating plate and reported detection by pressing a response paddle. Sensitivity thresholds were determined according to well-established psychophysical testing procedures. When the peak values from each of the whisker recordings obtained in the present study are plotted against the sensitivity threshold for the seal (Fig. 8), some conclusions may be drawn regarding detectability. It can be seen that all the frequencies and most of the amplitudes from the excised whisker recordings fall within the sensitive range of the seal. Therefore, we assume that the vibrations are a salient signal to the animal. We hypothesize that changes in the vibration would be detectable to the animal and convey information about the environment. Interestingly, the majority of the excised whisker recordings have peak frequency content near the upper end of the sensitive frequency range for the animal. While the behavioral data show 80 Hz as the best sensitivity for the seal, the peaks of the excised whisker signals are distributed at higher frequencies. The recordings at the 90◦ orientation fall closer to the peak behavioral sensitivity, and all are well above the sensitivity threshold. It should be noted, however, that the shape of the whisker signal is complex and energy exists at frequencies outside of the dominant peak. This is especially true in the 90 degree case, in which multiple peaks in the signal appear (Fig. 9). When positioned at 0◦ from the flow with respect to the ellipse axis (Fig. 7), the flow is more streamlined and thus in a lower drag configuration. At the 90◦ condition, the larger projected area and blunt cross-sectional area contribute to a more dramatic ◦ flow separation. More flow separation increases the drag with respect to the 0 configuration and also further excites the torsional and flexural modes of the

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Fig. 8 Peak vibration (from power spectral density) of all recorded whisker signals plotted against the published vibrotactile sensitivity curve for the harbor seal (black line). Sensitivity line represents a threshold, or minimum detectable level. Signals above the black line are within the sensitive range of the seal. Inset figure shows the data with frequency (x-axis) in logarithmic scale. This allows better visualization of point under the curve and highlights the characteristic U shape of the behavioral sensitivity curve. (Seal sensitivity data is adapted from [32])

cantilevered whisker. This is confirmed by visual inspection of the recordings. The combination of these phenomena results in a higher whisker vibration amplitude. When comparing the whisker recordings to the seal’s vibrotactile thresholds, some of the 0◦ recordings fell below the threshold line. This could indicate that the signals are not detectable by the seal or could be an effect of the recording methodology. We hypothesize that clamping the whisker at the base may slightly dampen the signal amplitude. To test this, we re-sampled a subset of two of the whiskers with a compliant base made a 60 duro silicone sheeting (Fig. 10). The results demonstrated that the hard clamped experimental setup was faithfully capturing the frequency characteristics and overall signal shape of the whisker vibration, but was slightly reducing the amplitude (Fig. 11). This suggests that the amplitudes of the whisker vibrations are slightly higher than recorded and are likely all within the detectable range of the seal. The biological tissue that holds the whisker in the seal is certainly more complex than our test material, and thus the amplitude of the whisker vibration on a live seal may be higher than recorded here. Though still an artificial setup, and unable to fully mimic the properties of the biological tissue, the compliant base condition more closely mimics the follicle

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Fig. 9 Example signals from a single whisker at three flow speeds at 0◦ (top) and 90◦ (bottom) angle of attack. At the 0◦ condition, the peak frequency of the vibration increases with speed. The signal at 90◦ is more complex, with peaks at multiple frequencies. These peaks are overall higher amplitude than at the 0◦ condition and remain centered in the same frequency region, with the dominant frequency peak shifting with speed

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Fig. 10 Whisker with compliant base. Compliant material surrounds the base of the whisker. Material is encapsulated by hard metal casing

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structure that holds the whisker base. Compliance testing demonstrated that with either a rigid or a flexible clamping, the shape of the vibration signal is captured. Looking at the trends in the whisker vibrations across conditions, we can attempt to infer what role the undulated structure may have in the signal response. In the 0◦ condition, a linear relationship can be inferred between flow speed and peak vibration frequency (Fig. 12) for each whisker. As we assume that the changes in these signals are detectable by the seal, we hypothesize that the vibrations can encode swim speed information for the animal. In the 90◦ condition, the vibration frequency stays almost constant with respect to speed, except for a transition point at which the frequency jumps up slightly. The speed at which this transition occurs varies between whiskers, but all occurred at or above 1.25 m/s. This jump in

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Fig. 12 Peak vibration for each whisker at all speeds at the 0◦ angle of attack (top) and 90◦ angle of attack. Each line represents a different whisker sample

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frequency indicates that the VIV is exciting modal behavior in the whisker. As was shown in Fig. 9, the whisker signal at 0◦ angle of attack is characterized by one dominant frequency peak, while the 90◦ condition has a more complex spectra with multiple peaks. The frequency shift observed at higher speeds in Fig. 12 appears to be indicative of the higher-frequency portion of the signal becoming the dominant in the spectra. These vibration data recorded from the whiskers can also be viewed in a nondimensionalized manner, to normalize the relationship between speed, whisker size, and frequency of vibration (Fig. 13). In this view, the hydrodynamic relationships between conditions can be more clearly observed. The 0◦ condition shows a relatively consistent St response, which remains nearly constant across Reynolds number. The mean St for the 0◦ condition is 0.27. In the 90◦ condition, the jump between vibration modes noted above is also clearly visible. A physical cylinder was not tested in the experiment, as it would have differed from the experimental samples in material properties, curvature, and taper. However, the shedding response on a smooth cylinder with circular cross section is well documented [9, 20, 36] and known to have a Strouhal number of approximately 0.2 within this Re range, providing some basis for comparison. Interestingly, all of the 0◦ recordings fall

Fig. 13 Nondimensionalized plot of whisker recordings. Individual whisker sample is noted by line color with 0◦ and 90◦ for each presented. Line color corresponds to matching values in Fig. 12. Re for the tested flow speed was calculated based on the characteristic diameter of each individual sample

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above that point, and all the 90◦ values fall below it, indicating that the presence of the undulations and the orientation of the whisker both impact frequency of vibration. These experimental measurements very closely mimic the physics of the whisker motion in all its complexity. Testing with a real whisker sample incorporates all variables of the structure, including the curvature of the whisker, the full series of non-uniform bumps along its tapered length, and its material properties. Although measurements are taken at the base where the whisker is nominally perpendicular to the mount, the vibrations are still affected by flexural and torsional modes and are influenced by taper, swept position, and variations in the flexural stiffness (due to cross-sectional changes). These experiments provide a comprehensive picture of the whisker response. However, the complexity makes it challenging to decipher which attributes of the frequency response can be attributed to aspects of the bumpy morphology. In an effort to parse out these effects, a complimentary effort was completed using physical and numerical modeling to isolate the effects of geometry.

6.2 Effect of Seal Whisker Geometry Explained by Fluid Dynamics Using computational fluid dynamics (CFD), the intricate structure of the vortices shed by the whisker can be visualized and compared with a baseline, smooth elliptical geometry. Figure 14 displays iso-surfaces of Q-criterion, a quantity computed from CFD flow that highlights the vortex-like structures in the vorticity field [16]. Parameters for the seal whisker geometry were based on values published in by Hanke et al. [17]. Simulations are performed using direct numerical simulations in OpenFOAM, a finite volume-based numerical solver. Even with the streamlined, smooth geometry of the ellipse in the top of Fig. 14, there exist alternating pairs of vortices that remain coherent in structure along the length of the model. In contrast, a model containing the topology of an average harbor seal whisker (Fig. 14 bottom) demonstrates a more complex and interweaving arrangement of vortical structures. The structures, in turn, dissipate faster and are weaker than the coherent ones in the smooth elliptical geometry. The result is that the oscillatory and mean forces are reduced due to the chaotic flow downstream of the whisker and the dominant frequency response is shifted to a lower Strouhal number and lower amplitude. Similar visualization can be obtained in the laboratory by injecting dye into the boundary layer surrounding the whisker. Large-scale models of the whisker morphology and an ellipse were 3D printed for dye visualization. Models were approximately 29 cm in length with chord length dimensions at the undulation peak of 22 mm × 8.8 mm for the whisker and 20 mm × 10.5 mm for the ellipse. Red and blue dyes were released upstream of the model from two small dye nozzles that

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Fig. 14 Iso-surfaces of Q-criterion for an ellipse (top) and seal whisker geometry (bottom)

were gravity fed from dye wells fixed above the tank. This setup released blue dye over the top of the model and red dye under the bottom of the model. The vorticity, or the rotational motion of fluid particles, forms the vortex structures downstream, and this vorticity is all created at the surface of the whisker where the dye is injected. Thus, the two colored contours in Fig. 15 represent the rotational fluid particles as they travel downstream from the body. Representing a smooth elliptical geometry, Fig. 15a has alternating and coherent vortices, similar to the CFD image. On the other hand, dye injected into the undulated seal whisker model (Fig. 15b, c) is much more disperse, breaking up into smaller structures and dissipating into the mean flow at a faster rate. This presents some preliminary insight into the effects. Continued dye visualization work will further characterize the shedding from these and additional geometries.

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Fig. 15 (a) Dye visualization of flow over an ellipse (Re = 500 for all panels). (b) Dye visualization of flow over 3D printed seal whisker geometry with the dye flowing over peak of an undulation (Re = 500 for all panels). (c) Dye visualization of flow over 3D printed seal whisker geometry with the dye flowing over trough of an undulation (Re = 500 for all panels)

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6.3 Effect of Undulation Geometry on Whisker Response This work converges with that of a separate study [27], in which we used CFD to perturb the various features of the seal whisker in such a way that amplifies the effect in the flow field. Undulations occur in two modes along the length or span of the whisker. One set of undulations has an amplitude in the streamwise direction, the nominal direction of the flow, or chord length amplitude. The second set of undulations has an amplitude in the normal direction to the flow, or thickness amplitude. These two amplitudes alternate with one another, however are not completely anti-symmetric or perfectly sinusoidal. In addition to the two undulation amplitudes and asymmetry parameters, the frequency of undulation and the mean aspect ratio are also explored. Using a carefully designed two-factor factorial design of experiments, a screening matrix of 16 large-eddy simulations varied the geometric features to assess which features most highly influenced drag reduction, frequency response, and amplitude of oscillation. The results of the simulations confirm prior studies that that both amplitudes are needed for the decrease in drag. The wavelength and the aspect ratio are also critical components, as well as the interactions between these four major parameters. The effects of the chord length and thickness amplitudes were explained by different mechanisms. The variation in chord length varied the aspect ratio of the whisker from a slender to a bluff-body mode, and thus the flow was able to stay attached over some cross sections of the simulation. On average, this reduced drag and decreased the overall strength of the unsteady vortices formed by separation over the bluff-bodied cross sections. The thickness amplitude added spanwise transport of vorticity and velocity almost immediately upon flow separation. This enabled the vortices to form into more three-dimensional structures, promoting a faster breakup and less coherence than the comparable smooth cylinder. Through a separate and more detailed direct numerical simulation (DNS), the effect of wavelength is explored in terms of its impact on the unsteady amplitude of lift force and its resulting frequency spectra. The simulations directly solve the Navier-Stokes equations using a second-order finite volume method and orient the whiskers at an angle of attack of zero. These simulations allow control over the specific geometry of the whisker and enable a detailed flow visualization of the resulting vortex flow structures that develop downstream. A canonical seal whisker model is developed based on the parameters proposed by Hanke et al. Keeping all other geometric parameters the same, the whisker undulation wavelength, λ, normalized by the mean thickness of the whisker, was varied from 1.0 to 6.8. Within this range, λ = 3.4 is the characteristic whisker wavelength previously defined as a baseline value [17]. The results of the simulations are shown in Fig. 16, with the Q-criterion displaying the vortex structure immediately downstream of the whisker at Reynolds number (based on hydraulic diameter) or Re = 250. The whisker on the left side of Fig. 16 has the lowest wavelength (λ = 1) and results in the highest mean drag, with Cd = 0.95, nondimensionalized by the frontal

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Fig. 16 Vortex structure downstream of three whisker geometry models

area. The mean lift force is zero around the whisker; however, the amplitude of the oscillating lift force scales with the vibration amplitude in VIV, and thus the root-mean-squared fluctuation in the lift coefficient, , is an important quantity. The low wavelength response is very similar to a smooth cylinder; the Q-criterion contours display coherent vortex structures over the entire length of the simulation that persist downstream. Due to this, remains relatively high, at 0.09. For comparison, a smooth cylinder under similar flow conditions has Cd = 1.19 and = 0.35. The low wavelength on the left of Fig. 16 can be directly contrasted to the middle frame (λ = 3.4) which is the nominal seal whisker wavelength. The iso-surfaces downstream of the whisker display a clear breakup of coherent vortices and formation of three-dimensional horseshoe-shaped structures. These structures enable faster dissipation of the vortices and result in a decrease in drag (Cd = 0.87). Most notable is the drop in lift oscillations by an order of magnitude ( = 8.9e4). As the frequency is increased further to λ = 6.8 (right frame), these structures become more two-dimensional, accompanied by a slight rise in drag (Cd = 0.88) and RMS lift ( = 0.1). Thus, it is clear that the wavelength regime of the whisker is optimal for reduction in both mean and fluctuating components. However, these simulations are performed at a fixed angle of attack of zero. As indicated by experimental results, changes in the angle of attack will modify the hydrodynamics and may be one mechanism that seals utilize in sensing. In addition to the mean and RMS forces, the frequency of vibration is also shifted due to the change in wavelength. Spectra for two different non-dimensional flow speeds, or Reynolds number, are presented in Fig. 17. At low wavelength, the Reynolds number difference is quite small, and the single peak is very similar to a smooth cylinder. The dominant frequency, or Strouhal number, is approximately St = 0.24. At the nominal seal whisker wavelength (middle frame), Re = 250 has a dramatic drop in amplitude and also a shift to a lower Strouhal number of 0.18. As the Reynolds number increases, the signal amplitude increases but has a broadband response and dominant peaks at approximately St = 0.22 and a smaller one at

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4

amplitude

10

2

Re=250 Re=250

10

0

10

10

-2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

non-dimensional frequency, fDh/U

Fig. 17 Spectra for three wavelength models at Reynolds numbers (based on hydraulic diameter) 250 and 500

0.36. As the wavelength is increased to λ = 6.8, this second peak is amplified, and Reynolds number effects are more prevalent. These simulations indicate that the dominant shedding frequency that drives the VIV on the seal whisker is largely dependent on both wavelength and flow speed and that the wavelength regime in which the drag reduction is beneficial spans from λ = 3.4 to 6.8, with slight shifts in the dominant frequency of oscillation. The conclusions drawn from the seal whisker investigations tell us that they are indeed a unique geometry, with very beneficial drag manipulation and frequency selection properties. There is a range of wavelengths for which this regime is most amplified, but there is also a strong influence on flow speed (Reynolds number) as well as orientation of the whisker. Compared with previous drag and vortex mitigation techniques, the whisker employs undulations in two alternating directions; however, the precise range of amplitudes and angles of these undulations have yet to be thoroughly investigated. As shown by Fig. 17, the amplitude of vibration is reduced when the wavelength of undulation is within the range of that measured on the seal whiskers. Furthermore, as demonstrated by Fig. 16, the VIV is no longer a coherent streak of vortices as one would expect on a cylinder, but rather it has been broken up into smaller segments which exert a much weaker force on the whisker and thus reduce the oscillations and modify the frequency of vibration as shown in the experimental data. Although bio-inspired whisker applications have large potential within the Navy and elsewhere, there is still much unknown about how these complex undulations interact with the fluid environment. The causal linkages between the geometry and frequency response, the interaction of multiple whiskers in array arrangements, and how these phenomena may scale up to much larger flow speeds and diameters have not yet been determined. Acknowledgments We thank the NISE 219 and NUWC ILIR funding programs and Dr. Anthony Ruffa and Dr. Elizabeth Magliula for their support of this work. We thank Kate Lyons for contribution of CFD content to this manuscript. We thank Yenny Cardona Quintero for assistance with manuscript items and James Travassos, Dave Stoehr, Michael Seaman, and Keith McClenning for graphics support. We thank Erin LaBrecque for manuscript support. We thank Dr. Aren Hellum, Dana Hrubes, Dr. Charles Henoch, and David Joe Wade for their assistance with water tunnel testing. The use of marine mammal samples was authorized under the National Marine Fisheries Service, letter of authorization to C. Murphy. We thank Dr. Colleen Reichmuth of the Pinniped Cognition and Sensory Systems Laboratory for photographs and data from the live seal (National

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Marine Fisheries Service permit 14535) and for continued collaborative engagement. We thank Sprouts the seal from the Pinniped Cognition and Sensory Systems Laboratory for his numerous contributions to the field and for inspiring this work. Distribution Statement A: Approved for public release. Distribution is unlimited.

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21. S.G. Horcas, M.H.A. Madsen, N.N. Sørensen, F. Zahle, Suppressing vortex induced vibrations of wind turbine blades with flaps, in Recent Advances in CFD for Wind and Tidal Offshore Turbines, (Springer, Cham, 2019), pp. 11–24 22. T.Y. Hubel, C. Tropea, The importance of leading edge vortices under simplified flapping flight conditions at the size scale of birds. J. Exp. Biol. 213(11), 1930–1939 (2010) 23. D. Kim, B. Strom, S. Mandre, K. Breuer, Energy harvesting performance and flow structure of an oscillating hydrofoil with finite span. J. Fluids Struct 70, 314–326 (2017) 24. R.A. Kroeger, H.D. Grushka, T.C. Helvey, Low speed aerodynamics for ultra-quiet flight. Technical Report Affdl-Tr-71-75. Tennessee Univ Space Inst Tullahoma (1972) 25. R.A. Kumar, C.H. Sohn, B.H. Gowda, Passive control of vortex-induced vibrations: An overview. Recent Pat. Mech. Eng 1(1), 1–11 (2008) 26. W.J. Kwon, J.H. Kim, S.W. Seo, Y.I. Shin, Observation of von Kármán vortex street in an atomic superfluid gas. Phys. Rev. Lett. 117(24), 245301 (2016) 27. K.M. Lyons, C.T. Murphy, J.A. Franck, Flow over seal whiskers: importance of geometric features for force and frequency response. PLOS ONE 15(10), e0241142 (2020) 28. L.A. Miller, A. Santhanakrishnan, S. Jones, C. Hamlet, K. Mertens, L. Zhu, Reconfiguration and the reduction of vortex-induced vibrations in broad leaves. J. Exp. Biol. 215(15), 2716– 2727 (2012) 29. D.S. Miklosovic, M.M. Murray, L.E. Howle, F.E. Fish, Leading-edge tubercles delay stall on humpback whale (Megaptera novaeangliae) flippers. Phys. Fluids 16(5), L39–L42 (2004) 30. C.T. Murphy, W.C. Eberhardt, B.H. Calhoun, K.A. Mann, D.A. Mann, Effect of angle on flowinduced vibrations of pinniped vibrissae. PLoS One 8(7), e69872 (2013) 31. C.T. Murphy, C. Reichmuth, W.C. Eberhardt, B.H. Calhoun, D.A. Mann, Seal whiskers vibrate over broad frequencies during hydrodynamic tracking. Sci. Rep. 7(1), 1–6 (2017) 32. C.T. Murphy, C. Reichmuth, D. Mann, Vibrissal sensitivity in a harbor seal (Phoca vitulina). J. Exp. Biol. 218(15), 2463–2471 (2015) 33. E. Naudascher, D. Rockwell, Flow Induced Vibrations: An Engineering Guide (Dover Publications, Mineola, 2005) 34. R.L. Panton, Incompressible Flow (Wiley, New York, 1984) 35. B.L.R. Ribeiro, S.L. Frank, J.A. Franck, Vortex dynamics and Reynolds number effects of an oscillating hydrofoil in energy harvesting mode. J. Fluids Struct 94, 102888 (2020) 36. A. Roshko, On the drag and shedding frequency of two-dimensional bluff bodies. Technical note 3169. National advisory committee for aeronautics (1954) 37. T. Schnipper, A. Andersen, T. Bohr, Vortex wakes of a flapping foil. J. Fluid Mech. 633, 411– 423 (2009) 38. N. Schulte-Pelkum, S. Wieskotten, W. Hanke, G. Dehnhardt, B. Mauck, Tracking of biogenic hydrodynamic trails in harbour seals (Phoca vitulina). J. Exp. Biol. 210(5), 781–787 (2007) 39. W. Shyy, H. Aono, S.K. Chimakurthi, P. Trizila, C.K. Kang, C.E. Cesnik, H. Liu, Recent progress in flapping wing aerodynamics and aeroelasticity. Prog. Aerosp. Sci. 46(7), 284–327 (2010) 40. Y. Su, K. Breuer, Resonant response and optimal energy harvesting of an elastically mounted pitching and heaving hydrofoil. Phys. Rev. Fluids 4(6), 064701 (2019) 41. A.D. Trim, H. Braaten, H. Lie, M.A. Tognarelli, Experimental investigation of vortex-induced vibration of long marine risers. J. Fluids Struct 21(3), 335–361 (2005) 42. T. Von Kármán, Aerodynamics, vol 9 (McGraw-Hill, New York, 1963) 43. S. Wieskotten, G. Dehnhardt, B. Mauck, L. Miersch, W. Hanke, Hydrodynamic determination of the moving direction of an artificial fin by a harbour seal (Phoca vitulina). J. Exp. Biol. 213(13), 2194–2200 (2010) 44. C.H.K. Williamson, R. Govardhan, A brief review of recent results in vortex-induced vibrations. J. Wind Eng. Ind. Aerodyn. 96(6-7), 713–735 (2008) 45. T. Zhou, S.M. Razali, Z. Hao, L. Cheng, On the study of vortex-induced vibration of a cylinder with helical strakes. J. Fluids Struct 27(7), 903–917 (2011)

A Series of Multidimensional Integral Identities with Applications to Multivariate Weighted Generalized Gaussian Distributions Anthony A. Ruffa and Bourama Toni

1 Introduction In this work we develop and provide a series of new integral identities using extensively the computer algebra system Mathematica; these identities complement known ones available in handbooks [1, 2, 8]; the identities are expressed in terms of common special functions such as gamma and beta functions, Bessel functions, error and complementary error functions, and hypergeometric functions. The approach uses a power transformation or change of variables to decouple the integration variables, along with some famous “tricks” by early mathematicians such as Euler, Laplace, and Legendre. The identities developed involve infinite n-dimensional integrals, encountered often in mathematical physics, engineering, signal processing, probability, and statistics theory. We are mainly concerned with integration in the sense of Riemann, which inherently refers to a function defined on a topological space and has been both in a general and a mathematical sense completed by Lebesgue integral.1 Recall that integration assigns a number, the integral, to certain functions defined on certain

1 Distribution

theory by Schwartz and measure theory by Lebesgue can be thought of as the completion of differential calculus and integral calculus, respectively.

A. A. Ruffa () Naval Undersea Warfare Center, Newport, RI, USA e-mail: [email protected] B. Toni Department of Mathematics, Howard University, Washington, DC, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_9

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sets, both function and set satisfying regularity conditions, such as continuity and boundedness. The basic set could be a Riemannian manifold (e.g., Euclidean space) or the phase space of a dynamical system or a probability space. A notion of ndimensional volume in n-space determines the formation of the integral, which could represent the notion of length, area, or volume in Euclidean space, or, in general, volume, pressure, potential, or expectation. That is, the set involved is endowed with a notion of measure. Integration, since its formation, has found important applications, e.g., allowing probability to make explicit the notion of random variable, extending matrix theory to infinite dimensions. A basic measure space consists of a set, a ring of its subsets, and a countably additive real nonnegative function on the ring. We derive multiple higher-dimensional integral identities from the following approach. Given the multiple infinite integral 

∞ 0



∞

···

F (x1 , x2 , . . . , xn )dx1 dx2 · · · dxn ,

(1)

0

one might first try to compute it with Mathematica if available; when unsuccessful, one might resort to numerical integration. We are proposing the use of a power change of variable to convert the integration process into a process of successive iterated integrals. That is, set xi =xnαi ui ,

i = 1, . . . , n − 1

xn =xn .

(2)

This power change of variable leads to a decoupling of the original variables, yielding the iterated integrals 

∞ 0

 ··· 0

∞

α

F (xnα1 u1 , xnα2 u2 , . . . , xn n−1 un−1 , xn )

n−1 *

xnαi du1 du2 · · · dun−1 dxn .

i=1

(3) Such decomposition effectively eases the integration process. The overall end result is a comprehensive list of new integral identities expressed in compact mathematical formulas, in closed-form or partially closed-form, involving special functions widely used in applied sciences. Compact mathematical solutions are often more insightful than the equivalent numerical formulations/approximations, allowing “true understanding” of real-world applications. The work is organized as follows: Sect. 1 features relevant known special functions involved in the compact expressions of the identities, such as gamma and Gaussian functions. In Sect. 2, we discuss identities involving exponential terms and their applications. Section 3 presents integrals involving the complementary error function, while integrals involving the logarithms appear in Sect. 4. In Sect. 5 we derive some important compact formulas respectively for generalized and weighted Gaussian functional integrals.

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1.1 The Gamma Function We recall the first of our special functions: the gamma function, defined and initially studied by the Swiss mathematician Leonhard Euler (1707–1783).2 The gamma function Γ is one of the so-called transcendental functions first introduced by Euler in an attempt to extend in a continuous way the factorial function to non-integers. The function is also thought of as one of the solutions to the functional equation f (x + 1) = xf (x),

(4)

a unique solution under the additional assumptions that the self-mapping f (x) on (0, ∞) satisfies the condition that ln(f (x)) is convex and f (1) = 1. (See also [4].3 Initially defined as  Γ (x) =

1

(− ln t)x−1 dt, x > 0

(5)

0

through some simple changes of variables, e.g., s = − ln t, and s 2 = − ln t, the function can be written in its most common expression as  Γ (x) =

∞

t x−1 e−t dt,

(6)

t 2x−1 e−t dt.

(7)

0

or 

∞

Γ (x) = 2

2

0

Note also that the gamma function could be derived as n!nx , n→∞ x(x + 1) · · · (x + n)

Γ (x) = lim t x−1 (1 − t/n)n dt = lim n→∞

(8)

or defining the factorial function with the Pochhammer symbols (α)0 =1, (α)n =α(α + 1) · · · (α + n − 1), (α = 0)

2 The

(9)

development of the gamma function has received great contributions from many eminent mathematicians, to include Adrien-Marie Legendre (1752–1833), Carl Friedrich Gauss (1777– 1855), Christopher Gudermann (1798–1852), Joseph Liouville (1809–1882), Karl Weierstrass (1815–1897), and Charles Hermite (1822–1901). + 3 The notation Γ (x) is due to Legendre in 1809, Gauss preferring (x), which is actually Γ (x+1).)

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we have (1)n = n! and again Γ (α + n) = (α)n Γ (α).

(10)

The following properties of the gamma function are well-known. 1. Γ (1) = 1. Γ (n + 1) = n!. Γ (x + 1) = xΓ (x) ! 1 2. Γ (x)Γ (y) = Γ (x + y)B(x, y) with B(x, y) = 0 t x−1 (1 − t)y−1 dt (beta function) + (1+1/k)z n!nz 3. Γ (z) = limn→∞ z(z+1)···(z+n) = 1z ∞ k=1 1+z/k 4. Γ (z)Γ (1 − z) = sinππ z (complement/reflection formula) √ 5. Γ (z)Γ (z + 12 ) = 21−2z π Γ (2z) (Legendre duplication formula) (k−1)/2 k 1/2−kz Γ (kz) (Gauss 6. Γ (z)Γ (z + k1 )Γ (z + k2 ) · · · Γ (z + k−1 k ) = (2π ) multiplication formula) (n−1)/2 (2π )√ 7. Γ ( n1 )Γ ( n2 ) · · · Γ ( n−1 (Euler multiplication formula) n )= n √ n −n 8. Γ (n + 1) = n! ≈ 2π nn e (Sterling-de Moivre asymptotic formula for n tending to infinity) !∞ n Γ (x) 9. d dx = 0 e−t t x−1 (ln t)n dt (polygamma function) n Finally, note that the gamma function is infinitely differentiable on its domain of definition, and offsprings of the gamma function include Ψ (i.e., the digamma function), defined as the logarithmic derivative of Γ (x) for any nonzero or negative integer, i.e., Ψ (x) =

d ln(Γ (x)), dx

(11)

and the polygamma function Ψn (x) = Ψ (n) (x) =

d n+1 ln(Γ (x)). dx n+1

(12)

Note that the zeros of the digamma function are also the extrema of the gamma function. The relation to the famous Riemann zeta function could be derived as follows. With the change of variable t = ks (where k is a positive integer), one may write  ∞ Γ (x) = k x s x−1 e−ks ds, (13) 0

yielding    ∞ ∞  1 1 1 x−1 ζ (x) = = s − 1 ds, kx Γ (x) 0 1 − e−s k=1

(14)

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hence the beautiful formula 

∞

ζ (x)Γ (x) =

t x−1 dt, x > 1 et − 1

0

(15)

from which one could derive, e.g., 

∞ 0

t π2 dt = . et − 1 6

(16)

The gamma function appears in the expression of many other special functions, in particular: !∞ !∞ !∞ 2 z 1. Γ (x) = 2 0 t 2x−1 e−t dt = a x 0 t x−1 e−at = −∞ exz e−e dz = 1 ! 1 ∞ −y x dy x 0 e !∞ a 1 2. Γ (1 + a ) = a1 Γ ( a1 ) = 0 e−u du; a > 0 (c−a−b) 3. ΓΓ (c)Γ (c−a)Γ (c−b) = 2 F1 (a, b; c; 1), where the Gauss hypergeometric function ,∞ (a)k (b)k zk 2 F1 (a, b; c; z) = k=0 (c)k k!

1.2 Multidimensional Gaussian Integrals In many areas of applied sciences, the Gaussian function is very important because of its significance as the probability density function of the normal distribution: it serves as weighting function to convey the idea that relevant data points concentrate close to the center/mean and are more desirable, whereas those far away (i.e., in the “tails”) are relatively insignificant, i.e., a “light-tailed” function. Its one-dimensional form is given by G(x) = e

− (x−μ) 2

2

2σ

,

(17)

where μ, the mean, is a measure of the central tendency of the data and σ , the standard deviation, is a measure of the dispersion of the data points in terms of the probability distribution. Here we are interested in multivariate Gaussian integrals. A multidimensional Gaussian integral has the form  Gn =

∞ −∞

 ···

∞ −∞

e−x

T Ax

dx1 · · · dxn ,

(18)

also simply written, with the obvious short-hand 

∞

−∞

e−x

T Ax

d n x,

(19)

168

A. A. Ruffa and B. Toni

'T & where x = x1 x2 x3 · · · xn ∈ Rn , d n x = dx1 · · · dxn denotes the volume element dV and A is a n × n matrix such that x T Ax is a real positive definite quadratic form of n variables satisfying x T Ax > 0 to ensure the convergence of the integrals. We now recall the so-called Laplace’s trick that led to the famous identity 

∞

e

−x 2

0

1 dx = 2



∞ −∞

e

−x 2

√

π . 2

dx =

(20)

That is, consider the power change of variable y = xs, with dy = xds, leading to 

∞

e−x dx 2

2

 =

0

0

 =

∞

e−x dx × 2

e 

=

∞

e−y dy = 2

0

∞  ∞

0



−x 2 (1+s 2 )





∞ ∞

0

e−(x

2 +y 2 )

dxdy

0

xdx ds

0 ∞

0

π 1 ds = . 4 2(1 + s 2 )

(21)

Hence the identity, which is generalized as follows, for ai > 0  Gn =

∞

−∞

 ···

∞ −∞

1

e− 2

,n

2 i=1 ai xi

dx1 · · · dxn = √

(2π )n/2 . a1 a2 · · · an

(22)

Note here that in relation to the previous gamma function, we have Γ

   ∞ √ 1 2 = e−t dt = π . 2 −∞

(23)

2 Identities Involving Exponential Terms Inspiration from work by Ruffa on the method of exhaustion [10] has led to a special use of power change of variables to decouple multiple variables of integration for infinite n-dimensional integrals in order to develop new integral identities. First we recall the method of exhaustion as presented in Ruffa’s work.

Multidimensional Integral Identities

169

2.1 The Method of Exhaustion The method of exhaustion4 can support the evaluation of improper integrals with the formula [10] 

∞

∞ ∞ 2 −1   n

f (x)dx = b

0

(−1)

m+1 −n

n=1 m=1 p=0

2

  mb f pb + n , 2

(24)

where 0 < b < ∞. To wit consider the double infinite integral 

∞ ∞ 0

0

' & exp −(x1a1 + x2a2 )a3 dx1 dx2 .

(25)

A direct attempt with Mathematica would not produce a result. However, through a2 a

the method of exhaustion with b = x2 1 , we obtain the following: 

∞ ∞ 0

& ' exp −(x1a1 + x2a2 )a3 dx1 dx2

0



∞

= 0

a2 a1

x2

∞ −1  ∞ 2  n

(−1)

m+1 −n

2

n=1 m=1 p=0

  a  m  a1 a2 a2 3 dx2 . exp − p + n x2 + x2 2

(26) Equation (24) is then used again, this time with b = 1, to convert (26) back to integral form, i.e., 

∞ 0

a2 a1

x2

∞ −1  ∞ 2  n

(−1)

m+1 −n

2

n=1 m=1 p=0



  a  m  a1 a2 a2 3 dx2 exp − p + n x2 + x2 2

∞ ∞

= 0

0

a2 ' & a x2 1 exp −(ua11 x2a2 + x2a2 )a3 du1 dx2 .

(27)

Remark 1 Equation (27) is identical to (33) presented below. Note that the idea of the power change of variable to compute multidimensional infinite integrals stemmed from the above procedure.

4 The

Method of Exhaustion, Méthode des Anciens, Methodus exhaustionibus, was used by Greek mathematicians to find the area of a shape by inscribing a sequence of n-sided polygons of known areas inside the shape and then “exhausting” the remaining area as n increases.

170

A. A. Ruffa and B. Toni

2.2 The Power Substitution Using the power substitution, we first develop identities for the n-dimensional infinite integrals, i.e., 

∞



∞

···

0

e−f (x1 ,...,xn ) dx1 · · · dxn ,

(28)

0

for certain expressions of the multivariate function f (x1 , . . . , xn ) that allow a decoupling of the variables under integration through the power change of variable. We start with the double infinite integral with a1 a +x2 2 )a3

f (x) = f (x1 , x2 ) = e−(x1

(29)

.

Theorem 1 (Double Integral Identity)  

∞ ∞



e 0

a a −(x1 1 +x2 2 )a3

dx1 dx2 =

Γ

1 a1

  1 a2

Γ

a1 a2 a3 Γ

0

 Γ 

1 a1

1 a3



+

+ 

1 a1

1 a2

1 a2

 ,

(30)

where a1 > 0, a2 > 0, a3 > 0.

(31)

Proof Consider the power substitution given by a2 a

x1 = x2 1 u1 ;

(32)

a2 a1

dx1 = x2 du1 ; so that  

∞ ∞

= 0

0

∞ ∞

0

0

' & exp −(x1a1 + x2a2 )a3 dx1 dx2

a2 a1

' & exp −x2a2 a3 (1 + ua11 )a3 du1 dx2 .

∞

a2 ' & a x2 1 exp −x2a2 a3 (1 + ua11 )a3 dx2

x2

(33)

Noting that  0

− 1 − 1 1  1 + ua11 a1 a2 Γ = a2 a3



1 a3



1 1 + a1 a2



(34) ,

Multidimensional Integral Identities

171

and  



∞

1 + u1

0



1 1 a1 − a1 − a2

du1 =

Γ

 

1 a1



a1 Γ

1 a2

Γ +

1 a1

1 a2

,

(35)

it follows that 

 

∞ ∞

0

' & exp −(x1a1 + x2a2 )a3 dx1 dx2 =

0

Γ

 

1 a1



1 a2

Γ

Γ 

a1 a2 a3 Γ

1 a3

1 a1



+

+ 

1 a1

1 a2

1 a2

 , (36)  

for a1 > 0, a2 > 0, a3 > 0, which is the required result. The case a3 = 1 leads to a known identity. Indeed Corollary 1 When a3 = 1, we obtain ∞ ∞

 0

0

  Γ & ' exp −(x1a1 + x2a2 ) dx1 dx2 =

1 a1

  Γ

1 a2

,

a1 a2

(37)

a1 > 0, a2 > 0, a3 > 0. We hence recover the well-known result in [1, 2]. One may also consider various expressions of a3 in terms of a1 and a2 . For example, assume a3 =

a1 + a2 , a1 + a2 + a1 a2

(38)

and derive by direct computation  Γ

1 a3



1 1 + a1 a2



 =Γ

     1 1 1 1 1 1 Γ , + +1 = + + a1 a2 a1 a2 a1 a2 (39)

which proves Corollary 2  0

∞ ∞ 0

  Γ a1 +a2 exp −(x1a1 + x2a2 ) a1 +a2 +a1 a2 dx1 dx2 =

  1 a1

  Γ

1 a2

a1 a2

1 a1

+

1 a2

 +1 . (40)

Similarly, assuming a1 + a1 > a1 a2 ,

(41)

172

A. A. Ruffa and B. Toni

and a3 =

a1 + a2 , a1 + a2 − a1 a2

(42)

results in  Γ

1 a3



1 1 + a1 a2



 =Γ

 1 1 + −1 , a1 a2

(43)

proving Corollary 3 

∞ ∞

0

0

  Γ a1 +a2 exp −(x1a1 + x2a2 ) a1 +a2 −a1 a2 dx1 dx2 =

  1 a1

  Γ

1 a2

a1 + a2

.

(44)

Now we extend the power substitution approach to more than two infinite integrals, starting from a triple integral to n-dimensional multiple integrals. Taking f (x) = f (x1 , x2 , x3 ) = (x1a1 + x2a2 + x3a3 )a4 ,

(45)

we prove Theorem 2 (Triple Integral Identity) 

∞ ∞ ∞ 0

=

0

Γ

0

  1 a1

' & exp −(x1a1 + x2a2 + x3a3 )a4 dx1 dx2 dx3  

Γ

1 a2

  Γ

1 a3

a1 a2 a3 a4 Γ



 Γ

1 a1

1 a4

+



1 a2

1 a1

+

+ a12 + 

1 a3

1 a3



(46) ,

where a1 > 0, a2 > 0, a3 > 0, a4 > 0.

(47)

Proof For this case, the following power substitutions are used: a3 a

a3 a

a3 a

a3 a

x1 = x3 1 u1 ; dx1 = x3 1 du1 x2 = x3 2 u2 ; dx2 = x3 2 du2

(48) (49)

Multidimensional Integral Identities

so that  ∞ 0

∞ ∞ 0



0

' & exp −(x1a1 + x2a2 + x3a3 )a4 dx1 dx2 dx3

∞ ∞ ∞

= 0

0

173

a3 a1

x3 x3

0

(50)

' & exp −x3a3 a4 (1 + ua11 + ua22 )a4 du1 du2 dx3 .

a3 a2

Noting that 

∞ 0

a3 a3 ' & a a x3 1 x3 2 exp −x3a3 a4 (1 + ua11 + ua22 )a4 dx3



− 1 − 1 − 1 1  1 + ua11 + ua22 a1 a2 a3 Γ = a3 a4

1 a4



1 1 1 + + a1 a2 a3



(51) ,

and 

 

∞ ∞ 0

1 + ua11

0

− − − + ua22 a1 a2 a3 1

1

1

Γ

du1 du2 =

1 a1

 

Γ 

a1 a2 Γ

1 a1

1 a2

+

  Γ

1 a2

+

1 a3

1 a3

. (52)  

Hence the claim.

To generalize, we now consider an arbitrary number of dimensions with the corresponding n-dimensional integral and prove i = 1 . . . n + 1, and setting

Theorem 3 For ai > 0, ∞ ∞

 I= 0



∞

···

0

-  exp −

0

n 

an+1 . xiai

dx1 dx2 · · · dxn ,

(53)

i=1

we obtain  

+n

i=1 Γ

I=

Γ

1 ai

,

 Γ

n 1 i=1 ai

,n

1 an+1

+

1 j =1 aj

 .

(54)

dx1 dx2 · · · dxn .

(55)

n+1 i=j aj

Proof Consider the integral 

∞ ∞

Ip = 0

0



∞

··· 0

-  exp −

n  i=1

p . xiai

174

A. A. Ruffa and B. Toni

A series of n − 1 substitutions are made for 1 ≤ i ≤ n − 1, i.e., an a

xi = xn i ui ,

(56)

an ai

dxi = xn dui , leading to 

∞ ∞

Ip = 0

0



∞ n−1 *

··· 0

⎡

an aj



xn exp ⎣−xnn

a p

n−1 

1+

p ⎤ uai ⎦ dxn du1 du2 · · · dun−1 . i

j =1

i=1

(57) Evaluating just the integral over xn in (57) leads to

Ip =

 ∞ ∞ 0

0

⎛ ⎛ ⎞− 1 ⎞ aj  ∞ n n−1 n  a  1 1 *⎝ 1 i⎠ ⎠ du1 du2 · · · dun−1 ··· ui Γ⎝ 1+ p ai 0 an p j =1

i=1

i=1

(58)

and Γ

 , n 1

1 j =1 aj

Ip p , = n I1 pΓ i=1

1 ai



.

(59)

Here I1 is the integral (55) or (58) when p = 1. Since  

+n

i=1 Γ +n

I1 =

1 ai

j =1 aj

(60)

,

it follows that +n Ip =

 

i=1 Γ +n

1 ai

j =1 aj

·

Γ

 , n 1

pΓ

p

1 j =1 aj

, n

1 i=1 ai



.

Equation (61) is identical to (54) when p = an+1 .

2.3 Applications We derive several other integral identities for the integral in standard form, i.e.,

(61)

 

Multidimensional Integral Identities



∞

0

175



∞

···

e−f (x1 ,...,xn ) dx1 · · · dxn .

(62)

0

First take f (x) = f (x1 , x2 , x3 ) = x12 + x1 x2n + x22n . That is, consider integrals of the type 

∞ ∞ 0

0

   exp − x12 + x1 x2n + x22n dx1 dx2 ,

(63)

where n ≥ 1. Again using the substitution x1 = x2n u and dx1 = x2n du leads to 

∞ ∞

0

0

   exp −x22n u2 + u + 1 x2n dx2 du.

(64)

We start with some numerical values for n, compute the corresponding integrals using Mathematica, and observe an emerging pattern involving the hypergeometric function p Fq (a1 , . . . , ap ; b1 , . . . , bq ; z), in particular the Gauss’ hypergeometric expression 2 F1 (a, b; c; z)

=

∞  (a)n (b)n zn

(c)n

n=0

n!

(65)

,

where (x)n is the Pochhammer symbol as defined previously [8]. 1. For n = 2 we obtain  0

∞ ∞

   Γ exp − x12 + x1 x22 + x24 dx1 dx2 =

3 4

 2 F1



1 3 5 4 , 4 ; 4 ; −3

√

2      √  √ 2 3+i 5 3 1 3 5 −(−1) · 3 · π · Γ 4 + (−1) 3 Γ 4 2 F1 4 , 4 ; 4 ; 6 3 0

7 8

=

 

1 8

1

3

(1 + (−1) 3 ) 4

. (66)

2. For n = 3 we get  ∞ ∞ 0

0

   Γ exp − x12 + x1 x23 + x26 dx1 dx2 = 2

=−

(−1) 3 Γ

  2 3

  2 3

 2 F1



1 , 2 ; 7 ; −3 6 3 6 2

23

√       5 1 1 √ (−1) 18 · 2 3 · 3 6 Γ 16 Γ 43 − 2 π 2 F1 13 , 23 ; 43 ; 3+i6 3 . 1 2 √ 4 π(1 + (−1) 3 ) 3

(67)

176

A. A. Ruffa and B. Toni

3. n =

3 2

leads to

∞ ∞

 0

0

   Γ 3 2 3 2 exp − x1 + x1 x2 + x2 dx1 dx2 =

29

=

2

1

−(−1) 36 · 2 3 · 3 12 ·

√ π ·Γ

  4 3

2

+ 2 · (−1) 3 Γ 1

  5 6

 2 F1



1 5 4 3 , 6 ; 3 ; −3 1

  5 6

 2 F1

23

√  1 5 7 3+i 3 , ; ; 6 6 6 6

5

(1 + (−1) 3 ) 6

.

(68)

These patterns allow us the following generalization which we formally prove Theorem 4  In =

∞ ∞

0

0

   exp − x12 + x1 x2n + x22n dx1 dx2  = 

Γ 

n+1 2n



 2 F1



1 n+1 2n+1 2n , 2n ; 2n ; −3

2

n−1 n

  i − n1     1 1 1 1 × 2√ 3− 2n Γ − = Γ 1+n 2n 2 2n 2 π  √  2in n − 1 3n − 1 3 + i 3 ; ; +√ ; n > 0. 2 F1 1, n 2n 6 3(n − 1) Γ

3 2

+

1 2n

1

1

Proof We first make the change of variable x2 = u2n with dx2 = n1 u2n 

∞ ∞

In = 0

0

−1

(69)

du2 , to get

 1 1 −1   u n du2 dx1 , exp − x12 + x1 u2 + u22 n 2

(70)

then, using Mathematica to integrate (70) leads to the first expression in (69). Next, we apply the change of variable x1 = x2n u to the inner integral in (69), and then we use Mathematica again to obtain  In = 0

∞ ∞ 0

   exp − (ux2n )2 + (ux2n )x2n + x22n x2n dx2 du 

∞

= 0

  − 1 − 1   1 + u + u2 2 2n Γ 12 3 + n1 1+n

(71) du.

Using Mathematica to integrate (71) yields the second expression in (69).

 

Multidimensional Integral Identities

177

1 Remark 2 Setting x = 2n in (69) leads to a new property of the gamma function as follows:   1 −x Γ (x) Γ 2   √   √ √ 3 + i 1 1 3 1 = π B −3; x, − x − 2ieiπ x 4−x π B ; − x, − x , 2 6 2 2 (72) where B(z; a, b) is the incomplete beta function, i.e.,

B(z; a, b) =

za 2 F1 (a, 1 − b; a + 1; z) a ∞  (1 − b)n n z . = za n!(a + 1)

(73)

n=0

Note that if x = 14 , then   √   2  √ √ 3+i 3 1 1 1 1 1 + (1 − i) π B ; , Γ = π B −3; , . 4 4 4 6 4 4

(74)

Theorem 5 

p

In =

∞ ∞

0

=

0

p    dx1 dx2 exp − x12 + x1 x2n + x22n

1 n+1 2n+1 Γ ( n+1 2np )2 F1 ( 2n , 2n ; 2n ; −3)

2

n−1 n

×p

(75) ; n > 0; p > 0.

Proof The change of variables x1 = x2n u and integration over x2 leads to p





∞

In = 0

Γ

1+n 2np



2np(1 + u + u2 )

1+n 2n

du.

(76)

This allows us to form the ratio   p Γ 1+n 2np In .  = In1 pΓ 1+n 2n

(77)

178

A. A. Ruffa and B. Toni p

Here In1 is In when p = 1, i.e., (69). Since In1 =

1 n+1 2n+1 Γ ( n+1 2n )2 F1 ( 2n , 2n ; 2n ; −3)

2

n−1 n

(78)

,

 

The claim (75) follows. Theorem 6  ∞ q In = 0

∞

x12 + x1 x2n + x22n

0

Γ (q +

=

q

   exp − x12 + x1 x2n + x22n dx1 dx2

n+1 1 n+1 2n+1 2n )2 F1 ( 2n , 2n ; 2n ; −3)

2

n−1 n

n+1 ; n > 0; q > − . 2n

(79)

Proof The change of variables x1 = x2n u and integration over x2 leads to q



In = 0

∞

 Γ q+

1+n 2n



2n(1 + u + u2 )

1+n 2n

(80)

du.

This allows us to form the ratio   q Γ q + 1+n 2n In  .  = In0 Γ 1+n 2n

(81)

q

Here In0 is In when q = 0, i.e., (69). Since In0 =

1 n+1 2n+1 Γ ( n+1 2n )2 F1 ( 2n , 2n ; 2n ; −3)

2

n−1 n

(82)

,

 

The claim (79) follows. 1

1

1

1

Now consider the function f (x) = f (x1 , x2 ) = (x1z + x2z ) + (x1z + x2z )−1 . With the same substitution approach, we first compute for several values for n the integral  In = 0

∞ ∞ 0

    −1  dx1 dx2 . exp − x1n + x2n − x1n + x2n

(83)

1. For n = 3 we have 1

2 · 2 3 K 2 (2)Γ ( 13 )Γ ( 76 ) 3 I3 = . √ 3 π

(84)

Multidimensional Integral Identities

2. For n = 4 we have I4 = 3. For n = 5 we get

179

Γ ( 45 )2 . e2

3

I5 =

2 · 25

√

π K 2 (2)Γ ( 65 ) 5

7 5Γ ( 10 )

.

(85)

;

(86)

;

(87)

Similarly, 4. 2

I6 =

2 · 23

√

π K 1 (2)Γ ( 76 ) 3

6Γ ( 23 )

5. 5

I7 =

2 · 27

√

π K 2 (2)Γ ( 87 ) 7

9 7Γ ( 14 )

6. 9

I11 =

2 · 2 11

√

π K 2 (2)Γ ( 12 11 ) 11

11Γ ( 13 22 )

.

(88)

Here Kn (z) represents the modified Bessel function of the second kind. We generalize the above pattern and prove Theorem 7 

∞ ∞ 0

0

-    1  . 1 1 1 −1 z z z z exp − x1 + x2 − x1 + x2 dx1 dx2 (89) 2z2 Γ 2 (z) K2z (2) . = Γ (2z)

The proof is based on the following proposition. Proposition 1 

∞ 0

e−x

n −x −n

xdx =

2 K 2 (2). n n

Proof The Wolfram functions package (i.e., functions.wolfram.com) defines

(90)

180

A. A. Ruffa and B. Toni

1 Kν (z) = 2



∞

z

e− 2 (e +e t

−t )

&

' eνt + e−νt dt,

(91)

 2t  2t e n + e− n dt.

(92)

0

from which we derive 2 1 K 2 (2) = n n n



∞

e−(e +e t

−t )

0

Therefore the changes of variables x n = et ,

dx =

1 t e n dtandx n = e−t , n

1 t dx = − e− n dt n

respectively lead to 1 n

I1 =

1 I2 = n

 

∞

e−(e +e t

−t )

 e2t/n dt =

0

∞

e−(x

n +x −n )

xdx;

1

∞

e

−(et +e−t ) −2t/n

e

 dt =

0

(93)

1

e

−(x n +x −n )

xdx.

0

Consequently we obtain  I1 + I2 =

∞

e−(x

n +x −n )

xdx =

0

2 K 2 (2). n n

(94)  

We now proceed to the proof of the theorem. Proof The usual change of variables, i.e., x1 = x2 u1 ,  I3 = 0

 =

0

∞ ∞ 0 ∞ ∞ 0

dx1 = x2 du1 leads to

    −1  dx1 dx2 exp − x1n + x2n − x1n + x2n     exp − un1 + 1 x2n − (un1 + 1)−1 x2−n x2 dx2 du1 .

Computing I3 after another transformation, i.e., 1

v = (1 + un1 ) n x2 ,

1

dv = (1 + un1 ) n dx2 ,

(95)

Multidimensional Integral Identities

181

we obtain 2 I3 = K 2 (2) n n



∞

du1 2

(1 + un1 ) n

0

1 2 √ Γ (1 + n ) 2 = K 2 (2)21− n π n n Γ ( 12 + n1 )

=

(96)

Γ 2 ( n1 ) 2 . K 2 (2) n2 n Γ ( n2 )  

Hence the claim. Similar procedures prove the following corollaries: 1

1

1

1

1

1

1. f (x) = f (x1 , x2 , x3 ) = (x1z + x2z + x3z ) + (x1z + x2z + x3z )−1 . Corollary 4  0

=

∞ ∞ ∞ 0

0

-    1  . 1 1 1 1 1 −1 exp − x1z + x2z + x3z − x1z + x2z + x3z dx1 dx2 dx3

2z3 Γ 3 (z) K3z (2) . Γ (3z) (97) 1 z

1 z

1 z

1 z

1 z

1 z

1 z

1 z

2. f (x) = f (x1 , x2 , x3 , x4 ) = (x1 + x2 + x3 + x4 ) + (x1 + x2 + x3 + x4 )−1 . Corollary 5  0

∞ ∞ ∞ ∞ 0

0

0

   1 1 1 1 exp − x1z + x2z + x3z + x4z

   1 1 1 1 −1 z z z z dx1 dx2 dx3 dx4 − x1 + x2 + x3 + x4 =

2z4 Γ 4 (z) K4z (2) . Γ (4z) 1

1

1

1

3. f (x) = f (x1 , x2 ) = (x1z + x2z )p + (x1z + x2z )−p .

(98)

182

A. A. Ruffa and B. Toni

Corollary 6  0

=

∞ ∞ 0

-    .  1 1 1 p 1 −p z z z z exp − x1 + x2 − x1 + x2 dx1 dx2 (99)

2z2 Γ 2 (z) K 2z (2) p

.

pΓ (2z)

3 Integrals Involving the Complementary Error Function We were also able to establish new integral identities involving the complementary error function, denoted and defined by 2 erfc(x) = 1 − erf(x) = √ π



∞

e−t dt, 2

(100)

x

where erf(x) is the error function [8]. We prove Theorem 8 With the simplest power substitution, we establish 

∞ ∞ 0

  Γ ( n1 ) erfc x1n + x2n dx1 dx2 = . 1 n · 4n

0

(101)

Proof Consider the integral 

∞ ∞

In = 0

0

  erfc x1n + x2n dx1 dx2 .

Making the substitution x1 = x2 u1 and dx1 = x2 du1 (and setting z = integrating by parts leads to 4z Γ (z) = z



∞ ∞

0

0

  1 1 z z erfc x1 + x2 dx1 dx2 .

(102) 1 n)

and

(103)  

Similarly, we obtain the following identities: 1. Γ 3 (z) 3 8z  = · 2 2 z Γ 3z 2



∞ ∞ ∞ 0

0

0

  1 1 1 erfc x1z + x2z + x3z dx1 dx2 dx3 ;

(104)

Multidimensional Integral Identities

183

2. 16z Γ 4 (z) = 2· 3 Γ (2z) z



  1 1 1 1 erfc x1z + x2z + x3z + x4z dx1 dx2 dx3 dx4 ;

∞ ∞ ∞ ∞ 0

0

0

0

(105)

3. 2z4 Γ 3 (z) = √ π Γ (2 + 2z)



∞ ∞

0

1

1



x1z + x2z

0

 1  1 erfc x1z + x2z dx1 dx2 ;

(106)

4.   5 Γ 3 (z) Γ 3z   ∞ ∞ ∞ 1 3z 1 1 2 3 z z z x1 + x2 + x3 × · √ = 2 π Γ (2 + 3z) 0 0 0  1  1 1 erfc x1z + x2z + x3z dx1 dx2 dx3 ;

(107)

5. ∞ ∞ ∞ ∞



4z6 Γ 4 (z) Γ (2z) = 2× √ π Γ (2 + 4z)

0

0

0

1 z

1 z

1 z

1 z

x1 + x2 + x3 + x4

0

 ×

 1  1 1 1 z z z z erfc x1 + x2 + x3 + x4 dx1 dx2 dx3 dx4 ; (108)

6. 



∞ ∞

erfc 0

0

1 z

1 z

p 

x1 + x2

  zΓ (z) Γ 12 + pz   . dx1 dx2 = 4z Γ 12 + z

(109)

We next study the following integral through successive transformations. That is, setting x1 = x2 u, dx1 = x2 du leads to 

∞ ∞

I1 = 0 ∞  ∞

 = 0

0

0

 1  1 erfc2 x1z + x2z dx1 dx2

   1 1 z z erfc (ux2 ) + x2 x2 dx2 du 2



(110)

∞

=

I2 du. 0 1

Then we transform the inner integral I2 through v = x2z , get, using again Mathematica,

dv =

1

1 z −1 dx2 , z x2

to

184

A. A. Ruffa and B. Toni

I2 =

 ∞ 0

  1 erfc2 u z v + v zv 2z−1 dv

1    1 (1 + u z )−2z √ +z π (1 + 2z)Γ = 2π(1 + 2z) 2       3 3 1 1 , 1 + z; ; −1 − 2 F1 + z, 1 + z; + z; −1 . − 2zΓ (z) (1 + 2z)2 F1 2 2 2 2

(111) Once again using Mathematica on 

∞

I1 =

(112)

I2 du 0

leads to the desired identity proving Theorem 9  1  1 z2 Γ (z)2 erfc2 x1z + x2z dx1 dx2 = × π Γ (2 + 2z) 0 0         √ 1 3 1 + z + iz(1 + 2z)Γ (z) B −1; , −z − i −2z B −1; + z, −z . 2 πΓ 2 2 2  ∞ ∞

(113) Similarly, we derive the following identities: Corollary 7  0

∞ ∞ ∞ 0

0

 1  1 1 z z z erfc x1 + x2 + x3 dx1 dx2 dx3 2

   √ z3 Γ (z)3 3 3z 2 πΓ + = π Γ (2 + 3z) 2 2        3z 1 3z 3z 3 1 3z B −1; , − − i −3z B −1; + , − ; + iz(1 + 3z)Γ 2 2 2 2 2 2 2 (114) Corollary 8 

∞ ∞ ∞ ∞

 2

erfc 0

0

0

0

1 z

1 z

1 z

1 z

x1 + x2 + x3 + x4

 dx1 dx2 dx3 dx4

   √ z4 Γ (z)4 3 2 πΓ + 2z = π Γ (2 + 4z) 2

Multidimensional Integral Identities

185

      1 1 −4z . + 2iz(1 + 4z)Γ (2z) B −1; , −2z − i B −1; + 2z, −2z 2 2 (115)

4 Integrals Involving the Logarithm The same substitution process described in the previous sections also leads to the following results. The proofs are straightforward from the above. 1. Proposition 2 

∞ ∞

0

0

-  1 . 1 1 a z z exp − x1 + x2 ln (x2 ) dx1 dx2 = (116)

az3 Γ (2az) 2 Γ (z) [Ψ (z) − Ψ (2z) + aΨ (2az)] . Γ (2z) 2. Proposition 3 

∞ ∞ ∞

0

0

0

-  1 . 1 1 1 a z z z exp − x1 + x2 + x3 ln (x3 ) dx1 dx2 dx3 = (117) az4 Γ (3az) 3 Γ (z) [Ψ (z) − Ψ (3z) + aΨ (3az)] . Γ (3z)

3. Proposition 4 

∞ ∞ ∞ ∞ 0

0

0

0

-  1 . 1 1 1 1 a z z z z exp − x1 + x2 + x3 + x4 ln (x4 ) dx1 dx2 dx3 dx4 = az5 Γ (4az) 4 Γ (z) [Ψ (z) − Ψ (4z) + aΨ (4az)] . Γ (4z) (118)

4. Proposition 5  0

∞ ∞ 0

-  1 . 1  1 1 1 a z z x1z + x2z ln (x2 ) dx1 dx2 = exp − x1 + x2

az3 Γ (a + 2az) 2 Γ (z) [Ψ (z) − Ψ (2z) + aΨ (a + 2az)] . Γ (2z)

(119)

186

A. A. Ruffa and B. Toni

5. Proposition 6 

∞ ∞ ∞ 0

0

0

-  1 . 1  1 1 1 1 1 a z z z x1z + x2z + x3z ln (x2 ) dx1 dx2 dx3 = exp − x1 + x2 + x3 az4 Γ (a + 3az) 3 Γ (z) [Ψ (z) − Ψ (3z) + aΨ (a + 3az)] . Γ (3z) (120)

6. Proposition 7 

∞ ∞ ∞ ∞ 0

0

=

0

0

-  1 . 1 1 1 1 a z z z z exp − x1 + x2 + x3 + x4 ×

  1 1 1 1 x1z + x2z + x3z + x4z ln (x2 ) dx1 dx2 dx3 dx4

az5 Γ (a + 4az) 4 Γ (z) [Ψ (z) − Ψ (4z) + aΨ (a + 4az)] . Γ (4z)

(121)

7. Proposition 8 ∞ ∞

 0

0

-  1 . 1 1 a z z exp − x1 + x2 ln (x1 ) ln (x2 ) dx1 dx2 =

  az4 Γ (2az) 2 Γ (z) (Ψ (z) − Ψ (2z) + aΨ (2az))2 − Ψ1 (2z) + a 2 Ψ1 (2az) . Γ (2z) (122) 8. Proposition 9 -  1 . 1 1 a az4 Γ (2az) 2 z z Γ (z) × exp − x1 + x2 (ln (x2 ))2 dx1 dx2 = Γ (2z) 0 0   (123) (Ψ (z) − Ψ (2z) + aΨ (2az))2 + Ψ1 (z) − Ψ1 (2z) + a 2 Ψ1 (2az) . 

∞ ∞

5 Identities Resulting from Power Substitution Variants We now explore variants of the power substitution; in particular, instead of xi = ui xnαi , we first translate the power substitution. In two dimensions, this leads to some very interesting results, e.g.,

Multidimensional Integral Identities

187

Theorem 10      ∞ 1 1 1 1 −ua a , 1 + + , 1 + u du = Γ 1 + ; a > 0, b > 0, I= e U b a b a 0 (124) where U is the confluent hypergeometric function of the second kind [8] defined by ! ∞ −xt α−1 1 c−α−1 dt α > the integral representation U(α, c, x) := Γ (α) e t (1 + t) 0 0, x > 0. We recall that, in the additional properties of the gamma  as stated! previously ∞ −ua 1 function, Γ 1 + a = 0 e du. In other words, weighting this integral with the confluent hypergeometric function having specific arguments returns the same expression of gamma, a surprising and interesting formula, noting also that b > 0 is a free parameter. Proof Using Mathematica, it can be shown that     1 1 I1 = e 1+ Γ 1+ ; a > 0, b > 0. a b 0 0 (125) The integral (125) can also be evaluated via a variant of the power substitution, 1 1 i.e., x1 = u1 (1 + x2b ) a and dx1 = du1 (1 + x2b ) a , so that 

∞ ∞

−x1a −x2b −1



∞ ∞

I1 = 0

0

1 dx1 dx2 = Γ e

1

e−u1 (1+x2 )−(1+x2 ) (1 + x2b ) a dx2 du1 . a

b

b

(126)

To evaluate (126) via Mathematica requires a second substitution, i.e., u2 = x2b and du2 = bx2b−1 dx2 , which leads to 1 I1 = b

∞ ∞

 0

0

1

1

e−u1 (1+u2 )−(1+u2 ) (1 + u2 ) a u2b a

−1

du2 du1 .

(127)

Mathematica then returns the following for (127):     1 1 1 1 U , 1 + + , 1 + ua1 du1 ; a > 0, b > 0. 1+ b b a b 0 (128) Equating (128) to (125) then leads to the claimed result.   

I1 =

∞

e−1−u1 Γ a

Remark 3 Consider the above integral representation of the confluent hypergeometric function [8], i.e., U(α, c, x) =

1 Γ (α)



∞ 0

e−xt t α−1 (1 + t)c−α−1 dt; α > 0, x > 0.

(129)

188

A. A. Ruffa and B. Toni

Substituting (129) into (128) with α = b1 , c = 1 + to the following: 1 I1 = b

∞ ∞

 0

1 a

+ b1 , and x = 1 + ua1 leads

1

1

e−1−u1 e−(1+u1 )t t b −1 (1 + t) a dtdu1 , a

a

(130)

0

which is identical to (127). We next recover some of the particular properties of the gamma function recalled above along with some additional variations in the corollaries below. Also recall the upper incomplete gamma given here by its definition and relation to the confluent hypergeometric function as: 

∞

Γ (a, x) :=

t a−1 e−t dt = e−x U(1 − a, 1 − a, x),

(131)

x

whereas the lower incomplete gamma function is given by  γ (a, x) :=

x

t a−1 e−t dt = a −1 x a 1 F1 (a, 1 + a; −x),

(132)

0

leading to Γ (a) = Γ (a, x) + γ (a, x). Corollary 9 When b → ∞, U  I→

∞



(133)

 + b1 , 1 + ua → 1, so that

1 1 b,1 + a

e−u du = Γ a

0

  1 ; a > 0. 1+ a

(134)

Corollary 10 When b → 1,  U

1 1 1 , 1 + + , 1 + ua b a b



1

→ e1+u (1 + ua )−1− a Γ a

  1 1 + , 1 + ua , a (135)

which leads to 

∞ 0

  Γ 1 + a1 , 1 + ua 1

(1 + ua )1+ a

du =

1 Γ e

  1 1+ ; a > 0. a

(136)

Multidimensional Integral Identities

Corollary 11 When b =

189

a 1+a ,

 1 1 1 a e ,1 + + ,1 + u U b a b   1 1 − 1 ua 1 + ua a − 12 − a1 2 2 =√ e , (1 + u ) K1+1 2 a 2 π 

−ua

(137)

which leads to 



1 a

e− 2 u

∞

1

1

(1 + ua ) 2 + a

0

K1+1 2

a

1 + ua 2

#

 du =

π Γ e

  1 1+ ; a > 0. a

Theorem 11      ∞ 1 1 Γ , 1 ; a > 0, b > 0, E1− 1 − 1 (1 + ua )du = Γ 1 + a b a b 0

(138)

(139)

where En (x) is the exponential integral function, i.e., 

∞

En (x) = 1

e−xt dt. tn

(140)

Proof Using Mathematica, it can be shown that  I2 =

∞ ∞

e 0

1 dx1 dx2 = Γ b

−x1a −(1+x2 )b

0

    1 1 1+ Γ ,1 ; a b

(141)

a > 0, b > 0. b

We can also evaluate the integral in (141) via the substitution x1 = u1 (1 + x2 ) a b and dx1 = du1 (1 + x2 ) a , which leads to 1 I2 = b

 0

∞

E1− 1 − 1 (1 + ua1 )du1 . a

b

Equating (142) to (141) leads to the claimed result.

(142)  

Theorem 12 

∞ 0

    1 1 1 5 erfc(1 + u2 ) π 5 4 du = F Γ − , ; , ; −1 √ 2 2 2 4 4 4 2 4 1 + u2 π     3 3 3 7 3 2 F , ; , ; −1 . + √ Γ 2 2 4 4 4 2 4 3 π

(143)

190

A. A. Ruffa and B. Toni

Proof Using Mathematica, it can be shown that     3 1 1 1 5 5 π2 −Γ , ; , ; −1 2 F2 8 4 4 4 2 4 0 0     3 3 3 3 7 1 , ; , ; −1 . + Γ 2 F2 6 4 4 4 2 4 (144) We can also evaluate (144) via the substitution x1 = u1 (1 + x2 ) and dx1 = du1 (1 + x2 ), which leads to 

I3 =

∞ ∞

e−(x1 +(1+x2 ) 2

2 )2

dx1 dx2 =

√  ∞ erfc(1 + u21 ) π du1 . I3 = 4 0 1 + u21

(145)  

Equating (145) to (144) leads to the claimed result.

6 Modified Forms of Multivariate Gaussian Distributions An interesting application of our method involves generalized and weighted probability distributions [5, 6, 9, 11, 12], commonly encountered in statistical physics, econometrics, engineering, medicine, and psychology. These distributions are milestones for effective modeling of statistical data interpretation and prediction when standard distributions become inadequate, as they modulate the probabilities of the events as observed and transcribed.

6.1 Generalized Gaussian Functional Integrals In statistical physics, probability theory as in the computation of the path integrals of stochastic processes or in the path integral approach to quantum field theory, one has to often evaluate Gaussian functional integrals given by 

∞

−∞

e

−x T Ax

 dx :=

∞ −∞

 ···

∞ −∞

e−x

T Ax

dx1 · · · dxn .

(146)

In quantum field theory, the exponent may have both quadratic and linear terms. As explained in the introductory sections, for a symmetric nonsingular matrix A satisfying x T Ax > 0, the integral identity has been derived as 

∞ −∞

 ···

∞ −∞

n

e−x

T Ax

dx1 · · · dxn =

where |A| denotes the determinant of the matrix A.

π2

1

|A| 2

,

(147)

Multidimensional Integral Identities

191

We make use of the power substitution procedure described previously to propose a generalized Gaussian integral and establish its corresponding identity. That is, we consider: Definition 1 (Generalized Gaussian Integral) The generalized Gaussian functional integral is defined by 

∞ −∞

 ···

∞ −∞

e−(x

T Ax)b

dx1 · · · dxn ,

(148)

with A = [Aij ]1≤i,j ≤n a nonsingular matrix, symmetric, and real positive definite to ensure convergence. We start with n = 3, and from the previous results we have Lemma 1  3 1 1 Γ ; 8 2 0 0 0  3   1 3  ∞ ∞ ∞ Γ Γ 2b 2 −(x12 +x22 +x32 )b   I2 = e dx1 dx2 dx3 = . 0 0 0 8bΓ 32 

∞ ∞ ∞

I1 =

e−(x1 +x2 +x3 ) dx1 dx2 dx3 = 2

2

2

(149)

Consequently we derive the ratio   3 Γ 2b I2  ; = I1 bΓ 32

b > 0.

(150)

In the general case we prove Theorem 13 Iˆb =



∞

−∞

 ···

n

∞

−∞

e

−(x T Ax)b

dx1 · · · dxn =

π2

1

|A| 2

·

Γ



bΓ



n 2bn  ; 2

b > 0.

(151)

Proof To perform the required substitutions, (151) must first be broken down into 2n integrals so that all of the integration limits are 0 to ∞ for each. (Integrals having integration limits from −∞ to 0 can be rewritten to have integration limits from 0 to ∞ via substitution.)

192

A. A. Ruffa and B. Toni (k)

We call these integrals Ib , where 1 ≤ k ≤ 2n , so that n

Iˆb =

2 

(k)

(152)

Ib .

k=1

Consider one such integral, i.e., (1)

Ib



∞

=



∞

···

0

e−(x

T Ax)b

dx1 · · · dxn ,

(153)

0

& 'T where x = x1 x2 x3 · · · xn and A = [Ai,j ]n×n . The following substitutions are made for 1 ≤ i ≤ n − 1: xi = xn ui ; dxi = xn dui .

(154)

This leads to (1)

Ib



∞

= 0



∞

···

e−(v

T Av)b x 2b n

0

xnn−1 dxn du1 · · · dun−1 ,

(155)

'T & where v = u1 u2 u3 · · · un−1 1 . Evaluating just the integral over xn in (155) leads to  ∞  ∞  n 1 T n (1) du1 du2 · · · dun−1 . Ib = ··· (156) (v Av)− 2 Γ 1 + n 2b 0 0 From (156) we can derive the ratio Ib(1) (1)

I1

=

Γ



bΓ



n 2bn  ; 2

b > 0,

(157)

(1)

where I1 is given by (156) by setting b = 1. We can also show that the ratio (157) is the same for all k, i.e., (k)

Ib

(k)

I1 Noting that

=

Γ



bΓ



n 2bn  ; 2

b > 0.

(158)

Multidimensional Integral Identities



n

Iˆ1 =

2 

(k) I1

=



∞ −∞

k=1

193

···

∞ −∞

 n 2 T n (v Av)− 2 Γ 1 + du1 du2 · · · dun−1 n 2

 =

∞

−∞

 ···

∞

−∞

n

e−(x

T Ax)

dx1 · · · dxn =

π2

1

|A| 2

, (159)

it follows that n

Iˆb =

2 

(k) Ib

=

k=1



Γ

bΓ

 2n n  (k) 2b n I1 2 k=1

n

=

π2

1

|A| 2

·

Γ



bΓ



n 2bn  ; 2

b > 0.

(160)  

Remark 4 Our integral identity (151), obtained without the use of polar/cylindrical change of coordinates, provides a compact and beautiful formula for the generalized Gaussian functional integral, generalizing indeed the usual double integral appearing in joint probability density theory, in particular in the computation of the second moment. See [7].

6.2 Weighted Gaussian Integrals We now consider the following general weighted Gaussian integral: 



∞

···

−∞

∞

−∞

(x T Ax)q e−x

T Ax

dx1 · · · dxn .

(161)

A common practice in the evaluation of such integral relies on approximation methods [3]. Here again our approach provides a compact mathematical formula. Indeed we prove Theorem 14 Iˆq =



∞ −∞

 ···



∞ −∞

(x T Ax)q e−x

T Ax

dx1 · · · dxn =

π

n 2

|A|

1 2

·

Γ

n+2q 2   Γ n2

 n ;q > − . 2 (162)

Proof We follow the same approach, i.e., breaking down (162) into 2n integrals, and we focus on the integral  Iq(1) =

0

∞

 ··· 0

∞

(x T Ax)q e−x

T Ax

dx1 · · · dxn .

(163)

194

A. A. Ruffa and B. Toni

Using the substitutions in (154) and integrating over xn leads to  Iq(1) =

∞



∞

···

0

0

n 1 T (v Av)− 2 Γ 2



n + 2q 2



n du1 du2 · · · dun−1 ; q > − . 2 (164)

This allows us to form the ratio 

Γ

(1)

Iq

(1) I0

=

n+2q 2   Γ n2

 n q>− , 2

;

(165)

(1)

where I0 is given by (164) by setting q = 0. We can again show that 

Iq(k) I0(k)

=

Γ



n+2q 2 n Γ 2

;

n q>− . 2

(166)

Since 

n

Iˆ0 =

2 

(k)

I0

=

k=1



∞ −∞

··· 

=

∞ −∞

∞

−∞

n

(v T Av)− 2 Γ 

···

n 2

du1 du2 · · · dun−1 (167)

n

∞ −∞

e

−(x T Ax)

π2

dx1 · · · dxn =

1

|A| 2

,

It follows then that Iˆq =

2n 



Iq(k) =

Γ

k=1

n+2q 2 n Γ 2



2n 



(k)

I0

=

k=1

π

n 2 1

|A| 2

·

Γ

n+2q 2 n Γ 2

 .

(168)  

Using the same methods for a more general result, it can further be proved that Theorem 15 

∞

−∞

 ···



∞

−∞

(x T Ax)q e−(x

T Ax)p

dx1 · · · dxn =

π

n 2

|A|

1 2

·



n+2q 2p   pΓ n2

Γ

n ; p > 0; q > − . 2 (169)

As a further application of our approach, we consider a simplified weighted Gaussian integral of interest identified by Lu and Darmofal [3], and we prove

Multidimensional Integral Identities

195

Corollary 12 

∞



∞

···

0

√

0

1

e

1 + xT x

−x T x

  n π2 n 1 n , + ,1 . dx1 · · · dxn = n · U 2 2 2 2

(170)

Proof Consider the integral 



∞

∞

···

0

√

0

1 1 + xT x

e−x

Tx

dx1 · · · dxn .

(171)

We can evaluate (171) by starting with the following (more general) integral: 



∞

Iq =

∞

···

0

(1 + x T x)q e−x

Tx

dx1 · · · dxn .

(172)

0

Again using the substitutions in (154) and integrating over xn leads to  Iq =

∞



∞

···

0

0

1 T −n (v v) 2 (R1 + R2 ) du1 du2 · · · dun−1 , 2

(173)

where R1 =

Γ

n 2

   n Γ − n2 − q n ; 1 + + q; 1 , 1 F1 Γ (−q) 2 2

(174)

and R2 = Γ

n 2

+q

 1 F1

  n −q; 1 − − q; 1 . 2

(175)

Here 1 F1 is the Kummer confluent hypergeometric function [8]. We can again establish an identity for a ratio of integrals, i.e.,  n Iq n = U , 1 + + q, 1 , I0 2 2 so that when q = − 12 , the claim follows immediately.

(176)  

Corollary 13     n n n 2Γ   π 2p Ψ 2p T p Iˆp = ··· ln x T Ax e−(x Ax) dx1 · · · dxn = ; p > 0.   1 −∞ −∞ p2 · Γ n2 · |A| 2 (177) 

∞



∞

196

A. A. Ruffa and B. Toni

Proof We again break down (177) into 2n integrals, and focus on  Ip(1) =

∞



∞

···

0

  T p ln x T Ax e−(x Ax) dx1 · · · dxn .

(178)

0

We use the substitutions in (154) and again evaluate only the integral over xn to obtain     n n  ∞  ∞ Ψ 2p Γ 2p n (1) T −2 ··· (v Av) du1 du2 · · · dun−1 ; p > 0, Ip = 2p2 0 0 (179) and after applying the same procedure to the remaining integrals, we get 

n

Iˆp =

2 

Ip(k) =

···

−∞

k=1





∞

∞

Γ n

−∞

(v T Av)− 2

n 2p



 Ψ

p2

n 2p

 du1 du2 · · · dun−1 ; p > 0. (180)

Next, it follows from (159) that 



∞

···

−∞

n

∞

−∞

T

(v Av)

− n2

du1 du2 · · · dun−1

π2 =   . 1 n Γ 2 · |A| 2

Substituting (181) into (180) leads to the claimed result.

(181)  

Corollary 14 Iˆq =



∞

−∞

 ···

∞



−∞

x T Ax n

=

q

  T p ln x T Ax e−(x Ax) dx1 · · · dxn

  Ψ n+2q 2p n ; p > 0; q > − . n 1 2 2 p · Γ 2 · |A| 2 

π2Γ

n+2q 2p



(182)

Proof We break down (182) into 2n integrals as before, and starting with the integral  Iq(1) =

∞ 0



∞

···

x T Ax

q

  T p ln x T Ax e−(x Ax) dx1 · · · dxn ,

(183)

0

we again use the substitutions in (154) and again evaluate only the integral over xn to obtain     n+2q n+2q  ∞  ∞ Ψ Γ n 2p 2p ··· (v T Av)− 2 du1 du2 · · · dun−1 ; p > 0, Iq(1) = 2 2p 0 0 (184)

Multidimensional Integral Identities

197

and, by extension, 

n

Iˆq =

2 

Iq(k) =

···

−∞

k=1





∞

∞ −∞

(v T Av)

− n2

Γ

n+2q 2p



 Ψ

n+2q 2p

p2

 du1 du2 · · · dun−1 ; p > 0. (185)  

Substituting (181) into (185) leads to the claimed result. Corollary 15 Iˆp =



∞

−∞

 ···

∞

−∞

  T p erfc (x T Ax)p e−(x Ax) dx1 · · · dxn 



   1 n n 1 n 1 ˜ + , ; ,1 + ; =   8p2 F2 1 3+ n 2 4p 4p 2 4p 4 2 2p p2 |A| 2 Γ n2 π

n+1 2

Γ

− n2 F˜2

n 2p



1 n n 3 3 n 1 + ,1 + ; , + ; 2 4p 4p 2 2 4p 4

 ; p > 0,

(186)

where 2 F˜2 is the regularized generalized hypergeometric function [8]. Proof Again, we break down (186) into 2n integrals and begin with  Ip(1) =

∞



∞

···

0

  T p erfc (x T Ax)p e−(x Ax) dx1 · · · dxn .

(187)

0

After the substitutions in (154), and integrating over xn , we get Ip(1) =

1 2



∞

 ···

0

∞

n

(v T Av)− 2 Rdu1 du2 · · · dun−1 ,

(188)

0

and, by extension, 

n

Iˆp =

2  k=1

Ip(k)

=

∞ −∞

 ···

∞ −∞

n

(v T Av)− 2 Rdu1 du2 · · · dun−1 ,

(189)

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where 

1

R=

π2Γ n 3+ 2p

2



n 2p



p2



 1 n n 1 n 1 + , ; ,1 + ; 2 4p 4p 2 4p 4   1 n n 3 3 n 1 ˜ + ,1 + ; , + ; . − n2 F2 2 4p 4p 2 2 4p 4

8p2 F˜2

Substituting (181) into (189) then leads to the claimed result.

(190)  

Corollary 16 Iˆp =





∞

···

−∞

∞

−∞

  (x T Ax)q erfc (x T Ax)p dx1 · · · dxn

 q + 2p n  n  ; p > 0; q > − . 1 2 (n + 2q) · |A| 2 · Γ 2

2π

=

n−1 2

·Γ



1 2

+

n 4p

(191)

Proof We break down (191) into 2n integrals as before and consider  Ip(1) =

∞



∞

···

0

  (x T Ax)q erfc (x T Ax)p dx1 · · · dxn .

(192)

0

After the substitutions in (154), and evaluating over xn , we get  Ip(1) =

∞

∞

···

0

 q + 2p du1 du2 · · · dun−1 , √ (n + 2q) π





Γ n

(v T Av)− 2

0

1 2

+

n 4p

(193)

and, by extension,  q + 2p Iˆp = Ip(k) = ··· (v T Av)− 2 du1 du2 · · · dun−1 . √ (n + 2q) π −∞ −∞ k=1 (194) Substituting (181) into (194) then leads to the claimed result.   

n

2 

∞





∞

2Γ n

1 2

+

n 4p

Corollary 17 Iˆp =



∞

−∞

 ···

n

∞

−∞

e

−(x T Ax)p −(x T Ax)−p

2π 2 · K 2pn (2) dx1 · · · dxn =   ; p > 0. 1 p · |A| 2 · Γ n2 (195)

Multidimensional Integral Identities

199

Proof We break down (195) into 2n integrals as before and consider  Ip(1) =

∞



∞

···

0

e−(x

T Ax)p −(x T Ax)−p

dx1 · · · dxn .

(196)

0

After making the substitutions in (154), we get 

∞

=

Ip(1)



∞

···

0

e−(v

T Av)p x 2p −(v T Av)−p x −2p n n

0

xnn−1 dxn du1 du2 · · · dun−1 . (197)

Next, making use of (90), we can show that 1 p

Ip(1) =





∞

∞

···

0

0

n

(v T Av)− 2 K 2pn (2)du1 du2 · · · dun−1 .

(198)

Similarly, we can show that 

n

Iˆp =

2 

Ip(k) =

k=1

2 p



∞

···

−∞

∞

−∞

n

(v T Av)− 2 K 2pn (2)du1 du2 · · · dun−1 .

(199)  

Substituting (181) into (199) then leads to the claimed result. Corollary 18 Iˆp =



∞ −∞

 ···

∞ −∞

(x T Ax)q e−(x

T Ax)p −(x T Ax)−p

dx1 · · · dxn

n

2π 2 · K n+2q (2) 2p =   ; p > 0. 1 p · |A| 2 · Γ n2

(200)

Proof We break down (200) into 2n integrals as before and consider  Ip(1) =

∞

 ···

0

∞

(x T Ax)q e−(x

T Ax)p −(x T Ax)−p

dx1 · · · dxn .

(201)

0

After making the substitutions in (154), we get  Ip(1)

∞

=



∞

···

0

(v T Av)q e−(v

T Av)p x 2p −(v T Av)−p x −2p n n

n+2q−1

xn

dxn du1 · · · dun−1 .

0

(202)

Again making use of (90), we can show that Ip(1) =

1 p



∞ 0



∞

··· 0

n

(v T Av)− 2 K n+2q (2)du1 du2 · · · dun−1 . 2p

(203)

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and n

Iˆp =

2 

Ip(k)

k=1

2 = p





∞

−∞

···

∞

−∞

n

(v T Av)− 2 K n+2q (2)du1 du2 · · · dun−1 .

(204)

2p

Substituting (181) into (204) then leads to the claimed result.

 

Corollary 19 

∞ −∞

 ···

∞

−∞

# =

(x T x + y T y)q e−αx

T x−y T y

dx1 · · · dxn dy1 · · · dym

   m n π n+m Γ n2 + q  n  · 2 F1 , −q; 1 − − q; α ; · n+2q 2 2 α Γ 2

(205)

α > 0; q = {0, 1, 2, . . .}. Proof We break down (205) into 2n integrals as before and perform the following substitutions for 1 ≤ i ≤ n − 1 and for 1 ≤ j ≤ m − 1: xi = xn ui ; dxi = xn dui ; yj = ym uˆ j ; dyj = ym d uˆ j .

(206)

Considering one such integral, these substitutions lead to  Iq =

∞

 ···

0

0

∞

2 q −αv (v T v · xn2 + w T w · ym ) e

T v·x 2 −w T w·y 2 n m

(207)

m−1 ×xnn−1 ym dxn dym du1 · · · dun−1 d uˆ 1 · · · d uˆ m−1 ,

'T & where w = uˆ 1 uˆ 2 uˆ 3 · · · uˆ m−1 1 . Evaluating just the integrals over xn and ym in (207) leads to  Iq = 0

∞



∞

···

n

m

(v T v)− 2 (w T w)− 2 Rdu1 · · · dun−1 d uˆ 1 · · · d uˆ m−1 ,

(208)

0

where R=

 m  m n 1 − n −q  n α 2 Γ +q Γ × 2 F1 , −q; 1 − − q; α ; 4 2 2 2 2 α > 0; q = {0, 1, 2, . . .}.

(209)

Multidimensional Integral Identities

201

Next, we use (208) to form the ratio    m Γ n2 + q Iq n −q   , −q; 1 − − q; α , =α × × 2 F1 n I0 2 2 Γ 2

(210)

where I0 is given by (208) when q = 0. Since it can be shown that (210) holds for all 2n such integrals, and since 

∞ −∞

 ···

#

∞

−∞

e

−αx T x−y T y

dx1 · · · dxn dy1 · · · dym =

π n+m ; α > 0, αn

(211)  

the claim follows immediately.

Remark 5 The integral (205) can also be evaluated via (162) and the binomial theorem as follows (John Polcari, private communication): 

∞

−∞

 ···

∞

−∞

(x T x + y T y)q e−αx

T x−y T y

dx1 · · · dxn dy1 · · · dym

q    q T T (x T x)q−k (y T y)k e−αx x−y y dx1 · · · dxn dy1 · · · dym = ··· k −∞ −∞





∞

∞

k=0

=

q   k=0



n

q α k−q− 2 π k

m+n 2

    Γ k + m2 Γ q − k + n2 m n ; α > 0; q = {0, 1, 2, . . .}. Γ 2 Γ 2 (212)

7 Summary and Conclusions The approach described here can generate a wide variety of results, many more than shown here. The initial results presented here have mainly focused on integrals with exponential terms, the complementary error function, and the logarithm. We have also introduced applications to generalized, weighted multivariate Gaussian integrals, possibly scaled themselves by a weighted distribution. Future work will contribute toward evaluating in compact form more realistic heavy-tailed distributions. The efficiency of our approach is clearly evident when one compares it to the approximation methods in [3]. Probabilistic simulations should greatly benefit from our compact evaluations of higher-dimensional integration of weighted Gaussian distributions. There could be another interesting application in statistical theory, in particular when one attempts to Gaussianize heavy-tailed data through some appropriate transformation. The Gaussianized form is an idealistic approach, as in practice, there often exists both asymmetry and heavy tails for data and/or noise, e.g., in speech signals, speed data, human dynamics, Internet traffic, financial

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data, etc., for which accurate inference is needed. One way could be to optimally transform a heavy-tailed random variable into a Gaussian one, weighted if required.

References 1. M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing (Dover, New York, 1972) 2. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products, 6th edn. (Academic, San Diego, 2000) 3. J. Lu, D.L. Darmofal, Higher-dimensional integration with Gaussian weight for applications in probabilistic design. SIAM J. Sci. Comput. 26(2), 613–624 (2004) 4. J. Mollerup, H. Bohr, Lærebog i Kompleks Analyse vol. III, Copenhagen (1922) 5. R.P. Mondaini, S.C. de Albuquerque Neto, Revisiting the evaluation of a multidimensional Gaussian integral. J. Appl. Math. Phys. 5, 449–452 (2017) 6. E. Ng, M. Geller, A table of integrals of the error functions. J. Res. Natl. Bureau Standards-B Math. Sci. 73B(1), 191–210 (1969) 7. M. Novey, T. Adah, A. Roy, A complex generalized Gaussian distribution: characterization, generation, and estimation. IEEE Trans. Signal Process. 58(3), 1427–1433 (2010) 8. F. Olver, D. Lozier, R. Boisvert, C, Clark (eds.), NIST Handbook of Mathematical Functions (U.S. Department of Commerce and Cambridge University Press, 2010) 9. F. Pascal, L. Bombrun, J.Y. Tourneret, Y. Berthoumieu, Parameter estimation for multivariate generalized Gaussian distribution. IEEE Trans. Signal Process. 61(23), 5960–5971 (2013) 10. A.A. Ruffa, The generalized method of exhaustion. Int. J. Math. Math. Sci. 31(6), 345–351 (2002) 11. W.O. Straub, A brief look at Gaussian integrals. Tech. Rep. (2009) 12. A. van den Bos, The multivariate complex normal distribution-A generalization. IEEE Trans. Inf. Theory 41(2), 537–539 (1995)

Minimum Uniform Search Track Placement for Rectangular Regions Richard D. Tatum, John C. Hyland, and Jeremy Hatcher

1 Introduction In general, the problem of search involves the allocation of effort to a given field F ⊂ R2 to detect the location of missing or unknown targets. Typically, search plans are generated by approximating solutions to optimization problems. These approximate solutions describe the allocation of effort to either maximize the effective coverage or minimize the time spent in F, subject to achieving a minimum effective coverage. The notion of effort is an abstract concept reflecting the application of sensor performance applied over F and depends upon several important factors. Clearly, sensor performance is an important factor, which is subject to environmental variations. A priori information, such as the last known whereabouts of the search target under consideration, may also be available for planning. Note that environmental variations can induce drift effects that will also impact the search target. Additional factors for planning may also include the number of available assets or the search geometry. We present an approach that considers a priori target information to maximize the effective coverage of F. While our approach can be augmented to handle the case of multi-asset work, for the sake of general exposition, we assume only a single asset. An approach to the multi-asset search problem is to first determine the number of required tracks and then assign assets to execute the search tracks. This is beyond

Distribution Statement A: Approved for public release; distribution is unlimited. R. D. Tatum () · J. C. Hyland · J. Hatcher Naval Surface Warfare Center Panama City Division, Upper Grand Lagoon, FL, USA e-mail: [email protected]; [email protected]; [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_10

203

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the scope of our work here, as we are primarily concerned with the first part of the problem, which is to determine the placement of tracks to achieve a desired coverage. We invite the reader to examine the non-exhaustive list of multi-asset works [1–4]. Our approach allocates uniformly spaced parallel tracks over sections of homogeneous sensor performance. Our use of parallel tracks is a relevant technique as can be seen in more recent applications to problems in agriculture [5], as well as search and rescue (SAR) [6]. Non-uniformly spaced tracks can be considered but require explicit representation of track position and, consequently, may have scaling issues associated with the number of tracks. Although uniformly spaced tracks approximate the optimal solution, they do not require explicit position representation, which is an observation we use to construct our track placement algorithm that we will discuss later. Our main contribution is to solve the search planning problem as a MILP optimization problem that is subject to a priori target distribution constraints, as well as minimum sectional and total effective coverage constraints. By the effective coverage constraint, we are referring to the cumulative probability of detection weighted by a priori target tendency distribution. The target tendency distribution is the expected distribution of targets, which does not necessarily mean the actual spatial target distribution. The sectional constraints are imposed to maintain a threshold of effective coverage, while the total effective coverage depends upon the sectional constraints as weighted by the a priori target tendency distribution. As we require our total effective coverage to be larger than each sectional effective coverage value, solving the optimization problem as posed is similar to determining where additional tracks will be placed to satisfy a total effective coverage constraint, while not sacrificing the effective coverage of each section. In other words, if m additional tracks are available after the sectional effective coverage constraint is met, our approach provides a method to place those tracks to account for the target tendency distribution. Our additional contributions include the derivation of a general method to evaluate the effective coverage that remains agnostic to the assumption of dependence or independence between tracks, as well as account for the inevitable overlap of parallel tracks. This establishes a quick performance estimator for uniform track coverage that we use for solving the MILP optimization problem. We also show that asymptotically, the placement of uniform tracks will approach the upper bound of sensor performance, which provides validation of our performance estimator for the theoretical case of a perfect sensor. The remainder of our chapter is organized as follows. Section 2 describes how our approach relates to existing work. Section 3 gives our general performance estimator and proofs of its asymptotic behavior for a large number of tracks. Section 4 describes our algorithm for track placement. Section 5 describes the simulation assumptions. Section 6 presents the results of the simulation. We conclude with a summary in Sect. 7.

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205

2 Related Work The theory of search can be traced to the pioneering work of Bernard Koopman, who developed the mathematical framework required to design and evaluate aerial flight plans used to patrol for hostile submarines in protection of a convoy of ships during World War II [7]. Expanding upon the work of Koopman, Stone gives a clear, concise description of how to plan a search [8], which was applied to the problem of guided local searches [9]. In particular, for parallel search tracks, Koopman derives the form P (x, x) = 1 − e−(x,x) , (x, x) =

n=∞ 

G(|x − nx|),

(1) (2)

n=−∞

where n is the lateral range curve from the n-th track, x is the track spacing, and G is a potential function. The ideas of Koopman with regard to search theory can be found in recent advances in disparate fields of study, including archaeology [10], resource allocation [11], and sweep width calculations for detecting wildlife using wildlife detector dogs [12]. The use of an exponential detection function, especially as applied to the problem of the optimal distribution of effort, can be found in several approaches. In [7, 13], use of an exponential detection law was given and used to determine estimates for the optimal distribution of search effort over multiple regions. Expanding upon some of Koopman’s earlier work, Iida [14] and Stone [15] also used the exponential search function to determine results for optimal search distribution. While not necessarily concerned with the optimal distribution of effort, more recently, Lai et al. have also used an exponential detection function in their work with pooling methods for convolutional neural networks [16]. However, as shown by [17], it is not necessary to assume any particular form for the probability of detection function to solve for the optimal distribution of effort. Likewise, our approach does not make any such assumption. We instead derive an effective coverage function that assumes a symmetric lateral range curve that describes the sensor performance. Due to the correlated nature of the same sensor system having multiple looks (detection opportunities) at the same target, the assumption of independence may not be valid as shown in [18]. Our detection function is general in that no assumption of dependence or independence with respect to different tracks is required. Furthermore, using the uniform placement of tracks coupled with the symmetry of the lateral range curves allows us to account for overlap of neighboring tracks and to provide an explicit expression for the effective coverage as a function of the number of tracks. To include navigation error, one approach is to convolve a symmetric navigation-error kernel with the lateral range function [19]. As this will result in a symmetric lateral range function, our approach is also applicable for such a scenario.

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Whenever a priori target information is available, most search approaches take advantage of the information and use it to improve the search performance. Kriheli et al. developed a priority-based strategy that assumes knowledge of probabilities of target locations to determine an optimal search approach [20]. Washburn offers an analytic optimal solution to the problem of how to search for a single, stationary target with a normal spatial target density [21]. Our approach also makes use of a priori target tendency distribution but requires the information be encoded into a probability distribution. Most importantly, we do not require a particular form of the target tendency distribution. To incorporate a priori target information, we impose the following constraint on the effective coverage in F. The effective coverage must be at least as large as the weighted sum of the effective coverage of each section, where the weights are defined by the a priori target tendency distribution. We constrain our problem further to consider an effective coverage for each section. Thus, we can plan using a priori target information but maintain a minimum effective coverage of each section for the case of incorrect target information. However, if the target tendency distribution is correct, then our approach will be rewarded by placing any additional tracks in the correctly weighted sections and less tracks in the sections that are less likely to contain targets. While much work has been done with regard to the abstract nature of optimal distribution of effort [13–15], our approach determines how to explicitly apportion tracks uniformly to sections of the field F which are partitioned based upon homogeneous sensor performance. Since we are assuming a uniform placement of tracks within each section, there is no need to explicitly represent each track position as a variable within our method, which is a critical observation that allows for our method to be highly computationally efficient. All that is required is a fast method for the evaluation of the effective coverage as a function of search effort, which we will discuss in the next section. These observations allow our problem to be approached and solved as a MILP problem. Others have considered MILP with regard to similar problems related to optimal search distribution. For example, Morin used a MILP formulation to determine how to allocate resources for search and rescue missions [22]. As for the case in which performance is not homogeneous, one approach is to construct the worst-case lateral range curve by selecting the smallest probability of detection values for all ranges in each section of a partition. The partition can be altered, producing new lateral range curves for the new section partition. The partition is selected that minimizes the sum of the variances between each sectional lateral range curve and the lateral curves in its section. For an approach to automated area segmentation, see [23]. We compare our method to two other methods through Monte Carlo simulations. For the purpose of comparison, we provide the effective coverage function under the assumption of independence using a simple range and effective coverage value. We compare this to two other methods, one of which also uses uniformly spaced tracks and the other which is a random path search method. The uniform track method that we compare uses the worst probability value for detection and a single value of track spacing. In this sense, it is a very conservative approach. The other approach

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207

plans paths based on a random direction and step size that extends from one side of the field to another side. Thus, the step size is not explicitly sampled as in the case of Lévy flights used for random changes of position for multi-agent searches [24, 25]. For all three methods, we use a homotopy method that continuously deforms the target tendency distribution from a known, biased distribution to an unknown, uniform distribution. As the homotopy method generates targets weighted by the target tendency distribution and the uniform distribution, we are able to provide computational results that show our method will not sacrifice minimum effective coverage when the underlying target tendency distribution is clearly incorrect. To compare all three methods, we measure the ratio of the number of detected targets to the total number of actual targets, as well as the efficiency of each method as measured by ratio of the number of detections to the total path length. In the next section, we provide a derivation of our effective coverage as a function of the number of placed tracks. Again, note that this approach does not make any assumptions about the resulting form of the effective coverage. Rather, it assumes symmetry about the lateral range curve.

3 Effective Coverage Intuitively, effective coverage is the probability of a moving sensor detecting a stationary target located in region  when the target’s positional probability is described by the target tendency distribution. If the target tendency distribution ψ(τ ) : τ ∈  ⊂ Rn → [0, 1]

(3)

is a normalized probability density function that represents the relative probability of a target being present at d and ρ(τ ) : τ ∈  ⊂ Rn → [0, 1] represents our conditional probability to detect a target at d, then the effective coverage in  is given as  P =

ρ(τ )ψ(τ )d.

(4)



Note that the authors in [7, 26, 27] have a similar form. The function ψ provides for a general mechanism to encode information including environmental and historical data as a heat map. For example, in the case where a target is using favorable environmental conditions to attempt to evade detection, a high value on the heat map would represent an area where the probability of detecting the target would be low due to environmental conditions. In the case where the target itself is a stationary sensor attempting to detect a moving target, a high value on the heat map would represent an increased ability for the detection device to be successful. In the

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context of this work, we do not explicitly address this type of scenario. However, the approach taken for the heat map construction is still valid. Furthermore, our approach is useful in that we do not require a distribution of the number of targets. Now specifically, suppose that we want to place tracks with width w to be parallel to the y-axis of the field F ⊂ R2 , with dimensions L × W. Note that w is dependent upon the sensors and largely chosen by the sensor developers through experimentation. Thus, for our purposes, we assume that w is chosen based upon some thresholding function. For a fixed x, we have ˆ ψ(x) =

W ψ(x, y)dy,

(5)

0

ρ(x, y) = ρ(x). ˆ

(6)

Using Equation 4, we have L P =

ˆ ρ(x) ˆ ψ(x)dx.

(7)

0

If ψ(x, y) is uniform and ρˆ is perfect, then P = 1 as expected. Using Equation 7, we turn to the problem of determining the effective coverage for placing tracks to cover all of F. To ensure that we do not waste any tracks by looking beyond the field, we can restrict the first and last track to be completely contained within the field. Thus, we can uniformly distribute n tracks over the range L − w, with a spacing given by x = L−w n−1 . We define xi = (i − 1)x, where i = 1, · · · , N = n + (k − 1) and k=2

3 w 4 . 2x

(8)

We will assume 2x ≤ w < L, which prevents analysis of cases in which tracks are bigger than the field, as well as gaps in coverage. The value of k importantly represents the number of x values required to completely cover the length of the lateral range curve. Figure 1 shows an example of the relationships between n, x, w, and k. In this case, we can easily see the presence of three phases with regard to the count of overlapping tracks: ascension, plateau, and descension. These phases are dictated w by k = 2 2x , which gives the number of x intervals that a track occupies. The ascension period corresponds to i = 1 · · · k − 1, the plateau corresponds to i = k, · · · n, and the descension period corresponds to i = n + 1, · · · , n + k − 1. Since we know the length of intervals measuring the overlap of tracks, and we know which part of the sensor performance function contributes to the overlap, then we can compute the effective coverage. Figure 1 shows that due to the symmetry of the lateral range curve and the periodicity of uniformly placed tracks, we should be able to compute the effective coverage over all of L with remarkably few computations. However, to perform

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209

Fig. 1 Track intersection counts for uniformly placed tracks Fig. 2 Ascension and descension track configurations. The configuration on the left occurs when w is an integer multiple of x. The configuration on the right occurs when w is not an integer multiple of x

this computation, we must be able to identify all of the possible track overlap configurations. If we define the distance of how far a track extends into an interval whose length is x as r = .5w −

5 w 6 x, 2x

(9)

then we can identify the configurations for all three phases. For the ascension and descension phases, Fig. 2 shows that there are only two configurations. This corresponds to the case in which r = 0 (no overlap) and r = 0 (overlap). We observe that r = 0 occurs when w is an integer multiple of x, while r = 0 occurs when w is not an integer multiple of x.

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Fig. 3 Plateau track configurations. When r = 0, we have the configuration on the left. When r = 0, the middle configuration and left configuration are possible

In contrast, the plateau phase contains three configurations, as shown in Fig. 3. As with the ascension and descension phases, the non-overlap case occurs when r = 0. However, when r = 0, there are now two configurations. The first occurs when the two tracks with terminating points within the range of x do not overlap. The second occurs when the two tracks do in fact overlap. Since we know the different track configurations, we can now compute the effective coverage. We measure the effective coverage related to overlapping track coverage as pn : N×R → [0, 1]. As we change the number of tracks, a new set of pn values must be computed. For p(1, τ ), this represents the track sensor performance with no overlap and is symmetric as indicated by the relationship p(1, τ ) = p(1, −τ ). For the ascension and descension phases, we can use symmetry and express part of the effective coverage as

An =

⎧ k−1 , x!i+1 ⎪ ⎪ pn (i, τ )dτ ⎪ ⎨

i=1 xi 2 k−1 xi+1 ⎪ , ! −r

⎪ ⎪ ⎩

i=1

pn (i − 1, τ )dτ +

xi

r=0 k−1 , x!i+1 i=1 xi+1 −r

pn (i, τ )dτ

r = 0

The effective coverage for the plateau period is given as ⎧ xk!+1 ⎪ ⎪ ⎪ pn (k, τ )dτ ⎪ ⎪ ⎪ ⎪ xk ⎪ ⎨ xk!+r xk+1 ! −r pn (k − 1, τ )dτ + pn (k − 2, τ )dτ Bn = (n−k +1) 2 ⎪ xk xk +r ⎪ ⎪ ⎪ xk+1 xk!+r ⎪ ! −r ⎪ ⎪ ⎪ 2 p (k − 1, τ )dτ + pn (k, τ )dτ n ⎩ xk

Combining An and Bn yields

xk+1 −r

r=0 0 < r ≤ .5x r > .5x.

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Pn =

1 (An + Bn ). L

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We now show how Pn behaves for large values of n. The first lemma is a statement about the growth of kx as n → ∞. Lemma 3.1 As n → ∞, kx → w. Proof Since we have x

w w ≤ kx ≤ 2x( + 1), x 2x

then w ≤ kx ≤ w + x. As n → ∞, kx → w.

 

We can use the previous lemma to prove two more lemmas about the asymptotic behavior of An and Bn , which are then easily used to show the asymptotic behavior of Pn . This result shows how the ascension, descension, and plateau phases contribute to the effective coverage as a function of effort. Lemma 3.2 If pn (k, τ ) → γ uniformly in τ as n → ∞, then An → 2wγ . Proof Let  > 0. Using the uniform convergence of pn (k, τ ), we can construct a neighborhood around γ that can be made arbitrarily small to contain most of the sequence pn (k, τ ) for all τ. For nj ∈ N, there exists Nj , Nj +1 ∈ N where Nj < Nj +1 such that |pn − γ | < Nj for all n > Nj and all τ. Of course, as n → ∞, Nj → ∞. We use this bound to prove our result. For both cases, we achieve the same upper and lower bounds, which are given as 2x(γ −

  )(k − 1) ≤ An ≤ 2x(γ + )(k − 1). Nj Nj

As n → ∞, all terms on the left and right side of the inequality go to zero, except for 2kx, which by the previous lemma shows that 2kxγ → 2wγ . Thus, 2wγ ≤ An ≤ 2wγ , which implies that An → 2wγ . Of course, a sequence is not guaranteed to exclusively belong to the case r = 0 or r = 0. However, as n → ∞, we are selecting elements from either sequence that are arbitrarily close to 2w, which implies convergence to 2w.   Lemma 3.3 If pn (k, τ ) → γ uniformly in τ as n → ∞, then Bn → (L − 2w)γ . Proof Let  > 0. As before, we use uniform convergence of pn (k, τ ) in τ to obtain the same bound for pn (k, τ ) as before in the previous lemma. Again, we only need to consider one pair of bounds for multiple cases, which is given as

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(n − k + 1)x(γ −

  ) ≤ Bn ≤ (n − k + 1)x(γ + ), Nj Nj

for all τ and Nj ≤ Nj +1 . As n → ∞, only two terms on the left and right side of the inequality contribute to the limiting value. In particular, they are nxγ and −kxγ . Since nx = n(L−w) n−1 , we see that nxγ → (L − w)γ . By the previous lemma, we know that −kx → −w, which combines with nxγ to show that (L − 2w)γ ≤ Bn ≤ (L − 2w)γ . Thus, Bn → (L − 2w)γ . Since all three cases will result in convergence to (L − 2w)γ , we note that as n → ∞, sampling from all three cases will result in a sequence that also converges to (L − 2w)γ , since all three sequences converge to (L − 2w)γ .   Theorem 3.4 If pn (k, τ ) → γ uniformly in τ, then P → γ . Proof Since Sn = (An + Bn )/L, we see that as n → ∞, Sn → (L − 2w + 2w)γ /L = γ by application of the previous lemmas.   Note that if γ = 1, as in the case of a perfect sensor, then P → 1, which means that we will detect all objects that are within F as to be expected.

4 Minimum Uniform Track Spacing Algorithm Uniformly placing tracks over an entire region to be surveyed is problematic as there would be no reasonable choice other than to let the worst sensor performance guide how the tracks are to be placed. However, if we partition the field into sections of relatively homogeneous sensor performance, then we are able to apply uniform track spacing to each section. With the sections identified, we assign weights to each section signifying our confidence in the presence of targets within each section based on some a priori information. Furthermore, we impose an additional constraint to each section to require a minimum effective coverage for each section for the case in which the target tendency distribution is incorrect. As we want the minimum number of tracks to cover all of the sections, we now describe the mathematics representing this problem. Let S1 , · · · , Sn be the sections of field F with length L and width W. Each section Si has a number of tracks ni that is required to produce an effective coverage ci that is bounded by βi ≤ ci ≤ γi < 1.

(11)

The lower bound βi is chosen to guarantee a minimum effective coverage is achieved in each section. If γi = 1, note that the effective coverage function will require an infinite number of tracks. To avoid this problem, an upper bound strictly less than 1 is required. For demonstration purposes, we discretize ψ into

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the αi values over the sections si . Since ψ is a probability distribution, we have the following constraint: n 

αi = 1.

(12)

i=1

To incorporate the a priori target tendency distribution, we use the αi as weights and recognize the total effective coverage condition as n 

αi ci ≥ tα ,

(13)

i=1

where tα is the desired effective coverage. Note that by imposing a minimum constraint for the effective coverage and total effective coverage constraint that considers a priori information, we are essentially maintaining a minimum effective coverage across all sections while allowing for additional tracks to be placed where we expect the targets to appear. Of course, we need to tie the effective coverage to some actual estimate of effective coverage based upon uniform track spacing. Using the uniform effective coverage formulation described in the previous section, we can represent the effective coverage as ci = f (ni ) =

1 (An + Bn ), L

(14)

where f : N → R is the effective coverage function. Observe that this function can be computed well before any optimization routine is invoked. This is powerful, as the effective coverage function is nonlinear and can be computed to arbitrary precision as governed by the computational resources available. Given all of these constraints, we can solve the uniform track placement problem by using a MILP solver to minimize the following objective function: C=

n 

ni .

(15)

i=1

Solutions for this problem generate tracks that guarantee a minimum effective coverage in each section yet will allocate additional tracks to those sections that require more effective coverage to accommodate the desired weighted effective coverage based upon a priori information about the targets. For the case in which no a prior knowledge is available, the weights can all be set to uniform weights, meaning any additional tracks will be distributed evenly to accomplish the desired total effective coverage.

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5 Simulation Description We consider the case in which the detection opportunities from each track are independent. Furthermore, while our method assumes a continuous lateral range curve, we use simple linear function ρ as defined as ρ(x) = p, −b ≤ x ≤ b.

(16)

Again, this is a simple, constant function defined over a finite range and not an overly complex lateral range curve. We note that assuming independence when the events are in fact dependent results in overestimating the performance of the sensor. Alternatively, the assumption of dependence provides a lower-bound estimate to sensor performance by using the maximum of all the probability detection opportunities as demonstrated in [28]. Of course, a weighted average of the two estimates can be computed, but how to weight the two bounds is beyond the scope of this work. To solve the optimization problem of the previous section, we need to determine the An and Bn values. Let fk = 1 − (1 − p)k . For this case An and Bn are evaluated as previously described. As for our environments,we consider a field that is partitioned into three sections, s1 , s2 , and s3 , as demonstrated in Fig. 4. Over the sections, we define three distinctive sets of sensor performances with varying performance variability given as ppoor = {.1, .5, .9}, pmedium = {.5, .7, .9}, pgood = {.88, .89, .9}, Fig. 4 Field sections that contain different target densities and sensor performance

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Fig. 5 Realizations of target tendency distributions. (a) Left target concentration. (b) Middle target concentration. (c) Right target concentration

where the position of each element corresponds to the section with the same position. For example, .7 in pmedium corresponds to s2 . Here, we are assuming a simple lateral range function to simplify the complexity of the computation to provide a clear explanation of our overall approach. We also consider three types of target tendency distributions, given as (α1 , α2 , α3 ) = (.7, .15, .15), (α1 , α2 , α3 ) = (.15, .7, .15), (α1 , α2 , α3 ) = (.15, .15, .7), where α1 , α2 , and α3 correspond to sections s1 , s2 , and s3 , respectively. Figure 5 contains target pattern realizations created using these αi target ratios defined over the three sections in conjunction with 500 random targets; the targets are uniformly distributed within each section as we are assuming no bias of the target tendency distribution within each section. We compare our solution to the optimization problem to two other methods. For the first comparison, we selected a uniform track spacing method that assumes a single-pd value for the entire field; we choose the worst available pd . Observe that this is a conservative approach, and as such, for fields that contain areas with very low sensor performance, we expect this method to generate several tracks in response to the selection of the worst pd value. For the second comparison, we selected a random search method that places random tracks in the field that emanate and terminate along the boundaries of the field. To facilitate a fair comparison between our multi-pd approach and the “worst-case,” single-pd uniform method, we use the same desired effective coverage of tα ≥ .93 for both methods, and for the multi-pd method, we require that each section has a minimum effective coverage of at least ci ≥ .9 Additionally, we ensure a fair comparison between our approach and the random search approach by using the same path length for the random search method as computed by the multi-pd approach. To understand how the track plans will look relative to each other, consider (α1 , α2 , α3 ) = (.15, .7.15) as an example. For ppoor , note that while most of the targets will be detected using a pd = .5, the single-pd method will use a value of .1, which will generate a large number of tracks. While single-pd and

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Fig. 6 Examples of search plans using target densities defined as α1 = .7, α2 = .15, α3 = .15, for sections s1 , s2 , and s3 , respectively. The single-pd method requires more tracks than the multipd method, given its conservative approximation of sensor performance. By design, the random search method has the same path length as the multi-pd method. (a) pd = .1, pd = .5, pd = .9. (b) Single-pd search. (c) Multi-pd search. (d) Random pd search. (e) pd = .5, pd = .7, pd = .9. (f) Single-pd search. (g) Multi-pd search. (h) Random pd search. (i) pd = .88, pd = .89, pd = .9. (j) Single-pd search. (k) Multi-pd search. (l) Random pd search

multi-pd methods will use the same desired effective coverage values, the multi-pd approach will require fewer tracks, since it is not restricted to use the worst pd value. The random search approach will use the same track length as computed by the multi-pd approach. Figure 6 shows the impact of ppoor , pmedium , and pgood on track plans produced by each of the three methods. We also want to demonstrate how our method performs as the underlying target tendency distribution gradually deforms to a uniform distribution of targets. Under such a deformation, we are able to test the response of our approach to an increasingly incorrect target tendency distribution as compared to the actual spatial target distribution. We generate sets of random targets by sampling from the target spatial distribution

Minimum Uniform Search Track Placement for Rectangular Regions

T = βB + (1 − β)U,

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where B is the target tendency distribution, U is a uniform distribution of targets over the field F = [0, 1500] × [0, 1500], and β ∈ {0.0, .1, · · · , .9, 1.0}. Effectively, β controls the noise introduced into the target tendency distribution B and represents our incorrect perception of the true target spatial distribution. For each β value, we create 30 runs for the simulation that allows us to compute statistics that we use for the comparison of each method. The main idea is to randomly generate sets of targets and apply the track plans generated from all three methods to determine the fraction of targets detected and efficiency values for each method, as respectively defined as D=

Nd , NT

(18)

where Nd is the number of targets detected and NT is the total number of actual targets and E=

Nd , Lpath

(19)

where Lpath is the path length of the tracks. Observe that the probability of the number of detections is related to the following binomial distribution given as P (Nd ) =

  NT ¯ Nd ¯ NT −Nd , P Q Nd

(20)

¯ = 1 − P¯ . where P¯ is the average probability of detection and Q

6 Simulation Results The results are given in Figs. 7 and 8. To understand how to interpret the figures, let us consider the top row in Fig. 7. When β = 0, the targets are sampled from a target distribution that concentrates about 70% of the targets in the left section. As β → 1, the targets will be uniformly distributed over all three sections. For all three cases of sensor performances considered, namely, ppoor , pmedium , and pgood , the targets will be increasingly sampled from regions of better sensor performance as β → 1. In the middle row, as β → 1, the targets will be sampled from regions of mixed sensor performance. In the final row, as β → 1, the targets will be sampled from regions of decidedly worse sensor performance. In each row, as we move from left to right or from ppoor to pgood , we see that while the overall sensor performance for the field increases, the variance of the sensor performance decreases.

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Fig. 7 Detections of targets undergoing continuous deformation from a biased distribution (β = 0) to a uniform distribution (β = 1). In the plots, the blue color represents the single-pd method, the magenta color represents the multi-pd method, and the green color represents the random search method. Both the single-pd and multi-pd methods outperform the random search method. (a) Targets, β = 0. (b) ppoor . (c) pmedium . (d) pgood . (e) Targets, β = 0. (f) ppoor . (g) pmedium . (h) pgood . (i) Targets, β = 0. (j) ppoor . (k) pmedium . (l) pgood

In Fig. 7, we observe that both single-pd and multi-pd methods outperform the random search, while both maintain a percentage of detection rate close to the desired value of 93%. If we look at the results for ppoor , we see that the random search method performs well when targets are located in regions of good sensor performance as expected. This corresponds to the top row when β = 1 and middle and bottom row when β = 0. That trend is maintained for both pmedium and pgood as well. In Fig. 8, we see that although the random search method is more efficient than the single-pd method, the multi-pd method dominates. This is clearly demonstrable as we observe the efficiencies of all of the methods as the variance of the sensor performance decreases. In fact, we see that the multi-pd method always outperforms both the single-pd method and random search method in all cases. As to why the multi-pd method performs better than the random search method, consider the following. Previously, Fig. 6 shows that as the sensor probabilities change, the multi-pd method requires a smaller number of tracks. Since the random search method is using the same path length as the multi-pd search method, it too requires a smaller path length. The multi-pd path length is ensured to maintain a particular

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Fig. 8 Detection efficiency of targets undergoing continuous deformation from a biased distribution (β = 0) to a uniform distribution (β = 1). In the plots, the blue color represents the single-pd method, the magenta color represents the multi-pd method, and the green color represents the random search method. The multi-pd method outperforms both the single-pd method and random search method. (a) Targets, β = 0. (b) ppoor . (c) pmedium . (d) pgood . (e) Targets, β = 0. (f) ppoor . (g) pmedium . (h) pgood . (i) Targets, β = 0. (j) ppoor . (k) pmedium . (l) pgood

effective coverage rate by design, as verified by our simulation results. The random search method, however, does not have any mathematical guarantee of this and as such will make less detections as the overall path length decreases. As for why the multi-pd method outperforms the single-pd method, we note that the single-pd method lacks the fidelity to assign more or less tracks to the field based upon sensor performance. Since it uses the worst-case sensor performance, it will always require more tracks than the multi-pd method, which of course reduces the efficiency of the single-pd method as compared to the multi-pd method.

7 Summary We have provided a MILP formulation for the problem of uniform allocation of tracks to sections with homogeneous sensor performance. To reduce the computational complexity, we created an effective coverage method that explicitly

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depends upon the tracks placed in each section. We compared our method to another uniform track spacing method, as well as a random path planning method. In both cases, our homotopy experiments showed that our method is indeed more efficient. It may be possible to extend the mathematical framework presented here to compute track plans that overlap non-parallel tracks and as such is the subject of future efforts.

References 1. W. Burgard, M. Moors, D. Fox, R. Simmons, S. Thrun, Collaborative multi-robot exploration, in ICRA (2000), pp. 476–481 2. W. Burgard, M. Moors, C. Stachniss, F.E. Schneider, Coordinated multi-robot exploration. IEEE Trans. Robot. 21(3), 376–386 (2005) 3. J. Baylog, T. Wettergren, A ROC-based approach for developing optimal strategies in UUV search planning. IEEE J. Oceanic Eng. 43(4), 843–855 (2017) 4. J.G. Baylog, T.A. Wettergren, Extended search games for UUV mission planning, in Oceans 2017-Anchorage (IEEE, 2017), pp. 1–9 5. I.A. Hameed, Coverage path planning software for autonomous robotic lawn mower using Dubins’ curve, in 2017 IEEE International Conference on Real-time Computing and Robotics (RCAR) (IEEE, 2017), pp. 517–522 6. I. Abi-Zeid, M. Morin, O. Nilo, Decision support for planning maritime search and rescue operations in Canada, in ICEIS 2019 (2019), pp. 328–339 7. B.O. Koopman, Search and Screening: General Principles with Historical Applications (Pergamon Press, New York, 1980) 8. L.D. Stone, The process of search planning: current approaches and continuing problems. Oper. Res. 31(2), 207–233 (1983) 9. A. Alsheddy, C. Voudouris, E.P. Tsang, A. Alhindi, Guided Local Search pp. 1–37 (Springer, Cham, Switzerland, 2016). https://doi.org/10.1007/978-3-319-07153-4_2-1 10. E.B. Banning, A.L. Hawkins, S.T. Stewart, P. Hitchings, S. Edwards, Quality assurance in archaeological survey. J. Archaeol. Method Theory 24(2), 466–488 (2017) 11. M. Patriksson, C. Strömberg, Algorithms for the continuous nonlinear resource allocation problem–new implementations and numerical studies. Eur. J. Oper. Res. 243(3), 703–722 (2015) 12. A.S. Glen, J.C. Russell, C.J. Veltman, R.M. Fewster, I smell a rat! Estimating effective sweep width for searches using wildlife-detector dogs. Wildlife Res. 45(6), 500–504 (2018) 13. B.O. Koopman, The theory of search: III. The optimum distribution of searching effort. Oper. Res. 5(5), 613–626 (1957) 14. K. Iida, Studies on the Optimal Search Plan, Lecture Notes in Statistics (Springer, New York, 2012) 15. L. Stone, Theory of Optimal Search, vol. 118 (Academic Press, New York, 1975) 16. X. Lai, L. Zhou, Z. Fu, S.M. Naqvi, J. Chambers, Enhanced pooling method for convolutional neural networks based on optimal search theory. IET Image Process. 13(12), 2152–2161 (2019) 17. J. De Guenin, Optimum distribution of effort: an extension of the Koopman basic theory. Oper. Res. 9(1), 1–7 (1961) 18. J. Hyland, R. Tatum, J. Hatcher, Analysis of harbor protection systems with strongly correlated sensors, in OCEANS 2017-Anchorage (IEEE, 2017), pp. 1–4 19. A.M. Mood, F.A. Graybill, D.C. Boes, Introduction to the Theory of Statistics (McGraw Hill, New York, 1974) 20. B. Kriheli, E. Levner, A. Spivak, Optimal search for hidden targets by unmanned aerial vehicles under imperfect inspections. Am. J. Oper. Res. 6(2), 153 (2016)

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21. A. Washburn, Piled-slab searches. Oper. Res. 54(6), 1193–1200 (2006) 22. M. Morin, I. Abi-Zeid, C.-G. Quimper, O. Nilo, Decision support for search and rescue response planning, in ISCRAM (2017) 23. J.C. Hyland, C. Smith, Automated area segmentation for ocean bottom surveys, in SPIE-DDS Symposium (2015) 24. J. Ni, L. Yang, P. Shi, C. Luo, An improved DSA-based approach for multi-AUV cooperative search. Comput. Intell. Neurosci. 2018, 2186574 (2018) 25. Y. Khaluf, S. Van Havermaet, P. Simoens, Collective Lévy walk for efficient exploration in unknown environments, in International Conference on Artificial Intelligence: Methodology, Systems, and Applications (Springer, 2018), pp. 260–264 26. T.A. Wettergren, J.G. Baylog, Modeling sequential searches with ancillary target dependencies. Adv. Decis. Sci. 2010, 1–26 (2010) 27. T. Pham-Gia, N. Turkkan, An optimal two-stage graphical search planning procedure for submerged targets. Math. Comput. Modell. 36(1–2), 217–230 (2002) 28. J. Hyland, C. Smith, Effects of stochastic traffic flow model on expected system performance, in Winter Simulation Conference (2012)

Antennas in the Maritime Environment David A. Tonn

1 Introduction Wireless communication is something that we almost take for granted today living in the twenty-first century. We have all become very accustomed to having the world at our fingertips through our smartphones and through our laptops. But in the history of human communications, radio technology is a relative newcomer. This is also true in the field of maritime communications. Prior to the advent of radio in the early twentieth century, ships had to be within visual sight of each other in order to be able to communicate with each other. This communication was in the form of either flags (semaphore) [1] or a lamp [2] that flashed Morse code. Communication over the horizon and long-haul communication were simply not possible until the advent of radio. Marconi’s groundbreaking experiment in 1904 [3] changed the landscape forever. With the advent of radio, originally referred to as wireless, ships no longer had to be within sight of each other in order to communicate. Nor do they have to be was inside of sure in order to receive messages from shore stations. Marconi’s work showed that radio signals could travel great distances and even span the Atlantic Ocean. This opened the door to long-haul wireless communication. Eventually, radio would also find uses in the field of navigation. And even today, radio technology is key to the safety of ships sea in terms of both their ability to communicate and their ability to navigate. Early maritime radio sets often consisted of simple spark gap generators that were tied to Morse code telegraph keys with crystal receivers. These sets were

D. A. Tonn () Submarine Electromagnetic Systems Department, Naval Undersea Warfare Center, Division Newport, Newport, RI, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_11

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capable of sending and receiving messages over distances of hundreds of miles [4, 5]. However useful these wireless sets were though, the commercial maritime community was often slow to adopt this new technology. All that changed on a cold April night in 1912 off the coast of Newfoundland, when the British liner Titanic sank after striking an iceberg. Titanic’s wireless operator was able to call for help and give the ship’s position, allowing rescue ships to be able to find her survivors in their lifeboats. After the Titanic tragedy, the usefulness of radio technology became apparent, and governments began to mandate that ships at sea carry radio transmitters for safety reasons. Standards were also adopted in terms of signals and frequencies [6]. To this day, radio technology continues to play an important role in the operation of ships at sea, including both commercial, military, and recreational vessels. A key component of any radio system is its antenna, and in this chapter, we shall examine some of the issues associated with operating antennas in the presence of the air-sea interface and some recent advances in antenna technology that allow for increased capability.

2 Basic Principles of Antennas In its most basic form, an antenna can be thought of as a passive device that, when used as a transmit component, converts a current applied to its input terminals (also known as the feed of the antenna) into a radiating electromagnetic wave. An antenna is also a reciprocal device, meaning that it can also be used in a receive mode where it converts an electromagnetic wave that impinges on it into a current at the antenna feed. In fact, an antenna’s properties when used as a transmit device can be shown to be identical to those when it is used as a receiving device [7]. The basic mechanism by which an antenna converts current to a radiated electromagnetic field is accelerated charge. In this chapter, we shall confine our attention to wire antennas. (Other antennas such as horns, dishes, etc. are treated in the classic texts on antenna theory found in [7–9].)

2.1 Fields Produced by a Simple Antenna One of the many outcomes of Einstein’s work on the theory of relativity was the discovery that when a charge is accelerated, it emits an electromagnetic wave [10]. Since wire antennas are constructed from metallic conductors – and these conductors contain a large number of conduction electrons – when a time-varying current is made to flow in the antenna, these charges accelerate, and a radiated field will be produced. The radiated field can be computed by knowing the amount of

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current flowing at each point along the length of the antenna and integrating to obtain the total field. From a mathematical point of view, it is often convenient to deal with the socalled vector magnetic potential that the antenna produces, rather than directly with the fields themselves. The vector potential is a mathematical construct that can be shown to contain all of the information needed to compute the fields via the Maxwell equations [7]: − → ∇· B = 0 − → ∇· D = ρV − → ∂B − → ∇× E =− ∂t − → − → − → ∂D ∇×H = J + ∂t In rationalized MKS units, B is the magnetic flux density in webers/m2 (or teslas), H is the magnetic field strength in amperes/meter, D is the electric flux density in coulombs/m2 , E is the electric field strength in volts/meter, and J is the current density in amperes/m2 . The magnetic vector potential A is defined such that: − → − → B =∇× A The Maxwell equations can then be used to show that (for ejωt time dependence) [7]: 1 − → − → − → ∇∇· A E = −j ω A + j ω while the overall vector potential obeys a standard wave equation: − → − → − → ∇ 2 A + k02 A = −μ0 J where k is the wavenumber which for free space is: k0 =

2π √ = ω μ0 0 λ

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Here, J is once again the current density, λ is the wavelength, μ0 is the permeability of free space (4π × 10−7 henry/meter), and ε0 is the permittivity of free space (8.85 × 10−12 farads/meter). The solution to this wave equation in standard spherical coordinates for a unit impulse of current at the origin can easily be shown to be: e−j k0 r − → u A = μ0 4π r This equation tells us that the vector potential comprises a spherical wave radiating outward from the source, where u is a unit vector in the direction of the current. This unit vector helps to determine the polarization or vector orientation of the fields in space. For example, if we had a unit impulse of current in the zdirection: e−j k0 r − → A = Az zˆ = μ0 zˆ 4π r The corresponding fields will be: 1 − → − → e−j k0 r H = ∇× A = μ0 4π r − → e−j k0 r E = 4π r

  1 j k0 + sinθ ϕˆ r

     1 1 1 1 ˆ + 2 η0 + sinθ θ cosθ r ˆ j ωμ0 + η0 + r r j ω0 r 2 j ω0 r 2

Far from the antenna – in what is known as the far-field zone – kr  1, and we obtain the simpler expressions: e−j k0 r − → H = j k0 sinθ ϕˆ 4π r e−j k0 r − → E = j ωμ0 sinθ θˆ 4π r This unit impulse of current produces a spherical wave radiating outward from the origin where the electric and magnetic fields are perpendicular to each other and both are perpendicular to the direction of propagation; this is a so-called transverse electromagnetic (TEM) wave and is the most basic type of radiated field from an antenna [7]. The ratio of the E and H fields is the impedance of free space: # η0 =

μ0 ≈ 120π Ω 0

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and is a frequency-independent term: an important property of a TEM wave. The total power radiated by the unit impulse of current can be found using the complex Poynting vector [7, 8]: S=

1− k0 ωμ0 2 → − →∗ E ×H = sin θ rˆ 2 8π r 2

This expression gives the power density in watts/m2 , and we see the familiar spherical wave 1/r2 dependency from other branches of physics and engineering. The sin2 term that appears is known as the power pattern function of this antenna and shows that the antenna is directional. It radiates its maximum amount of power toward the horizon and radiates zero power at the local zenith and nadir. To find the total power, we integrate over a closed sphere of radius r enclosing the source to get: 2π π P =

k0 ωμ0 2 2 k0 ωμ0 sin θ r sinθ dθ dϕ = 12π 8π r 2

0 0

If instead we had a more general current distribution on the antenna, we would need to integrate in order to find the vector potential and then repeat the steps shown here to find the fields and the radiated power. The integration would need to be over the entire volume containing the current; this is often a very difficult if not impossible integration to perform. − → A = μ0



e−j k0 R − → J dV 4π R

Here, R is the distance from the point on the antenna where the current density is J and the point of observation.

2.2 Basic Antenna Properties Input Impedance and Radiation Resistance The input impedance of an antenna is a complex valued quantity comprised of two portions – a real portion related to radiation and heat losses and an imaginary portion related to energy storage in the fields around the antenna. The radiation portion of the input resistance can be estimated if a relationship exists between the total radiated power and the current at the feed of the antenna. Suppose for a moment that our unit impulse of current was replaced by a short dipole of current where the length of the dipole was dL and the current on it was a uniform value I. If dL is much shorter than the wavelength of operation, it approximates the

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unit impulse of current we discussed in the last section in terms of what the fields look like, with the exception that the dipole moment of the antenna IdL comes into play. We refer to this type of antenna as a Hertzian dipole. The power radiated by the Hertzian dipole can easily be shown to be: k0 ωμ0 → − →∗ − → 1− S = E × H = (I dL)2 sin2 θ rˆ 2 32π 2 r 2 So that the total power radiated is: P = (I dL)2

k0 ωμ0 12π

If we define a radiation resistance to be a value R such that: P =

1 2 I Rr 2

We can find that: Rr = (dL)2

  dL 2 k0 ωμ0 = 80π 2 6π λ

Here, λ is the wavelength of operation. The remaining portion of the input resistance is comprised of ohmic losses in the conductors from which the antenna is made and can be estimated for a linear antenna as [7]: ROH M ≈

L 2π a

#

ωμ0 2σ

and a total resistance and efficiency given by: RI N = Rr + ROH M

η=

Rr Rr + ROH M

Note that for all but the most basic types of wire antennas, these quantities must often be obtained through 3D simulation software owing to the difficulty in obtaining analytical expressions for the current and the fields involved. The input reactance depends on the reactive fields near in to the antenna and is often very difficult to compute. Balanis gives some useful charts for estimating the input reactance of some basic wire antennas. He also gives a formula based on the induced EMF method for estimating the input impedance of a lossless wire dipole antenna of radius a and arm length l operating at a frequency corresponding to a wavenumber k which is [8]:

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Z=

229

% j 60 $ 2 kl − S cos2kl − sin2kl − C 4 S(kl)cos (2kl) (2kl)] [2C(kl) sin2 kl

where: C(ky) = ln

S(ky) =

2y 1 j − Cin (2ky) − Si (2ky) a 2 2 j 1 Si (2ky) − ka − Ci (2ky) 2 2

and:  Si(u) =

u sint 0

 Ci(u) =

t

dt

u 1 − cost 0

t

dt

The above formulas allow a good engineering estimate of the input impedance of a center-fed dipole antenna; for example, if we set kl = π /2 which corresponds to a dipole whose overall length is one-half a wavelength, we get the classic result that the input impedance is 73 + j42 ohms. This type of antenna is known as a halfwave dipole and is a very common type of antenna. However, its input impedance is highly frequency dependent, and so the bandwidth of the antenna – that is, the range of frequencies over which it remains well matched to its feeding transmission line – is limited. Another popular type of wire antenna is the quarter-wave monopole; this is basically half of a half-wave dipole and has an input impedance of 36.5 + j21 ohms. Note that when the antenna is at physical resonance (e.g., its physical length is a multiple of a quarter wavelength), the reactance is non-zero. This happens because electrical resonance and physical resonance do not occur at the same frequency as a result of field fringing around the ends of the antenna.

Directivity and Gain The gain and directivity of an antenna are two key performance metrics of an antenna that are closely related to each other. These are properties of an antenna that have to deal with its ability to focus – or direct – radiated energy in a given direction. This focusing effect is different than in an isotropic radiator which would radiate energy equally in all directions. (Note: physically there is no such thing as a truly isotropic antenna, but it is useful for the sake of theoretical comparison.)

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Let’s once again consider the unit impulse of current for a moment. We saw that its radiated power density was: k0 ωμ0 → − →∗ − → 1− S = E ×H = sin2 θ rˆ 2 32π 2 r 2 giving a total radiated power of: P =

k0 ωμ0 12π

Now, if this power was radiated by an isotropic source, it would be broadcast equally in all directions, and so its power density would be: SI SO =

P k0 ωμ0 = 2 4π r 48π 2 r 2

The directivity of the antenna is defined as the ratio of the maximum power density that it radiates to the average power density that an isotropic source would radiate. So for the case of the Hertzian dipole: D=

SMAX 3 = SI SO 2

In other words, this unit impulse of current radiates 1.5 more power in the direction of maximum radiation (in this case toward the horizon) than an isotropic source does. This is the basis of the gain of an antenna. The directivity is sometimes expressed in dB; the Hertzian dipole has a directivity of 1.76 dB. If we were to repeat this calculation for a half-wave dipole antenna, we would obtain a value of 1.64 (or 2.15 dB). The quarter-wave monopole has a directivity that is double that of the half-wave dipole (e.g., 5.15 dB) due to the fact that it radiates into a half-space and not the full sphere surrounding the antenna. The term gain is perhaps a bit misleading since we often associate it with active components such as amplifiers, etc. But here it is being used for a passive component. The law of conservation of energy reminds us that a passive component cannot add energy to the system, so what then do we mean by the gain of the antenna? What we mean is that the gain is a measure of the ability of the antenna to deliver electromagnetic energy on a direction in space that coincides with the direction of the antenna’s maximum radiation. In the example of our unit impulse, that direction is the horizon plane, and if the antenna were ideal (no losses, etc.), its gain would be a factor of 1.5 or 1.76 dB. In reality, an antenna will have losses due to its own electrical resistance and will not be perfectly matched to the transmission line it is fed from, so we in practice use the term “realized gain” to be defined to be:

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  G = Dη 1 − |Γ |2 Here, D is the directivity of the ideal antenna, η is the efficiency of the antenna, and ! is the voltage reflection coefficient between the antenna and the transmission line that feeds it. Since antennas are reciprocal devices, the gain and directivity of an antenna when used as a transmit device will be identical to those when it is used as a receive device. In this situation, the gain measures the ability of the antenna to convert the power in an incoming electromagnetic wave into power at its feed terminals that can be delivered to a transmission line, amplifier, etc.

The Friis Equation To understand the role that the antenna plays in a communications link, its ability to radiate and to capture energy – as expressed in terms of realized gain – is used in the well-known Friis equation. This equation allows the user to estimate the amount of power that an antenna will receive in terms of the amount of power another antenna some distance away has applied to its inputs. The expression is: PRx = PT x

GRx GT x λ2 (4π R)2

Here, the G terms are the realized gains of the transmit and receive antennas, respectively, R is the line of sight distance between them, l is the wavelength of operation, and PTx is the amount of power that the transmitter applies to the input of the transmitting antenna. This expression is extremely useful in estimating performance in many communications links but assumes that both antennas are operating in free space with no obstructions between or near them that could cause blockage or scattering of the radiated fields. Sometimes instead of gain, we speak of the effective aperture of an antenna; this is related to the gain of the antenna by the simple relationship: A=

λ2 G 4π

Here, λ is again the wavelength of operation. The effective aperture is sometimes useful in radio link analysis if the complex Poynting vector of the field is known. The Poynting vector has units of watts per square meter (i.e., power density), and so the effective aperture of the antenna can be used to compute the amount of power that the antenna will deliver to its feed from the power density that impinges on it. In other words, the effective aperture is a measure of the antenna’s ability to “capture” power from an incoming radio wave. The Friis equation can thus be expressed as:

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PRx = SARx =

GT x PT x ARx 4π R 2

Note: With aperture antennas such as horns and dishes, this form of the Friis equation is sometime preferred.

2.3 Basic Antenna Types It is possible to imagine many different types of wire or conducting antennas, but for practical purposes, there are only several kinds used in practice. The formulations given in this section allow the fields and directivity of an antenna to be found provided that the current distribution on the antenna is known. This is often not the case, and so numerical methods are frequently employed to determine the current distributions, and from that information, the fields and directivity and impedance can be computed. In practice, though, the types of antennas used in maritime applications (with the exception of aperture antennas such as horns and dishes) fall into several basic categories. (a) Dipole antennas: The classic example of this is the half-wave dipole mentioned earlier. This type of antenna is fed at its geometric center and consists of two arms of equal length that extend in opposite directions, as shown in Fig. 1. The antenna can be mounted horizontally or vertically, depending on the polarization of the field that is desired. The input impedance is ideally 73 + j42 ohms which makes it a good match to a 75 ohm transmission line. In a 50 ohm system, there is approximately 0.64 dB which is mismatch between the antenna and the feed, giving a maximum realized gain on the order of 1 dB. Fig. 1 Basic dipole antenna. (Wikipedia public domain image)

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Fig. 2 Folded dipole antenna. (Image source: https://www.quora.com/ What-is-a-half-wavelengthdipole-antenna)

(b) Folded dipole antennas: These antennas resemble a squashed loop of wire, but are actually a dipole-type antenna. It is designed so that the currents in the two vertical portions of the antenna add in phase; it has an input impedance of around 300 ohms making it useful in situations where older-style twowire transmission lines are employed. It also can be shown to have a higher bandwidth than a resonant half-wave dipole making it useful in radio and television communications and in wireless data links (Fig. 2). (c) Monopole antennas: Also known popularly as “whip antennas,” monopoles are basically half of a dipole antenna that are operated above a ground plane or other metal surface. They are lightweight and easy to install on ships, etc. but are limited in their bandwidth like their dipole cousins (d) J-pole antennas: These are very popular antennas in maritime applications since they are end-fed like a monopole, but do not require a ground plane or metal surface underneath them in order to operate. This makes them ideal for mounting on boats or other watercraft made from rubber or plastic or fiberglass: materials which are non-conducting. A basic J-pole is depicted schematically in Fig. 3. It consists of a J-shaped conductor that is fed by a coaxial line. The radiator is roughly half a wavelength long, and the feed is matched by means of a quarter-wavelength stub as suggested in the figure. It has a gain comparable to the half-wave dipole but can have improved bandwidth. This type of antenna finds wide application in commercial and private maritime situations where it is often used for VHF radio communications. (e) Electrically small antennas: Sometimes the need to operate at very long wavelengths combined with space and weight constraints forces the use of electrically small antennas. These antennas can approximate the Hertzian dipole discussed earlier but will have very low realized gains due to the large mismatch between the antenna impedance (which will look like a capacitive open circuit) and the transmission line feeding the system.

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Fig. 3 Basic J-pole antenna. (Image source: Wikipedia public domain image)

3 Antennas Operating Above the Ocean Surface Now that we’ve discussed some basics of antennas, we turn next to the effect that operating them in the proximity of the ocean has on the radiated fields. But first, we need to understand the electrical properties of seawater. Then we’ll examine how the presence of the air-sea interface impacts antenna performance.

3.1 The Electrical Properties of Seawater Many students of electromagnetics open their textbooks (e.g., [11]) to find that seawater has a conductivity of 4 Mhos/meter and a relative dielectric constant of 81. While these values are nominal, it is important to remember that the properties of the ocean’s water varies with position on the Earth’s surface. For example, the conductivity in some parts of the Pacific Ocean can give values as high as 5–6 Mhos/meter, owing to the higher salinity of the water. In river estuaries, this figure can be lower owing to the lower salinity caused by the mixing of fresh water with the ocean water. In any case, it is important to work with the values that corresponds to the salinity of the region which is under consideration. Additionally, the dielectric constant of the water is not a constant. Rather, it is a function of the frequency at which the dielectric constant is being computed or measured. The variation of the dielectric constant with respect to frequency is referred to as dispersion. Pure water by itself exhibits a dielectric constant that obeys the familiar Debye dispersion model [12]. However, this assumes zero salinity in the water. When salt is present, a more detailed model is needed. One such model is the Rivera-Bansal model [13]. Consider a cube of seawater measuring 1 meter on a side as depicted in Fig. 4.

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R1

C1

1m

R2

C2 1m

1m

Fig. 4 Rivera-Bansal equivalent circuit model for seawater dispersion [13] Table 1 Parameters of Rivera-Bansal model of seawater dispersion [13]

Parameter R1 R2 C1 C2

Value 250 mOhm-meter 21 mOhm-meter 43 pF/meter 583 pF/meter

If one were to measure the input admittance Y of the equivalent circuit shown, the following relationships can be used to extract the complex dielectric constant and the conductivity of the seawater comprising the cube: ε − j ε =

377 Y j k0

Here, Y is the admittance of the cube, and k0 is the free space wave number: k0 =

2πf c

where c is the speed of light. The parameters of the Rivera-Bansal model assume seawater at room temperature and nominal salinity of 4 siemens/meter and are given in Table 1. A plot of the real portion of the dielectric constant according to this model is given in Fig. 5. Note the Debye-like relaxation effect occurring in the UHF portions of the spectrum. This effect is important when discussing the operation of radars and other microwave systems in the maritime environment. Note also that this model predicts a static (low-frequency) dielectric constant closer to 70 than it is to the often reported value of 81.

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Re(ε)

50

20

10

1000

106

109

Frequency (Hz)

Fig. 5 Rivera-Bansal model for seawater dielectric constant Fig. 6 Geometry of Sommerfeld problem

3.2 Antennas Operating Above the Air-Sea Interface When an antenna operates above a lossy half-space, the fields produced by it differ from the fields that would be produced if the antenna was operating in free space. The problem of an antenna elevated above the surface of the ocean is such a situation. This type of problem was first solved by the mathematician Arnold Sommerfeld in 1909 [14, 15] and later expounded upon by Balthasar van der Pol in 1935 [16] and Kenneth Norton in 1936 [17]. Sommerfeld began his analysis by considering the potential that a unit impulse of current operating a distance h above the surface of a lossy half-space would produce. This is depicted in Fig. 6. By employing what is basically a two-dimensional Fourier transform in the x– y plane, Sommerfeld was able to derive an exact solution for the vector potential produced by the unit impulse of current in terms of a complex integral [18]:

Antennas in the Maritime Environment

μ0 A= 4π

∞ 0

237

    0 γ1 − 1 γ0 −2hγ0 −γ0 (z−h) J0 (αρ) αdα 1− e e γ0 0 γ 1 + 1 γ 0

Here, the  0 is the permittivity of free space,  1 is the complex permittivity of the ocean water, z is the altitude of the point of observation, and the gamma terms are: γ0 =

 α 2 − k02

γ1 =

 α 2 − k12

where k0 and k1 are the wavenumbers in the air and seawater regions, respectively: √ k0 = ω μ0 0 √ k1 = ω μ0 1 Finally, α is the variable of integration, and the ρ term in the Bessel function J0 is the distance from the origin in the x–y plane. (It is also the radial coordinate in standard cylindrical coordinates.) This integral gives the total potential as the sum of an infinite spectrum of plane waves weighted by the Bessel function of the first kind of order zero. The fact that this potential must reduce to the one shown previously in the absence of the seawater half-space requires that: e−j k0 R1 = R1

∞ 0

J0 (αρ) −γ0 (z−h) e αdα γ0

R12 = ρ 2 + (z − h)2 This is the famous Sommerfeld identity. By using it, we can reduce the expression for the potential due to the unit impulse of current to: ⎫ ⎧   ∞ ⎬ 1 μ0 ⎨ e−j k0 R1 J0 (αρ) e−j k0 R2 e−γ0 (z+h) αdα A= − + 21 ⎭ 4π ⎩ R1 R2 γ0 0 γ 1 + 1 γ 0 0

R22 = ρ 2 + (z + h)2

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What we see here is that the total potential is the sum of three terms: • The potential due to the impulse of current by itself at a height z = h • The potential due to an identical mirror image of the impulse reflected in the ocean surface at a depth z = − h • A third term that depends on the complex dielectric constant of the ocean In the event that both the source and point of observation are at (or very close to) the surface of the ocean, the first two terms cancel each other, leaving only the third term. Since this scenario is very common in maritime communications, we need to understand what this third term tells us about the potential produced by the unit impulse. The integral: ∞ 0

  1 J0 (αρ) e−γ0 (z+h) αdα γ0 0 γ 1 + 1 γ 0

is well known to be an intractable one and is in a class of integrals popularly known as Sommerfeld integrals. Numerical evaluation of it presents significant computational challenges since the integrand can easily be shown to be rapidly oscillating for situations involving common radio frequencies in the HF and VHF radio bands. However, if the point of observation is “far” from the source, an asymptotic approximation can be made. The most widely known of these far-field solutions is the Norton solution, derived by Kenneth Norton in the 1930s [17, 18]. Norton based his solution on a simplification proposed earlier by van der Pol (and later shown to be more generally applicable by Kuebler [19]) for the potential given by our third term. Norton was able to show that the potential (using our notation) reduces to: (   √ √ ) μ0 −j k0 ρ 1 − 0  μ0 −j k0 ρ −Φ A= 1 − j e erfc j Φ e e πΦ = Ls 2πρ 1 2πρ  Φ = −jk0 ρ

1 0 − 02



212

What we see here is at the surface, the third term gives us a cylindrical wave of energy that spreads out but also which attenuates as it travels. In fact, it can be shown that this wave is bound to the air-sea interface and diminishes rapidly as the point of observation moves away from the surface. As a result, this term gives rise to a special type of surface wave known as a ground wave. The term Ls is referred to as the Norton ground wave attenuation factor and is plotted in decibel units in the figure below for frequencies in the HF and lower VHF radio bands at a range of 1 mile.

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The data in this figure show that at very low frequencies and short ranges, the surface wave attenuation is not significant, but that the amount of attenuation increases as the frequency increases. The data in the plot further suggest a simpler form for the surface wave attenuation term; for the case of propagation over seawater, it can readily be shown that the potential reduces to: A=

μ0 −j k0 ρ − ρk03 δ2 e e 8 2πρ

where k0 is again the free space wave number and δ is the skin depth in the seawater region given by the familiar expression [7, 11]: " δ=

2 ωμ0 σ

The ground wave is very important in maritime communications in the HF band and lower portions of the VHF radio bands; since the ground wave is bound to the air-sea interface, it follows the curvature of the Earth as it propagates. This allows for communication beyond the traditional “line of sight” limit between two antennas. The ground wave should not be confused with the type of long-distance communication that is possible using so-called “sky wave” modes, where signals in the HF band are reflected off of the ionosphere. Sky wave signals alternate bouncing off the ionosphere and the Earth’s surface, allowing for communications ranges on thousands of kilometers to be obtained. This is the mechanism by which amateur radio operators and world band radio stations communicate. The ground wave does not give communications ranges as great as sky wave modes, but it does allow for communication beyond the horizon. Normally, when we perform computations about a communications link that uses a ground wave mode of propagation, we use the Friis equation discussed earlier, but we modify the path loss term to include the Norton ground wave attenuation function given in Fig. 7. This is normally done in dB units as part of a larger link budget calculation. Note that if we are operating in the UHF or one of the microwave bands, the ground wave is of no practical value since it attenuates so rapidly as we move away from the source. In situations like this, the antenna produces a total field that is the sum of the field of the antenna by itself, plus the field due to the image of the antenna that the air-sea interface creates. At these higher frequencies, this image may be taken to be approximately an exact image, and as such, the field can be easily determined using the image theory [7]. While Norton’s formulation is widely used, it assumes a flat Earth. To account for the curvature of the Earth, a more detailed formulation is needed, which is beyond the scope of this work. Fortunately, the results of this more detailed formulation have already been computed and are freely available from the International Telecommunication Union (ITU). In ITU Recommendation P.368-9 [20], a series of curves are

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Ls (dB) 5

10

50

100

f (MHz)

-1

-2

-3

-4

-5

Fig. 7 Plot of the Norton ground wave attenuation factor

presented that can be used to estimate the total path loss between two antennas operating over varying types of lossy half-spaces. Figure 2 in this document gives data for propagation over seawater. This figure is reproduced in Fig. 8. The data assume a short monopole antenna radiating 1 kW of power so that the field at 1 km is 300 mV/meter (109.5 dB//1 μV/m). This data in the ITU document is of the total field, and so they include both the spreading loss and the ground wave attenuation. To estimate the ground wave attenuation, a dashed line is provided on the plot that represents the spreading loss alone. By measuring the dB difference between this line and the curve for the frequency of interest, the ground wave attenuation can be determined.

3.3 Application: Measuring Antenna Gain on an Overwater Range One challenge of operating antennas in the presence of the air-sea interface is the task of measuring the antenna’s realized gain in situ. At the Naval Undersea Warfare Center, Division Newport, a novel technique has been developed and used over the years for this purpose. It has become known as the ground wave correction method [21]. In convention antenna gain measurements – such as those specified in IEEE Standard 149-1979 – two antennas are measured. One is a standard reference antenna whose gain is already known, and the second is the unknown antenna. A third antenna – whose gain need not be known – is used as a transmit antenna. The

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Fig. 8 ITU propagation curve for radio signals traveling over seawater

measurement consists of measuring the power that the standard receives from the transmit antenna when fed with a known amount of power and then repeating that measurement with the unknown antenna. The gain of the unknown can be found by comparing the received power levels and the gain of the standard: G = GST AN DARD + (PU N KN OW N − PST AN DARD ) where all quantities are in dB units. This approach is also known as the gain-transfer method. The ground wave correction method differs significantly from this method since it is sometimes hard to define the gain of the standard in the presence of the air-sea interface. (The interaction of the antenna with its image in the air-sea interface can affect its input impedance, thus changing its realized gain.) A standard gain antenna is not used in this approach. Rather, the method is based on an understanding of the surface wave mode of propagation. The ground wave correction method has been implemented at NUWC Newport’s Fishers Island antenna test range. This facility exists on a small island off the coast of New London CT and was once part of Fort H.G. Wright – a US Army fort that was dis-established after the Second World War [22]. The location of Fishers Island is nearly ideal for an antenna test range since it is far from any industrial activity which leads to low man-made noise levels.

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The test range at Fishers Island consists of two locations: a transmit site at Wilderness Point and a receive site 1 mile away near Race Point. An obstacle-free overwater propagation path exists between these two points. The antenna under test is installed at the transmit site – either in a test pool that can be filled with salt water from the ocean and that has buried ground radials that connect the pool to the ocean or in a tidal pond. An underground conduit allows the cable connecting the antenna under test to the transmit equipment to be shielded to prevent it from contributing to the radiated field. The test range is shown in Fig. 9 with a closer view of the test pool at the transmit site shown in Fig. 10. (Note the tidal pond is visible in both photographs.) The range is operated from a shielded control room located under the test pool. At the receive site, a calibrated loop antenna is used as a field sensor. The loop antenna is preferred in this application as a sensor since its behavior does not depend on how it is grounded, unlike a monopole antenna would. The loop antenna is connected to a selective level meter (basically a radio frequency voltmeter). To determine the gain of the antenna under test, a known amount of power is delivered to the antenna under test in a continuous tone at a single test frequency. A directional coupler or a power meter is used to measure the forward power that is delivered. The loop antenna is positioned 1 mile away on the beach; the voltage that it detects is used along with its calibration factor to determine the electric field strength that was present at the loop’s position on the beach. The present measurement system determines this value in dB relative to 1 microvolt/meter. Next, the amount of forward power measured is used to compute the electric field strength that an ideal quarter-wave monopole would produce 1 mile away across the seawater path. The Norton ground wave attenuation function is used to “correct” the field strength to account for the presence of the seawater path, giving this method its name. The field expression is [22]: Fig. 9 Fishers Island antenna test range overview

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243

Fig. 10 Fishers Island transmit site with a monopole antenna mounted

 EQW = 20log10

#

P 195 1000

 + 20log10 Ls + 60 dB//1μV /m

Here, P is the forward power in watts. By taking the ratio of these field strengths, the gain of the unknown antenna relative to that of a quarter-wave monopole can be determined. If we assume that the gain of an ideal quarter-wave monopole is 4.65 dBi1, the gain of the unknown can be computed. G = EMEAS − EQW + 4.65 dBi All quantities are again in dB units. This method has been used and documented for testing primarily in the 2–30 MHz frequency range (limited mostly by the operating band of the calibrated loop antenna). To illustrate this process in action, let us consider an example of a commercially available antenna, in this case a Shakespeare Model 120 multi-segment monopole antenna. In this specific case, five of the antenna segments were mounted onto a Shakespeare type 120-28 base which was bolted to the mount in the center of the Fishers Island transmit site pool. The pool was then flooded with seawater. Gain measurements were performed in the vicinity of the antenna’s quarter-wave

1 This

assumes a directivity of 5.15 dBi and an impedance mismatch of 0.5 dB.

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D. A. Tonn 5 Segment Shakespeare Antenna

Gain Relave to O4 Monopole (dB)

0 -1 -2 -3 -4 -5 -6

9

9.5

10

10.5 Frequency (MHz)

11

11.5

12

Fig. 11 Measured gain of 5-segment Shakespeare monopole antenna

resonance near 10 MHz; these data are shown in Fig. 11. The data give the measured gain relative to that of an ideal quarter-wave monopole at each frequency point. Ideally, the gain of the antenna should be 0 dB if it were an ideal quarter-wave resonator. The fact that the peak measured gain is in fact around −0.3 dB suggests ohmic losses in the antenna feed and in conducting material in the antenna itself, which is to be expected. The measurement has a margin of error of approximately 0.5 dB. This approach to gain measurement is unique in that it allows for measurement of the antenna gain, while the antenna under test is operated in situ in the presence of the air-sea interface. Other methods require the antenna to be mounted in an anechoic chamber; such an arrangement does not allow the impact that the proximity to the ocean has on the antenna’s gain performance.

4 Recent Advances in Antenna Engineering Electromagnetics as a whole is often viewed as a mature field of research, and many universities are no longer offering it as an area of concentration in electrical engineering. But this does not mean that advances in the science of electromagnetics or in antennas are no longer happening. One important area of research and development in antenna engineering has been in the use of advanced materials in the design and fabrication of improved antennas. The research area leverages some of the latest development in anisotropic dielectric materials along with the so-called metamaterials (i.e., materials which are

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designed to have properties not found in nature) to give improvements in antenna gain, bandwidth, or both. We shall consider several examples in this section.

4.1 Antennas Based on Anisotropic Dielectric Materials There have been a variety of papers published on the use of anisotropic media in the presence of radiating structures such as antennas; many of these papers, though, are purely academic in nature and show how to solve for the fields in and around the antenna when the antenna is operating in or near a material whose dielectric constant is a tensor rather than a scalar. Other papers, though, show how these anisotropic materials can be used to improve the performance of antennas. One notable example of this improvement was published by Jiang, Gregory, and Werner [23]. They proposed surrounding an ordinary resonant monopole antenna operating over a metal ground plane with a cylinder of anisotropic material. This material was implemented as a series of metal strips printed onto a flexible plastic substrate to make a planar sheet. Several of these sheets were bent into a cylindrical form and placed over the monopole; what resulted was an increase in bandwidth. The results reported by Jiang et al. were verified by the author in independent experiments. An example of the antenna used in one such experiment is shown in Fig. 12. The author’s model was scaled downward in frequency to the L microwave band. The figure of merit that we will use to compare antenna performance is the bandwidth. For the purposes of our discussion, the bandwidth shall be the range of frequencies over which the input voltage standing wave ratio (VSWR) is less than a specified amount, often 3:1 or 2:1. The VSWR is an indirect measure of the reflection coefficient at the terminals of the antenna. A 1:1 VSWR indicates a perfect match, while an infinite value denotes a total mismatch.

Fig. 12 Model used to verify Jiang predictions (left) and VSWR plot (right)

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Returning to the experiment, the monopole by itself (without the anisotropic cylindrical layer) exhibits a VSWR curve shown in red in the plot; when the engineered anisotropic material was placed over the antenna, the bandwidth defined by a 3:1 VSWR increased to nearly a full octave. While the exact analysis of this structure is beyond the scope of this chapter, its operation can be understood qualitatively. If the monopole is aligned with the “z”axis of a right cylindrical coordinate system, the stripes on the sheets are also aligned in that direction. If we treat the combination of the sheets plus the plastic tube used to hold them in place (see the figure) as an effective homogeneous medium (a valid assumption since all of the dimensions are small compared to the wavelength of operation in L band), we can think of the sheets+tube system as an equivalent homogeneous anisotropic medium. Based on CST Microwave Studio ™ modeling of this structure, the dielectric tensor of this equivalent medium is well approximated as a Cartesian tensor of the form: ⎛

⎞ 1.1 0 0 ε = ⎝ 0 1.1 0 ⎠ 0 0 8.5 This tensor tells us that the material is more polarizable in the z-direction than in the x–y plane. This is a key point. The monopole antenna radiates its far zone electric field polarized in the z-direction, if we place our point of observation on the x–y plane. This means that the radiated electric field “sees” the 8.5 term of the tensor which causes its guided wavelength to contract, in essence making the antenna “look” longer than it is and thus increasing its radiation resistance. But the reactive near field is in the x–y plane, and this field sees a tensor that is nearly equal to that of free space. Together, then, this leads to a reduction in the Q-factor of the antenna because we’ve increased the radiation resistance without substantially increasing the stored energy in the system. (Note: if we were to repeat this experiment with a cylinder of isotropic dielectric material whose dielectric constant√was 8.5, we would merely shift the operating frequency downward by a factor of 8.5 without impacting the Q-factor or the bandwidth since we would be increasing the radiation resistance and the stored energy by the same amount.) For the materials used in the verification, it was found that the bandwidth was optimal if the anisotropic cylinder extended just beyond the end of the antenna and its outer circumference matched the length of the monopole inside.

4.2 Slotted Cylinder Antennas Based on the results of the experiments using monopole antennas, the natural question becomes whether or not the use of anisotropic media can improve the bandwidth of other resonant antennas. A slotted cylinder antenna was tested next; slot antennas are sometimes used in the cellular telephone industry as part of the

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Fig. 13 Basic slotted cylinder antennas

antenna towers and also in digital television applications. A vertically disposed slotted cylinder basically is the dual of a vertically disposed monopole antenna and can be modeled as a magnetic monopole [7, 8]. Examples of this kind of antenna are shown in Fig. 13. According to the duality theorem from electromagnetics, this type of slot antenna will have a radiated E-field in the x–y plane and a reactive near field in the zdirection [7]. So a different kind of anisotropic dielectric tensor is desired from that used with the monopole antenna. This “slot monopole” differs from a conventional monopole in an important respect; there exists a lower cutoff frequency below which the slot will not support the desired field configuration needed to produce the magnetic monopole effect. As Rivera and Josypenko disclose in their patent [24], this cutoff frequency is defined in terms of an equivalent electric circuit for the antenna in which the slot is represented by a per unit length capacitance and the circumferential loop formed by the cylinder by a per unit length inductance. This capacitance and inductance are then used to compute the cutoff frequency in a manner similar to a resonant LC circuit. Specifically, the formulas that Rivera and Josypenko [24] disclose are: ϕ = arcsin  Z = 1 + 10

k=

w 2a

 π t  ϕ 2 + 2 (1 − lnϕ) + w 6 1 a

√

1 + 10Z − 1 Z

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f =

kc 2π

Here, the slot width is w, the thickness of the wall of the cylinder is t, c is the speed of light, and a is the radius of the cylinder. The resulting frequency f is computed in hertz. When operated above this cutoff frequency, the slot supports an electric field that produces radiation in the far-field zone of the antenna. When operated below this cutoff frequency, it does not. The data in Fig. 14 give the measured VSWR of a 4 inch long slotted cylinder made from a 3/4 inch diameter copper pipe that has a 1/16 inch wide slot cut into it (the center antenna in Fig. 13). The slot mode cutoff frequency for this antenna can be computed to be approximately 1.8 GHz. The measured data show a strong resonance at 609.5 MHz – this is where the cylinder itself is roughly a quarter of a wavelength long. At this frequency, the slot is cut off, but at the other two markers seen in the figure, we see the actual resonances of the slotted antenna. These first two resonant modes each have a small bandwidth as shown in the figure. To improve the bandwidth of the antenna, we shall surround it with an anisotropic dielectric material. In this situation, we want the anisotropic dielectric material to be highly polarizable in the x–y plane but weakly so in the z-direction. This orientation is

Fig. 14 Measured VSWR of basic slotted cylinder antenna 3/4 inch diameter by 4 inch long

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desired owing to the fields that the slot mode produces which are orthogonal to the ones that the monopole produces. The desired polarizability can be accomplished using the sheets of metal stripes used previously on the monopole antenna, provided that they are oriented horizontally. This arrangement is shown in Fig. 15. The measured bandwidth – based on a 3:1 VSWR – was over 100%. It is worth noting, though, that the resonance at 609.5 MHz is not affected by the covering; again this resonance has to do with the entire cylinder acting as a monopole, and so the orientation of the anisotropic material tensor is incorrect for producing improved bandwidth. However, the horizontal arrangement of the stripes does make a marked improvement in the bandwidth obtained in the slot modes. Another way to realize the anisotropic material is by stacking alternating layers of high and low dielectric constant material along the z-axis and placing this structure over the antenna. This configuration is depicted in Fig. 16 [from 25]. The anisotropic structure was fabricated using zirconium oxide (ZrO) washers (ε ~ 29.0ε0 ) with an inner diameter of 1 inch and an outer diameter of 2 inch and a 1/8 inch thickness alternating with foam spacers also 1 inch inner diameter but with a 1.5 inch outer diameter to minimize the amount of material in the spaces without compromising the mechanical integrity of the stack. The resulting VSWR plot shows a 3:1 bandwidth that easily exceeds an octave. Note that the high dielectric constant seen in the z-direction caused the 609.5 MHz resonance of the cylinder to shift downward in frequency and out of the frequency range of the measurement. These VSWR results show that the same sorts of improvement in bandwidth that can be obtained on a monopole antenna can also be obtained on that antenna’s dual radiating structure suing the same type of approach.

Fig. 15 Measured VSWR of 3/4 inch diameter – slotted cylinder with anisotropic cover layer

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Fig. 16 Slotted cylinder with anisotropic covering (left) and measured VSWR plot (right)

4.3 Patch Antennas Microstrip patch antennas – popularly referred to simply as patch antennas – are popular antennas where a lightweight, low-profile antenna is desired. As its name implies, a patch antenna consists of a patch of metal that is printed onto a grounded dielectric substrate. The feed for the antenna is a coaxial probe whose outer conductor connects to the ground on the substrate and whose inner conductor penetrates the dielectric to make contact with the patch above. A patch behaves much as a monopole does in that it supports a resonant current distribution, so it stands to reason that an anisotropic material would be useful in improving its performance. Experiments by the author have confirmed this effect. The same sheets of stripe material used in the monopole experiment described in Sect. 4.1 were re-used in planar form as a superstrate for a patch antenna designed to receive GPS signals at 1225 MHz and 1575 MHz, respectively. Plots of the VSWR before and after the application of the anisotropic superstrate are shown in Fig. 17. It is worth noting here, though, that the octave of bandwidth obtained with the monopole and slotted cylinder is not seen here. However, a useful improvement in the bandwidth of the antenna where the 1575 MHz resonance once was is now obtained [26].

4.4 Metamaterials In the early 2000s, one of the hottest topics in the field of electromagnetics research centered on the so-called metamaterials. These materials are so-called because

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Fig. 17 Measured VSWR of GPS patch antenna without anisotropic superstrate (left) and with anisotropic superstrate (right)

they are engineered to have properties – such as dielectric constant, magnetic permeability, etc. – that are not found in nature. Examples of metamaterials would include single negative materials (e.g., materials possessing a negative dielectric constant), double negative metamaterials (e.g., materials having a negative permittivity and permeability), near-zero index materials (e.g., materials having an optical refractive index that is nearly zero), and artificial impedance surfaces (e.g., materials whose surface impedance mimics a perfect magnetic conductor). Many papers and dissertations were produced (including the author’s [27]), some of which made amazing predictions about improvements that could be made to antennas [28]. For example, in 2003, Ziolkowski and Kipple [29] published a paper that predicted that surrounding a short Hertzian dipole (of the type we considered earlier) with a shell comprised of a double negative metamaterial would result in a dramatic increase in the amount of radiated power from the antenna. (In other words, an increase in radiation resistance.) However, the predictions made by researchers in this area were not universally embraced. Many of the early predictions neglected the inventible losses that must occur in metamaterials. This is a fundamental mistake. In order for there to be a causal solution to the Maxwell equations within a region filled with a particular material, that material’s permittivity and permeability must obey the KramersKronig relations. (In fact, Toll [30] has shown that the Kramers-Kronig relations form a necessary and sufficient condition for the existence of a causal solution.) The Kramers-Kronig relations for the relative permittivity of a material are [11]: 2 r = 1 + π 

∞ 0

ζ r  dζ ω2 − ζ 2

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r



2ω =− π

∞ 0

r  − 1 dζ ω2 − ζ 2

and they tell us that there is an intimate connection between energy storage in a material and energy dissipation in the material. In fact, they predict that the only truly lossless material is free space where r = 1. All other materials must have some losses, and these losses will be maximized where the dielectric constant is changing most rapidly. For example, if we apply the Kramers-Kronig relations to the Debye model for dispersion in water, an absorption peak will appear in the region of the spectrum where the dielectric constant is decreasing most rapidly. Some metamaterials are fabricated using FSS-like structures involving split ring resonators and stripes [27], similar to the structures described earlier to obtain anisotropic behavior. The difference with a metamaterial is that the structures are operated at or near their resonances. This means that the effective medium parameters of permittivity and permeability will be prone to rapid change near the resonance in the same way that the transfer function of an RLC filter changes rapidly near the natural resonant frequency of the RLC circuit. What this leads to is increased losses in the metamaterial, since the Kramers-Kronig relations predict the presence of absorption peaks precisely in the same portion of the spectrum where rapid changes in permittivity or permeability are taking place. It is a fundamental error, then, to neglect the losses in metamaterials. But some researchers have, and this omission has led to overly optimistic predictions of improved antenna performance. For instance, the Ziolkowski and Kipple paper already cited makes its basic predictions with no allowances for losses in the materials they assume. The results they predict are therefore not obtainable in practice. One notable work that raises objections to the very existence of metamaterials in electromagnetics came from the late Prof. Ben Munk of the Ohio State University. In a book [31] published shortly before his death, Munk launches into a scathing critique of metamaterials and of the bold predictions being made about their performance. His objections basically boil down to two issues: 1. Devices made from metamaterials would violate Foster’s reactance theorem. 2. Some predicted phenomena – such as negative refraction and near-zero index effects – violate causality and are unphysical. The first objection centers on an often forgotten theorem from circuit analysis. Foster’s reactance theorem [32] states that in a reactive component, the reactance will increase monotonically with increasing frequency. In other words, ∂X >0 ∂ω

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where X is the reactance (imaginary portion of the complex valued impedance) of the component. Munk’s argument against metamaterials proceeds thusly: suppose that a metamaterial has a negative permittivity and we make a parallel plate capacitor out of it. The resulting capacitance would therefore be negative, and the reactance of the capacitor would be (for exp(jωt) time dependence): C =  r 0

X=

A 0 ω |C|

Here, A is the plate area of the capacitor, and d is the spacing between the plates. Now, Foster’s reactance theorem would require: 1 ∂X =− 2 >0 ∂ω ω |C| which is impossible for all real values of frequency, and so Munk argues that metamaterials of this type are physically impossible. However, Munk’s argument is missing a key point: all metamaterials are lossy and dispersive as required by the Kramers-Kronig relations. If we account for the dispersion in the material, for instance, by assuming a simple lossless Drude model for the dielectric constant: r = 1 −

ωp2 ω2

it becomes a rather simple task to show that Foster’s reactance theorem is satisfied. So, Munk’s first objection is not valid, so long as we properly account for dispersion in the material. His second objection is harder to refute, though. The concept of causality is fundamental in physics: that is, the effect of an event cannot precede the cause of the event. For example, the light bulb in a room does not begin to shine before one turns on the light switch! But this simple concept has not prevented some from proposing non-causal situations involving metamaterials. These situations are the focus of Munk’s second objection. Consider the case of a near-zero index metamaterial for a moment. This is a material in which the familiar optical index of refraction takes on a numerical value near zero. (Recall that free space has a refractive index of 1.0 and that this is the smallest value of the refractive index found in nature, so the definition of a metamaterial applies.) Suppose that a source was placed inside the metamaterial and allowed to radiate into the half-space above it by the presence of a ground plane as suggested in Fig. 18. The situation on the left side shows the expected effect of radiating from within a material whose refractive index is >1. Taking a ray optics approach, rays leaving

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Fig. 18 Refraction from a conventional material (left) NZI metamaterial (right)

the source arrive at the interface and refract away from the normal, since they are passing from a more optically dense medium into a less dense one. But in the figure on the right, if the index of refraction of the material is nearly zero, a collimated beam of energy results – despite the fact that the optical path lengths are unequal in violation of Fermat’s principle. This objection appears to be sound. But what is to be made of instances where researchers have presented experimental data purporting to show that when an NZI material is placed in front of an antenna, the directivity increases? (Due to the collimation of the energy implied by the figure above.) Obviously, causality cannot be violated, so some other effect must be at work. The author has examined this effect and conducted an experiment using a frequency selective surface (FSS) sheet consisting of a pattern of rings in a rectangular grid. The FSS was modeled as having a Drude-like effective dielectric constant; near the plasma frequency seen before, the dielectric constant and refractive index both vanish, in principle, and should lead to a near-zero index effect. This sheet was placed on a low dielectric constant foam backer and placed in front of a broadband horn antenna to act as an NZI lens. An identical horn antenna was placed some distance away, enough to be in the far-field region. A vector network analyzer was used to measure the total path loss experienced by a signal entering the feed of antenna #1 and being radiated, then passing through space and through the FSS sheet, and then entering the second horn, and departing its feed. This path loss was measured with and without the FSS sheet present. What was remarkable was that a decrease of several dB was noted near the plasma frequency of the FSS sheet. It would seem that the effect is confirmed. However, another explanation – one that does not violate causality – is possible. The FSS sheet used to make the “lens” had a larger surface area than the aperture of the horn antenna. The plasma frequency predicted for the FSS sheet is also the frequency where the sheet resonates. At this frequency, the beam from the horn causes the sheet to “light up,” and it is the scattering of the fields from the sheet that are captured by the second horn. Since the area of the lens is greater than that of the horn, we can posit that the effective aperture of the horn has been increased due to the fact that the aperture doing the radiating is really the FSS and not the horn itself. The larger aperture brings with it higher directivity, which according to

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the Friis equation leads to a lower path loss. This explanation relies solely on sound antenna engineering principles and does not require any metamaterials or possible violations of fundamental laws of physics to support it. In summary, while there has been a great deal written about the use of metamaterials in antenna design, claims regarding the use of metamaterials for significant improvements in performance should be met with a healthy dose of skepticism.

5 Summary The world of wireless communications could not exist without antennas. Antennas form the basic interface that converts voltages and currents into radiated electromagnetic waves and vice versa and are essential components in all forms of radio communication, including systems that operate at or near the air-sea interface. In this chapter, we’ve considered some of the basics of antenna theory and shown how, when an antenna operates above the air-sea interface, the presence of the surface wave must be taken into account provided that the frequency of operation is below roughly the middle of the VHF band. We’ve also seen how recent developments in the field of anisotropic materials can be employed to improve the bandwidths of several basic antenna types, all of which have potential application to communications in the maritime environment.

References 1. H.P. Mead, The history of the international code. U.S. Naval Institute Proceedings 60(378), 1083–8 (1934) 2. http://www.jproc.ca/rrp/rrp2/visual_lights.html 3. G. Bussey, Marconi’s Atlantic Leap (Marconi Communications, New Century Park, 2000). ISBN 0-9538967-0-6 4. C. Hempstead, W.E. Worthington, Radio transmitters early, in Encyclopedia of 20th-Century Technology, (Routledge, New York, 2005), pp. 649–650. ISBN 978-1135455514 5. F.E. Terman, Radio Engineering, 2nd edn. (McGraw-Hill, New York, 1937), pp. 6–9 6. A.B. Magoun, The Titanic’s role in radio reform. IEEE Spectrum. Institute of Electrical and Electronics Engineers Web 16 (2014) 7. W.L. Stutzman, G.A. Thiele, Antenna Theory and Design (Wiley, Hoboken, 2012) 8. C.A. Balanis, Antenna Theory: Analysis and Design (Wiley, New York, 2016) 9. R.S. Elliot, Antenna Theory and Design (Wiley, Hoboken, 2006) 10. R.S. Elliott, Electromagnetics: History, Theory and Applications (IEEE Computer Society Press, Piscataway, 1993) 11. C.A. Balanis, Advanced Engineering Electromagnetics (Wiley, Hoboken, 1999) 12. A.R. Von Hippel, Dielectric Materials and Applications: Papers (Technology Press of MIT, Cambridge, MA, 1954) 13. D.F. Rivera, R. Bansal, Submarine antennas, in Encyclopedia of RF and Microwave Engineering, (Wiley, Hoboken, 2005), pp. 4937–4951

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14. A. Sommerfeld, Über die Ausbreitung der Wellen in der drahtlosen Telegraphie (Verlag der Königlich Bayerischen Akademie der Wissenschaften, München, 1909) 15. A. Sommerfeld, Partial Differential Equations in Physics (Academic Press, New York, 1949) 16. B. Van Der Pol, Theory of the reflection of the light from a point source by a finitely conducting flat mirror, with an application to radiotelegraphy. Physica 2(1–12), 843–853 (1935) 17. K.A. Norton, The propagation of radio waves over the surface of the earth and in the upper atmosphere. Proc. I.R.E. 24(10), 1367–1387 (1936) 18. R.E. Collin, Antennas and Radiowave Propagation (McGraw-Hill, New York, 1985) 19. W. Kuebler, A note concerning the evaluation of the Sommerfeld integral. IEEE Trans. Antennas Propag. 27(2), 254–256 (1979) 20. https://www.itu.int/rec/R-REC-P.368-9-200702-I/en 21. D. Tonn, P. Gilles, P. Mileski, A Ground-wave Correction Technique for Low Frequency Antenna Measurements over a Seawater Ground (AMTA 2002 Proceedings, Cleveland, 2002) 22. P. Rafferty, J. Wilton, Guardian of the Sound: A Pictorial History of Fort H.G. Wright (Mount Mercer Press, Fishers Island, 1998) 23. Z.H. Jiang, M.D. Gregory, D.H. Werner, A broadband monopole antenna enabled by an ultrathin anisotropic metamaterial coating. EEE Antennas Wirel. Propag. Lett. 10, 1543–1546 (2011) 24. D. Rivera, M. Josypenko, U.S. Patent #6,127,983 25. D. Tonn, S. Safford, M. Lanagan, E. Furman, S. Perini, Implementation and Testing of Engineered Anisotropic Dielectric Materials, Proceedings of the 38th Annual Meeting of the Antenna Measurement Techniques Association (AMTA ‘16), Austin (2016) 26. D.A. Tonn, Apparatus and method for improving the gain and bandwidth of a microstrip patch antenna. U.S. Patent 9,281,568, issued March 8 2016 27. D.A. Tonn, Application of double negative metamaterials for improving the performance of maritime antennas, Doctoral Dissertation, University of Connecticut (2007) 28. N. Engheta, R.W. Ziolkowski, A positive future for double-negative metamaterials. IEEE Trans. Microwave Theory Tech. 53(4), 1535–1556 (2005) 29. R.W. Ziolkowski, A.D. Kipple, Application of double negative materials to increase the power radiated by electrically small antennas. IEEE Trans. Antennas Propag. 51(10), 2626–2640 (2003) 30. J.S. Toll, Causality and the dispersion relation: Logical foundations. Phys. Rev. 104(6), 1760 (1956) 31. B.A. Munk, Metamaterials: Critique and Alternatives (Wiley, Hoboken, 2009) 32. R.M. Foster, A reactance theorem. Bell Syst. Tech. J. 3(2), 259–267 (1924)

The Destabilizing Impact of Non-performers in Multi-agent Groups Thomas A. Wettergren

1 Introduction Understanding how groups perform as a collective has been a topic of study for mathematical ecologists for decades [21]. In particular, when groups are comprised of individuals performing actions based only upon their own perceptions and with limited interaction, there is often some noticeable group behavior that emerges from the interaction. Such evolving natural systems often have group performance that exceeds what is expected from that of the individuals, and thus groups are often seen as performing better than the simple aggregation of their constituent members. Based on insights from these natural systems, it can be envisioned that man-made systems may be one-day engineered to mimic the behaviors found in these evolving natural systems, and these behaviors can allow the group to develop emergent behaviors that solve difficult problems without changing the individuals for each problem that emerges [14]. We consider the mathematical analysis of these types of systems and consider how the emergent behavior depends on the individual behavior, the number of individuals, and the nature of the group interaction. In particular we are interested in examining the impact of a small number of individuals not behaving as expected (socalled non-performers) on the overall group. The development of emergent behavior from a group of individuals following simple behaviors has been oftentimes referred to as swarm intelligence. In describing the observation of these behaviors in nature (in particular in insect groups), Bonabeau et al. [2] define swarm intelligence as “the emergent collective intelligence of groups of simple agents.” While we examine

T. A. Wettergren () Naval Undersea Warfare Center, Newport, RI, USA e-mail: [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_12

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similar types of emergence, we hesitate to use the term swarm intelligence as that is often referred to in specific applications such as the particle swarms that are at the core of the algorithmic development approaches that were pioneered by Kennedy and Eberhart [13]. We do not seek to develop an algorithm for performing swarmlike group behavior, but instead seek an analytical understanding of the mechanisms that lead to more basic emergent behavior, and also examine how a small number of non-performers may impact that performance.

2 Background on Emergent Group Behavior It has been observed in nature [21] that many types of animal behavior, in particular those concerned with foraging [8, 20], exhibit group behaviors that evolutionarily emerge from the simple behaviors of individuals. These collective group behaviors include such common phenomena as flocking, herding, and schooling, among others. While there are many specific details that lead to the various forms of these groups, one common aspect is that these groups have very limited interaction between the individuals that is based only on passive communications within the group. That is, the only interaction between individuals is due to what they can observe about the behavior of others; thus this typically limits the interactions to those individuals within a small neighborhood of one another. In one of the earliest thorough mathematical examinations of these grouping processes, Okubo [16] showed that the grouping results from the natural advectiondiffusion processes employed by individual animals only when there is an attractive force that biases motion of individuals based upon the action of neighbors. This advection-diffusion process modeling approach has been shown to admit a macroscopic modeling view of the group behavior of individuals [1]. Using the partial differential equations from such process models, aggregate behaviors of large groups can be described in terms of the movement of a density (or mass) of the individuals, which makes it beneficial when describing group behavior. Unfortunately, such models are only accurate for relatively large numbers of individuals and are of limited use in heterogeneous situations, such as those when some individuals behave differently than the others. One of the benefits of the macroscopic view is that it allows one to consider the group as an entity that is driven by the individuals (rather than just observing the aggregate behaviors of individuals). This naturally leads to questions concerning the evolution of group size and shape; one of the most basic of these questions is how many individuals should make up a group, and does this number stabilize under evolutionary group joining processes? Sibly [19] argued from an individual fitness perspective that there should be an optimal group size for animals based on diminishing returns for larger groups, yet empirical evidence shows that groups are usually larger than that. He thus argued that the notion of an optimal group size must lead to an unstable optimum and that stable group size is larger than that which optimizes individual fitness. However, Giraldeau and Gillis [9] point out that the

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argument is tied to the assumption of a fitness of joining a larger-than-optimal group being less than that of staying alone, and that is not observed in most natural settings. Thus, the selection of a proper function representing individual fitness based upon group membership is of paramount importance if one is to assess the existence of optimal numbers of individuals in a group. The Niwa model [15] is a model that performs a split-and-merge modeling paradigm to allow individuals to evolve to sets of groups that have a distribution of group sizes that matches those found in nature. It has been shown to work for both fish and some mammal populations. As the first part of this chapter, we look at this model as one example of the potential influence of non-performing individuals on group behavior. Reynolds [17] was one of the first to show that distributed emergent behavior could be accurately simulated through simple computer simulation. He used a model of bird behavior to show that flocking can emerge from the simulation as an aggregate behavior of individuals performing simple individual behaviors. The Vicsek model [22] extended this type of computational model to show that selfordered motion can emerge from systems with very simple interactions. In this case the influence between individuals is that individuals maintain uniform speed but adjust their direction to match the average movement direction of all neighbors within some range. Following that work, Jadbabaie et al. [11] developed a graph theoretical approach for assessing the convergence of Vicsek-style groups. Their method uses a computation of the average heading based on the graph Laplacian of the group. Han and colleagues [10, 23] have shown that the introduction of a small number of lightly controlled agents (called “shills” in their studies) can improve the performance of traditional group evolution in difficult scenarios. They achieve this level of soft control for group intelligence by steering the controlled agents toward the goal. Further work on the shill concept by Duan and Sun [7] showed that increasing the number of these controlled agents does not increase the positive benefits of cooperation, and thus they conclude that only a small number of controlled agents are necessary to achieve good group adaptation due to the cooperative nature of evolutionary processes. These models have even been applied to human behavior. When examining the behavior of humans in groups, Charness et al. [3] show that aggregate group behavior emerges when there is a payoff to individuals that includes a group component (as expected) but also show a stronger tendency toward group behavior when the group is more “salient,” that is, when individuals have a stronger awareness of the rest of the group. As autonomous vehicles have become popularized in recent years, the application of these behavioral models to robotic systems has been performed in order to create groups of robots that perform with limited communications [14]. This leads to the extension of the behaviors exhibited by biological individuals to more general “agents” in a multi-agent system. Under the limited communication paradigm, these agents can be programmed to perform in a similar manner to biological individuals, and thus the emergent group behaviors seen in biology can be created in engineered systems. Recent investigations [24] have examined the impact of non-performing agents (called “cheaters” in that study) on cooperative actions of social networks

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of multi-agent systems. In that study it was seen that the cooperative memory of a network, when the network is large enough, can be used to create actions that are robust to the influence of small numbers of bad agents. The resilience of robots in groups operating under distributed consensus has been examined as another engineering example of the impact of small numbers of non-performing agents [18]. In that study they make a clear distinction between robustness of a system and its resilience, the former dealing with errors that are modeled in an uncertainty framework and the latter dealing with errors that are not modeled. Thus, their consensus framework may be robust if it is tolerable to the modeled uncertainties yet is resilient only if it can handle the unexpected (such as the presence of a nonperforming agent). We utilize the Couzin model [4, 5] to model the observed process of animal aggregation as a combination of repulsion, attraction, and alignment of the position and orientation of individual agents relative to their neighbors. In this way the individual behavior leads to group formation and collective group behaviors based on the social interactions of individuals. An interesting observation from this model is that the results exhibit a form of collective memory of the group, whereby the previous history of group structure influences the collective behavior even when individuals exhibit none of this memory effect [4]. One benefit of the Couzin model is that it is amenable to the inclusion of informed agents, which are a small subgroup of the total set of agents that possess extra knowledge of the group goal. These agents can effectively control the group of other agents without direct communication, as the social behavior of group motion is the only mechanism for information sharing in the model. The effectiveness of informed agents has been observed in nature in the behavior of honeybee swarms [12], as the scout bees in the swarm effectively behave as informed agents. In this chapter we examine the question of how a group of non-performing agents may impact a multi-agent system that is operating as a group of independent agents with collective emergent behavior. In this context we examine the emergent behavior as a bias toward following the directions of neighboring agents, in particular when a group of neighbors is moving in a common direction. We first examine the effects on the distribution of group sizes under emergent behavior in order to see how the stable group size distribution is affected. Then we examine the kinematic problem of evolutionarily moving the group toward a specific goal location. In order to add purpose to the goal, we include soft control of the group via a small number of informed agents that know the goal location. All other agents appear identical to any given agent, such that an individual cannot tell if a neighbor is a regular agent, an informed agent, or a non-performing agent (i.e., a rogue agent). Our goal is to demonstrate the impact of the various combinations of numbers of non-performing and informed agents on group performance and to see how the size of an individual agent’s neighborhood of influence affects these results.

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3 The Distribution of Emergent Groups There are many examples of how groups can emerge from simple models of basic behavior with non-interacting agents. In particular, we look at a model of emergent groups that occur when a set of agents are in close proximity to one another. This notion of following the behaviors of those around you is a common instinct found in many natural systems and has potential in engineered systems as it does not require communication nor negotiation between agents. Assuming that once such groups are formed they have a tendency to move together, the overall system exhibits a trend toward large groups over time as new individuals are encountered regularly and join with the group. To mimic natural systems within the model, we consider that some subset of the agents will occasionally splinter off of the group to “go their own way” and form a new group. Over time, it can be expected that these joining and splitting phenomena will lead to some group sizing equilibrium. We show through a mathematical model that this is the case, and we then examine the stability of this equilibrium to perturbations due to a small number of agents violating the joining and splitting rules.

3.1 The Niwa Model for Dynamic Groups The Niwa model [15] is a stochastic differential equation model that is based on empirical observations of fish school sizes over time. In the model individual agents move between groups according to a specific random process. We begin with a set of n agents A = {A1 , A2 , . . . , An } with corresponding positions {x1 (t), x2 (t), . . . , xn (t)} that are located in a workspace W ⊂ R2 . The space W is partitioned into  distinct locations {L1 , L2 , . . . , L } such that Li ∩ Lj = 0 and ∪i Li = W. At each time step, the model assumes that all of the agents Ai that are in the same location Lj comprise a group. Specifically, this means that the group Gj (t) corresponding to location Lj at time t is given by the following: Gj (t) = {Ai : xi (t) ∈ Lj }

(1)

Over time, the agents move from group to group in a stochastic manner. Specifically, the agents move by choosing to leave their group according to a breakup rate of p (where p is the probability of leaving the group in unit time step) and join with any agents in the new location to form new groups. In addition, any intact groups move randomly to a new location at each time step. The dynamics of the combined individual behavior (leaving a group) and group-following behavior (moving with the group) create a process that has important stability properties. Let gi (t) represent the size of whichever group contains agent Ai at time t. It has been shown [15] that the Niwa process can be represented by the following Ito stochastic differential equation for gi (t):

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  2 dgi (t) = −p gi (t) − gp dt + 2σ (gi (t)) dB(t)

(2)

where dB(t) is a Wiener process noise with zero mean and variance dt and σ (gi (t)) is the standard deviation of group-size changes. The term gp represents the mean of the distribution of group sizes, which is a function of both the population size n and the breakup probability p. This mean group size is the expected group size that would be experienced by any individual agent that is selected at random from the population and thus is different from the observed mean group size. In particular, for a scenario with m different sizes of groups with a set of group sizes {g1 , g2 , . . . , gm } and corresponding frequency distribution of groups {w1 , w2 , . . . , wm }, the expected group size seen by an individual is given by: ,m

2 j =1 gj wj

gp = ,m

j =1 gj wj

(3)

, whereas the mean observed group size is given by gj wj . As any specific individual agent follows the dynamics of the stochastic differential equation (2), it will observe a change in group size about the mean from Equation (3) as other agents join, leave, and re-join with the agent’s group. Niwa showed [15] that Equation (2) admits a stationary solution which implies that the probability that an individual agent Ai is to be found in a group of size g is proportional to a stationary distribution of group sizes P (g). Specifically, the probability of being in a group of size g is given by a distribution of the form: P (g) ∝ g

−1



g exp − gp



  1 −g 1 − exp 2 gp

(4)

where we recall that the mean expected group size parameter gp depends on both the breakup rate p and the number of agents n. Thus we can conclude that the group sizes will stabilize even though the agents continue to move freely between groups. This stabilizing effect mimics what is observed in nature and has been seen in computational experiments of the Niwa process model [21]. Given that a set of agents all performing in this manner achieve a stable group-size distribution, we perform a set of simulation experiments in order to examine the effect that a small number of non-performing agents (ones with behavior different from the stated splitand-join rules) have on the resulting group-size distribution. For simulation experiments on the Niwa model, we consider a group of n = 2000 agents that are operating in a workspace W comprised of  = 2000 locations. We initialize the location of all 2000 agents by distributing them uniformly random across the locations. The split-and-merge process of forming, moving, and splitting groups is performed over t = 50,000 iterations using a group breakup probability of p = 0.3. Thus, at each iteration, each group of size g is split with probability p = 0.3; if a split occurs, we randomly choose a number of the agents (chosen uniformly over [1, g − 1]) to break away from the group and move to a new (randomly chosen)

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location. The remaining component of the group is then randomly moved as a group to another new (randomly chosen) location. At the end of the time step (i.e., once this process has completed for each group), the agents at each location are grouped together to form the new set of groups, and the algorithm iterates to the next time step. A time series of the number of groups obtained of each size g over the iterations is shown in Fig. 1. From the figure, it is clear that after very few iterations, the number of groups of various sizes stabilizes, even though individual agents continue to be randomly moved both separate from and within their respective groups. The final numbers of groups for the various group sizes g are marked as black x’s in the figure. In Fig. 2 we show the distribution of the sizes of these final numbers of groups as a function of group size g (shown on a log-log scale). Using these computed group sizes, the expected group size that is seen by an individual agent (computed as in Equation (3)) is found to be gp = 11.29. Additionally, the corresponding expected group sizes from the probability distribution of agent group sizes (given by P (g) from Equation (4)) is shown by the line in the graph, which clearly matches with the numerical experiment. The result in Fig. 2 only shows the sizes of groups at a single time step, yet agents continue to move into and out of groups at every time step. However, the analysis of the stochastic differential equation model of Equation (2) shows that it leads to an equilibrium value for the number of agents in each group (as shown in Equation (4)). To show that the solutions from the simulation do indeed appear to

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Fig. 2 Resulting group sizes after t = 50,000 iterations of the Niwa model with n = 2000 agents over  = 2000 locations. The curve represents the distribution from Equation (4)

reach an equilibrium, we collected the results for the 10 iterations leading to the conclusion of the simulation run shown in Fig. 1 (iterations from t = 49,990 to t = 49,999) and computed the average of the number of agents in each group of the sizes g. The resulting numbers for each group size are shown in Fig. 3. The resulting trend is similar to that shown in Fig. 2 (although note that this graph has a different vertical scale because the averaging process allows some groups with average numbers smaller than one). To form a more rigorous comparison, we compute the expected group size from the averaged group numbers and find it to be gp = 11.06. The curves of Equation (4) for both the averaged gp and the final gp are also plotted in Fig. 3, and they are virtually indistinguishable from one another. Thus, we conclude that the stochastic differential equation formulation for the discontinuous agent group change process is an appropriate model, as the stationary solution holds up to the averaging of multiple solutions near the final time step.

3.2 Impacts of Non-performers on Stable Group Sizes To examine the effects of non-performers on the Niwa model, we first consider the situation when non-performance is due to the agent leaving the scenario (such as what occurs with a complete loss of power). For this situation, we take the n =

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Fig. 3 Resulting group sizes averaged from iteration t = 49,990 to t = 49,999 from the Niwa model with n = 2000 agents over  = 2000 locations. The solid curve represents the distribution from Equation (4) with gp = 11.29 and dashed curve for gp = 11.06

2000 agent case that was run as shown in Fig. 2 and reduce the number of agents by 5%; effectively this is the same as running a scenario with n = 1900 agents. The resulting distribution of group sizes is shown as the triangles in Fig. 4. For comparison, we have included the results of the full n = 2000 agent scenario in the figure as well (shown as x’s). From this figure it is clear that the primary effect of reducing the number of agents by a small amount is to reduce the size of the large groups, as the distribution of groups of small sizes overlays with that from the full agent scenario. A second form of non-performer to consider in the Niwa model occurs when a subset of the agents performs in a manner inconsistent with the stated split-andmerge rules. In this case, we again consider 5% of the original n = 2000 agents to be non-performing, but now these 100 agents stay active and violate the behavior. For this simulation, these agents are assumed to all leave from their group at every time step, rather than obeying the group breakup probability p. As they make a random switch to another location, they likely arrive at a location with other agents in which case they join that group temporarily and then make another random switch on the next time step. These non-social agents are in every other way identical to the 1900 performing agents. The distribution of group sizes from a simulation of this scenario is shown in Fig. 5 where the scenario with non-performing agents is represented by the squares and, once again, the nominal scenario with all agents performing is shown by the x’s. From this figure, it is clear that the small number of

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Fig. 4 Resulting group sizes from the Niwa model with n = 2000 working agents (x’s) and with n = 1900 working agents (triangles). Both cases are for t = 50,000 iterations over  = 2000 locations

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Fig. 5 Resulting group sizes from the Niwa model with n = 2000 working agents (x’s) and with n = 1900 working agents plus n = 100 non-performers (squares). Both cases are for t = 50,000 iterations over  = 2000 locations

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Table 1 Numbers of resulting groups of various sizes from simulations of the Niwa model with n = 2000 total agents over  = 2000 locations Group size range g ≥ 30 g ≥ 25 g ≥ 20 g ≥ 15 g ≥ 10 g≥5

Nominal 4 7 11 21 51 129

5% Agent loss 4 7 7 22 44 125

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Fig. 6 Resulting group sizes from the Niwa model with n = 2000 working agents (x’s) and with varying percentages of non-performing agents (squares are 5% non-performing, circles are 10% non-performing, and diamonds are 20% non-performing). All cases are for t = 50,000 iterations over  = 2000 locations

non-performing agents leads to not just a decrease in the number and size of large groups but also to a noticeable increase in the number of small groups. To further illustrate the difference between non-performing and merely missing agents, Table 1 shows the number of groups above various size thresholds that are found in the various cases. From the table, it can be seen that the nominal case of all performing agents and the case with 5% missing/lost agents both have similar numbers of large groups. However, the case of 5% non-performing agents (the non-social agents) has noticeably smaller numbers of large groups with many more small groups. The effect of increasing small groups while decreasing large groups remains as the number of non-performing agents is increased beyond 5%. In Fig. 6 we show

268 Table 2 Expected group sizes based upon simulations of the Niwa model with n = 2000 total agents over  = 2000 locations

T. A. Wettergren Scenario Nominal 5% Agent loss 5% Non-performers 10% Non-performers 20% Non-performers

Expected group size gp 11.29 10.31 7.32 6.66 6.70

the results of having the numbers of non-performers at both 10% and 20% (for comparison we include the nominal and 5% cases from Fig. 5 on the plot as well). It is clear from the figure that increasing the number of non-performers has an effect on further increasing the number of small groups, although not as large of an effect as the initial change for adding the small number (5%) of non-performers. This is further seen by computing the expected mean group size gp that is seen by an individual agent as found in Equation (3). Table 2 shows these expected group sizes for the cases computed. From the table, it is seen that the loss of 5% of agents has a small effect on the expected group size (change from 11.29 to 10.31), whereas when those agents instead are non-performing, it dramatically lowers the expected group size (down to 7.32). Interestingly, the change to this expected group size for further increasing the number of non-performers has a minimal effect on the expected group size. Thus, we conclude that only a small number of non-performers is required to change the properties of the resulting group structure, and the magnitude of this change is relatively insensitive to increases in the number of these non-performers.

4 Kinematic Groups of Individuals with Limited Information While groups formed under the Niwa model are representative of natural systems, that model does not consider the kinematic motion of specific agents. Thus, we next describe a model that examines a kinematic group of moving agents that inform their intended direction of motion by following the directions of those around them. This passive type of group behavior again requires only observation of neighbors and no real communication between individuals. When there appears to be consensus among the members around an agent, it is then better off biasing its motion toward that consensus. An additional complication to this model is that some number (presumably small) of agents should be more informed than the others, so that this small number of agents can passively lead others to the goal. Without communication these informed agents appear no different than any other agents; however, it is known that a small number of these informed agents can drive a large non-informed group. In fact, this is the basic behavior that is found in groups like honeybees. We modify an existing model of this type of group behavior to include the effects of a small number of rogue leaders, that is, a group of agents that is heading to an errant goal location. Numerical experiments demonstrate the effect of

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the rogue agent behaviors on the group and how that effect depends on the number of agents and the size of the neighborhood groups.

4.1 The Couzin Model for Non-informed Following The Couzin model [4] is a mathematical model of groups of kinematic agents whose individual behaviors are to follow the aggregate motion of those agents within close proximity. It allows for both regular agents that only perform a following behavior and informed agents that are driven toward a known (or perceived) goal. We consider a group of n agents A = {A1 , A2 , . . . , An } of identical capability moving in a workspace W ⊂ R2 . Let xi (t) ∈ W represent the position of agent Ai at time t and vi (t) ∈ R2 represent its current velocity (with |vi (t)| = v0 = constant). Since all agents are moving at the same constant speed v0 , we normalize any dimensional units such that v0 = 1 for the sequel. At each time instant, agent Ai determines a desired direction di (t + Δt) for the next time step that moves the agent at a constant speed v0 to best align its direction with neighbors within a given neighborhood Ni (t) ⊂ W around the agent. We refer to these neighbors within a the neighborhood of agent Ai as the agent’s group Gi (t). The neighborhood Ni (t) is a disc of radius dgroup that is centered at the agent location xi (t). Specifically, the group is given by the following: Gi (t) = {Aj : xj (t) ∈ Ni (t), Aj ∈ A} = {Aj : ||xi (t) − xj (t)|| ≤ dgroup , Aj ∈ A}

(5)

where dgroup is a group distance range, which is given by the distance at which agents recognize others. The group distance range dgroup must be less than the physically observable range dsense of the agents, as agents in this model can only passively observe others to determine their own group. We note that these groups are dynamic, as agents may move in and out of the neighborhoods of one another. However it is clear that the following lemma holds true: Lemma 1 For any agent Ai ∈ A and any j = 1, . . . , n, Ai ∈ Gj if and only if Aj ∈ Gi . This lemma shows that agent groups are complementary; that is, an agent is a member of another agent’s group only if that agent is a member of its group. This feature of the grouping definition allows an individual agent to belong to multiple groups; however, agents that are alone (group of size one) do not belong to any other group. The desire to move in a direction aligned with neighbors and the need to avoid collision are often in conflict. As such, the Couzin model [4] defines the desired direction by considering collision avoidance a priority over all else, such that the desired direction at the next time interval di (t + Δt) is given by:

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di (t + Δt) =

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

, ,

j :Aj ∈Ci (t),j =i

k:Ak ∈Gi (t),k=i

xi (t)−xj (t) |xj (t)−xi (t)| ,

xk (t)−xi (t) |xk (t)−xi (t)|

+

, k:Ak ∈Gi (t)

if |Ci (t)| > 1 vk (t) |vk (t)| ,

(6) otherwise

where Ci (t) = {Aj : ||xi (t) − xj (t)|| ≤ dcoll , Aj ∈ A} is the set of agents that are within the collision distance dcoll of agent Ai . The first choice in this calculation creates a repulsion of the agent away from those within its collision range, and the second choice creates an attraction of the agent in the directions of those within its group range when there are none (other than itself) within collision range. Because of the additions in Equation (6), the desired direction di (t + Δt) is not necessarily normalized. Note that for a group of one (the smallest group size for any agent), this reverts to setting di (t + Δt) to the agent’s present velocity vi (t). We next consider three types of agents to interact in the Couzin model: regular (non-informed) agents, informed agents, and rogue agents. The regular agents are the standard Couzin model agents, and they move according to the agents within their group (along with the collision avoidance behavior). The motion direction of the regular agents is given by simple normalization of the desired motion direction computed by Equation (6). The informed agents are a small set of agents that know the goal location x∗ . These agents do not move directly toward the goal, but use a weighted combination of moving toward the goal and behaving as a regular Couzin agent, with a weighting parameter γ where 0 < γ < ∞. As γ → 0, the agent approaches the behavior of a regular non-informed agent, and as γ → ∞, the agent moves directly toward the goal. We generally consider a value of γ = 1 as a good comprise behavior for the informed agents. The rogue agents represent a third type of agent who have misinterpreted the goal location; because of this, they behave like an informed agent but pursue a goal location of xR = x∗ . No agents can tell if any neighboring agent is a regular agent, an informed agent, or a rogue agent. They merely follow according to their own behavior rules. This set of behaviors leads to the following set of normalized motion directions: Regular agents: di (t + Δt) dˆ i (t + Δt) = |di (t + Δt)|

(7)

di (t + Δt) + γ (x∗ − xi (t))|di (t + Δt)| dˆ i (t + Δt) = |di (t + Δt) + γ (x∗ − xi (t))|di (t + Δt)||

(8)

Informed agents:

Rogue agents: di (t + Δt) + γ (xR − xi (t))|di (t + Δt)| dˆ i (t + Δt) = di (t + Δt) + γ (xR − xi (t))|di (t + Δt)|

(9)

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These normalized motion directions dˆ i (t + Δt) yield a desired direction for each agent at each time step that is based upon their individual behavior rule (i.e., whether they are regular, informed, or rogue) and the observed motions of any agents in their group neighborhood Ni (t) (through the direction di (t + Δt)). All agents examine their local neighborhood Ni (t) at each time step and determine their desired normalized motion. As a practical matter, the physical normal acceleration of any agent is bounded and thus can only change its direction a certain amount over the discrete time step, and thus the desired direction may not be obtainable. This runs the risk of having some agents enter into the next time step at a non-standard speed (i.e. |vi (t + Δt)| = 1). To avoid this potential situation, each agent checks if the desired change to its course is beyond its normal acceleration limit. Assuming the maximum acceleration corresponds to a maximum angular change of θm per time step, the directional change allowed is limited to θm Δt. Then, the agent velocity vector is finalized according to the following:

vi (t + Δt) =

⎧ ⎪ ⎪ ⎨  ⎪ ⎪ ⎩ vi (t) +

dˆ i (t + Δt), dˆ i (t+Δt)−vi (t) |dˆ i (t+Δt)−vi (t)|

if |dˆ i (t + Δt) − vi (t)| ≤ θm Δt  θm Δt, otherwise

(10) This modified implementation of the Couzin model accounts for multiple types of agents with differing information, as well as some of the practical considerations that are found with physical implementations of mathematical behaviors. Given this modeling framework, we use numerical simulations of various conditions to examine the impact of different numbers of non-performing rogue agents on groups with small numbers of informed agents.

4.2 Impacts of Non-performers on Non-informed Followers To examine the impact of non-performing rogue agents on a group comprised primarily of passive follower agents, we develop simulations of the modified Couzin model described above. In this model, we consider a set of n agents comprised of ninf informed agents (using the behavior of Equation (8)), nr non-performing rogue agents (using the behavior of Equation (9)), and nnoninf = n − ninf − nr regular non-informed agents (using the behavior of Equation (7)). Agents all move with a speed v0 = 1 (covering one unit distance in one time step), and the simulations are run for t = 2000 time steps. The agent collision distance used in Equation (6) is dcoll = 5. The agents are initially distributed uniformly over a domain of size 800 × 800 units, and both the goal and rogue-goal locations are placed outside of the initialization region. After distributing all n agent initial locations, a random set of ninf are chosen to be the informed agents, and a random set of nr are chosen to be the rogue agents. All agents are initialized with a random initial motion direction.

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Fig. 7 Sample initialization for the Couzin model for n = 20 agents that are initially positioned in a domain of size 800 × 800 units (bounded by the dashed line). In this scenario there are nnoninf = 13 non-informed agents shown by x’s, ninf = 5 informed agents shown by ∗’s, and nr = 2 rogue agents shown by +’s. The goal location is shown by the star at (600,600), and the rogue location is shown by the square at (0,600)

An example initialization for n = 20 agents with ninf = 5 and nr = 2 is shown in Fig. 7. For each simulation scenario that is run, we perform 100 random realizations of the scenario and run each realization for a complete t = 2000 time steps. At the end of the simulation, the distance of each agent from the goal location is determined. Following Dong et al. [6], we utilize a measure of “relative size” Sgoal as the measure of performance. The relative size is simply the fraction of the agents that reach convergence; in this case that is considered to be coming within a distance dgoal of the goal location. In these simulations that distance is taken to be dgoal = 100 units. Goal locations are chosen so as to be at least this far from the initialization region, such that no agents are initialized to be in the convergence region. Finally, we report the average value of relative size, averaged over all 100 realizations as Sgoal , in the reported results. In Fig. 8 we show the results of a scenario with n = 200 total agents with a relatively large number of ninf = 50 informed agents. The curves in the figure are for varying numbers of rogue agents (showing nr = 0, nr = 10, and nr = 20), and the resulting relative size of agents converging to the goal is given as a function of the group size dgroup . For the largest group size of dgroup = 400, nearly all agents are in the same group. The figure shows that increasing the neighborhood group size improves the convergence to the goal, but also increases the impact of the rogue

The Destabilizing Impact of Non-performers in Multi-agent Groups 1

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Fig. 8 Relative size Sgoal of agents reaching the goal for a set of n = 200 agents with ninf = 50 informed agents. Results are averaged over 100 simulations, and plotted as a function of neighborhood group size dgroup . The x’s are for no rogues (nr = 0), the circles are for 5% rogues (nr = 10), and the triangles are for 10% rogues (nr = 20)

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Fig. 9 Relative size Sgoal of agents reaching the goal for a set of n = 200 agents with ninf = 20 informed agents. Results are averaged over 100 simulations, and plotted as a function of neighborhood group size dgroup . The x’s are for no rogues (nr = 0), the circles are for 5% rogues (nr = 10), and the triangles are for 10% rogues (nr = 20)

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agents. And the impact of the rogue agents increases with increasing numbers of rogue agents in a nearly linear impact (i.e., doubling the number of rogues doubles the impact). In Fig. 9 we show the results of a scenario with n = 200 total agents with a more moderate number of ninf = 20 informed agents. These results still show an increase in the impact of rogues as the group size increases and also a nearly linear relationship between the number of rogues and their impact. However, in this case, the largest neighborhood group sizes have a decrease in the performance. This is a consequence of agent confusion within groups as there are not enough informed agents to create a strong consensus in the proper direction. With smaller-sized groups, however, the groups that have strong leadership (due to larger fractions of informed agents) tend to “pick up” weaker groups as they come in proximity to them and pull them along in the proper direction. The behavior of these followers is an

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Fig. 10 Relative size Sgoal of agents reaching the goal for a set of n = 20 agents with ninf = 5 informed agents. Results are averaged over 100 simulations, and plotted as a function of neighborhood group size dgroup . The x’s are for no rogues (nr = 0), the circles are for 5% rogues (nr = 1), and the triangles are for 10% rogues (nr = 2)

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Fig. 11 Relative size Sgoal of agents reaching the goal for a set of n = 20 agents with ninf = 2 informed agents. Results are averaged over 100 simulations, and plotted as a function of neighborhood group size dgroup . The x’s are for no rogues (nr = 0), the circles are for 5% rogues (nr = 1), and the triangles are for 10% rogues (nr = 2)

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example of the competing components that drive the group: informed agents pulling to the goal, rogue agents pulling to the alternate location, and other following agents moving in a variety of directions. For similar numbers of informed and rogue agents, along with large group sizes, the effect of following a group of agents moving in different directions leads to confusion, causing significant numbers of agents not reaching the goal. Determining the precise combinations of parameters that create this confused state is a subject of ongoing study. As a final set of scenarios, we repeat the simulations from Figs. 8 and 9 with a significantly smaller number of overall agents. This is to see if the effects observed are limited to large populations. Specifically we consider n = 20 total agents with ninf = 5 and ninf = 2, the results of which are shown in Figs. 10 and 11, respectively. From these plots it is seen that there is less of an impact of the rogue behavior, yet the impact that does exist is still larger for bigger neighborhood group sizes.

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5 Conclusion In this chapter we have described a mathematical modeling study of the impact of non-performing agents on the group behaviors that emerge in multi-agent systems with no inter-agent communications. Both simple geographic membership systems and kinematic motion-following systems were examined, and small numbers of nonperforming agents were seen to have an effect on the aggregate performance in both types of system. For the geographic membership systems, the impact of the non-performers was to decrease the resulting size of groups, and the magnitude of that effect was the same for very small numbers of non-performers as well as larger numbers of non-performers. For the kinematic motion-following systems, the impact of non-performers was to decrease the number of agents reaching a goal location, and that effect was more pronounced with larger systems (more agents) as well as for systems where neighbors at greater distances influence an agent’s behavior. These types of effects must be considered when designing engineered systems that model emergent natural behaviors, as reliability limitations on physical systems will create some small numbers of non-performing agents. Other types of non-performing behaviors that may be considered include agents that stop moving completely as well as agents that blindly follow a specific other agent rather than making any decisions of their own. Acknowledgments This work has been supported by the Office of Naval Research.

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11. A. Jadbabaie, J. Lin, A.S. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans. Autom. Control 48(6), 988–1001 (2003) 12. S. Janson, M. Middendorft, M. Beekman, Honeybee swarms: how do scouts guide a swarm of uninformed bees? Anim. Behav. 70(2), 349–358 (2005) 13. J. Kennedy, R.C. Eberhart, Swarm Intelligence (Morgan Kaufmann, San Francisco, 2001) 14. J. Long, Darwin’s Devices: What Evolving Robots Can Teach Us About the History of Life and the Future of Technology (Basic Books, New York, 2012) 15. H.-S. Niwa, Power-law versus exponential distributions of animal group sizes. J. Theor. Biol. 224(4), 451–457 (2003) 16. A. Okubo, Dynamical aspects of animal grouping: swarms, schools, flocks, and herds. Adv. Biophys. 22, 1–94 (1986) 17. C.W. Reynolds, Flocks, herds, and schools: a distributed behavioral model. Comput. Graph. 21(4), 25–34 (1987) 18. K. Saulnier, D. Saldana, A. Prorok, G.J. Pappas, V. Kumar, Resilient flocking for mobile robot teams. IEEE Robot. Autom. Lett. 2(3), 1039–1046 (2017) 19. R.M. Sibly, Optimal group size is unstable. Anim. Behav. 31(3), 947–948 (1983) 20. D.W. Stephens, J.R. Krebs, Foraging Theory (Princeton University Press, Princeton, 1986) 21. D.J.T. Sumpter, Collective Animal Behavior (Princeton University Press, Princeton, 2010) 22. T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, O. Shochet, Novel type of phase transitions in a system of self-driven particles. Phys. Rev. Lett. 75(6), 1226–1229 (1995) 23. X. Wang, J. Han, H. Han, Special agents can promote cooperation in the population. PLoS One 6(12), e29182 (2011) 24. T. Winke, J.R. Stevens, Is cooperative memory special? The role of costly errors, context, and social network size when remembering cooperative actions. Front. Robot. AI 4, 52 (2017)

Improving Inertial Navigation Accuracy with Bias Modeling Tod E. Luginbuhl, Ahmed Zaki, and Eugene Chabot

1 Introduction Autonomous underwater vehicles (AUVs) have been in operation with worldwide acceptance by civilian and military organizations [1]. Accurate navigation and positional accuracy have been a challenge that limits the duration of long missions for AUVs. The challenges that limit AUV operation have been summarized in [1] as follows: multivehicle navigation, improved near-bottom navigation, optimal survey, and environmental estimation. An inertial navigation system (INS) determines a vehicle’s position using precision measurements from accelerometers and gyroscopes. Due to sensor drift, the resulting position estimates are biased because these precision measurements must be integrated multiple times to calculate position, i.e., gyros are integrated once for orientation estimate, and accelerometers are integrated twice for position estimate. One solution to this problem is to provide the INS with additional sources of unbiased information such as Doppler velocity logger (DVL) measurements and heading measurements for underwater vehicles (UUVs). However, this can be difficult and slow to converge if a Kalman filter is implemented at the sensors’ frame since the INS Kalman filter will typically have a large state vector, e.g., a minimum of 15 states of Kalman filter is required for typical INS. These states are, three positions, three velocities, three orientations, three gyro bias and three accelerometer bias. Note that both the state and measurement equations for the INS are nonlinear [2].

T. E. Luginbuhl · A. Zaki () · E. Chabot Naval Undersea Warfare Center, Newport, RI, USA e-mail: [email protected]; [email protected]; [email protected] © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1_13

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In [3], Hegrenæs and Hallingstad presented an approach for aiding the inertial navigation system (INS) of an underwater vehicle using velocity measurements provided by an experimentally validated kinetic vehicle model. As illustrated in this work, the integration of the model velocities in the navigation system is an effective and inexpensive approach toward the solution to several of the aforementioned challenges. Additional aiding is provided by an ultrashort base line (USBL) acoustic positioning system, ambient pressure readings, and velocity measurements from a Doppler velocity log (DVL) with bottom track. Global vehicle position measurements are obtained by combining differential global positioning system (DGPS) and USBL. In many practical cases, the position measurements will be unavailable for extended periods of time, and the INS then chiefly depends on external velocity aiding. Similarly, even when including a DVL, situations may arise where it fails to work or measurements are discarded due to decreased quality. This will, for instance, occur when operating in the midwater zone or over very rough bathymetry due to loss of bottom track. In [4], Sarma proposed debiasing an inertial navigation system (INS) by fusing the outputs from the INS with external unbiased measurements using a Kalman filter. The proposed algorithm combines the unbiased measurements from a DVL with outputs from the INS in a separate low-order Kalman filter. His approach includes bias correction terms to the x and y coordinates of the vehicle’s position. Depth was not considered, which is typical in this case because depth is measured by a pressure sensor that is absolute and accurate. In Sarma’s approach, the elements of the Kalman filter’s state vector are unbiased, and the INS measurements are biased. Bias correction is performed in the output equation of the Kalman filter. The INS is assumed to update far more rapidly than the sampling rate of the DVL, so the debiasing Kalman filter runs at the DVL rate and the INS estimates are decimated to the DVL rate. This allows the correlation between INS estimates to be ignored. The bias correction terms are included in the state vector but are assumed to be piecewise constant over intervals significantly longer than the filter update rate. When implementing the debiasing method proposed in [4], the authors of this paper noticed that system equations defining the Kalman filter were not observable. Consequently, a different output matrix was chosen to make the system observable. To improve the performance of this algorithm when a vehicle is turning, a coordinated turn model [5] was added. Switching between nominally straight line motion and turning is accomplished by monitoring the turn rate estimate from the INS. Debiasing the turn rate estimate from the INS is a further refinement reported in this paper. In order to make the system observable when debiasing turn rate, it is necessary to have an unbiased measurement of heading. The turn rate may be corrected directly, or the heading may be correct. Both approaches are presented. Two different approaches to debiasing the INS position estimates of the x and y coordinates are compared via simulation. These approaches are the method from [4] but with an observable system, switching between the method from [4] and a coordinated turn model. The switching mechanism is outlined in details and simulated under two different simulation scenarios to compare performance.

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2 Motion Models The vehicle’s motion is modeled using two motion models. Nearly straight line motion is modeled by a continuous white noise acceleration model [6]. This is the only model used in [4]. In the models considered in this paper, the vehicle turns are modeled using a coordinated turn model [5, 6]. When two motion models are used, models are selected by comparing the turn rate estimate from the INS to a threshold. The bias correction is incorporated by augmenting the state with bias correction terms and implemented in the system output equations as proposed in [4]. Doppler velocity logger (DVL) and heading sensors (compass) can provide independent and unbiased measurements so that there is sufficient information to estimate the bias correction terms in the augmented state. Model 1 only uses a continuous white noise acceleration model (nearly straight line motion) in x and y and DVL measurements. Model 1 is a linear model that corrects the bias in the x and y position [4]. Model 2 switches between the earlier continuous white noise acceleration model and a coordinated turn model [6]. It uses only DVL measurements to estimate the bias correction terms and also corrects the bias in the x and y position. A detailed description of all models appears later in this section. The external unbiased sources produce measurements every T seconds which is much larger than the time interval between INS estimates. Typically INS update rate is 200 Hz, while DVL update rate is 10 Hz maximum. The Kalman filter performing the debiasing updates every T seconds ( 0.1 s) when new measurements are available from the DVL and compass. T is assumed to be sufficiently larger than the update interval of the INS so that samples taken from the INS every T seconds are approximately independent. Following Sarma [4], the bias terms are modeled as constant values over an interval of length KT where K ≥ 1. Typically, K is much larger than 1. The debiasing interval KT value is usually in minutes, but it depends on INS specifications and trajectory of motion. At the end of each of these intervals [4], a hard reset of the Kalman filter is performed to enforce the pairwise constant bias model by setting the elements in the state covariance matrix corresponding to the bias correction terms to zero. Simulation experiments showed that the approach of hard resetting the Kalman filter degraded performance and led to short debiasing intervals. Experiments also showed that performance was dramatically improved by leaving the state covariance matrix unchanged between debiasing intervals. Rather than implementing the hard reset of the Kalman filter by zeroing the elements in the state covariance matrix corresponding to the bias correction terms every KT seconds as discussed in [4], two alternate approaches are presented in this paper. The first approach inflates the process noise for the bias correction terms as the end of an interval is approached and deflates them back to their normal values in the first part of the next interval. This allows the filter to easily adapt the bias

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correction terms without destroying the history contained in the state covariance matrix. The second approach is more in line with the intent in [4]. The second approach zeros out the rows and columns of inverse state covariance corresponding to the bias correction terms. This erases all prior information about the bias correction terms forcing the Kalman filter to re-estimate these terms. Note that the Kalman filter is implemented using a square root information filter, so zeroing particular rows and columns of the inverse state covariance matrix is straightforward.

2.1 Model 1 Let n represent the update index and m represent the index of the block over which the bias correction terms are nominally constant. The state vector at time t = nT equals & ' xn = x(nT ), x(nT ˙ ), y(nT ), y(nT ˙ ), μx (nT ), μy (nT )

(2.1)

where μx (nT ) and μy (nT ) are the bias correction terms for values of x(nT ) and y(nT ) produced by the INS. The elements in the state vector, xn , are unbiased. x is the matrix transpose of x. While the bias correction terms μx (nT ) and μy (nT ) are considered constant over blocks of length KT , their estimates are computed every update by the Kalman filter. The state vector, xn evolves according to the equations xn = A xn−1 + qn

(2.2)

where A is the state transition (state feedback) matrix and qn is the zero mean, process noise vector with covariance matrix Qnm . The covariance matrix depends on the update n and the block m because it is inflated and deflated at the end of and beginning of blocks. ql is independent of qn for all l = n. The state transition matrix, A, is given by the matrix ⎡

⎤ Ax 02×2 02×2 A = ⎣02×2 Ay 02×2 ⎦ 02×2 02×2 I2

(2.3)

where Ax = Ay =

  1T , 0 1

0k×l is the k × l zero matrix, and Ik is the k dimensional identity matrix.

(2.4)

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The process noise covariance matrix equals ⎤ 02×2 σx2 Qx 02×2 ⎥ ⎢ 02×2 = ⎣ 02×2 σy2 Qy ⎦ 02×2 02×2 σμ2 (n, m)T I2 ⎡

Qnm

(2.5)

where  Qx = Qy =

 T 3 /3 T 2 /2 , T 2 /2 T

(2.6)

σx2 and σy2 scale the process noise for the x and y coordinates. For the first method of resetting the bias terms in the state, σμ2 (n, m) scales the process noise for the bias correction terms and changes a state earlier around the end/beginning of a bias correction interval. To define this process noise scaling for the bias terms, let τm− = n − mK

(2.7)

τm+ = (m + 1)K − n.

(2.8)

and

Then

σμ2 (n, m)

=

σμ2

⎧ −(τ − )2 /2γμ ⎪ ⎪ ⎨cμ e m −(τm+ )2 /2γμ

cμ e ⎪ ⎪ ⎩1

if0 ≤ τm− andτm+ ≤ lμ , if0 ≤ τm+ andτm− ≤ lμ ,

(2.9)

otherwise.

This inflates the process noise variance for the bias correction terms using a Gaussian window parameterized by lμ and γμ . The scaling constant, cμ , sets the maximum inflation of σμ2 when the update, n, equals a bias interval start, mK. The window allows the process noise to smoothly ramp up from σμ2 to cμ σμ2 over lμ updates and then smooth ramp back down over the next lμ updates. For the second method of resetting the bias terms in the state, the process noise scaling is constant, so σμ (n, m)2 = σμ2 .

(2.10)

The measurement equation equals zn = Bnm xn + rn

(2.11)

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where rn is the zero mean measurement noise vector with covariance R. rl is independent of rn for all l = n. The measurement vector at time nT is given by  zn =

zins (nT ) zdvl (nT )

 (2.12)

where zins (nT ) and zdvl (nT ) are the vectors of measurements from the INS and DVL, respectively. The measurements from the INS are zins (nT ) = [xins (nT ), x˙ins (nT ), yins (nT ), y˙ins (nT )] .

(2.13)

The DVL measurements equal zdvl (nT ) = [x˙dvl (nT ), y˙dvl (nT )] .

(2.14)

The output matrix, Bnm , equals Bnm

  Bins (n, m) = Bdvl

(2.15)

where ⎡

⎤ 1 0 0 0 (n − mK)T 0 ⎢0 1 0 0 ⎥ 1 0 ⎥ Bins (n, m) = ⎢ ⎣0 0 1 0 0 (n − mK)T ⎦ 0001 0 1

(2.16)

and  Bdvl =

010000 000100

 (2.17)

where m = 0, 1, . . . is the debiasing interval index. It is straightforward to show that the observability matrix obtained from A and Bnm has rank 6, so the system is observable. The measurement covariance matrix, R, is the diagonal matrix % $ R = diag rx2 , rx2˙ , ry2 , ry2˙ , dx2˙ , dy2˙

(2.18)

where ra2 and db2 represent the measurement variances for component a from the INS and component b from the DVL, respectively. If the covariance matrices produced by the INS are available then, these could replace the first four rows and columns of R; however, this is not always desirable. If rx2˙ and ry2˙ are significantly smaller than dx2˙ and dy2˙ , then the Kalman filter will ignore the DVL measurements, and no bias

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correction will occur. Often, it is necessary to inflate the variances corresponding to the INS measurements so that the DVL measurements strongly influence the state estimate.

2.2 Model 2 Let s represent the discrete model selection variable. When s equals 1, the continuous white noise acceleration model is used. When s equals 2, the coordinated turn model is used. s equals 1 when the turn rate, ω, is less than some threshold (e.g., 1 degree per second); otherwise s equals 2. A stochastic model switching algorithm such as the interacting multiple model (IMM) [6] could be used instead, but since the variance on the turn rate from the INS is very small, the deterministic approach described here is used. For Model 2, the state vector is expanded to include turn rate: & ' xn = x(nT ), x(nT ˙ ), y(nT ), y(nT ˙ ), ω(nT ), μx (nT ), μy (nT )

(2.19)

where μx (nT ) and μy (nT ) are the bias correction terms for values of x(nT ) and y(nT ) as before and ω(nT ) is the down-sampled turn rate from the INS. The state vector xn evolves according to the equations xn = A(s) xn−1 + qn

(2.20)

where A(s) is the state transition matrix selected by the discrete variable, s, and qn is the zero mean, process noise vector with covariance matrix Qnm . qn is independent of ql for all n = l. Like Model 1, the process noise covariance depends on n and m because it is inflated and deflated at the boundaries of debiasing intervals as in the first model. The state transition matrix for nearly linear motion (s = 1) is essentially the same as in Model 1, but it has an additional row and column to account for the turn rate, ω : ⎡

⎤ Ax 02×2 02×3 A(s = 1) = ⎣02×2 Ay 02×3 ⎦ . 03×2 03×2 I3

(2.21)

A drift model is used for the turn rate and the bias correction terms. The transition matrix for coordinated turn model (s = 2) equals   Aturn 04×3 A(s = 2) = 03×4 I3

(2.22)

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where [6] ⎤ 0 −ω−1 (1 − cos[ωnT ]) 1 ω−1 sin[ωnT ] ⎥ ⎢0 cos[ωnT ] 0 − sin[ωnT ] ⎥. =⎢ −1 −1 ⎦ ⎣0 ω (1 − cos[ωnT ]) 1 ω sin[ωnT ] 0 sin[ωnT ] 0 cos[ωnT ] ⎡

Aturn

(2.23)

The drift model for turn rate and the bias correction terms remains the same. The process noise covariance matrix is the same for both values of s and is nearly identical to the one for Model 1. The only difference is the addition of an additional row and column for turn rate: ⎡ ⎤ σx2 Qx 02×2 02×1 02×2 ⎢0 ⎥ 02×2 ⎢ 2×2 σy2 Qy 02×1 ⎥ Qnm = ⎢ (2.24) ⎥. ⎣ 01×2 01×2 σω2 T ⎦ 01×2 02×2 02×2 02×1 σμ2 (n, m)T I2 σω2 is the process noise variance scale for the turn rate. σμ (n, m) implements the same process noise inflation scheme described by (2.9) at the boundaries of the bias correction interval or is the constant in (2.10) depending on which resetting scheme is being used. The measurement equation has the same form as (2.11), but the measurement vector is expanded to include the turn rate, ω, from the INS. The measurement vector from the INS now becomes zins (nT ) = [xins (nT ), x˙ins (nT ), yins (nT ), y˙ins (nT ), ωins (nT )] .

(2.25)

The DVL measurements in zdvl (nT ) are unchanged from (2.14). The part of the output matrix, Bnm , corresponding the INS now has an additional row for the turn rate. So, the output matrix, Bins , corresponding to the INS equals ⎡

⎤ 1 0 0 0 0 (n − mK)T 0 ⎢0 1 0 0 0 ⎥ 1 0 ⎢ ⎥ ⎢ ⎥ Bins (n, m) = ⎢0 0 1 0 0 0 (n − mK)T ⎥ . ⎢ ⎥ ⎣0 0 0 1 0 ⎦ 0 1 00001 0 0

(2.26)

The part of Bnm corresponding to the DVL, Bdvl , is largely unchanged from (2.17), but it needs an additional column of zeros to account for turn rate: Bdvl =

  0100000 0001000

(2.27)

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It is straightforward to show that the observability matrices constructed from A(s = 1) and Bnm or A(s = 2) and Bnm have full rank. The measurement covariance, R, is increased by one row and column to accommodate the turn rate as well and becomes % $ (2.28) R = diag rx2 , rx2˙ , ry2 , ry2˙ , rω2 , dx2˙ , dy2˙ where rω2 is the variance of the turn rate from the INS.

3 Kalman Filter As in [4], a standard linear Kalman filter performs the bias correction for Model 1 from the previous section. However, the coordinated turn model used in Model 2 requires an extended Kalman filter [5, 6]. Due to the extreme differences in the measure variances and the process noise for the difference components of the models, standard implementations for the Kalman filter and the extended Kalman filter are numerically unstable in this application. Consequently, a square root information form of both filters is implemented using the methods described in [7, 8]. Standard implementations of the Kalman filter and the extended Kalman filter are presented first for clarity followed by the square root information filter. The first step of the Kalman recursion is called the time update. The time update of the Kalman filter and the extended Kalman filter predicts the state estimate for the current update for all the models using xn|n−1 = A(s)xn−1|n−1

(3.1)

where xn−1|n−1 is the previous state estimate from the filter and xn|n−1 is the predicted state estimate. For the continuous white noise acceleration model, the Kalman filter computes the covariance for the predicted state estimate using Pn|n−1 = A(s = 1) Pn−1|n−1 A(s = 1) + Qnm

(3.2)

where Pn−1|n−1 and Pn|n−1 are the covariance matrices for xn−1|n−1 and xn|n−1 , respectively. For the coordinated turn model (s = 2), the covariance of the predicted state, Pn|n−1 , is calculated by Pn|n−1 = (∇A) Pn−1|n−1 (∇A) + Qnm

(3.3)

where ∇A is the gradient matrix of A(s = 2) [6]. The gradient matrix for Model 2 equals

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⎤ Aturn ∂A∂ωturn 04×2 ∇A = ⎣ 01×4 1 01×2 ⎦ 02×4 02×1 I2 ⎡

(3.4)

where the gradient of Aturn with respect to the turn rate equals

⎡

∂Aturn = ∂ω

⎤ x(2) ω−2 (ωT cos[ωT ] − sin[ωT ]) − x(4) ω−2 (ωT sin[ωT ] + 1 − cos[ωT ]) ⎢ ⎥ −x(2) T sin[ωT ] − x(4) T cos[ωT ] ⎢ ⎥ ⎣x(2) ω−2 (ωT sin[ωT ] + cos[ωT ] − 1) + x(4) ω−2 (ωT cos[ωT ] − sin[ωT ])⎦ . x(2) T cos[ωT ] − x(4) T sin[ωT ] (3.5) The second step of the Kalman filter and extended Kalman filter is called the measurement update and corrects the predicted state with the new measurement. The new state estimate equals xn|n = (I − Gn Bnm ) xn|n−1 + Gn zn

(3.6)

where the gain matrix Gn equals Gn = Pn|n−1 Bnm S−1 n .

(3.7)

The matrix Sn is the innovation covariance matrix and equals Sn = Bnm Pn|n−1 Bnm + R.

(3.8)

The covariance of the new state estimate, xn|n , equals Pn|n = (I − Gn Bnm ) Pn|n−1 .

(3.9)

Since the nonlinear parts of the coordinated turn model are confined to the state transition matrix, A(s = 2), the measurement step of the Kalman filter and the extended Kalman filter are identical. Consequently, (3.6) and (3.9) work for all the models described in the previous section. The square root information filter from [8] and [7] propagates the state estimate, xn|n , and an inverse square root matrix of the state covariance, Pn|n . The discussion presented here is a summary of Section 7 from [8]. The square root covariance matrix used in this paper is defined by Pn|n = Pn|n (Pn|n ) 1/2

1/2

(3.10)

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where Pn|n is a lower triangular matrix and (Pn|n ) is an upper triangular matrix. 1/2

1/2

Pn|n and (Pn|n ) are the Cholesky factors of the matrix Pn|n . The inverse state covariance matrix then is equal to 1/2

1/2

−1/2

−1/2

 P−1 n|n = (Pn|n ) Pn|n −1/2

(3.11)

−1/2

where Pn|n is a lower triangular matrix and (Pn|n ) is an upper triangular matrix. −1/2

The square root information filter propagates the upper triangular matrix (Pn|n ) . The time update of the square root information filter begins by whitening the prior state estimate using the prior inverse square root covariance matrix: −1/2

x¯ n−1|n−1 = (Pn−1|n−1 ) xn−1|n−1 .

(3.12)

The time update for the inverse square root state covariance matrix is computed first by taking the QR decomposition of an appropriate matrix. Let the matrix, T, be defined as  −1/2   ) 0 (Q T= (3.13) PA PA where (Q−1/2 ) is the upper triangular Cholesky factor of the process noise covariance matrix, Q. The matrix PA depends on the model and the switching state, s. For Model 1 and Model 2 when s = 1, −1/2

PA = (Pn−1|n−1 ) A(s = 1).

(3.14)

For Model 2 when s = 2 (coordinated turn model), the matrix PA equals −1/2

PA = (Pn−1|n−1 ) ∇A.

(3.15)

The matrix T is factored into a orthogonal matrix, U, and an upper triangular matrix, C, using the QR decomposition so that U C = [U1 , U2 ]

  C11 C12 = T. 0 C22

(3.16)

If N is the length of the state vector x, then U and T are 2N × 2N matrices, U1 and U2 are 2N × N matrices, and C11 , C12 , and C22 are all N × N matrices. The −1/2 predicted inverse square root covariance matrix, (Pn|n−1 ) , equals −1/2

(Pn|n−1 ) = U2 .

(3.17)

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The whitened predicted state estimate, x¯ n|n−1 , is given by x¯ n|n−1 =

U2





0N ×1

.

x¯ n−1|n−1

(3.18)

The unwhitened predicted state estimate is obtained by multiplying the whitened predicted state by the predicted square root covariance matrix. The measurement update for the square root information filter is largely performed by taking the QR decomposition of the matrix -

−1/2

(Pn|n−1 ) M= (R−1/2 ) Bnm

. (3.19)

where (R−1/2 ) is the upper triangular Cholesky factor of the measurement covariance matrix R. The QR decomposition of M yields the orthonormal matrix, V, and the upper triangular matrix, D, such that   D V D = [V1 , V2 ] 1 = M D2

(3.20)

where the matrices V1 , V2 , D1 , and D2 are all N × N matrices. The whited state estimate, x¯ n|n , equals x¯ n|n =

V1



x¯ n|n−1 (R−1/2 ) zn

 (3.21)

The corresponding inverse square root matrix of the state covariance matrix equals −1/2

(Pn|n ) = D1 .

(3.22)

The second resetting scheme is easily implemented by zeroing out the columns −1/2 of (Pn|n ) that correspond to the bias correction terms in the state vector. Finally, the unwhitened or “colored” state estimate is obtained by multiplying by the square root matrix of the state covariance matrix: xn|n = (Pn|n ) x¯ n|n . 1/2

(3.23)

A complete explanation and derivation of this square root information filter measurement update is given in Section 7 of [8].

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4 Examples Two simulated scenarios were generated to demonstrate the performance of the debiasing filters. In the first scenario, the vehicle’s path looks kind of like a “u” on its side and is shown in Fig. 1. The vehicle’s path in the second scenario forms a square and is shown in Fig. 2. The vehicle’s starting point is marked with a green square. The end point is marked with a red circle. In the second scenario, the vehicle moves around the square counter clockwise, so the vehicle passes its starting position before stopping. In both scenarios, the time between updates, T , is constant and equals 1/250. The two scenario each contain 1,22,250 updates, or equivalently, last 489 s. In both examples, the first 2500 updates (10 s) are to discarded the transient response of the INS.

Vehicle Motion

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Fig. 1 The vehicle actual trajectory in Scenario 1. The green square marks the start of the run, and the red circle marks the end of the run

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Fig. 2 The vehicle actual trajectory in Scenario 2. The green square marks the start of the run, and the red circle marks the end of the run. The vehicle moves counterclockwise

The two scenarios were generated using a framework for strapdown inertial navigation systems developed at the Applied Research Laboratory (ARL), Warminster, Pennsylvania. The framework was developed in Simulink and has been used to model inertial navigation units and test navigation augmentation algorithms such as map-matching and velocity correction. The data generated included the ground truth model position, orientation, velocity, and orientation rates. The ground truth data was then perturbed at the sensor level to simulate an inertial measurement unit (IMU) that included bias repeatability, stability, scale factor repeatability, cross-axis error, etc. The INS output integrates the accelerometer and gyros measurements in the wander frame and provides position and orientation. The INS is implemented using an error state Kalman filter. Simulated DVL measurements are also perturbed with noise and a small bias to match actual DVL specifications. The ground truth model output, the INS output, and the DVL measurements are all time-tagged and exported to the debiasing Kalman filter described in this article. The parameter values for the models described previously used to process the two scenarios are given here in roughly the order that they appeared previously. Optimization of the debiasing interval, KT , will be discussed afterward. Common

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parameters between the four models were the same for each model so that the performance comparisons were consistent. The one exception to this is the measurement variance of the turn rate from the INS.

4.1 Model 1 Parameters The process noise power spectral densities σx2 and σy2 for the x and y coordinates, respectively, were equal to σx2 = σy2 = 0.82 = 0.64

(4.1)

for all four models. The procedure used to select these values for the process noise spectral density is described in the Appendix. The three parameters defining the power spectral density, σμ (n, m), of the process noise in (2.9) for the bias correction terms μx , μy , and μω were as follows: σμ2 = 10−8 ,

(4.2)

cμ = 100,

(4.3)

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γμ = (20/6)2 .

(4.5)

and

The measurement noise variance for the x and y coordinates equaled rx2 = 100

(4.6)

ry2 = 100

(4.7)

and

respectively. These are inflated to force the Kalman filter to trust the DVL measurements. Similarly, the velocity component variances were artificially high as well: rx2˙ = 1

(4.8)

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and ry2˙ = 1.

(4.9)

The DVL measurement variances equaled dx2˙ = 4 × 10−6

(4.10)

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(4.11)

and

The difference between the INS and DVL velocity measurement variances insures that the velocity estimates from the INS are ignored which allows the bias in position and velocity to be corrected. For Model 1 the initial state vector for the Kalman filter was set equal to the first measurement vector from the INS for the Cartesian coordinates, while the two bias correction terms were initialized to zero: x0|0 = [zins (0), 0, 0] .

(4.12)

The covariance corresponding to the initial state vector equaled % $ P0|0 = diag 104 , 102 , 104 , 102 , 10−2 , 10−2 .

(4.13)

As a result the Kalman filter rapidly forgets the initial state.

4.2 Model 2 Parameters As stated earlier, the parameters common to Models 1 and 2 are the same, so only the additional parameters required by Model 2 are given in this subsection. The turn rate threshold used to switch from the continuous white noise acceleration model to the coordinated turn model was set to 1 degree (π/180 radians). The turn rate process noise power spectral density equaled σω2 = 3.34 × 10−3 .

(4.14)

The turn rate process noise must be large enough so that the filters used in Model 2 can track turn rate accurately. The method used to select the value for σω2 is described in the Appendix. The measurement variance for the turn rate equaled

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rω2 = 1.96 × 10−10 .

(4.15)

The initial state value corresponding to the turn rate was set to ωins (0). The corresponding value in the initial state covariance matrix equaled 4 × 10−2 .

4.3 Optimizing the Debiasing Interval Since the bias correction terms are modeled as approximately constant over intervals of KT , the selection of this interval significantly affects performance. Intuitively, KT should be roughly equal to the duration of the long straight runs in an event because these parts of a run best fit the bias correction model. The simulations will demonstrate that this is true for the second method of resetting the Kalman filter (zeroing parts of the inverse state covariance matrix) and the behavior of the first method (process noise inflation) is more nuanced. This may indicate that the process noise was not inflated enough at the end of the debiasing intervals. The value of the debiasing interval, KT , was optimized by performing a two-pass grid search to find the minimum mean square error over the entire run. The first pass of the grid search started with an initial value of KT = 10 s and increased with steps of 10 s. The first pass terminated when KT exceeded the time of the run. The second pass evaluated the mean square error over the interval [Ko T − 10, Ko T + 10] using a step size of 1 s where Ko T was the value from the first pass that minimized the mean square error. Results of this optimization on the two scenarios for Model 2 are shown in Figs. 3, 4, 5, and 6. Results for the other three models are not shown because the mean square error curves are nearly identical. While at first it may be surprising that there is little difference in the shape of the mean square error curves as a function of the debiasing interval for the four models, it makes sense because the error is driven by the run geometry. The minima in Figs. 3, 4, 5, and 6 correspond to the duration of the legs in the two runs. This is particularly clear with the bias correction term parts of the inverse state covariance matrix zeroed in Figs. 4 and 6. In Fig. 4, the minimum mean square error occurs for a debiasing interval of 122 s which is approximately the duration of the longest two legs in Fig. 1. The same debiasing interval is nearly the minimum in Fig. 6, but due to the symmetry of the square in Fig. 2, it is more advantageous to use the entire run as the debiasing interval so that biases for motion in opposite directions cancel out. In Fig. 3 the minimum mean square error for Model 2 on Scenario 1 was 166.35 m2 for a debiasing interval of 476 s which is essentially the length of the run. The minimum mean square error in Fig. 4 equaled 105.45 m2 for a debiasing interval of 122 s. For Scenario 2, the minimum mean square error in Fig. 5 equaled 42.21 m2 for a debiasing interval equal to 475 s which is essentially the entire run. The minimum mean square error for Fig. 6 equals 42.16 m2 with a debiasing interval of 474 s.

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Table 1 The mean square error in meters2 for the vehicle’s xand y position over the entire run of Scenarios 1 and 2 achieved by both models after optimizing the debiasing interval. The debiasing interval for Scenario 1 was 476 and 122 s for process noise inflation and zeroing the inverse state covariance matrix, respectively. The debiasing for Scenario 2 was 474 and 475 s for process noise inflation and zeroing the inverse state covariance matrix, respectively

Model 1 2

Scenario 1 Process noise inflation 166.35 166.35

Zero inverse covariance 105.45 105.45

Scenario 2 Process noise inflation 42.22 42.22

Zero inverse covariance 42.16 42.16

4.4 Results The data from the two scenarios was run through the two models using both methods of resetting the filter at the end of the debiasing interval. The mean square error position error (i.e., x and y coordinates) is shown in Table 1. The table shows there is no measurable difference in performance between both models; however, the results show that resetting the filter by zeroing elements of the inverse state covariance matrix that correspond to the bias correction terms produces better results. This may be because process noise is not inflated enough in the process noise inflation method. Figures 7 through 11 compare the estimates obtained by Model 2 on Scenario 1 by zeroing the inverse state covariance matrix to estimates from the INS. The effect of zeroing of the inverse state covariance matrix is clearly visible in Fig. 11. Comparative figures for Model 2 on Scenario 2 using the same resetting technique appear in Fig. 12 through 16. In Fig. 16, the effect of resetting the bias interval is visible as well, but since the bias estimation interval is nearly the same as the run length, it appears almost at the end of the run. More importantly, Fig. 16 shows that the symmetry of Scenario 2 removes most of the bias in position. Consequently, the bias estimates in Fig. 16 are nearly zero for most of the run (Figs. 8, 9, 10, 11, 12, 13, 14, 15, and 16).

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Fig. 7 Estimates of the vehicle’s position and velocity for Scenario 1 from the INS (blue) and Model 2 (green) zeroing the inverse state covariance matrix at the end of the debiasing interval. The true position is shown in red in the top graph

Fig. 8 The difference between the vehicle’s position and velocity estimates in Scenario 1 obtained by the INS and Model 2 when it zeros the inverse state covariance matrix at the end of the debiasing interval

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Fig. 11 Estimates of the bias in the vehicle’s position in the x and y directions for Scenario 1 obtained by Model 2 zeroing the inverse state covariance matrix at the end of the debiasing interval (upper plot). The standard deviation of the bias estimate is shown in the lower plot. The effect of zero in the inverse state covariance matrix is clearly visible in the lower plot

5 Summary This article extended INS debiasing method developed in [4] using a Kalman filter switching mechanism between a continuous white noise acceleration model and a coordinated turn model. In addition to debiasing the x and y coordinates using unbiased measurements from a DVL, methods to debias turn rate and heading were explored using unbiased compass measurements. In all, both models were described starting with a version of Sarma’s original model that is observable. The second model uses turn rate from the INS to switch between nearly straight line motion and turning. Both models were implementing using a square root information form of the Kalman filter [7, 8] for numerical stability. The bias correction terms are part of the Kalman filter’s state vector and are modeled as constant for some interval. At the end of this interval, the Kalman filter must be reinitialized. Two methods of reinitialization were used: process noise inflation and zeroing the inverse state covariance matrix. The process noise inflation method smoothly inflates and deflates the process noise spectral power density for the debiasing elements in the state vector at the end of a debiasing interval and the beginning of the next debiasing interval. Zeroing the inverse state covariance matrix

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Fig. 12 Estimates of the vehicle’s position and velocity for Scenario 2 from the INS (blue) and Model 2 (green) zeroing the inverse state covariance matrix at the end of the debiasing interval. The true position is shown in red in the top graph

Fig. 13 The difference between the vehicle’s position and velocity estimates in Scenario 2 obtained by the INS and Model 2 when it zeros the inverse state covariance matrix at the end of the debiasing interval

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Fig. 16 Estimates of the bias in the vehicle’s velocity in the x and y directions for Scenario 1 obtained by Model 2 zeroing the inverse state covariance matrix at the end of the debiasing interval (upper plot). The standard deviation of the bias estimate is shown in the lower plot. The effect of zero in the inverse state covariance matrix is clearly visible in the lower plot

sets the rows and columns in the inverse state covariance matrix corresponding to the debiasing elements in the state vector to zero. The results from the previous section show that zeroing the inverse state covariance matrix produces better results. This may be because the process noise for the debiasing terms was not inflated enough to fully reset the Kalman filter. The debiasing interval is unknown, but the results show that it is related to the longest linear leg in a scenario. The debiasing interval was optimized using a twopass grid search that minimizes the mean square error in position over an entire run. Because the optimal debiasing interval (in the sense of minimizing the mean square error) is related to the longest linear leg in a scenario, the debiasing interval was independent of the models. Optimal debiasing intervals were determined for two scenarios and were found to depend on the method used to reinitialize the Kalman filter at the end of a debiasing interval. The two scenarios were used to evaluate the mean square error of the models using these optimal values for the debiasing interval for both methods of reinitializing the Kalman filter. The results show that there is no measurable difference in the mean square error in position for both models; however, reinitializing the filter by zeroing the appropriate rows and columns of the inverse state covariance matrix worked better. Performance differences between the models may not be measurable

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in the two scenarios because the scenarios are short. Much longer scenarios probably are needed to allow more error to build up in the INS to differentiate the models.

Appendix: Setting the Process Noise Power Spectral Density On page 270 of [6], the authors recommend setting the process noise such that changes in x(nT ˙ √ ) and y(nT ˙ ) over the sampling interval, T , are on the order of √ σx T and σy T , respectively. To turn this statement into practical √ choices for σx and σy , the variability of the DVL measurements divided by T needs to√be studied. Figure A.1 shows the DVL measurements of x˙ and y˙ divided by T from approximately 260 to 330 s of Scenario 2. What is important is the random deviation about the mean because the velocity is an input to the filters. Figure A.1 suggests reasonable values for σx and σy are between 0.5 and 1.5. The DVL data from Scenario 1 yields the same results, so the chosen value of 0.8 is reasonable. Setting the turn rate process noise, σω , requires a different approach because filters in Model 2 need enough bandwidth to accurately follow the turn rate during turns. The performance of Model 2 tended to lag when only the random deviation in the turn rate process noise during straight legs was taken into account. In order to accurately follow the turn rate during turns, the process noise needs to be large

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Fig. A.2 A least squares line fit to the turn rate at the beginning of the first turn in Scenario 2. The blue circles are turn rate measurements from the INS. The green line is the least squares fit to the INS turn rate measurements

enough to catch the changing turn rate. Figure A.2 shows a segment from Scenario 2 at the start of the first turn along with a least square linear fit to the INS turn rate 2 . Consequently, measurements. The slope of the line is 3.66 × 10−3 radians/second √ −3 the turn rate process noise, σω , was set to 3.66 × 10 / T or 5.78 × 10−2 .

References 1. J.C. Kinsey, R.M. Eustice, L.L. Whitcomb, in Proceedings of the 7th IFAC Conference on Manoeuvring Control of Marine Craft Lisbon (2006), pp. 1–13 2. P. Groves, Principles of Navigation Systems (Artech House, Inc., Boston, 2015) 3. O. Hegrenaes, O. Hallingstad, IEEE J. Oceanic Eng. 36(2), 316 (2011) 4. A. Sarma, in IEEE/ION Position, Location, and Navigation Symposium (Monterey, 2014), pp. 136–146 5. G.A. Watson, W.D. Blair, in Proceedings of the SPIE 1698, Signal and Data Processing of Small Targets (1992), pp. 236–247 6. Y. Bar-Shalom, X.R. Li, T. Kirubarajan, Estimation with Applications to Tracking and Navigation: Theory, Algorithms, and Software (Wiley, New York, 2001) 7. B.D.O. Anderson, J.B. Moore, Optimal Filtering (Prentice Hall, Englewood Cliffs, 1979) 8. P.L. Ainsleigh, Parameter estimation in dynamical models for application in signal classification. Technical Memorandum 00–014, Naval Undersea Warfare Center Division Newport, Newport (2000)

Index

A Acoustic impedance, 16, 72 Acoustic modeling, 3 Acoustics finite and boundary element, 65 numerical results and discussion hydrodynamic stagnation point on a rigid body, 75–80 Piston source, 75–80 point acoustic source, 74–75 uniform homogeneous flow, 74–75 technical papers, 65 theory boundary conditions, 71–72 far-field pressures, 69–71 finite element formulation, 72–74 hydrodynamic mean flow, 67–68 inhomogeneous flow region, 68–69 velocity potential and pressure, 68–69 α − β filter, 95, 96, 121 Anisotropic materials, 5, 245, 246, 249, 250, 255 Antenna measurements, 5 Antennas engineering anisotropic dielectric materials, 245–246 metamaterials, 250–255 patch, 250 slotted cylinder, 246–250 fields produced, simple antenna, 224–227

operating above the ocean surface air-sea interface, 236–240 electrical properties of seawater, 234–236 overwater range, 240–244 properties directivity and gain, 229–231 The Friis equation, 231–232 input impedance, 227–229 radiation resistance, 227–229 radio technology, 224 types, 232–234 wireless communication, 223 Autonomous underwater vehicles (AUVs), 277 AUVs, see Autonomous underwater vehicles (AUVs)

B Bandlimited signal, 48, 55–56, 62 Beam pattern, 3, 70, 74, 78–80 Bessel function, 5, 179, 237 Bias estimation, 296 Bio-inspired model biomimetic systems and processes, 139 engineering, 4 flow-induced vibrations, 140 Naval applications, 139–140 seal whiskers, 144–159 technological challenges, 139 VIV, 140–144

© This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2021 A. A. Ruffa, B. Toni (eds.), Recent Trends in Naval Engineering Research, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health, https://doi.org/10.1007/978-3-030-64151-1

305

306 C Cagniard method convolution method, 48–49 cylindrical coordinates, 49–50 impulse response, 50 MFP, 47–48 multiple-image Green’s function, 52 numerical models, 47 Pekeris shallow-water case, 53 single-frequency, 51 acoustic models, 47, 48 step-function response, 51–52 time domain, 48 travelling waves, 50–51 Cavity ventilation, 4 Cellular solid, 7, 9, 11, 12, 14, 21, 41 CFD, see Computational fluid dynamics (CFD) Complementary error function, 5, 163, 182–185, 201 Computational fluid dynamics (CFD), 154–157 Computational methods dynamic dispersion surfaces, 36–38 finite element implementation, 31–36 homogenization, 29, 31 spatial-temporal analog, 29, 30 mathematical and, 3 static, 24–29 Convolution bandlimited signals, 55–56, 60 Fourier synthesis method, 48, 53 time-domain procedure, 3

D Damping, 20–21 and isolation characteristics, 4 properties, 130 ratio vs. dynamic viscosity, 130, 131 vibration, 123 Differential cavity ventilation cross-sectional view, 84 with gas, 83–85 local pressure, 83 pressure coefficient, 85 results apparatus, 85 force produced, 91, 92 hysteresis, 93–94 installed setup, 85, 86 length scale, 93 minor differences, 86 preliminary investigation, 91 scaling, 91

Index Digitally-designed, 9, 12, 14, 16, 19, 41 Doppler velocity logger (DVL) INS, 278 measurements, 277, 279, 282–284 DVL, see Doppler velocity logger (DVL)

E Effective coverage, 207–212 constraints, 5, 204 priori target information, 203 simulation results, 219 symmetric lateral range curve, 205 target tendency distribution, 207 Emergent behaviors, 257–260

F Fabrication methods, 3, 7, 10 additive manufacturing, 16–17 other methods hollow microlattice, 18–19 self-propagating photopolymerization, 17–18 and parent materials, 10 FD, see Finite difference (FD) FE, see Finite element (FE) Finite difference (FD), 23 Finite element (FE) and boundary element methods, 65, 80 computational domain, 76 method, 3 formulation, 72–74 free-wave propagation, 31–36 normalized magnitude, 78 point acoustic source, 74 stiffness trends, 13 Finite volume (FV), 23, 154, 157 Foam bend-dominated, 13 dielectric constant, 254 fluid-saturated aluminum, 126–127 metallic, 123 open-cell aluminum, 124, 125 20 PPI aluminum, 130 stochastic open-cell, 12 transient excitation, 4 FV, see Finite volume (FV)

G Gamma function, 5, 165–168, 177, 188 Gaussian distribution

Index functional integrals, 190–193 weighted Gaussian integrals, 193–201 Gaussian integrals, 167–168, 191, 193–201

H Hydrodynamic control, 3, 66 mean flow, 67–68 stagnation, 74–80 Hydrofoil, 4, 83–85 Hypergeometric function, 5, 163, 167, 187, 188, 195, 197

I Inertial navigation system (INS) examples debiasing interval, 293–296 model 1 parameters, 291–292 model 2 parameters, 292–293 results, 296–302 low-order Kalman filter, 278 motion models model 1, 280–283 model 2, 283–285 underwater vehicle, 278 vehicle’s position, 277 INS, see Inertial navigation system (INS) Integral identities complementary error function, 182–185 computer algebra system, 163 exponential terms applications, 174–182 method of exhaustion, 169 power substitution, 170–174 Gamma function, 165–167 logarithm, 185–186 multidimensional Gaussian integrals, 167–168 multiple higher-dimensional, 164 multivariate Gaussian distributions, 190–201 power substitution variants, 186–190 sense of Riemann, 163 Interval length, 4, 95, 96, 102–104, 109, 121, 122

K Kalman filter, 285–288 INS, 278 sensors, 277 square root information filter, 280 Kalman smoother

307 block tridiagonal system, 96 continuous-time kinematic models, 95 derivation, 96–98 solution approach state vector one-dimensional, 102–107 two-dimensional, 107–120 two special cases state vector one-dimensional, 99–100 two-dimensional, 100–102

L Lattice material, 9, 11, 12, 19, 41 Lift production cross-sectional view, 84 with gas, 83–85 local pressure, 83 pressure coefficient, 85 results apparatus, 85 force produced, 91, 92 hysteresis, 93–94 installed setup, 85, 86 length scale, 93 minor differences, 86 preliminary investigation, 91 scaling, 91 ventilated foil, 85, 87–90 Locally resonant lattice materials acoustics, 39–40 vibrations, 40–41

M Mathematical sciences information starved, 2 multivariate/multidimensional, 1 numerical simulations, 6 RHS, 4 steady-state hydrodynamic field, 3 ultrametricity, 2 VIV, 5 wireless communications, 5 Metallic foam metamaterials, 123 open-cell, 124 petroleum jelly-filled, 135 transmissibility, 131 Metamaterials, 250–255 background information, 123 elastic modulus, 124, 125 properties, 124

308

Index

Metamaterials (cont.) geometry, 123 literature review, 126 microlattice, 14 open-cell aluminum, 123, 124 steady-state vibration experimental and analytical results, 128–131 experimental setup, 126–128 stiffness, 125 transient vibration, 132–136 Microlattice materials computational methods, 23–38 influential factors connectivity, 11–13 phase material, 11 relative density, 11 lattice materials, 38–41 nomenclature, 7–9 property evaluation, 19–23 MILP, 5, 204, 206, 213, 219 Minimum uniform coverage effective coverage, 207–212 factors, 203 MILP optimization, 204 related work, 205–207 simulation description, 214–217 results, 217–219 spaced parallel tracks, 204 target information, 203 track spacing algorithm, 212–213 Multi-agent system distribution of emergent groups Niwa model for dynamic groups, 261–264 non-performers on stable group sizes, 264–268 emergent group behavior, 258–260 individual behavior, 257 kinematic groups of individuals Couzin model, 269–271 non-performers on non-informed followers, 271–274 mathematical ecologists, 257 Multifunctional metamaterial, 14, 41

P Parallel search track, 205 Pekeris shallow-water environment, 4 Pekeris waveguide environmental parameters, 58 frequency-independent fluid-fluid, 58–59 image theory numerical example, 59–62 numerical example, 59–62 single-frequency image, 58 Performance analysis air-sea interface, 234 anisotropic material, 250 fabrication methods, 10 fixed-lag smoother, 96 homogeneous sensor, 5 microlattice, 16 primary structure, 41 priority-based strategy, 206 sensor, 208, 210, 212, 214–218 simulation scenarios, 278 Periodic microstructure, 23 Power substitution, 170–174, 182, 186–191 variants, 186–190 Pressure distribution coefficient, 85 cylindrical coordinates, 49 far-field, 69–71 inhomogeneous flow region, 68–69 spherical wave, 22 structural stability, 14 structure-borne vibrations, 38 Property evaluation acoustic absorption, 21–23 damping, 20–21 mechanical properties, 19–20 physical properties, 19

N Naval engineering, 2, 3 Naval interests structural and Buoyancy, 14–15 vibrations and acoustics, 15–16 Noise power spectral density, 303–304

S Seal whisker array, 144, 145 CFD, 154–156 fluid-structure interaction, 145 sea lion, 144, 145

R Radio-wave propagation, 231 Right-hand side (RHS), 4, 102–104, 107, 113–115, 119, 120 Resilience, 260

Index undulation geometry, 157–159 vibrational responses, 146 water tunnel experiments, 146–154 Sectional coverage, 5, 204 Similitude source, 48, 57, 61 Sommerfeld integral, 238 Square-root filtering, 280, 285–289, 299 Stretch-dominated, 12–14 Subsonic flow, 65 Supercavitation, 83 Surface wave propagation, 5, 238, 239, 241, 255

T Target tendency distribution, 204, 206, 207, 212, 213, 215–217 Tracking index, 4, 96, 98, 99, 102, 104, 121, 122 Transient modeling convolution bandlimited signals, 55–56 quasi-monochromatic bandwidth, 56–57 Fourier synthesis method, 53 Transient vibration experimental and analytical results, 133–136 setup, 132, 133 metallic foam, 125 Tridiagonal Toeplitz system, 102, 107

V Ventilated cavitation, 85–87, 91, 94 Vibration, 40–41 and acoustics, 15–16 damping and isolation characteristics, 4 flow-induced, 140 saturating foams, 126 and shock, 126

309 steady-state experimental and analytical results, 128–131 experimental setup, 126–128 suppression, 15 transient experimental and analytical results, 133–136 experimental setup, 132, 133 VIV, 140–144 Viscosity, 11, 126, 129–131, 136 VIV, see Vortex-induced vibration (VIV) Vortex-induced vibration (VIV) fluid dynamics, 146 impact on engineering, 141–142 mitigation in Navy applications, 142–143 physics, 140–141 positive feedback loop, 140 reduction in nature, 143–144 shed vortices, 5 vibration amplitude, 158 whisker geometry models, 5 W Water tunnel experiments, 146–154 Weighted generalized Gaussian function, 5 complementary error function, 182–185 computer algebra system, 163 exponential terms applications, 174–182 method of exhaustion, 169 power substitution, 170–174 Gamma function, 165–167 logarithm, 185–186 multidimensional Gaussian integrals, 167–168 multiple higher-dimensional, 164 multivariate Gaussian distributions, 190–201 power substitution variants, 186–190 sense of Riemann, 163 Weighted multivariate Gaussian, 201