Advances in Trefftz Methods and Their Applications [1st ed.] 9783030528034, 9783030528041

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23 Carlos Alves · Andreas Karageorghis Vitor Leitão · Svilen Valtchev  Editors

Advances in Trefftz Methods and Their Applications

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SEMA SIMAI Springer Series Volume 23

Editors-in-Chief Luca Formaggia, MOX-Department of Mathematics, Politecnico di Milano, Milano, Italy Pablo Pedregal, ETSI Industriales, University of Castilla-La Mancha, Ciudad Real, Spain Series Editors Mats G. Larson, Department of Mathematics, Umeå University, Umeå, Sweden Tere Martínez-Seara Alonso, Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain Carlos Parés, Facultad de Ciencias, Universidad de Málaga, Málaga, Spain Lorenzo Pareschi, Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Ferrara, Italy Andrea Tosin, Dipartimento di Scienze Matematiche “G. L. Lagrange”, Politecnico di Torino, Torino, Italy Elena Vázquez-Cendón, Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, A Coruña, Spain Paolo Zunino, Dipartimento di Matemática, Politecnico di Milano, Milano, Italy

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Carlos Alves • Andreas Karageorghis • Vitor Leitão • Svilen Valtchev Editors

Advances in Trefftz Methods and Their Applications

Editors Carlos Alves Mathematics, CEMAT Instituto Superior Técnico, University of Lisbon Lisboa, Portugal Vitor Leitão Instituto Superior Técnico University of Lisbon Lisboa, Portugal

Andreas Karageorghis Mathematics and Statistics University of Cyprus Nicosia, Cyprus

Svilen Valtchev CEMAT Instituto Superior Técnico University of Lisbon Lisboa, Portugal

ISSN 2199-3041 ISSN 2199-305X (electronic) SEMA SIMAI Springer Series ISBN 978-3-030-52803-4 ISBN 978-3-030-52804-1 (eBook) https://doi.org/10.1007/978-3-030-52804-1 © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed 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

Preface

The use of Trefftz methods to solve boundary value problems for partial differential equations dates back to the work of Erich Trefftz (1888–1937) and his seminal paper “A Counterpart of the Ritz Method” (in German “Ein Gegenstück zum Ritzschen Verfahren”), published in 1926. The general idea behind Trefftz methods is the following: Consider a generic boundary value problem (P)

 Du = 0 Bu = g

in , on ∂,

where D is a linear differential operator acting in the domain  and B is a linear operator acting on the corresponding boundary ∂. Using a set of particular solutions φ1 , . . . , φn , that satisfy the homogeneous equation, Dφj = 0 (j = 1, . . . , n), an approximate solution of problem (P) may be obtained by taking the linear combination un =

n 

a j φj .

j =1

The unknown coefficients aj , j = 1, . . . , n, may be determined after fitting the boundary data Bun ≈ g.

v

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Preface

When problem (P) is well-posed, the bound ||u − un || ≤ C ||g − Bun ||∂ , in appropriate norms, allows to control the approximation error in the domain by estimating the error on the boundary. One way of determining the unknown coefficients aj , j = 1, . . . , n, in the Trefftz approximation is to satisfy the boundary condition on a set of collocation points x1 , . . . , xm , Bun (xi ) = g(xi )

(i = 1, . . . , m)

which yields the linear system Ba = g, where a is the vector of the unknown coefficients a1 , . . . , an , B is a dense matrix with entries given by Bij = Bφj (xi ), and g = (g(x1 ), . . . , g(xm )). Taking m = n, this approach corresponds to the solution of an interpolation system, and if m > n, the system is solved in the least squares sense, usually considering Tikhonov regularization with parameter τ ≈ 0, (τ I + B B)a = B g. A regularization scheme, such as Tikhonov’s, is sometimes required to overcome the fact that the Trefftz approach often leads to linear systems that are illconditioned. Even for low dimensional matrices, the condition number grows rapidly and a Tikhonov pseudo-inverse technique may be required to overcome the instabilities. It should be also mentioned that simple interpolation (using m = n) yields another type of problem, illustrated by Runge’s example—the function Bun is equal to g at a finite number of points, but the two functions differ considerably in norm. Thus, the decrease of the residual r = g − Ba does not always ensure a good fit of the boundary data. Instead, a different set of boundary points zi = xi should be used to evaluate the approximation, and the error terms g(zi ) − Bun (zi ) should be calculated. An alternative approach is to fit the boundary condition in a weak sense, Bun , vi ∂ = g, vi ∂

(i = 1, . . . , m),

using an inner product (usually in the L2 space) and a set of test functions v1 , . . . , vm .

Preface

vii

˜ with entries This results in a similar type of matrix, B,   B˜ ij = Bφj , vi ∂ . In the limiting case when the vi are mollifiers tending to the Dirac functionals δxi , centered at xi (with respect to the boundary measure), this would correspond exactly to the collocation identities     Bφj , vi ∂ → Bφj , δxi ∂ = Bφj (xi ). Also, with Dirichlet boundary conditions, when the test functions are the particular solutions, this leads to the entries   B˜ ij = φj , φi ∂ , and we have the continuous least squares version of the data fitting, with g˜i = g, φi ∂ . The method of fundamental solutions (MFS) is a particular case of the class of Trefftz methods, in which shifted fundamental solutions of the differential operator are used as particular solutions, Dφj = δyj with the Dirac delta distributions centered at point sources exterior to the domain, ¯ y1 , . . . , yn ∈ / . A central question regarding the MFS approach is the establishment of density results that justify the completeness of the basis φ1 , . . . , φn , . . . , usually with a dense sequence of source points y1 , . . . , yn placed on an artificial boundary outside the domain and surrounding the physical boundary of the problem. Initial density results were obtained by Kupradze and Alekside in 1964, but the use of the MFS as a numerical method can only be traced back to Mathon and Johnston in 1977. The MFS can also be viewed as a variant of the indirect boundary element method (IBEM). However, the IBEM requires numerical integration on the artificial boundary, while the MFS approximations do not require the existence of such integrals. For instance, the Dirichlet data g = φz with the source point z located between the true and the artificial boundary cannot be represented by an IBEM integral, but it may be accurately recovered by the MFS due to the relevant density results. Trefftz methods, and in particular the MFS and its variants, have been applied to a vast number of engineering problems, especially in the last two decades. In

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Preface

this brief presentation, we mention just the generic formulation for homogeneous linear partial differential equations (PDEs), but there have been significant advances concerning the application of the method to other cases, such as nonhomogeneous and nonlinear PDE problems. These methods are truly meshless in the sense that they have the advantage of completely avoiding the need to set up a mesh, even on the boundary, in contrast to boundary integral equation methods. Trefftz methods present advantages in several engineering applications such as inverse problems and free boundary problems, where the domain is unknown. For such problems, other numerical methods would require an intensive remeshing approach. Trefftz methods are also known to perform remarkably well for regular domains and regular data in boundary value problems, achieving exponential convergence. However, they may also, under certain circumstances, exhibit numerical instability and lead to poorly conditioned linear systems. Recent variants of the MFS have been shown to circumvent some of these difficulties, but open problems related to the positioning of the source points, among others, still persist. This book is divided into ten chapters that illustrate recent advances in Trefftz methods and their application to engineering problems. The first eight chapters are devoted to the MFS and related variants, while the last two chapters are devoted to related meshless engineering applications. The chapters included in this book are based on research results presented at the 9th Conference on Trefftz Methods (also 5th Conference on Method of Fundamental Solutions) that took place in Lisbon, July 29–31, 2019. In the first chapter, Chen, Cheung, and Ling use an extension of the MFS for nonhomogeneous equations (called the domain MFS) to solve surface partial differential equations. Akhmouch, Naji, Duan, and Fu, in the second chapter, use the MFS to solve steady magnetohydrodynamic flow problems, by considering the coupled equations in the velocity and induced magnetic field. In the third chapter, Gaspar considers an automatic way of defining the source points in the MFS using a quadtree cell system controlled by the boundary of the domain. Liu and Sarler, in the fourth chapter, consider an improved non-singular MFS to solve problems in 2D anisotropic linear elasticity. In the fifth chapter, Barbeiro and Serranho address the numerical simulation of wave propagation and induced displacements in the human retina, for elastography imaging, using the MFS. Martins, in the sixth chapter, considers an inverse source problem for the Brinkman system of equations, proving uniqueness of recovery, uses the MFS to simulate the direct problem data, and applies a superposition of complex shear waves to reconstruct the body force term. In the seventh chapter, Marin investigates the application of the fading regularization method in conjunction with the MFS to a Cauchy problem in 2D anisotropic heat conduction.

Preface

ix

Bozzoli, Mocerino, Cattani, Vocale, and Rainieri, in the eighth chapter, address an inverse heat flux problem in coiled ducts using the reciprocity functional approach, which requires the solution of a sequence of auxiliary problems, solved by the MFS. In the ninth chapter, Moldovan, Cismasiu, and Freitas consider a hybrid Trefftz method using finite elements that combine favorable features of the finite and boundary element methods. Fu, Li, and Zhang, in the tenth chapter, present a meshless collocation method, using generalized finite differences, using longitudinal waves which are Trefftz functions, to calculate the acoustic band gaps of 2D liquid phononic crystals with square and triangular lattices. The editors would like to dedicate this book to Professor Ching Shyang Chen (better known as C. S. Chen) on the occasion of his 67th birthday, celebrated during the conference, and in recognition of his pioneering work on the MFS. Lisboa, Portugal Nicosia, Cyprus Lisboa, Portugal Lisboa, Portugal March 4 2020

Carlos Alves Andreas Karageorghis Vitor Leitão Svilen Valtchev

Contents

Solving Partial Differential Equations on Surfaces with Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Meng Chen, Ka Chun Cheung, and Leevan Ling

1

Solving Magneto-Hydrodynamic (MHD) Channel Flows at Large Hartmann Numbers by Using the Method of Fundamental Solutions . . . . . Latifa Akhmouch, Ahmed Naji, Yong Duan and Zhuojia Fu

13

Application of Quadtrees in the Method of Fundamental Solutions Using Multi-Level Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Csaba Gáspár

41

Method of Fundamental Solutions Without Fictitious Boundary for Anisotropic Elasticity Problems Based on Mechanical Equilibrium Desingularization.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Qingguo Liu and Božidar Šarler The Method of Fundamental Solutions for the Direct Elastography Problem in the Human Retina .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Sílvia Barbeiro and Pedro Serranho

59

87

Identification and Reconstruction of Body Forces in a Stokes System Using Shear Waves .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103 Nuno F. M. Martins MFS-Fading Regularization Method for Inverse BVPs in Anisotropic Heat Conduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 121 Liviu Marin Non-intrusive Estimate of Spatially Varying Internal Heat Flux in Coiled Ducts: Method of Fundamental Solutions Applied to the Reciprocity Functional Approach . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 139 Andrea Mocerino, Fabio Bozzoli, Luca Cattani, Pamela Vocale, and Sara Rainieri xi

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Unified Hybrid-Trefftz Finite Element Formulation for Dynamic Problems . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 157 Ionu¸t Drago¸s Moldovan, Ildi Cisma¸siu, and João António Teixeira de Freitas Acoustic Bandgap Calculation of Liquid Phononic Crystals via the Meshless Generalized Finite Difference Method . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 189 Zhuo-Jia Fu, Ai-Lun Li, and Han Zhang

About the Editors

Carlos Alves graduated from the University of Lisbon and received his PhD in Applied Mathematics from Ecole Polytechnique, France. He is Professor in the Department of Mathematics of Instituto Superior Técnico, University of Lisbon. He has published more than 50 papers on numerical analysis, meshless methods, and inverse problems in partial differential equations. As an editor, he has published two books in Springer, and he is on the editorial board of Applied Mathematics and Computation, Inverse Problems in Science and Engineering, and Engineering Analysis with Boundary Elements. Andreas Karageorghis completed both his undergraduate and graduate studies at the University of Oxford. After holding positions at the University of Kentucky, the University of Wales, and Southern Methodist University, he joined the Department of Mathematics and Statistics of the University of Cyprus where he is currently a Professor. He has published over 150 research papers in international journals and his research interests include numerical algorithms and scientific computing. Vitor Leitão graduated from Instituto Superior Técnico of the Technical University of Lisbon, Portugal, and received his Ph.D. from the University of Portsmouth, UK. He is currently an Associate Professor of Civil Engineering at Instituto Superior Técnico of the Technical University of Lisbon. He is the author or co-author of around 100 research works and co-editor of 8 conference proceedings books and special issues, and he is on the editorial board of Engineering Analysis with Boundary Elements. His current research interests include meshless methods, in particular radial basis functions and fundamental solutions for structural mechanics applications.

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About the Editors

Svilen S. Valtchev graduated from the Instituto Superior Técnico of the University of Lisbon, where he was also a teaching assistant, and received his Ph.D. in Mathematics from the same university. He is currently an Assistant Professor at the Polytechnic of Leiria and a Researcher at Center for Computational and Stochastic Mathematics of the University of Lisbon. He has authored and coauthored more than 20 research papers and book chapters on meshfree methods for partial differential equations, with applications in acoustic and elastic wave propagation.

Solving Partial Differential Equations on Surfaces with Fundamental Solutions Meng Chen, Ka Chun Cheung, and Leevan Ling

Abstract The aim of this paper is to present partial differential equations (PDEs) on surface to the community of methods of fundamental solutions (MFS). First, we present an embedding formulation to embed surface PDEs into a domain so that MFS can be applied after the PDEs is homogenized with a particular solution. Next, we discuss how the domain-MFS method can be used to directly collocate surface PDEs. Some numerical demonstrations were included to study the effect of basis functions and source point locations. Keywords Laplace-Beltrami · Embedding method · Collocation · Dual reciprocity method

1 Partial Differential Equations on Surfaces In this paper, we focus on second-order elliptic partial differential equations (PDEs) posed on some sufficiently smooth, connected, and compact surface S ⊂ Rd with bounded geometry. Without loss of generality, we assume dim(S) = d − 1, a.k.a., S has co-dimension 1. We denote the unit outward normal vector at x ∈ S as n = n(x) and the corresponding projection matrix to the tangent space of S at x as P(x) = [P1 , . . . , Pd ](x) := Id − nnT ∈ Rd×d ,

(1.1)

where Id is the d × d identity matrix.

M. Chen Department of Mathematics, Nanchang University, Nanchang, Jiangxi, China K. C. Cheung NVIDIA AI Technology Center (NVAITC), NVIDIA, Santa Clara, CA, USA L. Ling () Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 C. Alves et al. (eds.), Advances in Trefftz Methods and Their Applications, SEMA SIMAI Springer Series 23, https://doi.org/10.1007/978-3-030-52804-1_1

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M. Chen et al.

Example 1 Let S be the unit circle in R2 . Then, we have n = (x, y)T = x for every x ∈ S and the projection matrix is  P(x) =

1 − x 2 −xy −xy 1 − y 2



 =

y 2 −xy −xy x 2

 .

Unlike standard PDEs posed in some bounded domains with flat geometry, curvatures of our computational domain S plays key roles in solution behaviours of the PDEs. The surface gradient ∇S can be defined in terms of the standard Euclidean gradient ∇ for Rd via projection P as ∇S := P∇

(1.2)

and the Laplace-Beltrami S operators (a.k.a. the surface Laplacian) can then be defined using the surface gradient operator by S := ∇S • ∇S .

(1.3)

Surface gradient is straightforward to compute, whereas the analytic expression of surface Laplacian involves derivatives of normal vector n. In [2], one can find the following formula: for any sufficient smooth function uS : S → R, we have  S uS =

d  d  ∂uS ∂ 2 uS  ∂ 2 uS trace(P · J (Pi )T ) + + Pii P , ij ∂xi ∂xi ∂xj ∂xi2

(1.4)

i,j =1 i=j

i=1

where J denotes the Jacobian operator, Pi and Pij denotes the i-th column and ij -th entries of the projection matrix P in (1.1) respectively. Example 2 Let S be the unit circle in R2 . The surface gradient operator takes the form  2    2  y −xy ∂x y ∂x − xy∂y ∇S = = . ∂y −xy∂x + x 2 ∂y −xy x 2 We can also compute the coefficients of surface Laplacian with simplifications based on the parametric equation x 2 + y 2 = 1 as follows trace(P · J (P1 )T ) = −xy 2 − x 3 = −x, trace(P · J (P2 )T ) = −x 2 y − y 3 = −y. Together, we have S = y 2 circle.

∂ ∂2 ∂2 ∂2 ∂ + x2 2 − x −y on the unit − 2xy 2 ∂x ∂x∂y ∂y ∂x ∂y

Solving Partial Differential Equations on Surfaces with Funda. . .

3

Any linear second-order elliptic differential operators on surfaces S ⊂ Rd can be expressed in the form of LS := aS S + bS • ∇S + cS ,

(1.5)

where aS , cS : S → R, and bS : S → Rd are bounded coefficients that can be functions of x ∈ S. The elliptic surface PDEs is then defined as LS uS = fS ,

(1.6)

for some sufficiently smooth right hand function fS , and, because S is closed, without any other conditions. Under some standard smoothness assumptions [14], we know (1.6) has classical solutions u∗S : S → R.

2 Traditional MFS Approach Since the surface PDE (1.6) is inhomogeneous, the traditional approach is to find a particular solution in order to obtain a homogeneous PDE. However, finding the particular solution to (1.6) is equivalent to solving the uniquely solvable PDE; see [2] for some intrinsic algorithms. This leaves no room for applying MFS. We now present an embedding framework to solve (1.6) by dual reciprocity method (DRM) [4, 9] and MFS. Our approach is based on the closest point method [13]. Let the closest point map be defined as cp(x) = arg inf x − x2 (Rd ) . x∈S

The differentiability of the closest point map is directly related to the smoothness of S, see [3]. For smooth S, there exits ε > 0 depending on the curvature of S so that the closest point map is well-defined in the narrow band domain

  := x ∈ Rd : inf s − x2 (Rd ) < ε ⊂ Rd . s∈S

The first aim is to embed (1.6) to another PDE in . To do so, we define the constantalong-normal extension operator En := En,S → so that, for every function wS : S → R, its extension has the property that w (x) := (En wS )(x) = wS (cp(x))

for all x ∈ .

(2.1)

L := a  + b • ∇ + c ,

(2.2)

Now we consider an embedding PDE to (1.5)–(1.6) L u = f

with

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where f = En fS , a = En aS , c = En cS , and b = En bS with componentwise extension. The governing equation (2.2) alone is not yet well-posed and we need the following embedding conditions ∂n u = 0

∂n(2)u := nT J (∇u )n = 0

and

on S,

(2.3)

to ensure the existence of unique solution. The connection between surface PDE (1.6) and embedding PDE (2.2)–(2.3) is that the restriction of the embedding PDE solution u on S coincides with the surface PDE solution uS . This is an immediate consequence of the following equalities [7]: ∇S u := ∇u − n∂n u

and

S u := u − HS ∂n u − ∂n(2) u

on S,

 where HS (x) = trace J (n)(Id − nnT ) . In the same article, readers can find Kansa-type algorithms for solving (2.2)–(2.3) directly. At this point, we have (2.2)–(2.3) posed in the narrow band domain (2.1), which is analogue to the standard elliptic PDEs and is DRM-MFS ready. We can decompose the embedding PDE into

(I ) L up = f ,

and

⎧ in , ⎨ L uh = 0 (I I ) ∂n uh = −∂n up on S, ⎩ (2) ∂n uh = −∂n(2)up on S.

Solving the above by some appropriate means, e.g., (I) by DRM and (II) by MFS, yields the embedding PDE solution u = uh + up , whose restriction is the surface PDE solution uS = u|S . An Dirichlet-type alternative to problem (II) is (I I  )



L uh = 0 in , uh − uh ◦ cp = up ◦ cp − up on ∂,

in which we require u = u ◦ cp on the whole boundaries ∂. When the width ε of the narrow band domain is small, we can easily see that (II’) is a finite difference approximations to the embedding conditions (2.3). This idea is essentially the orthogonal gradient method [12] and helps avoid differentiation to up .

2.1 DRM for Solving (I) Although problem (I) is posed in , our interest is only on S and, thus, it is sufficient to have interpolation conditions solely on S in DRM. Suppose we use nI interpolation points X = {x1 , . . . , xnI } on S to solve (I) by DRM. The DRM

Solving Partial Differential Equations on Surfaces with Funda. . .

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I I requires two couple sets of basis functions ({ψk }nk=1 , {ϕk }nk=1 ) that satisfy

L ψk = ϕk ,

1 ≤ k ≤ nI .

One can find some closed form formulas of analytic particular solution ψ in [5] of commonly used RBFs ϕ for some standard elliptic operators L . In cases of RBF symmetric interpolation, we want to center RBFs at the same set of points, i.e., ϕk = ϕ( · −xk ), and solve nI 

ak ϕ(xi − xk ) = f (xi ),

1 ≤ i ≤ nI .

k=1 I Once the coefficients a = {ak }nk=1 were obtained, the approximate particular solution is given as

Up =

nI 

ak ψk (·) =

k=1

nI 

ak ψ( · −xk ).

k=1

The computation cost here is dominated by the cost of solving an nI ×nI symmetric matrix system.

2.2 MFS for Solving (II) When applying MFS to (II), all collocation points for the two embedding conditions are on S and there is real role for . Let Y = {y1 , . . . , ynI I /2 } be a set of nI I /2 points on S. Going after a square system, one can simply put nI I source points Z = {z1 , . . . , znI I } in ∂ in order to avoid singularity. Let G denote the fundamental solution of L such that L G( · −z) = δ( · −z),

z ∈ Rd .

The standard approach is to numerically expand the approximate homogenous solution by Uh =

nI I 

λj G( · −zj ),

j =1

 = {λj }nI I : and solve the following collocation matrix system for λ j =1 

   nT ∇G(yi − zj ) −nT ∇ψ(yi − xk ) = λ a , nT J (∇G(yi − zj ))n −nT J (∇ψ(yi − xk ))n

(2.4)

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for 1 ≤ i ≤ nI I /2, 1 ≤ j ≤ nI I , and 1 ≤ k ≤ nI . Note that all differential operators here act upon the variable y and the right hand vector is computed based on the approximate particular solution Up . The computation cost here is dominated by the cost of solving an nI I × nI I asymmetric matrix system.

2.3 MFS for Solving (II’) When MFS is applied to solve (II’), four layers of points are required. First, we have to distribute collocations points on ∂. Here, we want ε as small as possible to minimize the finite difference error. Their respective closest points have to be identified so that we can evaluate uh ◦ cp, i.e., fundamental solutions, and up ◦ cp on S. A simpler approach is to put a set of nI I /2 points Y  = {y1 , . . . , ynI I /2 } on S. Then, we extend Y  by ±εn as in the orthogonal gradient method to generate collocation points on ∂, i.e., the set of nI I collocation points is given by Y = {y1 ± εn, . . . , ynI I /2 ± εn}. By construction, if we operate on sets, we have cp(Y ) = Y  and no closest point search is required. For convenience, let Y  =< Y  , Y  > be a set of nI I entries that is ordered in such a way that cp(yj ) = yj for all collocation points yj ∈ Y . In terms of source points placement, there is now a requirement on the fictitious boundary imposed from the narrow band domain, i.e., our choice of ε. Away from , we have to place the inner and outer source points to complete the set up of MFS. One can, of course, repeat the same data points extension technique to complete the job. When S is nonconvex, however, feasible choice of extension may be limited. For now, we assume the set of source points Z is fixed by some appropriate means. The collocation system to be solved is in the form of      = ϕ(yi − xk ) − ϕ(yi − xk ) a . G(yi − zj ) − G(yi − zj ) λ  The costs The approximate homogenous solution is given by (2.4) once we found λ. of (II) and (II’) are similar, but (II’) requires no numerical differentiations. The above algorithms inherit all limitations in the traditional MFS approach that heavily relies on our knowledge of the fundamental solution to L . Method in the next section will circumvent the problem.

Solving Partial Differential Equations on Surfaces with Funda. . .

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3 Using Fundamental Solutions for Collocations In this section, we focus on Kansa-type collocation method for solving surface PDEs in the form of (1.5)–(1.6). The idea is to analytically carry out some calculations, similar to our Example 2 above, so that we can collocate the surface operator directly. Our convergent analysis in [2] applies to kernels that satisfy 2 −m  c m (1 + ω22 )−m ≤ m (ω) ≤ C m (1 + ω2 )

for all ω ∈ Rd ,

(3.1)

for some constants 0 < c m ≤ C m and smoothness order m > d/2. Simply by restricting the global kernels m on S, we have a surface kernel m−1/2 := m|S ×S : S × S → R that reproduces Hm−1/2 (S) provided the smoothness assumption on S stated in [8, 11] are satisfied. Commonly used kernels in this class include Whittle-Matérnm−d/2 Sobolev kernels m (x) := x2 Km−d/2 (x2 ) and the family of compactly supported piecewise polynomial Wendland functions. The m > d/2 requirement is commonly seen in RBF theories. In both R2 and R3 , we must take m ≥ 2 if we insist on integer order smoothness. This means that H3/2 (S) is the largest solution space on which our theories applied. This is the motivation of using fundamental solutions as basis is a density result in [1]. In our notation, it reads as follows: let ϒk be the fundamental solutions of the modified Helmholtz equation −( − k 2 )ϒk ( · −z) = δ( · −z),

z ∈ Rd ,

and Sˆ ⊃ S be some sufficiently smooth artificial surface containing S. Then, ˆ span{ϒk ( · −s)|S : s ∈ S} is dense in H1/2 (S). This is beneficial for surface PDEs with low regularity, i.e., when there are near-singularity [6] on S. In other words, we are considering a collocation method using basis functions ⎧ 1 ⎪ ⎨ in 2D, K0 (kr) ϒk (r) := 2π exp(−kr) 1 ⎪ ⎩ in 3D, 4π r for some k > 0. As we do not require ϒk to satisfy any governing equations, one can drop the constant for simplicity. The dimension here should match with that of the embedding space, i.e., 2D and 3D basis are for solving PDEs on curve and surface respectively. This collocation approach comes with a computational cost of solving an nZ × nZ asymmetric matrix system.

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Example 3 Consider modified Helmholtz LS uS := (S − I )uS = fS on the unit circle. Define sets of source points Z ⊂  \ S and collocation points Y ⊂ S. By fixing some k > 0 and therefore a basis function ϒk , we can obtain a collocation system   LS ϒk (yi − zj ) α = f (yi ),

yi ∈ Y, zj ∈ Z

Z . Based on the results in Example 2, we know that for the unknown α = {αj }nj =1

LS = y 2

∂ ∂2 ∂2 ∂2 ∂ + x2 2 − x −y − I, − 2xy 2 ∂x ∂x∂y ∂y ∂x ∂y

(x, y) ∈ S,

which can be used to analytically evaluate LS ϒk ((x, y) − ·). The numerical solution is of the form US =

nZ 

αj ϒk ( · −zj ).

j =1

Note that the selection of wavenumber k in ϒk does not need to match with the wavenumber in LS , which may not even be of Helmholtz in general. Example 4 We now consider a modified Helmholtz LS uS := (S − I )uS = fS on the unit sphere. In this example, we shall use the 3D basis function ϒk (r) =

1 exp(−kr) . 4π r

A smooth solution uS = 10xyz + 5xy + z is considered; we aim to study the effects of wavenumber k and source points distribution. For nZ = {250, 500}, the set of nZ collocation points Y ⊂ S is quasi-uniformly distributed on the unit sphere. The set of source points Z is uniformly distributed on a sphere with radius R > 0. Taking advantage of the simple geometry, we simply generate Z by an extension of the set Y along normal direction by Z = RY . Figure 1 shows the resulting maximum error over a range of R for various values of k, i.e., basis functions. When R = 100 , source points are right on the surface and, as expected, we see error blow up nearby. When R < 100, i.e., source points are placed inside the unit sphere, the selection of wavenumber k has nearly no effect on the accuracy and only the value of nZ matters. Like the traditional MFS, nZ also affect the location of the Goldilocks zone: R ≈ 0.1 and R ≈ 0.2 for nZ = 250 and 500 respectively. The error profiles on the R > 100 sides is less consistent. In this example, we can see that large k has better conditioning but lower accuracy.

Solving Partial Differential Equations on Surfaces with Funda. . .

9

nZ = 250

nZ = 500 0.1

k =

0.1

1

1

2

2

5

5

L∞ error

L∞ error

k =

R

R

Fig. 1 Maximum error against source points location R on one side of S of collocation methods using modified Helmholtz fundamental solution with various wavenumbers k

nZ = 250 k =

0.1 1 2

L∞ error

5

direct greedy

R Fig. 2 Maximum error against source points location R on both sides of S of collocation methods using modified Helmholtz fundamental solution with various wavenumbers k

Example 5 It is commonly seen that MFS source points are placed on both sides of annulus domains. We repeat Example 4 with source points distributed as Z = {Rz1 , (2 − R)z1 , . . . , Rzn Z /2 , (2 − R)zn Z /2 }

0 0 is a fixed number, independent of N. That is, if N increases (the discretization is finer) then δ decreases (the sources are located closer to the original boundary R ). In other words, δ is proportional to the characteristic distance of the sources. Let u(N) be the approximate solution of the problem (1)–(2): u(N) (x) :=

N−1 

αj · (x − x˜j )

(7)

j =0

(x ∈ R , the closure of R ), where α := (α0 , α1 , . . . , αN−1 ) is the a priori unknown vector of coefficients. Introduce the operator A(N) mapping CN into the function space L2 (0, 2π) by A(N) α := u(N) |R , i.e. in polar coordiates: (A(N) α)(t) := u(N) (R, t) Standard calculations show that A(N) α can be expressed in terms of complex Fourier series by: (A(N) α)(t) = αˆ 0 log(R + δ) −

 |k| 1  1 R · αˆ k e−ikt , 2 |k| R+δ k=0

(8)

Quadtrees and Multi-Level Tools in the MFS

45

where αˆ denotes the discrete Fourier transform of the vector α (extended to the set of the integer numbers Z in an N-periodic way): N−1 

αˆ k =

αj e

2π ikj N

j =0

The coefficients can be calculated by taking the boundary condition (2) into account: A(N) α = u0

(9)

This equation has no solution in general, but can be solved in the sense of least squares by solving the Gaussian normal equations of (9): (A(N) )∗ A(N) α = (A(N) )∗ u0

(10)

where (A(N) )∗ A(N) is an N-by-N matrix. The Gaussian normal equations have a unique solution since A(N) is one-to-one (provided that R + δ = 1). In the orthonormal basis spanned by the vectors α (p) (p = − N2 + 1, − N2 + 2, . . . , N2 ), where (p)

αj

2π ipj 1 = √ e− N N

(j = 0, 1, . . . , N − 1),

(A(N) )∗ A(N) is a diagonal matrix. The diagonal elements (the eigenvalues of the matrix) are: ⎛ ((A(N) )∗ A(N) )0,0 = 2Nπ · ⎝(log(R + δ))2 +

 =0

1 4|N|2



R R+δ

2|N|

⎞ ⎠,

and, for k = 1, 2, . . . , N − 1: ((A(N) )∗ A(N) )k,k = 2Nπ ·

∞  =−∞

1 4|k + N|2



R R+δ

2|k+N|

The condition number of (A(N) )∗ A(N) can now be estimated easily. If the distance δ > 0 is fixed, independently of N, then the condition number increases expo 2N  R nentially with respect to N (due to the exponentially decreasing factor R+δ ). In contrast to this, if the sources are near-boundary sources, i.e. the relation (6) is satisfied, then the condition number increases much more moderately: cond((A(N) )∗ A(N) ) = O(N 2 ). This implies that the system (10) becomes moderately ill-conditioned when N increases. Nevertheless, this moderately ill-condi-

46

C. Gáspár

tioned character of the matrix still makes the usual iterative methods e.g. the (conjugate) gradient method slow. However, in the restricted finite dimensional operator (A(N) )∗ A(N) to the ‘highfrequency subspace’ spanned by the vectors α (p) , where |p| > N4 , the matrices remain uniformly well-conditioned, i.e. on this subspace, the condition number does not exceed a constant C, independently of N. Consequently, the (conjugate) gradient iteration damps the high-frequency error components much more efficiently than the low-frequency ones, since the speed of convergence of the (conjugate) gradient method depends on the condition number. This makes it possible to create two- and multi-level methods, where the smoothing procedure of the method is defined to be the classical (conjugate) gradient iteration. In practice, this means that one has to define fine-level sources (x˜0(F ), x˜1(F ) , . . . , (F ) x˜N−1 ∈ R+δ in our model problem) and coarse-level sources (x˜0(C), x˜1(C), . . . , x˜ (C) N 2

∈ R+2δ ). The approximate solution is defined as follows: uapprox(x) :=

N−1 

−1

N 2

(F ) αj (x

(F ) − x˜j ) +

j =0

−1 

(C)

(C)

αj (x − x˜j )

(11)

j =0

To enforce the Dirichlet boundary condition (2), the following system of equations has to be solved: N−1 

N 2 −1

αj(F ) (x

− x˜j(F ))

j =0

+



αj(C) (x − x˜j(C)) = u0 (x)

(12)

j =0

(x ∈ R ) in the sense of least squares. This can be done by splitting the system into a coarse-level subproblem: N 2 −1



(C)

(C)

αj (x − x˜j ) = u0 (x) −

j =0

N−1 

(F )

(F )

(13)

(C)

(C)

(14)

αj (x − x˜j )

j =0

and a fine level subproblem: N−1  j =0

N 2 −1

(F ) αj (x

(F ) − x˜j )

= u0 (x) −



αj (x − x˜ j )

j =0

The solution of the subproblems has to be repeated iteratively until convergence. Both subproblems should be solved in the sense of least squares, i.e. by solving the corresponding Gaussian normal equations. The benefit of this two-level approach is that the fine level subproblem can efficiently be solved by performing a few (conjugate) gradient iteration steps, since the (conjugate) gradient iteration damps the high-frequency error components much more efficiently. This significantly reduces the computational cost. The coarse level subproblems can be solved directly,

Quadtrees and Multi-Level Tools in the MFS

47

but introducing even coarser levels, the approach can be extended to a real multilevel technique in a straightforward way. Remark The above technique can be generalized to Neumann and mixed boundary conditions as well as to multiply connected domains without difficulty. For details, see Gáspár [13].

2.1 Source Definition Using Quadtrees Perhaps the most natural way to define near-boundary sources is to shift some boundary points in the outward normal direction with a distance δ > 0 (fine level) and 2δ, respectively (coarse level). However, if the shape of the domain is complicated and/or the domain is multiply connected, this may be uncomfortable. A completely automated strategy is the use of the quadtree subdivision procedure. Recall that the quadtree algorithm is a systematic subdivision of an initial square controlled by a finite set of points (controlling points). If the number of the controlling points contained in the actual square is more than a predefined value Nmin , and the level of subdivision is less than a predefined value Lmax , then the actual square is split into four congruent subsquares, and the subdivision process is recursively repeated also for the subsquares. The algorithm results in a cell system, the spatial density distribution of which follows the spatial density distribution of the controlling points, i.e. local refinements are generated in the vicinity of the controlling points in an automatic way. By performing some additional subdivisions, the quadtree cell system can be made regular; in this case the ratio of the neighbouring cell sizes is at most 2, which assures that no abrupt changes in cell sizes occur. See e.g. Gáspár [7] for details. In the presented method, the quadtree cell systems are controlled by the boundary of the domain . This is also the basis of the method of inner collocation points detailed in the next section. Note that the quadtree cell system gives a comfortable tool to create simple finite volume methods as well (embedded in a natural multi-level context), see Gáspár [7]. The external sources are now defined to be the centers of the external cells. The cell centers of bigger (resp. smaller) cells form the ‘coarser-level’ (resp. the ‘finerlevel’) sources. Thus, the spatial density distribution of sources decreases rapidly when going far away from the boundary. To illustrate the quadtree subdivision technique, two examples are shown. The first one is a domain, the boundary of which is a smooth, star-like curve parametrized by the following pair of functions: x(t) = 0.5 + 0.3 · (1 + 0.2 · sin(5t)) · cos(t) y(t) = 0.5 + 0.3 · (1 + 0.2 · sin(5t)) · sin(t)

(15)

where (0 ≤ t < 2π). Figures 1 and 2 show the boundary and the generated regular quadtree cell system.

48 Fig. 1 A smooth star-like domain

Fig. 2 Quadtree cell system generated by a smooth star-like domain

C. Gáspár

Quadtrees and Multi-Level Tools in the MFS

49

Fig. 3 An amoeba-like domain

The second example is a more complicated amoeba-shaped domain. The boundary is parametrized by the pair of functions: x(t) :=

1 6

  · 2.5 + (esin t · sin2 2t + ecos t · cos2 2t) · cos t

y(t) :=

1 5

  · 2.0 + (esin t · sin2 2t + ecos t · cos2 2t) · sin t

(16)

where (0 ≤ t < 2π). Figures 3 and 4 show the boundary and the generated regular quadtree cell system, respectively. Figures 5 and 6 show the external source locations generated by the quadtree algorithm. In both examples, the initial square is the unit square, and the maximal level of 1 subdivision is 8, i.e. the size of the finest (eight-level) cells is 256 .

2.2 Numerical Examples The above outlined multi-level method is illustrated through two examples. In both examples, the pure Dirichlet problem (1)–(2) is solved in different domains but with the same test solution: u(x, y) := sin

πy πx sinh , 2 2

(17)

50 Fig. 4 Quadtree cell system generated by an amoeba-like domain

Fig. 5 Star-like domain, external sources

C. Gáspár

Quadtrees and Multi-Level Tools in the MFS

51

Fig. 6 Amoeba-like domain, external sources

where the more familiar notations x, y are used for the spatial variables. In the first example, the Dirichlet problem (1)–(2) is considered in the smooth star-shaped domain parametrized by (15). In the second example, the domain was the amoebashaped domain parametrized by (16) (see also Figs. 1 and 3). The Dirichlet boundary conditions are consistent with the test solution (17). A two-level method was performed on different levels: Lfine , Lcoarse denote the level of the fine and the coarse cells, respectively, where the 0th level corresponds to the initial unit square. The maximal level is 8. The number of collocation points was always equal to 500. The coarse subproblem was solved directly, while on the fine level, 16 conjugate gradient steps were performed. The accuracy was characterized by the relative L2 -errors, i.e. by the ratio ||u − u0 ||L2 () ||u0 ||L2 () (expressed in %), where u denotes the approximate solution. The L2 ()-norms were approximated by the root mean square taken at the boundary collocation points. Table 1 shows these relative errors with different coarse and fine levels. Ncoarse (resp. Nfine ) denotes the number of sources at the coarse (resp. fine) level.

52

C. Gáspár

Table 1 Two-level method, based on external sources, accuracy. Test solution: (17) Star-shaped domain Lcoarse /Lfine Ncoarse /Nfine Rel. L2 -error (%) Amoeba-shaped domain Lcoarse /Lfine Ncoarse /Nfine Rel. L2 -error (%)

3/4 14/70 0.0303

4/5 70/124 7.62E − 6

5/6 124/240 2.72E − 5

6/7 240/486 3.02E − 5

3/4 19/69 0.0064

4/5 69/124 2.27E − 6

5/6 124/220 5.86E − 5

6/7 220/480 6.86E − 5

3 Near-Boundary Internal Collocation Points An alternative approach to define MFS-like methods is to locate the sources along the boundary . If the collocation points are located also along , this leads to (weakly or strongly) singular problems due to the singularity of the fundamental solution and its derivatives. As pointed out in the Introduction, a lot of special methods have been developed, but almost all of them are based on the proper evaluation of the appearing singular terms. Here another technique is presented which preserves the simplicity of the original MFS. The main idea is to define new collocation points in the interior of the domain, in the vicinity of the boundary. Thus, the problem of singularity is avoided and so is the automatic generation of sources. Instead, a new problem arises, namely, the proper definition of the values of the solution in the inner collocation points. If the inner collocation points are defined by shifting certain boundary points in inward normal direction, the MFS principle can be combined with classical finite difference schemes. For instance, if xk ∈  and nk is the outward normal unit vector at xk , then (xk − δnk ) is in the interior of  (for sufficiently small values of the distance δ > 0). Let u be the MFS-solution of the original problem: u(x) =

N 

αj (x − x˜j ),

j =1

as usual, where x˜1 , x˜2 , . . . , x˜N are the boundary source points. Performing a finite Taylor series expansion around the point (xk − δnk ), we have: u0 (xk ) = u(xk − δnk ) + δ ·

∂u (xk − δnk ) + O(δ 2 ), ∂nk

Quadtrees and Multi-Level Tools in the MFS

53

which defines the scheme: N  j =1

  ∂ αj (xk − δnk − x˜j ) + δ · (xk − δnk − x˜j ) = u0 (xk ) ∂nk

The scheme is of second order with respect to δ. Now, instead of the boundary condition, this equation is required for sufficiently many boundary points xk . For further details, see Gáspár [11]. ˜ ⊂ . If the values of In contrast to this approach, consider a smaller domain  ˜ (say, u) u along ∂  ˜ are calculated correctly, then it is sufficient to solve the new problem ˜ u = 0 in ,

u|∂ ˜ = u˜

in exactly the same way as in the previous section. Now the role of the nearboundary sources is played by the original boundary sources x˜1 , x˜2 , . . . , x˜N and the role of the boundary collocation points is played by the near-boundary inner collocation points. The only difference is that both the coarse and the fine level sources are located along the original boundary; the coarse level sources x˜ 1(C), x˜2(C), . . . , x˜ (C) N (F ) may form a subset of the fine level sources x˜1(F ), x˜2(F ) , . . . , x˜N . The approximate solution is again sought in the following form:

uapprox(x) :=

N  j =1

2

N

(F ) αj (x

−

(F ) x˜ j ) +

2 

(C)

(C)

αj (x − x˜ j )

(18)

j =1

which can be split into a coarse-level and a fine-level subproblem, cf. (13) and (14). The same iterative technique can be applied to this case. Again, the coarse level subproblems should be solved directly, while the fine level subproblems can (approximately) be solved by performing a few (conjugate) gradient iteration steps. The crucial point is to properly approximate the values of the solution at the inner collocation points. It should be pointed out that this can be regarded as a “black box”; in principle, an arbitrary (but meshless) technique can be applied, provided that it is sufficiently economical from computational point of view.

3.1 Collocation Points Definition Using Quadtrees The quadtree algorithm is suitable to define near-boundary internal points (without knowing the normal vectors) in a completely automatic way. This can be done by simply selecting the centers of the internal cells. As an illustration, Figs. 7 and 8 show the quadtree-defined cell centers for the previous examples, when the

54

C. Gáspár

Fig. 7 Smooth star-like domain, internal collocation points

Fig. 8 Amoeba-like domain, internal collocation points

original domain is a smooth star-shaped domain and an amoeba-shaped domain, respectively. For the internal collocation points, the use of the cell centers of some fixed level L is recommended, where L < Lmax , the maximal level of subdivision.

Quadtrees and Multi-Level Tools in the MFS

55

At the selected cell centers, the boundary conditions should be redefined. Here a multi-level solution method of the original problem is utilized based on a simple multi-level finite volume schemes defined on quadtree cell systems Gáspár [7]. Though the finite volume method is a domain-type method, it can be considered meshless, since the cell system generation as well as the whole solution process is fully automated and controlled by the boundary of the domain  only. The accuracy of the applied scheme is moderate; however, more accurate schemes can also be applied without difficulty. The essential difference of the MFS-based methods and the quadtree-based finite volume methods is that the latter ones produce the values of the approximate solution in predefined points (i.e. in the cell centers) only, while the MFS-based methods provide a simple formulation to be evaluated in arbitrary points.

3.2 Numerical Examples The above method based on the internal collocation points is illustrated through the same two test examples as in the previous section. The Dirichlet problem (1)–(2) is solved in a smooth star-shaped and an amoeba-shaped domain using the test solution (17) again, cf. Figs. 7 and 8. The maximal level of subdivision was 8 in both cases, and the inner collocation points are defined by the cell centers belonging to the 7th level. The number of source points was 8, 16, 32, 64, 128 and 256. The number of collocation points was constant, 434 in the case of the star-shaped domain and 451 in the case of the amoeba-shaped domain. The two-level technique was applied based on internal collocation points. The coarse subproblems were solved directly, while on the fine levels, 16 conjugate gradient steps were performed. The accuracy was characterized again by the following ratio: ||u − u0 ||L2 () ||u0 ||L2 () (in %; here u0 denotes the redefined boundary condition at the internal collocation points). The L2 ()-norm was approximated by the root mean square taken at the internal collocation points. Table 2 shows these relative errors with different coarse and fine levels having Ncoarse (resp. Nfine ) sources. Table 2 Two-level method, based on internal collocation points, accuracy. Test solution: (17) Ncoarse /Nfine Rel. L2 -error (%) Star-shaped domain Amoeba-shaped domain

8/16

16/32

32/64

64/128

128/256

5.7161 7.9572

1.8728 2.7699

0.4183 0.7732

0.0675 0.2366

0.0491 0.1931

56

C. Gáspár

Comparing the results of Tables 1 and 2, it can be seen that the method based on external sources is more accurate. The reason is the modest accuracy of the applied simple finite volume technique. Using a more accurate method to predict the values of the solution in the inner collocation points, however, the accuracy can be improved.

4 Summary and Conclusions The traditional version of the Method of Fundamental Solutions is revisited. The external near-boundary sources are defined in an automated way by using a quadtree cell system controlled by the boundary of the domain. The spatial density distribution of sources is rapidly decreasing when going far away from the boundary, thus giving a natural multi-level context for the MFS. The smoothing procedure of the multi-level method is a traditional (conjugate) gradient iteration. A similar multi-level technique has also been presented where the sources are located along the boundary, and the collocation points are defined in the interior of the domain by using quadtree subdivision again. The boundary condition was redefined at the inner collocation points applying a quadtree-based finite volume technique. The accuracy of the resulting methods are less than that of the traditional MFS, but it is still acceptable. However, due to the multi-level character of the method, the computational cost is more moderate (only a few iteration steps are performed at each level); moreover, the problem of direct solution of large linear systems with extremely ill-conditioned matrices is avoided. Acknowledgments The research for this paper was financially supported by the European Union and the Hungarian Government from the project ‘Intensification of the activities of HU-MATHSIN-Hungarian Service Network of Mathematics for Industry and Innovation’ under grant number EFOP-3.6.2-16-2017-00015.

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5. Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9, 69–95 (1998) 6. Fam, G.S.A., Rashed, Y.F.: A study on the source points locations in the method of fundamental solutions. In: Brebbia, C.A., Tadeu, A., Popov, V. (eds.) Proceedings of the 24th International Conference on the Boundary Element Method incorporating Meshless Solution Seminar (17– 19 June 2002 Sintra, Portugal). International Series on Advances in Boundary Elements, vol. 13, pp. 297–312. WIT, Southampton (2002) 7. Gáspár, C.: A meshless polyharmonic-type boundary interpolation method for solving boundary integral equations. Eng. Anal. Boundary Elem. 28, 1207–1216 (2004) 8. Gáspár, C.: Some variants of the method of fundamental solutions: regularization using radial and nearly radial basis functions. Cent. Eur. J. Math. 11(8), 1429–1440 (2013) 9. Gáspár, C.: A regularized multi-level technique for solving potential problems by the method of fundamental solutions. Eng. Anal. Boundary Elem. 57, 66–71 (2015) 10. Gáspár, C.: A multi-level method of fundamental solutions using quadtree generated sources. In: Proceedings of the 9th International Conference on Computational Methods, 6th–10th August 2018. ScienTech, Rome, Paper ID: 3227 (2018). (ISSN 2374–3948, online) 11. Gáspár, C.: The method of fundamental solutions combined with a multi-level technique. In: Dimov, I., Faragó, I., Vulkov, L. (eds.) Finite Difference Methods. Theory and Applications (LNCS), vol. 11386, pp. 241–249. Springer, Heidelberg (2019) 12. Gáspár, C.: A fast and stable multi-level solution technique for the method of fundamental solutions. In: Schweitzer, M.A., Griebel, M. (eds.) Meshfree Methods for Partial Differential Equations IX. Lecture Notes in Computational Science and Engineering (LNCSE), vol. 129, pp. 19–42. Springer, Heidelberg (2019) 13. Gáspár, C.: A multi-level solution technique for the method of fundamental solutions without regularization and desingularization. Eng. Anal. Boundary Elem. 103, 145–159 (2019) 14. Golberg, M.A.: The method of fundamental solutions for Poisson’s equation. Eng. Anal. Boundary Elem. 16, 205–213 (1995) 15. Gu, Y., Chen, W., Zhang, J.: Investigation on near-boundary solutions by singular boundary method. Eng. Anal. Boundary Elem. 36, 1173–1182 (2012) 16. Jopek, H., Kołodziej, J.A.: Application of genetic algorithms for optimal positions of source points in method of fundamental solutions. In: Bergen, B., De Munck, M., Desmet, W., Moens, D., Pluymers, B., Schueller, G.I., Vandepitte, D. (eds.) Proceedings of the Leuven Symposium on Applied Mechanics in Engineering 2008 (LSAME 08), and 5th International Workshop on Trefftz Methods (Trefftz 08). Katholieke Universiteit Leuven Department of Mechanical Engineering, pp. 229–239. Leuven, Belgium (2008) 17. Liu, Y.J.: A new boundary meshfree method with distributed sources. Eng. Anal. Boundary Elem. 34, 914–919 (2010) 18. Mitic, P., Rashed, Y.F.: Convergence and stability of the method of meshless fundamental solutions using an array of randomly distributed sources. Eng. Anal. Boundary Elem. 28, 143– 153 (2004) 19. Nishimura, R., Nishimori, K., Ishihara, N.: Determining the arrangement of fictitious charges in charge simulation method using genetic algorithms. J. Electrostat. 49, 95–105 (2000) 20. Poullikkas, A., Karageorghis, A., Georgiou, G.: The method of fundamental solutions for inhomogeneous elliptic problems. Comput. Mech. 22, 100–107 (1998) 21. Šarler, B.: Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions. Eng. Anal. Boundary Elem. 33, 1374–1382 (2009) 22. Schaback, R.: Adaptive numerical solution of MFS systems. In: Chen, C.S., Karageorghis, A., Smyrlis, Y.S. (eds.) The Method of Fundamental Solutions—A Meshless Method, pp. 1–27. Dynamic Publishers, Atlanta (2008) 23. Young, D.L., Chen, K.H., Lee, C.W.: Novel meshless method for solving the potential problems with arbitrary domain. J. Comput. Phys. 209, 290–321 (2005)

Method of Fundamental Solutions Without Fictitious Boundary for Anisotropic Elasticity Problems Based on Mechanical Equilibrium Desingularization Qingguo Liu and Božidar Šarler

Abstract In this chapter, an Improved Non-singular Method of Fundamental Solutions (INMFS) is developed for solving the 2D anisotropic linear elasticity problems. In the INMFS, the artificial boundary, present in the classical Method of Fundamental Solutions (MFS), is removed. The singularities are substituted by the normalized area integrals of the Fundamental Solution (FS) over small squares, covering the source points that intersect with the collocation points. The singularities of the fundamental traction are dealt with considering the mechanical equilibrium, calculated by the boundary integration of the forces on the considered body. This more appropriate approach avoids solving the problem two times as in Non-singular MFS (NMFS), developed by Liu and Šarler in 2014. The integral over a small disk in NMFS is replaced by a small square in INMFS, amenable to be solved analytically. The viability and superiority of INMFS in comparison with the MFS and the NMFS is assessed in details. The advantage of having no artificial boundary and the straightforward implementation of the INMFS on problems with different materials in contact is demonstrated. Keywords Anisotropic elasticity · Displacement and traction boundary conditions · Fundamental solution · Desingularization · Force balance · Bi-material

Q. Liu · B. Šarler () Laboratory for Simulation of Materials and Processes, Institute of Metals and Technology, Ljubljana, Slovenia Department for Fluid Dynamics and Thermodynamics, Faculty of Mechanical Engineering University of Ljubljana, Ljubljana, Slovenia e-mail: [email protected]; [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 C. Alves et al. (eds.), Advances in Trefftz Methods and Their Applications, SEMA SIMAI Springer Series 23, https://doi.org/10.1007/978-3-030-52804-1_4

59

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1 Introduction There are many materials which are studied through the linear elastic model, but are not even nearly isotropic. Examples are wood, composite and many biological materials. The mechanical properties of these materials differ in different directions. In other words, their stiffness depends on the direction in which the stress is applied. Materials with this direction dependence property are called anisotropic. The plane elasticity theory of isotropic materials has been well established [1]. Both the stress and displacement formulations have been successfully applied to solve various problems in [2, 3]. On the other hand, the theory of planar anisotropic elasticity is still an active research topic. The method of fundamental solutions (MFS) is a simple, accurate and efficient meshless method [4]. The basis of the classic MFS is the representation of the solution of homogeneous linear differential equation by a linear combination of the Fundamental Solutions (FS). The FS contains sets of sources and collocation points located outside and on the problem boundary, respectively. The MFS was first proposed in 1964 by Kupradze and Aleksidze [5] for boundary value problems. Then the MFS became increasingly popular in science and engineering problems, i.e. potential, Helmholtz and diffusion problems [6], Poisson equation [7], etc. [8–10]. The MFS can also be applied on the nonhomogeneous equations when combined with other methods [11]. Comprehensive reviews for the applications of the MFS are available in the literature [12, 13]. There are still some unresolved issues in implementation of MFS on problems defined in complicated domains, the large-scale problems, the optimal location of the sources, and so on. References [14–16] are dedicated to dealing with the mentioned challenges. However, all the proposed approaches are computationally expensive. The positioning of the sources to the actual boundary is receiving increasing attention and various modifications are available, like the boundary knot method (BKM) [17] with the non-singular general solution of the equation as the basis function, the singular boundary method (SBM) [18] based on the concept of origin intensity factors, etc. [19–22]. The Non-singular MFS (NMFS) based on Boundary Distributed Source Method (BDSM) [23] is presented by Liu and Šarler for 2D thermoelasticity [24], isotropic [25] and anisotropic elasticity [26], and 3D isotropic [27] problems. In NMFS, the singular FS is replaced by a normalization of the integral of FS over a circular disc or a sphere, covering the source points, in 2D or 3D problems, respectively. The method in [19], involving additional reference solutions, is used in NMFS to calculate the derivatives (required by the traction Boundary Condition (BC)) of the FS in the source points coincided with the collocation points. The NMFS is recently extended to solve problems of Darcy porous media flow with free and moving boundaries (Stefan problems) arising in systems with wetted and unwetted regions of porous media [28] and Stokes flow problems [29]. The MFS is not very stable because of the singularity

Method of Fundamental Solutions Without Fictitious Boundary for Anisotropic. . .

61

of the FS and geometric restrictions [30]. The NMFS is more stable than MFS, because the sources are overlapping with the collocation points. However, when the Neumann BC are present, two or three additional solutions are required to calculate the diagonal elements in 2D or 3D, respectively. Additional computing time is thus needed. Kim [31] proposed a simpler way of dealing with Neumann BC by suggesting that the boundary integration of the normal gradient of the potential should vanish. Liu and Šarler extended Kim’s approach [31] from the potential problems to 2D [32] and 3D [33] linear elasticity problems. The purpose of this paper is to improve the formulation of NMFS [26] for 2D anisotropic elasticity problems by considering the mechanical equilibrium. The non-singular anisotropic FS is in the present paper made more efficient by analytical integration over squares instead of numerical integration over circles, as in [26]. In Sect. 2, the governing equations are introduced, while in Sect. 3 INMFS procedure for 2D anisotropic elasticity problems is developed. In Sect. 4, two numerical test examples, including void problems and elastic/rigid inclusion problems are investigated. Comparison of the results with MFS, NMFS and analytical solutions is reported. The feasibility and the accuracy of INMFS is demonstrated. The conclusions and further research directions are included in the final section.

2 Governing Equations for 2D Anisotropic Elasticity Problems For keeping the consistent subscript of the 2D coefficients with 3D, we begin from 3D problem description. First, we have a 3D domain , surrounded by a boundary , filled with an anisotropic elastic solid. A 3D Cartesian coordinate system, described by base vectors (ix , iy ,iz ) and coordinates (px , py ,pz ), is used to mark the position of point p = px ix + py iy + pz iz . To simplify the calculations, we shall assume that (i) the solid is free of body forces, (ii) the thermal strains can be neglected. Under these conditions, the general equation of elasticity for the displacement u [34], is a result of the equilibrium of the stresses σ in the form 2

(p) Cςξ υτ ∂∂puςυ∂p = 0; τ

ς, ξ, υ, τ = x, y, z,

where Cςξ υτ are the components of the elastic stiffness matrix C [35].

(1)

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⎡

C ≡ Cςξ υτ

Cxxxx Cxxyy ⎢C ⎢ xxyy Cyyyy ⎢ Cyyzz ⎢C = ⎢ xxzz ⎢ Cxxyz Cyyyz ⎢ ⎣ Cxxxz Cxzyy Cxxxy Cxyyy ⎡ c11 c12 ⎢ c c22 12 ⎢ ⎢ ⎢ c13 c23 =⎢ ⎢ c14 c24 ⎢ ⎣ c15 c25 c16 c26

⎤ Cxxxz Cxxxy Cxzyy Cxyyy ⎥ ⎥ ⎥ Cxzzz Cxyzz ⎥ ⎥ Cxzyz Cxyyz ⎥ ⎥ Cxzxz Cxyxz ⎦ Cxyxz Cxyxy ⎤ c16 c26 ⎥ ⎥ ⎥ c36 ⎥ ⎥. c46 ⎥ ⎥ c56 ⎦ c66

Cxxzz Cxxyz Cyyzz Cyyyz Czzzz Cyzzz Cyzzz Cyzyz Cxzzz Cxzyz Cxyzz Cxyyz c13 c23 c33 c34 c35 c36

c14 c24 c34 c44 c45 c46

c15 c25 c35 c45 c55 c56

(2)

Cςξ υτ satisfy the full symmetry conditions Cςξ υτ = Cξ ςυτ , Cςξ υτ = Cςξ τ υ , Cςξ υτ = Cυτ ςξ .

(3)

Here, Hooke’s law is used to connect the strains ε and stress σ σ = Cε, The relation of the strains εςξ , displacement is εςξ

1 = 2

(4)

ς, ξ = x, y, z and the gradients of the 

∂uξ ∂uς + ∂pξ ∂pς

 .

(5)

Under plane strain (εzz = εxz = εyz = 0) conditions, the equilibrium equations reduce to c11

∂ 2 uy ∂ 2 uy ∂ 2 uy ∂ 2 ux ∂ 2 ux ∂ 2 ux + c + 2c + c + c + (c + c ) = 0, 66 16 16 26 12 66 ∂px ∂py ∂px ∂py ∂px2 ∂py2 ∂px2 ∂py2

(6) c16

∂ 2 uy ∂ 2 uy ∂ 2 uy ∂ 2 ux ∂ 2 ux ∂ 2 ux + c26 + (c12 + c66 ) + c66 + c22 + 2c26 = 0, 2 2 2 2 ∂px ∂py ∂px ∂py ∂px ∂py ∂px ∂py

(7) c15

∂ 2 uy ∂ 2 uy ∂ 2 uy ∂ 2 ux ∂ 2 ux ∂ 2 ux + c + (c + c ) + c + c + (c + c ) = 0. 46 56 14 56 24 25 46 ∂px2 ∂py2 ∂px ∂py ∂px2 ∂py2 ∂px ∂py

(8) uς , ς = x, y only need to satisfy two, but not all three, of Eqs. (6)–(8). c11 > 0, c22 > 0, c66 > 0 should be satisfied. Consequently, the third equation (8)

Method of Fundamental Solutions Without Fictitious Boundary for Anisotropic. . .

63

can only be satisfied by setting c15 = c46 = c14 = c56 = c24 = c25 = 0. The stiffness matrix thus reduces to ⎡

⎤ c11 c12 c16 C = ⎣ c12 c22 c26 ⎦ . c16 c26 c66

(9)

The formulas for plane stress problems can be obtained from the formulas of plane strain problems by replacing the components cij by cij = cij −

ci3 c3j c33

, j = 3, 4, 5.

(10)

The inverse of (4) is ε = Sσ,

(11)

where Sςξ υτ are the components of matrix S. The tractions tς , ς = x, y come from the stresses, as follows tς (p) = σςx (p)nx + σςy (p)ny ,

(12)

where n is the outward normal vector with the coordinates nς , ς = x, y at the boundary point p. The boundary  is structured into parts, on which the displacement (Dirichlet) BC  D and traction (Neumann) BC  T (see Fig. 1) are defined.

s p ΓD Ω

iy

n

ix

Γ

T

• : collocation points : source points

Fig. 1 A schematic view of domain  and boundary . The solid is exposed to displacement (Dirichlet)  D and (dashed line) traction (Neumann)  T boundary conditions

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uς (p) = u¯ ς (p); p ∈  D ,

(13)

tς (p) = t¯ς (p);

(14)

p ∈ T ,

where u¯ ς and t¯ς , ς = x, y represent the known BC’ definition functions.

3 INMFS Procedure for 2D Anisotropic Elasticity Problems 3.1 Fundamental Solution of 2D Anisotropic Elasticity Problems for the Displacement and Traction The definition [36] of the following complex variables zm , m = 1, 2, 3 are needed in analytical expression of the anisotropic FS zm (p) = px + ρm py .

(15)

The complex constants ρα , α = 1, 2, 3 are the roots with positive imaginary parts of the sixth-order determinant [35] |Q+ρ(R + RT ) + ρ 2 T| = 0,

(16)

where | · | is the determinant of matrix, and the matrices Q, R and T are defined as Qςξ = Cςxξ x , Rςξ = Cςxξy , Tςξ = Cςyξy .

(17)

For a general anisotropic material, the sextic equation in ρ from Eq. (16) is then ' ' ' c11 + 2ρc16 + ρ 2 c66 c16 + ρ(c12 + c66 ) + ρ 2 c26 c15 + ρ(c14 + c56 ) + ρ 2 c46 '' ' ' ' ' c16 + ρ(c12 + c66 ) + ρ 2 c26 c66 + 2ρc26 + ρ 2 c22 c56 + ρ(c46 + c25 ) + ρ 2 c24 ' = 0. ' ' ' c15 + ρ(c14 + c56 ) + ρ 2 c46 c56 + ρ(c46 + c25 ) + ρ 2 c24 ' c55 + 2ρc45 + ρ 2 c44

(18) Our interest is primarily the plane strain deformation of anisotropic materials. As represented in [35], in-plane and anti-plane deformations are uncoupled only for anisotropic materials, which satisfy c14 = c15 = c24 = c25 = c46 = c56 = 0.

(19)

Under these conditions, Eq. (18) is ' ' ' ' c16 + ρ(c12 + c66 ) + ρ 2 c26 ' c11 + 2ρc16 + ρ 2 c66 ' (c55 + 2ρc45 + ρ c44 ) ' ' = 0. ' c16 + ρ(c12 + c66 ) + ρ 2 c26 ' c66 + 2ρc26 + ρ 2 c22 2

(20)

Method of Fundamental Solutions Without Fictitious Boundary for Anisotropic. . .

65

So c55 + 2ρc45 + ρ 2 c44 = 0,

(21)

or ' ' ' c11 + 2ρc16 + ρ 2 c66 c16 + ρ(c12 + c66 ) + ρ 2 c26 '' ' ' = 0. ' c16 + ρ(c12 + c66 ) + ρ 2 c26 c66 + 2ρc26 + ρ 2 c22

(22)

The roots of Eq. (22) with positive imaginary part, ρ1 andρ2, are those required for the analysis of plane strain deformations. The root of Eq. (21) with positive imaginary part is needed for problem of anti-plane strain. But not for plane strain deformations. m We define the matrix γ with the elements γςξ , ς, ξ = x, y as in [36] m = γςξ

ιΓςξ 2 )(ρ −ρ¯ ) ( (ρ −ρ )(ρ −ρ¯ ) , (c11 c66 −c16 m m m n m n

m, n = 1, 2

(23)

m=n

where ρ¯m is the conjugate complex of ρm , and 2 2 ,Γxy = Γyx = −(c44 + c12 )ρm ,Γyy = c11 + c44 ρm . Γxx = c44 + c11 ρm

(24)

A FS for the system (1) and (2) is given by Ting [35, 37]:  1 m Uςξ (p, s) = − Re γςξ log (zm (p) − zm (s)), π 2

(25)

m=1

with s representing the source point. The related fundamental tractions are given by 1 Txς (p, s) = − Re π * +

+

m + c ρ γm c11 γxς 12 m yς

m=1

(px − sx ) + ρm (py − sy )

m + γm) c66 (ρm γxς yς

m=1

(px − sx ) + ρm (py − sy )

)*

+

2 

2 

1 Tyς (p, s) = − Re π *

)*

+

,

+

m + γm) c66 (ρm γxς yς

m=1

(px − sx ) + ρm (py − sy )

m + c ρ γm c21 γxς 22 m yς

m=1

(px − sx ) + ρm (py − sy )

(26)

ny ,

2 

2 

nx

+

nx

, ny .

(27)

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3.2 Non-singular Fundamental Solution of 2D Anisotropic Elasticity Problems for Displacement The sources are located external to the real domain in MFS, so that there is no singularity (see Fig. 1). The singularity of the FS (when the source s located at the same position of the collocation points p) is removed by scaling the singularity with respect to the area of the square, for which the desingularization integration is made (see Fig. 2). Here, the desingularization integration is over a small square instead over a small circular disc as in [26]. This enables the analytical calculation of the expression for desingularized FS. To maintain the consistency of the desingularized FS U˜ ςξ (p, s)with the singular FS Uςξ (p, s), the integration of the desingularization is normalized with respect to the area of the square U˜ ςξ (p, s) =

⎧ ⎨ Uςξ (p, s), ⎩

1 4a 2

' ' ' ' 'pζ − sζ ' > a, ' ' Uςξ (p, s) dA, pζ − sζ ≤ a,

ς, ξ, ζ = x, y,

p∈S(s,a)

(28) where S (s,a) (see Fig. 2) is a square, with length 2a,' that is 'centred at s. The analytical integration gives the following expression for 'pζ − sζ ' ≤ a U˜ ςξ (p, s) =

⎧ ⎪ ⎨− ⎪ ⎩

1 (4π ) Re

2 . m=1

0 0 / 1 12 m −12ρ −(ρ −1)2 log −a 2 (ρ −1)2 +(ρ +1)2 log −a 2 (ρ +1)2 γςξ m m m m m , 2ρm

ς = ξ, ς = ξ.

0,

(29) Fig. 2 The source, distributed on a square S (s,a) with the side length 2a

y S(s, a) s p

O

a

x

Method of Fundamental Solutions Without Fictitious Boundary for Anisotropic. . .

67

3.3 Discretisation and Calculation of the Unknown Coefficients A solution of the discussed problem is searched for as uς (p) =

N 

Uςx (p, sn ) αn +

n=1

N 

Uςy (p, sn ) βn , ς = x, y.

(30)

n=1

As a result of Eqs. (4), (5), and (12), are the tractions expressed as tς (p) =

N 

Tςx (p, sn ) αn +

n=1

N 

Tςy (p, sn ) βn , ς = x, y.

(31)

n=1

sn are the sources located on an artificial boundary in MFS or on the physical boundary in NMFS/INMFS, respectively. The expression Uςξ (p, sn ) in MFS, is substituted by U˜ ςξ (p, sn ) in NMFS/INMFS. The coefficients αn and βn result from 2N algebraic equations  Ax =

Axx Axy Ayx Ayy

    α bx = b, = by β

(32)

where (Aςξ )ij = χςD (pi )Uςξ (pi , sj ) + χςT (pi )Tςξ (pi , sj ), ς, ξ = x, y. i, j = 1, · · · N, (33) (bς )i = χςD (pi )uς (pi ) + χςT (pi )tς (pi ), ς = x, y. i = 1, · · · N.

(34)

The displacement χςD , ς = x, y and the traction χςT , ς = x, y type of BC indicators are as follows   1; p ∈  D in iς direction, T 1; p ∈  T in iς direction, D χς (p) = χς (p) = D 0; p ∈ /  in iς direction, 0; p ∈ /  T in iς direction. (35)

3.4 Desingularization of the Tractions for Anisotropic Elasticity Problems Using Force-Balance When the source locates at the same position as the collocation point, the terms T˜ςξ (pn , pn ),ς, ξ = x, y, n = 1, . . . , N in Eq. (32) are determined indirectly

68

Q. Liu and B. Šarler

in case of NMFS [26]. Two reference displacement solutions of Eqs. (6) and (7) are required. System (32) are solved twice. By invoking the fact that the boundary integration of the forces on the body should vanish at a mechanical equilibrium for saving the computing time, we improve the NMFS [26] by solving the following two integral equations  tς (p)d =

N  

T˜ς x (p, pn )dαn +

n=1 



N  

T˜ςy (p, pn )dβn = 0,

ς, ξ = x, y.

n=1 

(36) Since Eq. (36) should be satisfied for arbitrary conditions or source density distributions, we have 

T˜ςξ (p, pn )d ≈



N 

T˜ςξ (pm , pn )lm = 0,

ς, ξ = x, y,

(37)

m=1,

and T˜ςξ (pn , pn ), ς, ξ = x, y can be evaluated approximately as 1 T˜ςξ (pn , pn ) = − ln

N 

T˜ςξ (pm , pn )lm .

(38)

m=1,m=n

The integrals were evaluated by assuming a constant value of the fundamental tractions on the boundary segments (see Fig. 3), where ln is the length of the boundary segment that covers the collocation point pn . For problems where only one domain is considered, the boundary length of the segment is estimated as follows d(pn ) l, N . d(pm )

ln =

(39)

m=1

lN

Fig. 3 Scheme of the boundary segments ln

pN

p1

l1 p2

Ω

iy

pn ix

ln

l2

Method of Fundamental Solutions Without Fictitious Boundary for Anisotropic. . .

69

where l is the total perimeter of the problem domain. d(p) is the smallest distance between p and the neighbouring nodes. For problems, where one domain is embedded in the other domain, the length of the segments is estimated as follows

ln =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

d(pn ) lin , N . d(pm )

pn on the inner boundary,

m=1

d(pn ) lout , N . d(pm )

pn on the outer boundary,

(40)

m=1

where lin is the perimeter of the inner boundary and lout is the perimeter of outer boundary. All values of xk are obtained from the inverse of the system (32). Then, we can pose a solution to the governing Eqs. (6) and (7) in the following form uς (p) =

N 

U˜ ςx (p, pn ) αn +

n=1

N 

U˜ ςy (p, pn ) βn , ς = x, y,

(41)

n=1

where p can be any point in the problem domain or on the boundary.

3.5 Boundary Conditions on Interface Between Two Different Materials We extend the previous discussion to two materials with different properties in mechanical contact. The domain  is now composed of two parts, I and II . Their boundaries are denoted by  I and  II and the common contact boundary by  I∩II , as illustrated in Fig. 4. The properties of the materials in the two domains can in general be different. The mechanics of anisotropic materials I and II are governed by the Eqs. (6) and (7) with the properties CI and CII , where the superscripts I and II indicate the material in I and II , respectively  C=

CI , CII ,

p ∈ I , p ∈ II .

(42)

The BC for  I and  II are formally in the layout, defined by Eqs. (13) and (14). The BC at the interface  I∩II can experience the following ternary concurrences see Table 1.

70 Fig. 4 A domain with two different materials in the domains I and II in mechanical contact

Q. Liu and B. Šarler

(a)

I

Γ

pm

I

Γ

Ω

I ∩ II

n

I

II

I pn pm

I

iy

n

Ω

Γ

II II

pn

n o

II

ix

(b) I

pm Γ

I

pn

I

Ω Ω

I

iy

n

I

Γ n

o

n

II

II

pm

II

I ∩ II

I

ix

Table 1 The BC at  I∩II for different situations

Boundary condition at  I∩II No inclusion (Fig. 4a)/

uςI (p) − uςII (p) = 0, ς = x, y

Elastic inclusion (Fig. 4b)

tςI (p) + tςII (p) = 0 , ς = x, y

Rigid inclusion (Fig. 4b)

uςI (p) = uςII (p) = 0, ς = x, y

Void inclusion (Fig. 4b)

tς (p) = 0 , ς = x, y

II

Method of Fundamental Solutions Without Fictitious Boundary for Anisotropic. . .

71

4 Numerical Examples 4.1 Numerical Example 1 The problem of a square with the side length 2 m, centred at px = py = 0 m is employed in Example 1. The analytical solution is elementary ux = px , uy = py .

(43)

The properties of crystal copper c11 = c22 = 16.84, c12 = 12.14, and c44 = c55 = c66 = 7.54 are considered. The example was previously studied in [38] for plane strain problems using MFS. The error with respect to the solution in relative terms, is given as   M 1  (uςn − uςn )2 , eς =  M u2ςn

ς = x, y.

(44)

n=1

where uςn and uςn , ς = x, y are the analytical and the numerical solution, respectively. M is the number of the points used in the assessment of the error.

4.1.1 Case 1: Dirichlet Boundary Conditions The analytical solution (43) of the boundary points is used as the displacement BC for all the four boundaries (see Fig. 5). M = 18 domain points that are distributed equidistantly along the line −0.85 m ≤ px = py ≤ 0.85 m, are used for error assessment. Figure 6 presents the relative errors for INMFS with different a, in the case with N = 200 boundary points. The relative error in y-direction is growing with decrease in a(p). The relative error in x-direction is best when a is 1/3 of Fig. 5 Example 1, Case 1. Scheme of a square subject to displacement BC on entire boundary

(-1,1)

iy u x = px , u y = 1

(1,1)

Ω u x = −1 u y = py

(-1,-1)

o

ux = 1 u y = py

u x = px , u y = −1

ix

(1,-1)

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Fig. 6 Example 1, Case 1. Relative errors vs. length of desingularized square, calculated by INMFS

d(p) as shown in Fig. 6 denoted by ♦, where d(p) is the smallest distance between p and the neighbour point of p at the boundary. Considering the errors in Fig. 6, a(p) = d(p)/2 is chosen for all the following cases. RN (p) = d(p)/5 is the radius of the desingularization circle for NMFS. RM = 5d is the distance between the collocation point and the source point for MFS, where d is the minimum of d(p). The relative errors, obtained for MFS, NMFS and INMFS are plotted in Fig. 7 as a function of the number of the boundary nodes from 200 to 2000. The accuracy of the method in the domain is evaluated with M = 18 domain points, the same as in Fig. 6. The errors of the INMFS, resulting in the points in the domain, is plotted in Fig. 7 and compared with the respective MFS and NMFS errors. The NMFS results are better than INMFS results in x-direction and worse in y-direction in this example. However, both of the relative errors at the domain points are less than 10−2 for NMFS and INMFS. The NMFS and INMFS solutions of the domain points both converge to the analytical solution as the number of the boundary nodes increases. Here it should be noted that the MFS solution errors are rather small, however the convergence is not uniform. This fact is due to the choice of the artificial boundary position, that was for all node arrangements RM = 5d, and thus most probably not optimal.

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Fig. 7 Example 1, Case 1. Relative error at equidistantly positioned domain points vs. number of boundary nodes, calculated by MFS, NMFS and INMFS

74 Fig. 8 Example 1, Case 2. Scheme of a square subject to mixed BC

Q. Liu and B. Šarler

iy

(-1,1)

(1,1)

tx = 0, u y = 1

Ω u x = −1 uy = py

(-1,-1)

o

ux = 1 ty = 0

u x = px , u y = −1

ix

(1,-1)

4.1.2 Case 2: Mixed Boundary Conditions We consider the solution of the governing equations (6) and (7) at the previously defined square, subject to the BC t¯y = 0 at the points of the side px = 1, and t¯x = 0 at the side py = 1. The analytical solution (43) of the boundary points is employed as the displacement BC on the other boundary points (see Fig. 8). The discretization is the same as in Case 1. The plots with the error with respect to the boundary and domain points, obtained with MFS, NMFS and INMFS, are presented in Figs. 9 and 10. The errors obtained for the MFS, NMFS and INMFS are plotted in Fig. 9 for the nodes at the boundary and in Fig. 10 for the domain points. In this case, INMFS results are better than NMFS in both x- and y-direction for boundary points. INMFS results are better than NMFS results in y-direction and worse in x-direction. However, both of the relative errors of the boundary and domain points are less than 10−2 for NMFS and INMFS. The NMFS and INMFS solutions at the boundary nodes and the domain points both converge to the analytical solution as the number of the boundary nodes increases.

4.2 Numerical Example 2 In this example, a small circular domain with radius rin = 0.1 m, centred at px = 0 m, py = 0 m, is considered inside the square  = (−1, 1) × (−1, 1) (see Fig. 11). This example is divided into four variants. The data for them are provided in Table 2. The square is considered to be subject to u¯ x = 0 m and u¯ y = −0.1 m on the north side py = 1 m and u¯ x = 0 m and u¯ y = 0.1 m on the south side py = −1 m, t¯x = 0 N/m2 and t¯y = 0 N/m2 on the other sides in all four cases (see Fig. 11). The following parameters are used a I (p) = d I (p)/2, a II (p) = d II (p)/2. In case

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Fig. 9 Example 1, Case 2. Relative error at the boundary points vs. the boundary nodes, calculated by MFS, NMFS and INMFS

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Fig. 10 Example 1, Case 2. Relative error in equidistantly positioned domain points vs. the boundary nodes, calculated by MFS, NMFS and INMFS

Method of Fundamental Solutions Without Fictitious Boundary for Anisotropic. . . Fig. 11 Example 2. Scheme of a square with two different material parts

77

iy

(-1,1)

(1,1)

u x = 0, u y = −0.1 ΩI

Ω II ro = 0.1 m o tx = 0

tx = 0 ty = 0 (-1,-1)

ix

ty = 0 u x = 0, u y = 0.1

(1,-1)

Table 2 Example 3. The features of the four different cases of Example 3 Case Square domain 1

I = 16.84, c11

2

I c11

3 4

= 16.84,

Inclusion properties I = 12.14, c12

I = 7.54 Void c66

= 12.14,

I = 7.54 cI = 16.84, c66 11

I = 12.14, c12

I = 7.54 c66

I = 16.84, c11

I = 12.14, c12

I = 7.54 cI I = 24.65, c66 11

I I = 14.73, c12

I I = 12.47 c66

I = 16.84, c11

I = 12.14, c12

I = 7.54 Rigid c66

I c12

of MFS, the distance between the fictitious and the physical boundary is equal to I = 5d I , R II = 5d II . RM M 4.2.1 Case 1 In this case is the number of boundary nodes N = Ncube +Nsphere = 1500, Ncube = 1400 located on the square and Nsphere = 100 located on the circle. The INMFS results of selected domain points (see Fig. 12) are compared with MFS results as shown in Fig. 13. The artificial dotted lines are the contour lines of the selected points’ radius in Figs. 12b, 15, and 17 and contour lines of the selected points’ ycoordinates py in Figs. 13, 14, 16, and 18. The INMFS results are close and consistent with MFS results, as displayed in Fig. 13. The maximum differences between the MFS and INMFS results in x- and y-directions are ux = 5.2823 × 10−4 m, uy = 5.6701 × 10−4 m. We know from Example 1 that the INMFS solution converge with the increasing number of the boundary nodes. The difference between the MFS and INMFS solutions is expected to decrease further with the increasing number of the nodes. The condition number of A for INMFS is much lower than the condition number for MFS, i.e., the condition number of A is 5.5212 × 1019 for MFS and 5.3208 × 103 for INMFS. The INMFS is much more stable than MFS because of the different location of the source points.

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Fig. 12 Example 2. Scheme of the position of the selected domain points for I and II

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Fig. 13 Example 2, Case 1. The MFS and INMFS result for selected domain points from Fig. 12a

4.2.2 Case 2 Before discretisation of this case, we use the Eq. (43) as the analytical solution for both I and II to compare the errors of the MFS, NMFS and INMFS. The MFS result is most accurate. However, the NMFS and INMFS results converge to the analytical solution as the number of the boundary nodes increases, which is similar to the conclusion of Example 1. The discretisation in this case is the same as in Case 1. The INMFS results of I and II are compared with MFS results as displayed in Figs. 14 and 15. The INMFS results are close and consistent with the MFS results in both I and II , as displayed in Figs. 14 and 15. The maximum differences between the MFS and NMFS results in x- and y-direction are ux = 1.0889 × 10−4 m, uy = 2.0767 × 10−4 m in I , and ux = 3.4848 × 10−5 m, uy = 5.8559 × 10−5 m in II .

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Fig. 14 Example 2, Case 2. The MFS and INMFS results for selected domain points of I from Fig. 12a

4.2.3 Case 3 The discretisation in this case is the same as in Case 1. The INMFS results of I and II are compared with MFS results as displayed in Figs. 16 and 17. The INMFS results are close and consistent with MFS results in both I and II as displayed in Figs. 16 and 17. The maximum of differences between the MFS and INMFS results in x- and y-direction are ux = 1.0919 × 10−4 m, uy = 2.0992 × 10−4 m. uz = 7.4858 × 10−3 m in I and ux = 7.2076 × 10−5 m, uy = 4.9177 × 10−5 m in II .

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Fig. 15 Example 2, Case 2. The MFS and INMFS results for selected domain points of II from Fig. 12b

4.2.4 Case 4 The discretisation in this case is the same as in Case 1. The INMFS results of I are compared with MFS results as displayed in Fig. 18. The INMFS results are close and consistent with MFS results in I (see Fig. 18). The maximum differences between the MFS and INMFS results in x- and ydirection are ux = 1.0481 × 10−4 m, uy = 2.0611 × 10−4 m.

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Fig. 16 Example 2, Case 3. The MFS and INMFS results for selected domain points of I from Fig. 12a

5 Conclusions In the current work, we extend a novel INMFS to solve the problems associated with 2D linear anisotropic elasticity problems. It differs from the previously developed NMFS formulation in two aspects. First, squares are used to replace the circular discs as used in NMFS in desingularization. The singular fundamental solution for anisotropic elasticity can therefore be integrated analytically in INMFS. Second, the singularities of the fundamental traction are solved by considering the mechanical equilibrium, calculated by the boundary integration of the forces on the considered body, instead of with the two reference solutions. Both listed modifications make the INMFS much more efficient as compared with the NMFS. The numerical examples

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Fig. 17 Example 2, Case 3. The MFS and INMFS results for selected domain points of II from Fig. 12b

show that the NMFS and the INMFS results have the similar accuracy and are more stable than the MFS results. At the same time, the artificial boundary is removed in INMFS approach, so that the coding is simplified and the issue with the positioning of the artificial boundary is resolved completely. Several numerical examples have shown the simplicity, accuracy and effectiveness of the proposed method. The development of INMFS for 3D anisotropic elasticity is underway.

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Fig. 18 Example 2, Case 4. The MFS and INMFS results for selected domain points of I from Fig. 12a

Acknowledgments This work was funded by the Slovenian Research Agency (ARRS) in the framework of the basic research project J2-1718 and program group P2-0162.

References 1. Teodorescu, P.P.: One hundred years of investigations in the plane problem of the theory of elasticity. Appl. Mech. Rev. 17, 175–186 (1964) 2. Boresi, A.P., Chong, K.P.: Elasticity in Engineering Mechanics. Elsevier, New York (1987) 3. Gao, X.L.: A general solution of an infinite elastic plate with an elliptic hole under biaxial loading. Int. J. Press. Vessels Pip. 67, 95–104 (1996) 4. Chen, C.S., Karageorghis, A., Smyrlis, Y.S.: The Method of Fundamental Solutions—A Meshless Method. Dynamic Publishers, Atlanta (2008) 5. Kupradze, V.D., Aleksidze, M.A.: The method of functional equations for the approximate solution of certain boundary value problems. USSR Comput. Math. Math. Phys. 4, 82–126 (1964) 6. Golberg, M.A., Chen, C.S.: The method of fundamental solutions for potential, Helmholtz and diffusion problems. Boundary Integral Methods-Numer. Math. Aspects 1, 103–176 (1998)

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7. Golberg, M.A.: The method of fundamental solutions for Poisson’s equation. Eng. Anal. Boundary Elem. 16, 205–213 (1995) 8. Young, D.L., Jane, S.J., Fan, J.M., Murugesan, K., Tsai, C.C.: The method of fundamental solutions for 2D and 3D Stokes problems. J. Comput. Phys. 211, 1–8 (2006) 9. Balakrishnan, K., Ramachandran, P.A.: The method of fundamental solutions for the liner diffusion-reaction equations. Math. Comput. Modell. 31, 221–237 (2000) 10. Karageorghis, A., Fairweather, G.: The method of fundamental solutions for the numerical solution of the biharmonic equation. J. Comput. Phys. 69, 434–459 (1987) 11. Fu, Z.J., Chen, W., Yang, H.T.: Boundary particle method for Laplace transformed time fractional diffusion equations. J. Comput. Phys. 235, 52–66 (2013) 12. Kołodziej, J.A.: Review of application of boundary collocation methods in mechanics of continuous media. SM Arch. 12, 187–231 (1987) 13. Karageorghis, A., Lesnic, D., Marin, L.: A survey of applications of the MFS to inverse problems. Inverse Prob. Sci. Eng. 19, 309–336 (2011) 14. Fan, C.M., Huang, Y.K., Chen, C.S., Kuo, S.R.: Localized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equations. Eng. Anal. Boundary Elem. 101, 188–197 (2019) 15. Alves, C.J.S.: On the choice of source points in the method of fundamental solutions. Eng. Anal. Boundary Elem. 33, 1348–1361 (2009) 16. Chen, C.S., Karageorghis, A., Li, Y.: On choosing the location of the sources in MFS. Numer. Algorithms. 72, 107–130 (2016) 17. Fu, Z.J., Xi, Q., Chen, W., Chen, A.H.D.: A boundary-type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations. Comput. Math. Appl. 76, 760–773 (2018) 18. Fu, Z.J., Chen, W., Wen, P.H., Zhang, C.Z.: Singular boundary method for wave propagation analysis in periodic structures. J. Sound Vib. 425, 170–188 (2018) 19. Šarler, B.: Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions. Eng. Anal. Boundary Elem. 33, 1374–1382 (2009) 20. Young, D.L., Chen, K.H., Chen, J.T., Kao, J.H.: A modified method of fundamental solutions with source on the boundary for solving Laplace equations with circular and arbitrary domains. CMES: Comput. Model. Eng. Sci. 19, 197–222 (2007) 21. Wen, S.T., Wang, K., Zahoor, R., Li, M., Šarler, B.: Method of regularized sources for twodimensional Stokes flow problems based on rational or exponential blobs. Comput. Assisted Methods Eng. Sci. 22, 289–300 (2017) 22. Wang, K., Wen, S., Zahoor, R., Li, M., Šarler, B.: Method of regularized sources for axisymmetric Stokes flow problems. Int. J. Numer. Methods Heat Fluid Flow 26, 1226–1239 (2016) 23. Liu. Y.J.: A new boundary meshfree method with distributed sources. Eng. Anal. Boundary Elem. 34, 914–919 (2010) 24. Liu, Q.G., Šarler, B.: A non-singular method of fundamental solutions for two-dimensional steady-state isotropic thermoelasticity problems. Eng. Anal. Boundary Elem. 75, 89–102 (2017) 25. Liu, Q.G., Šarler, B.: Non-singular method of fundamental solutions for two-dimensional isotropic elasticity problems. CMES Comput. Model. Eng. Sci. 91, 235–266 (2013) 26. Liu, Q.G., Šarler, B.: Non-singular method of fundamental solutions for anisotropic elasticity. Eng. Anal. Boundary Elem. 45, 68–78 (2014) 27. Liu, Q.G., Šarler, B.: Non-singular method of fundamental solutions for elasticity problems in three-dimensions. Eng. Anal. Boundary Elem. 96, 23–35 (2018) 28. Perne, M., Šarler, B., Gabrovšek, F.: Calculating transport of water from a conduit to the porous matrix by boundary distributed source method. Eng. Anal. Boundary Elem. 36, 1649–1659 (2012) 29. Sincich, E., Šarler, B.: Non-singular method of fundamental solutions based on Laplace decomposition for 2D Stokes flow problems. Eng. Anal. Boundary Elem. 99, 393–415 (2014)

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30. Kuo, S.R., Chen, J.T., Kao, S.K.: Linkage between the unit logarithmic capacity in the theory of complex variables and the degenerate scale in the BEM/BIEMs. Appl. Math. Lett. 26, 929–938 (2013) 31. Kim, S.: An improved boundary distributed source method for two-dimensional Laplace equations. Eng. Anal. Boundary Elem. 37, 997–1003 (2013) 32. Liu, Q.G., Šarler, B.: Improved non-singular method of fundamental solutions for twodimensional isotropic elasticity problems with elastic/rigid inclusions or voids. Eng. Anal. Boundary Elem. 68, 24–34 (2016) 33. Liu, Q.G., Šarler, B.: Method of fundamental solutions without fictitious boundary for three dimensional elasticity problems based on force-balance desingularization. Eng. Anal. Boundary Elem. 108, 244–253 (2019) 34. Bower, A.F.: Applied Mechanics of Solids. CRC Press, USA (2009) 35. Ting, T.C.T.: Anisotropic Elasticity. Oxford Science, Oxford (1996) 36. Stroh, A.S.: Dislocations and cracks in anisotropic elasticity. Philos. Mag. 30, 625–646 (1958) 37. Teway, V.K., Wagoner, R.H., Hirth, J.P.: Elastic Green’s function for a composite solid with a planar interface. J. Mater. Res. 1, 113–123 (1989) 38. Berger, J.R., Karageorghis, A.: The method of fundamental solutions for layered elastic materials. Eng. Anal. Boundary Elem. 25, 877–886 (2001)

The Method of Fundamental Solutions for the Direct Elastography Problem in the Human Retina Sílvia Barbeiro and Pedro Serranho

Abstract This paper addresses the numerical simulation of the mechanical waves propagation and induced displacements in the human retina, for the elastography imaging modality. In this way, we use a model for the human eye and numerically approximate the propagation of time-harmonic acoustic waves through the different media of the eye and the respective elastic excitation in the retina, through a layered representation approach based on the method of fundamental solutions. We present numerical results showing the feasibility of the method. Keywords Elastography · Method of fundamental solutions · Coupled acoustic wave propagation · Elastic displacement model

1 Introduction This work was done in the framework of ElastoOCT project whose objective is to develop an Optical Coherence Elastography (OCE) technique for imaging in vivo the mechanical properties of the retina [1]. OCE is an imaging modality which combines mechanical excitation of the retina with optical coherence tomography (OCT) for measuring the corresponding elastic displacement [2–5]. When using acoustic loading, an ultrasound source is coupled with an OCT device to this end. Applications of this technique vary from skin to the retina, being the latter the

S. Barbeiro Department of Mathematics, University of Coimbra, CMUC, Coimbra, Portugal e-mail: [email protected] P. Serranho () Coimbra Institute for Biomedical Image and Translational Research, Faculty of Medicine, University of Coimbra, Coimbra, Portugal Mathematics Section, Department of Science and Technology, Universidade Aberta, Lisbon, Portugal e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 C. Alves et al. (eds.), Advances in Trefftz Methods and Their Applications, SEMA SIMAI Springer Series 23, https://doi.org/10.1007/978-3-030-52804-1_5

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one of most interest to this work. The importance of elastography is that usually pathological tissue has mechanical properties distinct from healthy tissue, therefore imaging the elastic properties of the tissue gives a map of the more affected regions by the disease. We are interested in applying OCE to the human retina, therefore we will consider the geometry and physical properties of the retina [6] to model the problem. The mathematical simulation of this process includes the propagation of the acoustic wave from the source through the eye to the retina, the interaction of the acoustic pressure to generate an elastic wave in the retina and the propagation of the elastic wave in the retina, namely the induced displacements in the retinal tissue. In this way, we consider a layered medium corresponding to the different media within the eye. As the excitation is taken to be time-harmonic, we propose to approximate the solution in each layer by the method of fundamental solutions (MFS). Transmission conditions between layers establish the way the different layers interact between them. MFS has increased its popularity in the engineering community [7–14], due to its simple implementation and good accuracy. In an hand-waiving argument, the method consists in representing the solution of a partial differential equation as a linear combination of fundamental solutions with source points outside the domain. Hence, the approximation automatically satisfies the differential equation, being the only concern to determine the weights of the linear combination to approximate the boundary conditions. However, the choice of source points is crucial for the method, since it has high influence in the conditioning and accuracy of the method. There are studies in 2D that show that the choice of points vary the quality of the approximated solution from excellent to numerically unsolvable [15]. In this way, researchers usually consider the source points in a smooth boundary obtained by moving the real boundary outside the domain (for the interior problem) in the normal direction [16]. The distance between the boundary and these source points should be taken into account, since they should not be too far, due to accuracy, nor too close, due to the ill-conditioning. Also, it has been illustrated in 3D that the use of quasi-equidistant uniformly distributed points is favourable in terms of accuracy and conditioning [17]. The paper is organized as follows. In Sect. 2 we present the mathematical model for the problem at hand. In particular, we will detail the considered geometry, the system of partial differential equations to be solved and the boundary and transmission conditions for each layer. In Sect. 3 we present the method of fundamental solutions, namely the ansatz that we consider for each layer and we establish the linear system to be solved in order to determine the approximation. In Sect. 4 we present the numerical results for the problem at hand. Finally, in Sect. 5, we summarize the results and suggest some future perspectives for this work.

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2 Mathematical Setting We start by describing the assumptions that were made to model this problem. Based on the fact that the ultrasound source is directional and the optical coherence tomography A-scan that measures the displacements is very narrow, we consider as domain of interest a narrow cylindrical layered domain, as in Fig. 1. We also consider the interface boundaries in the eye to be plane within the cylinder. Since we are modelling the problem within a very narrow cylinder, it is natural to expect that the curve boundaries of the eye components are close to planar within the cylinder. Let us denote the domain of interest by , which we assume to be the union of the subdomains j , j = 1, . . . , n, =

3

j .

j =1,...,n

Each layer j represents different parts of the eye with different acoustic and elastic properties. The interface between the layer j and the layer j +1 is denoted by j ,

Fig. 1 Geometry of the problem (left) and scheme of the eye and the depth of layers in mm (right)

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for j = 1, 2, . . . , n − 1. We use the notation ∂ for the boundary of  and ∂j for the boundary of j , j = 1, . . . , n. The upper layer n represents the retina, so we are interested in simulating acoustic propagation in the layers 1 , 2 , . . . , n−1 and elastic propagation in n . The interaction between the acoustic pressure and the elastic displacement occurs in the boundary n−1 which will be detailed later. We assume time-harmonic acoustic emission in the source of the form  (1) P (x, t) = Re p(x)eiωt , for some angular frequency ω, so the acoustic pressure field p satisfies the Helmholtz equation in each layer [18]. Therefore, we consider that the timeharmonic acoustic pressure pj in the layer j satisfies [19, 20] pj + κj pj = 0 in j ,

j = 1, 2, . . . , n − 1

(2)

with wavenumber κj = ω/cj , where ω is the angular frequency and cj is the speed of sound in the layer j . Also, we consider a plane wave excitation with incident direction d and amplitude A and therefore the condition in the lower boundary 0 is given by p1 = Aeiκ1 x·d on 0 .

(3)

Between acoustic layers we assume continuity of the acoustic pressure and of the particles’ velocity, respectively, given by

ρj

pj = pj +1 on j , j = 1, 2, . . . , n − 2,

(4)

∂pj ∂pj +1 = −ρj +1 on j , j = 1, 2, . . . , n − 2, ∂νj ∂νj +1

(5)

where ρj is the density in the layer j and νj is the outward unit normal to ∂j . Finally, we assume that there is no pressure flux in the lateral boundary of the cylinder layer, namely   ∂ pj = 0 on ∂ ∪j =1,2,...,n−1 j \ (0 ∪ n−1 ). ∂ν

(6)

As for time-harmonic elastic propagation, the elastic displacement field u in n satisfies the Lamé equation [21, 22] μu + (λ + μ) grad div u + ω2 ρu = 0 in n ,

(7)

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where the Lamé constants are given by μ=

E , 2(1 + υ)

λ=

υE , (1 + υ)(1 − 2υ)

where E is the Young’s Modulus and υ is the Poisson’s ratio. The acoustic-elastic transmission condition is given by Ito et al. [19], and Ito and Toivanen [20] 1 ∂ pn−1 = ω2 u · ν on n−1 , ρ ∂ν

(8)

pn−1 νn−1 = σ (u)νn on n−1 ,

(9)

where the stress tensor is given by σ (u) = μ(∇u + (∇u)T ) + λ div uI. We also consider no stress in the normal direction for the elastic field on the remaining boundaries, that is, σ (u)ν = 0 on ∂n \ n−1 .

(10)

The interior direct problem of obtaining the acoustic and elastic field is wellposed if we do not consider resonance frequencies for the domains at hand. For the acoustic case this is equivalent to saying that the considered wavenumber κj is not an eigenvalue of the negative Laplace operator for the interior domain j [18] for j = 1, 2, n − 1. Throughout this paper we assume that this is the case.

3 The Method of Fundamental Solutions For each layer j we consider the ansatz for the approximated acoustic field j

pj (x) =

ns 

 (j ) (j ) αk κj x − sk

(11)

k=1 (j )

(j )

∈ / j and weights αk ∈ C for k = for some given source points sk j 1, 2, . . . , ns , j = 1, 2, . . . , n − 1, where the fundamental solution of the Helmholtz equation in R3 is [18]

κ (x) =

eiκ|x| , 4π|x|

x ∈ R3 \ {0}.

(12)

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Again we stress that by the properties of the fundamental solution [18], the approximations of the previous form automatically satisfy the Helmholtz equa(j ) tion (2) for pj , j = 1, 2, . . . , n − 1, regardless of the value of the weights αk , as (j ) long as the source points satisfy sk ∈ / j . Therefore, the solution of the problem (j ) might be approximated by the MFS considering collocation points x ∈ ∂j , j = (j ) 0, 1, 2, . . . , n − 1,  = 1, 2, . . . , nc for the boundary conditions (3) and (6) and the interface transmission conditions (4) and (5). As ansatz, we consider the approximated elastic field n

u(x) =

ns 

(n)

(n)

E (|x − sk |)αk

(13)

k=1 (n)

(n)

/ n and weights αk ∈ C3 for k = 1, 2, . . . , nns , for some given source points sk ∈ where the fundamental solution for the Lamé equation (7) is given by Alves and Kress [23] eiκs |x| 1 κs2

E (x) = . I+ grad gradT 2 4πω ρ |x| 4πω2 ρ



eiκp |x| eiκs |x| − |x| |x|

 (14)

for x ∈ R3 \ {0}, where κp2 =

ω2 ρ , λ + 2μ

κs2 =

ω2 ρ . μ

Again we stress that by the properties of the fundamental solution, the approximations of the previous forms automatically satisfy the Lamé equation (7) in n , regardless of the value of the weights αk(n) . Therefore, the solution of the problem can be approximated by the MFS considering collocation points for the acoustic-elastic coupling transmission conditions (8) and (9) and the boundary condition (10). (j ) j Altogether, we obtain a linear system on the unknowns αk for k = 1, 2, . . . , ns , j = 1, 2, . . . , n, that will then determine the approximated solution. Remark 1 (Denseness Results) The denseness of the approximations (11) and (13) (as linear combination of fundamental solutions) in the space of solutions of the respective elliptic equations is shown in [13, 24, 25] and in particular for the Helmholtz equation in [26] and for Lamé equations in [27].

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3.1 Linear System In this section we translate the boundary and transmission conditions (3), (4), (5), (6), (8), (9) and (10) in terms of the representations (11) and (13) as a linear system on the weights α (j ) for j = 1, 2, . . . , n.

3.1.1 Acoustic Component Since   1 iκ ∂ κ − , (|x − y|) = ν · (x − y) κ (x − y) ∂νx |x − y| |x − y|2 from (4)–(5) one gets '  ' ' ' ' (j ) ' (j ) (j ) (j ) ' (j +1) (j +1) ' αk κj 'x − sk ' = αl

κj +1 'x − sl ' ,

 k

ρj

(15)

l

 k

' ' ' ' '  ' iκj 'x(j ) − sk(j ) ' − 1 ' (j ) (j ) (j ) ' (j ) (j ) αk κj 'x − sk ' νx (j ) · x − sk ' '2  ' (j ) (j ) ' 'x − sk '

= ρj +1

 l

(j )

for x

(16)

' ' ' ' '  ' iκj +1 'x(j ) − sl(j +1) ' − 1 ' (j ) (j +1) (j +1) ' (j ) (j +1) αl

κj +1 'x − sl , ' νx (j ) · x − sl ' '  ' (j ) (j +1) '2 'x − sl '

∈ j , from (3) one gets 

' ' (1) ' (1) (1) (1) ' αk κ1 'x − sk ' = eiκ1 x ·d ,

(1)

x ∈ 0

(17)

k

and from (6) one gets  k

⎡ ⎤ ' '  iκj 1 ' (j ) ⎥ (j ) (j ) ' (j ) (j ) ⎢ '−' αk κj 'x − sk ' νx (j) · x − sk ⎣ ' '2 ⎦ = 0, ' (j ) (j ) '  ' ' (j ) (j ) 'x − sk ' 'x − sk '

(18) (j )

for x

∈ ∂j \ (j −1 ∪ j ).

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3.1.2 Elastic Component We start by noting that for each x ∈ R3 the fundamental solution is a 3 × 3 complex matrix. Therefore, in the ansatz ⎤

1p (|x − sk |)αk,p ⎣ 2p (|x − sk |)αk,p ⎦ , u(x) =

E (|x − sk |)αk = k=1 k=1 p=1 3p (|x − sk |)αk,p ns 

ns  3 

⎡

(19)

where the weights αk are vectors in R3 . For the boundary coupling condition, one needs to characterize σ (u)ν = μ(∇uν + (∇u)T ν) + λ div uI ν

(20)

for u given by (19), in terms of a matrix multiplication by the weight vector α. Since ⎡ ∂ u1 ⎢ (∇u)ν = ⎢ ⎣

∂x1 ν1 ∂ u2 ∂x1 ν1 ∂ u3 ∂x ν1 1

+ + +

∂ u1 ∂x2 ν2 ∂ u2 ∂x2 ν2 ∂ u3 ∂x ν2 2

+ + +

⎤ ∂ u1 ∂x3 ν3 ⎥ ∂ u2 ⎥ ∂x3 ν3 ⎦ , ∂ u3 ∂x ν3

⎡ ∂ u1 ⎢ (∇u)T ν = ⎢ ⎣

3

∂x1 ν1 ∂ u1 ∂x2 ν1 ∂ u1 ∂x ν1 3

+ + +

∂ u2 ∂x1 ν2 ∂ u2 ∂x2 ν2 ∂ u2 ∂x ν2 3

+ + +

⎤ ∂ u3 ∂x1 ν3 ⎥ ∂ u3 ⎥ ∂x2 ν3 ⎦ , ∂ u3 ∂x ν3 3

one has ⎡

⎤ ∂ 11  ∂ 12  ∂ 13 νj νj νj ⎢ ⎥⎡ ⎤ ∂xj ∂xj ⎢ j ∂xj ⎥ α1 j j ⎢ ∂  ∂ 22  ∂ 23 ⎥ 21 ⎢ ⎥⎢ ⎥ νj νj νj ⎥ ⎢ α2 ⎥ (∇ E α)ν = ⎢ ⎢ ⎥⎣ ⎦ ∂xj ∂xj ∂xj j j ⎢ j ⎥ ⎢  ∂ 31  ∂ 32  ∂ 33 ⎥ α3 ⎣ νj νj νj ⎦ ∂xj ∂xj ∂xj j

j

(21)

j

and ⎡ ∂

j1

νj

 ∂ j 2

νj

 ∂ j 3

⎤ νj

⎢ ⎥⎡ ⎤ ∂x1 ∂x1 ⎢ j ∂x1 ⎥ α1 j j ⎢ ∂   ∂ j 2 ∂ j 3 ⎥ j1 ⎢ ⎥⎢ ⎥ νj νj νj ⎥ ⎢ α2 ⎥ . (∇ E α)T ν = ⎢ ⎢ ⎥⎣ ⎦ ∂x2 ∂x2 ∂x2 j j ⎢ j ⎥ ⎢  ∂ j 1  ∂ j 2  ∂ j 3 ⎥ α3 ⎣ νj νj νj ⎦ ∂x3 ∂x3 ∂x3 j

j

j

(22)

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Similarly, one has that  div uI =

∂ u2 ∂ u3 ∂ u1 + + ∂x1 ∂x2 ∂x3

 I

and ⎛ ⎞  ∂ uj ⎠ν div uI ν = ⎝ ∂xj j

so therefore ⎡⎛ ⎞  ∂ j 1 ⎢⎝ ⎠ ν1 ⎢ ⎢ j ∂xj ⎢⎛ ⎞ ⎢ ⎢  ∂ j 1 ⎝ ⎠ ν2 div( E α)I ν = ⎢ ⎢ ⎢ j ∂xj ⎢⎛ ⎞ ⎢ ⎢  ∂ j 1 ⎣⎝ ⎠ ν3 ∂xj j

⎛ ⎞  ∂ j 2 ⎝ ⎠ ν1 ∂xj j ⎛ ⎞  ∂ j 2 ⎝ ⎠ ν2 ∂xj j ⎛ ⎞  ∂ j 2 ⎝ ⎠ ν3 ∂xj j

⎛ ⎞ ⎤  ∂ j 3 ⎝ ⎠ ν1 ⎥ ⎥ ∂xj ⎥⎡ ⎤ j ⎛ ⎞ ⎥ α1 ⎥  ∂ j 3 ⎥⎢ ⎥ ⎝ ⎠ ν2 ⎥ ⎢ α2 ⎥ . ⎥⎣ ⎦ ∂xj ⎥ ⎛ j ⎞ ⎥ α3 ⎥  ∂ j 3 ⎥ ⎝ ⎠ ν3 ⎦ ∂xj j

(23) Combining (21), (22), and (23), we get a linear system for (20) in terms of the weights vector α, that can then be used to express the transmission condition (9) and the boundary condition (10) as a linear system.

4 Numerical Results To illustrate the performance of the method proposed, we present the results of some numerical experiments using parameters compatible with the real application. For a cylinder of radius r = 0.0005 m and height h = 0.0346 m, we considered the interfaces at heights z1 = 0.010,

z2 = 0.0105,

z3 = 0.0136,

z4 = 0.0172,

z5 = 0.0344,(24)

according to the scheme in Fig. 1 based on [6]. The wavenumber in each layer j , j = 1, 2, . . . , 6, is given by κj = ω/cj with the velocity of sound cj (m/s) given by c1 = 1506,

c2 = 1553,

c3 = 1506,

c4 = 1620,

c5 = 1506,

(25)

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according to the characteristics of the eye in [6]. Taking into account the properties of the source used in elastography, we consider the angular frequency ω = 2π ×106 rad/s. Therefore one gets the wavenumbers (rad/m) in each layer given by κ1 = 4172.1,

κ2 = 4045.8,

κ3 = 4172.1,

κ4 = 3878.5,

κ5 = 4172.1.

The elastic constants Young’s modulus E and Poisson’s ratio υ are defined respectively by Jones et al. [28] E = 2 × 104 Pa, υ = 0.498, and in (3) the incident acoustic wave was defined as pi (x) = eiκ1 x·(0,0,−1) MPa.

(26)

We considered total of nc = 17, 516 collocation points and nS = 15, 976 source points as displayed in Figs. 2 and 3. The results of the MFS approximation of the acoustic and elastic fields are shown in Fig. 4. The linear system was directly solved in a least squares sense, using MatLab.

Fig. 2 Collocation points on the boundary (red) and outer source points (blue) in each layer, for a total of nC = 17,516 collocation points and nS = 15,976 source points

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Fig. 3 Collocation points on the boundary (red) and outer source points (blue) for the total domain, for a total of nC = 17,516 collocation points and nS = 15,976 source points

In Table 1 we present the measured errors ei in the boundary and transmissions conditions for i = 1, 2, . . . , 22735. The errors are computed in points distributed on the boundaries ∪j ∂j for j = 1, 2, . . . , 6, for each of the layers, that differ from the collocation points. The error ei is the difference between the imposed boundary condition value and the approximated value obtained by the MFS, for each of the conditions (3), (4), (5), (6), (8), (9), and (10). For instance, for a point x˜ in the boundary j , j = 1, . . . , n − 2, one has to evaluate the following two errors ∂p ∂p given by pj (x) ˜ − pj +1 (x) ˜ and ∂νj (x) ˜ − ∂νj+1 (x) ˜ corresponding to the boundary conditions (4) and (5), respectively. A similar procedure is taken to evaluate the errors on the coupling transmission conditions (8) and (9), while for the boundary conditions (3), (6), and (10) only one error is computed for each point. Due to the denseness results (see Remark 1) and the fact that the problem is wellposed (as resonance frequencies are not considered) one has that if the error on the

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Fig. 4 MFS approximation of the acoustic field (left) and elastic field (right) considering a total of nC = 17,516 collocation points and nS = 15,976 source points. Interfaces between layers are represented in red

Table 1 Numerical error ei , i = 1, 2, . . . , 22735 on the boundary, according to the imposed boundary and transmission conditions, distinct from the used collocation points, for nC collocation points and nS source points nC nS ||e||∞ ||e||2

4644 4368 24.560 0.013268

8934 8260 0.034792 1.15712 ×10−5

13,254 12,082 0.028352 1.02184 ×10−5

17,516 15,976 0.025371 9.8271 ×10−6

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boundary goes to zero, so does the error in the interior of the domain. We illustrate the behaviour in the boundary of the error with the increase of the number of points, both in the maximum norm .∞ and in the 2 -norm .2 given by e∞ =

max |ei |,

i=1,...,nE

e2

  nE  1  = ei · e¯i , nE i=1

with nE = 22,735 in this case.

5 Conclusions and Future Perspective The application of MFS to simulate the process of elastography seems feasible, even in the presence of high frequencies. However, the accuracy seems to be unable to improve significantly with the increase of the number of points after some stage, probably due to the ill-conditioning of the linear system, as usual in MFS. Future perspectives for this work may include generalizing the model for the curved interfaces of the eye, validate the results against real data and solve the inverse problem, namely to numerically determine the elastic properties of the retina from the displacement field obtained by optical coherence elastography. Acknowledgments The authors would like to acknowledge their work is partially supported by FCT (Portugal) research project PTDC/EMD-EMD/32162/2017, COMPETE and Portugal2020. The first author acknowledges her work is partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. The second author acknowledges his work is partially supported by CIBIT and CNC.IBILI of the University of Coimbra—UID/NEU/04539/2019, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.

References 1. Morgado, A.M., Barbeiro, S., Bernardes, R., Cardoso, J.M., Domingues, J., Loureiro, C., Santos, M., Serranho, P.: Optical coherence elastography for imaging retina mechanical properties, FCT project. http://miguelmorgado.net/research/projects/on-going/elastooct.html 2. Claus, D., Mlikota, M., Geibel, J., Reichenbach, T., Pedrini, G., Mischinger, J., Schmauder, S., Osten, W.: Large-field-of-view optical elastography using digital image correlation for biological soft tissue investigation. J. Med. Imaging 4(1), 1–14 (2017) 3. Kennedy, B.F., Liang, X., Adie, S.G., Gerstmann, D.K., Quirk, B.C., Boppart, S.A., Sampson, D.D.: In vivo three-dimensional optical coherence elastography. Opt. Express 19(7), 6623– 6634 (2011) 4. Qu, Y., He, Y., Zhang, Y., Ma, T., Zhu, J., Miao, Y., Dai, C., Humayun, M., Zhou, Q., Chen, Z.: Quantified elasticity mapping of retinal layers using synchronized acoustic radiation force optical coherence elastography. Biomed. Opt. Express 9(9), 4054–4063 (2018)

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5. Zhu, J., Miao, Y., Qi, L., Qu, Y., He, Y., Yang, Q., Chen, Z.: Longitudinal shear wave imaging for elasticity mapping using optical coherence elastography. Appl. Phys. Lett. 110(20), 201101 (2017) 6. Thijssen, J., Mol, H., Timmer, M.: Acoustic parameters of ocular tissues. Ultrasound Med. Biol. 11(1), 157–161 (1985) 7. António, J., Tadeu, A., Godinho, L.: A three-dimensional acoustics model using the method of fundamental solutions. Eng. Anal. Boundary Elem. 32(6), 525–531 (2008). Meshless Methods Meshless Methods 8. Bin-Mohsin, B., Lesnic, D.: The method of fundamental solutions for Helmholtz-type equations in composite materials. Comput. Math. Appl. 62(12), 4377–4390 (2011) 9. Fairweather, G., Karageorghis, A., Martin, P.: The method of fundamental solutions for scattering and radiation problems. Eng. Anal. Boundary Elem. 27(7), 759–769 (2003). Special issue on Acoustics 10. Fam, G.S., Rashed, Y.F.: The method of fundamental solutions applied to 3d elasticity problems using a continuous collocation scheme. Eng. Anal. Boundary Elem. 33(3), 330–341 (2009) 11. Karageorghis, A., Lesnic, D., Marin, L.: The method of fundamental solutions for threedimensional inverse geometric elasticity problems. Comput. Struct. 166, 51–59 (2016) 12. Marin, L., Karageorghis, A., Lesnic, D.: Regularized {MFS} solution of inverse boundary value problems in three-dimensional steady-state linear thermoelasticity. Int. J. Solids Struct. 91, 127–142 (2016) 13. Smyrlis, Y.-S.: Applicability and applications of the method of fundamental solutions. Math. Comput. 78(267), 1399–1434 (2009) 14. Smyrlis, Y.-S., Karageorghis, A.: Efficient implementation of the MFS: the three scenarios. J. Comput. Appl. Math. 227(1), 83–92 (2009). Special Issue of Proceedings of {NUMAN} 2007 Conference: Recent Approaches to Numerical Analysis: Theory, Methods and Applications 15. Alves, C.J.: On the choice of source points in the method of fundamental solutions. Eng. Anal. Boundary Elem. 33(12), 1348–1361 (2009). Special Issue on the Method of Fundamental Solutions in honour of Professor Michael Golberg 16. Barnett, A., Betcke, T.: Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. J. Comput. Phys. 227(14), 7003–7026 (2008) 17. Araújo, A., Serranho, P.: On the use of quasi-equidistant source points over the sphere surface for the method of fundamental solutions. J. Comput. Appl. Math. 359, 55–68 (2019) 18. Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edn. Springer, Berlin (2013) 19. Ito, K., Qiao, Z., Toivanen, J.: A domain decomposition solver for acoustic scattering by elastic objects in layered media. J. Comput. Phys. 227(19), 8685–8698 (2008) 20. Ito, K., Toivanen, J.: A fast iterative solver for scattering by elastic objects in layered media. Appl. Numer. Math. 57(5), 811–820 (2007). Special Issue for the International Conference on Scientific Computing 21. Doyley, M.M.: Model-based elastography: a survey of approaches to the inverse elasticity problem. Phys. Med. Biol. 57(3), R35–R73 (2012) 22. Park, E., Maniatty, A.M.: Shear modulus reconstruction in dynamic elastography: time Harmonic case. Phys. Med. Biol. 51(15), 3697–3721 (2006) 23. Alves, C.J.S., Kress, R.: On the far-field operator in elastic obstacle scattering. IMA J. Appl. Math. 67(1), 1–21 (2002) 24. Bogomolny, A.: Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal. 22(4), 644–669 (1985) 25. Browder, F.E.: On approximation by solutions of partial differential equations. Bull. Am. Math. Soc. 68(1), 36–38 (1962)

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26. Alves, C.J.S.: Density results for the Helmholtz equation and the method of fundamental solutions. Adv. Comput. Eng. Sci. I, 45–50 (2000) 27. Alves, C.J., Martins, N.F., Valtchev, S.S.: Extending the method of fundamental solutions to non-homogeneous elastic wave problems. Appl. Numer. Math. 115, 299–313 (2017) 28. Jones, I.L., Warner, M., Stevens, J.D.: Mathematical modelling of the elastic properties of retina: a determination of Young’s modulus. Eye 6(15), 556–559 (1992)

Identification and Reconstruction of Body Forces in a Stokes System Using Shear Waves Nuno F. M. Martins

Abstract In this paper we consider an inverse source problem for the Brinkman system of equations (or unsteady Stokes equations). The uniqueness problem of recovering the source term from traction boundary data is studied and established from density properties of shear waves with complex frequency. This result is then applied to the reconstruction of the body force term as a superposition of shear waves. Some numerical results will be presented in order to illustrate the accuracy of the proposed method. Keywords Inverse source problems · Stokes systems · Shear waves method · Method of fundamental solutions

1 Introduction Inverse source problems belongs to a class of inverse problems that arises naturally in many scientific areas such as mechanics and medical imaging. Considered both in steady or unsteady problems using intrusive or non-intrusive data, they have been widely studied by many authors. We refer the reader to the monograph by Isakov [11] for further reading on the subject. The objective of this work is to study inverse source problems for a Brinkman system. Brinkman systems or unsteady Stokes systems have been studied both in the context of mathematical and engineering problems. These equations are the so called Stokes resolvent equations (eg. [18]) and can be obtained by applying the Laplace transform (with respect to time) to the Stokes system for time dependent problems (eg. [17]). They were proposed by H.C. Brinkman (cf. [6]) as a model

N. F. M. Martins () Department of Mathematics, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Lisboa, Portugal e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 C. Alves et al. (eds.), Advances in Trefftz Methods and Their Applications, SEMA SIMAI Springer Series 23, https://doi.org/10.1007/978-3-030-52804-1_6

103

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for fluid flow in porous media but have also been considered in several engineering problems (eg. [12]). Regarding inverse source problems for Stokes systems we refer the reader to the works by Imanuvilov and Yamamoto [10] and Choulli et al. [7] where identification results were obtained using intrusive data for time dependent Stokes systems and linearized Navier-Stokes equations, respectively. Here, we consider a non-intrusive framework for the inverse source problem and study the determination of the source term from boundary Cauchy data. It is well known that from a single measurement we cannot identify the source term (eg. [8]). Furthermore, the problem cannot be tackled by increasing the number of measurements (that is, by knowing the so called Dirichlet to Neumann map). One possibility to solve this problem is to take several measurements obtained by changing the frequency. Using Green’s formula, this enables us to recover the Fourier coefficients of the source from traction data thus leading to identification results as long as we have many of these measurements. This multi-frequency approach for inverse source problems was considered in [4] for acoustic sources and [1] also for acoustic sources but in a heterogeneous domain of propagation. For elastic sources we refer the reader to the paper [5] and a survey of these multifrequency methods can be found in [14]. For the unsteady Stokes equations, multi-frequency results were obtained in [13] for the reconstruction of both the body force and the divergence source. Here we consider only incompressible flows, that is with divergence source term equal to zero. Theoretical results concerning the reconstruction of the source term are established using elastic shear waves (eg. [16]) with complex frequency. This is a different approach when compared with that of [13]. These results are presented in Sect. 3. Then, a method for the numerical reconstruction of the source term is proposed in Sect. 4.2 and tested in Sect. 5. In Sect. 4.1 we propose a decomposition method for the direct problems. First we compute a particular solution and then solve an homogeneous Stokes boundary value problem. We develop a method of fundamental solutions for the domain problem (eg. [2]) and apply the classical method of fundamental solutions for the homogeneous problem using unsteady Stokeslets (cf. [15] and see also [3] for the Stokes system). We start with the formulation of the direct and inverse problems.

2 Direct and Inverse Problems Let  ⊂ R2 be an open, bounded, simply connected set with C 2 boundary  = ∂ and consider the following Brinkman system with Dirichlet boundary condition, ⎧ ⎨ ( − κ)u − ∇p = f in  ∇ ·u=0 in  , κ ≥ 0. ⎩ u=0 on 

(1)

Identification and Reconstruction of Body Forces in a Stokes System

105

Here we consider the usual L2 () := L2 ()2 functional framework for the body force term f. This problem has an unique solution u = (u1 , u2 ) ∈ H10 () for the velocity whilst the pressure p ∈ L2 () is determined up to a constant. This constant is uniquely determined by assuming, for instance, that

 p ∈ L20 () := p ∈ L2 () : < p, 1 >L2 () = 0 . Furthermore we can consider the more general boundary condition u = g on  with g ∈ H1/2 () satisfying the compatibility condition  g · ndS = 0,

(2)



where n(x) is the normal vector field at x ∈ , pointing outwards with respect to . Given f and knowing the domain of propagation , the direct problem consists in computing the traction boundary field gn = T (u, p)n| = (−pI + 2(u)) n| , where (u) is the strain tensor of u, 1  ∇u + ∇u = (u) = 2

)

∂u1 ∂x1

1 ∂u1 2 ( ∂x2

+

∂u2 ∂x1 )

∂u2 1 ∂u1 2 ( ∂x2 + ∂x1 ) ∂u2 ∂x2

, ,

I ∈ R2×2 is the identity matrix and (u, p) satisfies (1). The inverse problem consists in the determination of the source term f from the Cauchy data (0, gn )| . This can also be seen in terms of a linear equation $κ (f) = gn , where $κ : L2 () → H−1/2(), f → gn is the source to Neumann map. Remark 1 Note that it is sufficient to consider boundary velocities u = 0. In fact, if we consider the direct source problem ⎧ ⎨ ( − κ)w − ∇r = f in  ∇ ·w=0 in  ⎩ w=g on 

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N. F. M. Martins

then we can start by solving the following homogeneous problem (which is not related to f) ⎧ ⎨ ( − κ)v − ∇q = 0 in  ∇·v=0 in  ⎩ v=g on  and then, solving the problem ⎧ ⎨ ( − κ)u − ∇p = f in  ∇ ·u=0 in  ⎩ u=0 on  we obtain the pair (w, r) in terms of the decomposition (w, r) = (v, q) + (u, p).

3 Uniqueness Results In this section we address uniqueness results for the reconstruction of a source f from Neumann data $κ (f). In the single frequency setting, the uniqueness question is related to the injectivity of $κ . In the multi-frequency problem, the question is to decide if $κ (f) = $κ (g) for κ in some interval implies that f = g. We start with a non uniqueness result. Lemma 1 We have $κ (∇(H01 ())) = Rn| ∀κ ≥ 0. Furthermore, there exists f ∈ ∇H01 () \ {0} such that f∈

4

ker $κ .

κ≥0

Proof Let f be an irrotational field f = ∇p with p ∈ H01 (). Then T (0, −p)n = pn| = 0.

Identification and Reconstruction of Body Forces in a Stokes System

107

Hence, $κ (f) = −

 pdx

||

n|

and this shows that $κ (∇(H01 ())) ⊆ Rn| . Given a constant c ∈ R, let p be a smooth function with support contained in  and such that  pdx = −c||. 

Then, q = −p − c ∈ L20 () and $κ (−∇p) = cn| . Since p ∈ H01 (), the inclusion Rn| ⊆ $κ (∇(H01 ())) follows. In order to establish the last part of the Lemma, we take two bump functions p(x) = e

−

q(x) = ce

1 12 −|x−a|2

−

χR2 \B

1  2 −|x−b|2 2

1

χR2 \B

(a) (x),

2 (b)

(x),

2 / where B1 (a) = x ∈ R2 : |x − a| < 1 , with supports supp p = B 1 (a) ⊂ , supp q = B 2 (b) ⊂  such that supp p ∩ supp q = ∅ and c is a constant so that 

 pdx = 

qdx. 

Then p − q ∈ H01 () ∩ L20 () and, for f = ∇(p − q) ∈ ∇H01 () \ {0} we have $κ (f) = 0, ∀κ ≥ 0.

" !

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This last result and its proof shows, in particular, that given f ∈ ∇H01 () we can always find g ∈ ∇H01 () \ {f} such that $κ (f) = $κ (g), κ > 0. In this sense, there is no uniqueness in the reconstruction of irrotational fields f ∈ ∇H01 () from the Neumann data $κ (f) even if we take measurements for several frequencies κ. Furthermore, the space Rn| completely characterizes the multifrequency data space for ∇H01 () sources (Theorem 1 below). In order to establish this property we introduce the following results related to shear wave functions. Let H(∇·, ) denote the Hilbert space of L2 () functions with L2 () divergence (see for instance [9]). The divergence can be seen as a bounded linear operator ∇· : H(∇·, ) → L2 (), f → ∇ · f. The corresponding kernel is ker ∇· = {f ∈ H(∇·, ) : ∇ · f = 0 in } . This kernel is the set of solenoidal vector fields. Among these fields, we are interested in the so called shear wave functions. Given a plane wave function ψξ (x) = eix·ξ , (x, ξ ) ∈ R2 × R2 , we define a shear wave function as follows φξ = ξ ⊥ ψξ where, for ξ = (ξ1 , ξ2 ), ξ ⊥ = (−ξ2 , ξ1 ). Clearly, φ ∈ ker ∇·. Moreover, these functions can be seen as the (vector) curl of plane waves in the sense that ∇ ⊥ ψξ = iφξ . with ∇ ⊥ = (−

∂ ∂ , ) ∂x2 ∂x1

the two dimensional vector curl. Lemma 2 The set of shear waves with complex frequency / 2 φiξ | : ξ ∈ O where O ⊆ R2 is open, spans a dense subspace in ker ∇· .

Identification and Reconstruction of Body Forces in a Stokes System

109

Proof Let f ∈ H 1 () and consider the (analytic) function h(ξ ) =< f, ψiξ >H 1 () , ξ ∈ R2 . If h(ξ ) = 0 in an open set O ⊆ R2 then, by analytic continuation, h = 0 in C2 . In particular, h(iξ ) =< f, ψ−ξ >H 1 () = 0, ξ ∈ O and thus f = 0, because 2 / the set of plane waves ψξ | : ξ ∈ O spans a dense subspace in H 1 (). The density claim follows from the identities φiξ = i∇ ⊥ ψiξ and (see for example [9]) ∇ ⊥ H 1 () = ker ∇ · .

" !

Define the potential  S : L2 () → L2 (O), f →

f(x) · φiξ (x)dx. 

Then, Lemma 3 We have the following representation for S,  S(f)(ξ ) =

$|ξ |2 (f) · φiξ dS.

(3)

(u|ξ |2 , p|ξ |2 ) ∈ H1 () × L20 ()

(4)



Proof Let ξ ∈ R2 , f ∈ L2 () and

be the solution of (1) with κ = |ξ |2 . Consider the pair (v, q) = (φiξ , 0).

(5)

This pair satisfies ⎧ ⎨ ( − |ξ |2 )v − ∇q = 0 in  ∇ ·v =0 in  ⎩ on . v = φiξ The representation for S follows from applying Green’s formula,   (( − |ξ |2 )u − ∇p) · v − (( − |ξ |2 )v − ∇q) · u dx 



=

(T (u, p)n · v − T (v, q)n · u) dS 

to the pairs (4) and (5).

" !

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We can now establish the following result. Theorem 1 Let f ∈ L2 (). If there exists a constant c ∈ R such that $κ (f) = cn, ∀κ ∈ I =]0, r[ then f = ∇q ∈ ∇H01 () and

-

 qdx

= −c||.

Proof Given ξ ∈ O = B√r (0) we have, by Lemma 3,  φiξ · ndS.

S(f) = c

(6)



Since ∇ · φiξ = 0 in , then, from Gauss’s theorem, the right hand side of (6) vanishes. From the definition of S this reads  f · φiξ dx = 0, ∀ξ ∈ O 

and by Lemma 2 above, f ∈ ker ∇ ·⊥ . The conclusion that f ∈ ∇H01 () follows from the Helmholtz-Leray decomposition (eg. [9]) L2 () = ker ∇ · ⊕∇H01 ().

(7)

In particular, -

$κ (f) = $κ (∇(−q + and hence c = −

-

 qdx

||

 qdx

||

)) = −

 qdx

||

n " !

.

Note that from the Leray decomposition we can only aim for a uniqueness result in the space of solenoidal sources. This uniqueness result is a direct consequence of Theorem 1: Corollary 1 Given f ∈ L2 () satisfying the conditions of Theorem 1 then %(f) = 0,

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where % is the orthogonal projection of L2 () into ker ∇·. Therefore if $κ (f) = $κ (g) for κ ∈]0, r[ then %(f) = %(g). Remark 2 Suppose that f ∈ Hs (R2 ), s > 0 and let ˜f = F −1 (χB(0,r)F (f)) be a truncated Fourier expansion of f. It is well known that (see for example [5]) ||f − ˜f||L2 (R2 ) ≤ (1 + r 2 )−s/2||f||Hs (R2 ) . Hence, for f ∈ Hs (R2 ) such that ∇ · f = 0 in R2 we have the estimate ||f − %(˜f)||L2 () = ||%(f − ˜f)||L2 () ≤ C||%(f − ˜f)||L2 (R2 ) ≤ C||f − ˜f||L2 (R2 ) ≤ C(1 + r 2 )−s/2 ||f||Hs (R2 )

(8)

where % is the above orthogonal projection for functions defined in the whole space R2 and C > 0 is some constant depending only on . Remark 3 The previous identification results can be obtained for the 3D problem. In this case, we can consider three dimensional shear waves j

φξ = icurl(ψξ ej ), j = 1, 2, 3 with (e1 , e2 , e3 ) ∈ R3 the standard basis and, for u = (u1 , u2 , u3 ), curl(u) = (

∂u3 ∂u2 ∂u1 ∂u3 ∂u2 ∂u1 − , − , − ). ∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2

(9)

The three dimensional version of the above results can be established replacing ∇ ⊥ by the three dimensional curl (9) and using the identity (eg. [9]) curl(H1 ()) = ker ∇· where H1 () = (H 1 ())3 . Remark 4 Uniqueness results for solenoidal functions cannot be obtained when we consider only a single frequency. For instance, consider the function u(x) = 8(1 − |x|2)2 (2 + |x|2 )x ⊥

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and the domain  = B1 (0). This function satisfies ⎧ u(x) = 192(2|x|2 − 1)x ⊥ ⎪ ⎪ ⎨ ∇ ·u =0 ⎪ u=0 ⎪ ⎩ T (u, 0)n = 0

in R2 in R2 . on ∂B1 (0) on ∂B1 (0)

The source f(x) = 192(2|x|2 − 1)x ⊥ = 0 belongs to ker ∇· and $0 (f) = $0 (0).

4 Numerical Methods In this section we describe the methods applied for the numerical solution of both direct and inverse problems.

4.1 Numerical Methods for the Direct Problems In order to solve the direct problem (1) we apply a Trefftz method with unsteady Stokeslets as basis functions for the numerical approximation of Stokes systems. The two dimensional unsteady Stokeslet is given by the tensor

κ (x) =

  √ 1 x ⊗x √ d1 ( κ|x|)δij + d ( κ|x|) , κ>0 2 2π |x|2

where d1 (λ) =

1 λK0 (λ) + K1 (λ) 2 − 2 , d2 (λ) = 2 − K2 (λ) λ λ λ

and Kν is the modified Bessel function of order ν. We also define the pressure vector p(x) =

x . 2π|x|2

The pair ( κ (•−y), p(•−y)) is a fundamental solution for the Brinkman system of equations (eg. [18]), that is, 

( − κ) κ (• − y) − ∇p(• − y) = −δy I2 , y ∈ R2 . ∇ · κ (• − y) = 0

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j

In particular, when y ∈ / , the functions (uy , py ) = ( κ (• − y)ej , p(• − y) · ej ) with e1 = (1, 0), e2 = (0, 1) satisfy 

j

j

( − κ)uy − ∇py = 0 in  j in  ∇ · uy = 0

ˆ be a regular line, where  ˆ ⊂ R2 is an open, bounded, Theorem 2 Let ˆ = ∂  simply connected domain containing . The set

 ˆ j = 1, 2, κ ∈ I ⊆ R+

κ (• − y)ej | : y ∈ , spans a dense subspace in ker ∇·. Proof Let f ∈ ker ∇· such that 

ˆ j = 1, 2 f(x) · κ (x − y)ej dx = 0, y ∈ , 

and κ in an open interval I ⊆ R+ . Let (u, p) ∈ H10 () × L20 () be the solution of (1). From Green’s formula we obtain   0= f(x) · κ (x − y)ej dx = $κ (f) · κ (• − y)ej dS. 



 ˆ j = 1, 2 From [15], the set of boundary velocities κ (• − y)ej | : y ∈ , spans a dense subspace in the space of functions g ∈ H1/2 () satisfying (2). Hence, $κ (f) = 0, ∀κ ∈ I. Due to Corollary 1 we conclude that f = %(f) = 0, where the first identity holds because by hypothesis f is solenoidal. Considering u˜ part (x) =

P  2 M   j =1 k=1 l=1

αj,k,l κj (x − yk )el ,

" !

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with source points y1 , . . . , yP , placed on an artificial curve ˆ ⊂ R2 \  and κ1 , . . . , κM > 0 some frequencies, we compute the coefficients αj,k,l by imposing the linear equations ( − κ)u˜ part (x  ) − ∇ p˜part (x  ) =

M  P  2 

αj,k,l (κj − κ) κj (x  − yk )el = f(x  )

j=1 k=1 l=1

on some domain points x  ∈ . The function p˜ part is the corresponding pressure, p˜part (x) =

M  P  2 

αj,k,l p(x − yk ) · el .

j =1 k=1 l=1

It remains to approximate the solution of the homogeneous problem 

( − κ)uhomog − ∇phomog = 0, in  . ∈ uhomog = −u˜ part ,

This approximation, (u˜ homog , p˜homog ), is computed using the method of fundamental solutions for the unsteady Stokes system (cf. [15]). Therefore we obtain an approximation for the solution of the direct problem (1), (u, p) ≈ (u˜ homog + u˜ part , p˜homog + p˜ part ) with normalized pressure p˜ homog + p˜ part ∈ L20 (). The traction boundary data is approximated by $κ (f) ≈ T (u˜ homog + u˜ part , p˜ homog + p˜part )n

(10)

4.2 Numerical Method for the Source Reconstruction The numerical method for the reconstruction of a solenoidal source f is based on the representation for the operator S (Lemma 3) and Corollary 1 (uniqueness result). Let ˜f =

J  n=1

αn φν n , ν 1 , . . . ν J ∈ O

(11)

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with O an open set in R2 . Since the operator S is linear, S(˜f) =

J 

αn S(φiν n ).

n=1

We compute the coefficients αn ∈ C by imposing the conditions S(˜f) =

 

$|ξ m |2 (f) · φiξ m dS

(12)

with ξ m points in an open subset of R2 . In particular, we may consider the same points ν j = ξ j and the equations (12) can be written as AX = B with 

 φiξ m · φiξ n dx

A= 





, B= m,n



$|ξ m |2 (f) · φiξ m dS

(13)

and X = [αn ]. We take the function thus obtained, ˜f as a reconstruction for f.

5 Numerical Simulations In the following simulations we considered the domain of propagation  = B2 (0). For a given a frequency κ > 0 the Neumann data, $κ (f), was computed using the approximation (10) at 60 observation points uniformly distributed (in the sense of arc length) on the boundary  = ∂. Moreover, we considered data ˜ κ (f) = (1 + η)$κ (f) $ where, η is a random variable with uniform distribution U [−0.05, 0.05]. This simulates measurement errors by adding up to 5% of random pointwise noise to $κ (f). For the approximation of a solenoidal field f we used (11) with   2π − 0.01 2π − 0.01 ν ∈ λd : λ = 0.5, . . . , 10.5, d = (cos( j ), sin( j )), j = 0, . . . , 59 . 59 59

√ For each shear wave φν we evaluated the operator S(φν ) on a set of points κe, with 100 directions e as above. The domain integrations were performed using a quadrature formula from the software (Wolfram’s Mathematica, version 10). This gives an approximation for the coefficient matrix A defined in (13). For the approximation of vector B we applied a trapezoidal rule. A Tikhonov regularization scheme for this system was implemented with parameters 10−8 (respectively 10−3 )

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for data without noise (respectively with up to 5% of noise). These regularization parameters were obtained by trial and error. Example 1 We consider the solenoidal source (see Fig. 1) f1 (x1 , x2 ) = 2e−(x1 −0.5)

2 +(x

2 −0.8)

2

(x2 − 0.8, 0.5 − x1 ).

Considering only one boundary measurement (at 60 boundary observations) for κ = 1 we obtained the reconstructions plotted in Fig. 2. We also considered two measurements associated with the frequencies κ ∈ {1, 3}. From this data we obtained the results plotted in Fig. 3. We repeated the experience considering data with up to 5% of random noise. The corresponding results are plotted in Fig. 4.

0.5 0.0 -0.5 -2

1 0 -1

0

2 0.5 0.0 -0.5 -2

-1 1

2

2 1 0 -1

0

-2

-1 1

2

-2

Fig. 1 Plot of the source term f1 . Left—first coordinate. Right—second coordinate

0.5 0.0 -0.5 -2

2 1 0 -1

0

-1 1

2

-2

0.5

2

0.0

1

-0.5 -2

0 -1

0

-1 1

2

-2

Fig. 2 Reconstruction of the source f1 from one measurement without noise. Left—first coordinate. Right—second coordinate

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0.5 0.0 -0.5

1

-2

2 1 0 -1

0

0.5 0.0 -0.5 -2

-1 1

2

2 0 -1

0

-2

-1 1

2

-2

Fig. 3 Reconstruction of the source f1 from two measurements without noise. Left—first coordinate. Right—second coordinate

0.5 0.0 -0.5 -2

2 1 0 -1

0

-1 1

2

0.5 0.0 -0.5 -2

2 1 0 -1

0

-2

-1 1

2

-2

Fig. 4 Reconstruction of the source f1 from two measurements with up to 5% of random noise. Left—first coordinate. Right—second coordinate

Example 2 The source to be recovered is now (see Fig. 5)   5 5 3x2 sin(3x1 ) = ∇ ⊥ 2 + cos(3x1 )x2 . f2 (x1 , x2 ) = − 2 + cos(3x1 ), − √ 2 2 + cos(3x1 ) The numerical reconstruction for this source was obtained from three boundary measurements, associated with the frequencies κ ∈ {1, 3, 5}. Figure 6 shows the results for boundary data without noise and in Fig. 7 we plot the reconstruction for data with up to 5% of noise.

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-1.0 -1.2 -1.4 -1.6

2 1 0

-2

-1

0

2 1 0 -1 -2 -2

0 -1

-1 1

2 1 -1

0

1

-2

2

2

-2

Fig. 5 Plot of the source f2 . Left—first coordinate. Right—second coordinate

-0.5 -1.0 -1.5

2 1

-2

0

-1

0

2 1 0 -1 -2 -2

-1 1

2

2 1 0 -1

-1

0

1

-2

2

-2

Fig. 6 Reconstruction of the source f2 from three measurements without noise. Left—first coordinate. Right—second coordinate

2

-0.5 -1.0 -1.5 -2

0 1

-1

0

0

2

-2 -2

-1

1

1

-1

2 -2

Fig. 7 Same as last figure but for data with up to 5% of noise

0

0 -1

1 2 -2

2

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6 Conclusions In this work we studied the multi-frequency inverse source problem for the unsteady Stokes system. Uniqueness results were established using properties of shear waves basis functions with complex frequencies. The proposed numerical method is based on a weak formulation using these basis functions and some numerical results are presented in order to show its feasibility and accuracy. The numerical results show that the quality of the reconstruction is strongly influenced by the frequencies. This is expected since truncating the Fourier expansion of a function using a small number of frequencies may not give good approximation results (see Remark 2). Acknowledgments The financial support of the Portuguese Fundação para a Ciência e a Tecnologia (FCT), through the projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT/ISTID is gratefully acknowledged.

References 1. Acosta, S., Chow, S., Taylor J., Villamizar, V. : On the multi-frequency inverse source problems in heterogeneous media. Inverse Probl. 28(7), 075013 (2012) 2. Alves, C.J.S., Chen, C.S.: A new method of fundamental solutions applied to nonhomogeneous elliptic problems. Adv. Comput. Math. 23, 125–142 (2005) 3. Alves, C.J.S., Silvestre, A.L.: Density results using Stokeslets and a method of fundamental solutions for the Stokes equations. Eng. Anal. Bound. Elem. 28, 1245–1252 (2004) 4. Alves, C.J.S., Martins, N.F.M., Roberty, N.C. : Full identification of acoustic sources with multiple frequencies and boundary measurements. Inverse Probl. Imag. 3(2), 275–294 (2009) 5. Alves, C.J.S., Martins, N.F.M., Roberty, N.C. : Identification and reconstruction of elastic body forces. Inverse Probl. 30(5), 055015 (2014) 6. Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid in a dense swarm of particles. Appl. Sci. Res. A1, 27–34 (1949) 7. Choulli, M., Imanuvilov, O.Y., Puel, J.-P., Yamamoto, M.: Inverse source problem for linearized Navier-Stokes equations with data in arbitrary sub-domain. Appl. Anal. 92(10), 2127–2143 (2013) 8. El Badia, A., Ha Duong, T.: Some remarks on the problem of source identification from boundary measurements. Inverse Probl. 14, 651–663 (1998) 9. Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. In: Theory and Algorithms. Springer Series in Computational Mathematics, vol. 5. Springer, New York (1986) 10. Imanuvilov, O.Y., Yamamoto, M.: Inverse source problem for the Stokes system. In: Direct and Inverse problems of Mathematical Physics. International Society for Analysis, Applications and Computation, vol. 5, pp. 441–451. Springer, New York (2000) 11. Isakov, V. : Inverse source problems. In: Mathematical Surveys and Monographs, vol. 34. American Mathematical Society, Providence (1990) 12. Krotkiewski, M., Ligaarden, I.S., Lie, K.-A., Schmid, D.W.: On the importance of the Stokes-Brinkman equations for computing effective permeability in carbonate karst reservoirs. Commun. Comput. Phys. 10(5), 1315–1332 (2011) 13. Martins, N.F.M.: Identification results for inverse source problems in unsteady Stokes flows. Inverse Probl. 31(1), 015004 (2015)

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14. Martins, N.F.M.: Multi-frequency identification in inverse source problems. In: Inverse Problems and Computational Mechanics, vol. 2, pp. 225–249. Editura Academiei Române Bucarest, Bucure¸sti (2016) 15. Martins, N.F.M., Rebelo, M.: Meshfree methods for nonhomogeneous Brinkman flows. Comput. Math. Appl. 68(8), 872–886 (2014) 16. Nedelec, J.C.: Acoustic and electromagnetic equations. In: Integral Representations for Harmonic Problems. Applied Mathematical Sciences, vol. 144. Springer, New York (2001) 17. Pozrikidis, C.: Boundary Integral and Singularity Methods for Linearized Viscous Flows. Cambridge University Press, New York (1992) 18. Varnhorn, W. : Boundary Value Problems and Integral Equations for the Stokes Resolvent in Bounded and Exterior Domains of Rn , pp. 206–224. World Scientific Publishing, New York (1998)

MFS-Fading Regularization Method for Inverse BVPs in Anisotropic Heat Conduction Liviu Marin

Abstract We investigate the application of the fading regularization method, in conjunction with the method of fundamental solutions, to the Cauchy problem in 2D anisotropic heat conduction. More precisely, we present a numerical reconstruction of the missing data on an inaccessible part of the boundary from the knowledge of over-prescribed noisy data taken on the remaining accessible boundary part. The accuracy, convergence, stability and robustness of the proposed numerical algorithm, as well as its capability to deblur noisy data, are validated by considering a test example in a 2D simply connected domain.

1 Introduction For numerous natural and man-made materials the dependence of the thermal conductivity with direction, i.e. the anisotropy of the thermal conductivity, has to be accounted for in the modelling of the heat transfer. To exemplify some anisotropic materials, we mention crystals, wood, sedimentary rocks, metals that have undergone heavy cold pressing, laminated sheets, composites, cables, heat shielding materials for space vehicles, fibre reinforced structures, etc. The mathematical problems associated with the anisotropic heat conduction equation, also referred to as the Laplace-Beltrami equation, have been the subject of numerous studies using various numerical methods, e.g. the finite-difference method (FDM) [9, 20], the boundary

L. Marin () Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest, Romania Research Institute of the University of Bucharest (ICUB), University of Bucharest, Bucharest, Romania Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Bucharest, Romania e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 C. Alves et al. (eds.), Advances in Trefftz Methods and Their Applications, SEMA SIMAI Springer Series 23, https://doi.org/10.1007/978-3-030-52804-1_7

121

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element method (BEM) [17, 18], the finite element method (FEM) [11, 21], the method of fundamental solutions (MFS) [3, 14], etc. Direct problems in anisotropic heat conduction are characterised by the knowledge of the domain occupied by the solid, the boundary conditions on the entire boundary of the solution domain and the material properties of the solid, i.e. the anisotropic thermal conductivity tensor, and such problems have been extensively studied in the literature. However, in many real life problems, the boundary conditions available are often incomplete, either in the form of under- or overspecified boundary conditions on different parts of the boundary, or the solution is prescribed at some internal points in the domain. These problems are referred to as inverse problems and they are, in general, ill-posed [6], meaning that the existence, uniqueness and stability of their solutions are not always guaranteed. A classical example of inverse problems in anisotropic heat conduction is represented by the so-called Cauchy problem which is a particular case of inverse boundary value problems. Cauchy problems are characterised by the lack of boundary conditions on a part of the boundary of the solution domain, e.g. that boundary may be assumed to be inaccessible for measurements, whilst the remaining boundary is assumed to be over-specified, in the sense that both Dirichlet and Neumann (or Robin) boundary conditions are available. Since the Cauchy problem is ill-posed, one cannot use the same methods as those employed when solving a direct problem and, therefore, special methods should be designed to resolve these inverse boundary value problems in a stable and convergent manner. There are numerous and important theoretical and numerical studies in the literature devoted to the Cauchy problem in steady state heat conduction, however, most of these deal with the isotropic case, i.e. Laplace’s equation, whilst just a few are related to the anisotropic case, i.e. Laplace-Beltrami’s equation, see e.g. Karageorghis et al. [8] and the references therein. Mera et al. [17] implemented the alternating iterative algorithm of Kozlov et al. [12] together with the BEM to obtain a stable solution for the Cauchy problem in 2D anisotropic heat conduction. The same problem was investigated later by Jin et al. [7] and Marin [14], however by employing the MFS in conjunction with the truncated singular value decomposition method and the alternating iterative algorithm of Kozlov et al. [12], respectively. Relaxation procedures for the MFS-based alternating iterative algorithm analysed in [14] were also proposed by Marin [15]. The MFS is a meshless boundary collocation method which belongs to the family of so-called Trefftz methods [10] and is applicable to boundary value problems whose operator in the governing equation has a fundamental solution. In spite of this restriction, the MFS has become very popular primarily because of the ease with which it can be implemented, particularly for problems in complex geometries. The MFS was originally proposed by Kupradze and Aleksidze [13] and later introduced as a numerical method by Mathon and Johston [16]. Since then, it has been successfully applied to a large variety of physical problems, see e.g. the survey papers of Fairweather and Karageorghis [3], Fairweather et al. [4], and Karageorghis et al. [8].

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The aim of this work is to emphasize and combine the features of the fading regularization method, originally proposed by Cimetière et al. [1, 2] for Laplace’s equation, and the MFS in order to obtain a robust, versatile, accurate, stable and convergent iterative procedure for the numerical solution of inverse boundary value problems in 2D anisotropic heat conduction. The paper is organized as follows: In Sect. 2 we formulate mathematically the inverse problems under investigation. The fading regularization method is described in Sect. 3, whilst the MFS approach for inverse boundary value problems in 2D anisotropic heat conduction is presented in Sect. 4. The accuracy, convergence and stability of the numerical results obtained using the proposed MFS-fading regularization method are thoroughly analysed for an inverse Cauchy problem in a 2D simply connected domain with a smooth boundary in Sect. 5. Finally, some concluding remarks and possible future work are provided in Sect. 6.

2 Mathematical Formulation Consider an open bounded domain  ⊂ R2 that is bounded by a (piecewise) smooth curve ∂, such that ∂ = 0 ∪1 , where ∅ = i ⊆ ∂ for i = 1, 2, and 0 ∩1 = ∅. Assume that  is occupied by an anisotropic  solid that is characterised by the thermal conductivity tensor K(x) = Kij (x) i,j =1,2 ∈ R2×2 , x := (x1 , x2 ) ∈ , such that KT (x) = K(x) , ∃ k0 > 0 :

∀ x ∈ ;

  η · K(x) η ≥ k0 (η · η) > 0, Kij ∈ L∞ () ,

(1a)

∀ x ∈ , ∀ η ∈ R2 \ {0};

i, j = 1, 2.

(1b) (1c)

Under these assumptions, the temperature distribution u in the domain  satisfies the following elliptic partial differential equation, also referred to as the 2D anisotropic heat conduction equation or Laplace-Beltrami’s equation, see e.g. [20],   L u(x) := −∇ · K(x) ∇u(x) = 0,

x ∈ .

(2)

 T We now let n(x) := n1 (x), n2 (x) ∈ R2 be the outward unit normal vector at x ∈ ∂ and q(x) be the normal heat flux at a boundary point x ∈ ∂ defined by [20]   q(x) := n(x) · K(x) ∇u(x) ,

x ∈ ∂.

(3)

We further assume that the boundary part 1 is inaccessible and hence no information about the boundary temperature and the normal heat flux on 1 are available.

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However, the remaining part of the boundary 0 := ∂ \ 1 is assumed to be accessible and both the Dirichlet (i.e. temperature) and the Neumann (i.e. normal heat flux) boundary conditions can be measured on 0 , namely u(x) = u∗ (x),

x ∈ 0 ,

and q(x) = q ∗ (x),

x ∈ 0 ,

(4)

where u∗ ∈ H1/2 (0 ) and q ∗ ∈ H−1/2 (0 ) are the prescribed temperature and normal heat flux on 0 , respectively. These assumptions give rise to the following inverse boundary value problem for the 2D anisotropic heat conduction operator L , also referred to as the Cauchy problem: ' ' Given the compatible Cauchy data u∗ ∈ H1/2 (0 ) and q ∗ ∈ H−1/2 (0 ), ' ' determine u ∈ H1 () satisfying equations (2) and (4).

(5)

The inverse problem described by Eqs. (2) and (4) is much more difficult to solve both analytically and numerically than the corresponding direct problem since the solution does not satisfy the general conditions of well-posedness [6]. More specifically, although this problem may have a unique solution, its solution is unstable with respect to small perturbations in the data on the over-prescribed boundary 0 . Consequently, the problem is ill-posed and one cannot use the same approach as that employed for solving direct problems associated with the 2D anisotropic heat conduction operator L . Therefore, regularization methods are required in order to solve accurately the Cauchy problem given by Eqs. (2) and (4). The numerical solution to the inverse problem investigated herein is obtained by adapting and applying the fading regularization method, originally proposed by Cimetière et al. [1, 2] for Laplace’s equation, in conjunction with the method of fundamental solutions (MFS) [3] and this is described in the following two sections.

3 Fading Regularization Method According to the approach introduced by Cimetière et al. [1, 2], we define the space of solutions to the anisotropic heat conduction equation (2) as ' 

' H() := u ∈ H1 () ' L u = 0 in a weak sense ,

(6)

MFS-Fading Regularization Method for Inverse BVPs in Anisotropic Heat Conduction

125

where H1 () is the Sobolev space of functions defined on  that are L2 -integrable, along with their first-order derivatives. Next, we also define the space of compatible data on ∂ associated with the anisotropic heat conduction equation (2) by

 ' '   '' H(∂) := U := u, q(u) ' ∃ v ∈ H() s.t. v '∂ = u , q(v)'∂ = q(u) . (7a) ' Here, q(u)'∂ ∈ H−1/2 (∂) denotes the Neumann data on ∂ (i.e. the normal heat ' flux on ∂) associated with the Dirichlet data u'∂ ∈ H1/2 (∂) (i.e. the temperature on ∂), that is '   ' q(u)'∂ := n · K∇u '∂ .

(7b)

Clearly, the space of solutions H() is a closed subspace of the Sobolev space H1 () and becomes a Hilbert space if equipped with the norm induced by the scalar product on H1 (). At the same time, it can easily be showed that the space of compatible data H(∂) is a closed subspace of H1/2 (∂) × H−1/2 (∂) and it also becomes a Hilbert space when endowed with the norm  · H(∂) induced by the following scalar product U, VH(∂) := u, vH1/2 (∂) + q(u), q(v)H−1/2 (∂) ,     ∀ U = u, q(u) ∈ H(∂) , ∀ V = v, q(v) ∈ H(∂) .

(8)

For any  ⊂ ∂ one can also define H() as the space of restrictions on  of H(∂) endowed with the scalar product ·, ·H(∂) given by (8) and the corresponding induced norm  · H(∂) . With the function spaces, scalar product and norm defined by relations (6)–(8), the Cauchy problem (4) can be reformulated in an equivalent form as '     ' Given U∗ = u∗ , q ∗ ∈ H(0 ) , find U = u, q(u) ∈ H(∂) such that ' ' (9) ' ' U ' = U ∗ . 0

It is important to mention that problem (9) is in general not solvable unless the given Cauchy data are compatible, that is U∗ ∈ H(0 ). If a solution of problem (9) exists, then it is unique but does not depend continuously on the data, i.e. problem (9) is ill-posed. Therefore, in order to obtain a stable and physically meaningful solution of inverse problem (9), this should be regularized/stabilised and this is achieved by employing the fading regularization method.

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The main idea for solving inverse problem (9) consists of seeking the best solution satisfying the boundary conditions (4) on the accessible and over-prescribed boundary 0 , provided that the thermal equilibrium equation (2) is fulfilled. This leads to defining the solution of Cauchy problem (2) and (4) in terms of an approximate solution which solves the following optimisation problem: ' ' Given U∗ = u∗ , q ∗  ∈ H(0 ) , find U = u, q(u) ∈ H(∂) such that ' (10a) '   ' J (U) ≤ J (V) , ∀ V = v, q(v) ∈ H(∂) , where J (·) : H(∂) −→ [0, ∞) ,

J (V) = V − U∗ 2H( ) . 0

(10b)

It should be noted that the optimisation problem (10a) is also ill-posed. Moreover, ' it is always possible to find a solution U ∈ H(∂) for which U' approximates ' 0 very well the over-prescribed data U∗ , whilst at the same time U'∂\ is unstable. 0 This instability of the solution should be overcome and to do this, it is necessary to introduce a control term in the functional J . This is achieved by replacing the optimisation problem (10a) with the following one: ' ' Given U∗ = u∗ , q ∗  ∈ H() , find U = u, q(u) ∈ H(∂) such that ' (11a) '   ' Jλ (U) ≤ Jλ (V) , ∀ V = v, q(v) ∈ H(∂) , where 62 6 62 6 Jλ (·) : H(∂) −→ [0, ∞) , Jλ (V) = 6V − U∗ 6H( ) + λ 6V − 6H(∂) , 0 (11b) λ > 0 is a parameter to be specified and  ∈ H(∂). The functional Jλ contains a control term defined on the entire boundary of the solution domain, namely V − 2H(∂) , which may be regarded as a regularizing term. Such a control term makes problem (11a) well-posed in the sense of Hadamard [6] and, consequently, its solution U depends continuously on the data U∗ , the parameter λ and the choice of . In order to obtain a solution of the Cauchy problem (2) and (4) independent of the parameter λ and the choice of , we may regard it as the limit of a sequence of well-posed optimisation problems, i.e. instead of considering it as the solution of the optimisation problem (11). This approach actually yields an iterative regularizing strategy, see e.g. [1, 2], and makes the solution of problem (9) the fixed point of an operator defined on H(∂) and taking values in H(∂) or, equivalently, each iterate

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is the solution of a well-posed optimisation problem. More precisely, one obtains the following iterative algorithm: Step 1: Choose the initial guess U(0) = 0. Step 2: For each k ≥ 1, solve the following minimisation problem: ' ' Find U(k) ∈ H(∂) such that ' ' ' Jλ(k−1) (U(k) ) ≤ Jλ(k−1) (V) , ∀ V ∈ H(∂) ,

(12a)

Jλ(k−1) (·) : H(∂) −→ [0, ∞) , 6 6 62 62 Jλ(k−1) (V) = 6V − U∗ 6H( ) + λ 6V − U(k−1) 6H(∂) . 0

(12b)

where

The above iterative procedure (12) can also be written in the following equivalent form: Step 1: Choose the initial guess U(0) = 0. Step 2: For each k ≥ 1, solve the following problem: ' ' Find U(k) ∈ H(∂) such that ' (13) ' (k) ' U − U∗ , VH(0 ) + λ U(k) − U(k−1) , VH(∂) = 0 , ∀ V ∈ H(∂) . This iterative process accounts, in a very good and precise manner, for the heat conduction equation (2) since at each iteration the optimal element is sought in the space H(∂). The minimising functional in (11b) or (12b) consists of two terms, each of these having its very specific own role. The first term of the functional given by (11b) or (12b) is defined on 0 only and measures the gap between the sought optimal element in H(∂) and the given over-specified boundary conditions on 0 . The second term of the functional from equation (11b) or (12b) is a regularization term and is defined not only on the under-specified boundary 1 := ∂ \ 0 , where the boundary data is reconstructed, but on the entire boundary ∂ of the solution domain, with the mention that this term controls the distance between the new optimal element sought at the present iteration and that obtained at the previous iteration. It should be noted that the norm of the regularization term decreases and tends to zero as the number of iterations increases. Hence at each iteration, the optimal element satisfies the heat conduction equation ' '  as well as (2)' and agrees possible with the over-specified boundary data U∗ ' = u∗ ' , q ∗ ' , while at the 0 0 0 same time remains close to the optimal element obtained at the previous iteration. We also emphasize that the proposed fading regularization algorithm allows for both the reconstruction of the missing boundary data on the under-specified boundary 1 and the denoising of the perturbed over-specified boundary data on 0 . Analogous to the approach of Cimetière et al. [1, 2], one can show the following convergence result for the sequence of approximate solutions generated by the

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fading regularization method in the case of the Cauchy problem associated with the 2D anisotropic heat conduction:   Theorem 1 Let U∗ = u∗ , q ∗ ∈ H(0 ) be a pair of compatible and exact Cauchy   data and U(e) = u(e) , q (e) ∈ H(∂), where u(e) ∈ H1 () is the solution of the corresponding Cauchy problem (2) and (4), and q (e) := q(u(e)). 2 / (k) Then the sequence U k≥1 ⊂ H(∂) produced by the iterative procedure (12) ' ' converges strongly to U(e) ' := U∗ in H(0 ) and weakly to U(e) '∂ in H(∂), 0 respectively.

4 Method of Fundamental Solutions In the following, we assume that the anisotropic solid occupying the 2D domain  is homogeneous and hence characterised by a heat conductivity tensor K ∈ R2×2 . A fundamental solution for the temperature in 2D homogeneous anisotropic heat conduction is given by, see e.g. Fairweather and Karageoghis [3], F(x, ξ) = −

1 √ ln R(x, ξ) , 2π det(K)

x ∈ ,

ξ ∈ R2 \  ,

(14a)

where R(x, ξ) is the so-called geodesic distance between the collocation point x ∈  and the source point ξ ∈ R2 \  and is defined by R(x, ξ)2 := (x − ξ) · K−1 (x − ξ) ,

x ∈ ,

ξ ∈ R2 \  .

(14b)

By taking the collocation point x ∈ ∂ and differentiating Eq. (14a) with respect to i = 1, 2, one obtains, in a straightforward manner, the gradient of the fundamental solution ∇x F(x, ξ) for the temperature. Further, by combining these expressions for ∇x F(x, ξ) with the definition of the normal heat flux at x ∈ ∂ given by Eq. (3), one obtains the fundamental solution for the normal heat flux on ∂ in 2D homogeneous anisotropic heat conduction, i.e. G(x, ξ; n(x)) := n(x) · ∇x F(x, ξ) = − x ∈ ∂ ,

1 n(x) · K−1 (x − ξ) , √ R(x, ξ)2 2π det(K)

(14c)

ξ ∈ R2 \  .

The temperature is approximated, in the MFS, by/ a linear combination of 2 fundamental solutions (14a) with respect to N sources ξ(n) n=1,N ⊂ R2 \ , in the form [3] u(x) ≈ uN (c, ξ; x) =

N  n=1

F(x, ξ(n) ) cn ,

x ∈ ,

(15a)

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T  where the vector c := c1 , c2 , . . . , cN ∈ RN contains the unknown MFS 2N coefficients, cn ∈ R, n = 1, N , and ξ ∈ R is a vector containing the coordinates  (n) T ∈ R2 , n = 1, N. Analogous to Eq. (15a), one of the sources ξ(n) := ξ(n) 1 , ξ2 can approximate the normal heat flux on ∂ / by2a linear combination of fundamental solutions (14c) with respect to N sources ξ(n) n=1,N ⊂ R2 \  as q(x) ≈ qN (c, ξ; x, n(x)) =

N 

G(x, ξ(n) , n(x)) cn ,

x ∈ ∂ .

(15b)

n=1

At each step k ≥ 0 of the minimisation problem (11) ' or (12)' associated with the fading regularization method, both the known u' and q ' and the 0 0 ' ' unknown boundary data u' and q ' are approximated by employing the MFS 1 1 formulae (15a) and (15b), respectively, and also taking into account the given boundary conditions 2 (9). Therefore, we collocate the boundary conditions (9) / at the points x(m) m=1,M ⊂ 0 and also express the approximations (15a) 0 and 2 for the unknown boundary temperature and normal heat flux at the points / (m)(15b) x ⊂ 1 , where M := M0 + M1 . Consequently, at each step m=M0 +1,M0 +M1 k ≥ 0, the minimisation problem (11) or (12) is reduced to a linear minimisation problem with respect to the corresponding unknown MFS constants cn , n = 1, N .

5 Numerical Results and Discussion In this section, we apply the fading regularization method described in Sect. 3, in conjunction with the MFS presented in Sect. 4, to the Cauchy problem in 2D anisotropic heat conduction to one test example. More specifically, we consider  an anisotropic material that is characterised by the heat conduction tensor K = Kij i,j =1,2 ∈ R2×2 given by K11 = 2.0 ,

K12 = K21 = 0.5 ,

K22 = 1.0 ,

(16a)

and occupies the unit disk ' 2 /  = x ∈ R2 ' ρ(x) < 1, θ (x) ∈ [0, 2π) ,

(16b)



x12 + x22 and θ (x) := atan(x2 /x1 ) denote the polar radius and angle  T corresponding to x := x1 , x2 ∈ R2 , respectively. We further assume that Cauchy data is available on the accessible over-specified boundary where ρ(x) :=

' / 2 0 = x ∈ ∂ ' ρ(x) = 1, θ (x) ∈ (0, 3π/2) ,

(16c)

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whilst no data is available on the remaining boundary ' / 2 1 := ∂ \  0 = x ∈ ∂ ' ρ(x) = 1, θ (x) ∈ (3π/2, 2π) .

(16d)

We also consider the following analytical (exact) solutions for the temperature u(an) (x) =

1 1 K21 K22 x12 − K11 K22 x1 x2 + K12 K11 x22 , 2 2

x ∈ ,

(17a)

and the corresponding normal heat flux on the boundary 0 1 q (an) (x) = det(K) K11 x2 n1 (x) + K22 x1 n2 (x) ,

x ∈ ∂ .

(17b)

We further take M0 and M1 uniformly distributed collocation points on 0 and 1 , respectively, such that M = M0 + M1 and M1 = M0 /3, as well as N = M uniformly distributed sources associated with the entire boundary ∂ of the solution domain . It should be mentioned that the sources are preassigned and kept fixed throughout the solution process (i.e. the so-called static MFS approach) on a pseudo 7 of a similar shape to that of ∂ such that d := dist ∂ , 7 ∂ > 0 is a boundary ∂  fixed constant, see e.g. Gorzela´nczyk and Kołodziej [5]. To simulate the inherent measurement errors, we consider that the boundary data corresponding to the inverse problems' investigated' herein is noisy, ' i.e. we assume ' that the given exact boundary data u∗ ' = u(an) ' and/or q ∗ ' = q (an) ' has 0 0 0 0 been perturbed as   u∗ε (x) = 1 + pu ϕ u(an) (x) ,

x ∈ 0 ,

(18a)

  qε∗ (x) = 1 + pq ϕ q (an) (x) ,

x ∈ 0 ,

(18b)

' ' where pu and pq are the levels of percentage noise added to u∗ ' = u(an) ' and 0 0 ' ' q ∗ ' = q (an) ' , respectively, and ϕ is a pseudo-random number drawn from the 0

0

standard uniform distribution on the interval [−1, 1] generated using the MATLAB command −1 + 2 ∗ rand(·). In order to assess the accuracy and convergence of the combined MFS-fading regularization algorithm proposed herein, for  = /0 or2  = ∂ \  0 , any scalarvalued function f :  −→ R and any set of points x(n) n=1,N ⊂ , we introduce  the following relative root mean square (RMS) error of f on :  9   N N 82  1   1  0      2 (num) (n) (n)   f x −f x fj x(n) , e (f ) = N N n=1

(19)

n=1

where f (num) (x) denotes an approximate numerical value for f (x) with x ∈ .

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Figures 1 and 2 present the relative RMS errors e∂\ 0 (u) and e∂\0 (q), respectively, as functions of the number of iterations k, when applying the fading regularization method presented in Sect. 3, in conjunction with the MFS described in Sect. 4, using M = N = 80, d = 3.0, pu ∈ {1%, 3%, 5%} and λ = 10−3 , for the Cauchy problem given by Eqs. (2) and (4). It can be seen from these figures that both these errors attain their corresponding minimum for about the same number of iterations k, after which they start increasing and ultimately reach a plateau region. Consequently, the combined MFS-fading regularization algorithm should be regularized by stopping it at the aforementioned value of k and this is achieved by introducing an appropriate stopping criterion. As can be seen from Figs. 3 and 4, a good option for the required stopping criterion would be to cease the ' iterative procedure when'the relative RMS errors (19) for the known temperature u' and normal heat flux q ' , respectively, attain their 0

Fig. 1 The relative RMS error e∂\0 (u), obtained using M = N = 80, d = 3.0, pu ∈ {1%, 3%, 5%} and λ = 10−3

0

100

10-1

10-2

10-3

10-4

Fig. 2 The relative RMS error e∂\0 (q), obtained using M = N = 80, d = 3.0, pu ∈ {1%, 3%, 5%} and λ = 10−3

0

50

100

150

200

250

300

0

50

100

150

200

250

300

100

10-1

10-2

10-3

132 Fig. 3 The relative RMS error e0 (u), obtained using M = N = 80, d = 3.0, pu ∈ {1%, 3%, 5%} and λ = 10−3

L. Marin 100

10-1

10-2

10-3

10-4

Fig. 4 The relative RMS error e0 (q), obtained using M = N = 80, d = 3.0, pu ∈ {1%, 3%, 5%} and λ = 10−3

0

50

100

150

200

250

300

0

50

100

150

200

250

300

100

10-1

10-2

10-3

10-4

corresponding minimum. However, since the fading regularization method may be regarded as a generalization of the classical Tikhonov regularization method applied to the continuous inverse problem under investigation, we prefer to employ another stopping criterion, which is rigorous and, at the same time, has a mathematical rationale, namely the discrepancy principle of Morozov [19]. To do so, we introduce the following convergence error ' E(k) := U(k) ' − U∗ε 2 , 0

(20a)

' where U∗ε and U(k) ' are the measured noisy data and the computed data at step 0 k of the iterative procedure on the over-specified boundary 0 , respectively. Note that this convergence error actually measures the fit of the numerical solution at step k into the discrete MFS system and represents the discretised version of the

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(k−1)

first term in the minimisation functional Jλ given by Eq. (12). The discrepancy principle [19] states that the iterative procedure should be stopped at the first iteration, kopt , when the convergence error has about the same order as the noise in the data, namely ' 2 / kopt = min k ≥ 1 ' E(k) ≤ ε ,

(20b)

' ' ' ' where ε = u∗ε ' − u∗ ' 2 + qε∗ ' − q ∗ ' 2 is a discrete measure of the noise 0 0 0 0 in the data. Figure 5 displays the convergence error E defined by Eq. (20a) as a function of −3 the number of iterations k, obtained ' using M = N = 80, d = 3.0, λ = 10 and ' various levels of noise added to u  , for the Cauchy problem investigated herein, 0 as well as the corresponding values of ε (the dotted lines). By comparing Figs. 1, 2 and 5, it can be concluded that Morozov’s discrepancy principle (20b) yields a good approximation for the minimum of the relative RMS accuracy errors e∂\0 (u) and e∂\0 (q) and, consequently, has a stabilizing effect on the iterative procedure given by the fading regularization MFS algorithm presented herein. Figures 6 and 7 present the exact and numerical temperature u and normal heat flux q on the under-specified boundary ∂ \  0 , respectively, retrieved using the combined MFS-fading regularization algorithm, the stopping criterion (20b), M = −3 N = 80, ' d = 3.0, λ = 10 and various amounts of noise added to the Dirichlet ' data u  . It can be seen from Figs. 6 and 7 that the fading regularization algorithm 0 and the MFS, presented in Sects. 3 and 4, respectively, together with the discrepancy principle of Morozov [19] produce very accurate and stable approximations with ' respect to decreasing the level of noise added to the Dirichlet data u' for both the 0 unknown boundary temperature and normal heat flux on 1 . Although not presented here, it is reported that very accurate and stable reconstructions for the unknown Fig. 5 The convergence error ' E(k) = U(k) ' − U∗ε 2 , as 0 a function of the number of iterations k and the corresponding ε (· · · ) given by Morozov’s discrepancy principle, obtained using M = N = 80, d = 3.0, pu ∈ {1%, 3%, 5%} and λ = 10−3

101

100

10-1

10-2 0

50

100

150

200

250

300

134 Fig. 6 Exact and numerically recovered temperature ' u'∂\ , obtained using 0 Morozov’s discrepancy principle, M = N = 80, d = 3.0, pu ∈ {1%, 3%, 5%} and λ = 10−3

L. Marin 1.4 1.2 1 0.8 0.6 0.4 0.2 0.75

Fig. 7 Exact and numerically' recovered normal heat flux q '∂\ , obtained 0 using Morozov’s discrepancy principle, M = N = 80, d = 3.0, pu ∈ {1%, 3%, 5%} and λ = 10−3

0.8

0.85

0.9

0.95

1

0.8

0.85

0.9

0.95

1

3 2.5 2 1.5 1 0.5 0 -0.5 0.75

boundary temperature and normal heat flux on ∂ \  0 have been obtained when the Neumann data has been perturbed on the over-specified boundary 0 . Next, we discuss the sensitivity of the numerical results obtained with respect to the distance of the sources from the boundary of the solution domain, as well as the parameter λ associated with the minimisation functional (12). Table 1 presents the values of the relative RMS errors, e∂\0 (u) and e∂\ 0 (q), the convergence error E defined by Eq. (20a) and the number of iterations performed kopt , obtained when applying the MFS-fading regularization algorithm in conjunction with the stopping

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Table 1 The values of the errors, e∂\0 (u), e∂\0 (q) and E, and the number of iterations performed, kopt , obtained using M = N = 80, d = 3.0, pu ∈ {1%, 3%, 5%} and various values of λ λ

pu

e∂\0 (u)

e∂\ 0 (q)

E

kopt

1.0 × 10−2

1% 3% 5% 1% 3% 5% 1% 3% 5% 1% 3% 5%

3.67 × 10−3 1.31 × 10−2 2.44 × 10−2 5.24 × 10−3 1.55 × 10−2 2.63 × 10−2 4.30 × 10−3 1.32 × 10−2 2.11 × 10−2 3.82 × 10−3 1.29 × 10−2 1.44 × 10−2

1.55 × 10−2 5.02 × 10−2 9.03 × 10−2 1.97 × 10−2 5.81 × 10−2 9.80 × 10−2 1.57 × 10−2 4.80 × 10−2 7.65 × 10−2 1.43 × 10−2 4.72 × 10−2 5.35 × 10−2

3.15 × 10−2 9.46 × 10−2 1.58 × 10−1 3.15 × 10−2 9.46 × 10−2 1.58 × 10−1 3.15 × 10−2 9.46 × 10−2 1.58 × 10−1 3.15 × 10−2 9.46 × 10−2 1.58 × 10−1

333 155 91 125 66 41 24 14 10 12 7 6

5.0 × 10−3

1.0 × 10−3

5.0 × 10−4

criterion (20b), M = N = 80, d = 3.0, perturbed Dirichlet data on 0 and various values of λ. The following conclusions can be drawn from Table 1: (a) For all levels of noise pu and all values of λ considered, as expected, the inaccuracies in the numerical normal heat fluxes are higher than those in the numerical temperatures since the former contain derivatives of the latter. (b) For each fixed value of λ, the relative RMS errors, e∂\0 (u) and e∂\0 (q), which measure the accuracy of the numerically reconstructed boundary temperature and normal heat flux, respectively, decrease with respect to decreasing the ' amount of noise added to the prescribed temperature u' . 0 (c) There is a relatively wide range of admissible values of λ to be used in the functional (12), namely λ ∈ [5.0 × 10−4 , 1.0 × 10−2 ], for the inverse problem analysed in Table 1. (d) For each fixed amount of noise pu and all admissible values of λ, the relative RMS errors, e∂\0 (u) and e∂\0 (q), are almost independent of λ, hence showing the robustness of the MFS-fading regularization algorithm with respect to the admissible value of λ chosen. (e) The number of iterations, kopt , required for the MFS-fading regularization algorithm to attain the convergence of the numerical solution according to the discrepancy principle of Morozov [19] decreases with respect to decreasing the parameter λ, with the mention that there is a certain range for the admissible values of the latter, see also (c). (f) The best numerical results, in terms of accuracy, are obtained when the convergence error, E, has a value as close as possible to that of ε, i.e. the a priori estimate for the level of noise in the prescribed Cauchy data.

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Table 2 The values of the errors, e∂\0 (u), e∂\ 0 (q) and E, and the number of iterations performed, kopt , obtained using M = N = 80, d = 4.0, pu ∈ {1%, 3%, 5%} and various values of λ λ

pu

e∂\0 (u)

e∂\0 (q)

E

kopt

5.0 × 10−2

1% 3% 5% 1% 3% 5% 1% 3% 5% 1% 3% 5%

3.66 × 10−3 1.09 × 10−2 1.85 × 10−2 3.96 × 10−3 1.16 × 10−2 1.94 × 10−2 3.77 × 10−3 1.13 × 10−2 1.81 × 10−2 3.55 × 10−3 8.32 × 10−3 1.61 × 10−2

1.26 × 10−2 3.75 × 10−2 6.38 × 10−2 1.38 × 10−2 4.06 × 10−2 6.79 × 10−2 1.31 × 10−2 3.94 × 10−2 6.31 × 10−2 1.22 × 10−2 2.82 × 10−2 5.50 × 10−2

3.15 × 10−2 9.46 × 10−2 1.58 × 10−1 3.15 × 10−2 9.46 × 10−2 1.58 × 10−1 3.15 × 10−2 9.46 × 10−2 1.58 × 10−1 3.15 × 10−2 9.46 × 10−2 1.58 × 10−1

486 309 226 83 53 39 52 33 25 11 8 6

1.0 × 10−2

5.0 × 10−3

1.0 × 10−3

Similar conclusions can be drawn from Table 2, which tabulates the values of e∂\ 0 (u), e∂\ 0 (q), E and kopt , obtained when applying the MFS-fading regularization algorithm, the stopping criterion (20b), M = N = 80, d = 4.0, pu ∈ {1%, 3%, 5%} and various values of λ. By comparing Tables 1 and 2, it can be seen that, for the same number ' of MFS boundary collocation points, all levels of noise in the Dirichlet data u' and all admissible values of λ, the numerically 0 reconstructed solutions for the unknown temperature and normal heat flux on 1 become more accurate when increasing the distance d from the boundary of the 7 associated with the MFS sources. solution domain to the pseudo-boundary ∂ 

6 Concluding Remarks In this paper, we have studied the numerical reconstruction of the missing temperature and normal heat flux on an inaccessible part of the boundary for the 2D anisotropic heat conduction from over-prescribed noisy measurements taken on the remaining accessible boundary. This inverse problem was solved by combining the fading regularization method presented in Sect. 3 with the MFS described in Sect. 4. The iterative procedure was stopped according to the discrepancy principle of Morozov [19]. This inverse Cauchy problem has been thoroughly investigated for a 2D simply connected domain with a smooth boundary and it was showed that the numerical solutions retrieved by the MFS-fading regularization algorithm are accurate, convergent, stable and very robust with respect to the numerical parameters of the problem, i.e. M, N, d and λ. From the numerical results obtained and presented in Sect. 5, we can conclude that the MFS-fading regularization algorithm described, analysed and implemented

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in this paper provides us with very accurate, convergent and stable numerical results for boundary data reconstruction problems in 2D anisotropic heat conduction, for a wide range of values of the admissible parameter λ, at the same time being a very robust and versatile iterative algorithm for such inverse Cauchy problems. Future extensions of the fading regularization MFS algorithm are currently under investigation and they refer to inverse boundary value problems associated with 3D isotropic and anisotropic heat conduction, as well as 2D and 3D linear thermoelasticity. Acknowledgments This work was supported by a grant of Ministry of Research and Innovation, CNCS–UEFISCDI, project number PN–III–P4–ID–PCE–2016–0083, within PNCDI III.

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17. Mera, N.S., Elliott, L., Ingham, D.B., Lesnic, D.: The boundary element solution for the Cauchy steady heat conduction problem in an anisotropic medium. Int. J. Numer. Meth. Eng. 49, 481–499 (2000) 18. Mera, N.S., Elliott, L., Ingham, D.B., Lesnic, D.: A comparison of boundary element formulations for steady state anisotropic heat conduction problems. Eng. Anal. Bound. Elem. 25, 115–128 (2001) 19. Morozov, V.A.: On the solution of functional equations by the method of regularization. Doklady Math. 7, 414–417 (1966) 20. Özi¸sik, M.N.: Heat Conduction. Wiley, New York (1993) 21. Szabó, B., Babuška, I.: Introduction to Finite Element Analysis. Wiley, New York (2011)

Non-intrusive Estimate of Spatially Varying Internal Heat Flux in Coiled Ducts: Method of Fundamental Solutions Applied to the Reciprocity Functional Approach Andrea Mocerino, Fabio Bozzoli, Luca Cattani, Pamela Vocale, and Sara Rainieri

Abstract Wall curvature is a widely used technique to passively enhance convective heat transfer that has also proven to be effective in the thermal processing of highly viscous fluids. These geometries produce a highly uneven convective heatflux distribution at the wall along the circumferential coordinate, thus affecting the performance of the fluid thermal treatment. Although many authors have investigated the forced convective heat transfer in coiled tubes, most of them have presented the results only in terms of heat flux density averaged along the wall circumference. The estimation of the heat exchanger performances requires the proper knowledge of the thermo-fluid dynamic interaction between fluid and device. One of these aspects is related to the estimation of the internal convective heat flux: unfortunately, this quantity is really complicated to measure. One of the most challenging applications requires the solution of the inverse heat conduction problem. This approach deals with the estimation of the local internal properties, given the external temperature measurements, possibly by means of contactless experimental methodologies (i.e. infrared camera imaging). The solution strategy presented here is based on the reciprocity functional approach, which requires the solution of a sequence of auxiliary problems, solved in this paper by means of the method of fundamental solutions. Its adoption permits to obtain a very effective estimation of the local heat flux with a small computational effort. Keywords Inverse heat conduction problem · Reciprocity functional · Method of fundamental solutions · Local heat flux

A. Mocerino · F. Bozzoli () · S. Rainieri Department of Industrial Engineering, University of Parma, Parma, Italy e-mail: [email protected] L. Cattani · P. Vocale CIDEA Interdepartmental Centre, University of Parma, Parma, Italy © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 C. Alves et al. (eds.), Advances in Trefftz Methods and Their Applications, SEMA SIMAI Springer Series 23, https://doi.org/10.1007/978-3-030-52804-1_8

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Nomenclature h k n qg r F ,G K N S X Y γ Θ Γ ψ Ω

Convective heat transfer coefficient (W/m2 K) Number of harmonics of the orthonormal basis Outward-pointing unit normal vector Heat source per unit volume (W/m3 ) Radial coordinate (m) Auxiliary functions Thermal conductivity (W/m K) Total number of basis functions Numbers of external surface temperature measurements Sensitivity matrix Measured temperature available at the external surface (K) Trace of the auxiliary function at the internal boundary Circumferential coordinate (rad) Domain’s boundary Generic orthonormal basis function Cross sectional domain

Subscripts int ext b env avg

Internal External Bulk Environmental Average

1 Introduction In many industrial applications the estimation of local heat flux on internal surfaces of heat transfer devices represents a crucial issue to obtain good working performance or to improve existing apparatuses. While measuring temperature is usually an easy task, direct measurement of heat flux is not always straightforward and may not be even possible. An industrial sector that is for sure interested by the necessity of the estimation of the local heat transfer performance is constituted by the commercial heat exchangers. In fact, for savings in materials and energy use, passive heat transfer enhancement techniques are usually adopted (e.g., treated surfaces, rough surfaces, displaced enhancement devices, swirl-flow devices, surface-tension devices, coiled tubes, or flow additives) [1]. These methods originate a significant variation of the heat fluxes at the fluid-wall interface along the surfaces and, in some applications in which uniform thermal processes are required, this irregular distribution behaviour impacts negatively on the efficiency of these devices [2]. Among passive heat transfer enhancement techniques, wall curvature represents one of the most widely adopted and has proven to also be effective in the thermal processing of highly viscous fluids. This geometry produces a highly uneven convective heat-flux distribution at the wall along the circumferential coordinate, thus affecting the performance of the fluid thermal

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treatment [2]. Although many authors have investigated the forced convective heat transfer in coiled tubes, most of them have presented the results only in terms of heat flux density averaged along the wall circumference. The estimation of the heat exchanger performances requires the proper knowledge of the thermo-fluid dynamic interaction between fluid and device. One of these aspects is related to the estimation of the internal convective heat flux: unfortunately, this quantity is really complicated to measure. The hint for succeeding in the estimation of the local behaviour comes from some research papers in which the space-variable heat transfer coefficient on the inner/outer pipe surface was determined, given measurements of temperature at some points of the pipe wall. Some of these authors [3, 4] used a specific experimental setup, based on a thin metallic layer, applied to the tube wall, which was heated by Joule effect. Other authors [5–9] adopted the inverse heat conduction problem (IHCP) solution approach to estimate the local convective heat transfer coefficient starting from the temperature distribution measured on the other side of the pipe wall. Although the IHCP approach is very promising because it can be applied to any kind of heating configuration and pipe geometry, it presents some complications since it is characterised by an ill-posed nature and, consequently, it is very sensitive to small perturbations in the input data. Many methods have been developed for solving IHCPs using analytical and numerical techniques. Some of these methods are the conjugate gradient iterative method; Laplace transform method; sequential function specification method; regularization methods, such as Tikhonov regularization; mollification method; truncated singular value decomposition method; and filtering technique method [10, 11]. Particularly effective for this kind of application is the Reciprocity Function approach (RF), which needs low computational resources. Two important works by Andrieux and Abda [12, 13] described the core concepts and the working principles of reciprocity functional. Starting from these works, other studies based on the reciprocity functional approach began to emerge in different areas. Delbary et al.[14] developed a qualitative method for breast cancer detection by combining the reciprocity functional method with the linear sampling method. Colaço and Alves [15] estimated spatial variation of the thermal contact conductance by using a reciprocity functional approach with the method of fundamental solutions and nonintrusive temperature measurements. Shifrin and Shushpannikov [16] developed a method for identifying small defects in an anisotropic elastic body based on the reciprocity functional. According to the previous discussion, the estimation of the local heat flux is not an easy task. The aim of the present work is to present a procedure to determine the spatial variation of this quantity without intrusive measurements. The presented methodology is formulated in terms of a reciprocity functional approach coupled with the method of fundamental solutions [17] to solve a sequence of auxiliary problems. The Method of Fundamental Solutions (MFS) is a boundary-type meshfree technique for the solution of partial differential equations and it has become very popular in recent years due to its ease of implementation [18]. The meshfree feature is really attractive when a modification of the domain shape is required, as for instance occurs in shape optimization and inverse problems [19]. The MFS presents some benefits over classical boundary element methods: it

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doesn’t necessitate an elaborate discretization of the boundary; the determination of an approximation to the solution at a point in the interior of the domain of the problem only requires an evaluation of the approximate solution; its derivatives approximation can also be evaluated directly [20]. The new solution methodology is firstly tested using synthetic data and then it is applied to an experimental case with real measurements.

2 Direct Problem The objective of this work is to present and validate a procedure to estimate the local heat flux on the internal side of a coiled pipe (Fig. 1a), in a forced convection problem. The test section is modelled as a 2D solid domain (Fig. 1b) since the temperature gradient along the tube’s axis is assumed to be negligible with respect to the one along the angular direction. Consequently, the problem under test can be schematized as in Fig. 2 where the cross section of the pipe is shown together with the adopted coordinate system: the fluid, having bulk temperature Tb flows internally, while the external pipe’s wall is exposed to a uniform temperature environment Tenv and a uniform heat generation qg is considered within the solid wall domain Ω. Both external and internal surfaces are subjected to Neumann’s boundary condition: the exterior surface Γext is exposed to an overall convective heat transfer coefficient henv while the interior surface to a heat flux distribution qint . The circular section presents an internal radius rint , an external radius rext and the wall is characterized by a thermal conductivity K, assumed to be constant and uniform.

Fig. 1 (a) Example of a curved pipe; (b) particular of the coiled pipe

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Fig. 2 (a) Physical model; (b) direct problem boundary conditions

Under the conditions described above, the local steady state energy balance equation in the solid domain is expressed as follows: −qg in Ω ∇ 2T = K ' ∂T '' −K = qint on Γint ∂n 'Γint ' ∂T '' −K = henv (T − Tenv ) on Γext ∂n 'Γext

(1) (2) (3)

The direct problem presented in Eqs. (1)–(3) concerns the determination of the temperature field T within the domain Ω for a given distribution of the heat flux qint on the internal surface. On the other hand, the inverse problem consists in recovering the internal heat flux qint given temperature measurements Y taken on the external surface Γext .

3 Inverse Problem The methodology used to solve the inverse heat conduction problem in the pipe wall is based on the reciprocity functional approach [12, 13]; this technique, under the assumption of linear inverse problem, can restore the internal heat flux avoiding the use of iterative procedures, starting from experimental temperature measurements on the external surface (Fig. 3). In the inverse formulation Tb , K, henv and Tenv are

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Fig. 3 Inverse problem boundary conditions

supposed to be known values. The determination of the unknown internal heat flux qint is performed by solving auxiliary problems [19]. As suggested by Colaço et al. [21], the following well-posed auxiliary problem, given by the equations (4)–(6), can be used to obtain qint : ∇ 2G = 0 ' ∂Gk '' =0 ∂n 'Γint

in Ω

Gk = ψk

on Γext

(4)

on Γint

(5) (6)

where ψk is a set of orthogonal functions, which, for a 2D case, can be written as a standard orthonormal Fourier’s basis (7)–(9): 1 ψk = √ √ rext 2π 1 k ψk = √ √ cos (θ ) 2 rext π 1 k−1 ψk = √ √ sin (θ ) 2 rext π

f or

f or

k=1

(7)

k = 2, 4, 6, . . . .N − 1

(8)

k = 3, 5, 7, . . . .N

(9)

f or

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Defining the reciprocity functional and the integral of the internal heat source, respectively, as follows:  RG,k = Γext



 = Γext

  ∂Gk ∂T −Y dΓext = ψk ∂n ∂n

(10)

 ∂Gk henv (T − Tenv ) −Y dΓext − ψk K ∂n QG,k

   qg = Gk dΩ k Ω

(11)

it is possible to write the following identity:  Γint

  ∂T Gk dΓint = −RG,k − QG,k ∂n

(12)

More details about the equation above can be found in [21]. Defining γk as the trace of the solution Gk on Γint , from the basis ψ1 , . . . , ψN , defined in equations (7)–(9), it is possible to determine the generated basis γ1 , . . . , γN that are linear independent and complete in L2 , as demonstrated in [19]. Therefore, Eq. (12) can be written as:       ∂T ∂T dΓint = dΓint Gk γk − RG,k − QG,k = (13) ∂n ∂n Γint Γint The normal derivative of the temperature on the internal surface can be approximated by using the induced basis γ1 , . . . , γN [21] as follows: qint

' ∂T '' = = α1 γ1 + . . . + αn γn ∂n 'Γint

(14)

Therefore, truncating the expansion with N terms, Eq. (13) is reduced to the following linear system: N   k=1 Γint

  N   γk αj γj dΓint = k=1 Γint

  γk γj αj dΓint = −RG − QG

(15)

for j=1,2,3,. . . .N. Equation (15) can be written in matrix form as: Mα = −RG − QG

(16)

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The matrix M is invertible since γ1 , . . . , γN are linearly independent and therefore the aj coefficients can be found as: α = M −1 (−RG − QG )

(17)

After that, the estimation of qint is obtained by simply using the expansion presented in Eq. (14). The M matrix is ill-conditioned, therefore, it has to be regularized in order to make it invertible. In the present analysis, the criterion provided by the discrepancy principle, originally formulated by Morozov [22], was adopted. The auxiliary problems presented in Eqs. (4)–(6) are solved numerically by using the Method of Fundamental Solution (MFS). The method of fundamental solutions, based on the linear combination of a set of basis functions, was introduced in 1964, by Kupradze and Aleksidze [17]. This method is one of the simplest methods for solving boundary value problems for linear partial differential equations (PDEs). It is a meshfree method that presents remarkable results with a small computational effort [20]. A key point for the application of the MFS is the choice of collocation points on the boundary and of the source points, collocated outside the boundary. To obtain good approximations the number of the source points has to be high but not too high because it implies a worse conditioning [19]. Therefore, the optimal number of the source points is a compromise between these two issues. In the present analysis, the optimal number was found to be 60. The same value has been adopted also for the collocation points. As concern, the location of the source points, the singularity of the fundamental solution means that choosing source points very close to the boundary collocation points leads to big diagonal entries, and in the limit, to a diagonally dominant matrix. The distance between this fictitious boundary and the domain boundary should be chosen by considering that if the distance is high the problems become too ill-conditioning, while if the distance is small the accuracy is poor. In the present analysis, the optimal distances were equal to 2rext and to 0.5rint for the external and internal source points, respectively, as shown in Fig. 4. Further details on the method could be find in Fairweather and Karageophis [20] that realised a comprehensive review on this numerical method. In the present work, those basis functions were assumed to be the fundamental solution of the Laplace equation described in Eq. (4); therefore, it is possible to write the approximated solution as: Gk =

N 

αi Φ(dij )

(18)

i=1

where: dij = ||(xiC , yiC ) − (xjS , yjS )||2

(19)

represents the Euclidean distance between the collocation points and the source points. The αi term appearing in Eq. (18) are the unknown coefficients that are

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Fig. 4 Source and collocation points location

able to decompose Gk , using the fundamental solution Φ defined as: − ∇ 2Φ = δ

(20)

where δ is the Dirac delta function. It is possible to demonstrate that, the fundamental solution of the 2D Laplace equation is: Φ=−

1 log(dij ) 2π

(21)

Finally, it is possible to write the system of Eqs. (4)–(6) in a block matrix, such as: )

Φ(dijD ) ∂Φ(di j N ) ∂n

,

 αi =

Gk 0

 (22)

where dijD denotes the Euclidean distance between the collocation point, located on the boundary in which the Dirichlet boundary condition is applied, to each source point, while dijN denotes the Euclidean distance between the collocation point, located on the boundary in which the Neumann boundary condition is applied to each source point. Finally, the unknown αi coefficient could be simply determined by inverting the liner system of Eq. (22). Since the linear system in the MFS is ill-conditioned, the regularization scheme adopted, reduced the number of basis function until the Morozov’s discrepancy principle was respected.

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To verify the quality of the reconstruction test points on the boundary domain were selected. By considering the value of 10−10 K as accuracy of the numerical solution of the auxiliary problems, no difference between the boundary condition reconstructed on the collocation points and the one reconstructed on the test points was observed. Therefore, it is possible to consider the solution of the auxiliary problems stable. Once the auxiliary problems are solved the inverse problem is regularized by choosing a convenient number of basis functions. It is well known that the effectiveness of all regularization approaches strongly depends on the choice of a proper value of the regularization parameter [10, 11]. In the present analysis the criterion provided by the discrepancy principle, originally formulated by Morozov [22], was adopted to define the number of basis functions.Therefore, the number of the basis depended on the noise level: it was equal to 22, 8 and 4 for noise level equal to 0, 0.5 and 1 K, respectively.

4 Numerical Validation One of the main goals of the present paper was to validate the above described procedure based on the coupling of the Reciprocity Functional approach (RF) with the Method of Fundamental Solution (MFS). By imposing a known distribution of qint and by numerically solving the direct problem expressed by Eqs. (1)–(3) within the Comsol © Multiphysics environment, a synthetic temperature distribution on the external wall surface was obtained. The physical and geometrical parameters used in this work correspond to a stainless-steel tube AISI 304 with an internal radius of 0.007 m and an external radius of 0.008 m. The thermal conductivity of the pipe K = 15 W/m K was assumed to be constant such as the environmental temperature Tenv = 298.15 K and the bulk temperature Tb = 292.15 K of the fluid that flows inside the tube. The exterior surface Γext of the tube was subjected to an overall convective heat transfer coefficient henv = 5 W/m2 K and the domain Ω, that was assumed to be homogeneous and isotropic, was subjected to an internal heat generation per unit of volume qg = 106 W/m3 . For the internal convective heat flux, a meaningful distribution, suggested by the data of Bozzoli et al. [23], was adopted: q(θ ) = −2000(θ/π)2 + 2000

(23)

Then the synthetic temperature distribution over the external wall surface, deliberately spoiled by random noise was used as the input data of the inverse problem implemented within Matlab® environment. A white noise characterized by a standard deviation σ ranging from 0.01 to 1 K was introduced according to: ' Y = Texact 'Γ

ext

+ σ

(24)

where  is a random Gaussian variable with zero mean and unit variance. By solving the inverse problem, a convective heat flux distribution on the internal wall side qint

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was restored and compared to the distribution adopted to generate the input data. In order to quantify the effectiveness of the proposed approach at different noise levels an error analysis could be performed by plotting the global relative estimation error, defined as follows: E=

||qrest ored − qexact ||2 ||qexact ||2

(25)

versus the standard deviation of the measurement error. Since the added noise depends intrinsically on the random sequence generated by Matlab®, the estimation procedure was repeated for 100 different random sequences of noise added to the simulated measurements and an averaged value Eavg was calculated for each noise level. The average estimation error of the considered technique at different noise levels, is reported in Fig. 5. The graph highlights that the proposed method shows a good ability in restoring the heat flux distributions for noise levels lower than 0.5 K while increasing the noise value it presents some problems in the reconstruction of the original information. In order to improve the analysis on the adopted approach, in Fig. 6 the local heat flux distributions restored are compared with the exact one for two representative noise levels that are usually encountered in experimental applications. From the graph reported in Fig. 6 it is possible to notice that for the noise level σ = 0.01 K the here presented solution methodology reconstructs the local heat transfer coefficient almost perfectly. A good restoration of the original information is also guaranteed for the noise level σ = 0.1 K even if the estimated heat flux distribution suffers of ringing artifacts. Fig. 5 Average estimation error of the four considered techniques at different noise level, for the two considered numerical test case

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Fig. 6 (a) σ = 0.01 K; (b) σ = 0.1 K

5 Experimental Data The parameter estimation procedure described above was then applied to an experimental case, already considered in Bozzoli et al. [24] and regarding forced convection problem in coiled pipes. The pipe under test was characterized by a helical profile composed by eight coils: the diameter and the pitch of the helix were approximately 0.31 m and 0.20 m, respectively. The tube presents a circular cross-section having an internal radius rint = 0.007 m and an external radius rext = 0.008 m. The thermal conductivity of the pipe K = 15 W/m K is assumed to be constant. The pipe was kitted out with fin electrodes attached to a power supply, type HP 6671A, that operates in the ranges 0–8 V and 0–220 A. This arrangement was defined to analyse the heat transfer behaviour of the duct under the condition of heat generated by Joule effect in the wall. The temperature of the working fluid was maintained constant at the entrance of the pipe thanks to a secondary heat exchanger, supplied with tap water. The whole tube is thermally insulated with a double layer of expanded polyurethane with the aim of minimising the heat losses to the environment. A small portion of the external tube wall, near the downstream region of the heated section, was made accessible to an infrared imaging camera by removing the thermally insulating layer, and it was coated by a thin film of opaque paint of uniform and known emissivity. The surface temperature distribution was acquired experimentally by means of a FLIR SC7000 infrared camera, with a 640 × 512 pixel detector array. Its thermal sensitivity, as reported by the instrument manufacturer, is 20 mK at 303 K, while its accuracy is ±1 K. By moving the infrared camera around the tube, different images of the test section were acquired: then thanks to a position reference fixed on the tube wall, the different infrared images were conveniently cropped, processed by perspective algorithms [25] and merged together in Matlab ® environment. A sketch of the experimental setup is reported in Fig. 7, and Fig. 8 shows a representative infrared image of the test section.

Non-intrusive Estimate of Spatially Varying Internal Heat Flux in Coiled Ducts:. . .

Fig. 7 Sketch of the experimental setup

Fig. 8 Representative infrared image of the coil wall

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With this data processing procedure, a continuous temperature distribution on the tube wall versus the circumferential angular coordinate was obtained. In Fig. 9 the temperature circumferential distribution is reported for representative Reynolds number value (Re = 665). For the case reported in Fig. 9 the environmental temperature and the bulk temperature of the fluid that flows inside the tube were Tenv = 296.8 K and Tb = 294.7 K, respectively. The exterior surface Γext of the tube was subjected to an overall convective heat transfer coefficient henv = 5 W/m2 K with the environment

Fig. 9 Temperature distribution on the coil external wall (Re=665)

Fig. 10 Residuals between the experimental and the reconstructed temperature distribution (Re = 665)

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Fig. 11 Restored convective heat-flux distribution (Re = 665)

while the domain Ω, that is assumed to be homogeneous and isotropic, was subjected to an internal heat generation per unit of volume qg = 4.8 × 106 W/m3 . The residuals between the experimental and the reconstructed temperature, plotted in Fig. 10, could give additional useful information about the performances of the estimation procedure. The residuals between the experimental and the computed temperature values are randomly distributed: this confirms that the simplified numerical model used in this study adequately describes the physical problem being tested. For the same Reynolds number value considered in Figs. 9 and 10, the distribution of the convective heat flux restored by the proposed method is reported in Fig. 11. In Fig. 11 the restored convective heat flux obtained using the proposed method is compared with the one obtained with one of the most adopted regularization methods, Tikhonov regularization Method [26]. To make the comparison between the considered regularization techniques more straightforward, the criterion provided by the discrepancy principle, was adopted for both techniques. The comparison between the two restored distributions of the convective heat flux underlines that, for the case investigated in the present analysis, the two approaches give equivalent results: this fact confirms the robustness of the proposed approach.

6 Conclusions The objective of the present work was to present and verify a procedure to estimate the local convective heat flux on the internal wall surface of a coiled pipe, under a forced convection problem. The procedure was based on the solution of the

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Inverse Heat Conduction Problem within the solid wall by adopting the external wall temperature distribution as input data. The solution strategy, developed for a 2D model, is based on the Reciprocity Functional approach (RF), coupled with the Method of Fundamental Solution (MFS). This approach presents some practical advantages over the more classical inverse problem solution techniques since it is completely numerical and, for this reason, it is extremely versatile. To validate the presented procedure, synthetic temperature data were generated by solving the direct problem with a known distribution of convective heat flux qint at the internal wall surface. This temperature distribution, spoiled by a random noise, was then used as the input data of the inverse problem. An estimation error has been computed and the solution methodology shows good ability in reconstruct the original information for noise values lower than 0.5 K. The validation of the approach is completed by its application to experimental input data regarding a forced convection problem in a coiled pipe. The adopted solution technique is also compared to another well-known and consolidated approach: Tikhonov Regularization Method. The comparison highlights that, for the problem investigated in the present study, the two approaches give equivalent results. Thus concluding, the application of RF+MFS to numerical and experimental benchmarks highlighted its robustness and versatility, suggesting its application to other challenging inverse problems.

References 1. Webb, R., Kim, N.H.: Principles of Enhanced Heat Transfer, 2nd edn. Taylor & Francis, New York, NY (2005) 2. Bergles, A.E.: Techniques to Enhance Heat Transfer in: Handbook of Heat Transfer. McGrawHill, New-York, NY (1998) 3. Giedt, W.H.: Investigation of variation of point unit heat-transfer coefficient around a cylinder normal to an air stream. ASME Trans. 71, 375–381 (1949) 4. Aiba, S., Yamazaki, Y.: An experimental investigation of heat transfer around a tube in a bank. J. Heat Transf. 98, 503–508 (1976) 5. Taler, J.: Nonlinear steady-state inverse heat conduction problem with space-variable boundary conditions. J. Heat Transf. 114, 1048–1051 (1972) 6. Taler, J.: Determination of local heat transfer coefficient from the solution of the inverse heat conduction problem. Forschung im Ingenieurwesen 71, 69–78 (2007) 7. Bozzoli, F., Pagliarini, G., Rainieri, S.: Experimental validation of the filtering technique approach applied to the restoration of the heat source field. Exp. Therm. Fluid Sci. 44, 858–867 (2013) 8. Bozzoli, F., Cattani, L., Mocerino, A., Rainieri, S.: Turbulent flow regime in coiled tubes: local heat-transfer coefficient. Heat Mass Transf. 54, 2371–2381 (2018) 9. Carlomagno, G.M., Cardone, G.: Infrared thermography for convective heat transfer measurements. Exp. Fluids 49, 1187–1218 (2010) 10. Beck, J.V., Backwell, B., Clair, C.R.S.: Inverse Heat Conduction — Ill-posed Problems. WileyInterscience, New-York, NY (1985) 11. Orlande, H.R.B., Olivier, F., Maillet, D., Cotta, R.M.: Thermal Measurements and Inverse Techniques. Taylor & Francis, New York, NY (2011)

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12. Andrieux, S., Abda, A.B.: The reciprocity gap: a general concept for flaws identification problems. Mech. Res. Commun. 20, 415–420 (1993) 13. Andrieux, S., Abda, A.B.: Identification of planar cracks by complete overdetermined data: Inversion formulae. Inverse Probl. 12, 553–563 (1996) 14. Delbary, F., Aramini, R., Bozza, G., Brignone, M., Piana, M.: On the use of the reciprocity gap functional in inverse scattering with near-field data: an application to mammography. J. Phys. Conf. Ser. 135, 8 pp. (2008) 15. Colaço, M.J., Alves, C.J.S.: A fast non-intrusive method for estimating spatial thermal contact conductance by means of the reciprocity functional approach and the method of fundamental solutions. Int. J. Heat Mass Transf. 60, 653–663 (2013) 16. Shifrin, E.I., Shushpannikov, P.S.: Identification of a spheroidal defect in an elastic solid using a reciprocity gap functional. Inverse Probl. 26, 055001 (2010) 17. Kupradze, V.D., Aleksidze, M.A.: The method of functional equations for the approximate solution of certain boundary value problems. USSR Comput. Math. Math. Phys. 4, 82–126 (1964) 18. Jin, B., Marin, L.: The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction. Int. J. Numer. Meth. Eng. 69, 1570–1589 (2007) 19. Alves, C.J.S.: On the choice of source points in the method of fundamental solutions. Eng. Anal. Bound. Elem. 33, 1348–1361 (2009) 20. Fairweather, G., Karageophis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9, 69–95 (1998) 21. Colaço, M.J., Alves, C.J.S., Bozzoli, F.: The reciprocity function approach applied to the nonintrusive estimation of spatially varying internal heat transfer coefficients in ducts: numerical and experimental results. Int. J. Heat Mass Transf. 90, 1221–1231 (2015) 22. Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Springer, New York, NY (1984) 23. Bozzoli, F., Cattani, L., Mocerino, A., Rainieri, S., Bazán, F.S.V.: A novel method for estimating the distribution of convective heat flux in ducts: Gaussian filtered singular value decomposition. Inverse Probl. Sci. Eng. 27, 1595–1607 (2019) 24. Cattani, L., Bozzoli, F., Rainieri, S.: Experimental study of the transitional flow regime in coiled tubes by the estimation of local convective heat transfer coefficient. Int. J. Heat Mass Transf. 112, 825–836 (2017) 25. Cyganek, B., Siebert, J.P.: An introduction to 3D Computer Vision Techniques and Algorithms. Wiley, New York (2011) 26. Tikhonov, A.N., Arsenin, V.Y.: Solution of Ill-Posed Problems. Winston & Sons, Washington, DC (1997)

Unified Hybrid-Trefftz Finite Element Formulation for Dynamic Problems Ionu¸t Drago¸s Moldovan, Ildi Cisma¸siu, and João António Teixeira de Freitas

Abstract Hybrid-Trefftz finite elements combine favourable features of the Finite and Boundary Element methods. The domain of the problem is divided into finite elements, where the unknown quantities are approximated using bases that satisfy exactly the homogeneous form of the governing differential equation. The enforcement of the governing equations leads to sparse and Hermitian solving systems (as typical to Finite Element Method), with coefficients defined by boundary integrals (as typical to Boundary Element Method). Moreover, the physical information contained in the approximation bases renders hybrid-Trefftz elements insensitive to gross mesh distortion, nearly-incompressible materials, high frequency oscillations and large solution gradients. A unified formulation of hybrid-Trefftz finite elements for dynamic problems is presented in this chapter. The formulation reduces all types of dynamic problems to formally identical series of spectral equations, regardless of their nature (parabolic or hyperbolic) and method of time discretization (Fourier series, time-stepping procedures, or weighted residual methods). For non-homogeneous problems, two novel methods for approximating the particular solution are presented. The unified formulation supports the implementation of hybrid-Trefftz finite elements for a wide range of physical applications in the same computational framework.

I. D. Moldovan () · J. A. T. de Freitas CERIS Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal e-mail: [email protected]; [email protected] I. Cisma¸siu UNIC Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Caparica, Portugal e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 C. Alves et al. (eds.), Advances in Trefftz Methods and Their Applications, SEMA SIMAI Springer Series 23, https://doi.org/10.1007/978-3-030-52804-1_9

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1 Introduction Hybrid-Trefftz finite elements approximate the solution of a boundary value problem using trial functions that satisfy exactly the homogeneous form of the governing differential equation in the domain of the element. The trial functions are subsequently combined to enforce the initial and boundary conditions, either on average or by collocation. However, as opposed to conforming finite elements (also labelled here as conventional), none of the boundary conditions is exactly satisfied. Hybrid-Trefftz finite elements are designed to combine the best features of the boundary and finite elements, namely the boundary integral formulation that typifies the former and the sparse, Hermitian solving system featured by the latter. Moreover, all approximation bases are regular, as fundamental solutions of the governing equations are not required. Hybrid-Trefftz finite elements feature physically meaningful approximation bases, tailored specifically for the problem under analysis. This enables highly accurate solutions to be obtained with very coarse meshes with relatively few degrees of freedom, and renders Trefftz elements virtually insensitive to issues known to hinder the behaviour of conventional finite elements, such as gross mesh distortions, nearly-incompressible materials, high solution gradients and high frequency content in transient problems [1]. The first applications of the Trefftz concept in the context of the finite element method date back to 1973 [2], followed by significant contributions by Jirousek [3], Herrera [4], Piltner [5], Qin [6] and Freitas [7], among many other authors. Such research efforts have led to the development of variants of hybrid-Trefftz finite elements for a wide breadth of physical problems, ranging from heat conduction [8] to structural elasticity [3], and from plate bending [9] to poroelasticity [10]. A comprehensive review of some key contributions is given in Ref. [11]. In spite of the advantages they provide over conventional finite elements, hybrid-Trefftz elements have not been consistently included in public and userfriendly analysis software. A recent contribution in this field is the FreeHyTE platform, presented by the authors in [12]. FreeHyTE is a public, open-source and user-friendly software for the solution of initial boundary value problems using hybrid-Trefftz finite elements. The FreeHyTE platform is simple to use and amenable to expansion, as its development is grounded in a unified hybrid-Trefftz formulation, straightforwardly adaptable to all kinds of boundary values problems (elliptic, parabolic or hyperbolic) and time discretization techniques (Fourier series, time-stepping procedures, or weighted residual methods). This unified formulation enables the use of similar solution techniques for a wide range of physical problems, provides uniform procedures in all phases of the solving process (e.g. data input, construction and manipulation of the solving system, and post-processing of the results) and fosters the modularity and re-usability of the code. This chapter presents the unified formulation of hybrid-Trefftz finite elements for time-dependent problems, with a focus on transient problems, where the use of Fourier transforms for the time discretization is inadequate.

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The hybrid-Trefftz finite elements presented in this chapter are a generalisation of those proposed by Jirousek and, later, Qin. Here, the concept of nodes is abandoned and hierarchical bases are used for all domain and boundary approximations. This option leads to a greater freedom in choosing the domain and boundary (or frame, according to the nomenclature coined by Jirousek) bases, which are now nodeindependent. On the other hand, the physical meaning of the boundary variables is lost, unless the boundary basis is chosen such as to emulate the basis used in node-dependent formulations. A wider discussion regarding the contextualisation of our approach to hybrid-Trefftz finite elements is presented in Ref. [12].

2 Description of the Problem Consider the arbitrary domain V presented in Fig. 1, bounded by the complementary Dirichlet and Neumann boundaries σ and u ( = σ ∪ u and φ = σ ∩ u ). For simplicity, Robin boundaries are not included in this presentation. However, they pose no significant difficulty to the formulation and are explicitly considered, for instance, in Ref. [13]. The general description of parabolic and hyperbolic boundary value problems is given next. To preserve generality, the main physical quantity of the problem is referred to as the state field, while its (generalised) gradient as the flux field.

2.1 Parabolic Problems The general parabolic problem is defined as, D· [k Du(x, t)] + b(x, t) = c· u(x, ˙ t) Fig. 1 Domain, Neumann and Dirichlet boundaries

(1)

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where D and D· are some generalised gradient and divergence operators, b(x, t) is a source term defined on the domain, k and c are generalised conductivity and capacity terms, and u(x, ˙ t) is the first time derivative of the state field u(x, t). State and flux boundary conditions are enforced on Dirichlet (u ) and Neumann (σ ) boundaries, reading, u(x, t) = u (x, t), on u

(2)

n · σ (x, t) = t  (x, t), on σ

(3)

where σ (x, t) = k Du(x, t) is the general definition of the flux, u is the state field prescribed on the Dirichlet boundary u , t  is the normal flux prescribed on the Neumann boundary σ , and n collects the components of the outward normal to the Neumann boundary. Equations (1) to (3) are complemented by the initial condition, u(x, 0) = u0 (x)

(4)

where u0 (x) denotes the initial state field throughout domain V . In the case of transient heat conduction problems, which is arguably the epitome of parabolic problem defined above, the state and flux fields are the temperature and heat flux fields, k is the (scalar) thermal conductivity and c → c · ρ is the (scalar) product between the specific heat capacity and density of the material.

2.2 Hyperbolic Problems The general definition of a hyperbolic problem is, D · [k D u(x, t)] + b (x, t) = d · u˙ (x, t) + m · u¨ (x, t)

(5)

where b(x, t) is a source term defined on the domain V , and k, d and m are some generalised stiffness, damping and mass terms, and u(x, ˙ t) and u(x, ¨ t) are the first and second time derivatives of the state field u(x, t). Differential operators D and D· have the same meaning as in equation (1). Dirichlet and Neumann boundary conditions are consistent to those expressed for parabolic problems by equations (2) and (3), and the initial conditions are given by, u(x, 0) = u0 (x)

(6)

u(x, ˙ 0) = v 0 (x)

(7)

where u0 (x) denotes the initial state field, and v 0 (x) the initial values of its first time derivative. This general description is valid for most hyperbolic problems. For instance, for acoustic and electromagnetic application, D and D· are the gradient and divergence

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operators, u is the (scalar) acoustic pressure, electric or magnetic fields, k is a unit scalar, d = 0, and m = c−2 , where c is the speed of sound or the speed of light, respectively. Conversely, in solid mechanics applications, state field u represents the displacement field in the medium, and k, d and ρ are the material stiffness, damping and mass matrices, respectively. In the boundary conditions (2) and (3), u and t  represent the enforced displacements and tractions on the Dirichlet and Neumann boundaries.

3 Discretization in Time The unified approach to the formulation of hybrid-Trefftz finite elements is based on the fact that, regardless of the time discretization technique, the parabolic problem defined by equations (1) to (4) and hyperbolic problem defined by equations (5) to (7) can be cast as a series of non-homogeneous spectral problems in space variables only. The techniques for the time discretization of the parabolic and hyperbolic problems can be grouped into three categories: transform-based techniques (e.g. Fourier and Laplace transforms), weighted residual techniques (e.g. [14]), and finite difference techniques (e.g. Euler, Crank-Nicolson, Newmark). In the following sections, the unified spectral form of the time-discretized problem is obtained for parabolic and hyperbolic equations using each of these classes.

3.1 Unified Spectral Form of the Time-Discretized Problem The time discretization of the parabolic and hyperbolic problems described in Sects. 2.1 and 2.2 reduces the original problems in time and space to a series of spectral problems in space. Each of these problems is defined as follows:

Definition Find the state field un (x) that satisfies the (spectral type) differential equation, D · [kDun (x)] + ωn2 ρ n un (x) = f 0n (x)

(8)

and boundary conditions, un (x) = un (x), on u , and n · [kDun (x)] = n · σ n (x) = t n (x), on σ

(9) (10)

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In equations (8) to (10), index n denotes the nt h spectral component of the respective field, ωn is a generalised spectral frequency, f 0n is a generalised source term collecting, in general, the effects of the source term b and the initial conditions, and ρ n is, in parabolic (hyperbolic) problems, a generalised capacity (density) term that models the influence of the capacity matrix (mass and damping matrices) on the response of the structure. These terms have different definitions depending on the type of problem and on the method of time discretization. These definitions are derived below for some common discretization procedures.

3.2 Discrete Fourier Transform Periodic time-dependent problems can be decomposed in series of spectral components by taking Fourier expansions of the time-varying fields over a finite number of discrete frequencies. Let all time-dependent fields present in the definitions of the parabolic and hyperbolic problems (1) to (4) and (5) to (7), respectively, be decomposed into harmonic components using a Fourier expansion taken over the period T and a finite number N of spectral frequencies ωn = n · 2π T : α (x, t) =

N−1   1  α n (x) exp ıˆωn t T

(11)

n=0

In expression (11), α represents a generic time-dependent field, which can be the unknown state field u or flux field σ , or the known fields b, u or t  . The spectral forms of the parabolic and hyperbolic problems are obtained equating the coefficients corresponding to each of the N linearly independent terms of the Fourier expansion.

3.2.1 Parabolic Problem Following this procedure, the domain equation (1) coalesces to, D · [k Dun (x)] − ıˆωn c · un (x) = −bn (x)

(12)

meaning that the unified spectral form (8) is recovered by setting f 0n = −bn and ρ n = − ωıˆn c. It is noted that the initial conditions are not included in expression (12) as the problem is assumed periodic, and is defined and solved over the duration of one period. This is a typical feature of the Fourier decomposition method and renders its application to transient, non-periodic problems quite problematic.

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Boundary conditions (9) and (10) are trivially recovered by equating the spectral components of the involved weights.

3.2.2 Hyperbolic Problem Analogous treatment of the domain equation (5) yields,  D · [k Dun (x)] + ωn2 m − ıˆωn d un (x) = −bn (x)

(13)

and the unified spectral form (8) is recovered setting f 0n = −bn and ρ n =  m − ωıˆn d . The observations regarding the periodicity assumption made for the parabolic problem remain valid. Boundary conditions (9) and (10) are trivially recovered by equating the spectral components of the involved weights.

3.3 Weighted Residual Method Weighted residual methods can be seen as a generalisation of many time-stepping and transform-based time integration techniques, and quite a large set of such methods exist [14]. In this context, the weighted residual method proposed by Freitas (e.g. [15–17]) is presented. In general, the method is based on the division of the duration of the problem into time steps t, although it is quite common that a single time step covers the whole duration. On each time step, independent time approximations of the unknown fields are constructed, using N linearly-independent scalar functions Wn (t), whose choice is not constrained in any way to Fourier or any other function space. According to the Galerkin principle, the same functions are used to define the generalised weighting operator W (t) for the weak form enforcement of the governing equations.

3.3.1 Parabolic Problem Assume that the state field u, its first time derivative v, and flux field σ are independently approximated using basis W (t), u (x, t) =

N−1 

Wn (t) un (x)

(14)

Wn (t) v n (x)

(15)

n=0

v (x, t) =

N−1  n=0

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σ (x, t) =

N−1 

Wn (t) σ n (x)

(16)

n=0

Since the approximations of the state field (14) and of its ‘velocity’ (15) are independent, the condition of the latter to be the time derivative of the former must be explicitly enforced. This is done using the Galerkin weighted residual principle, 

t

W ∗ (v − u) ˙ dt = 0

(17)

0

where W ∗ denotes the conjugate transposed of basis W . The second term of expression (17) is integrated by parts, to force the emergence of initial condition terms, where initial condition (4) is explicitly inserted. After some mathematical manipulations, described in detail in Ref. [18], the following equation is obtained: t H v n = G un − W ∗ (0) u0  t 1 W ∗ W dt H = t 0  t ∗ ˙ ∗ W dt G = W (t) W (t) − W

(18) (19) (20)

0

The uncoupling of equation (18) can be achieved if the time approximation basis is constructed in such way that matrices H and G are proportional through a diagonal matrix of constants : H=G

(21)

This is the only constraint on the approximation basis W (t). It should not, however, be seen as a limitation of the method, as the algorithm used to secure it, detailed in Refs. [15, 18], can be applied to any function basis. Condition (21) uncouples system (18) into, t v n = n un − ψn u0 ψn =

N−1 

:m (0) H nm W

(22) (23)

m=0

:m is the complex conjugate of function Wm and H nm is the general term of where W the inverse of matrix H .

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The weak enforcement of the governing equation (1) follows, using the same time basis for weighting. 

t

W ∗ [D · (k Du) − c · u] ˙ dt = −



0

t

W ∗ b dt

(24)

0

Inserting approximations (14) and (15) into equation (24) and taking into account expression (22) yields,    t ψn 1 n un + c · u0 = − W ∗ b dt H · D · (k Dun ) − c · t t t 0

(25)

which uncouples to, ψn n un = −bn − c · u0 t t  t N−1 1  :m b dt H nm bn = W t 0 D · (k Dun ) − c ·

(26) (27)

m=0

The unified spectral form (8) is recovered from equation (26) setting the (generalised) spectral frequency, capacity matrix and source term to, n t ıˆ ρn = − c ωn ωn2 = −ˆı

f 0n = −bn − ıˆ

(28) (29) ψn ωn ρ n u0 t

(30)

Boundary conditions (9) and (10) are recovered by enforcing weakly equations (2) and (3) using the time basis W for weighting and substituting approximations (14) and (16) into the resulting expressions. The right hand side terms in the boundary conditions are defined as, un =

 t N−1 1  :m u dt H nm W t 0

(31)

 t N−1 1  :m t  dt W H nm t 0

(32)

m=0

t n =

m=0

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As compared to the definitions of the generalised frequency, capacity matrix and source term obtained for the Fourier discretization of the parabolic problem (Sect. 3.2.1), it is concluded that, • the generalised frequency, which was real in the Fourier approach, is now complex (in general). • the generalised capacity matrix preserves the same expression. • the generalised source term now includes the influence of the initial conditions, besides the source term b. The application of the weighted residual technique lifts the periodicity restriction present in the Fourier approach.

3.3.2 Hyperbolic Problem The derivation of the spectral form for the hyperbolic problem follows the same technique as for the parabolic problems. Besides the state field, its first time derivative, and the flux field, the second derivative of the state field, a, is also approximated independently here, reading, a (x, t) =

N−1 

(33)

Wn (t) a n (x)

n=0

The first derivative condition (17) is enforced similarly to Sect. 3.3.1 and complemented by a second derivative condition, reading, 

t

W ∗ (a − v˙ ) dt = 0

(34)

0

Proceeding in the same manner, an equation similar to (22) is obtained: t a n = n v n − ψn v 0

(35)

Substitution of v n given by equation (22) into expression (35) yields, t 2 a n = n2 un − n ψn u0 − ψn t v 0

(36)

The weak enforcement of the governing equation (5) follows, using the same time domain basis for weighting. 

t 0

W ∗ [D · (k Du) − d · u˙ − m · u] ¨ dt = −

 0

t

W ∗ b dt

(37)

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Inserting approximations (14), (15) and (33) into equation (37) and taking into account expressions (22) and (36) yields,     2   t n ψn n ψn ψn n 1 un − u0 −m v H · D · (k Dun)−d u − u − W ∗ b dt = − n 0 0 t t t 2 t 2 t t 0

(38) which uncouples to,     n2 n ψn ψn ψn n D · (k Dun )− d +m 2 un = −bn − d +m v0 u0 −m 2 t t t t t (39) bn =

 t N−1 1  :m b dt W H nm t 0

(40)

m=0

The unified spectral form (8) is recovered from equation (39) by setting the (generalised) spectral frequency, density matrix and source term, to, ωn2 = −ˆı

n t

ρn = m −

(41)

ıˆ d ωn

f 0n = −bn −

 ψn  ıˆ ωn ρ n u0 + m v 0 t

(42) (43)

Boundary conditions (9) and (10) are recovered by in the same way as in Sect. 3.3.1 and lead to definitions (31) and (32) for the respective right hand side terms. As compared to the definitions of the generalised frequency, density matrix and source term obtained for the Fourier discretization of the hyperbolic problem (Sect. 3.2.2), it is concluded that, • the generalised frequency, which was real in the Fourier approach, is now complex (in general). • the generalised density matrix preserves the same expression. • the generalised source term now includes the influence of the initial conditions, besides the source term b.

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3.4 Finite Difference Methods Finite difference techniques are based on the division of the total duration of the analysis into time steps t, on which the governing equations are collocated in some ‘adequate’ point, typically the end of the time step. The time derivatives of the state field are expressed using the finite difference method as functions of the values of the state field in the beginning and the end of the time step. As opposed to the weighted residual methods presented in Sect. 3.3, one single spectral problem is obtained for each time step using finite difference methods. The application of these methods is exemplified using a generalised mid-point technique for the parabolic problem and the Newmark method for the hyperbolic problem.

3.4.1 Parabolic Problem The time derivative of the state field at the end of each time interval, u˙ t , is approximated using the finite difference method as, u˙ t 

1 1−α u˙ 0 (ut − u0 ) − αt α

(44)

where u0 is the state field at the beginning of the current time step, known from the previous time step or is an initial condition if the current time step is the first. A variety of implicit time discretization schemes are recovered by setting different values for parameter α in  equation (44). The most well-known options  1 2 include the Crank-Nicolson scheme α = 2 , the Galerkin scheme α = 3 , and the backward Euler scheme (α = 1). The value of the time derivative of the state field u˙ 0 at t = 0 may be specified as an initial condition, or it can be computed by collocating equation (1) at t = 0 and using the initial state field specified by condition (4). Collocation of the parabolic equation (1) at the end of the time step and substitution of approximation (44) into the resulting expression recovers the unified spectral form (8), with ωn2 = −

1 αt

(45)

ρn = c f 0n = −bt − c

(46) 1 [u0 + (1 − α) t u˙ 0 ] αt

(47)

where bt is the source field b prescribed at the end of the time interval. Boundary conditions (9) and (10) are obtained by simply collocating definitions (2) and (3) at the end of the time step.

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As compared to the definitions of the generalised frequency, capacity matrix and source term obtained using the previous discretization techniques (Sects. 3.2.1 and 3.3.1), it is concluded that, • the generalised frequency, which was real in the Fourier approach and complex in the weighted residual case, is now purely imaginary. • the generalised capacity matrix has a different expression, being equal to the actual capacity matrix. • as for the weighted residual technique, the generalised source term now includes the influence of the initial conditions, besides the source term bt . Finite difference methods pose no periodicity constraint and can be used for the solution of transient problems. However, the constraints enforced on the size of the time step are much stricter than for the weighted residual method.

3.4.2 Hyperbolic Problem The time derivatives of the state field u at the end of a time step are approximated using the Newmark scheme as,       γ γ γ γ ut − u0 + − 1 v0 + − 1 u¨ 0 t βt βt β 2β     1 1 1 1 v0 + − 1 u¨ 0  ut − u0 + βt 2 βt 2 βt 2β

u˙ t 

(48)

u¨ t

(49)

where β and γ are calibration parameters controlling the stability and numerical damping of the procedure. The value of the second time derivative of the state field u¨ 0 at t = 0 may be specified as an initial condition, or it can be computed by collocating equation (5) at t = 0 and using the initial state and ‘velocity’ fields specified by conditions (6) and (7). The collocation of the governing equation (5) at the end of the time interval and substitution of approximations (48) and (49) in the resulting expressions recovers the spectral form (8) under the following definitions, ωn2 = −

1 βt 2

γ ıˆ ρn = m − √ d β ωn    1  f 0n = − bt + γ d · u0 + m · v 0 βt

(50) (51) (52)

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where,     β 1 β v 0 t + − u¨ 0 t 2 u0 = u0 + 1 − γ 2 γ   1 − β u¨ 0 t 2 v 0 t = u0 + v 0 t + 2

(53) (54)

Again, boundary conditions (9) and (10) are obtained by collocating definitions (2) and (3) at the end of the time step. As compared to the definitions of the generalised frequency, density matrix and source term obtained using the previous discretization techniques (Sects. 3.2.2 and 3.3.2), it is concluded that, • the generalised frequency, which was real in the Fourier approach and complex in the weighted residual case, is now purely imaginary. • the generalised density matrix has slightly different expression, including the Newmark coefficients. However,  it is noted that for the most common definition of the calibration parameters γ = 12 ; β = 14 , the same definition as in the previous cases is recovered. • as for the weighted residual technique, the generalised source term now includes the influence of the initial conditions, besides the source term bt .

4 Hybrid-Trefftz Finite Element Formulation The trademark feature of the hybrid-Trefftz finite elements is that the approximation basis in the domain of the element is constructed using trial functions that satisfy exactly the homogeneous form of the differential equation governing the problem. Consequently, the trial basis encompasses considerable physical information regarding the modelled phenomenon, which accounts for the insensitivity of hybrid-Trefftz elements to issues known to hinder the behaviour of conventional elements. For dynamic problems, the wavelength sensitivity of conventional elements is arguably the most critical issue, and it is a thumb rule that ten conventional elements should be used to span one wavelength for the wave to be correctly modelled. Conversely, Trefftz elements contain the wavelength information in their trial bases and can easily accommodate three or more wavelengths per element [1]. On the other hand, the fact that the trial bases are problem-dependent means they typically need to be derived anew when the physical or algorithmic conditions of the problem change. This difficulty is avoided here by using the unified spectral form (8)–(10), valid for a large variety of physical and algorithmic conditions. The formulation of the state model of the hybrid-Trefftz finite element for the solution of the spectral problem defined by equations (8)–(10) is given in this section. It is noted that hybrid-Trefftz elements can also be formulated in terms

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of fluxes (e.g. [10]), but the state formulation is deemed more consistent with equation (8). The index n designating the current spectral problem in equations (8) to (10) is dropped from this point on, to keep notations simple.

4.1 Approximation Bases Let the domain presented in Fig. 1 be divided into an arbitrary number of finite elements and let V e , ue , σe and ie designate the domain, Dirichlet, Neumann and interior boundaries of a generic finite element (Fig. 2). The state model of the hybrid-Trefftz finite element is constructed on the e independent approximation of the state field in the domain ; e (V ) of the element e e and of the flux field on its essential boundary, e = u i . The state field is approximated as, u = U c x c + up , in V e

(55)

where basis U c contains the trial functions that model the complementary solution of equation (8), vector x c collects their unknown weights, and up is a particular solution of equation (8). As typical of Trefftz methods, the functions contained in basis U c belong to the solution space of the homogeneous form of equation (8), D · (kDU c ) + ω2 ρ U c = 0

(56)

The particular solution up may have a closed-form for simple generalised source terms f 0 . This may happen, for instance, when the source term b present in equations (1) or (5) is constant (e.g. constant heat generation or constant own Fig. 2 Finite elements, Neumann, Dirichlet and interior boundaries

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weight), and the time discretization is performed using the Fourier transform, meaning that no initial condition terms are present in the spectral equation. For more complex problems, such as those where initial condition terms are present, the source term f 0 does not have a simple expression and closed-form particular solutions are not readily derivable. In such cases, the particular solution can be recovered in given points in the domain using Green’s functions, or can be approximated using additional trial functions. The latter approach is adopted here. The particular solution is approximated using a separate basis U p , to which correspond the generalised state variables x p , u = U c x c + U p x p , in V e

(57)

Two novel techniques have been proposed by the first author to construct approximate particular solutions. The first technique couples the particular and complementary solution approximations in the same basis and their weights are determined in a single step. Conversely, the two approximations are uncoupled in the second technique. The particular solution is approximated first, using a novel variant of the Dual Reciprocity Method, and the complementary solution is derived in a subsequent step. The two variants are fully discussed in Refs. [19–21]. An independent normal flux approximation is made on the essential boundary of the element, reading, nσ = Zy, on ee

(58)

The normal flux basis is denoted by Z and y collects its generalised flux variables. No restraints (besides completeness and linear independence) are enforced on the flux basis Z. This is an important feature of this hybrid-Trefftz formulation, distinguishing it from those typically proposed by Jirousek [3] and Qin [11], where the boundary (also known as ‘frame’) approximations are linked to the nodes of the elements. Such constraint is not strictly required, however, and its removal leads to superior flexibility in the definition of the domain and boundary bases which may be different on different elements and essential boundaries. It is noted, however, that the node-dependent formulations can be straightforwardly recovered by simply using nodal interpolation functions to construct the boundary basis Z. The finite element equations are presented next, assuming that a closed-form expression for the particular solution up cannot be found. The coupled and uncoupled techniques for the approximation of the particular solution are both covered.

4.2 Finite Element Equations Using the Coupled Approach Described at length in Ref. [19], the coupled approach integrates Trefftzcompliant, U c , and particular solution, U p , trial functions in the same basis, without

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making any distinction on their roles in the solution recovery process. This option enables Trefftz-compliant functions to improve the effectiveness of the particular solution basis, meaning that accurate total solutions can be obtained with relatively poor particular solution approximations. In order to avoid, to the largest extent possible, the emergence of boundary integrals in the formulation, the particular solution basis, U p , is built using functions that satisfy exactly the static problem,   D · kDU p = 0

(59)

The finite element equations in the domain are obtained by enforcing weakly equation (8), using bases U c and U p for weighting, 

 8 < U ∗i D · (kDu) + ω2 ρu dV e = U ∗i f 0 dV e

(60)

where i = {c, p} and U ∗i is the complex conjugate of the U i basis. The first term of equation (60) is integrated by parts, and the Neumann boundary conditions (10) and flux approximation (58) are inserted in the resulting boundary integrals, yielding, D ic x c + D ip x p − B i y = x i − x 0i

(61)

where the following definitions are used, D ∗ci

D ic =

 D pp =  Bi =  x i =  x 0i =

 =

U ∗i n (kDU c ) d e

  U ∗p n kDU p d e − ω2

(62) 

U ∗p ρ U p dV e

(63)

U ∗i Z dee

(64)

U ∗i t  dσe

(65)

U ∗i f 0 dV e

(66)

; ; and  e is the total boundary of the finite element,  e = ue σe ie . Domain equations (61) are complemented by the enforcement of the Dirichlet boundary condition (9) on each essential boundary of the element, using basis Z for weighting. 

Z ∗ (u − u ) dee = 0

(67)

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Substitution of the state field approximation (57) into equation (67) yields, B ∗c x c + B ∗p x p = y   y  = Z ∗ u dee

(68) (69)

The solving system of the (coupled) hybrid-Trefftz state element is obtained by coupling domain and boundary statements (61) and (68),

(70) The construction of the complementary and particular solution bases such as to satisfy the Trefftz conditions (56) and (59) secure the reduction of most terms in the solving system to boundary integrals. The exceptions are the dynamic matrix D pp and source vectors x 0i . Domain integration can be avoided by adopting the Dual Reciprocity approach described in the next section, at the price of losing the capacity of the two coupled bases to compensate for each other’s weaknesses. The solution of system (70) yields a unique estimate for the domain state field, calculated using approximation (57).

4.3 Finite Element Equations Using the Dual Reciprocity Approach The Dual Reciprocity Method finds a particular solution approximation in two steps. In the first step, the source function f 0 is approximated by collocating a trial basis F p into a set of collocation points in the domain of the element, x ∈ V e , F p (x) x p = f 0 (x)

(71)

In the second step, a consistent particular solution basis U p is obtained by solving analytically equation (8), having the basis F p as the non-homogeneous term, 0 1 D · k D U p (x) + ω2 ρ U p (x) = F p (x)

(72)

If equations (71) and (72) are simultaneously satisfied, then the particular solution up = U p x p satisfy precisely the spectral equation (8) in the collocation points,

 1 0 D · k D U p (x) + ω2 ρ U p (x) · x p = f 0 (x)

(73)

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The Dual Reciprocity method presents the advantage of completely avoiding the need for domain integration, but the expressions of the functions included in basis U p may be extremely complex and sometimes singular, in its conventional variants [22]. To avoid this issue, a novel Dual Reciprocity variant was recently developed [13, 21, 23]. It differs from the conventional Dual Reciprocity Method in that it uses the same trial functions (except for a scalar multiplier) for both source term and particular solution approximations, and all trial functions have simple analytic expressions. The particular solution basis is built using functions that satisfy exactly the Helmholtz equation,   D · kDU p + λ2 ρ U p = 0

(74)

where λ is some arbitrary, generalised wave number, and λ = ω. Equation (74) is a Trefftz-type expression, formally similar to equation (56). Therefore, no additional coding effort is required for its implementation and limited computational effort is needed for its calculation. The convergence and completeness properties of basis U p were investigated in Ref. [21]. It was concluded that the basis corresponds to a particular solution expansion in a spherical harmonic series in the angular direction, and in a Dini series in the radial direction. Further convergence and density results are reported in Ref. [23]. Multiple wave numbers λ can be used to improve the convergence and numerical robustness of the particular solution basis. Substitution of definition (74) into equation (72) yields,  F p (x) = ω2 − λ2 ρ U p (x)

(75)

meaning that  the source  term and particular solution bases only differ by the constant multiplier ω2 − λ2 ρ. With bases F p and U p completely defined, the solution of system (71) yields the particular solution approximation for each finite element. The complementary solution is subsequently obtained by enforcing weakly the spectral equation (8) in the domain of the element, using basis U c for weighting, 

 8 < U ∗c D · (kDu) + ω2 ρu dV e = U ∗c f 0 dV e

(76)

Equation (76) is worked out using the same type of manipulations as described in Sect. 4.2. As opposed to the coupled approach, however, substitution of property (73) into the resulting expressions cancel out all domain integrals, yielding, Dc x c − B c y = x  − Dp x p

(77)

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where the following definitions are used,  Dc =  Dp =  x =

U ∗c n (kDU c ) d e

(78)

  U ∗c n kDU p d e

(79)

U ∗c t  dσe

(80)

and B c is given by expression (64), with i ≡ c. The enforcement of the boundary state field is identical to equations (67) and (68) used in the coupled approach. The solving system of the hybrid-Trefftz state element is constructed on the domain and boundary statements (77) and (68). The weights corresponding to the particular solutions are now listed in the right hand side of system (81), as they are known form the solution of system (71).

(81) No domain integrals are present in system (81), which is a considerable advantage over the coupled technique presented in Sect. 4.2. As stated before, the price to be paid for this advantage is that complementary and particular solution bases can no longer compensate for each other’s weaknesses. Upon the solution of system (81) the domain state field is calculated according to definition (57).

5 Implementation Notes The unified formulation presented in this chapter enables the implementation of hybrid-Trefftz finite elements for a wide range of physical applications in the same computational platform. The formulation recovers, in good measure, the generality that typifies conventional finite elements (where the same type of approximation functions are used independently of the physical phenomenon), without compromising the physical information built in the trial basis that accounts for the exceptional robustness of the hybrid-Trefftz elements. The computational implementation of the unified formulation poses no major difficulties, although the boundary integration and system solution must be handled with care since the integrands may be highly oscillatory and the solving system

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ill-conditioned. However, since the integration is largely confined to the boundaries, using a more refined quadrature is not particularly time-consuming. Likewise, the dimensions of the hybrid-Trefftz solving systems are generally small as compared to conventional finite elements because very coarse meshes are affordable. Therefore, the use of a pre-conditioner is not computationally expensive [12]. A workflow for the implementation of the unified formulation is presented in Fig. 3. It consists of the following main steps: • Pre-processing. After reading the input data supplied by the user, the time discretization is performed according to Sect. 3. The spectral frequency, generalised material parameters and the spectral components of the enforced boundary conditions are computed and stored. Memory is stored for each block of the solving system and the data structures are created for the first (or current) spectral problem. • Processing. Depending on the Trefftz formulation (coupled or uncoupled), the particular and complementary solution computations may be performed in a single step or sequentially. The calibration of the integration quadrature according to the spectral frequencies is recommended, in order to mitigate the integration errors of the oscillatory integrands. To improve the conditioning of the solving system preconditioning is also provided. After the solution of the solving system, the spectral solution is recovered and stored for post-processing. The next spectral problem is launched. • Post-processing. The time domain solution generally needs to be reconstructed based on the spectral results. If a finite difference technique was adopted for the time discretization, this step is not required, as the spectral problems recover the solution at the end of each time step. The results are output according to the needs of the analyst. This workflow is implemented in a novel computational framework, registered under the brand FreeHyTE [24]. The implementation is supported by user-friendly graphical interface, unified data structures for system construction and manipulation, adaptive p-refinement, and post-processing procedures, and features a high level of generalisation and modularity [12]. FreeHyTE is open-source, and distributed under a GNU-GPL licence. The FreeHyTE modules available to date are listed in Table 1. For completeness, the list also includes hybrid-Trefftz flux models and direct boundary element models, which are not treated in this chapter. The modules which are not currently deployed online may be obtained from the authors upon request.

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Fig. 3 Workflow for the implementation of the unified formulation

Pre-processing

user input

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recovery of the spectral solution

recovery of the time domain solution

output

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memory allocation, initialisation of the data structures

Plane strain porodynamics (saturated porous media) Plane strain porodynamics (unsaturated porous media)

Parabolic, transient (e.g. transient heat diffusion) Hyperbolic, transient (e.g. transient acoustics)

Plane structural elastostatics

Helmholtz equation, homogeneous (e.g. frequency domain acoustics)

Description of the module Poisson’s equation with constant source term (e.g. steady-state heat diffusion)

Table 1 FreeHyTE modules available to date Model Hybrid-Trefftz state (temperature) Pure hybrid state (temperature) Direct boundary methods Hybrid-Trefftz state Hybrid-Trefftz flux Direct boundary methods Hybrid-Trefftz displacement Hybrid-Trefftz stress Hybrid-Trefftz state (temperature), dual reciprocity Hybrid-Trefftz state, coupled Hybrid-Trefftz state, dual reciprocity Hybrid-Trefftz flux, coupled Hybrid-Trefftz displacement Hybrid-Trefftz displacement

GUI Yes Yes Yes Yes No Yes Yes Yes Yes Yes Yes No Yes Yes

Deployment Web page On request Web page On request On request Web page Web page Web page Web page On request Web page On request Web page Web page

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6 Numerical Example The use of the plane strain porodynamics module of FreeHyTE for the solution of a problem involving the propagation of a shock wave through a semi-infinite saturated porous medium is illustrated in this section. The physical model of the problem is presented in Fig. 4. It consists of a semi-unbounded water-saturated Molsand soil subjected to a vertical shock load f (x, t) = fx (x) · ft (t) applied on a 2L0 = 16m strip of its surface. The force f (x, t) is defined by fx (x) = −1, and ft (t) = 100 if t ≤ 0.8msec and ft (t) = 0 otherwise, applied on the solid phase, meaning that the surface pore pressure is kept null at all times. The total duration of the analysis is 30.0msec. The domain of interest for the analysis is confined by a radius R = 20m around the origin of the referential. The domain beyond this radius is modelled using an absorbing boundary, which is a special type of elastic boundary designed to prevent spurious wave reflections into the interior domain. Due to symmetry, only half of the domain presented in Fig. 4 is included in the numerical model (Fig. 5). The definition of the problem in FreeHyTE is made in four Graphical User Interfaces (GUIs), complemented by a visualization interface and a verification interface. The process is described below. GUI 1: Structural and Algorithmic Definitions The layout of the first GUI is presented in Fig. 6. Its main zones are listed below: • Algorithmic definitions. The user must choose between the regular and nonregular mesh generators using the Mesh generation popup menu. The regular mesh generator is adequate for rectangular geometries, to be meshed using rectangular elements of equal sizes. The non-regular mesh generator is adequate for all geometries. Additionally, the user must specify the number of Gauss-Legendre quadrature points needed to compute the coefficients of the solving system. Less Gauss-Legendre points decrease the duration of the analysis, but increase the numerical integration errors. Two important algorithmic options correspond to checkboxes Parallel processing and Fig. 4 Physical model for the shock wave propagation problem

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Fig. 5 Numerical model for the shock wave propagation problem

Fig. 6 Layout of GUI 1

Use least norm solvers. The former enables and disables the parallel solution of the time-discretized problems. In the parallel processing mode, FreeHyTE summons all cores of the machine to perform the calculations, but their activation is not recommended for problems of low complexity. The Use least norm solvers checkbox enables FreeHyTE to use least norm iterative solvers on ill-conditioned solving systems, at a greater computational cost. Finally, the user can request the solutions to be plotted at a certain number of time steps. • Geometry and meshing. This area is only editable if the regular mesh generator is used to define the structure. Its fields correspond to the size of the rectangular domain in the Cartesian directions and the number of elements to be used in

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each direction. If the non-regular mesh generator is chosen in the algorithmic definitions, the Matlab-native pdetool interface is launched automatically after exiting the first GUI to enable the geometry and mesh definitions. • Boundary and domain orders. Here, the user should insert the orders of p-refinement of the Trefftz basis (55) in the domain of the element (Loops order) and of the Chebyshev basis (58) on its essential boundaries (Edges order). • Time integration. The saturated porodynamics module of FreeHyTE uses the weighted residual technique presented in Sect. 3.3.2 for the time discretization of the original problem. The time basis used for approximations (14), (15) and (33) is constructed using Daubechies wavelets, for whose definition the user must insert the family number, the order of the basis and the number of (dyadic) time points where the solution should be calculated. Null initial conditions are implicitly assumed. The total duration of the analysis must also be specified in this area. • Material parameters. The material parameters required for the execution of the analysis must be specified here. For the problem defined above, the geomechanical characteristics of the Molsand soil are taken from Ref. [18]. GUI 2: Structure and Mesh Generation The pdetool interface is automatically invoked during the execution of FreeHyTE if a non-regular mesh was selected in the first GUI. The geometry definition consists of the following steps: • Draw the geometry. Rectangular, elliptical and polyline shapes can be used to construct the geometry of the domain (Fig. 7). For this case, we use a circle and a polyline and take their intersection to obtain the domain. The choice of a polyline allows one to define the desired leading size of the finite elements in the various regions of the domain. The mesh is locally refined close to the origin, where large solution gradients are expected. • Define the finite element mesh. The mesh generation is controlled by the localized refinement of the polyline, and the maximum element size and mesh growth rate, specified in the Mesh menu of the pdetool interface. The resulting mesh is presented in Fig. 8. The local refinement is clearly visible in the region underneath the applied load. Despite the apparent density of the mesh (consisting of 502 finite elements), even the most refined elements are fairly large, with leading dimensions of 0.5m. The largest elements’ leading dimensions are close to 3m. GUI 3: Definition of Boundary Types The third GUI is used to define the type (Dirichlet, Neumann, Robin or absorbing) for each exterior side of the domain (Fig. 9). The definitions corresponding to the main zones of the third GUI are listed below: • Structure visualisation zone. The structure visualisation zone consists of an interactive plot of the mesh, with three buttons controlling the displayed information.

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Fig. 7 Constituent shape definition in pdetool

Fig. 8 Finite element mesh

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Fig. 9 Layout of the third GUI

For a detailed inspection of the structural data, the Enlarge image button can be used to open a separate, visualisation-only interface with zoom, pan and data cursor capabilities. • Definition of the external boundary types. The external boundaries of the structure are listed in the table located in the central zone of GUI 3. To define the boundary types, the user must select the boundaries in the table (multiple selection is possible), select the desired boundary type from the popup menu Define the type of external boundary and click the Assign type button. The boundary types in the table on the right should automatically adjust to reflect the changes. GUI 4: Definition of the Boundary Conditions The fourth GUI (Fig. 10) is used to define the boundary conditions according to the boundary types specified in the previous interface. The boundary conditions that can be input are of the Dirichlet, Neumann or Robin (elastic) types. Absorbing boundary conditions do not require input from the user. Instead, the flexibility of the absorbing boundary is computed automatically according to the material parameters such as to minimise the wave reflections from the boundary. The Dirichlet and Neumann boundary conditions must be specified in the boundary normal and tangential directions in the solid phase and in the boundary normal direction in the fluid phase. A generic boundary condition α is defined as α (s, t) = αs (s)·αt (t), where s is the local coordinate of the current boundary. The time variation αt (t) of the generic boundary condition is specified by any analytic expression involving the time variable t that can be interpreted by Matlab. The space variation αs (s) is defined by setting its values in as many equally spaced points on the boundary as necessary to specify a polynomial variation.

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Fig. 10 Layout of the fourth GUI

After the definition of the boundary conditions is completed, a verification interface is called before launching the execution. The total number of degrees of freedom (i.e. the dimension of each of the solving systems (81)) is 33,980. The execution in the parallel mode using four cores (16MB RAM) took roughly 4 hours, while in the sequential mode it took 8 hours. Results After the execution is completed, FreeHyTE provides plots of the displacement, stress and pore pressure fields in the domain of the problem at the dyadic times specified in GUI 1. However, since the colour plotting functions are fairly slow in Matlab, the recommended approach to the visualisation of the solution is to use external plotting software. To support this option, FreeHyTE saves a series of files with the values of the fields in the Gauss points of each element, at each dyadic time. The output files are formatted for direct loading in the post-processing software Tecplot, but can be used in other visualisation software as well, since they simply contain a list of points and the corresponding displacements, stresses and pore pressure values. Moreover, some open-source visualisation software (e.g. Paraview) feature Tecplot interpreters, meaning that they should read the input files generated by FreeHyTE seamlessly. The total stress field in the vertical direction is presented at nine points in time in Fig. 11. The solutions recover well the enforced boundary conditions and present excellent continuity between adjacent elements. It is recalled that the stress continuity between adjacent elements is not explicitly enforced in the model, so the stress continuity is only achieved upon convergence. The shock load applied on the surface of the medium generates two wavefronts that propagate downwards with very different velocities, damping ratios and wavelengths. The faster wavefront corresponds to the compression wave propagating through the solid phase, also known as the primary compression wave in the theory of porous media. The slower compression wave, clearly visible in Fig. 11a, is the

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Fig. 11 Vertical total stress at nine points in time. (a) t = 1.9msec. (b) t = 3.7msec. (c) t = 5.6msec. (d) t = 7.5msec. (e) t = 9.4msec. (f) t = 11.3msec. (g) t = 13.1msec. (h) t = 15.0msec. (i) t = 16.9msec.

secondary compression wave, that propagates through the fluid phase. As compared to the primary compression wave, it is considerably slower, with shorter wavelength, and more evanescent. When the compression wave reaches the far-field boundary (Fig. 11f), it does not reflect back into the interior domain as it would if the far-field boundary was

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rigid. Consequently, the fields in the interior domain are not subjected to any visible spurious perturbation (Figs. 11g–i).

7 Closure For the solution of dynamic problems, hybrid-Trefftz finite elements offer considerable advantages over conventional (conforming) finite elements. The physical information built into their approximation bases lifts the usual restriction of having to use at least ten finite elements to cover a wavelength and offers superior robustness to mesh distortion, near-incompressibility and high solution gradients. Moreover, Trefftz formulations reduce the coefficients of the solving system to boundary integral expressions. On the other hand, these advantages come at a price, as the Trefftz bases are typically computed anew for each type of physical problem and also depend on the algorithm used for its discretization in time. The unified hybrid-Trefftz formulation presented in this chapter is aimed at securing the key advantages of the Trefftz method while providing a very general framework for its computational implementation. It is shown how the same type of time-independent, spectral problems can be reached from both parabolic and hyperbolic problems defined in time and space, and irrespective of the time discretization method. Since the spectral problems generally result non-homogeneous, they require the calculation of both particular and complementary solutions. Two novel strategies are presented for the recovery of the particular solution. In the first strategy, the particular and complementary solution bases are coupled, and both solutions are calculated at the same time. This allows the two parts of the basis to compensate for each other’s weakness, but some of the coefficients in the solving system are defined by domain integrals. In the second strategy, the particular and complementary solutions are calculated sequentially, but the domain integrals are completely avoided. A general framework for the implementation of the unified formulation is also given. This framework is employed by FreeHyTE, an opensource and user-friendly computational platform dedicated to Trefftz and boundary methods. The platform is used to solve a shockwave propagation problem through a semi-infinite saturated porous medium. Acknowledgments This research was supported by Fundação para a Ciência e a Tecnologia through grants PTDC/EAM- GTC/29923/2017 and UID/ECI/04625/2019.

References 1. Moldovan, I.D., Freitas, J.A.T.: Hybrid-Trefftz displacement and stress elements for bounded poroelasticity problems. Comput. Geotech. 42, 129–144 (2012) 2. Ruoff, G.: Die praktische Berechnung der Kombination der Trefftzschen Methode und bei flachen Schalen, pp. 242–259. Finite Elemente in der Statik, Berlin (1973)

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3. Jirousek, J., Teodorescu, P.: Large finite elements method for the solution of problems in the theory of elasticity. Comput. Struct. 15, 575–587 (1982) 4. Herrera, I.: Boundary Methods - an Algebraic Theory. Pitman Advanced Publishing Program. Boston, London, Melbourne (1984) 5. Piltner, R.: Special finite elements with holes and internal cracks. Int. J. Numer. Methods Eng. 21, 1471–1485 (1985) 6. Qin, Q.H.: Postbuckling analysis of thin plates by a hybrid Trefftz finite element method. Comput. Methods Appl. Mech. Eng. 128, 123–136 (1995) 7. Freitas, J.A.T., Moldovan, I.D., Cisma¸siu, C.: Hybrid-Trefftz displacement element for bounded and unbounded poroelastic media. Comput. Mech. 48, 659–673 (2011) 8. Cheung, Y.K., Jin, W.G., Zienkiewicz, O.C.: Direct solution procedure for solution of harmonic problems using complete, non-singular, Trefftz functions. Commun. Appl. Numer. Methods 5(3), 159–169 (1989) 9. Piltner, R.: The application of a complex 3-dimensional elasticity solution representation for the analysis of a thick rectangular plate. Acta. Mech. 75, 77–91 (1988) 10. Moldovan, I.D., Cao, D.T., Freitas, J.A.T.: Elastic wave propagation in unsaturated porous media using hybrid-Trefftz stress element. Int. J. Numer. Methods Eng. 97, 32–67 (2014) 11. Qin, Q.H., Wang, H.: MATLAB and C Programming for Trefftz Finite Element Methods. CRC Press, Boca Raton, London, New York (2009) 12. Moldovan, I.D., Cisma¸siu, I.: FreeHyTE: a hybrid-Trefftz finite element platform. Adv. Eng. Softw. 121, 98–119 (2018) 13. Moldovan, I.D., Coutinho, A., Cisma¸siu, I.: Hybrid-Trefftz finite elements for nonhomogeneous parabolic problems using a novel dual reciprocity variant. Eng. Anal. Bound. Elem. 106, 228–242 (2019) 14. Tamma, K.K., Zhou, X., Sha, D.: The time dimension: A theory towards the evolution, classification, characterisation and design of computational algorithms for transient/dynamic applications. Arch. Comput. Meth. Eng. 7(2), 67–290 (2000) 15. Freitas, J.A.T.: Time integration and the Trefftz method. Part I - First-order and parabolic problems. CAMES 10(4), 453–463 (2003) 16. Freitas, J.A.T.: Time integration and the Trefftz method. Part II - Second-order and hyperbolic problems. CAMES 10(4), 465–477 (2003) 17. Arruda, M.R.T., Moldovan, I.D.: On a mixed time integration procedure for non-linear structural dynamics. Eng. Comput. 32(2), 329–369 (2015) 18. Moldovan, I.D.: Hybrid-Trefftz Finite Elements for Elastodynamic Analysis of Saturated Porous Media. PhD thesis, Universidade Técnica de Lisboa (2008) 19. Moldovan, I.D.: A new particular solution strategy for hyperbolic boundary value problems using hybrid-Trefftz displacement elements. Int. J. Numer. Methods Eng. 102, 1293–1315 (2015) 20. Moldovan, I.D.: A new approach to non-homogeneous hyperbolic boundary value problems using hybrid-Trefftz stress finite elements. Eng. Anal. Bound. Elem. 69, 57–71 (2016) 21. Moldovan, I.D., Radu, L.: Trefftz-based dual reciprocity method for hyperbolic boundary value problems. Int. J. Numer. Methods Eng. 106, 1043–1070 (2016) 22. Cho, H.A., Golberg, M.A., Muleshkov, A.S., Li, X.: Trefftz methods for time dependent partial differential equations. Comput. Mater. Continua 1(1), 1–37 (2004) 23. Alves, C.J.S., Martins, N.F.M., Valtchev, S.S.: Extending the method of fundamental solutions to non-homogeneous elastic wave problems. Appl. Numer. Math. 115, 299–313 (2017) 24. FreeHyTE distribution page (2015). https://www.sites.google.com/site/ionutdmoldovan/ freehyte. Cited 29 Sep 2019

Acoustic Bandgap Calculation of Liquid Phononic Crystals via the Meshless Generalized Finite Difference Method Zhuo-Jia Fu, Ai-Lun Li, and Han Zhang

Abstract This paper presents a recently developed meshless collocation method, the generalized finite difference method (GFDM), to calculate the acoustic bandgaps of 2D liquid phononic crystals with square and triangular lattices. The moving least squares theory and Taylor series expansion are used to construct the GFDM difference formulation for the generalized eigenvalue equations arising from acoustic bandgap analysis. In comparison with the finite element method (COMSOL software), the proposed GFDM can provide similar accurate results with less computational cost for calculating the band structures of the simple/complicated shape scatters in the square/triangular lattice.

1 Introduction In recent years there has been growing interest in the acoustic wave propagation in periodic composite materials known as phononic crystals. With the specific frequency bandwidths, wave cannot propagate through the phononic crystals, which is called as acoustic phononic bandgap. Due to this bandgap property, the phononic crystals have a broad range of applications, such as noise/vibration reduction,

Z.-J. Fu () Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing, China Institute of Continuum Mechanics, Leibniz University Hannover, Hannover, Germany e-mail: [email protected] A.-L. Li Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing, China e-mail: [email protected] H. Zhang () Institute of Acoustics, Chinese Academy of Sciences, Beijing, China e-mail: [email protected] © The Editor(s) (if applicable) and The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 C. Alves et al. (eds.), Advances in Trefftz Methods and Their Applications, SEMA SIMAI Springer Series 23, https://doi.org/10.1007/978-3-030-52804-1_10

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earthquake isolation and acoustic invisibility cloak and so on. To understand the bandgap distribution in phononic crystals, various numerical methods have been proposed, such as the plane wave expansion method (PWE) [1, 2], the wavelet method [3], the multiple scattering theory (MST) [4, 5], the finite difference time domain method (FDTD) [6, 7], the finite element method (FEM) [8–10], the boundary element method (BEM) [11, 12], the singular boundary method (SBM) [13] and the meshless methods [14], just to mention a few. All of the abovementioned numerical methods have their own merits and demerits. For examples, the PWE has the slow convergence rate, the MST is only available for some specific simple geometries, and the numerical stability and accuracy is sensitive to the generated meshes/grids used in the mesh-based methods (FDTD, FEM and BEM). Recently, more attention has been paid on the development of the localized meshless collocation methods [15, 16] including the localized radial basis function collocation method (LRBFCM) [17], the generalized finite difference method (GFDM) [18, 19], and the RBF-based finite difference method (RBF-FD) [20, 21]. These localized meshless collocation methods not only have the simple discretization formulation like the FDM, but also have the meshless property without the ill-conditioning resulting matrices [22, 23]. In this study, we focus on the meshless generalized finite difference method for acoustic bandgap analysis,taking into account longitudinal waves that are Trefftz particular solutions [24]. It has been first proposed by Benito and his collaborators in 2001 [18], and later they gave the theoretical analysis in the GFDM solution of the parabolic, the hyperbolic and the fourth-order PDEs [19, 25]. After that, the GFDM has been successfully applied to various engineering and science fields, such as the nonlinear obstacle problems [26], three-dimensional transient electromagnetic problems [27], double-diffusive natural convection [28] and so on. This paper presents the GFDM for acoustic bandgap analysis. The rest of this paper is briefly summarized as follows. Section 2 introduces the mathematical model of the acoustic wave propagation in 2D liquid phononic crystals, and the corresponding discretization formulation based on the generalized finite difference method. Section 3 provides several numerical examples to verify the efficiency of the proposed GFDM, and then discusses the effect of the scatter shapes on the bandgap properties in 2D liquid phononic crystals. Finally some conclusions are summarized in Sect. 4.

2 Numerical Model This section makes the first attempt to apply the GFDM to the bandgap calculation in liquid phononic crystals. Consider 2D liquid phononic crystals with the square lattice as shown in Fig. 1, where the lattice constant a represents the distance between two adjacent scatters. Both matrix and scatters are assumed to be isotropic, homogeneous media of liquid or gas. Therefore only longitudinal waves, that are Trefftz particular solutions of the Helmholtz equation, are allowed. Its governing

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Fig. 1 The arrangement of scatters, the unit-cell and the first Brillouin zone for the square lattice. (a) Arrangement. (b) Unit-cell. (c) First Brillouin zone

equation can be expressed as: λj ∇(∇uj ) + ρ j ω2 uj = 0,

(1)

where ω is the angular frequency, λ and ρ denotes, respectively, the bulk modulus and mass density of the fluid, u is the displacement. The superscript j represents the quantity related to the matrix (j = 0) or the scatter (j = 1). Due to the periodicity of the 2D phononic crystals, we can only consider the unit cell instead of the whole system as shown in Fig. 1(a.2). Γ1 − Γ4 denote the boundaries of the square unit cell, and Γ0 represents the interface between the matrix and the scatter. All the quantities of the acoustic wave field on periodic boundaries Γ1 − Γ4 should satisfy the Bloch theorem, namely, u0 (x + a) = eika u0 (x) , T 0 (x + a) = eika T 0 (x)

(2)

= where k = (kx1 , kx2 ) is the wave vector, T =λj ∂uj (x) ∂n, nrepresents the unit √ normal vector, i = −1, a = m1 a1 + m2 a2 , in which m1 and m2 are arbitrary integers, a1 and a2 represent the fundamental translation vectors of the lattice. And the following continuity conditions should be satisfied on the interface boundary Γ0 , u1 (x) = u0 (x), x ∈ Γ0 . T 1 (x) = T 0 (x), x ∈ Γ0

(3)

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By employing the governing equation (1) and the boundary conditions (2) and (3), the generalized eigenvalue equations can be represented in the uniform matrix 0 1T form AU = ω2 BU, where U= u0 , u1 and ⎤ ⎤ ⎡ eikx1 a u0 (xΓ 1 ) − u0 (xΓ 3 ) 0 0 0 ⎥ ⎢ eikx2 a u0 (xΓ 2 ) − u0 (xΓ 4 ) ⎢ 0 0 0 ⎥ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ikx a ∂u0 (xΓ 1 ) ∂u0 (xΓ 3 ) ⎢ 0 0 ⎥ ⎥ ⎥ ⎢e 1 ⎢ − 0 ∂n ∂n ⎥ ⎥ ⎢ ⎢ 0 0 ⎥ ⎥ ⎢ eikx2 a ∂u0 (xΓ 2 ) − ∂u0 (xΓ 4 ) ⎢ 0 ⎥ ⎥. ⎢ ⎢ ∂n ∂n , BS = ⎢ 0 AS = ⎢ ⎥ ⎥ 0 0 1 −u (xΓ 0 ) ⎥ u (xΓ 0 ) ⎥ ⎢ ⎢ 0 1 ⎥ ⎥ ⎢ ⎢ 0 0 ∂u (xΓ 0 ) ∂u (xΓ 0 ) ⎥ 0 1 ⎥ ⎢ ⎢ −λ λ ∂n ∂n ⎥ ⎢ ⎢ − ρ 0 u0 0 ⎥ 0 ⎦ ⎦ ⎣ ⎣ Δu 0 λ0 ρ1 1 1 0 − λ1 u 0 Δu ⎡

(4) According to the aforementioned generalized eigenvalue equations, the relationship between the wave vector k = (kx1 , kx2 ) and the eigen frequency ω2 can be determined. For the band gap calculations, it is only necessary to calculate the corresponding eigenfrequency ω2 by solving the generalized eigenvalue equations with sweeping the wave vector k = (kx1 , kx2 ) along the boundary of the irreducible Brillouin zone. The triangle Γ XM is the irreducible Brillouin zone as shown in Fig. 1(a.3). It should be mentioned that the generalized eigenvalue equations of 2D liquid phononic crystals with the triangular lattice can be similarly derived as the aforementioned way for the square lattice. In the numerical discretization of the aforementioned generalized eigenvalue equations, the GFDM introduces the moving least squares theory and Taylor series expansion to construct the numerical differentiation formulations to approximate the partial derivative terms of u (ξ ) in the matrix AS and AT , which can be represented by a linear combination of the displacements uat each discretization node in its stencil support Ξξ (Fig. 2). By using Taylor series expansion to second order, one may obtain      ∂u (ξ ) ∂u (ξ ) 1 ∂u (ξ ) ∂u (ξ ) 2 hj x1 u xj = u (ξ ) + hj x1 + hj x2 + + hj x2 + o ρ3 , ∂x1 ∂x2 2 ∂x1 ∂x2

(5) where xj ∈ Ξξ hj x1 = xj x1 − ξx1 , hj x2 = xj x2 − ξx2 , ρj = then possible to define the functionC (u)



h2j x1 + h2j x2 . It is

⎞ ⎤2   ) ∂u(ξ ) + h u (ξ ) − u xj + hj x1 ∂u(ξ j x   ∂x1  2 2 ∂x2 ⎠ γ ρj ⎦ , ⎣⎝ C (u) = ∂u(ξ ) ∂u(ξ ) 1 + 2! hj x1 ∂x1 + hj x2 ∂x2 j =1 N 

⎡⎛

(6)

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Fig. 2 Sketch of the sets of the discretization nodes Ξ and the related stencil support Ξξ selected by the nearest neighbours algorithm in the GFDM



where γ ρj



 =

1−6



ρj dj

2

+8



ρj dj

3

−3



ρj dj

4

, ρj ≤ dj

is the weighting 0, ρj > dj function used in this study [18], and dj denotes the maximum distance between central node ξ and each discretization node of its stencil support Ξξ . By minimizing the function C (u), we obtain the following linear equation system Ψ Du = b,

(7)

where ⎡

N .

⎢ ⎢ j=1 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ψ =⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

h2jx1 γ 2

N .

hjx1 hjx2 γ 2

j=1 N . j=1

h2jx2 γ 2

SY M

⎤ h2jx1 hjx2 γ 2 ⎥ ⎥ j=1 j=1 j=1 ⎥ N h2 hj x N h3 N . . . ⎥ j x1 j x 2 2 2 2 γ hjx1 h2jx2 γ 2 ⎥ 2 2 γ ⎥ j=1 j=1 j=1 ⎥ ⎥ N h2 h2 N h3 hj x N h4 . . . j x1 j x1 j x2 j x1 2 2 2 2 γ γ ⎥ ⎥, 4 γ 4 2 ⎥ j=1 j=1 j=1 ⎥ N hj x h3 N h4 ⎥ . . j x2 j x 1 2 2 2 γ ⎥ ⎥ 4 γ 2 ⎥ j=1 j=1 ⎥ N . ⎦ 2 2 2 hjx1 hjx2 γ N .

h3j x 1 2

γ2

N .

hj x1 h2j x 2 2

γ2

N .

j=1

(8)

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⎡ ⎢ ⎢ ⎢ ⎢ b= ⎢ ⎢ ⎢ ⎣

h1x1 u (x1 ) γ 2 h1x2 u (x1 ) γ 2

h2x1 u (x2 ) γ 2 h2x2 u (x2 ) γ 2

··· ···

u (x1 ) γ 2

u (x2 ) γ 2

···

h21x 1 2 h21x 2 2

h22x 1 2 h22x 2 2

hN x1 u (xN ) γ 2 hN x2 u (xN ) γ 2 h2N x 1 2 h2N x 2 2

u (xN ) γ 2

u (x1 ) γ 2 u (x2 ) γ 2 · · · u (xN ) γ 2 h1x1 h1x2 u (x1 ) γ 2 h2x1 h2x2 u (x2 ) γ 2 · · · hN x1 hN x2 u (xN ) γ

⎤

⎤ ⎡ ⎥ u (x1 ) ⎥ ⎥ ⎢ ⎥ ⎢ u (x2 ) ⎥ ⎥ ⎢ ⎥ = Z⎢ . ⎥ ⎥. ⎥ ⎣ .. ⎦ ⎥ ⎦ u (xN ) 2

(9) According to Eqs. (7) and (9), the partial derivative vector Du can be represented as follows Du = Ψ −1 b = Ψ −1 ZU = WU.

(10)

Namely, the weighting matrix W in the GFDM is determined as W = Ψ −1 B, which can be rewritten as follows ' N  ∂u '' x1 ,i = w u + wjx1 ,i uj , xi = ξ ∈ Ξ, i i ∂x1 'i

(11)

' N  ∂u '' x2 ,i = w u + wjx2 ,i uj , xi = ξ ∈ Ξ, i i ∂x2 'i

(12)

' N  ∂ 2 u '' x12 ,i x 2 ,i = w u + wj 1 uj , xi = ξ ∈ Ξ, ' i i 2 ' ∂x1 i j =1,j =i

(13)

' N  ∂ 2 u '' x22 ,i x22 ,i = w u + w uj , xi = ξ ∈ Ξ, ' i i j ∂x22 'i j =1,j =i

(14)

' N  ∂ 2 u '' x1 x2 ,i = w u + wjx1 x2 ,i uj , xi = ξ ∈ Ξ, i i ∂x1 ∂x2 'i

(15)

j =1,j =i

j =1,j =i

j =1,j =i

x1 ,i ⎤ w1x1 ,i · · · wix1 ,i · · · wN ⎢ .. .. ⎥, and it should be mentioned that the .. .. where W= ⎣ ... . . . . ⎦ x1 x2 ,i x1 x2 ,i x1 x2 ,i · · · wi · · · wN w1 weightingmatrix W is a sparse matrix with no more than ns nonzero elements in one wji = 0, xj ∈ Ξxi ⊂ Ξ row, i.e. . ns denotes the number of discretization nodes wji = 0, xj ∈ Ξ \Ξxi in the stencil support Ξxi .

⎡

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3 Numerical Results and Discussions In this section, the efficiency and accuracy of the proposed GFDM are firstly tested to compare with the FEM (COMSOL) on the phononic crystal system with circular scatters. Then several influence factors, including the filling fraction and the scatter shapes on the band structures of 2D phononic crystals, are investigated. Unless otherwise specified, the node spacing Δh = ρj = a/40 is employed in the proposed GFDM, the medium parameters used in this study are listed in Table 1, and all the following numerical examples are computed on a personal computer with Intel Core i7-7700, 3.60 GHz CPU.

3.1 Computational Efficiency Analysis First considers acoustic wave in 2D liquid phononic crystals with square or triangular lattice of the air matrix including the mercury/water circular cylinder, whose filling fraction f = 0.2827 for square lattice and f = 0.3265 for triangular lattice. The numerical accuracy is measured by the relative error Rerr defined as: j j N .ω ωnum −ωref

Rerr(ω) =

j =1

|

j

ωref

Nω

| ,

(16)

where Nω represents the total number of the eigenvalues calculated by the GFDM; j j ωnum and ωref represent the numerical solution and the reference solution of j t h eigenvalue, respectively. Figure 3 shows the convergence rates of the proposed GFDM for both the square and triangular lattices in Mercury/Air system. It can be observed that the smaller the node spacing, the more accurate the result will be. Figure 4 plots the band structures for Mercury/Air system calculated by GFDM and FEM (COMSOL). Figures 5 and 6 show the first eight calculated normalized eigenmodes for the square/triangular lattices with wave vector at Point M in the irreducible Brillouin  > √ zone (k = (π /a, π /a) for square lattice, k = π 3a , π /a for triangular lattice), respectively. The calculated band structures and their related normalized eigenmodes by the GFDM with the node spacing Δh = ρj = a/20 are in good agreement with the FEM (COMSOL) results. As shown in Table 2, the GFDM

Table 1 Medium parameters of the matrix and scatter in phononic crystals Medium Air Mercury

Mass density ρ (kg/m3 ) 1.21E+00 1.35E+04

Bulk modulus (N/m2 ) 1.40E+05 2.86E+10

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Fig. 3 Convergence rates of the square lattice and the triangular lattice by using the GFDM

(a)

(b)

Fig. 4 Band structures for Mercury/Air system calculated by GFDM and FEM. (a) Square lattice. (b) Triangular lattice

(a) GFDM results

(b) FEM(COMSOL) results Fig. 5 First eight normalized eigenmodes for square lattice with wave vector at Point M in the irreducible Brillouin zone (k = (π/a, π/a)) calculated by (a) the proposed GFDM and (b) the FEM (COMSOL)

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(a) GFDM results

(b) FEM(COMSOL) results Fig. 6 First eight normalized eigenmodes lattice with wave vector at Point M in the  √ for triangular irreducible Brillouin zone (k = π/ 3a , π/a ) calculated by (a) the proposed GFDM and (b) the FEM (COMSOL) Table 2 Computational costs by using the GFDM and the FEM(COMSOL) for the circular-shape scatterer case Arrangement Square lattice Triangular lattice

GFDM DOF 503 463

CPU time 2.49s 2.82s

FEM(COMSOL) DOF CPU time 1553 83.00s 1081 96.00s

requires less computational resource than the FEM (COMSOL) in acoustic bandgap analysis. Note that the degree of freedoms in the GFDM denotes the node number in Table 2.

3.2 Some Influence Factors on the Band Structures First the effects of the filling fractions with circular scatters in Mercury/Air phononic crystal system are numerically investigated. The calculated band structures of the circular scatters in the square/triangular lattices with filling fraction f = 0.05, 0.25, 0.45 are presented in Figs. 7 and 8, respectively. With an increase of the filling fraction, more bandgaps appear at the low frequency region, in particular the triangular lattice. Furthermore, we consider the following two complicated geometry shapes of scatters in Mercury/Air phononic crystal system, Geometry 1 (Kite-shape scatter, Fig. 9a): Its parametric equation is 

x1 =

a 6 × (2.35 + cos θ + 0.65 cos 2θ ) x2 = a × (0.5 + 0.25 sin θ )

θ ∈ [0, 2π].

(17)

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(a)

(b)

(c)

Fig. 7 Band structures for the phononic crystal system with square lattice under different filling fractions. (a) Filling fraction 0.05. (b) Filling fraction 0.25. (c) Filling fraction 0.45

(a)

(c)

(b)

Fig. 8 Band structures for the phononic crystal system with triangular lattice under different filling fractions. (a) Filling fraction 0.05. (b) Filling fraction 0.25. (c) Filling fraction 0.45

(a)

(b)

Fig. 9 Sketch of two complicated geometry shapes of scatters. (a) Kite-shape scatter. (b) Bladeshape scatter

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Geometry 2 (Blade-shape scatter, Fig. 9b): Its parametric equation is 

x1 = x2 =

a 10 a 10

× (3 + sin 4θ ) × cos(θ + sin44θ ) × (3 + sin 4θ ) × sin(θ + sin44θ )

θ ∈ [0, 2π].

(18)

Table 3 compares the DOFs used in the GFDM and the FEM (COMSOL) for the complicated-shape scatter cases. The calculated band structures for the aforementioned complicated-shape scatters in the square/triangular lattices are compared by using the proposed GFDM and the FEM (COMSOL), which are shown in Figs. 10 and 11. It can be observed that, in comparison with the FEM(COMSOL) results, the proposed GFDM performs equally well and requires slight coarser discretization in the complicated-shape scatter cases. Finally, we consider the effect of the arm widths with two cross-shaped scatters (Cross and Double-cross, Fig. 12) in Mercury/Air phononic crystal system. To make a fair comparison, the filling fraction is fixed as f = 0.2 in both these two crossshaped scatters. The arm widths Aw = 0.26a, 0.20a, 0.14a for cross scatter and Aw = 0.14a, 0.11a, 0.08a for double-cross scatter in the square/triangular lattice are calculated, respectively. The related band structures are plotted in Figs. 13, 14, 15, 16. It can be observed from these figures that, with the decreasing arm width,

Table 3 Computational costs by using the GFDM and the FEM(COMSOL) for the complicatedshape scatterer case Arrangement Square lattice Triangular lattice

Kite GFDM 1783 1589

(a)

FEM 2485 2083

Blade GFDM 1869 1679

FEM 2629 2427

(b)

Fig. 10 Band structures for the liquid phononic crystal with the square lattice calculated by the GFDM and the FEM (COMSOL). (a) Kite-shape scatter. (b) Blade-shape scatter

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(a)

(b)

Fig. 11 Band structures for the liquid phononic crystal with the triangular lattice calculated by the GFDM and the FEM (COMSOL). (a) Kite-shape scatter. (b) Blade-shape scatter

(a)

(b)

Fig. 12 Sketch of the cross-shaped scatters. (a) Cross-shape scatter. (b) Double-cross-shape scatter

(a)

(b)

(c)

Fig. 13 Band structures of cross scatter with different arm widths in square lattice. (a) Arm width 0.26a. (b) Arm width 0.20a. (c) Arm width 0.14a

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(a)

(b)

201

(c)

Fig. 14 Band structures of cross scatter with different arm widths in triangular lattice. (a) Arm width 0.26a. (b) Arm width 0.20a. (c) Arm width 0.14a

(a)

(b)

(c)

Fig. 15 Band structures of double-cross scatter with different arm lengths in square lattice. (a) Arm width 0.14a. (b) Arm width 0.11a. (c) Arm width 0.08a

(a)

(b)

(c)

Fig. 16 Band structures of double-cross scatter with different arm lengths in triangular lattice. (a) Arm width 0.14a. (b) Arm width 0.11a. (c) Arm width 0.08a

the more bandgaps appear at the lower frequency region, except the case about the square lattice including the cross scatter with Aw = 0.20a. The cross scatter may produce more obvious bandgaps than the double-cross scatter under the same filling fraction.

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4 Conclusions In this paper, the meshless generalized finite difference method (GFDM) is applied to calculate the acoustic bandgaps in 2D liquid phononic crystals. The proposed GFDM results are in good agreement with the FEM (COMSOL) results for calculating the band structures of the simple/complicated shape scatters in the square/triangular lattice. Under the present numerical investigations, the following conclusions are drawn: (1) The effect of the filling fraction is revisited under the circular scatters in Mercury/Air phononic crystal system. With the increasing filling fraction, more bandgaps appear at the low frequency region, in particular the triangular lattice. (2) With the decreasing arm width in the cross-shaped scatter cases, the more bandgaps appear at the lower frequency region, except the case about the square lattice including the cross scatter with Aw = 0.20a. The cross scatter may produce more obvious bandgaps than the double-cross scatter under the same filling fraction. Besides, it is worth noting that the proposed GFDM for elastic wave in 2D/3D phononic crystals are under intense study and will be reported in a subsequent paper. Acknowledgments The work described in this paper was supported by the National Science Fund of China (Grant Nos. 11772119, 11972354, 11772349), the Foundation for Open Project of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and astronautics) (Grant No. MCMS-E-0519G01), Alexander von Humboldt Research Fellowship (ID: 1195938) and the Six Talent Peaks Project in Jiangsu Province of China (Grant No. 2019-KTHY-009).

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