Modeling Simulation and Optimization of Fluid Dynamic Applications [148, 1 ed.] 9783031451577, 9783031451584

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148

Armin Iske · Thomas Rung  Editors

Modeling, Simulation and Optimization of Fluid Dynamic Applications Editorial Board T. J.Barth M.Griebel D.E.Keyes R.M.Nieminen D.Roose T.Schlick

Lecture Notes in Computational Science and Engineering Volume 148

Series Editors Timothy J. Barth, NASA Ames Research Center, Moffett Field, CA, USA Michael Griebel, Institut für Numerische Simulation, Universität Bonn, Bonn, Germany David E. Keyes, Applied Mathematics and Computational Science, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia Risto M. Nieminen, Department of Applied Physics, Aalto University School of Science & Technology, Aalto, Finland Dirk Roose, Department of Computer Science, Katholieke Universiteit Leuven, Leuven, Belgium Tamar Schlick, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

This series contains monographs of lecture notes type, lecture course material, and high-quality proceedings on topics described by the term “computational science and engineering”. This includes theoretical aspects of scientific computing such as mathematical modeling, optimization methods, discretization techniques, multiscale approaches, fast solution algorithms, parallelization, and visualization methods as well as the application of these approaches throughout the disciplines of biology, chemistry, physics, engineering, earth sciences, and economics.

Armin Iske • Thomas Rung Editors

Modeling, Simulation and Optimization of Fluid Dynamic Applications

Editors Armin Iske Department of Mathematics University of Hamburg Hamburg, Germany

Thomas Rung Institute for Fluid Dynamics and Ship Theory Hamburg University of Technology Hamburg, Germany

ISSN 1439-7358 ISSN 2197-7100 (electronic) Lecture Notes in Computational Science and Engineering ISBN 978-3-031-45157-7 ISBN 978-3-031-45158-4 (eBook) https://doi.org/10.1007/978-3-031-45158-4 Mathematics Subject Classification (2020): 41A05, 41A15, 49-XX, 65-XX, 76-XX © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 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 Paper in this product is recyclable.

Preface

This book describes recent efforts combining the expertise of applied mathematicians, engineers and geophysicists within a research training group (RTG 2583) on ”Modeling, Simulation and Optimization of Fluid Dynamic Applications” funded by the Deutsche Forschungsgemeinschaft (DFG). The RTG aims at the holistic research and education in the mathematical fields of Modeling, Simulation, and Optimization (MSO). The focus is on mathematical modeling, adaptive discretization and approximation strategies and shape optimization with PDEs. The balanced research program is based on the guiding principle that mathematics drives applications and is inspired by applications. With this leitmotif the RTG advances research in MSO by an interdisciplinary approach, i.e., to stimulate fundamental education and research by highly complex applications and at the simultaneously transfer tailored mathematical methods to applied sciences. The reported research involves nine projects and addresses challenging fluid dynamic problems inspired by applied sciences, such as climate research & meteorology (Chapters 3 and 7), energy, aerospace & marine engineering (Chapters 2, 8 and 9) or medicine (Chapter 6). More fundamental research concerning analysis, approximation and numerics is covered through Chapters 1, 4, 5. We believe that the material represents a successful attempt to exchange research paradigms between different disciplines and thus displays a modern approach to basic research into scientifically and societally relevant contemporary problems. Hamburg, September 2023

Armin Iske Thomas Rung

Acknowledgements The chapters of this work contribute to projects within the research training group ”Modeling, Simulation and Optimization of Fluid Dynamic Applications” (RTG 2583) funded by the Deutsche Forschungsgemeinschaft (DFG). Moreover, the encouraging support of Dr. Martin Peters (Springer Spektrum, Heidelberg), which led to the initiation of the book project, is gratefully acknowledged. v

Contents

1

1

Lower Bounds for the Advection-Hyperdiffusion Equation . . . . . . . . . Fabian Bleitner, Camilla Nobili

2

Modeling and Simulation of Parabolic Trough Collectors using Nanofluids: Application to NOOR I Plant in Ouarzazate, Morocco . . 21 Hamzah Bakhti, Ingenuin Gasser

3

Adaptive Discontinuous Galerkin Methods for 1D unsteady Convection-Diffusion Problems on a Moving Mesh . . . . . . . . . . . . . . . 35 Ezra Rozier, J¨orn Behrens

4

Anisotropic Kernels for Particle Flow Simulation . . . . . . . . . . . . . . . . . 57 Kristof Albrecht, Juliane Entzian, Armin Iske

5

An Error-Based Low-Rank Correction for Pressure Schur Complement Preconditioners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Rebekka S. Beddig, J¨orn Behrens, Sabine Le Borne, Konrad Simon

6

Radon-based Image Reconstruction for MPI using a continuously rotating FFL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Stephanie Blanke, Christina Brandt

7

Numerical Simulation of an idealized coupled Ocean-Atmosphere Climate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Kamal Sharma, Peter Korn

8

Application of p-Laplacian relaxed Steepest Descent to Shape Optimizations in Two-Phase Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Peter Marvin M¨uller, Martin Siebenborn, Thomas Rung

9

Towards Computing High-Order p-Harmonic Descent Directions and Their Limits in Shape Optimization . . . . . . . . . . . . . . . . . . . . . . . . 147 Henrik Wyschka, Martin Siebenborn vii

Chapter 1

Lower Bounds for the Advection-Hyperdiffusion Equation Fabian Bleitner, Camilla Nobili

Abstract Motivated by [7], we study the advection-hyperdiffusion equation in the whole space in two and three dimensions with the goal of understanding the decay in time of the H −1 - and L2 -norm of the solutions. We view the advection term as a perturbation of the hyperdiffusion equation and employ the Fourier-splitting method first introduced by Schonbek in [8] for scalar parabolic equations and later generalized to a broader class of equations including Navier-Stokes equations and magnetohydrodynamic systems. This approach consists of decomposing the Fourier space along a sphere with radius decreasing in time. Combining the Fourier-splitting method with classical PDE techniques applied to the hyperdiffusion equation we find a lower bound for the H −1 -norm by interpolation.

1.1 Introduction We study the advection-hyperdiffusion equation in Rn given by ∂t θ + u · ∇θ + κ∆ 2 θ = 0 ∇·u = 0 θ (x, 0) = θ0 (x)

in Rn × (0, ∞), in Rn × (0, ∞), in Rn ,

(1.1)

where κ > 0 is the molecular diffusion coefficient and n = 2, 3. This equation models a hyperdiffusive scalar concentration field θ advected by a time dependent incompressible vector field u. Analytical and numerical mixing estimates are derived by Fabian Bleitner Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] Camilla Nobili Department of Mathematics, University of Surrey, United Kingdom e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Iske, T. Rung (eds.), Modeling, Simulation and Optimization of Fluid Dynamic Applications, Lecture Notes in Computational Science and Engineering 148, https://doi.org/10.1007/978-3-031-45158-4_1

1

2

F. Bleitner, C. Nobili

Miles and Doering in [6] for the classical advection-diffusion equation, obtained by replacing ∆ 2 with −∆ in (1.1). Their numerical simulations indicate that the ”filamentation length” ∥∇−1 θ ∥2 λ= , ∥θ ∥2 where

∥∇−1 θ ∥22 =

Z Rn


2 |ξ |−1 |θˆ | dξ

reaches asymptotically (in time) a minimal scale, called the Batchelor scale. The filamentation length λ defined by Miles and Doering is a measure of mixing. Different mixing measures have been introduced in the last few years (see the excellent review paper [10] for an overview on this topic), but the H −1 -norm proposed in [4], is the one that better captures the mixing mechanism as it emphasises on the role of large wave lengths in comparison to small wave lengths. The rigorous proof for the convergence of λ to the Batchelor scale on bounded domains is a hard problem and only few partial results are available. In particular we want to cite the work in [9], where the author derives L1 estimates on Littlewood–Paley decompositions for linear advection–diffusion equations on the torus. Using the Fourier-splitting method introduced by Schonbeck [8], Pottel and the second author in [7] showed that in the whole space and under suitable conditions on the initial data, depending on decay rate assumptions on ∥u(t)∥2 and ∥∇u(t)∥∞ , the filamentation length λ either diverges to infinity or is bounded from below by a function that converges to zero. In this second scenario, mixing is possible. The problem of mixing for the advection-hyperdiffusion equation has already been addressed in a few works. Here we want to mention the work in [1], where the authors considered the advective Cahn-Hilliard equation ∂t θ + u · ∇θ + κ∆ 2 θ = ∆ (θ 3 − θ ),

(1.2)

on the n-dimensional torus, and study the effect of stirring on spontaneous phase separation. For u = 0 and small κ solutions to (1.2) can spontaneously separate into regions of high and low concentration. On the other hand, if u is sufficiently mixing, the authors show that the separation effect will be dominated and the solution converges exponentially to a homogeneous mixed state. In [2] the authors considered (1.1) where u is an incompressible flow with circular or cylindrical symmetry in 2 and 3 space dimensions, respectively, and studied enhanced dissipation, a phenomena intrinsically connected to mixing (see [11] where the authors establish a precise connection between quantitative mixing rates in terms of decay of negative Sobolev norms and enhanced dissipation time-scales). Exploiting the enhanced dissipation arising from the combined action of the hyper-diffusion and the advection to control both the nonlinearity as well as the destabilizing effect of the negative Laplacean at large scale, the authors in [5] prove global existence for the modified Kuramoto-Sivashinsky equation ν ∂t φ + u(y)∂x φ + |∇φ |2 + ν∆ 2 φ + ν∆ φ = 0 2

1 Lower Bounds for the Advection-Hyperdiffusion Equation

3

on the 2-dimensional torus. In this paper, we are interested in understanding how the lower bounds for λ change when the diffusion operator in the advection-diffusion equation is substituted by the bilaplacian. We apply the methods of [7] to bound solutions of the advectionhyperdiffusion equation. Our main result is the lower bound for the energy of the solution given suitable initial values and decay estimates on the flow. Theorem 1.1 (Lower Bound for the Energy). Let the initial data satisfy θ0 ∈ L 1 ∩ L 2 and

|θˆ0 (ξ )| ≥ M

for |ξ | ≤ δ and some constant M > 0. Moreover assume ∥u(t)∥2 ∼ (1 + t)−α

(1.3)

with α > 34 . Then for sufficiently large t the solution θ of (1.1) is bounded from below by 1

∥θ (t)∥2 ≥

where ωn−1

2 ωn−1 1 2

n

n

4

Mδ 2 e−κδ (1 + t)− 8 ,

2n is the measure of the (n − 1)-sphere.

As a consequence of the energy bound one gets a lower bound for the filamentation length and the H˚ −1 -norm. Theorem 1.2 (Mixing Bound). Let the assumptions of Theorem 1.1 be satisfied and additionally suppose θ0 ∈ H 1 and ∥∇u(t)∥∞ = c∇u (1 + t)−ν .

(1.4)

Then there exist constants c1 , c2 > 0 depending on c∇u , n, M, α, δ , κ, ∥θ0 ∥1 , ∥θ0 ∥2 and ∥∇θ0 ∥2 such that for sufficiently large t n

1

∥∇−1 θ (t)∥2 ≥ c1 (1 + t)− 8 + 4 f∇u (t), 1

λ ≥ c2 (1 + t) 4 f∇u (t),

where

 c∇u − ((1+t)1−ν −1)  e 1−ν f∇u (t) = 1  c∇u −(ν−1)  ) (1 + t)c∇u e− ν−1 (1−(1+t)

for 0 < ν < 1, for ν = 1, for ν > 1.

(1.5)

(1.6)

4

F. Bleitner, C. Nobili

The long time behaviour of these bounds is given in Table 1.1, where t→∞

∥∇−1 θ ∥2 ≥ f (t) −→ f∞ , t→∞

λ ≥ g(t) −→ g∞ .

0

1 8 1 8 1 8

Table 1.1: Long time behaviour of the mixing bounds.

Oragnization of the paper In Section 1.2 we introduce the Fourier-splitting method, which consists in splitting the whole Fourier space in a ball with decreasing in time radius and its complement. This allows us to estimate the time derivative of the L2 -norm of θ on the whole space by the L2 -norm in this ball. Integration in time then yields an upper bound on ∥θ (t)∥2 , under some specific decay assumptions on ∥u(t)∥2 . In Section 1.3 we prove lower bounds for ∥θ (t)∥2 by viewing the solution of (1.1) as a perturbation of the hyperdiffusion equation. In particular, in Subsection 1.3.3 we show that, under suitable decay assumptions on the stirring field, the perturbation decays at least as fast as the solution of the hyperdiffusion equation, yielding the result Theorem 1.1. Finally, in Section 1.4 we derive upper bounds for the gradient of the solution yielding the result in Theorem 1.2 by interpolation. Notation Norms are considered over the whole space unless differently stated. a ≲ b denotes a ≤ cb for a constant c > 0. Similarly a ∼ b and a ≳ b with a = cb and a ≥ cb. As we did not attempt to optimize the constant prefactors in the upper/lower bounds, the dependency of constants on parameters like the dimension n, the molecular diffusivity κ or the decay exponents α and ν (see (1.3),(1.4)) will often be hidden to increase the readability of the manuscript, unless tracking is required for the argu-

1 Lower Bounds for the Advection-Hyperdiffusion Equation

5

ment to work. Note that in few estimates these constants that we do not want to track may change from line to line, without being renamed.

1.2 Upper Bound for the Advection-Hyperdiffusion Equation In this section we introduce the Fourier-splitting method and prove upper bounds for the advection-hyperdiffusion equation under suitable decay of the flow field. This bound will later be used to derive upper bounds for the perturbation and the gradient of the solution. Lemma 1.1 (Upper Bound for the Advection-Hyperdiffusion Equation). Let the initial data satisfy θ0 ∈ L 1 ∩ L 2

and let u(·,t) ∈ L2 be a divergence-free vector field satisfying ∥u(t)∥2 ∼ (1 + t)−α

(1.7)

with α > 6−n 8 . Then there exists a constant c > 0 depending on n, α, κ, ∥θ0 ∥1 and ∥θ0 ∥2 such that n ∥θ (t)∥2 ≤ c(1 + t)− 8 . Proof. The proof consists of 3 steps. First we bound the derivative of the L2 -norm of θ over the whole space by the L2 -norm of θ over a ball with radius decreasing in time. Second we show, by integrating this ODE in time, that if θ decays with some rate in a specific range then it actually decays faster. Third we iterate to get the maximal decay rate of this range resulting in the final bound. Step 1: Testing (1.1) with θ , using partial integration, one gets 1 d ∥θ ∥22 = −κ 2 dt

Z Rn

θ ∆ 2 θ dx −

Z Rn

θ u · ∇θ dx = −κ∥∆ θ ∥22 ,

(1.8)

where the advection term vanishes by the incompressibility condition of u as Z Rn

θ u · ∇θ dx = −

Z Rn

θ 2 ∇ · u dx −

Z Rn

θ u · ∇θ dx = −

Z Rn

θ u · ∇θ dx.

By Plancherel (1.8) can be expressed in Fourier space via d ˆ 2 ∥θ ∥2 = −2κ∥ξ 2 θˆ ∥22 . dt Define the set

(1.9)

6

F. Bleitner, C. Nobili

S(t) = {ξ ∈ Rn | |ξ | ≤ r(t)}

 1 4 β with r = 2κ(1+t) , where β > 0 is a constant to be specified later and denote its complement by Sc . Then by (1.9) d ˆ 2 ∥θ ∥2 = −2κ∥ξ 2 θˆ ∥22 ≤ −2κ∥ξ 2 θˆ ∥2L2 (Sc ) = −2κ |ξ |4 |θˆ |2 dξ dt Sc Z   |θ |2 dξ = −2κr4 ∥θˆ ∥22 − ∥θˆ ∥2L2 (S) ≤ −2κr4 Z

Sc

 β  ˆ 2 ∥θ ∥2 − ∥θˆ ∥2L2 (S) =− 1+t

such that  d  d (1 + t)β ∥θˆ ∥22 = β (1 + t)β −1 ∥θˆ ∥22 + (1 + t)β ∥θˆ ∥22 dt dt  ≤ β (1 + t)β −1 ∥θˆ ∥22 − β (1 + t)β −1 ∥θˆ ∥22 − ∥θˆ ∥2 2



L (S)

(1.10)

= β (1 + t)β −1 ∥θˆ ∥2L2 (S) .

Step 2: Assume

∥θ ∥2 ≲ A(1 + t)−γ

(1.11)

θˆt = −κ|ξ |4 θˆ − u\ · ∇θ

(1.12)

with A depending on κ, n, ∥θ0 ∥1 and ∥θ0 ∥2 for some γ ≥ 0. Notice that by (1.9) the assumption is fulfilled for γ = 0. Writing (1.1) in Fourier space

and its solution is given by 4 θˆ (t, ξ ) = e−κ|ξ | t θˆ0 (ξ ) −

Z t 0

4 (t−s)

e−κ|ξ |

u\ · ∇θ (s, ξ ) ds.

Young’s inequality yields Z t

4 4 4 |θˆ (ξ )|2 = e−2κ|ξ | t |θˆ0 (ξ )|2 − 2e−κ|ξ | t θˆ0 (ξ ) e−κ|ξ | (t−s) u\ · ∇θ (s, ξ ) ds 0 Z t 2 4 + e−κ|ξ | (t−s) u\ · ∇θ (s, ξ ) ds

0

4

≤ 2e−2κ|ξ | t |θˆ0 (ξ )|2 + 2

Z 0

t

4 (t−s)

e−κ|ξ |

2 u\ · ∇θ (s, ξ ) ds

Z t 2 \ −2κ|ξ |4 t ˆ 2 ≤ 2e |θ0 (ξ )| + 2 u · ∇θ (s, ξ ) ds . 0

(1.13)

1 Lower Bounds for the Advection-Hyperdiffusion Equation

7

Using the incompressibility of u, the advection term can be estimated by Z Z Z \ iξ ·x iξ ·x iξ ·x u · ∇θ e dx = uθ · iξ e dx uθ · ∇e dx = u · ∇θ (ξ ) = Rn

Rn

Rn

−α−γ

≤ |ξ | ∥u∥2 ∥θ ∥2 ≲ A|ξ |(1 + t)

,

(1.14) where in the last inequality we used assumptions (1.7) and (1.11). Using ln(1 + t) ≲ ε1 (1 + t)ε with 0 < ε ≪ 1, the integral on the right-hand side of (1.13) can be bounded by Z t Z t \ u · ∇θ (s, ξ ) ds ≤ A|ξ | (1 + s)−α−γ ds 0

0    1 1−(α+γ) − 1    1−(α+γ) (1 + t) = A|ξ | ln(1 + t)      1 1 − (1 + t)−(γ+α)+1 α+γ−1

 1 1−(α+γ)   1−(α+γ) (1 + t) 1 ε ≤ A|ξ | ε (1 + t)   1 α+γ−1

for α + γ < 1 for α + γ = 1 for α + γ > 1

for α + γ < 1 for α + γ = 1 for α + γ > 1

≤ A|ξ |hα+γ (t), where

(1.15)

 1 1−(α+γ)   1−(α+γ) (1 + t) hα+γ (t) = ε1 (1 + t)ε   1 α+γ−1

for α + γ < 1 for α + γ = 1 for α + γ > 1

Calculating ∥ξ ∥2L2 (S) =

Z |ξ |≤r(t)

ωn−1 = n+2



|ξ |2 dξ = ωn−1 β 2κ(1 + t)

 n+2 4

Z r(t) 0

ρ n+1 dρ =

ωn−1 n+2 r n+2

(1.16)

we can integrate (1.13) over S and, using (1.15) and Lemma 1.5 (proven in the Appendix), find

8

F. Bleitner, C. Nobili

2 4

∥θˆ ∥2L2 (S) = 2 e−κ|ξ | t θˆ0 2

L (S)

4

2 ≲ e−κ|ξ | t θˆ0 2

L (S)

Z t

2



\

+2 ds u · ξ ) (s, ∇θ

0

L2 (S)

2 + A2 ∥ξ ∥L2 2 (S) hα+γ (t)

  n+2

2

n+2 4 β 4

≲ e−κ|ξ | t θˆ0 + A2 (1 + t)− 4 h2α+γ (t) κ 2   n+2 n n+2 β 4 ≲ (κt)− 4 ∥θ0 ∥21 + A2 (1 + t)− 4 h2α+γ (t). κ

(1.17)

Plugging (1.17) into (1.10) one gets  d  (1 + t)β ∥θˆ ∥22 = β (1 + t)β −1 ∥θˆ ∥2L2 (S) dt n n ≲ β κ − 4 (1 + t)β −1− 4 ∥θ0 ∥21 + A2 β

n+6 4

κ−

n+2 4

(1 + t)β −

n+6 4

h2α+γ (t).

Choosing β big enough such that the right-hand side becomes a polynomial with positive exponent and integrating in time yields (1 + t)β ∥θˆ ∥22 − ∥θˆ0 ∥22

  β β − n4 ˆ 2 −1 n ∥θ0 ∥1 (1 + t) β−4    1 1 β − n4 −2(α+γ)+ 32 − 1  (1 + t) n 3   2    1−(α+γ) β − 4 −2(α+γ)+ n 1 n+6 1 2 − n+2 4 4 ε −2 (1 + t)2ε+β − 4 − 2 − 1 +A κ β n 1 2ε+β − 4 − 2     n 1  1  1 (1 + t)β − 4 − 2 − 1 n 1 α+γ−1 n

≲ κ− 4

β− 4 − 2

such that dividing by (1 + t)β results in n β − n4 ˆ 2 ∥θˆ ∥22 ≲ ∥θˆ0 ∥22 (1 + t)−β + κ − 4 n ∥θ0 ∥1 (1 + t) β−4  1 1 − n4 −2(α+γ)+ 32    1−(α+γ) β − n4 −2(α+γ)+ 32 (1 + t)  n+2 n+6 1 2ε− n4 − 12 ε −2 + A2 κ − 4 β 4 n − 1 (1 + t) 2ε+β −  4 2  n 1  1  1 (1 + t)− 4 − 2

α+γ−1 β − n − 1 4 2

and for α + γ
in the desired decay rate.

6−n 8

n 3 − +α +γj 8 4

ensures γ j is a strictly increasing sequence resulting

1.3 Lower Bound for the Advection-Hyperdiffusion Equation For the lower bound on the solution of the advection-hyperdiffusion equation we first prove a lower bound for the hyperdiffusion equation and then show that the perturbation that arises because of the advection decays at least with the same rate.

1.3.1 Lower Bound for the Hyperdiffusion Equation Lemma 1.2 (Lower Bound for the Hyperdiffusion Equation). Let T solve ∂t T + κ∆ 2 T = 0 T (0, x) = θ0 (x) and θ0 ∈ L2 satisfy

in Rn × (0, ∞), in Rn

|θˆ0 (ξ )| ≥ M

for |ξ | ≤ δ . Then for t ≥ 1 1

1

n

(1.19)

4

n

2 n− 2 Mδ 2 e−κδ (1 + t)− 8 , ∥T (t)∥2 ≥ ωn−1

where ωn−1 is the measure of the n − 1-sphere. Proof. As the Fourier transform of (1.18) is given by ∂t Tˆ + κ|ξ |4 Tˆ = 0 Tˆ (0, ξ ) = θˆ0 (ξ )

(1.18)

in Rn × (0, ∞),

in Rn

10

F. Bleitner, C. Nobili

its solution can be represented by 4 Tˆ (t, ξ ) = θˆ0 (ξ )e−κ|ξ | t ,

which by Plancherel and assumption (1.19) yields Z Rn

|T |2 dx =

Z Rn

|Tˆ |2 dξ =

Z Rn

4 |θˆ0 |2 e−2κ|ξ | t dξ ≥ M 2

Z

4

|ξ |≤δ

e−2κ|ξ | t dξ .

(1.20) 1 By passing to spherical coordinates and the change of variables µ = (2κt) 4 r the integral can be expressed by Z

4

|ξ |≤δ

e−2κ|ξ | t dξ = ωn−1

Z δ 0

4

e−2κr t rn−1 dr

= ωn−1 (2κt)

− n4

Z (2κt) 14 δ 0

(1.21) e

−µ 4

µ n−1 dµ,

where ωn−1 is the measure of the n − 1-sphere. For t ≥ 1 we estimate this integral by Z (2κt) 14 δ 0

−µ 4

e

µ

n−1

dµ ≥

Z (2κ) 14 δ 0

e

−µ 4

µ

n−1

n 4 1 = e−2κδ (2κ) 4 δ n . n

dµ ≥ e

−2κδ 4

Z (2κ) 14 δ 0

µ n−1 dµ

(1.22)

Combining (1.20), (1.21) and (1.22) yields the claim as Z Rn

2

|T | dx ≥ M

2

≥ M2

Z |ξ |≤δ

e

−2κ|ξ |4 t

2

dξ = M ωn−1 (2κt)

n ωn−1 −2κδ 4 n e δ (1 + t)− 4 . n

− n4

Z (2κt) 14 δ 0

4

e−µ µ n−1 dµ

1.3.2 Upper Bound for the Perturbation Lemma 1.3. The perturbation η = θ − T solves ∂t η + κ∆ 2 η + u · ∇θ = 0 η(x, 0) = 0

in Rn × (0, ∞), in Rn .

(1.23)

Suppose the assumptions of Theorem 1.1 are fulfilled. Then there exists a constant c > 0 depending on n, α, κ, ∥θ0 ∥1 and ∥θ0 ∥2 such that

1 Lower Bounds for the Advection-Hyperdiffusion Equation 3

5 n 0. Then from Plancherel follows

2 d ∥η∥22 = −2κ |ξ |2 ηˆ 2 + 2∥∇T ∥∞ ∥u∥2 ∥θ ∥2 dt Z ≤ −2κ

Sc

ˆ 2 dξ + 2∥∇T ∥∞ ∥u∥2 ∥θ ∥2 |ξ |4 |η| Z

ˆ 2 dξ + 2∥∇T ∥∞ ∥u∥2 ∥θ ∥2 |η|   ˆ 22 − ∥η∥ ˆ 2L2 (S) + 2∥∇T ∥∞ ∥u∥2 ∥θ ∥2 = −β (1 + t)−1 ∥η∥ ≤ −β (1 + t)−1

such that

Sc

12

F. Bleitner, C. Nobili

 d  d (1 + t)β ∥η∥22 = β (1 + t)β −1 ∥η∥22 + (1 + t)β ∥η∥22 dt dt (1.24) ˆ 2L2 (S) + 2(1 + t)β ∥∇T ∥∞ ∥u∥2 ∥θ ∥2 . = β (1 + t)β −1 ∥η∥ In order to estimate the first term, the solution of (1.23) in Fourier space is given by Z t

ˆ ξ) = − η(t,

0

4 (t−s)

e−κ|ξ |

u\ · ∇θ (ξ ) ds

such that by (1.14), the decay assumption (1.7), Lemma 1.1 and (1.16) ˆ 2L2 (S) ∥η∥

= ≤

Z Z t S

0

Z Z t S

Z

≲

S

Z

0

|ξ |

2

2

2 u · ∇θ (ξ ) ds dξ

−κ|ξ |4 (t−s) \

e

4 (t−s)

e−κ|ξ |

Z 0

t

2 |ξ |∥u∥2 ∥θ ∥2 ds dξ

−α− 8n

(1 + s) 1



ds

2

dξ 2

−α− n8 +1



1 − (1 + t) α + n8 − 1  −2 Z n ≤ α + −1 |ξ |2 dξ 8 S  n+1 4 2 β ∼ κ(1 + t)

=

S

|ξ |

(1.25) dξ

as α > 1 − n8 . Combining (1.25) and (1.24), Lemma 1.5 (proven in the Appendix), the decay assumption (1.7) and Lemma 1.1 we find  d  ˆ 2L2 (S) + 2(1 + t)β ∥∇T ∥∞ ∥u∥2 ∥θ ∥2 (1 + t)β ∥η∥22 = β (1 + t)β −1 ∥η∥ dt n

3

n

3

n

1

3

n

≲ β 4 + 2 κ − 4 − 2 (1 + t)β − 2 − 4 + (1 + t)β ∥∇T ∥∞ ∥u∥2 ∥θ ∥2 3

n

1

n

≲ β 4 + 2 (1 + t)β − 2 − 4 + (1 + t)β − 4 − 4 −α . Choosing β > 12 + n4 , 34 + n4 + α and integrating in time yields (1 + t)β ∥η∥22 ≲

1



β − n4 + 12 +

 n 1 (1 + t)β − 4 − 2 − 1

  1 β − n4 −α+ 34 (1 + t) − 1 β − n4 − α + 34 n

1

n

3

≲ (1 + t)β − 4 − 2 + (1 + t)β − 4 −α+ 4 such that dividing by (1 + t)β results in the desired decay as

1 Lower Bounds for the Advection-Hyperdiffusion Equation n

1

13 3

n

∥η∥22 ≲ (1 + t)− 4 − 2 + (1 + t)− 4 −α+ 4 .

1.3.3 Lower Bound for the Energy We are now able to prove Theorem 1.1, i.e. the lower bound for the energy. Proof (Theorem 1.1). As η = θ − T and therefore ∥T ∥2 = ∥η − θ ∥2 ≤ ∥θ ∥2 + ∥η∥2 the energy of the solution of the advection-hyperdiffusion equation can be bounded by ∥θ ∥2 ≥ ∥T ∥2 − ∥η∥2 . Lemma 1.2 yields a lower bound for T , while Lemma 1.3 implies an upper bound on η such that n

n

n

4

3

α 5

2 e−κδ (1 + t)− 8 + c(1 + t)− 8 + 8 −min{ 2 , 8 } ∥θ ∥2 ≥ ∥T ∥2 − ∥η∥2 ≥ cMδ ˜   α 5 3 n n 4 ≥ c˜ Mδ 2 e−κδ − c(1 + t) 8 −min{ 2 , 8 } (1 + t)− 8 , 1

1

2 where c˜ = ωn−1 n− 2 . By assumption α >

3 4

such that for all t > t1 given by

3 α 5 n 4 c Mδ 2 e−κδ = (1 + t1 ) 8 −min{ 2 , 8 } 2

the solution decays at most by 1

∥θ ∥2 ≥

2 ωn−1

2n

1 2

n

4

n

Mδ 2 e−κδ (1 + t)− 8 .

1.4 Lower Bounds on the Mixing Norms In order to get lower bounds for the mixing norms we need an upper bound for the gradient of the solution as the interpolation takes the reciprocal of this bound.

1.4.1 Upper Bound for the Gradient of the Advection-Hyperdiffusion Equation Lemma 1.4 (Upper Bound for the Gradient of the Advection-Hyperdiffusion Equation). Let the assumptions of Theorem 1.1 be fulfilled and additionally suppose ∥∇u∥∞ = c∇u (1 + t)−ν .

(1.26)

Then there exists a constant c > 0 depending on c∇u , n, α, κ, ∥θ0 ∥1 , ∥θ0 ∥2 and ∥∇θ0 ∥2 such that

14

F. Bleitner, C. Nobili n

c∇u

1

1−ν −1

∥∇θ ∥2 ≤ c(1 + t)− 8 − 4 e 1−ν ((1+t) ∥∇θ ∥2 ≤ c(1 + t) ∥∇θ ∥2 ≤

)

if 0 < ν < 1,

− n8 − 14

if ν = 1,

c

n 1 −(ν−1) ∇u ) c(1 + t)− 8 − 4 −c∇u e ν−1 (1−(1+t)

if ν > 1.

Proof. Differentiating (1.1) yields ∂t ∇θ + κ∇∆ 2 θ + ∇(u · ∇θ ) = 0 such that testing with ∇θ and partial integration implies 1 d ∥∇θ ∥22 = −κ 2 dt

Z Rn

∇θ · ∇∆ 2 θ dx −

= −κ∥∆ ∇θ ∥22 − ∑

Z Rn

i, j

Z Rn

∇θ · ∇(u · ∇θ ) dx

∂i θ ∂i u j ∂ j θ dx − ∑

Z Rn

i, j

∂i θ u j ∂i ∂ j θ dx.

By partial integration and the impressibility condition on u the third term on the right-hand side vanishes as

∑

Z

i, j

Rn

∂i θ u j ∂i ∂ j θ dx = − ∑

Z

= −∑

Z

i, j

i, j

Rn

Rn

∂ j ∂i θ u j ∂i θ dx − ∑ i, j

Z Rn

∂i θ ∂ j u j ∂i θ dx

∂i θ u j ∂i ∂ j θ dx.

Therefore H¨older’s inequality implies 1 d ∥∇θ ∥22 = −κ∥∆ ∇θ ∥22 − ∑ 2 dt i, j

Z Rn

∂i θ ∂i u j ∂ j θ dx

(1.27)

≤ −κ∥∆ ∇θ ∥22 + ∥∇u∥∞ ∥∇θ ∥22 . Similar to before define S(t) = {ξ ∈ Rn | |ξ | ≤ r(t)} , where r =



β 2κ(1+t)

Z

1 4

with a constant β > 0. Then Plancherel and Lemma 1.1 imply Z

Z

|ξ |6 |θˆ |2 dξ ≥ |ξ |6 |θˆ |2 dξ ≥ r4 |ξ |2 |θˆ |2 dξ Rn Sc Sc     Z Z 4 2 2 ˆ 2 4 2 2 2 ˆ = r ∥∇θ ∥2 − |ξ | |θ | dξ ≥ r ∥∇θ ∥2 − r |θ | dξ S S  
n ≥ r4 ∥∇θ ∥22 − r2 ∥θ ∥22 ≥ r4 ∥∇θ ∥22 − cr2 (1 + t)− 4 .

∥∆ ∇θ ∥22 =

Combining (1.27) and (1.28), the definition of r yields

(1.28)

1 Lower Bounds for the Advection-Hyperdiffusion Equation

15

  n d ∥∇θ ∥22 ≤ −2κr4 ∥∇θ ∥22 − cr2 (1 + t)− 4 + 2∥∇u∥∞ ∥∇θ ∥22 dt
3 n 3 = 2 ∥∇u∥∞ − β (1 + t)−1 ∥∇θ ∥22 + cβ 2 (1 + t)− 4 − 2 . By Gr¨onwall’s inequality and the decay assumption (1.26) ∥∇θ ∥22 ≤ e2

Rt

−1 0 ∥∇u∥∞ −β (1+τ) 3

+ cβ 2 ≤ e2

Rt

Z t 0

e2

Rt τ

dτ

∥∇θ0 ∥22

∥∇u∥∞ −β (1+s)−1 ds

−ν −1 0 c∇u (1+τ) −β (1+τ) 3

+ cβ 2

Z t 0

e2

Rt

τ c∇u (1+s)

dτ

n

3

(1 + τ)− 4 − 2 dτ (1.29)

∥∇θ0 ∥22

−ν −β (1+s)−1

ds

3

n

(1 + τ)− 4 − 2 dτ,

where the second exponentiated integral can be calculated by  c∇u 1−ν 1−ν  e2( 1−ν ((1+t) −(1+τ) )−β (ln(1+t)−ln(1+τ)))  Rt −ν −1 e2 τ c∇u (1+s) −β (1+s) ds = e2(c∇u −β )(ln(1+t)−ln(1+τ))  c∇u  2( ν−1 ((1+τ)−(ν−1) −(1+t)−(ν−1) )−β (ln(1+t)−ln(1+τ))) e  2 c∇u (1+t)1−ν −(1+τ)1−ν ( ) if ν < 1 2β   e 1−ν
1+τ 2c∇u 1+t = if ν = 1 1+τ  1+t  2 c∇u ((1+τ)−(ν−1) −(1+t)−(ν−1) ) e ν−1 if ν > 1 Choosing β > 14 + n8 + c∇u we can calculate the whole expression on the right-hand side of (1.28). In case 0 < ν < 1 c∇u

1−ν −1

∥∇θ ∥22 ≤ (1 + t)−2β e2 1−ν ((1+t) 3

+ cβ 2 (1 + t)−2β

Z t

) ∥∇θ ∥2 0 2

0 c∇u −2β 2 1−ν ((1+t)1−ν −1)

≤ (1 + t)

e

c∇u

3

n

∥∇θ0 ∥22

1−ν −1

+ cβ 2 (1 + t)−2β e2 1−ν ((1+t) c

∇u (1+t)1−ν −1 −2β 2 1−ν ( )

= (1 + t)

e

c∇u

3

1−ν −(1+τ)1−ν

(1 + τ)2β − 4 − 2 e2 1−ν ((1+t)

∥∇θ0 ∥22

)

Z t 0

n

) dτ

3

(1 + τ)2β − 4 − 2 dτ

3   c∇u cβ 2 −2β 2 1−ν ((1+t)1−ν −1) (1 + τ)2β − n4 − 12 − 1 (1 + t) e 2β − n4 − 12 c∇u n 1 1−ν ≤ c(1 + t)− 4 − 2 e2 1−ν ((1+t) −1) .

+

In case ν = 1

16

F. Bleitner, C. Nobili 3

∥∇θ ∥22 ≤ (1 + t)2(c∇u −β ) ∥∇θ0 ∥22 + cβ 2 (1 + t)2(c∇u −β )

Z t 0

n

3

(1 + τ)2β −2c∇u − 4 − 2 dτ

= (1 + t)2(c∇u −β ) ∥∇θ0 ∥22 3

+

  cβ 2 2(c∇u −β ) 2β −2c∇u − 4n − 12 (1 + t) (1 + t) − 1 2β − 2c∇u − n4 − 12 n

1

≤ c(1 + t)− 4 − 2 . In case ν > 1

c∇u

−(ν−1)

∥∇θ ∥22 ≤ (1 + t)−2β e2 ν−1 (1−(1+t)

c∇u

3

) ∥∇θ ∥2 0 2

−(ν−1)

+ cβ 2 (1 + t)−2β e2 ν−1 (1−(1+t) ·

c∇u 2 ν−1

≲ (1 + t)−2β e

Z t 0

c∇u

e2 ν−1 ((1+τ)

(1−(1+t)−(ν−1) ) ∥∇θ ∥2 0 2 c∇u

3

+ cβ 2 (1 + t)−2β e2 ν−1 (1−(1+t) c

∇u 1−(1+t)−(ν−1) −2β 2 ν−1 ( )

= (1 + t)

e

−(ν−1) −1

3

−(ν−1)

)

) (1 + τ)2β − n4 − 32 dτ

Z t 0

)

n

3

(1 + τ)2β −2c∇u − 4 − 2 dτ

∥∇θ0 ∥22

c∇u −(ν−1) cβ 2 ) + e2 ν−1 (1−(1+t) 2β − 2c∇u − n4 − 12   n 1 · (1 + t)−2β (1 + τ)2β −2c∇u − 4 − 2 − 1 c∇u

−(ν−1)

≤ ce2 ν−1 (1−(1+t)

) (1 + t)−2c∇u − n4 − 12 .

Combining Theorem 1.1 and Lemma 1.4 we are now able to prove the lower bounds for the mixing norms. Proof (Theorem 1.2). By Plancherel and H¨older’s inequality





∥θ ∥22 = ∥θˆ ∥22 = |ξ |θˆ |ξ |−1 θˆ 1 ≤ |ξ |θˆ 2 |ξ |−1 θˆ 2 = ∥∇θ ∥2 ∥∇−1 θ ∥2 such that using the lower bound on θ of Theorem 1.1 and the upper bound on ∇θ of Lemma 1.4 one finds ∥∇−1 θ ∥2 ≥ ∥θ ∥22 ∥∇θ ∥−1 2  c∇u 1−ν  e− 1−ν ((1+t) −1)  n 1 ≳ (1 + t)− 8 + 4 1  c∇u −(ν−1)  ) (1 + t)c∇u e− ν−1 (1−(1+t) and for the filamentation length

if 0 < ν < 1, if ν = 1, if ν > 1

1 Lower Bounds for the Advection-Hyperdiffusion Equation

17

∥∇−1 θ ∥2 ≥ ∥θ ∥2 ∥∇θ ∥−1 2 ∥θ ∥2  c∇u 1−ν − ((1+t) −1)  e 1−ν 1 ≳ (1 + t) 4 1  c∇u −(ν−1)  ) (1 + t)c∇u e− ν−1 (1−(1+t)

λ=

if 0 < ν < 1, if ν = 1, if ν > 1.

1.5 Conclusion Using the Fourier-splitting method we were able to prove upper bounds for the advection-hyperdiffusion equation under suitable decay assumptions on the velocity field. In turn, viewing this equation as a perturbation of the hyperdiffusion equation provides lower bounds on the energy decay. Interpolation then results in bounds on the mixing norms. In particular if the velocity field is decaying sufficiently fast the filamentation length diverges to infinity.

1.6 Appendix The following properties of the hyperdiffusion kernel can be found in [3]. Proposition 1.1. The hyperdiffusion kernel is given by   |x| − n4 , G(t, x) = αnt fn 1 t4 where αn is a normalization constant and • fn (η) = η

1−n

Z ∞ 0

4

n

e−s (ηs) 2 J n−2 (ηs) ds, 2

where Jν denotes the ν-tv Bessel function of the first kind. • there exists Kn > 0 and µn > 0 such that | fn (η)| ≤ Kn e−µn η •

4 3

for all η ≥ 0. for all n ≥ 1.

fn′ (η) = −η fn+2 (η)

Lemma 1.5 (Bounds for the hyperdiffusion equation). For 1 ≤ p < ∞ and t > 0 the hyperdiffusion kernel fulfills

18

F. Bleitner, C. Nobili

∥G∥ p ≲n,p t ∥∇G∥ p ≲n,p t

  − 4n 1− 1p   − n4 1− 1p − 14

and the solution of the hyperdiffusion equation (1.18) can be bounded by n

∥T ∥2 ≲ (κt)− 8 ∥θ0 ∥1 n

1

n

1

∥∇T ∥2 ≲ (κt)− 8 − 4 ∥θ0 ∥1

∥∇T ∥∞ ≲ (κt)− 8 − 4 ∥θ0 ∥2 and

n

−κ|ξ |4 t ˆ θ ≲ (κt)− 8 ∥θ0 ∥1 .

e 2

1

Proof. By passing to spherical coordinates, the change of variables ρ = rt − 4 and Proposition 1.1 one gets   p Z   p Z ∞ |x| r n−1 p p − pn − pn 4 4 fn dx ∼ t r fn 1 dr ∥G∥ p = an t 1 n 0 R t4 t4 =t

− n4 (p−1)

Z ∞ 0

ρ

n−1

p

| fn (ρ)| dρ ≲ t

− n4 (p−1)

Z ∞ 0

4 3

n

ρ n−1 e−pµn ρ dρ ≲ t − 4 (p−1)

and similarly ∥∇G∥ pp

  p Z   p |x| − 4p (n+1) fn′ |x| dx ∇ fn dx ∼ t = 1 1 n n R R 4 t4 t   p Z ∞ Z ∞ p p n r rn−1 fn′ dr ∼ t − 4 (p−1)− 4 ρ n−1 | fn′ (ρ)| p dρ ∼ t − 4 (n+1) 1 0 0 t4 pn anpt − 4

n

Z

p

= t − 4 (p−1)− 4 ≲t

− n4 (p−1)− 4p

Z ∞ 0

p

n

ρ n | fn+2 (ρ)| p dρ ≲ t − 4 (p−1)− 4

Z ∞ 0

.

The bounds on T follow directly as T (t, ·) = G(κt, ·) ⋆ T0 (·), Young’s inequality for convolution and the previous bounds on G yield n

∥T ∥2 ≤ ∥G ⋆ θ0 ∥2 ≤ ∥G∥2 ∥θ0 ∥1 ≲ (κt)− 8 ∥θ0 ∥1 n

1

n

1

4 3

ρ n e−pµn+2 η dρ

∥∇T ∥2 ≤ ∥∇G ⋆ θ0 ∥2 ≤ ∥∇G∥2 ∥θ0 ∥1 ≲ (κt)− 8 − 4 ∥θ0 ∥1

∥∇T ∥∞ ≤ ∥∇G ⋆ θ0 ∥∞ ≤ ∥∇G∥2 ∥θ0 ∥2 ≲ (κt)− 8 − 4 ∥θ0 ∥2 .

1 Lower Bounds for the Advection-Hyperdiffusion Equation

19

Analogously by Plancherel

n 4

e−κ|ξ | t θˆ0 = ∥G(κt, ·) ⋆ θ0 (·)∥2 ≤ ∥G∥2 ∥θ0 ∥1 ≲ (κt) 8 ∥θ ∥1 . 2

Acknowledgment The authors acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 ”Modeling, Simulation and Optimization of Fluid Dynamic Applications”.

References 1. Y. Feng, Y. Feng, G. Iyer, and J.-L. Thiffeault. Phase separation in the advective Cahn-Hilliard equation. Journal of Nonlinear Science 30(6): 2821–2845, 2020. 2. Y. Feng, A. Mazzucato, and C. Nobili. Enhanced dissipation by circularly symmetric and parallel pipe flows. Physica D: Nonlinear Phenomena 445: 133640, 2023. 3. F. Gazzola and H.-C. Grunau. Some new properties of biharmonic heat kernels. Nonlinear Analysis: Theory, Methods & Applications 70(8): 2965–2973, 2009. 4. G. Mathew, I. Mezi´c, and L. Petzold. A multiscale measure for mixing. Physica D: Nonlinear Phenomena 211(1): 23–46, 2005. 5. M. Coti Zelati, M. Dolce, Y. Feng, and A. Mazzucato. Global existence for the two-dimensional Kuramoto-Sivashinsky equation with a shear flow. Journal of Evolution Equations 21(4): 5079– 5099, 2021. 6. C. Miles and C.R. Doering. Diffusion-limited mixing by incompressible flows. Nonlinearity 31(5): 2346–2359, 2018. 7. C. Nobili and S. Pottel. Lower bounds on mixing norms for the advection diffusion equation in Rd . Nonlinear Differential Equations and Applications 29(2): 1–32, 2022. 8. M. Schonbek. Decay of solution to parabolic conservation laws. Communications in Partial Differential Equations 5(4): 449–473, 1980. 9. C. Seis. On the Littlewood-Paley spectrum for passive scalar transport equations. Journal of Nonlinear Science 30(2): 645–656, 2020. 10. J.-L. Thiffeault. Using multiscale norms to quantify mixing and transport. Nonlinearity 25(2): R1–R44, 2012. 11. M. Coti Zelati, M.G. Delgadino, and T.M. Elgindi. On the relation between enhanced dissipation timescales and mixing rates. Communications on Pure and Applied Mathematics 73(6): 1205–1244, 2020.

Chapter 2

Modeling and Simulation of Parabolic Trough Collectors using Nanofluids: Application to NOOR I Plant in Ouarzazate, Morocco Hamzah Bakhti, Ingenuin Gasser

Abstract A mathematical model is built to study the performance of parabolic trough collectors with the assumption of added nanoparticles to thermal oil used as the base fluid of the heat transfer fluid. In this work, the use of different nanoparticles, namely Copper Oxide, Alumina and Titanium Oxide, is investigated where a numerical approach is presented for a single parabolic collector model. The effects of the nanoparticles on the model parameters as well as the energy efficiency and exergetic efficiency are analyzed. The proposed mathematical model takes into account the system data and external climatic conditions from the NOOR I power plant in Ouarzazate region of Morocco. It is clearly shown that the presence of 1, 2 and up to 5% volume fraction of the nanoparticles in the base fluid (Therminol VP-1) enhances the energetic and exergetic efficiencies of the parabolic trough collector. On the other hand, a slight increase in the friction loss is also noticed when considering nanofluids.

2.1 Introduction Solar radiation is one of the most sustainable and renewable sources of energy, where in most power plants, the radiation is often used to heat a medium in order to produce electrical energy. Parabolic trough solar plants are one of the best known and most established technologies for generating electricity from solar radiation and it uses long, trough-shaped solar concentrators to collect solar heat and focus it onto a linear heat absorber. These reflectors track the Sun across the sky Hamzah Bakhti Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] Ingenuin Gasser Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Iske, T. Rung (eds.), Modeling, Simulation and Optimization of Fluid Dynamic Applications, Lecture Notes in Computational Science and Engineering 148, https://doi.org/10.1007/978-3-031-45158-4_2

21

22

H. Bakhti, I. Gasser

for maximum efficiency. The heat transfer fluid (HTF) flows through the system of collectors tubes and stores the heat to release the steam into the turbine and generate the electricity [1, 2]. One of the largest known parabolic trough power plants is the Moroccan solar plant NOOR I in the region of Ourzazate constructed in 2016, which is owned by Moroccan Agency For Solar Energy (MASEN) as part of the different solar power plants implimented in the country. The power plant uses solar radiations to generate electricity by using thermal oil as the HTF [3, 4, 5]. In recent years, researchers suggested various types of improvements to increase the performance of parabolic trough collectors. Only limited literature can be found concerning the effects of the use of nanofluids as HTCs. Most recent studies include the work of Allouhi et. al. [6], where a one-dimensional flow model is presented to investigate the effect of three-different nanoparticles on the thermal efficiency of parabolic trough collectors using climatic data from the region of Ouarzazate, Morocco. The main observation is the slight increase in efficiency with respect to the volume fraction of the nanoparticles in the HTC. In this paper, by using Copper Oxide, Alumina and Titanium Oxide as the three types of nanoparticles in the HTC, a one-dimensional scaled mathematical model is set up using asymptotic analysis in order to analyze the flow parameters in a parabolic trough collector, namely fluid velocity, pressure, temperature, heat capacity and density. An adaptive numerical technique is then presented for the given model. Finally, the model is applied using real world data from NOOR I power plant in order to obtain the results for the different fluid parameters and the effect of the nanoparticles on the energetic and exergetic performances of the parabolic trough collector.

2.2 Mathematical Model

Radiation loss Base Fluid

Beam radiation

Convection loss Nanoparticles Mirror

Cover

Fig. 2.1: Schematic representation of the operation of a parabolic solar collector using nanofluids as a heat transfer fluid.

2 Modeling and Simulation of Parabolic Trough Collectors using Nanofluids

23

In the following, a unidirectional system of equations is developed representing HTF flow in a solar parabolic tough collector involving different fluid quantities, namely fluid density, velocity, temperature, specific heat capacity, viscosity and pressure are described as functions of time and space, while other quantities are obtained such as Reynolds number, friction factor, energetic and exergetic efficiencies. Moreover, in this study, nanofluids are used as the HTF in our system with the classical Therminol VP-1 considered as the base fluid, which is suitable for such purposes, mixed with three types of oxide nanopaticles at a time, namely copper oxide (CuO), alumina (Al2 O3 ) and titanium oxide (TiO2 ). The scaled unidirectional model is obtained by assuming the cross-sectional area of the collector flow tube is very small compared to its length. Based on that, a onedimensional flow model is derived by averaging variables over the cross-section in y˜ and z˜ directions. Based on the conservation and balance laws, a system of ˜ x, equations is derived where the unknowns are: the density ρ˜ = ρ( ˜ t˜), the velocity ˜ x, ˜ t˜), specific heat capacity c˜ p = c˜ p (x, u˜ = u( ˜ t˜), viscosity µ˜ = µ( ˜ t˜), the tempera˜ x, ˜ x, ˜ t˜) and the pressure p˜ = p( ture T˜ = T˜ (x, ˜ t˜), all intended as cross-sectional mean values. Thus, the one-dimensional governing equations are given by: ˜ x=0 ρ˜t + (ρ˜ u)

(2.1)
ξ 4 A˜ (ρ˜ u) ˜ t + A˜ ρ˜ u˜2 + p˜ x = −U˜ i ρ˜ u| ˜ u| ˜ + A˜ (µ˜ u˜x )x (2.2) 8 3      1 1 A˜ c˜ p ρ˜ T˜ + ρ˜ u˜2 + A˜ u˜ c˜ p ρ˜ T˜ + ρ˜ u˜2 + p˜ 2 2 x t
ξ 4 = U˜ i ρ˜ u˜2 |u| ˜ + A˜ (µ˜ u˜x u) ˜ x + A˜ k˜ T˜x x + A˜ q˜˙s − U˜ o q˜˙rad − U˜ o q˜˙conv (2.3) 8 3 where ξ denotes the coefficient of friction, q˙˜s = q˜˙s (t˜) the beam solar radiation, q˜˙conv = q˜˙conv (x, ˜ t˜) the convective heat loss and q˜˙rad = q˜˙rad (x, ˜ t˜) the radiation heat  ˜ 2 ˜ ˜ x) is the exchange between the absorber and the atmosphere, A˜ = A( ˜ = π Do (x) 2

cross-sectional area, U˜ i = U˜ i (x) ˜ = π D˜ i (x) ˜ is the inner circumference (D˜ i inner diameter) of the tube and U˜ o = U˜ o (x) ˜ = π D˜ o (x) ˜ is the outer absorber circumference (D˜ o outer diameter). In order to calculate the friction factor, the Colebrook equation [7, 8] is used, given as follows: ! 2.51 1 ε p = −2 log10 p + (2.4) 3.7D˜ i Re f ξ ξ where, ε is the roughness of the pipe and Re f is the fluid Reynolds number which can be determined by: ρ˜ u˜D˜ i Re f = µ˜ The solar radiation received is considered as a heat flux/rate and can be expressed as [6]: q˜˙s (t˜) = γαg rmWa kθ G˜ bt (t˜)

24

H. Bakhti, I. Gasser

where G˜ bt is the beam solar radiation, Wa is the collector width, γ is the intercept factor, αg is the absorbance of glass cover, rm is the specular reflectance of the mirror and kθ is the incident angle modifier. The heat loss by convection and radiation are given by [6]:
q˜˙conv = hw T˜ − T˜a
4 q˜˙rad = εrad σ T˜ 4 − T˜sky where T˜a is the ambient temperature around the solar field near the ground, while T˜sky = 0.0552 T˜a is the sky temperature in high altitudes [6], σ˜ as Stefan-Boltzmann constant and given by σ = 5.67 × 10−8 W /(m2 · K 4 ) with εrad the emittance and hw the convective heat transfer coefficient given as: hw =

Nuair kair D˜ o

where kair = 0.024 W /(m · K) is the air thermal conductivity and Nuair is the air Nusselt number obtained as follows: ( 0.4 + 0.54Re0.52 air if 0.1 < Reair < 1000 Nuair = 0.3Re0air .6 if 1000 < Reair < 50000 ˜

air Do is air Reynolds number expressed with respect to air density with Reair = ρairµ˜u˜air ρ˜ air , wind velocity u˜air and air viscosity µ˜ air .

˜

Table 2.1: Properties of the used nanomaterials. Material

Specific heat Thermal conductivity Density (J/(kg K)) (W/(m K)) (kg/m3 ) Copper Oxide (CuO) 551 33 6000 Alumina (Al2 O3 ) 773 40 3960 Titanium Oxide (TiO2 ) 692 8.4 4230 In order to complete the mathematical model, temperature dependent thermal properties are required for a more accurate modeling of the system. Hence, the thermal properties varying with the temperature were extracted from the manufacturer datasheet. Integrating nanoparticles in the base fluid will induce an enhancement in its thermal properties. These properties are influenced by the volume fraction of the nanoparticles and their typology. Generally, this volume fraction does not exceed 5% [6]. The nanofluid thermal properties (i.e. density, specific heat capacity and dynamic viscosity) as functions of the volume fraction of nanoparticles (φ ), are derived from the following expressions:

2 Modeling and Simulation of Parabolic Trough Collectors using Nanofluids

25

ρ n f = (1 − φ )ρ b f + φ ρ s cnp f =

(1 − φ )ρ b f cbp f + φ ρ s csp ρn f

µ n f = µ b f 1 + 2.5φ + 6.25φ 2




where the subscript n f denotes nanofluid, b f base fluid and s solid nanoparticles. The thermal properties of these nanoparticles are given in Table 2.1. For more simplicity, the density and specific heat capacity are fitted under first order polynomial equations to be appropriately coupled with the presented system of equations, Eqs. (2.1)-(2.3), while the viscosity is fitted with higher order which does not add any complexity to our system, see Fig. 2.2. The scaled expressions of the density and specific heat capacity can be expressed as follow: ρ n f = α1 T + α0 cnp f = β1 T + β0 where the fitting coefficients α1 , α0 , β1 and β0 depend on the volume fraction of the nanoparticles φ . In order to identify the order of magnitude of each term, a scaled system of equations is obtained, where the reference values are denoted by an index r and the dimensionless quantities are introduced as follow: f (x,t) =

f˜(xxr ,ttr ) f˜(x, ˜ t˜) = . fr fr

Moreover, the scaling of the nanofluid constitutive law leads to the following nondimensional parameters: α0 = β0 =

α˜ 0 , ρr β˜0 c p,r

α1 = α˜ 1 ,

Tr ρr

(2.5)

Tr β1 = β˜1 c p,r

(2.6)

The reference values are associated with the real-world data of the considered power plant, in our case the Ouarzazate Noor I [6]. Table 2.2 shows the typical reference values that we use for scaling and Table 2.3 contains the Noor I model parameters. Ref. xr Tr tr

Unit Value Ref. Unit Value Ref. Unit m L q˙r W · m−1 103 pr Pa m˙ −1 ρ kg K 320 ur m·s · m−3 r Aρr xr 2 −2 −1 2 c p,r m · s · K c p (Tr ) µr m · s−1 s ur Table 2.2: Reference values used for the scaling

Value 3.3 · 106 ρ(Tr ) µ(Tr )

26

H. Bakhti, I. Gasser 1400 BF+5%TiO2 BF+5%Al2O3

1300

BF+5%CuO BF+3%TiO2 BF+3%Al2O3

1200

Density (kg/m3)

BF+3%CuO BF+1%TiO2

1100

BF+1%Al2O3 BF+1%CuO BF

1000

900

800

700

600

300

350

400

450

500

550

600

650

700

550

600

650

700

Temperature (K)

2800 BF+5%TiO2

Specific Heat Capacity (J/(kg K))

2600

2400

BF+5%Al2O3 BF+5%CuO BF+3%TiO2 BF+3%Al2O3 BF+3%CuO BF+1%TiO2

2200

2000

BF+1%Al2O3 BF+1%CuO BF

1800

1600

1400

1200

300

350

400

450

500

Temperature (K)

7

10-3 BF+5%TiO2 BF+5%Al2O3 BF+5%CuO BF+3%TiO2

6

BF+3%Al2O3

Viscosity (Pa s)

5

BF+3%CuO BF+1%TiO2 BF+1%Al2O3

4

BF+1%CuO BF

3

2

1

0

300

350

400

450

500

550

600

650

700

Temperature (K)

Fig. 2.2: Thermal properties of base fluid and nanofluids with their fittings.

2 Modeling and Simulation of Parabolic Trough Collectors using Nanofluids

27

In the following, the cross-sectional area A is considered to be constant. After scaling the governing equations (2.1)-(2.3) together with (2.5)-(2.6) can be written in dimensionless form as follows (dropping the index (·)n f ): ρt + (ρu)x = 0

1 η (ρu)t + (ρu2 + p)x = − ρu|u| + (µux )x ε Di 1 2 1 η (ρc p T + εδ ρu )t + (ρuc p T + εδ ρuu2 + δ pu)x = εδ ρu2 |u| 2 2 Di
κ Uo 4 Uo 4 + (µuux )x + (kTx )x + q˙s − κς T − Tsky (T − Ta ) − κρ A A A ρ = α1 T + α0 c p = β1 T + β0 A scaled asymptotic model can then introduced by neglecting the terms with very low order of magnitude in the governing equations and applying asymptotic analysis on the remaining terms (further details can be found in [9, 10]). The model includes the most relevant terms, namely friction loss in the momentum balance, the solar input and the radiative and convective heat losses in the energy balance. Therefore, the final form of the simplified model is given as follows: ρt + (ρu)x = 0  
1 Uo 4 α1 κ Uo 4 q ˙ − ς T − T (T − T ) ux = − − ρ s a sky (c p + β1 T )ρ 2 A A A εϑ px = − ρu|u| Di ρ − α0 T= α1 c p = β1 T + β0

Parameter Collector Length L Collector Width Wa Inner Diameter D˜ i Outer Diameter D˜o Intercept Factor γ Absorbance of Glass Cover αg Specular Reflectance rm Incident Angle Modifier kθ Emittance of the Cover εrad

Unit m m m m − − − − −

Value 12.27 5.76 0.066 0.07 0.867 0.94 0.94 1 0.15

Table 2.3: Model parameters of NOOR I parabolic trough collector [6]

(2.7) (2.8) (2.9) (2.10) (2.11)

28

H. Bakhti, I. Gasser

Coeff. Expression Order Coeff. Expression Order ε µ k κ

ρr u2r pr 4 µ˜ 3 xr ur k˜ c p,r ρr ur xr xr q˙r ρr c p,r ur Tr Ar

10−6 10−8

η δ

10−8

ρ

10−2

ς

ξ xr Ui,r Ar pr c p,r ρr Tr Tr Ur h˜ w q˙r Tr4Ur εrad σ q˙r

10−2 100 100 10−2

Table 2.4: Dimensionless parameters with order of magnitude. The presented system of equations is coupled with the following initial and boundary conditions: T (x, 0) = Tint ,

p(0,t) = pl ,

u(0,t) = ul ,

T (0,t) = Tl .

The instantaneous energetic efficiency refers to the ratio between the useful thermal energy gained by the working fluid to the available solar beam energy falling onto the PTC reflector. It is expressed as: η=

Aρu

R Tr Tl

c p (T )dT

Wa LGbt

(2.12)

where Wa is the width of the collector, L the length of the collector and Gbt is the beam incident radiation. The exergetic efficiency can be defined as the ratio of gain exergy to available solar radiation exergy and can be expressed as: R  R c (T ) Aρu TTlr c p (T )dT − Ta TTlr pT dT  (2.13) ηex =    4  Ta Ta 1 4 Wa LGbt 1 − 3 + Tsun + 3 Tsun where Tsun is the sun’s apparent temperature taken to be 6000 K.

2.3 Numerical Simulation In order to solve Eqs. (2.7)-(2.11), a temporal and spatial discretization is required. The discrete grid points for space and time are denoted, respectively, by x1 , . . . , xNx and t1 , ..., tNt . The discrete variables are accordingly provided with the indexing (n, i), where the first index is relating to the time coordinate, the second to the spatial coordinate. The semi-implicit scheme is used to solve the mass conservation, Eq. (2.7), as follows: n+1 n ui−1 ρin+1 − ρin ρin+1 uni − ρi−1 + =0 ∆t ∆x

2 Modeling and Simulation of Parabolic Trough Collectors using Nanofluids

29

Calculation of the discrete temperature and specific heat capacity values result from the linear relationships, Eqs. (2.10)-(2.11). On the other hand, the velocity solution can be obtained by integrating (2.8)   Z x
α1 κ 1 Uo 4 Uo 4 u(t, x) = ul − q˙s − ς T − Tsky − ρ (T − Ta ) ds 2 A A A 0 (c p + β1 T )ρ The nonlinear fitting of the viscosity is used to calculate the discrete values in order to calculate the fluid Reynolds number and obtain the friction factor ξ by solving the Colebrook’s equation, Eq. (2.4). The pressure values are then obtained by solving Eq. (2.9) using the backward finite difference scheme. Finally, the energetic and exergetic efficiencies are calculated from Eqs. (2.12)(2.13) using the Trapezoidal rule to approximate the integrals.

2.4 Results and Discussion Adaptive numerical schemes are implemented in the software MATLAB to solve the system of equations (2.7)-(2.11). The solar radiations and ambient temperatures for a typical sunny day in the region of Ouarzazate (Morocco) are considered [6]. The hourly variation of the considered data are plotted in Fig. 2.3 between the sunrise and the sunset, where the observed peak solar radiation is around 1000 W /m2 and the peak ambient temperature is around 308 K.

310

1000

800

600 300

Ta (K)

2

Gbt (W/m )

305

400 295 200

0

6

8

10

12

14

16

18

290

Time (h)

Fig. 2.3: Typical hourly climatic conditions in the region of Ouarzazate-Morocco. The obtained results are presented to validate the derived model by comparing the predicted temperature variations with existing values in the literature. The inlet temperature is set at 320 K (46, 85 ◦C). Fig. 2.4, shows the time variation of the obtained results for the fluid properties, namely the density, velocity, pressure, temperature, specific heat capacity and viscosity as function of space (along the pipe)

30

H. Bakhti, I. Gasser Density (kg/m3)

Velocity (m/s)

18 16

1070

18

1060

16

0.127 0.1265

1050 1040

12

1030

10

1020

14

t (h)

t (h)

14

1010

8

0.1275

0.126

12

0.1255

10

0.125 0.1245

8

1000

6 0

0.124

6

990

2

4

6

8

10

12

0

0.1235

2

4

x (m)

6

8

10

12

x (m) Temperature (K) 380

18

370

t (h)

16

360

14

350

12

340 330

10

320

8

310 300

6 0

2

4

6

8

10

12

x (m) Specific Heat Capacity (J/(kg K))

Viscosity (mPa s) 1800

18

18 2.1 1750

16

16 2

14

1700

12 1650

t (h)

t (h)

14

10

1.8

10

8 6 0

1.9

12

2

4

6

8

10

1600

8

1550

6

12

0

1.7 1.6

2

4

x (m)

6

8

10

12

x (m)

Fig. 2.4: Evolution of system variables for the base fluid Therminol VP-1. Reynolds Number (-)

Friction Factor (-)

18 16

t (h)

t (h)

4800 4600

10

4400

10

8

4200

8

6

4000

6

2

4

6

x (m)

8

10

12

0.039

14

12

0

0.0395

16

5000

14

0.04

18

5200

0.0385

12

0

0.038 0.0375 0.037

2

4

6

8

10

12

x (m)

Fig. 2.5: Evolution of Reynolds number and friction factor for the base fluid Therminol VP-1.

and time (hours of the day). It is noticed from the different plotted results that the minimum or maximum values of the different parameters are obtained at the peak value of the solar radiation during the day. Furthermore, the collector generates a maximum temperature of around 380 K at the outlet of the pipe when the solar radiation is at the peak value during the day. Similar results are shown in Fig. 8 from

31

2 Modeling and Simulation of Parabolic Trough Collectors using Nanofluids

the work of Allouhi et al. [6], where it is observed that the maximum temperature of around 375 K is reached at midday. The author used only heat equations to predict the output temperature for the same NOOR I model parameters. Moreover, the spatial and temporal variations of the Reynolds number and friction factors are obtained and plotted in Fig. 2.5. At the outlet of the collector, it is observed that the Reynolds number reaches the peak value when the solar radiation is at the peak value during the day, while the friction factor is at the lowest value at the same period of time. Thus, the fluid temperatures correlate with the variations of the fluid Reynolds number and friction factor. The next set of results illustrates the effect of using nanofluids as working fluids in the PTC. The same previous operating conditions were considered. The temporary evolution of the outlet temperature, thermal efficiency and exergy efficiency are depicted in Fig. 2.6. The nanoparticle concentration was set to vary between φ ∈ {1%, 2%, 5%}. One can see clearly that the nanofluids reach higher temperatures than the base fluid, especially at high radiation levels inducing greater heat propagation in the working fluid. The maximal energy efficiency reaches 43% for the base fluid and increase with up to 5% when nanofluids are employed. Due to its high density values, CuO based nanofluid leads to the most significant increase in thermal efficiency, where the maximal exergy efficiency reaches up to 5.5%. Outlet Temperature (K)

420

Energetic Efficiency (%)

100

400 80

360

data1 +1% CuO +1% Al2O3

340

+1% TiO

+2% TiO

40

20

+5% TiO2

7

9

11

13

15

17

2

+5% CuO +5% Al2O3 +5% TiO2

0 5

19

2

+2% CuO +2% Al2O3 +2% TiO

2

+5% CuO +5% Al2O3

300 280 5

+1% TiO

2

+2% CuO +2% Al2O3

320

data1 +1% CuO +1% Al2O3

60

(%)

Temperature (K)

380

7

9

11

t (h)

13

15

17

19

t (h) Exergetic Efficiency (%)

15

10

ex

(%)

data1 +1% CuO +1% Al2O3 +1% TiO

2

+2% CuO +2% Al2O3

5

+2% TiO

2

+5% CuO +5% Al2O3

0 5

+5% TiO2

7

9

11

13

15

17

19

t (h)

Fig. 2.6: Temporary evolution of outlet temperature, thermal efficiency and exergy efficiency.

32

H. Bakhti, I. Gasser

Finally, Table 2.5 presents the variations of the friction factor with respect to the volume fraction of the nanoparticles in the basefluid for three different materials. In contrast to the thermal gain, it is noticed from the presented table that with the presence of nanparticles in the base fluid the friction factor increases with respect to the volume fraction which leads to an increase in the friction loss inside the parabolic collector. Table 2.5: Friction factor with respect to nanoparticles volume fraction. Volume Fraction 1% 2% 5% Copper Oxide (CuO) 0.036628 0.036710 0.036985 Alumina (Al2 O3 ) 0.036704 0.036863 0.037377 Titanium Oxide (TiO2 ) 0.036692 0.036838 0.037311

2.5 Conclusions The presented model is a significant step towards accurate modeling of parabolic trough power plants. The main physical phenomena are included. Nevertheless the model is simple enough to allow for fast and robust simulations tasks. The presented model was validated and the results are in good agreement with the numerical studies using data from existing parabolic trough power plants. The purpose of this study is to examine the benefits of using nanofluids as working fluids in parabolic trough collectors with application to the Moroccan NOOR I power plant in Ouarzazate. It is observed that, for a nanoparticle concentration of 1,2 and 5%, improvement up to 5% of the energy efficiency and up to 1.3% of the exergy efficiency have been shown. Slight increase of the friction loss is also observed when increasing the nanoparticle concentration. The proposed model can be of great use with regards to large systems of collector tubes in order to improve the design and optimization of such power plant systems.

Acknowledgment The authors acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 ”Modeling, Simulation and Optimization of Fluid Dynamic Applications”.

2 Modeling and Simulation of Parabolic Trough Collectors using Nanofluids

33

References 1. A. Jaber Abdulhamed, N. Mariah Adam, M. Z. Abidin Ab-Kadir, and A.A. Hairuddin. Review of solar parabolic-trough collector geometrical and thermal analyses, performance, and applications. Renewable and Sustainable Energy Reviews 91, 822–831, 2018. 2. Y. Krishna, M. Faizal, R. Saidur, K.C. Ng, and N. Aslfattahi. State-of-the-art heat transfer fluids for parabolic trough collector. International Journal of Heat and Mass Transfer 152, 119541, 2020. 3. B. El Ghazzani, D. Martinez Plaza, R. Ait El Cadi, A. Ihlal, B. Abnay, and K. Bouabid. Thermal plant based on parabolic trough collectors for industrial process heat generation in Morocco. Renewable Energy 113, 1261–1275, December 2017. 4. T. Bouhal, Y. Agrouaz, T. Kousksou, A. Allouhi, T. El Rhafiki, A. Jamil, and M. Bakkas. Technical feasibility of a sustainable concentrated solar power in Morocco through an energy analysis. Renewable and Sustainable Energy Reviews 81, 1087–1095, January 2018. 5. Z. Aqachmar, A. Allouhi, A. Jamil, B. Gagouch, and T. Kousksou. Parabolic trough solar thermal power plant Noor I in Morocco. Energy 178, 572–584, July 2019. 6. A. Allouhi, M. Benzakour Amine, R. Saidur, T. Kousksou, and A. Jamil. Energy and exergy analyses of a parabolic trough collector operated with nanofluids for medium and high temperature applications. Energy Conversion and Management 155, 201–217, January 2018. 7. C.F. Colebrook, T. Blench, H. Chatley, E.H. Essex, J.R. Finniecome, G. Lacey, J. Williamson, and G.G. MacDonald. Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. Journal of the Institution of Civil Engineers 12(8), 393–422, June 1939. 8. U. S¸ahin. A new non-iterative friction factor correlation for heat transfer fluids in absorber tube of parabolic trough collector. Engineering Science and Technology, an International Journal 21(1), 89–98, February 2018. 9. H. Bakhti, I. Gasser, S. Schuster, and E. Parfenov. Modelling, simulation and optimisation of parabolic trough power plants. European Journal of Applied Mathematics, 592–615, June 2023. 10. E. Parfenov and I. Gasser. Zur Modellierung und Simulation eines Parabolrinnenkraftwerkes. Master Thesis, Department of Mathematics, Universit¨at Hamburg, 2011.

Chapter 3

Adaptive Discontinuous Galerkin Methods for 1D unsteady Convection-Diffusion Problems on a Moving Mesh Ezra Rozier, J¨orn Behrens

Abstract In convection-dominated flows, large scale trends necessarily coexist with small-scale effects. While reducing the convection-dominance by moving the mesh, also called Arbitrary Lagrangian-Eulerian (ALE), already proved efficient, Adaptive Mesh Refinement (AMR) is able to catch the small scale effects. But, ALE introduces uncertainties that cannot be neglected in front of the small scale effects, therefore it is unsatisfying to use AMR the same way in an ALE situation as we do on static meshes. In this paper, an h-refinement criterion is built up and tested by studying the approximation error of a moving mesh, interior penalty discontinuous Galerkin (DG) semi-discretization of the 1D nonstationary convection-diffusion equation. It is done through three hotspots: the uncertainties due to the mesh movement, the error sources and the error propagation. Whereas the uncertainties as well as the error sources are easily measurable, having a precise understanding of the error propagation remains difficult. The cheap and efficient way to have a faithful picture of the error propagation in a dynamic situation, is by measuring the residual of the approximation solution. In addition to a stabilizing effect of the moving mesh, this method provides interesting results: while ALE approximates in terms of the L2 -norm, the developed refinement criterion spots early where the H 1 -norm of the approximation error will explode.

Ezra Rozier Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] J¨orn Behrens Department of Mathematics/CEN, Universit¨at Hamburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Iske, T. Rung (eds.), Modeling, Simulation and Optimization of Fluid Dynamic Applications, Lecture Notes in Computational Science and Engineering 148, https://doi.org/10.1007/978-3-031-45158-4_3

35

36

E. Rozier, J. Behrens

3.1 Introduction When measuring the error of a convection-diffusion problem’s DG approximation, 2 a term proportional to ∥Vε∥∞ appears (see [21], V is the convection speed and ε the diffusion coefficient). In the situation of convection-dominated flows (where ∥V ∥ ≫ ε) this term, that occurs in the interaction between transport and diffusion effect on the edges of the mesh, becomes significant. Reducing this effect implies to understand what composes the nonlinearity V : in turbulent cases, it is composed of a large scale trend and small scale turbulences. Moving the mesh by a large scale speed V˜ can help to reduce this term of interaction ˜ 2 to ∥V −εV ∥∞ (see [7]) and focus our study on the interactions between small scale turbulences and diffusion. To resolve these small scale effects, one can imagine a tailoring strategy on the cells’ level. AMR proposes this approach ([19]), which consists in splitting or merging the mesh’s cells depending on a local error indicator. In the situation here, the ˜ mesh movement introduces uncertainties of order e∥∇V ∥∞ T that become significant after some time T . Therefore, the AMR strategy has to be adapted to the moving mesh situation. AMR implies to develop a refinement strategy that goes along with a refinement criterion depending on some error indicators. Three strategies exist: featured-based refinement that is based on the values of the features and benefit from apriori error estimates ([10], [15]), goal-oriented estimators that are based on the resolution of an optimization problem ([14]) and residual-error based estimators that is coping with the residual of the problem ([11], [21], [22], [19], [1], [16]). [18] offers a comparison of those three paradigms in the situation of p-refinement. Because this study aims to develop an efficient and cheap h-refinement strategy, a residual-error based estimator (analogous to [5] on static meshes) will be developed here. Building this strategy will imply to develop and use tools to approximate the error in the situation of moving mesh and end up with an aposteriori error estimator. To build those tools implies to first understand the very basis of DG for convectiondiffusion (introduced in [8], supplemented by [9] and [12]) and see how to connect it with moving meshes ([17], [20]). The derived formulation will be studied in extension to the existing ALE DG studies through stability ([4], [2], [13], [23]) and apriori error estimates ([6], [3]) and completed by the work done for aposteriori error estimation for static meshes ([19], [5]). In this paper, the DG method proposed uses upwind discretization of the transport term and a classical interior penalty method for the diffusive term. Based on a study of the error propagation for this semidiscretization, a complete aposteriori error estimator is derived that can help to parametrize the mesh movement and to define a refinement strategy. This paper will present this work process as follows: Section 2 will introduce the framework of ALE as a flow map as well as the DG method for a convectiondiffusion equation. In Section 3, an aposteriori error estimator is derived for the steady-state problem and Section 4 will use this error estimator to build an error

3 Adaptive Discontinuous Galerkin Methods on a Moving Mesh

37

estimator for the unsteady problem. A test of the error criteria will be held in Section 5. Finally, Section 6 will gather the theoretical and computational discussions before concluding.

3.2 Flow map and interior penalty discretization 3.2.1 Model problem We consider the nonstationary convection-diffusion equation: ∂u ∂t

2u= f +V · ∂x u − ε∂xx

u = uD ε ∂∂ nu

= uN

u(x, 0) = u0 (x)

[0, T ] × Ω

[0, T ] × ΓD

[0, T ] × ΓN

(3.1)

Ω

in Ω = [y0 , y1 ] ⊂ R, Γ = {y0 , y1 } = ΓD ∪ ΓN . The final time T is arbitrary, but kept fixed in what follows. We assume that the data satisfy the conditions (A1) f ∈ C(0, T ; L2 (Ω )), V ∈ C(0, T ;W 1,∞ (Ω )). (A2) 0 < ε ≪ 1. R (A3) ∀t ∈ [0, T ], − Ω ∂xV (t, x)dx := α(t) > 0. (A4) For all t, the Dirichlet boundary ΓD includes the inflow boundary Γ in . Assumption (A2) means that we are interested in the convection-dominated regime. Assumption (A1) can be replaced by weaker conditions concerning the temporal smoothness.

3.2.2 Flow maps We will consider the ALE method as a classical DG method defined on a deforming space. In order a smooth velocity V˜ = V˜ (t, x) s.t.  to do so, define ˜ ∈ C(0, T ;C1 (Ω )) V   ˜ V = 0 on Γ × [0, T ] (A) − 1 ∂ (V (t, x) − V˜ (t, x)) := β (t) > 0    2 x ||∂x (V (t, x) − V˜ (t, x))|| ≤ c∗ β for a constant c∗ independent of time. The existence of such a function V˜ is ensured by assumption (A3). We will introduce the flow map to distinguish a Lagrangian (or reference) variable X and an Eulerian (or spatial) variable x. X lives in a space Ωr that we define

38

E. Rozier, J. Behrens

later (and x lives in Ω ). We will do all the computations in Ωr and express them back in Ω . Given V˜ the associated flow map, x = χ(t, X), X ∈ Ωr , satisfies : ˙ X) = V˜ (t, x(t, X)), x(t,

x(0, X) = X

As V˜ smooth, χ(t, .) : R → R is a C1 -diffeomorphism, we denote by J the Jacobian J = J(t, X) := ∂∂Xx . ˙ X) = (∂xV˜ (t, x))J(t, X), J(t,

x = χ(t, X).

J(0, X) = 1,

Because of the value of V˜ on Γ , Ωr = Ω (t) = χ(t, Ω ) = Ω , χ(t,ΓD ) = ΓD and χ(t,ΓN ) = ΓN . For any function v(t, x) we introduce the ˆ notation s.t. v(t, ˆ X) := v(t, x(t, X)) and reciprocally v can be defined thanks to v. ˆ Remark 3.2.1 Thisˆnotation works also for functions independent of time : v(t, ˆ X) := v(x) and v(t, x) := v(X). ˆ We have uˆt = ut + V˜ ∂x u,

∂x u =

∂X uˆ J ,

∆ u = 1J ∂X { ∂XJ uˆ }.

nr (X) is the outward normal vector of Ωr in X (X = y0 or y1 ) defined by nr (X) = 1X=y1 − 1X=y0 The problem becomes : ∂t uˆ + (V − V˜ ) ∂XJ uˆ − εJ ∂X { ∂XJ uˆ } = fˆ uˆ = uˆD εnr (X) ∂XJ uˆ = uˆN u(0, ˆ X) = uˆ0 (X)

[0, T ] × Ωr [0, T ] × ΓD

[0, T ] × ΓN Ωr

(3.2)

Finally V − V˜ ∈ C(0, T ;W 1,∞ (Ω )), then for ω ⊂ Ωr let αω (t) = ∥(V − V˜ )(t, ·)∥L∞ (ω) .

3.2.3 Semi-discretization and function spaces 3.2.3.1 Notation and weak form For any bounded open subset ω of Ω with boundary γ, we denote by H k (ω), k ∈ N, L2 (ω) = H 0 (ω) the usual Sobolev and Lebesgue spaces equipped with the standard norm ∥ · ∥k;ω = ∥ · ∥H k (ω) as well as the standard semi-norm | · |H 1 (ω) = ∥∂x · ∥0,ω . Similarly, (·, ·)ω denote the scalar product of L2 (ω). If ω = Ω , we will omit the index Ω .

3 Adaptive Discontinuous Galerkin Methods on a Moving Mesh

39

Let ωr ⊂ Ωr with boundary γr (recall that Ωr = Ω ), for a fixed t, denoting ω(t) := χ(t, ωr ) and γ(t) its boundary (γ(t) = χ(t, γr )) we have ω bounded, open and we see that theˆoperator defines a bijection from H k (ω(t)) into H k (ωr ). Defining ∥v∥ ˆ 2Hω

r (t)

:=

R

ωr

vˆ2 J(t, ·), |v| ˆ U2 ω

r (t)

:=

(∇X v) ˆ2 ωr J(t,·)

R

we have ∥v(t, ·)∥0;ω = ∥v∥ ˆ Hωr (t) , and |v(t, ·)|H 1 (ω) = |v| ˆ Uωr (t) . By this property we see that we will do approximations in the reference variable, by denoting e the approximation error, we will give bounds of ∥e∥ ˆ H(t) and |e| ˆ U(t) to have L2 and H 1 bounds of e. We define the spaces L p (0, T ; X) (with X a Banach space) that consists of mesurable functions v : [0, T ] → X for which : ∥v∥Lp p (0,T ;X) :=

RT 0

∥v(t)∥Xp dt < +∞ for 1 ≤ p < +∞

∥v∥L∞ (0,T ;X) := ess sup∥v(t)∥Y < +∞ for p = +∞ 0≤t≤T

Additionally H 1 (0, T ; X) := {u ∈ L2 (0, T ; X) : ut ∈ L2 (0, T ; X)}. Set

HD1 (Ωr ) := {vˆ ∈ H 1 (Ωr ) : vˆ = 0 on ΓD } and H01 (Ωr ) := {vˆ ∈ H 1 (Ωr ) : vˆ = 0 on Γ } Therefore we can define the weak form of (2) Find uˆ ∈ L2 (0, T ; HD1 (Ωr )) ∩ H 1 (0, T ; L2 (Ωr )) s.t. ∀t ∈ [0, T ], ∀vˆ ∈ HD1 (Ωr ) A(u, ˆ v) ˆ := l(v) ˆ :=

R

R

Ωr

Ωr

Z Ωr

J

∂ uˆ vˆ = l(v) ˆ − A(u, ˆ v) ˆ (3.3) ∂t

J[ε ∂XJ uˆ ∂XJ vˆ + (V − V˜ ) ∂XJ uˆ v] ˆ

J fˆvˆ + ∑ uˆN vˆ ΓN

Remark 3.2.2 Each bilinear form in this paper are time-dependent but for clarity it will not always be explicitly stated. By integration by part A(u, ˆ v) ˆ :=

R

Ωr

ˆ + ∑ nrV uˆvˆ J[ε ∂XJ uˆ ∂XJ vˆ − (V − V˜ ) ∂XJ vˆ uˆ − ∂x (V − V˜ )uˆv] ΓN

40

E. Rozier, J. Behrens

3.2.3.2 Bilinear forms and function spaces for the semi-discretization To discretize (3.3) we consider (n + 1) dots y0 = X0 < · · · < Xn = y1 that meshes Ω into Th = {K1 , . . . , Kn } with Ki = [Xi−1 , Xi ] locally quasi-uniform and introduce the notation  − Xi if (V − V˜ )(t, x(t, Xi )) > 0 out Xi = and hi = max(|Ki |, |Ki+1 |). Xi+ if (V − V˜ )(t, x(t, Xi )) < 0   0 if V (t, y0 ) > 0 ˆ 1 ) if V (t, y1 ) > 0 u(y out out . ˆ n )= u(X and u(X ˆ 0 )= u(y ˆ 0 ) if V (t, y0 ) < 0 0 if V (t, y1 ) < 0 We will also write : HK = |K| and Hi = max(|Xi − Xi−1 |, |Xi − Xi+1 |), and for the lagrangian elements : hK = hKi = |xi−1 − xi | and hi = max(|xi − xi−1 |, |xi − xi+1 |). And in addition we will write : δK =

R 1 K J

HK

, δi = max(δKi , δKi+1 )

(3.4)

Remark 3.2.3 We have the following relations between hK , HK and δK : hK =

Z K

J and

HK ≤ δK hK

(3.5)

The broken Sobolev spaces associated with the mesh Th : H k (Th ) = {ϕ ∈ L2 (Ωr ) : ∀K ∈ Th , ϕ|K ∈ H k (K)}

Denote :

Vh := {ϕ ∈ H 1 (Th ) : ∀K ∈ Th ϕ|K ∈ S p (K)} with the set S p is the space of polynomials of degree ≤ p. Finally we denote Uh := Vh + HD1 (Ωr ) and Vhc := Vh ∩ HD1 (Ωr ). In an inner mesh point Xi , the average and jump of a function vˆ ∈ H 0 (Th ) across the point are defined as : ˆ i+ ) + v(X {{v}} ˆ = 12 (v(X ˆ i− )), [[v]] ˆ = v(X ˆ i+ ) − v(X ˆ i− ) and [[v]] ˆ = v(X) ˆ on Γ . We consider the discontinuous Galerkin method that is based on an upwind discretization for the convective term and on a symmetric interior penalty discretization for the Laplacian : 1

∂ uˆh vˆh = lh (vˆh ) − Ah (uˆh , vˆh ) ∂t Ωr where uˆh (0, ·) ∈ Vh is a projection of uˆ0 (·) onto Vh . (3.6)

Find uˆh ∈ C (0, T ;Vh ) s.t. ∀t ∈ [0, T ], ∀vˆh ∈ Vh

Z

J

3 Adaptive Discontinuous Galerkin Methods on a Moving Mesh

ˆ v) ˆ := Ah (u,

Z

∑

+

∑

Jε(

i=1,...,n−1

∂X uˆ ∂X vˆ ∂X vˆ − (V − V˜ ) uˆ − ∂x (V − V˜ )uˆv] ˆ J J J

J[ε

i=1,...,n Ki

{{∂X v}} ˆ [[u]] γε [[u]][[ {{∂X u}} ˆ [[v]] ˆ ˆ ˆ v]] ˆ − )|Xi + |Xi J J J J Hi J

+ ∑(−1)δni Jε( ΓD

−

41

∑

∂X uˆ vˆ ∂X vˆ uˆ γε uˆvˆ − )|Xi + |X J J J J Hi J i

(V − V˜ )[[v]] ˆ u(X ˆ iout ) +V v| ˆ Xn u(X ˆ nout ) −V v| ˆ X0 u(X ˆ 0out )

i=1,...,n−1

uˆD vˆ − ˆ uˆ +V + v| ˆ X0 uˆD . lh (v) ˆ := l(v) ˆ − ∑ (−1)δni Jε ∂XJ vˆ uˆJD |Xi − γε Xn D Hi J |Xi −V v| ΓD

V + and V − are respectively the positive and negative parts of V (V − < 0 < V + ) and γ > 0 the interior penalty parameter that is described in the literature to be depending on the degree of the polynomials. Finally, for u, ˆ vˆ ∈ H 1 (Th ) : R ˆ Dh (u, ˆ v) ˆ := ∑ Ki J[ε ∂XJ uˆ ∂XJ vˆ − ∂x (V − V˜ )uˆv] i=1,...,n

Oh (u, ˆ v) ˆ := − ∑

R

i=1,...,n

Ki (V

− V˜ )u∂ ˆ X vˆ

−(V − V˜ )[[v]]| ˆ Xi u(X ˆ iout ) +V v| ˆ Xn u(X ˆ nout ) −V v| ˆ X0 u(X ˆ 0out )

Jh (u, ˆ v) ˆ :=

ˆ v]] ˆ γε [[u]][[ |Xi ∑ H J i=1,...,n−1 i

uˆvˆ + ∑ γε Hi J |Xi ΓD

A˜ h (u, ˆ v) ˆ := Dh (u, ˆ v) ˆ + Oh (u, ˆ v) ˆ + Jh (u, ˆ v) ˆ K˜ h (u, ˆ v) ˆ :=

∑

i=1,...,n−1

{{∂X v}} ˆ [[u]] ˆ [[v]] ˆ ˆ Jε( {{∂XJ u}} J − J J )|Xi

+ ∑ (−1)δni Jε( ∂XJ uˆ Jvˆ − ∂XJ vˆ uJˆ )|Xi ΓD

Ah (u, ˆ v) ˆ := A˜ h (u, ˆ v) ˆ − K˜ h (u, ˆ v) ˆ Note that ∀t ∈ [0, T ], ∀u, ˆ vˆ ∈ HD1 (Ωr ) A˜ h (u, ˆ v) ˆ = A(u, ˆ v) ˆ

We will call uˆ (resp. uˆh ) the solution to (3.3) (resp. (3.6)) and define uˆs (t) ∈ HD1 (Ωr ) and uˆsh (t) ∈ Vh for each t such that : (

∀vˆ ∈ HD1 (Ωr ) l(v) ˆ −

∀vˆh ∈ Vh

R

Ωr

lh (vˆh ) −

R

J ∂∂tuˆh vˆ − A(t; uˆs (t), v) ˆ =0

Ωr

J ∂∂tuˆh vˆh − Ah (t; uˆsh (t), vˆh ) = 0

In the following, we will find error estimates for the spatial operator by studying the stationary problem defining uˆs and uˆsh and then use these error bounds to build refinement criteria for the nonstationary problem.

42

E. Rozier, J. Behrens

3.3 Error bound for the stationary problem We define for Y ⊂ Ωr , ωS := {K ∈ Th : S ∩ K ̸= 0}. / To analyse the spatial error, we introduce the following quantities for vˆ ∈ H 1 (Th ) and q ∈ H 0 (Ωr ) |||v||| ˆ U2 K (t) + β (t)∥v∥ ˆ 2HK (t) ] + ˆ t2 := ∑ [ε|v| K∈Th

|q|t;∗ :=

sup

R

q∂X vˆ |||v||| ˆ t

∑

i=1,...,n−1

ˆ 2 γε [[v]] Hi J |Xi

vˆ + ∑ γε Hi J |Xi 2

ΓD

Ωr

v∈H ˆ 01 (Ωr )−{0}

2 2 + := |(V − V˜ )v| |v| ˆ t;A ˆ t,∗

∑

(β +

i=1,...,n−1

αX2 i ˆ 2Xi ε )hi [[v]]|

+ ∑ (β + ΓD

αX2 i ˆ 2Xi ε )hi v|

Remark 3.3.1 As well as for the bilinear forms, the t will be omitted in the notation. The first norm can be viewed as the energy norm associated with the discontinuous Galerkin discretization of the convection–diffusion problem (3.1). The third norm measures the error of the transport behaviour. Property 3.3.2 Coercivity : Let t ∈ [0, T ] fixed and vˆ ∈ HD1 (Ωr ). A˜ h (v, ˆ v) ˆ ≥ |||v||| ˆ 2

(3.7)

Proof. By addition of the first and second formulations of A Z ∂X vˆ ∂X vˆ 1 ∂X vˆ J[ε · + (V − V˜ ) v] ˆ A˜ h (v, ˆ v) ˆ = A(v, ˆ v) ˆ = 2 Ωr J J J  Z Z ∂X vˆ ∂X vˆ ∂X vˆ + J[ε · − v(V ˆ − V˜ ) − ∂x (V − V˜ )vˆ2 ] + nr (V − V˜ )vˆ2 J J J Ωr ΓN Thus

A(v, ˆ v) ˆ =

Z Ωr

ε

1 (∂X v) ˆ2 J − ∂x (V − V˜ )vˆ2 + J 2 2 ≥

∑

K∈Th

Z

nr (V − V˜ )vˆ2 ΓN [ε|v| ˆ U2 K (t) + β (t)∥v∥ ˆ 2HK (t) ] = |||v||| ˆ 2

Remark 3.3.3 The condition β (t) ≥ 0 ensures the stability of the Galerkin semidiscretization. Properties 3.3.4 For vˆ ∈ H01 (Ωr ), wˆ 1 , wˆ 2 ∈ Uh |Dh (wˆ 1 , wˆ 2 )| ≲ |||wˆ 1 ||| · |||wˆ 2 |||

(3.8)

3 Adaptive Discontinuous Galerkin Methods on a Moving Mesh

43

|Jh (wˆ 1 , wˆ 2 )| ≲ |||wˆ 1 ||| · |||wˆ 2 |||

(3.9)

ˆ ≤ |(V − V˜ )wˆ 1 |∗ · |||v||| ˆ |Oh (wˆ 1 , v)|

(3.10)

Proof. (4) and (5) are direct consequenses of Cauchy-Schwarz and the fact that β (t) > 0. R ˆ ≤ | ∑ K J wˆ 1 (V − V˜ ) ∂XJ vˆ | |Oh (wˆ 1 , v)| K∈Th

|

R

∂ vˆ J wˆ (V −V˜ ) X |

1 Ωr J ≤ |||v||| ˆ |||v||| ˆ ≤ |(V − V˜ )wˆ 1 |∗ · |||v||| ˆ

The litterature gives us the inverse and trace inequalities used for eulerian problem in (3.41) [12]. Lemma 3.3.5 Let vˆ ∈ Vh , K ∈ Th ˆ HK (t) . - Inverse inequality : |v| ˆ UK (0) ≲ H1K ∥v∥ - Trace inequality : for a mesh point X of K, v(X) ˆ 2≲

1 ˆ 2HK (t) . HK ∥v∥

The outline of the demonstration for the stationary case is as follow : separate our solution into a continuous and a discontinuous part (Lemma 3.3.8), give a bound to the discontinuous part (Lemma 3.3.9), give a bound to the bilinear forms as an estimate times the triple norm of the continuous function (Lemma 3.3.11, Lemma 3.3.12) and conclude with Property 3.3.6. Property 3.3.6 We have the following inf-sup inequality : A˜ h (v, ˆ w) ˆ inf sup ≥C >0 (|||v|||+|(V ˆ −V˜ )v| ˆ )·|||w||| ˆ 1 (Ω )\{0} v∈H ˆ D1 (Ωr )\{0}w∈H ˆ r

∗

D

Proof. Let vˆ ∈ HD1 (Ωr )\{0} and θ ∈]0; 1[. Then by definition of the * norm  |||wˆ θ ||| = 1 R ∃wˆ θ ∈ H01 (Ωr ) s.t. ˆ∗ Oh (v, ˆ wˆ θ ) = − Ωr J(V − V˜ )vˆ ∂XJwˆ θ ≥ θ |(V − V˜ )v| A˜ h (v, ˆ wˆ θ ) ≥ −C1 |||v||| ˆ · |||wˆ θ ||| + θ |(V − V˜ )v| ˆ ∗ = θ |(V − V˜ )v| ˆ ∗ −C1 |||v||| ˆ for a constant C1 > 0. Let us then define : |||v||| ˆ wˆ θ vˆθ = vˆ + 1+C 1 1 s.t. |||vˆθ ||| ≤ |||v|||(1 ) and vˆθ ∈ HD1 (Ωr ). ˆ + 1+C 1 Thus by (3.8), (3.9) and (3.10) : ˜ ˆ ] A˜ h (v, ˆ vˆθ ) = A(v, ˆ vˆθ ) = A(v, ˆ v) ˆ + |||v||| ˆ A(v,ˆ wˆ θ ) ≥ |||v||| ˆ · [|||v||| ˆ + θ |(V −V )v|ˆ ∗ −C1 |||v||| 1+C1

Finally :

sup

w∈H ˆ D1 (Ωr )\{0}

A˜ h (v, ˆ w) ˆ |||w||| ˆ

≥

A˜ h (v, ˆ vˆθ ) |||vˆθ |||

≥

θ |(V −V˜ )v| ˆ ∗ +|||v||| ˆ 2C1

1+C1

44

E. Rozier, J. Behrens

Remark 3.3.7 C1 = O(c∗ ) We will now study an approximation of elements of Vh by elements of Vhc . A similar theorem for the eulerian problem is stated in Theorem 2.2 in [16]. Lemma 3.3.8 There exists an approximation operator Ah : Vh → Vhc satisfying :  2 hi [[vˆh ]]2  ∑  ∑ ||vˆh − Ah vˆh ||HK (t) ≲ K∈Th {1,...,n−1}∪ΓD ∀vˆh ∈ Vh δi 2  ∑ |vˆh − Ah vˆh |U2 (t) ≲ ∑  Hi [[vˆh ]] K K∈Th

{1,...,n−1}∪ΓD

with δ defined in (3.4). ( j)

Proof. For each K ∈ Th , NK := {xK : j = 1, . . . , m} set of distinct nodes of K ( j) with nodes on both ends and {φK : j = 1, . . . , m} local basis function defined by (i) ( j) φK (xK ) = S δi j . Let N := NK the set of nodes. Let K∈Th

ND := {ν ∈ N : ν ∈ ΓD } NN := {ν ∈ N : ν ∈ ΓN } Ni := {ν ∈ N − ND : |ων | = 1} Nv := N − (Ni ∪ ND )

and for each ν ∈ N , let ων := {K ∈ Th : ν ∈ K}. To each ν ∈ N we associate a basis function φ (ν) : supp φ (ν) =

S K∈ων m

( j)

( j)

K, φ (ν) |K = φK , xK = ν. ( j) ( j)

Let vˆh ∈ Vh written as vˆh = ∑ ∑ αK φK we define : K∈Th j=1

( Ah vˆh := ∑ β (ν) φ (ν) , where β (ν) := ν∈N

( j)

0 1 2

( j) ∑x( j) =ν αK K

( j)

We define now βK := β (ν) if xK = ν. ( j)

We have ||φK ||∞ ≲ 1, Thus :

KJ

R

( j)

= hK and ||∂X φK ||∞ ≲ HK−1 .

if ν ∈ ND o

if ν ∈ N − ND

3 Adaptive Discontinuous Galerkin Methods on a Moving Mesh ( j)

∑ |vˆh − Ah vˆh |U2 K (t) ≲ ∑ ||∂X φK ||2∞ ≲ ≲ ≲

∑ ||vˆh − Ah vˆh ||2HK (t) ≲

K∈Th

≲ ≲ ≲ ( j)

R 1 m ( j) ( j) 2 K J ∑ |αK − βK |

j=1 ( j) ( j) 2 ∑ ∑ |αK − βK | j=1 K∈Th ( j) δ (Hν = max (HK )) ∑ Hνν ∑ |αK − β (ν) |2 ων ( j) ν∈N xK =ν ( j) ( j) δ δ ∑ Hνν ∑ |αK − β (ν) |2 + ∑ Hνν ∑ |αK |2 ( j) ( j) ν∈Nv ν∈ND xK =ν xK =ν m ( j) 2 R ( j) ( j) 2 ∑ ||φK ||∞ K J ∑ |αK − βK | j=1 K∈Th m ( j) ( j) ∑ hK ∑ |αK − βK |2 j=1 K∈Th ( j) ∑ hν ∑ |αK − β (ν) |2 ( j) ν∈N xK =ν ( j) ( j) ∑ hν ∑ |αK − β (ν) |2 + ∑ hν ∑ |αK |2 ν∈Nv x( j) =ν ν∈ND x( j) =ν K K

K∈Th

K∈Th

45

δK HK

m

because αK = β (ν) for ν ∈ Ni . For ν ∈ Nv we write ων = {K + , K − }. Finally (j ) (j ) ( j) δ δ ∑ |vˆh − Ah vˆh |U2 K (t) ≲ ∑ Hνν |αK ++ − αK −− |2 + ∑ Hνν |αK |2 ν∈Nv

K∈Th

∑

K∈Th

||vˆh − Ah vˆh ||2HK (t) (j )

(j )

≲ ∑

ν∈Nv

( j+ ) ( j− ) hν |αK + − αK − |2 +

Knowing |αK ++ − αK −− | = |[[vˆh ]]|. Then :  ∑ ||vˆh − Ah vˆh ||2HK (t) ≲ ∑  K∈Th

{1,...,n−1}∪ΓD

K∈Th

{1,...,n−1}∪ΓD

2   ∑ |vˆh − Ah vˆh |UK (t) ≲

Let’s work with

∑

ν∈ND

( j)

∑ hν |αK |2

ν∈ND

hi [[vˆh ]]2 δi 2 Hi [[vˆh ]]

uˆsh = uˆch + uˆrh with uˆch = Ah uˆsh

(3.11)

Here, uˆch is a continuous projection of uˆsh and uˆrh catches the jumps. With this we will try to find a bound to κ := |||uˆs − uˆsh ||| + |uˆs − uˆsh |A

By definition κ ≤ |||uˆs − uˆch ||| + |uˆs − uˆch |A + |||uˆrh ||| + |uˆrh |A . 1 1 Let ρS (t) := min(hS ε − 2 , β − 2 ), S = i or Ki . Let

(3.12)

46

E. Rozier, J. Behrens

 2  ηJ := 1 [(β (t) + αωi−1 )hi−1 + γε δi−1 (1 + 1 )][[uˆs ]]|2   i h Xi−1 2 ε Hi−1 J2    αω2 i γε  1 1 s 2  + 2 [(β (t) + ε )hi + Hi δi (1 + J 2 )][[uˆh ]]|Xi  ρ  [[ ε ∂ uˆs ]]|2 + 12 √ρiε [[ εJ ∂X uˆsh ]]|2Xi  ηEi := 12 √i−1  ε J X h Xi−1      ∂ uˆs ∂ uˆs  ηRi := ρKi || fˆ − ∂t uˆh + εJ { XJ h } − (V − V˜ ) XJ h ||2HK (t)

(3.13)

i

And for the boundary points i = 0 or i = n :  α2  1 s 2  ηJi := 1{Xi ∈ΓD } [(β (t) + εωi )h0 + γε  Hi δi (1 + J 2 )](uˆh − uˆD )|Xi   2 α ω δi±1 (1 + J12 )][[uˆsh ]]|2Xi±1 + 12 [(β (t) + εi±1 )hi±1 + Hγε i±1     ρi ε 1 ρi±1 ε  η := 1 s 2 s 2 Ei {Xi ∈ΓN } √ε ( J ∂X uˆh − uˆN )|Xi + 2 √ε [[ J ∂X uˆh ]]|Xi±1

(3.14)

We then bound κ with these error estimators. 1

Lemma 3.3.9 |||uˆrh ||| + |uˆrh |A ≲ (∑[ 1γ + 1]ηJi ) 2 with uˆrh defined in (3.11). i

Proof. Knowing that |||uˆrh |||2 + |uˆrh |2A =

[[uˆrh ]]

∑

K∈Th

= [[uˆsh ]]

on ΓD ∪ {1, . . . , n − 1} :

[ε|uˆrh |U2 K (t) + β ||uˆrh ||2HK (t) ] + |(V − V˜ )uˆrh |2∗

∑

+

[(β +

ΓD ∪{1,...,n−1}

By Lemma 3.3.8  r 2   ∑ ε|uˆh |UK (t)

≲ γ −1

K∈Th

r 2   ∑ β ||uˆh ||HK (t) ≲ K∈Th

|(V − V˜ )uˆrh |2∗ ≤ ≤

∑

ΓD ∪{1,...,n−1}

∑

ΓD ∪{1,...,n−1}

sup

v∈H ˆ 01 (Ωr ) : 1 ε

αX2i γε )hi + ][[uˆsh ]]|2Xi ε JHi

γε r 2 Hi δi [[uˆh ]]|Xi

≲ γ −1 ∑ηJi

i

2 ) ≤ 1 ||(V − V ˜ )uˆr ||2 (||(V − V˜ )uˆrh ||2H(t) · |v| ˆ U(t) h H(t) ε

|||v|||=1 ˆ

∑ αK (t)2 ||uˆrh ||2HK (t) ≲

K∈Th

≲ ∑ηJi

∑

ΓD ∪{1,...,n−1}

i

Then for i ∈ {1, . . . , n − 1}

i

β hi [[uˆrh ]]|2Xi ≲ ∑ηJi

 X−X i−1   Xi −Xi−1 if X ∈ [Xi−1 ; Xi ] X−X Λi = X −Xi+1 if X ∈ [Xi ; Xi+1 ]   i i+1 0 else

hi αω2 i r 2 ε [[uˆh ]]|Xi

3 Adaptive Discontinuous Galerkin Methods on a Moving Mesh

Λ0 = 1{X0 ̸∈ΓD }



0

( Λn = 1{Xn ̸∈ΓD }

X−X1 X0 −X1

X−Xn−1 Xn −Xn−1

if X ∈ [X0 ; X1 ] else if X ∈ [Xn−1 ; Xn ] . else

0

And then we denote : ˆ K ∈ S1 (K), ϕˆ = 0 on ΓD } Ih (t) : L1 (Ωr ) → {ϕˆ∈ C(Ωr ) : ϕ| vˆ 7→ ∑ i

R1

J vˆ }Λi . ωi J ω i R

Lemma 3.3.10 We have ε||∂X Ih v|| ˆ 2HK (0) ≲

hK ˆ 2 HK |||v|||

and

 −2 ˆ 2HK (t) ≲ |||v||| ˆ 2  ∑ [ρK ] ||vˆ − Ih (t)v|| K∈T √h ε ˆ 2Xi ρ (vˆ − Ih (t)v)| i i

≲ |||v||| ˆ 2

∑

∀vˆ ∈ HD1 (Ωr ). Proof. We first build the same operator in the lagrangian variable : For i ∈ {1, . . . , n − 1}  x−x i−1   xi −xi−1 if x ∈ [xi−1 ; xi ] i+1 λi = xx−x if x ∈ [xi ; xi+1 ] −x   i i+1 0 else  x−x1 if x ∈ [x0 ; x1 ] λ0 = 1{x0 ̸∈ΓD } x0 −x1 0 else ( x−xn−1 if X ∈ [xn−1 ; xn ] λn = 1{xn ̸∈ΓD } xn −xn−1 . 0 else And denote : Jh (t) : L1 (Ω ) → {ϕˆ∈ C(Ω ) : ϕ|K ∈ S1 (K), ϕ = 0 on ΓD } v 7→ ∑ i

R 1 ωi

dx ωi vdx }λi .

R

Lemma 5.3 [22] for Jh (t) gives the following estimates :   ε|Jhˆ(t)v|UK (t) ≲ |||v||| ˆ ωK    −2 ||vˆ − J ˆ(t)v||2 [ρ ] ˆ 2ωK ∑ K h HK (t) ≲ |||v||| K∈T  √εh    ∑ ρi (v − Jh (t)v)|2Xi ≲ |||v||| ˆ 2ωK

R

vdx ω dx

Rωi

i

i

=

R

Rωi

ωi

J vˆ J

gives the first and last inequalities.

For the second inequality we notice that

47

48

E. Rozier, J. Behrens

 Then (

||vˆ − Ih (t)v|| ˆ HK (t) ≤ ||vˆ − Jhˆ(t)v||HK (t) + ||Ih (t)vˆ − Jhˆ(t)v||HK (t) |λ − Λ | < 1 ε ||Ih (t)vˆ − Jhˆ(t)v||2 ≤ |||Jhˆ(t)v|||2 h2 β ||Ih (t)vˆ − Jhˆ(t)v||2 ≤ |||v||| ˆ 2

We will now bound K˜ h . Lemma 3.3.11 ∀vˆ ∈ Vh ∀sˆ ∈ HD1 (Ωr ) with ( wˆ = Ih (t)sˆ Bh (v, ˆ w) ˆ := K˜ h (v, ˆ w) ˆ + ∑ (−1)δni ε ∂XJwˆ uˆD |Xi ΓD

1

|Bh (v, ˆ w)| ˆ ≲ γ− 2 (

∑

{1,...,n−1}

Proof. Bh (v, ˆ w) ˆ =−

∑

i=1,...,n−1

And by Cauchy-Schwarz : 1

|Bh (v, ˆ w)| ˆ ≤ γ− 2 (

∑

i ˆ − uˆD )2 ) 2 |||s||| + ∑ Jεγδ ˆ 2 H (v 1

i

ΓD

ˆ [[v]] ˆ ∂X wˆ v− ˆ uˆD δni Jε {{∂XJw}} J |Xi − ∑ (−1) Jε J J |Xi

δi

{1,...,n−1}

εγδi [[v]]| ˆ 2Hi J 2 Hi

ΓD

1 εγ εγ [[v]] ˆ 2 + ∑δi 2 (vˆ − uˆD )2 ) 2 J 2 Hi J H i ΓN

(

∑

{1,...,n−1}∪ΓD

1 εHi (∂X w) ˆ 2) 2 δi

With the Inverse inequality, lemma 3.3.10 and (3.5) : ∑

{1,...,n−1}∪ΓD

εHi ˆ 2 δi (∂X w)

ε ˆ 2HK (0) δ ||∂X w|| K∈Th K

≲ ∑

≲

hK ˆ 2 HK δK |||s|||

≲ |||s||| ˆ 2

proves the lemma. We can now bound |||uˆs − uˆch ||| + |uˆs − uˆch |A . 1

Lemma 3.3.12 |||uˆs − uˆch ||| + |uˆs − uˆch |A ≲ (∑ηi ) 2 with uˆch defined in (3.11). with ηi := [(1 + 1γ )ηJi + ηEi ] + ηRi .

i

Proof. We will first bound : R ˆ − A˜ h (uˆsh , vˆ − Ih v) ˆ for vˆ ∈ HD1 (Ωr ). T (v) ˆ := l(vˆ − Ih v) ˆ − Ωr J ∂∂tuˆh (vˆ − Ih v) We have T = T1 + T2 + T3 with :  R ∂ uˆs ∂ uˆs  T1 (v) ˆ := ∑ K J( fˆ − ∂t uˆh + εJ { XJ h } − (V − V˜ ) XJ h )(vˆ − Ih v) ˆ    K∈Th   ε s T2 (v) ˆ := ∑ (uˆN − εJ ∂X uˆsh )(vˆ − Ih v) ˆ + ˆ Xi ∑ J [[∂X uˆh ]](vˆ − Ih v)| i=1,...,n−1 Γ  N     ˆ := − ∑ (V − V˜ )[[uˆsh ]](vˆ − Ih v)| ˆ Xi  T3 (v) i=1,...,n−1

3 Adaptive Discontinuous Galerkin Methods on a Moving Mesh

49

We have : 1 1 |T1 | ≤ (∑ηRi ) 2 ( ∑ [ρK ||vˆ − Ih v|| ˆ 2HK (t) ) 2 i

1

K∈Th

≲ (∑ηRi ) 2 |||v||| ˆ i

1

|T2 | ≲ (∑ηEi ) 2 |||v||| ˆ i

1

|T3 | ≲ (∑ηJi ) 2 |||v||| ˆ i

We have : ∀vˆ ∈ HD1 (Ωr ) R ˆ A˜ h (uˆs − uˆch , v) ˆ = l(v) ˆ − Ωr J ∂∂tuˆh vˆ − A˜ h (uˆch , v) R ∂ uˆh s ˜ = l(v) ˆ − Ωr J ∂t vˆ − Ah (uˆh , v) ˆ + A˜ h (uˆrh , v) ˆ And : R R h vˆ ˆ + ∑ (−1)δni ε ∂X I ˆ − Ωr J ∂∂tuˆh Ih v] l(Ih v) ˆ − Ωr J ∂∂tuˆh Ih vˆ = [lh (Ih v) J uˆD |Xi ΓD

h vˆ = A˜ h (uˆsh , Ih v) ˆ + K˜ h (uˆrh , Ih v) ˆ + ∑ (−1)δni ε ∂X I J uˆD |Xi

= A˜ h (uˆsh , Ih v) ˆ + Bh (uˆrh , Ih v) ˆ

ΓD

Then : A˜ h (uˆs − uˆch , v) ˆ = T (v) ˆ + A˜ h (uˆrh , v) ˆ + Bh (uˆsh , Ih v) ˆ 1

|A˜ h (uˆs − uˆch , v)| ˆ ≲ (∑ηi ) 2 |||v||| ˆ i

And by noticing that : |uˆs − uˆch |A = |(V − V˜ )(uˆs − uˆch )|∗ : |||uˆs − uˆch ||| + |uˆs − uˆch |A ≲

sup

v∈H ˆ D1 (Ωr )−{0}

A˜ h (uˆs −uˆch ,v) ˆ |||v||| ˆ

Theorem 3.3.13 κ 2 ≲ ∑ηi with κ defined in (3.12). i

Proof. The result follows directly from Lemma 3.3.9 and Lemma 3.3.12.

3.4 Error bound for the semidiscrete nonstationary problem To go from the stationary to the nonstationary problem, we write uˆs (t, ·) := uˆs (t) and uˆsh (t, ·) := uˆsh (t). Now for every t ∈ [0, T ] we have uˆh (t, ·) the unique solution of the same problem as uˆsh (t) then uˆh = uˆsh . We define eˆ := uˆ − uˆh = ρˆ + θˆ with ρˆ := uˆ − uˆs and θˆ := uˆs − uˆsh = uˆs − uˆh . Let

50

E. Rozier, J. Behrens

 2  η t := 1 [(β (t) + αωi−1 )hi−1 + γε δi−1 (1 + 12 )][[uˆh ]]|2  Xi−1 J  2 ε Hi−1 i J   αω2 i  γε 1 1  2  + 2 [(β (t) + ε )hi + Hi δi (1 + J 2 )][[uˆh ]]|Xi    ρ [[ ε ∂ uˆ ]]|2 + 12 √ρiε [[ εJ ∂X uˆh ]]|2Xi ηEt i := 12 √i−1  ε J X h Xi−1       η t := ρK || fˆ − ∂t uˆh + ε { ∂X uˆh } − (V − V˜ ) ∂X uˆh ||2  i  J J J HKi (t)  Ri  ηit := [(1 + 1γ )ηJt i + ηEt i ] + ηRt i

(3.15)

Similarly for the boundary points i = 0 or i = n :  α2  1 2  ηJt i := 1{Xi ∈ΓD } [(β (t) + εωi )hi + γε  Hi δi (1 + J 2 )](uˆh − uˆD )|Xi   2 αω δi±1 (1 + J12 )][[uˆh ]]|2Xi±1 + 12 [(β (t) + εi±1 )hi±1 + Hγε i±1     ρi ε 1 ρi±1 ε  η t := 1 2 2 {Xi ∈ΓN } √ε ( J ∂X uˆh − uˆN )|Xi + 2 √ε [[ J ∂X uˆh ]]|Xi±1 Ei

(3.16)

Such that by Theorem 3.3.13 : ∀t ∈ [0, T ], κ 2 = (|||θˆ ||| + |θˆ |A )2 ≲ ∑ (1 + 1γ )ηit . K∈Th

Notice that the Ah operator that we built in Lemma 3.3.8 is preserving the smoothness in time, we can define uˆrh and uˆch as space-time functions: uˆrh ∈ C1 (0, T ;Vh ) and uˆch ∈ C1 (0, T ;Vhc ). And for all t ∈ [0, T ] |||uˆrh |||2 + |uˆrh |2A ≲ ∑ [ 1γ + 1]ηJt i K∈Th

∥uˆrh ∥2H(t) ≲ ∥

∂ uˆrh 2 ∂t ∥H(t)

≲

∑

hi [[uˆh ]]|2Xi

∑

hi [[ ∂∂tuˆh ]]|2Xi

ΓD ∪{0,...,n−1} ΓD ∪{0,...,n−1}

We already did the demonstration for the first inequality, the last two follow ∂ (Ah vˆh ) = Ah ∂∂tvˆh , and applying the Lemma 3.3.8 to since, ∀vˆh ∈ C1 (0, T ;Vh ), ∂t the function

∂ uˆh ∂t .

Lemma 3.4.1 ∀vˆ ∈ HD1 (Ωr ),

R

Ωr

ˆ v) ˆ =0 J ∂∂teˆ vˆ + A(t; ρ,

Proof. The Lemma is a direct consequence of the definition of uˆ and uˆs . To state the final global error estimate we need some criteria :

3 Adaptive Discontinuous Galerkin Methods on a Moving Mesh

51

 t η1 := ∑ηKt i    i   t := η hi [[ ∂∂tuˆh ]]|2Xi . ∑ 2 Let ∪{0,...,n−1} Γ D   t   η3 := hi [[uˆh ]]|2Xi ∑  ΓD ∪{0,...,n−1}

For vˆ ∈ L∞ (0, T ; H 1 (Th )) and v(t, x) := v(t, ˆ X) we define : ||v||2# := ||v||2L∞ (0,T ;L2 (Ω )) +

RT 0

|||v||| ˆ t2 dt

The final estimate follows. Theorem 3.4.2 ||e||2# ≲ S(t){||e(0)||2L2 (Ω ) +

RT 0

η1t + T

with S(t) = exp(∥∂X V˜ ∥∞t)

RT 0

η2t + max (η3t )} t∈[0,T ]

Proof. Let θˆc := uˆs − uˆch ∈ C1 (0, T ; HD1 (Ωr )) and eˆc := uˆ − uˆch ∈ C1 (0, T ; HD1 (Ωr )). Taking Lemma 3.4.1 with eˆc we have R

Ωr

J ∂∂teˆc eˆc + A(t; eˆc , eˆc ) =

∂ uˆrh ˆ ∂t eˆc + A(t; θc , eˆc )

Ωr

J

≥

∥∂xV˜ ∥∞ 1 d 2 ∥ec ∥2L2 (Ω ) 2 dt ∥ec ∥L2 (Ω ) − 2

R

We have the following inequalities : R

Ωr

J ∂∂teˆc eˆc =

R ∂xV˜ 2 1 d 2 2 dt ∥ec ∥L2 (Ω ) − Ωr 2 J eˆc

A(t; eˆc , eˆc ) ≥ |||eˆc |||t2 R

Ωr

J

∂ uˆrh ∂t eˆc

≤ T2 ∥

A(t; θˆc , eˆc ) ≤ C · (|||θˆc |||t + |θˆc |A )|||eˆc |||t ≤

Then

C2 2

∂ uˆrh 2 1 2 ∂t ∥H(t) + 2T ∥ec ∥L2 (Ω ) 2

· (|||θˆc |||t + |θˆc |A )2 + |||eˆ2c |||t

∂ uˆr d ∥ec ∥2L2 (Ω ) + (|||eˆc |||t2 −C2 · (|||θˆc |||t + |θˆc |A )2 − T · ∥ h ∥2H(t) ) dt ∂t

1 ≤ (∥∂X V˜ ∥∞ + )||ec ||2L2 (Ω ) T

And by Gronwall’s lemma R R ∂ uˆr ∥ec ∥2# ≲ S(t){∥ec (0)∥2L2 (Ω ) + 0T (|||θˆc |||t + |θˆc |A )2 + T 0T ∥ ∂th ∥H(t) } Then by definition of η3t ∥e∥2# ≲ S(t){∥e(0)∥2L2 (Ω ) +

RT 0

η1t + T

RT 0

η2t + max (η3t )} t∈[0,T ]

52

E. Rozier, J. Behrens

3.5 Test case This test case will be presented to see how the error behaves in time and we will see that the mesh movement has an effect on the error. First the speed of the flow of the test cases have to be globally compressible (meaning that V (t, y0 ) > V (t, y1 )). We will test the error criteria developed here by comparing them with the error in the energy-norm and the exact error. The test case is as follows : Ω = [0, 1], ΓD = {0, 1}, ΓN = 0/ ε = 0.01, V = 2HM + H(t, x) − (x − 21 ) V˜ (t, X) = V (t, x) − cos(πx)7V (t, 0) y(x)) x−0.5 H(t, x) = − 32 · tanh(β , HM (t) = max(H(t, x)), y(x) = (c+x)(1+c−x) y′ (x) β = 20, c = 0.1 u(t, x) = (x − 12 )(1 − 4(x − 12 )2 ) We notice here that ∂xV (t, 0) > 0 and ∂x (V − V˜ )(t, x) < 0, the DG method would be unstable in a static mesh’s situation, whereas here it is stabilized by the moving mesh.

Fig. 3.1: In red the energy error, in green the aposteriori estimate after one time step The right hand side f is computed such that the equation is verified. In this test case we can see two very interesting effects : ˜ 2 - for the ||V −εV ||∞ : we can see in Fig. 3.1 that the energy-error is proportional to ||V −V˜ ||2∞ ε

and that it is well measured by the error estimator.

3 Adaptive Discontinuous Galerkin Methods on a Moving Mesh

53

Fig. 3.2: In red the energy error, in green the aposteriori estimate after 41 time steps

- for the error in the central tile : the energy-norm of the exact solution is proportional to the size of the tile and is decreasing proportionally to exp(||∂xV˜ ||t), the energy error is as well proportional to the size of the tile in this case and this is well measured by the error estimator as well (in Fig. 3.2 , exp(||∂xV˜ ||t) ≈ 2.5).

3.6 Conclusion The mesh movement reduces the complexity of the PDE by splitting our problem into an ODE and a PDE. This can be first seen in the range of stability for this method: the pointwise compressibility that requires the classical DG method is replaced by an averaged compressibility. This also comes with a reduction of the error that takes place in the L2 -norm. Yet, the large-scale trend of the flow still dominates the transport of the solution. This relaxation can also be found in the error analysis. The study of the error propagation, which is the pivot of any study of the unsteady problem, is relaxed by the large-scale speed and we built the quantity ||∗ to study only the remaining scale. Just like in the study of static meshes, we find that the error propagation has to be studied on the boundaries of the cells and that there is an interaction effect between the diffusion and the remaining transport. To some extent, this error estimator can be used to focus our analysis only on the error propagation term:

αω2 i hi 2 ε [[uˆh ]]|Xi .

In a context where the large-scale trend of

54

E. Rozier, J. Behrens

the transport is significant compared to its small-scale variations, this term mainly determines the error for a large range of T . In conclusion, the study of the complete picture of the error criteria could help to choose the parametrization of the mesh speed V˜ by finding a balance between the contributions of ||V − V˜ || and ||∇V˜ || as well as the finishing time T.

Acknowledgment The authors acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 ”Modeling, Simulation and Optimization of Fluid Dynamic Applications”.

References 1. R. Araya and P. Venegas. An a posteriori error estimator for an unsteady advection-diffusionreaction problem. Computers and Mathematics with Applications 66, 2456–2476, 2014. 2. M. Bal´aszov´a, M. Feistauer, and M. Vlas´ak. Stability of the ALE space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains. ESAIM: M2AN 52(6), 2327–2356, 2018. 3. A. Bonito, I. Kyza, and R.H. Nochetto. Time-discrete high-order ALE formulations: a priori error analysis. Numer. Math. 125, 225–257, 2013. 4. A. Bonito, I. Kyza, and R.H. Nochetto. Time-discrete high-order ALE formulations: stability. SIAM J. Numer. Anal. 51(1), 577–604, 2013. 5. A. Cangiani, E.H. Georgoulis, and S. Metcalfe. Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems. IMA Journal of Numerical Analysis 34, 1578– 1597, 2014. 6. K. Chrysafinos and N.J. Walkington. Error estimates for the discontinuous Galerkin methods for parabolic equations. SIAM J. Numer. Anal. 44(1), 349–366, 2006. 7. K. Chrysafinos and N.J. Walkington. Lagrangian and moving mesh methods for the convectiondiffusion equation. ESAIM: Mathematical Modeling and Numerical Analysis 42, 22–55, 2008. 8. B. Cockburn and C.-W. Shu. Runge-Kutta discontinuous Galerkin methods for convectiondiffusion problems. Journal of Scientific Computing 16, 173–261, 2001. 9. V. Dolejˇs´ı, M. Feistauer, and C. Schwab. A finite volume discontinuous Galerkin scheme for nonlinear convection-diffusion problems. Calcolo 39, 1–40, 2002. 10. V. Dolejˇs´ı and M. Feistauer. Error estimates of the discontinuous Galerkin method for nonstationary convection-diffusion problems. Numerical Functional Analysis and Optimization 26(3), 349–383, 2005. 11. V. Dolejˇs´ı. hp-DGFEM for nonlinear convection-diffusion problems. Mathematics and Computers in Simulation 87, 87–118, 2013. ˇ Space-time discontinuous Galerkin method for 12. M. Feistauer, J. H´ajek, and K. Svadlenka, solving nonstationary convection-diffusion-reaction problems. Applications of Mathematics 52, 197–233, 2007. 13. S. Ganesan and S. Srivastava. Local projection stabilization with discontinuous Galerkin method in time applied to convection dominated problems in time-dependant domains. BIT Numerical Mathematics 60, 481–506, 2020. 14. P. Houston and E. S¨uli. hp-adaptive discontinuous Galerkin finite element methods for firstorder hyperbolic problems. SIAM J. Sci. Comput. 23, 1226–1252, 2001.

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15. P. Houston, C. Schwab, and E. S¨uli. Stabilized hp-finite element methods for first-order hyperbolic problems. SIAM J. Numer. Anal. 37, 1618–1643, 2000. 16. A. Karakashian and F. Pascal. A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problem. SIAM J. Numer. Anal. 41, 2374–2399, 2003. 17. V. Kuˇcera, M. Feistauer, and J. Prokopov´a. The discontiunuous Galerkin method for convection-diffusion problems on time-dependant domains. In: Numerical Mathematics and Advanced Applications, G. Kreiss, P. L¨otstedt, A. M˚alqvist, and M. Neytcheva (eds.), Springer, Berlin, 551–559, 2009. 18. F. Naddei, M. de la Llave Plata, V. Couaillier, and F. Coquel. A comparison of refinement indicators for p-adaptive simulations of steady and unsteady flows with discontinuous Galerkin methods. Journal of Computational Physics 376(1), 508–533, 2018. 19. D. Sch¨otzau and L. Zhu. A robust a posteriori error estimator for discontinuous Galerkin methods for convection-diffusion equations. Applied Numerical Mathematics 59(9), 2236–2255, 2009. 20. J.J. Sudirham, J.J.W. van der Vegt, and R.M.J. van Damme. Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependant domains. Applied Numerical Mathematics 56(12), 1491–1518, 2006. 21. R. Verf¨urth. A posteriori error estimators for convection-diffusion equations. Numer. Math. 80, 641–663, 1998. 22. R. Verf¨urth. Robust a posteriori error estimates for nonstationary convection-diffusion equations. SIAM J. Numer. Anal. 43, 1783–1802, 2005. 23. L. Zhou and Y. Xia. Arbitrary Lagrangian-Eulerian local discontinuous Galerkin method for linear convection-diffusion equations. Journal on Scientific Computing 90, 2022.

Chapter 4

Anisotropic Kernels for Particle Flow Simulation Kristof Albrecht, Juliane Entzian, Armin Iske

Abstract This contribution discusses the construction and the utility of anisotropic kernels for numerical fluid flow simulation. So far, commonly used radial kernels, such as Gaussians, (inverse) multiquadrics and polyharmonic splines, were proven to be powerful tools in various applications of multivariate scattered data approximation. Due to the well-known uncertainty principle, however, their resulting reconstruction methods are often critical when it comes to combine high order approximation with numerical stability. In many cases this leads to severe limitations, especially when it comes to fluid flow simulations. Therefore, more sophisticated kernel methods are required. In this paper, we show how to obtain anisotropic positive definite kernels from standard kernels rather directly. Our proposed construction yields a new class of more flexible kernels that are particularly useful for fluid flow simulations. To this end, the finite volume particle method is used as a prototype of our discussion, where scattered data approximation is needed in the recovery step of weighted essentially non-oscillatory (WENO) reconstructions. Supporting numerical examples and comparisons are provided.

Kristof Albrecht Institute of Mathematics, Technische Universit¨at Hamburg, Germany e-mail: [email protected] Juliane Entzian Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] Armin Iske Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Iske, T. Rung (eds.), Modeling, Simulation and Optimization of Fluid Dynamic Applications, Lecture Notes in Computational Science and Engineering 148, https://doi.org/10.1007/978-3-031-45158-4_4

57

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K. Albrecht, J. Entzian, A. Iske

4.1 Introduction Particle methods provide flexible discretization schemes for numerical simulations of multiscale phenomena in various applications from computational science and engineering. In the modelling of time-dependent evolution processes, for instance, adaptive particle-based algorithms are particularly well-suited to cope with rapid variation of domain geometries and anisotropic large-scale deformations. To briefly recall their basic concept within fluid flow simulation, particle methods work with a finite set of scattered particles, where some specific physical properties are attached to the individual particles. In numerical simulations of timedependent evolution processes, the particles are dynamically modified during the simulation. Particle flow simulations require high order approximations for local scattered data reconstruction. To this end, kernel-based approximation methods — often referred to as radial basis function (RBF) methods — are powerful tools. We remark that RBF methods are meshfree and particularly easy to implement. On the down side, standard RBF kernels, such as Gaussians and polyharmonic splines, are radially symmetric, which often leads to severe restrictions when it comes to capture anisotropic features in fluid flow simulations. This in turn requires the construction of more flexible kernels, where anisotropic kernels are primarily important. In this contribution, we show how to construct anisotropic kernels (cf. [3], [6]). Our straight forward construction embarks on radially symmetric positive definite kernels, which allows us to maintain the celebrated advantages of meshfree RBF methods. We will give supporting arguments in favour of anisotropic kernel methods, where we focus on their numerical stability and their approximation behaviour. We show that our achievements support adaptive concepts of particle simulation particularly well. The outline of this work is as follows. In Section 4.2, we introduce the Eulerian finite volume particle method (FVPM), which is used as a prototypical concept for the numerical solution of flow simulation model problems that we wish to address, in particular of hyperbolic conservation laws. This includes a short discussion on the concept of weighted essentially non-oscillatory (WENO) reconstruction. Then, in Section 4.3, we explain the general framework of kernel-based reconstruction in particle flow simulations. This then leads us to the radially symmetric polyharmonic spline kernels in Section 4.4, before we turn to the construction of adaptive kernels in Section 4.5. Supporting numerical results and comparisons are finally presented in Section 4.6.

4.2 The Finite Volume Particle Method In this section, we explain the main ingredients of the FVPM [13], which we use as a prototype for an Eulerian particle-based concept in numerical flow simulation. Before we detail on this, let us recall that numerical flow simulation requires suitable approximation algorithms, in particular for the solution of time-dependent hyper-

59

4 Anisotropic Kernels for Particle Flow Simulation

bolic conservation laws

∂u + ∇ f (u) = 0, ∂t

(4.1)

where for some domain Ω ⊂ Rd , d ≥ 1, and a compact time interval [0, T ], T > 0, the solution u : [0, T ] × Ω → R of (4.1) is sought, and where f (u) = ( f1 (u), . . . , fd (u))T is a given flux tensor, and it is usually assumed that initial conditions u(0, x) = u0 (x)

for x ∈ Ω

(4.2)

at time t = 0 are given. We recall that nonlinear fluxes f lead to discontinuities in the solution u, shocks, as observed in many relevant applications of computational fluid dynamics. Such discontinuities of the solution u in (4.1) can easily develop spontaneously even from smooth initial data u0 in (4.2). Therefore, nonlinear flow simulation requires more sophisticated and flexible mathematical and computational methods to numerically solve the Cauchy problem (4.1), (4.2). For a comprehensive introduction to numerical methods for hyperbolic problems we recommend the textbook [19]. To explain basic features of the FVPM, let Ξ = {ξ1 , . . . , ξn } ⊂ Ω denote a finite point set of particles (i.e., particle positions). Moreover, for any particle ξ ∈ Ξ we denote its influence area by VΞ (ξ ) ⊂ Ω . To make a rather straight forward example, the particles’ influence areas may, for instance, be given by the Voronoi tiles   VΞ (ξ ) = x ∈ Ω : ∥x − ξ ∥ = min ∥x − ν∥ ⊂ Ω for ξ ∈ Ξ ν∈Ξ

of the Voronoi diagram VΞ = {VΞ (ξ )}ξ ∈Ξ for Ξ , in which case VΞ yields a decomposition of Ω into convex and closed subdomains VΞ (ξ ) ⊂ Ω with pairwise disjoint interior, see Figure 4.1 for illustration.

11 00 111111111 000000000 00 11

Fig. 4.1: A finite set Ξ of scattered particles ξ (displayed •) and their influence areas, here given by their Voronoi tiles VΞ (ξ ), are shown.

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K. Albrecht, J. Entzian, A. Iske

Note that the Voronoi diagram VΞ is entirely determined by the geometry of the particle distribution in Ξ . We remark that there are efficient algorithms from computational geometry [21] for the construction and maintenance of the Voronoi diagram VΞ . The combination between Voronoi diagrams and finite volumes yields through the basic concept of the FVPM a flexible particle method for the numerical solution of (4.1), (4.2). Now, for any particle located at ξ ∈ Ξ at time t its particle average is defined by u¯ξ (t) =

1 |VΞ (ξ )|

Z VΞ (ξ )

u(t, x) dx

for ξ ∈ Ξ and t ∈ [0, T ].

According to the classical concept of finite volume methods [19], for each ξ ∈ Ξ the average value u¯ξ (t) is, at time step t → t + τ, updated by an explicit numerical method of the form u¯ξ (t + τ) = u¯ξ (t) −

τ Fξ ,ν , |VΞ (ξ )| ∑ ν

(4.3)

where Fξ ,ν denotes the numerical flux between particle ξ and a neighbouring particle ν ∈ Ξ \ ξ . The required exchange of information between neighbouring particles is modelled via a generic numerical flux function, which may be implemented by using any suitable FV flux evaluation scheme, e.g. by the generalized Godunov approach of high order ADER flux evaluation [22, 23]. For the sake of brevity, we prefer to omit further details concerning ADER flux evaluation, but rather refer to the up-to-date survey [24]. The following algorithm reflects one basic time step of the FVPM. Finite Volume Particle Method. INPUT: Time step τ > 0, particles Ξ , particle averages {u¯ξ (t)}ξ ∈Ξ at time t. FOR each ξ ∈ Ξ DO

(a) Determine set Nξ ⊂ Ξ \ ξ of neighbouring particles around ξ ; (b) Compute numerical flux Fξ ,ν for each ν ∈ Nξ ; (c) Update particle average u¯ξ for ξ by (4.3). OUTPUT: Particle averages {u¯ξ (t + τ)}ξ ∈Ξ at time t + τ. Finally, we briefly explain how FVPM can be combined with essentially nonoscillatory (ENO) [12] and WENO [20] reconstruction. To this end, let us view the influence area VΞ (ξ ) of any particle ξ ∈ Ξ as the control volume of ξ , where the control volume VΞ (ξ ) is uniquely represented by ξ . The basic idea of the ENO method is to first select, for each particle ξ ∈ Ξ , a small set {Si }ki=1 of k stencils, where any stencil Si ⊂ Ξ is given by a set of particles lying in the neighbourhood of ξ . Then, for each stencil Si , 1 ≤ i ≤ k, a reconstruction

4 Anisotropic Kernels for Particle Flow Simulation

61

si ≡ sSi is computed, which interpolates the given particle averages {u¯ν (t) : ν ∈ Si } over the control volumes {VΞ (ν)}ν∈Si of the stencil Si . Among the k different reconstructions si , 1 ≤ i ≤ k, for the k different stencils Si , ENO reconstruction takes the smoothest (i.e., the least oscillatory) – according to some suitable oscillation indicator – as numerical solution on control volume VΞ (ξ ). In WENO reconstruction a specific convex combination of the reconstructions si is taken. For more details on ENO/WENO schemes we refer to [16]. We further remark that a more general concept of the FVPM [13, 18], allows for overlapping influence areas {VΞ (ξ )}ξ ∈Ξ in which case, however, the FVPM needs to be combined with a partition of unity method (PUM). This provides more flexibility but it leads to a more complicated FVPM discretization. For more details we refer to [18].

4.3 Kernel-based Reconstruction in Particle Flow Simulations In a generic formulation of particle methods [15, 16] we are essentially concerned with the reconstruction of a numerical solution u ≡ u(t, ·) for fixed time t ∈ [0, T ] from its discrete values uΞ = (u(ξ1 ), . . . , u(ξn ))T ∈ Rn , taken at a scattered set Ξ = {ξ1 , . . . , ξn } ⊂ Rd of particles, cf. Figure 4.2 for illustration. In the Eulerian FVPM of the previous section, for instance, the discrete values in uΞ may be regarded as particle averages of u attached to the particles’ positions in Ξ . In a semi-Lagrangian particle method (SLPM) the values uΞ may reflect the concentration of the solution u at upstream locations on the characteristic curves (streamlines) of the backward flow. For more details concerning relevant Lagrangian and Eulerian particle methods, we refer to our previous work [15], where suitable prototypes for a finite volume particle method (FVPM) and a semi-Lagrangian particle method (SLPM) have been developed. To explain the basic features of such kernel-based reconstructions, we restrict ourselves to the special case of Lagrange interpolation where we seek to compute a suitable interpolant s : Ω → R satisfying uΞ = sΞ , i.e., u(ξk ) = s(ξk )

for all k = 1, . . . , n.

(4.4)

According to the general formulation of kernel-based interpolation, we assume that the reconstruction s has the form s(x) =

n

∑ c j ϕ(x − ξ j ) + p(x)

j=1

d , for p ∈ Pm−1

(4.5)

for some coefficients c1 , . . . , cn ∈ R, where ϕ : Ω → R is a fixed (conditionally posd is the linear space of all d-variate polynoitive definite) kernel function and Pm−1

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K. Albrecht, J. Entzian, A. Iske

Fig. 4.2: A finite scattered set Ξ = {ξ1 , . . . , ξn } ⊂ R2 of particles. Each particle ξ (displayed •) bears a scalar function value u(ξ ) ≡ u(t, ξ ) of the numerical solution u at time t ∈ [0, T ]. mials of degree less than m ∈ N0 . The required order m in (4.5) is determined by the choice of ϕ. If m = 0, then the polynomial part in (4.5) is empty, in this case the reconstruction s has the form s(x) =

n

∑ c j ϕ(x − ξ j ).

(4.6)

j=1

Rather than dwelling much on explaining conditionally positive definite kernel functions and the structure of their native reproducing kernel Hilbert spaces we refer to the text books [5, 10, 17, 25]. For the following of our discussion, it is sufficient to say that scattered data interpolation by positive definite kernels (where m = 0) leads to a unique reconstruction of the form (4.6). Moreover, for conditionally positive definite kernels of order m ∈ N, we obtain under vanishing moment conditions n

∑ c j p(ξ j ) = 0

j=1

for all p ∈ Pmd

(4.7)

a reconstruction s of the form (4.5), where s is unique, if any polynomial p ∈ Pmd can uniquely be reconstructed from its values at the points Ξ , i.e., pΞ = 0 implies p ≡ 0. Let us give examples of commonly used radial kernel functions ϕ(x) = φ (∥x∥), along with their orders m ≡ m(φ ), where r = ∥x∥ ∈ [0, ∞) is, for x ∈ Rd , the radial variable with respect to the Euclidean norm ∥ · ∥ on Rd . Example 4.1. The positive definite Gaussian function φ (r) = e−r

2

for r ∈ [0, ∞)

is a radial kernel of order m = 0, so that the reconstruction s has the form (4.6).

4 Anisotropic Kernels for Particle Flow Simulation

63

Example 4.2. The multiquadric φ (r) = 1 + r2


β

for β > 0 and β ∈ /N

is a conditionally positive definite kernel of order m = ⌈β ⌉. The inverse multiquadric φ (r) = 1 + r2


β

for β < 0

is positive definite, and so m = 0. In this case, the reconstruction s has the form (4.6). Example 4.3. The radial characteristic functions [2]  (1 − r)β for r < 1 β φ (r) = (1 − r)+ = 0 for r ≥ 1 are for d ≥ 2 positive definite on Rd , provided that β ≥ (d + 1)/2. In this case, m = 0, and so the reconstruction s has the form (4.6).

4.4 Reconstruction by Polyharmonic Splines Polyharmonic splines, due to Duchon [7], are traditional tools for Lagrange interpolation from multivariate scattered data. According to the polyharmonic spline interpolation scheme, the reconstruction s has the form (4.5), where the radial polyharmonic spline kernel ϕ(x) = φd,m (r), for r = ∥x∥, is given as ( φd,m (r) =

r2m−d log(r)

for d even

r2m−d

for d odd

) for 2m > d,

with m being the order of the kernel φd,m , i.e., m is the order of the polynomial in (4.5). According to [7], scattered data interpolation by polyharmonic splines is optimal in its native reproducing kernel Hilbert space, as given by the Beppo Levi space o n BLm (Rd ) = u : Dα u ∈ L2 (Rd ) for all |α| = m ⊂ C (Rd ), being equipped with the semi-norm |u|2BLm

  m = ∑ ∥Dα u∥L2 2 (Rd ) . α |α|=m

In other words, the reconstruction s in (4.5) minimizes the Beppo Levi energy functional | · |BLm among all recovery functions u in BLm (Rd ), i.e.,

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K. Albrecht, J. Entzian, A. Iske

for all u ∈ BLm (Rd ) with uΞ = sΞ .

|s|BLm ≤ |u|BLm ,

Therefore, the energy functional |·|BLm is a natural choice for the aforementioned oscillation indicator of WENO reconstruction (cf. [1]). Let us finally discuss the popular special case of thin plate spline reconstruction. In this case, d = m = 2, so that the thin plate spline kernel is φ2,2 (r) = r2 log(r). Therefore, the reconstruction in (4.5) has the form s(x) =

n

∑ c j ∥x − ξ j ∥2 log(∥x − ξ j ∥) + d0 + d1 x1 + d2 x2

j=1

where

for x = (x1 , x2 ),

n o BL2 (R2 ) = u : Dα u ∈ L2 (Rd ) for all |α| = 2 ⊂ C (R2 )

is the Beppo-Levi space of second order over R2 , whose semi-norm |u|2BL2 =

Z R2


u2x1 x1 + 2ux21 x2 + ux22 x2 dx

reflects the bending energy for a thin plate of infinite extent. Since the resulting reconstruction minimizes the bending energy |·|BL2 among all interpolants in BL2 (R2 ) this motivates the naming thin plate spline.

4.5 Reconstruction by Anisotropic Kernels We keep in mind that (W)ENO reconstructions can lead to severe numerical instabilities, especially in situations with anisotropic particle distributions. In order to stabilize (W)ENO schemes, we propose to work with anisotropic kernels that are well-adapted to the geometry of the underlying problem. We explain two possibilities to adapt kernels. Firstly, the anisotropy can be adapted to the particle distribution, i.e., we rely on the geometry of a given data point set, see Figure 4.4. Secondly, we can adapt the anisotropy using a priori knowledge on the problem’s geometry, see the numerical example in Section 4.6.1. Let ϕ : Rd −→ R be a positive definite function that is radially symmetric, i.e., ϕ ≡ φ (∥ · ∥), with respect to the Euclidean norm ∥ · ∥ on Rd . Moreover, let Ξ = {ξ1 , . . . , ξn } ⊂ Rd denote a set of pairwise distinct particles. The basic idea to construct an anisotropic kernel from ϕ is to replace the Euclidean norm in ϕ ≡ φ (∥ · ∥) by an anisotropic norm. Let A = V ΛV T ∈ Rd×d

(4.8)

be the singular value decomposition of a symmetric positive definite matrix A, where V is unitary and Λ is a diagonal matrix containing the eigenvalues of A. In the following we denote

4 Anisotropic Kernels for Particle Flow Simulation

65

L = Λ 1/2V T , hence A =

LT L.

(4.9)

Note that the matrix A defines an (anisotropic) norm on ∥x∥2A := xT Ax = ∥Lx∥2

for x ∈ Rd

Rd

by (4.10)

which coincides with the Euclidean norm ∥ · ∥ after the transformation L. Now, our aim is to construct a transformation matrix L which is well-adapted to the geometry of the particle set Ξ , whereby radial kernel functions ϕ(x) = φ (∥x∥) are transformed into anisotropic kernel functions of the form ϕL (x) = φ (∥x∥A ) = φ (∥Lx∥). Example 4.4. Let β ≥ (d + 1)/2 ≥ 3/2. Then the anisotropic characteristic function ϕL : Rd −→ R resulting from the positive definite kernel φ of Example 4.3 is defined by β ϕL (x) = (1 − ∥x∥A )+ for x ∈ Rd .

We remark that ϕL is positive definite, i.e., for any finite particle set Ξ = {ξ1 , . . . , ξn }, the corresponding kernel matrix (ϕL (ξ j − ξk ))1≤ j,k≤n ∈ Rn×n is symmetric positive definite. This simple observation follows directly from the properties of the singular value decomposition. Figure 4.3 displays a contour map of the radial characteristic function of Example 4.3 and that of the corresponding anisotropic function ϕL .

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K. Albrecht, J. Entzian, A. Iske

Radial Characteristic Function

Anisotropic Characteristic Function

2.0

2.0

1.5

1.5 1.0

0.1

1.0

1.0

1.5

1.5

2.0

0.4

0.5

50

0.5

0 0.0 0

0.7

50

0.0

00

0.6

0.000

0.0

00

0.6

0

0.5

50

0.30

0.3 00

0.5

0.4

50 0. 750 0.1 50

1.0

2.0 2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

2.0

2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

2.0

Fig. 4.3: Contour plot of the radially symmetric kernel φ in Example 4.3 (left) and the corresponding anisotropic kernel ϕL in Example 4.4 (right).

Anisotropic Particles

3

10

2

3

1 1

8

4

3

2

913 2

1

9 13

2

7 5

0

11

2

15

3

1

6

1

6 15

10

v1

v1

0

3

2

12 14

v0

75

Uncorrelated Particles

3

v0

12

1

14

8 11 4

2

1

0

1

2

3

3

3

2

1

0

1

2

3

Fig. 4.4: Anisotropic particle set Ξ with eigenvectors and eigenvalues of its covariance matrix (left); linear transformation of Ξ by L chosen as in (4.11) (right).

Our first approach is to construct the transformation matrix L depending on the geometry of the particle distribution. To do this, we apply standard principle component analysis (PCA) to the particle set Ξ . Following along the lines of [6], we work with the covariance matrix C=

1 QQT ∈ Rd×d , n

4 Anisotropic Kernels for Particle Flow Simulation

where

67

  Q = ξ1 − ξ , . . . , ξn − ξ ∈ Rd×n

with center ξ=

1 n

n

∑ ξ j ∈ Rd .

j=1

Note that C is symmetric and positive semi-definite. Furthermore, note that C will be (strictly) positive definite, if we chose a subset of linearly independent particles from Ξ , in which case Q had full rank. According to the basic properties of PCA, an eigenvector v0 corresponding to the largest eigenvalue of C points into the direction of the largest deviation of the particles Ξ , see Figure 4.4 (left). We use this simple observation to transform the points in the anisotropic particle set Ξ , see Figure 4.4 (left), to an isotropic particle set LΞ , see Figure 4.4 (right). This is done by using the inverse covariance matrix C−1 for the matrix A in (4.10), i.e. we work with A := C−1 , in which case the linear transform L in (4.9) is given by L = Λ −1/2V T , where

(4.11)

C = V ΛV T

denotes the singular value decomposition of the covariance matrix.

4.6 Numerical Examples and Comparisons We consider the two-dimensional Burgers equation ut + u · ux1 + u · ux2 = 0 with initial condition u(x, 0) = u0 (x) =

  ( ∥x−c∥2 exp ∥x−c∥ 2 −R2 0

for ∥x − c∥ < R,

otherwise,

where c = (−0.25, −0.25)T , R = 0.25 and ∥ · ∥ denotes the Euclidean norm on R2 (cf. [14, 11]). For the following numerical comparisons we calculated a numerical reference solution on a 100 × 100 grid in [−0.5, 0.5] × [−0.5, 0.5] using a first order 2d finite volume method (FVM) with explicit forward Euler time-stepping and a Rusanov numerical flux [19]. To demonstrate the effect of anisotropic kernel functions in multivariate interpolation, we focus on the solution at times t0 = 0 and t1 = 1.2 as shown in Figure 4.5.

68

K. Albrecht, J. Entzian, A. Iske Solution at t = 0

0.4

0.2

0.0

0.2

Solution at t = 1.2

0.0 0.2 0.4

0.4

0.2

1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

0.4 0.4

0.2

0.0

0.2

0.0 0.2 0.4

0.2

0.4

0.4

Fig. 4.5: Numerical solution of Burgers equation at t0 = 0 (left) and t1 = 1.2 (right).

4.6.1 Anisotropic Kernels adapted to underlying Geometry To capture the preference direction of the Burgers solution’s flow along the diagonal of the Cartesian coordinate system (cf. Figure 4.5), we adapted the kernels to the support of the numerical solution plotted in Figure 4.6. As the support at initial time t0 = 0 is a circle, we chose the unit vectors (1, 0)T and (0, 1)T as the eigenvectors of A0 in (4.8), in which case V0 in (4.8) is the identity, i.e., V0 = I. Yet it remains to choose the entries of the diagonal matrix Λ0 . To achieve a good trade-off between 1 · I. This yields the transformation matrix error and stability, we chose Λ0 = 0.15 (g)

L0 =

   √  2.582 0 1/ 0.15 √0 ≈ 0 2.582 0 1/ 0.15

at t0 = 0. Figure 4.6 (right) shows an egg-shaped support of Burgers solution at t1 = 1.2 with the tip pointing to the upper right. To capture this behaviour we chose   1 1 1 V1 = √ , 2 1 −1 and√the eigenvalues √ in the diagonal matrix Λ1 were calculated from the dilations as 1/ 0.3 and 1/ 0.16. It all adds up to transformation matrix  √     1 1/ 0.3 √0 1 1 1.29 1.29 (g) L1 = √ ≈ 1 −1 1.768 −1.768 0 1/ 0.16 2

4 Anisotropic Kernels for Particle Flow Simulation

69

t=0

t = 1.2 0.4

0.2

0.2

0.0

0.0

v0

0.4

v1

v1

0.2

0.2 0.0

00

v0

0.4

0.4

00

0.4

0.2

0.0

0.0

0.2

0.4

0.4

0.2

0.0

0.2

0.4

Fig. 4.6: Support of Burgers’ solution at initial time t0 = 0 (left) and t1 = 1.2 (right). at t1 = 1.2. The two transformation matrices L0 and L1 yield the two anisotroic characteristic kernels adapted to the underlying geometry of the Burgers’ equation (g) (g) ϕ0 = ϕ (g) at t0 = 0 and ϕ1 = ϕ (g) at t1 = 1.2, see Example 4.4. L0

L1

4.6.2 Data Point Selection The selection of data points (i.e., the particle positions) is highly critical for the accuracy and the numerical stability of kernel-based interpolation. For the sake of fair comparison we used the psr-greedy method (cf. [8]) to extract the most significant 2500 data points from a 100 × 100 grid for each kernel and time-step. This greedy method was designed to provide a good trade-off between approximation quality and stability of the interpolation method depending on the kernel function. Hence, the data points are optimized for each individual kernel. (r) (r) Figure 4.7 shows the selected data point sets Ξ0 and Ξ1 for the radial characteristic function (Example 4.3) with parameter β = 2 at times t0 = 0 and t1 = 1.2, (g) (g) whereas Figure 4.8 shows the data point selections Ξ0 and Ξ1 for the anisotropic (g)

(g)

kernels ϕ0 and ϕ1 at times t0 = 0 and t1 = 1.2. In every case the greedy method manages to capture the support of the numerical solution (see Figure 4.5). As the greedy data selection depends on the kernel the (r) (g) (r) (g) data point set Ξ0 differs from Ξ0 and Ξ1 from Ξ1 respectively. Note that the choice of the data points takes significant impact on the reconstruction quality.

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K. Albrecht, J. Entzian, A. Iske

characteristic function: t = 0.0

characteristic function: t = 1.2

0.4

0.4

0.2

0.2

0.0

0.0

0.2

0.2

0.4

0.4 0.4

0.2

0.0

0.2

(r)

0.4

0.4

0.2

0.0

0.2

0.4

(r)

Fig. 4.7: Point sets Ξ0 and Ξ1 for the radial characteristic function φ at times t0 = 0 and t1 = 1.2. anisotropic characteristic function: t = 0.0

anisotropic characteristic function: t = 1.2

0.4

0.4

0.2

0.2

0.0

0.0

0.2

0.2

0.4

0.4 0.4

0.2

0.0

0.2

(g)

(g)

Fig. 4.8: Point sets Ξ0 , Ξ1 t0 = 0 and t1 = 1.2.

0.4

0.4

0.2

0.0

0.2

0.4

(g)

(g)

for anisotropic characteristic functions ϕ0 , ϕ1

at

4.6.3 Anisotropic Kernels adapted to Particle Distribution In Section 4.5 we discussed the adaptation of the kernel to the geometry of a given particle set. This is done by using the covariance of the data. The application of this (r) (r) scheme to the 2500 selected data points in Ξ0 and Ξ0 , for the special case of the

4 Anisotropic Kernels for Particle Flow Simulation

71

radial characteristic kernel function, yields the two similar transformation matrices     −3.196 3.195 −3.321 3.249 (p) (p) , and L1 = L0 = −2.373 −2.374 −2.403 −2.457 (p)

(p)

where L0 is the transformation matrix at time t0 = 0 and L1 at time t1 = 1.2. For these two time steps, the eigenvalues are σ1 ≈ 0.05 and σ2 ≈ 0.09 with eigenvectors v1 ≈ (−0.7, 0.7)T and v2 ≈ (−0.7, −0.7)T . This yields two anisotropic charac(p) (p) teristic kernels ϕ0 = ϕ (p) and ϕ1 = ϕ (p) adapted to the particle distribution, see Example 4.4.

L0

L1

4.6.4 Numerical Comparisons In our numerical experiments, we worked with the radial characteristic function φ (r) (r) (cf. Example 4.3) with β = 2 on its fitted data points Ξ0 and Ξ1 and with its (g)

(g)

(g)

(g)

anisotropic version ϕ0 and ϕ1 on its fitted data points Ξ1 and Ξ0 , see Section 4.6.1. To make a fair comparison (on the same data set), we constructed the (p) (p) anisotropic characteristic kernel ϕ0 and ϕ1 , see Section 4.6.3, and chose to inter(r)

(r)

polate on the data points Ξ0 and Ξ1 , respectively. For the purpose of numerical comparison, we also applied interpolation by using thin-plate splines (Section 4.4) (r) on the interpolation points Ξi , i ∈ {0, 1}. But we did not generate an optimized interpolation point set for the thin-plate splines, as there is no psr-greedy method available for conditionally positive definite functions so far. First of all, we see that all four kernels are able to provide sufficiently good reconstructions, cf. Figures 4.5, 4.9. In Figures 4.10, 4.11 and 4.12, the error between the interpolants and the numerical solution of Burgers’ equation is visualized. It can be seen that the approximation quality is similar for the radial characteristic function φ (g) (p) and its anisotropic version ϕi , i ∈ {0, 1}. In contrast, the anisotropic version ϕi , i ∈ {0, 1}, based on the particle distribution leads to a worser approximation quality at both time steps. This observation is also supported through Figure 4.13, where the L∞ -errors are plotted depending on different numbers of interpolation points. (p) Recall that the anisotropic kernels ϕi , i ∈ {0, 1}, are operating on the data sets (r) Ξi , i ∈ {0, 1}, which were selected to achieve good approximation results with the radial characteristic function φ . It cannot be expected that the same approximation quality is achieved with the modified kernel on the same point set. For example, the greedy algorithm selected a large collection of points in the middle of the egg(g) shaped support at t1 = 1.2 for the anisotropic kernel ϕ1 (Figure 4.8), which do not (r) occur in the point set Ξ1 for the radial characteristic function φ (Figure 4.7). Note (p) that the approximation error for ϕi is particularly bad in this area (Figure 4.12). But the large approximation error cannot only be due to a bad choice of interpolation

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K. Albrecht, J. Entzian, A. Iske

Interpolation at t = 0.0

Interpolation at t = 1.2

1.0 0.8 0.6 0.4 0.2 0.0 0.4 0.2

0.0

0.2

1.0 0.8 0.6 0.4 0.2 0.0

0.4 0.2 0.0 0.2 0.4

0.4

0.4 0.2

0.0

0.2

0.4

0.4 0.2 0.0 0.2 0.4

(g)

Fig. 4.9: Interpolation of Burgers’ solution at t0 = 0 with ϕ0 (left) and at t1 = 1.2 with

(g) ϕ1

(right). Error for characteristic kernel using 2500 points at t = 0.0 at t = 1.2 0.00020

0.4

0.00015 0.00010

0.2

0.2

0.0002

0.2

0.0001

0.00000 0.0 0.00005 0.000100.2

0.0000

0.00005

0.0

0.4

0.0001 0.0002

0.00015 0.4 0.00020

0.4 0.4

0.2

0.0

0.2

0.4

0.4

0.2

0.0

0.2

0.4

0.0003

(r)

Fig. 4.10: Interpolation error for radial characteristic kernel φ on data points Ξi , i ∈ {0, 1}, at t0 = 0 (left) and t1 = 1.2 (right). points. However, both anisotropic kernels yield significantly better condition numbers at both time steps, see Figure 4.14. The spectral condition number is reduced by up to two decimal powers. We remark that thin plate spline interpolation is inferior in comparison with interpolation by the radial and the anisotropic characteristic function. This is due to the lack of greedy selection. Nevertheless, except for the L∞ -error of the anistropic (p) kernel ϕi , every L∞ -error has the same order of magnitude at t0 = 0 and only the ∞ L -error for the thin plate spline at t1 = 1.2 is one decimal power poorer than the anisotropic one. But the thin plate spline cannot maintain its good performance re-

4 Anisotropic Kernels for Particle Flow Simulation

73

Error for anisotropic characteristic kernel using 2500 points at t = 0.0 at t = 1.2 0.4

0.0002 0.4

0.2

0.0001 0.2

0.0

0.0000 0.0

0.2

0.0001 0.2 0.0002

0.4 0.4

0.2

0.0

0.2

0.0003 0.0002 0.0001 0.0000 0.0001 0.0002

0.4

0.4

0.0003 0.4

0.2

0.0

0.2

0.4

(g)

Fig. 4.11: Interpolation error for anisotropic characteristic kernel ϕi (g) Ξi , i ∈ {0, 1}, at t0 = 0 (left) and t1 = 1.2 (right).

on data points

Error for anisotropic kernel depending on particle distribution using 2500 points at t = 0.0

at t = 1.2

0.006

0.4

0.4

0.005

0.2

0.002 0.2

0.000

0.0

0.000 0.0

0.2

0.002 0.2 0.004

0.4

0.0060.4

0.004

0.4

0.2

0.0

0.2

0.4

0.005 0.010

0.4

0.2

0.0

0.2

0.4

(p)

Fig. 4.12: Interpolation error for anisotropic characteristic kernel ϕi (r) Ξi , i ∈ {0, 1}, at t0 = 0 (left) and t1 = 1.2 (right).

0.015

on data points

garding stability. At both time steps, its condition number is at least two decimal powers worse than the one of the radial characteristic function and its anistropic versions. We can finally conclude that the anisotropic kernels increase, in contrast to the isotropic kernels, the numerical stability at comparable approximation quality. Hence, anisotropic kernels serve to stabilize numerical reconstructions.

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Errors

at t = 0.0 10

10

radial anisotropic (geometry of problem) anisotropic (particle distribution) thin plate spline

2

10

1

10

2

10

3

at t = 1.2 radial anisotropic (geometry of problem) anisotropic (particle distribution) thin plate spline

3

500

1000

1500

2000

2500

500

1000

1500

2000

2500

Fig. 4.13: Decay of L∞ -error with respect to the number of interpolation points.

Condition numbers

at t = 0.0

107

radial anisotropic (geometry of problem) anisotropic (particle distribution) thin plate spline

106

at t = 1.2

107

106

105

radial anisotropic (geometry of problem) anisotropic (particle distribution) thin plate spline

105

104

104

103 500

1000

1500

2000

2500

103

500

1000

1500

2000

2500

Fig. 4.14: Growth of spectral condition number w.r.t. number of interpolation points.

4.7 Outlook In this contribution, we discussed anisotropic kernels which were constructed by replacing the standard Euclidean norm with an anisotropic norm ∥x∥2A := xT Ax

for x ∈ Rd

4 Anisotropic Kernels for Particle Flow Simulation

75

in the argument of a fixed radial kernel function. For the standard Gaussian kernel and a positive definite diagonal matrix A = diag(ε1 , ..., εd ), this construction yields the kernel 2

e−∥x∥A = e−x

T Ax

d

= ∏ e−εi xi

2

i=1

for x = (x1 , ..., xd )T ∈ Rd ,

see also [9]. Hence, this anisotropic version of the standard Gaussian kernel is given by a product of lower-dimensional kernels, each of them equipped with an own shape parameter εi > 0. This observation gives rise to investigate kernels of the form d

Φ(x) = ∏ Φi (xi ) i=1

for x ∈ Rd ,

where each Φi is positive definite definite (cf. [4]). In this approach, it is possible to choose different kernels for the different components, which yields even more flexibility.

Acknowledgment The authors acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 ”Modeling, Simulation and Optimization of Fluid Dynamic Applications”. We thank Dr. Claus R. Goetz for his assistance with the numerical reference solution of Burgers equation.

References 1. T. Aboiyar, E.H. Georgoulis, and A. Iske. Adaptive ADER methods using kernel-based polyharmonic spline WENO reconstruction. SIAM J. Scient. Computing 32(6), 2010, 3251–3277. 2. R. Askey. Radial Characteristic Functions. Technical report TSR # 1262, University of Wisconsin, Madison, 1973. 3. R. Beatson, O. Davydov and J. Levesley. Error bounds for anisotropic RBF interpolation. Journal of Approximation Theory 162(3), 2010, 512–527 4. A. Berlinet and T. Agnan. Reproducing Kernel Hilbert Spaces in Probability and Statistics. Springer, New York, 2004 5. M.D. Buhmann. Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge, UK, 2003. 6. G. Casciola, D. Lazzaro, L.B. Montefusco, and S. Morigi. Shape preserving surface reconstruction using locally anisotropic radial basis function interpolants. Computers & Mathematics with Applications 51(8), 2006, 1185–1198. 7. J. Duchon. Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In: Constructive Theory of Functions of Several Variables, W. Schempp and K. Zeller (eds.), Springer, Heidelberg, 1977, 85–100.

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8. S. Dutta, M.W. Farthing, E. Perracchione, G. Savant, and M. Putti. A greedy non-intrusive reduced order model for shallow water equations. Journal of Computational Physics 439, 2021. 9. G. Fasshauer. Positive Definite Kernels: Past, Present and Future. Illinois Institute of Technology. http://www.math.iit.edu/fass/PDKernels.pdf 10. G.E. Fasshauer. Meshfree Approximation Methods with Matlab. Interdisciplinary Mathematical Sciences 6, World Scientific Publishing, Singapore, 2007. 11. J. F¨urst and T. Sonar. On meshless collocation approximations of conservation laws: positive schemes and dissipation models. ZAMM - Journal of Applied Mathematics and Mechanics 81, 2001, 403–415. 12. A. Harten, B. Engquist, S. Osher, and S. Chakravarthy. Uniformly high order essentially nonoscillatory schemes, III. J. Comput. Phys. 71, 1987, 231–303. 13. D. Hietel, K. Steiner, and J. Struckmeier. A finite-volume particle method for compressible flows. Math. Mod. Meth. Appl. Sci. 10(9), 2000, 1363–1382. 14. A. Iske. Multiresolution Methods in Scattered Data Modelling. Lecture Notes in Computational Science and Engineering 37, Springer-Verlag, Berlin, 2004. 15. A. Iske. Polyharmonic spline reconstruction in adaptive particle flow simulation. In: Algorithms for Approximation, A. Iske and J. Levesley (eds.), Springer, Berlin, 2007, 83–102. 16. A. Iske. On the construction of kernel-based adaptive particle methods in numerical flow simulation. In: Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation, R. Ansorge, H. Bijl, A. Meister, and T. Sonar (eds.), Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), Springer-Verlag, Berlin, 2013, 197-221. 17. A. Iske. Approximation Theory and Algorithms for Data Analysis. Springer, 2018. 18. M. Junk. Do finite volume methods need a mesh? In: Meshfree Methods for Partial Differential Equations, M. Griebel, M.A. Schweitzer (eds.), Springer, 2003, 223–238. 19. R.L. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, UK, 2002. 20. X. Liu, S. Osher, and T. Chan. Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 1994, 200–212. 21. F.P. Preparata and M.I. Shamos. Computational Geometry. Springer, New York, 1988. 22. V.A. Titarev and E.F. Toro. ADER: arbitrary high order Godunov approach. J. Sci. Comput. 17, 2002, 609–618. 23. E.F. Toro, R.C. Millington, and L.A.M. Nejad. Towards very high order Godunov schemes. In: Godunov Methods (Oxford, 1999), Kluwer/Plenum, New York, 2001, 907–940. 24. E.F. Toro, V.A. Titarev, M. Dumbser, A. Iske, C.R. Goetz, C. Castro, G.I. Montecinos, and R. Dematt`e: The ADER approach for approximating hyperbolic equations to very high accuracy. In: XVIII International Conference on Hyperbolic Problems: Theory, Numerics, Applications. (HYP2022), 2023. 25. H. Wendland. Scattered Data Approximation. Cambridge Univ. Press, Cambridge, UK, 2005.

Chapter 5

An Error-Based Low-Rank Correction for Pressure Schur Complement Preconditioners Rebekka S. Beddig, J¨orn Behrens, Sabine Le Borne, Konrad Simon

Abstract We describe a multiplicative low-rank correction scheme for pressure Schur complement preconditioners to accelerate the iterative solution of the linearized Navier-Stokes equations. The application of interest is a model for buoyancydriven fluid flows described by the Boussinesq approximation which combines the Navier-Stokes equations enhanced with a Coriolis term and a temperature advection-diffusion equation. The update method is based on a low-rank approximation to the error between the identity and the preconditioned Schur complement. Numerical results on a cube and a shell geometry illustrate the action of the lowrank correction on spectra of preconditioned Schur complements using known preconditioning techniques, the least-squares commutator and the SIMPLE method. The computational costs of the update method are also investigated. The goal is to analyze whether such an update method can lead to accelerated solvers. Numerical experiments show that the update technique can reduce iteration counts in some cases but (counter-intutively) may increase iteration counts in other settings.

Rebekka Beddig Hamburg University of Technology, TUHH, Institute of Mathematics, Germany e-mail: [email protected] J¨orn Behrens Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] Sabine Le Borne Hamburg University of Technology, TUHH, Institute of Mathematics, Germany e-mail: [email protected] Konrad Simon Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Iske, T. Rung (eds.), Modeling, Simulation and Optimization of Fluid Dynamic Applications, Lecture Notes in Computational Science and Engineering 148, https://doi.org/10.1007/978-3-031-45158-4_5

77

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5.1 Introduction Buoyancy-driven atmospheric dynamics can be described with the Boussinesq approximation. Here, it is assumed that temperature-dependant density variations are only significant for the dynamics if scaled with gravity. This yields a model which combines the incompressible Navier-Stokes equations (NSEs) and a temperature advection-diffusion equation. The Navier-Stokes equations have the buoyancy as forcing and an additional Coriolis term that describes the deflection due to the earth’s rotation. The simulation time of the Boussinesq model is dominated by the numerical solution of the time-dependent incompressible Navier-Stokes equations. The efficient solution of the NSEs is challenging due to its nonlinearity and asymmetry of the linearized systems. The simulation of the NSEs requires the repeated solution of large-scale linear saddle-point systems of the type      f u A BT . = 0 p B 0 Solving these saddle-point systems with a (Krylov) subspace method within a reasonable amount of time demands efficient preconditioners. In fluid dynamic applications, the block triangular preconditioner is a common choice, exploited, for example, in [8, 9]. This block preconditioner requires approximations to the inverse of the velocity-related block A of the saddle-point matrix and the inverse of the (negative) pressure Schur complement S ··= BA−1 BT . An overview of the numerical solution of saddle-point problems and preconditioners in fluid flow applications is for example given in the book [7]. Well-known approaches are pressurecorrection methods, such as the SIMPLE(R) methods [25, 12], commutator-based preconditioners, such as the least-squares commutator (LSC) [6, 7] or the pressureconvection-diffusion commutator [16], and augmented Lagrangian preconditioners [12, 13] or grad-div preconditioners [5, 19, 10] and nullspace preconditioners [18, 21]. SIMPLE-type pressure-correction methods are based on splitting the coupling of the velocity and the pressure and are popular due to their simplicity. Algebraic commutator-based preconditioners such as the least-squares commutator directly approximate the inverse Schur complement. They avoid the evaluation and inversion of the dense Schur complement but require the solution of Poissontype problems. Augmented Lagrangian preconditioners and grad-div preconditioners simplify the preconditioning of the pressure Schur complement but shift the difficulty to the velocity block. Typical preconditioners for the velocity-related block of the saddle-point matrix are (algebraic or geometric) multigrid methods [24]. If the time step length is small enough, the block A is dominated by the velocity mass matrix. This article is concerned with a low-rank update to improve existing Schur complement preconditioners. Low-rank corrections aim to improve the clustering of eigenvalues of the preconditioned matrix around one. The survey in [4] gives an overview of popular low-rank techniques for preconditioners of linear systems: deflation, tuning, and spectral preconditioners. In [26], the tuning strategy is applied

5 A Low-Rank Correction for Schur Complement Preconditioners

79

to a preconditioner for the linearized Navier-Stokes equations. There, the update is only applied to the preconditioner for the velocity-related matrix block of the system matrix but not to the Schur complement. We discuss a low-rank update that is based on the error between the identity matrix and the preconditioned Schur complement. This idea is based on the paper [27] where it is applied to the Schur complement arising in a domain decomposition method for a power Schur complement low-rank correction preconditioner.

5.1.1 Model We consider the Boussinesq approximation which models buoyancy-driven fluid flows. The fluid dynamics, described by the fluid pressure p and the velocity field u, are modeled by the incompressible NSEs with an additional Coriolis term. The temperature field T that induces the fluid dynamics is modeled by an advectiondiffusion equation. This gives us the following set of non-dimensional partial differential equations   1 1 ∂t u + (u · ∇) u − ∇ · ε(u) + ∇p = ρ(T )g + ω × u, Re Ro ∇ · u = 0, 1 ∂t T − ∇ · ∇T = γ − u · ∇T. Pe U The non-dimensional numbers Re = ρrefµUL , Ro = Lω , Pe = UL κ denote the Reynolds number, the Rossby number, and the P´eclet number, respectively. Here, U, L denote the velocity and the length scale and the physical parameters ρ, µ, κ denote the density, the dynamic viscosity at the bottom reference temperature, and the heat diffusivity of the air. The parameter ω is the norm of the angular velocity of the earth. The expression ε(u) = 12 (∇u + (∇u)T ) is the strain rate tensor, ρ(T ) = 1 − β (T − Tref ) the temperature-dependent density with the heat expansion coefficient β and reference temperature Tref , g the gravity vector, ω the earth’ angular velocity, γ includes external heat sources, and ∂t denotes the partial time derivative. A time discretization with the semi-implicit Euler method leads to the time-discrete equations   un − un−1 1 + (un · ∇)un − ∇ · ε(un ) + ∇pn (5.1) kn Re 1 = ρ(Tn−1 )g + ω × un−1 , (5.2) Ro ∇ · un = 0, Tn − Tn−1 1 − ∇ · ∇Tn = γ − un−1 · ∇Tn−1 kn Pe

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where the indices n, n − 1 denote the solutions at time steps n, n − 1, and kn denotes the nth time step length. This time discretization is easy to implement and has the advantage that we can solve the rotating Navier-Stokes equations and the temperature advection-diffusion equation separately in each time step. In order to form a well-posed problem we need initial conditions u(x, 0) = u0 (x), T (x, 0) = T0 (x),

∀x ∈ Ω , ∀x ∈ Ω ,

on the domain Ω and boundary conditions for u and T for all t > 0. We solve the NSEs on two domains, a unit cube and a shell with an inner radius of 2 and an outer radius of 2.1. Example grids for both geometries are given in Figure 5.1. For both systems, we choose zero initial conditions for the velocity u(x, 0) = 0. For the cube, we choose no-slip boundary conditions for the velocity and Dirichlet conditions for the temperature at the bottom face u|∂ Ωno−slip = 0,

T |∂ Ωno−slip = Tb .

At the top face, we use no-flux conditions for the velocity and the temperature. n · ∇u|∂ Ωno−flux = 0,

n · ∇T |∂ Ωno−flux = 0.

On the sides, we use periodic conditions for the velocity and temperature. Here, ∂ Ω = Γno−slip ∪Γno−flux ∪Γperiodic denotes the boundary of the domain. For the shell, we choose no-flux conditions at the outer boundary Ωouter and no-slip conditions at the inner boundary Ωinner for the velocity. For the temperature, we use Dirichlet conditions at the inner boundary and insulated conditions at the outer boundary. The boundary conditions for the shell read u|∂ Ωinner = 0, T |∂ Ωinner = Tb ,

n · ∇u|∂ Ωouter = 0, n · ∇T |∂ Ωouter = 0.

The NSEs are spatially discretized with the finite element method using TaylorHood elements of the lowest order on hexahedral cells. The temperature equation is spatially discretized with quadratic elements. This choice of finite elements is also used in a model for earth mantle convection that is based on a similar set of equations [17]. For the spatial discretization with the finite element method and linearization with a Picard-type fixed-point iteration we follow the (standard) approach as described in [7] or [15] and obtain linear systems of the form

5 A Low-Rank Correction for Schur Complement Preconditioners

(a) Grid of a cube.

81

(b) Grid of a shell.

Fig. 5.1: Example grids. 

A BT B 0

    u f = , p 0 DT = g

(5.3) (5.4)

where A ∈ Rnu ×nu , B ∈ Rn p ×nu , D ∈ RnT ×nT , and nu , n p , nT are the degrees of freedom of the discrete velocity, pressure, temperature, respectively. We refer to the Picard linearized saddle-point systems as Oseen systems. We additionally consider Stokes systems that are obtained by using an explicit time discretization for the advection term, i.e. replacing the term (un · ∇)un by (un−1 · ∇)un−1 in Equation (5.2). We refer to the obtained symmetric saddle-point systems as Stokes systems. In each time step, the discrete NSEs (5.3) have to be solved repeatedly with varying saddlepoint matrices in the Picard iteration, followed by solving the discrete temperature equation (5.4). The main difficulty lies within the solution of the NSEs whereas the temperature equation can be solved efficiently with the conjugate gradient method preconditioned with the inverse diagonal of the system matrix. Hence, we focus on solvers of the NSEs in the following.

5.1.2 The initial preconditioner We solve the NSEs (5.3) with a flexible generalized minimum residual method (FGMRes) [22] and a right block-triangular preconditioner. The FGMRes method is a flexible variant of the restarted GMRes method that allows using nonlinear or varying preconditioners in the iterations. Right preconditioning with the ideal blocktriangular preconditioner  −1 −1 T −1  A A B S Pideal = , 0 −S−1 yields the preconditioned matrix

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   −1 −1 T −1    I 0 A A B S A BT = B 0 BA−1 I 0 −S−1 which has only eigenvalues of value one. To find a feasible preconditioner we reb−1 , Sb−1 . We obtain the place the exact inverses A−1 and S−1 by approximations A block-triangular preconditioner  −1 −1 T −1  b A b B Sb A P= . −Sb−1 0 b−1 denotes a preconditioner for the upper left block A and Sb−1 denotes a Here, A preconditioner for the pressure Schur complement S ··= BA−1 BT . We use the incomplete LU factorization (ILU(0)) from the Trilinos Ifpack package [23] b−1 since we need a preconditioner for which a transposed application to a vecfor A tor is defined (see Section 5.3). For the pressure Schur complement, we choose the LSC method [6] and the Schur complement approximation of the SIMPLE preconditioner in [25] T −1 SbLSC = BD−1 u B −1 SbSIMPLE


−1

T BDu−1 ADu−1 BT BD−1 u B
−1 = B diag(A)−1 BT


−1

,

(5.5) (5.6)

as initial preconditioners where Du is a diagonal approximation of the velocity mass matrix. We use the diagonal of the velocity mass matrix as an approximation. We

T −1 and B diag(A)−1 BT −1 with incomplete approximate the inverses BD−1 u B Cholesky factorizations. The remainder of this article is structured as follows: In Section 5.2 we derive the low-rank correction scheme. We discuss computational costs in Section 5.3. In Section 5.4 we illustrate the action of the update method on the eigenvalue distribution of the preconditioned Schur complement and convergence of the iterative solver. Section 5.5 concludes and gives an outlook on open questions.

5.2 Error-based low-rank correction The multiplicative low-rank correction technique is based on the error E between the identity and the right preconditioned Schur complement E ··= I − SSb−1

(5.7)

for an initial preconditioner Sb−1 . We rearrange and obtain the alternative formulation for the inverse Schur complement S−1 = Sb−1 (I − E)−1 .

(5.8)

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83

Note that the matrix I − E is invertible. To find an approximation to S−1 , we approximate E using a (best) rank-r approximation E = I − SSb−1 ≈ Ur DrVrT =·· Er and insert it into equation (5.8). We obtain S−1 ≈ Sb−1 (I − Er )−1 assuming that I − Er is invertible. This assumption is reasonable especially for a good low-rank approximation Er since I − Er approximates the invertible matrix I − E. This yields the update scheme −1 = Sb−1 (I − Er )−1 Supd

= Sb−1 (I + Eer )

(5.9)

by applying the Sherman-Morrison-Woodbury formula and defining T Eer ··= Ur D−1 r −Vr Ur


−1

VrT .

(5.10)

5.2.1 Construction of the update The update formula (5.9) requires a rank-r approximation of the error matrix E. We compute the low-rank approximation of E with a randomized singular value decomposition (SVD) [20] using the randomized subspace iteration from [11] to find an orthonormal matrix Q ∈ Rn×r whose range approximates the range of E T . The procedure is as follows: 1. We find the orthonormal matrix Q with a randomized subspace iteration using q power iterations. Furthermore, we orthogonalize the columns after each multiplication with E or E T to improve accuracy and robustness: a. Form M0 = E T G for a random test matrix G ∈ Rn×l , l = r + k, k ≥ 0, and compute its QR factorization M0 = Q0 R0 . b. For j = 1, . . . , q: e j−1 Re j−1 , e j = EQ j−1 and compute its QR factorization M ej = Q • Form M T e j−1 and compute its QR factorization M j = Q j R j . • form M j = E Q c. Set Q = Qq . A small number q of power iterations (q ≤ 3) is usually sufficient. We choose q = 3 and k = 10. The power iterations increase accuracy for slowly decaying singular values. 2. We compute the economy-size SVD of C ··= EQ = Ul Σl VblT and truncate the SVD to rank r. 3. We find the approximate truncated SVD of E as

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R.S. Beddig, J. Behrens, S. Le Borne, K. Simon

E = ET


T

≈ QQT E T


T

= QCT


T

= CQT

= Ur ΣrVbrT QT = Ur ΣrVrT . Note that we need the transposed application of E to a vector in step 1. This is further discussed in Section 5.3.

5.2.2 Analysis of the remainder In the following, we will analyze how the update acts on the preconditioned Schur −1 complement. Our goal is to reduce the error Eupd ··= I − SSbupd between the identity and the updated preconditioned Schur complement. To derive an expression for Eupd we need the approximation error R ··= E − Er ,

(5.11)

introduced by the low-rank approximation. We rearrange equation (5.10) to find an alternative formulation for Er and obtain

⇔

(I − Er )−1 = I + Eer ,

I = I − Er + (I − Er ) Eer , Er = (I − Er ) Eer .

⇔

(5.12)

Now, we discuss how the error E (5.7) is improved by the update. We obtain   (5.10) I − SSb−1 = I − SSb−1 (I − Er )−1 = I − SSb−1 I + Eer Eupd = upd

(5.7)

=

(5.11),(5.12)

=

E − (I − E) Eer = E − (I − Er + Er − E)Eer E − Er + REer

using the definitions (5.7), (5.10), (5.11), and equation (5.12). Hence, the error after applying the update is the sum of the approximation error E − Er and the additional term REer . This means that a good low-rank approximation to E does not guarantee to improve the initial preconditioner. Figure 5.2 shows the approximation error E − Er and the new error Eupd for two initial preconditioners, LSC and SIMPLE, on a cube and on a shell. The errors are computed in MATLAB with direct solvers for all arising inverses. We observe that Eupd is only slightly larger than E − Er for the LSC preconditioner but for the SIMPLE-type preconditioner there is a significant difference between E − Er and Eupd . Increasing the rank from r = 30 to r = 100 does not further reduce the new error Eupd .

5 A Low-Rank Correction for Schur Complement Preconditioners

2

kE − Er k2 kEupd k2

1.5

85

kE − Er k2 kEupd k2

4 3

1

2

0.5

1

0

20

40 60 rank r

80

100

(a) Values for the LSC preconditioner on a shell (nu = 10422, n p = 490).

kE − Er k2 kEupd k2

6 4

0

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40 60 rank r

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(b) Values for the SIMPLE-type preconditioner on a shell (nu = 10422, n p = 490).

kE − Er k2 kEupd k2

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40 60 rank r

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(c) Values for the LSC preconditioner on a cube (nu = 14739, n p = 729).

0

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40 60 rank r

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(d) Values for the SIMPLE-type preconditioner on a cube (nu = 14739, n p = 729).

Fig. 5.2: The approximation error ∥E − Er ∥2 and the error ∥Eupd ∥2 .

5.2.3 Repeated updates Another approach to (hopefully) further improve an updated preconditioner is to apply the update technique to the updated preconditioner. We obtain the new preconditioner −1 −1 Sbupd,2 = Sbupd (I − Er,2 )−1 = Sb−1 (I − Er )−1 (I − Er,2 )−1 −1 where Er,2 is a low-rank approximation of Eupd = I − SSbupd . The repeated update allows us to adapt the preconditioner in a flexible way. Furthermore, setting up two updates of rank r instead of one update of rank 2r can be cheaper depending on the parameters in the randomized SVD and the update rank r (see Section 5.3).

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5.3 Complexity In the following, we outline relevant computational costs for the setup and application of the low-rank update. The preliminary costs for the first update come mainly from the computation of the update vectors. To compute the randomized SVD, we need to draw a n p × l random matrix, compute matrix-vector products with E and E T , orthogonalize l vectors after each multiplication, and compute an economic SVD. Filling the random test matrix needs NfillG = O(n p l) operations. The matrix-vector products with E and E T require the sum of two vectors of size n p , and multiplication with the preconditioner Sˆ−1 and with the Schur complement S = BA−1 BT . Hence, we have NmultE = NmultS + NmultSb−1 + O(n p ). We use a modified Gram-Schmidt method with reorthogonalization if needed to orthogonalize the resulting vectors. This needs North = O(4n p l 2 ) operations if we reorthogonalize in each step. Computing the economic SVD requires Nsvd = O(n p l 2 ) operations. So, the computation of the update vectors requires Nsetup = NfillG + 2(q + 1)lNmultE + 2(q + 1)North + Nsvd
= O(n p l) + 2(q + 1)l NmultS + NmultSb−1 + O(n p )

+2(q + 1)O(4n p l 2 ) + O(n p l 2 )

operations. The dominant costs are the matrix-vector products with the error matrix E and its transpose. The most expensive part in computing   Ev = I − SSb−1 v = v − BA−1 BT Sb−1 v or

T  E T v = I − SSb−1 v = v − Sb−T BA−T BT v

is the application of A−1 . We construct the update by approximating the inverses b−1 )T ≈ A−T . So, we need to know b−1 ≈ A−1 and (A A−1 , A−T by an approximation A −1 b acts on a vector but also how its transposed acts on a vector. Hence, we how A cannot use a multigrid solver. Instead, we use an incomplete LU factorization. An advantage is that the incomplete LU factorization is more robust if we use larger time steps. In the setup of the second update, we can reuse the product of the preconditioned Schur complement with the initial random matrix. To compute the first and second updates, we need   E T G = I − Sb−T ST G = G − Sb−T ST G = G − X,   T −T T G = I − Sbupd S G = G − (I − Er )−1 Sb−T ST G Eupd = G − (I − Er )−1 X,

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with X ··= Sb−T ST G. We can precompute and store X needed to compute E T G in step 1 of the randomized SVD. Thus, we save l applications of Sb−T and ST to a vector in the setup of the second update compared to the setup of the first update. So, we need 2(q + 1)l + (2(q + 1)l − l) = (4q + 3)l = (4q + 3)(r + k) applications of Sb−T and ST to a vector to set up two updates of rank r. In the setup of one update of rank 2r, we need 2(q + 1)(2r + k) applications of Sb−T and ST . Hence, the setup of two updates of rank r is cheaper than the setup of one update of rank 2r if (4q + 3)(r + k) < 2(q + 1)(2r + k).

(5.13)

We rearrange the inequality (5.13) and obtain that the setup of two updates of rank r is cheaper than the setup of one update of rank 2r if r > (2q + 1)k. Hence, for our choice of parameters (q = 3, k = 10) the setup of two updates of rank r is only cheaper for r > 70. Now, we briefly discuss the application cost of the update technique. The application of one update requires two matrix-vector products with thin rectangular matrices, the application of an inverse of size r ×
r, and the sum of two vectors of T size n p . We precompute the matrix D−1 r −Vr Ur and its LU factorization. Hence, the application of the update is cheap.

5.4 Numerical results The model and solvers are implemented in C++ using the deal.II 9.3.3 library [1, 2] for the finite element discretization and iterative solvers and Trilinos 12.14.1 [14] for the linear algebra. We set the length scale to 1.0m and the velocity scale to 0.01ms−1 . With the density ρref = 1.29 kg m−3 , the dynamic viscosity µ = 1.82e-5 kg m−1 s−1 , the earth’ angular velocity ω = 7.272205e-5 s−1 , and the thermal conductivity κ = 2.62e-2 W m−1 K−1 , we obtain a Reynolds number of 708.8, a Rossby number of 137.5, and a P´eclet number of 494.8. Furthermore, we obtain the buoyancy forcing as defined in Section 5.1.1 with a heat expansion coefficient β of 0.003661 K−1 , a reference temperature Tref of 2 ◦ C, and a gravity constant of 9.81 m s−2 . We use the system matrices obtained after 5 Picard iterations in the first time step to investigate spectra and convergence results. We also consider the numerical solu-

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tion of Stokes systems obtained with an explicit time discretization of the advection (u · ∇) u. We solve the block system with FGMRes with a restart every 30 iterations. We increase the restart length to 50 if we do not achieve convergence within two restarts. We stop when the relative residual drops below 10−8 . The Poisson-type problems
−1
−1 involving BDu−1 BT are approximated with an incomand B diag(A)−1 BT plete Cholesky factorization (ic(0)). In the following, we analyze how the update technique acts on the eigenvalues of the preconditioned Schur complement. The numerical results are shown for one test run. Note that the randomness in the setup of the update can lead to varying results.

5.4.1 Action on the eigenvalues In this subsection, we analyze how the update method acts on the spectrum of the error matrix E. The eigenvalues are evaluated in MATLAB using direct solvers for all arising inverses in the evaluation of the error matrix and the initial preconditioners. Figure 5.3 shows eigenvalue distributions of E for one and two updates −1 −1 (5.6), on (5.5) and SbSIMPLE of rank r. We use the two initial preconditioners, SbLSC the cube (nu = 14739, n p = 729, kn = 0.005) and the shell geometry (nu = 10422, n p = 490, kn = 0.0005). We observe that the update improves the clustering of eigenvalues around one for the LSC preconditioner for both geometries. For the SIMPLErelated preconditioner, small eigenvalues of the preconditioned Schur complement are shifted towards one, but the largest real parts even increase. Besides, the update technique is not symmetric. So, the imaginary parts of some eigenvalues increase as can be seen for the SIMPLE-type preconditioner on the Oseen system on the shell and the LSC preconditioner on both Stokes systems.

5.4.2 Convergence This subsection is concerned with the effect of the update on the convergence behavior of the iterative solver. Table 5.1 shows the needed number of FGMRes iterations for different ranks for the Oseen and Stokes systems. We observe that the iteration counts are decreased for the LSC preconditioner. Here, an update with rank r = 15 already decreases iteration counts significantly. But for the SIMPLE-related preconditioner, iteration counts increase for the shell geometry. Furthermore, for the Oseen system for the cube geometry iteration counts increase for an update rank r from 15 to 45. Table 5.2 shows iteration counts, solver and setup times for the Oseen systems for the two geometries, each with two different problem sizes resulting from different refinement levels. The setup times include the computation of the low-rank approximation of E ≈ Ur DrVrT using the randomized SVD and the precomputation

−0.1

−3.0

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−0.1

−0.2 −1.0−0.5 0.0 0.5 1.0 Re(λ)

(a) Spectra for the Oseen system.

Im(λ)

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·10−3

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Im(λ)

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Im(λ)

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·10−3

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Im(λ)

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89

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Im(λ)

5 A Low-Rank Correction for Schur Complement Preconditioners

·10−3

1.0 0.0 −1.0

−2.0

−0.4

(b) Spectra for the Stokes system.

Fig. 5.3: Spectra of E without an update (left column), after one update (middle column), and after two updates (right column) of r = 30. The first row shows spectra −1 −1 for SˆLSC on the shell, the second row spectra for SˆSIMPLE on the shell, the third row −1 −1 spectra for SˆLSC on the cube, and the fourth row spectra for SˆSIMPLE on the cube. T of the factorization of D−1 r −Vr Ur . Table 5.3 shows results obtained for the Stokes systems. We compare results for the LSC and SIMPLE-type initial preconditioner. The CPU times are obtained on an Intel Xeon Gold 6240 processor with 2.60 GHz and 18 cores and 32 GB RAM. The setup times for the updated LSC preconditioner are approximately twice as large as the setup times for the updated SIMPLE-type preconditioner since we need to (approximately) solve twice as many Poisson-type problems. For some systems, we did not achieve convergence within 1000 iterations. This is marked with ’–’. We observe that the update reduces iteration counts and solver times for the LSC preconditioner. As discussed in the previous paragraph,

Table 5.1: Number of FGMRes iterations for the Oseen and the Stokes systems, solved on a cube (nu = 107811, n p = 4913, kn = 5e − 3) and on a shell (nu = 78438, n p = 3474, kn = 5e − 4) with one update for different ranks. Oseen Stokes r r 0 15 30 45 60 0 15 30 45 60 shell, LSC 24 16 15 14 13 21 11 11 10 10 SIMPLE 21 36 38 37 29 17 20 21 21 21 cube, LSC 83 62 52 32 28 58 19 15 14 14 SIMPLE 70 96 84 76 65 27 19 18 17 17

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we observe that the update may deteriorate the SIMPLE-related preconditioner for the smaller systems. But for the larger systems, the update decreases the needed number of iterations in all test cases. Due to high setup costs, the total CPU time, including setup and solver times, is higher than the solver time without an update in most of the shown cases. Table 5.2: Number of needed FGMRes iterations and solver times (seconds) for the Oseen systems. nu shell, kn = 5e−4 shell, kn = 2.5e−4 cube, kn = 5e−3 cube, kn = 2.5e−3

np

LSC 78438 3474 SIMPLE 78438 3474 LSC 608454 26146 SIMPLE 608454 26146 LSC 107811 4913 SIMPLE 107811 4913 LSC 823875 35937 SIMPLE 823875 35937

no update 1 update, r = 30 2 updates, r = 30 1 update, r = 60 solve iters. setup solve iters. setup solve iters. setup solve iters. 3.15 24 21.77 2.01 15 40.74 1.77 13 38.19 1.76 13 1.88 21 10.79 3.38 38 20.23 2.12 24 18.98 2.55 29 – – 180.85 94.74 87 336.56 52.65 48 313.66 43.95 40 50.48 68 90.76 40.09 54 169.22 17.62 24 160.31 25.64 34 15.07 83 30.58 9.63 52 57.56 5.17 28 54.39 5.18 28 9.01 70 15.18 10.41 84 28.49 8.95 72 26.70 8.06 65 – – 247.81 96.82 66 460.01 78.40 52 436.80 81.87 55 82.66 81 125.61 75.52 75 233.06 69.18 68 221.93 65.78 65

Table 5.3: Number of needed FGMRes iterations and solver times (seconds) for the Stokes systems.

shell, kn = 5e−4 shell, kn = 2.5e−4 cube, kn = 5e−3 cube, kn = 2.5e−3

nu np LSC 78438 3474 SIMPLE 78438 3474 LSC 608454 26146 SIMPLE 608454 26146 LSC 107811 4913 SIMPLE 107811 4913 LSC 823875 35937 SIMPLE 823875 35937

no update 1 update, r = 30 solve iters. setup solve iters. 2.84 21 21.74 1.51 11 1.53 17 10.71 1.85 21 – – 182.37 97.42 89 48.10 65 90.42 33.06 44 10.88 58 30.97 2.78 15 3.39 27 15.10 2.23 18 712.86 485 248.15 87.49 59 63.05 62 123.43 24.00 24

2 updates, r = 30 1 update, r = 60 setup solve iters. setup solve iters. 40.86 1.39 10 38.06 1.39 10 20.10 1.35 15 18.86 1.86 21 340.25 54.49 49 318.16 54.97 50 169.02 15.45 21 159.22 21.23 29 58.02 2.52 13 54.54 2.70 14 28.23 1.76 14 26.59 2.12 17 464.45 40.10 27 438.01 40.11 27 231.49 20.12 20 217.12 21.12 21

5.5 Outlook Intuitively, we would expect that the derived low-rank correction should improve any given preconditioner. However, we have observed that the update does not guarantee to improve the clustering of eigenvalues of the preconditioned Schur complement around one and thus does not guarantee to improve the initial preconditioner. So, the low-rank correction cannot be used as a blackbox method. However, we hope that the update yields similar results for similar settings such that observations from numerical experiments can also be applied to other systems. Nevertheless, the

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update method can improve given preconditioners concerning iteration counts and solver times but cannot yet reduce the total CPU times for most systems. In [3], we show that an update based on cheaper low-rank approximations including a scaling parameter reduces the needed number of FGMRes iterations as well. The current update technique is not symmetric. A symmetry-preserving update technique is under investigation that can be used for preconditioning the Stokes equations.

Data Availability The code used to produce the results in this article is provided in a repository at https://github.com/rbeddig/LowRankUpdates/tree/YRMCSE22-proceedings.

Acknowledgment The authors acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 ”Modeling, Simulation and Optimization of Fluid Dynamic Applications”.

References 1. D. Arndt, W. Bangerth, B. Blais, M. Fehling, R. Gassm¨oller, T. Heister, L. Heltai, U. K¨ocher, M. Kronbichler, M. Maier, P. Munch, J.-P. Pelteret, S. Proell, K. Simon, B. Turcksin, D. Wells, and J. Zhang. The deal.II library, version 9.3. J. Numer. Math. 29(3), 171–186, 2021. 2. D. Arndt, W. Bangerth, D. Davydov, T. Heister, L. Heltai, M. Kronbichler, M. Maier, J.-P. Pelteret, B. Turcksin, and D. Wells. The deal.II finite element library: Design, features, and insights. Comput. Math. Appl. 81, 407–422, 2021. 3. R.S. Beddig, J. Behrens, S. Le Borne. A low-rank update for relaxed Schur complement preconditioners in fluid flow problems. Numer. Algorithms, Preprint available at Research Square [https://doi.org/10.21203/rs.3.rs-2492974/v1]. 4. L. Bergamaschi. A survey of low-rank updates of preconditioners for sequences of symmetric linear systems. Algorithms 13(4), 2020. 5. S. B¨orm and S. Le Borne. H -LU factorization in preconditioners for augmented Lagrangian and grad-div stabilized saddle point systems. Internat. J. Numer. Methods Fluids 68, 83–98, 2010. 6. H.C. Elman, V. Howle, J. Shadid, R. Shuttleworth, and R. Tuminaro. Block preconditioners based on approximate commutators. SIAM J. Sci. Comput. 27, 1651–1668, 2006. 7. H.C. Elman, D.J. Silvester, and A.J. Wathen. Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics. Oxford University Press, 2014. 8. H.C. Elman. Preconditioning for the steady-state Navier-Stokes equations with low viscosity. SIAM J. Sci. Comput. 20(4),1299–1316, 1999. 9. H.C. Elman. Preconditioners for saddle point problems arising in computational fluid dynamics. Appl. Numer. Math. 43(1), 75–89, 2002. 10. J.A. Fiordilino, W. Layton, and Y. Rong. An efficient and modular grad–div stabilization. Comput. Methods Appl. Mech. Engrg. 335, 327–346, 2018.

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11. N. Halko, P.G. Martinsson, and J.A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217–288, 2011. 12. X. He and C. Vuik. Comparison of some preconditioners for the incompressible Navier-Stokes equations. Numer. Math., Theory Methods Appl. 9(2), 239–261, 2016. 13. X. He and C. Vuik. Efficient and robust schur complement approximations in the augmented lagrangian preconditioner for the incompressible laminar flows. J. Comput. Phys. 408, 109286, 2020. 14. M.A. Heroux, R.A. Bartlett, V.E. Howle, R.J. Hoekstra, J.J. Hu, T.G. Kolda, R.B. Lehoucq, K.R. Long, R.P. Pawlowski, E.T. Phipps, A.G. Salinger, H.K. Thornquist, R.S. Tuminaro, J.M. Willenbring, A. Williams, and K.S. Stanley. An overview of the Trilinos project. ACM Trans. Math. Softw. 31(3), 397–423, 2005. 15. V. John. Finite Element Methods for Incompressible Flow Problems. Springer Ser. Comput. Math., 2016. 16. D. Kay, D. Loghin, and A. Wathen. A preconditioner for the steady-state Navier–Stokes equations. SIAM J. Sci. Comput. 24, 237–256, 2002. 17. M. Kronbichler, T. Heister, and W. Bangerth. High accuracy mantle convection simulation through modern numerical methods. Geophys. J. Int. 191, 12–29, 2012. 18. S. Le Borne. Preconditioned nullspace method for the two-dimensional Oseen problem. SIAM J. Sci. Comput. 31, 2494–2509, 2009. 19. S. Le Borne and L. Rebholz. Preconditioning sparse grad-div/Augmented Lagrangian stabilized saddle point systems. Comput. Vis. Sci. 16, 259–269, 2015. 20. P.G. Martinsson, V. Rokhlin, and M. Tygert. A randomized algorithm for the decomposition of matrices. Appl. Comput. Harmon. Anal. 30(1):47–68, 2011. 21. J. Pestana and T. Rees. Null-space preconditioners for saddle point systems. SIAM J. Matrix Anal. Appl. 37(3): 1103—1128, 2016. 22. Y. Saad. A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2):461–469, 1993. 23. M. Sala and M. Heroux. Robust algebraic preconditioners with IFPACK 3.0. Technical Report SAND-0662, Sandia National Laboratories, 2005. 24. U. Trottenberg, C. Oosterlee, and A. Sch¨uller. Multigrid. Elsevier Academic Press, London, 2001. 25. C. Vuik and A. Saghir. The Krylov accelerated SIMPLE(R) method for incompressible flow. Reports of the Department of Applied Mathematical Analysis, 02-01, 2002. 26. F. Zanetti and L. Bergamaschi. Scalable block preconditioners for linearized Navier-Stokes equations at high Reynolds number. Algorithms 13(8), 2020. 27. Q. Zheng, Y. Xi, and Y. Saad. A power Schur complement low-rank correction preconditioner for general sparse linear systems. SIAM J. Matrix Anal. Appl. 42, 659–682, 2021.

Chapter 6

Radon-based Image Reconstruction for MPI using a continuously rotating FFL Stephanie Blanke, Christina Brandt

Abstract Magnetic particle imaging is a relatively new tracer-based medical imaging technique exploiting the non-linear magnetization response of magnetic nanoparticles to changing magnetic fields. If the data are generated by using a fieldfree line, the sampling geometry resembles the one in computerized tomography. Indeed, for an ideal field-free line rotating only in between measurements it was shown that the signal equation can be written as a convolution with the Radon transform of the particle concentration. In this work, we regard a continuously rotating field-free line and extend the forward operator accordingly. We obtain a similar result for the relation to the Radon data but with two additive terms resulting from the additional time-dependencies in the forward model. We jointly reconstruct particle concentration and corresponding Radon data by means of total variation regularization yielding promising results for synthetic data.

6.1 Introduction Reliable and fast medical imaging techniques are indispensable for diagnostics in clinical everyday life. One promising example is magnetic particle imaging (MPI) invented by Gleich and Weizenecker [8]. MPI is a tracer-based imaging modality aiming for the reconstruction of the spatial distribution of magnetic nanoparticles injected into the patient’s body. For data generation changing magnetic fields are applied to the field of view (FOV). These fields are composed of a selection field HS Stephanie Blanke Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] Christina Brandt Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Iske, T. Rung (eds.), Modeling, Simulation and Optimization of Fluid Dynamic Applications, Lecture Notes in Computational Science and Engineering 148, https://doi.org/10.1007/978-3-031-45158-4_6

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featuring a low-field volume (LFV), and a drive field HD moving the LFV through the FOV. The particles’ corresponding non-linear magnetization response is measured in terms of induced voltages via receive coils. Note that only particles within the LFV contribute to the signal as all other particles stay in magnetic saturation, guaranteeing spatial information to be encoded in the data. The LFV takes different shapes depending on the specific scanner implementation. Most commonly, a fieldfree point (FFP) is used for spatial encoding. However, alternatively a field-free line (FFL) can be used as it was first suggested by [27]. For a detailed introduction to MPI, we recommend to consult [17]. For the specific case of an FFL scanner we further refer to [4] and [7]. For the remainder, we restrict ourselves towards the FFL encoding scheme. The first FFL imaging systems were developed by [9] and [3]. For further steps and breakthroughs in MPI history we refer to [19]. In [20] the open-source project OS-MPI is presented. Thereby, information about a small-bore FFL imager is shared. Advantages of using an FFL scanner lie in a possible increase in sensitivity [27] as more particles are contributing to the signal. Further, in [15] it is stated that using an FFL may lead to a less ill-posed problem compared to the FFP scanner. During data acquisition the FFL is rotated and translated through the FOV resulting in scanning geometries resembling those in computerized tomography (CT). In CT the intensity loss of X-rays traversing the object under investigation is measured. Having access to data for a suitable choice of different positions for radiation source and detector panel, the attenuation coefficient of the specimen can be reconstructed. The forward operator is given by the Radon transform mapping a function to the set of its line integrals. See e.g. [22] for more information concerning CT. Since only particles in close vicinity to the FFL contribute to the induced voltage, it stands to reason that the MPI signal equation is linked to the Radon transform of the particle concentration. Indeed, it was shown in [18] that MPI data can be traced back to the Radon transform of the particle concentration. In [5] a similar result was derived using their newly developed 3D model applicable to magnetic fields approximately parallel to their velocity field. Hence, for concentration reconstruction, results and methods from the well-known imaging technique CT are accessible. Because of the aforementioned analogies, the idea aroused to combine the MPI with a CT scanner, thus combining quantitative information about the tracer distribution with structural information about the tissue itself. The first hybrid MPI-CT scanner was proposed in [25]. For the derivation of the relation between Radon and MPI data in [18], the FFL is assumed to be rotated in between measurements. In this work, we will use a slightly different scanning geometry, allowing the FFL to be rotated simultaneously with its translation as presented in the initial publication with respect to MPI using a field-free line [27]. In [5] and [18] it was proposed that the derived relation is still applicable in case that the rotation is sufficiently slow compared to the translation speed. In the remainder sections, we will investigate the supposition in some more details. Throughout the article we will apply the Langevin theory to model the forward operator. This corresponds to assuming the particles to be in thermal equilibrium [17].

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While being not a proper description of the particle dynamics, it is still the state of the art approach when using a model-based formulation of the signal equation. Alternatively, commonly a measurement-based approach is applied, where, before the actual data generation takes place, measurements are taken for a small probe being moved trough the FOV called calibration process [17]. However, in recent years the problem of finding more accurate modeling methods aroused further interest, cf. ([16], [13], [26]) to name a few examples. In [13] e.g. the modeling task was traced back to an inverse parameter identification problem. In [14] a survey regarding mathematical modeling of the signal chain is given. Utilizing methods based on total variation (TV) has a broad range of applications and is especially suitable for piecewise constant functions. See e.g. [6] for an introduction to TV regularization. Also in the setting of MPI TV has already been applied ([23], [28], [2]). The aforementioned works regard an FFP scanner. In contrast, we consider the FFL encoding scheme enabling a connection between the MPI forward operator and the Radon transform. In [12] the system-based approach combined with TV regularization has been compared with the projection-based approach using the Radon transform for an ideal FFL setup. For the latter, they first reconstruct the according Radon data and then determine the particle concentration afterwards using the inverse Radon transform. Inspired by [24] we propose an image reconstruction approach, which jointly solves for the particle concentration as well as its corresponding Radon data by combining TV regularization with the projectionbased formulation of the MPI signal equation. The article is organized as follows: In Section 6.2 we fix notation, introduce the MPI forward model for an FFL scanner, and review the link between MPI data and Radon transform of the particle concentration developed in [18]. Section 6.3 extends this relation to the setting of simultaneous line rotation. In Section 6.4 we propose a joint reconstruction of particle concentration and corresponding Radon data applying TV regularization. We close with numerical examples for synthetic data in Section 6.5.

6.2 Field-free Line Magnetic Particle Imaging According to [17] the particle concentration c : R3 → R0+ := R+ ∪ {0} can be linked to the voltage signal ul : R+ → R, l ∈ {1, . . . , L} induced in the l-th receive coil via ul (t) = −µ0

Z R3

c (r)

∂ m (r,t) · pl (r) d r. ∂t

(6.1)

Thereby, µ0 denotes the magnetic permeability, m : R3 × R+ → R3 the mean magnetic moment, and pl : R3 → R3 the receive coil sensitivities. Note that we omit signal filtering. In practice, the direct feed-through of the excitation signal needs to ∂ m (·,t) · pl (·) are L2 funcbe removed. In the following, we will assume that c and ∂t

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tions, i.e. square-integrable. Thus, (6.1) is well-defined. When the field-free region of the MPI implementation is a straight line and the magnetic fields are constant along lines parallel to this FFL, it is called ideal FFL scanner. The attribute ideal is added as in practice we are confronted with field imperfections leading to deformed LFVs ([4], [5]). In the following, we consider an ideal FFL scanner generating data for different positions and directions by moving the FFL through the xy-plane. For convenience, we will assume the particle concentration to be contained within this plane and regard the problem as two dimensional c : R2 → R+ 0 (cf. [15] for more details). Further, let supp (c) ⊂ BR ⊂ R2 with BR denoting the circle of radius R > 0 around the origin. Corresponding to [18] we model the magnetic fields as
H(r, ϕ,t) = HS (r, ϕ) + HD (ϕ,t) = −G r · eϕ + AΛ (t) eϕ with eϕ := (− sin ϕ, cos ϕ)T . Here, G is the gradient strength determining the width of the LFV, A denotes the drive peak amplitude, and Λ is a periodic excitation function usually chosen to be sinusoidal. A simple computation shows that the FFL, which is orthogonal to eϕ and with displacement st := GA Λ (t) to the origin, builds the center of the LFV
 FFL eϕ , st := r ∈ R2 : r · eϕ = st . The corresponding geometry is depicted in Figure 6.1. Looking at this visualization, by replacing the FFL with X-rays, similarities between the scanning process for MPI using an FFL scanner and the well-known medical imaging technique CT become obvious. Note that, as it was already mentioned in [18], when comparing with the notation in common CT literature ϕ needs to be shifted by π2 . This is because in CT the angle is usually measured from the axis to the orthogonal vector eϕ instead to the X-ray itself.

s

Fig. 6.1: Visualization of the FFL orthogonal to eϕ and with displacement s to the origin. Following [18] the mean magnetic moment can be rewritten using the Langevin model of paramagnetism

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6 Radon-based Image Reconstruction for MPI using a continuously rotating FFL

m (r, ϕ,t) = m (∥H (r, ϕ,t)∥)

H (r, ϕ,t) ∥H (r, ϕ,t)∥

  with m (H) = mL µkB0 m T H denoting the modulus of the mean magnetic moment. Further, m is the magnetic moment of a single particle, kB is the Boltzmann constant, T the particle temperature, and L : R → [−1, 1] the Langevin function defined as ( coth (λ ) − λ1 , λ = ̸ 0, L (λ ) := 0 , λ = 0. Thus, m (−H) = −m (H) and the signal equation (6.1) can be written as ul (ϕ,t) = −µ0

Z BR

c(r)


∂ m −G r · eϕ + AΛ (t) eϕ · pl (r) d r. ∂t

(6.2)

Different scanning geometries exist. We regard sequential line rotation (Figure 6.2a) i.e. the FFL is sequentially translated through the FOV and rotated in between measurements, and simultaneous line rotation (Figure 6.2b) i.e. the FFL is rotated simultaneously to the translation (cf. [27], [18]). In the case of simultaneous line rotation we need to replace ϕ via ϕt := 2π frott with line rotation frequency frot > 0 everywhere ϕ occurs [5].

FFL rotation

FFL rotation

FFL FFL (a) Sequential line rotation

(b) Simultaneous line rotation

Fig. 6.2: Different scanning geometries: (a) Sequential translation and rotation of the FFL, (b) Simultaneous translation and rotation of the FFL. We now review the definition of the Radon transform. Let S1 be the unit sphere
in 1 × R the unit cylinder in R3 . The Radon transform R : L B , R+ → R2 and Z := S R 2 0
L2 Z, R+ 0 of the particle concentration c is given as
Z Rc eϕ , s =

BR


c (r) δ r · eϕ − s d r.

(6.3)

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For sequential line rotation a link between MPI data and the Radon transform of the particle concentration was shown (see Theorem 6.1 below). Theorem 6.1. [18] Given spatially homogeneous receive coil sensitivities, sequential line rotation, and ideal magnetic fields, the signal equation (6.2) can be written as   
 A ul (ϕ,t) = −µ0 AΛ ′ (t) eϕ · pl m′ (G ·) ∗ Rc eϕ , · Λ (t) . (6.4) G For simultaneous line rotation it is reasonably proposed that (6.4) holds approximately true as long as the FFL rotation is sufficiently slow compared to the translation ([18], [5]). The aim of this article is to investigate the setting of simultaneous line rotation in more detail. To this end, we define the MPI forward operator for the special case of ideal FFL scanner.
+ Definition 6.1. Let AlFFL : L2 BR , R+ 0 → L2 (R , R) be defined as
∂ m −G r · eϕt + AΛ (t) eϕt · pl (r) d r. (6.5) ∂t BR

FFL : L B , R+ → L R+ , RL Then, for ideal magnetic fields, the mapping A R 2 2 0
with A FFL c (t) = AlFFL c (t) l=1,...,L is the forward operator for field-free line magnetic particle imaging. AlFFL c (t) := −µ0

Z

c(r)

Remark 6.1. Note that, by choosing ϕt to be piecewise constant, sequential line rotation is also contained in the previous definition. For concentration reconstruction we need to solve the linear ill-posed inverse problem A FFL c = u with measured data u = (ul )l=1,...,L and forward operator A FFL .

6.3 Relation of the MPI Forward Operator to the Radon Transform In the following, we examine the MPI forward operator in the ideal FFL setting for simultaneous line rotation. Therefore, let       cos ϕ − sin ϕ − sin ϕ cos ϕ ϕ ⊥ R = , eϕ = , eϕ = − . sin ϕ cos ϕ cos ϕ sin ϕ Further, we define a weighted Radon transform
Z e eϕ , s := Rc

BR


c (r) δ r · eϕ − s r · e⊥ ϕ d r.

(6.6)

6 Radon-based Image Reconstruction for MPI using a continuously rotating FFL

99

The next theorem states a link between the MPI forward operator (6.5), the Radon transform (6.3), and the weighted Radon transform (6.6). Theorem 6.2. Given spatially homogeneous receive coil sensitivities, simultaneous line rotation, and ideal magnetic fields, the MPI forward operator with respect to the l-th receive coil can be written as AlFFL = K1,l ◦ R + K2,l ◦ Re + K3,l ◦ R with convolution operators Ki,l : L2 (Z, R) → L2 (R+ , R) for i = 1, 2, 3 and l ∈ {1, . . . , L}
K1,l f (t) = −µ0 AΛ ′ (t) eϕt · pl m′ (G ·) ∗ f eϕt , · (st ) ,
K2,l f (t) = µ0 Gϕt′ eϕt · pl m′ (G ·) ∗ f eϕt , · (st ) ,
K3,l f (t) = −µ0 ϕt′ e⊥ ϕt · pl m (G ·) ∗ f eϕt , · (st ) . Proof. Computing the derivative in (6.5) and assuming spatially homogeneous receive coil sensitivities, we obtain Z

AlFFL c (t) = − µ0 AΛ ′ (t) eϕt · pl + µ0 Gϕt′ eϕt · pl −

µ0 ϕt′ e⊥ ϕt

· pl

BR

Z


c(r)m′ −G r · eϕt + AΛ (t) r · e⊥ ϕt d r

BR

Z


c(r)m′ −G r · eϕt + AΛ (t) d r


c(r) m −G r · eϕt + AΛ (t) d r ,

BR

where we used (eϕt )′ = ϕt′ e⊥ ϕt . We proceed similar to [18] and rotate the coordinate ′ −ϕ t system r := R r such that the x-axis gets parallel to the FFL. Using  ′ ϕt ′ ϕt v = s′ eϕt − v′ e⊥ R r =R ϕt s′ yields o
 n ′ FFL eϕt , s′ = r ∈ R2 : r · eϕt = s′ = s′ eϕt − v′ e⊥ ϕt : v ∈ R   ′  ϕt v ′ = R : v ∈R s′ and with R−ϕt BR = BR we obtain AlFFL c (t) = − µ0 AΛ ′ (t) eϕt · pl +

µ0 Gϕt′ eϕt

−

µ0 ϕt′ e⊥ ϕt

· pl

· pl

Z

Z R

R

Z R



Rc eϕt , s′ m′ −G s′ + AΛ (t) d s′



e eϕ , s′ m′ −G s′ + AΛ (t) d s′ Rc t



Rc eϕt , s′ m −G s′ + AΛ (t) d s′ .

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Hence, we finally get AlFFL = K1,l ◦ R + K2,l ◦ Re + K3,l ◦ R completing the proof.

⊓ ⊔

Remark 6.2. The operators K2,l and K3,l result from the extra derivatives due to the additional time dependencies in (6.5) for simultaneous line rotation. For sequential line rotation these terms vanish as we choose ϕt to be piecewise constant and Theorem 6.2 reduces to Theorem 6.1. Remark 6.3. Regarding an oscillating non-rotating FFL with a phantom continuously rotating in opposite direction to the FFL rotation in the simultaneous setting, we would obtain AlFFL = K1,l ◦ R + K2,l ◦ Re as forward operator. For simultaneous line rotation the term K3,l results from the temporal change of the magnetic field orientation with respect to the receive coil sensitivity pl and thus, is not present in the just considered case. Since the two additional terms scale with the FFL rotation speed, we assume, as proposed in ([5], [18]), that Theorem 6.1 can still be used in case ϕt′ is suitably small. we conclude this section by deriving upper bounds To further confirm this, e for K2,l Rc (t) respectively K3,l Rc (t) . Lemma 6.1. Let R = GA , i.e. we suppose that the particle concentration is completely located within the sampling region as usually GA is the maxium displacement of the FFL. Then, we have that ′ e (t) . K1,l Rc (t) ≥ |Λ (t)| K2,l Rc ϕt′ Proof. It holds
Z
e eϕ , s = Rc c (r) δ r · eϕ − s r · e⊥ ϕ dr B R √ − R √R2 −s2 c s e −v e⊥
v d v, for s ≤ R, ϕ ϕ − R2 −s2 =  0, for s > R,
≤ R Rc eϕ , s . Hence, we get

6 Radon-based Image Reconstruction for MPI using a continuously rotating FFL

101


K1,l Rc (t) = µ0 AΛ ′ (t) eϕ · pl m′ (G ·) ∗ Rc eϕ , · (st ) t t
= µ0 GRΛ ′ (t) eϕt · pl m′ (G ·) ∗ Rc eϕt , · (st )
e eϕ , · (st ) ≥ µ0 GΛ ′ (t) eϕt · pl m′ (G ·) ∗ Rc t ′ Λ (t)
e eϕ , · (st ) = ′ µ0 Gϕt′ eϕt · pl m′ (G ·) ∗ Rc t ϕt |Λ ′ (t)| e (t) . = K2,l Rc ϕt′ ⊓ ⊔ The ratio in the last lemma relates the FFL translation to the rotation speed and e (t) might be neglected in the image reconthus, if it is sufficiently large, K2,l Rc struction process. As already mentioned, usually the excitation function is chosen to be sinusoidal, e.g. Λ (t) = cos (2π fdt) with drive frequency fd > 0. Then, we obtain |Λ ′ (t)| fd |sin (2π fdt)| . = ′ ϕt frot

(6.7)

At each turning point of the FFL the first term K1,l Rc becomes zero. By (6.7) the e might overweight the first integral, time intervals, in which the second term K2,l Rc fd are the smaller the larger the ratio frot gets. According to [26] drive frequencies are around 1 kHz to 150 kHz. A typical upper bound for the rotation frequency is 100 Hz [5] leading to ffrotd ≥ 10. At last, we state an example of particle concentrae tions leading to a vanishing K2,l Rc. Example 6.1. Let c be radial symmetric, i.e. c(r) = c(|r|). For s > R we already
e eϕ , s = 0. We compute for s ≤ R know that Rc √ √ Z R2 −s2 p  
Z R2 −s2  ⊥ e c s eϕ −v eϕ v d v = √ c s2 + v2 v d v = 0. −Rc eϕ , s = √ −

R2 −s2

−

R2 −s2

Next, we give an estimate for the third integral K3,l Rc (t) . Lemma 6.2. With m denoting the magnetic moment of a single particle and Np the total amount of particles contained in the tracer injection, it holds that K3,l Rc (t) ≤ µ0 ϕt′ ∥pl ∥ m N p . Proof. According to the proof of Theorem 6.2 we have Z
. K3,l Rc (t) = µ0 ϕt′ e⊥ · p c(r) m −G r · e + AΛ (t) d r ϕt l ϕt B R

The modulus of the mean magnetic moment is bounded by m (cf. Langevin model in Section 6.2) yielding

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Z K3,l Rc (t) ≤ µ0 ϕt′ e⊥ c(r) d r ≤ µ0 ϕt′ ∥pl ∥ m N p . ϕt · pl m B R

⊔ ⊓ Remark 6.4. For practical reasons it might be convenient to state an upper bound in terms of the maximal particle concentration cmax . This can be easily obtained by using N p ≤ cmax πR2 . In case the phantom is fully located within the saturation area on one side of the FFL, the above inequality holds approximately with equality. The estimate in the last Lemma can be computed beforehand to measurements as all components are determined by the scanner setup and the choice of the injected tracer. Thus, this upper bound can be determined and compared to the magnitudes of measured data in order to evaluate whether incorporation is needed. From the s-shape of the Langevin is largest at turning points of the FFL, which Rc function it follows that K (t) 3,l are the zero crossings of K1,l Rc (t) .

6.4 TV regularized Image Reconstruction Inspired by [24] we reconstruct particle concentration and Radon data simultaneously via total variation regularization. To this end, we first introduce the space of functions of bounded variation BV on the domain BR BV (BR , R) := {c ∈ L1 (BR , R) : TV (c) < ∞} , with TV (c) := sup

Z R2


c (r) div (g) (r) d r : g ∈ C0∞ BR , R2 , |g (r)|2 < 1 for all r .

Equipped with the norm ∥·∥BV := ∥·∥L1 + TV (·) this space becomes a Banach space and it holds BV (BR , R) ⊂ L2 (BR , R). Additionally, the Poincar´e-Wirtinger inequality holds Z 1 ∥c − c∥L2 ≤ C TV (c) , c = c (r) d r (6.8) πR2 BR for some constant C > 0. For more information concerning TV we recommend to consult e.g. [1] and [6]. Define D := L2 (BR , R) × L2 (Z, R) and let Al : D → L2 (R+ , R) such that   Al (c, v) := K1,l v + K2,l ◦ Re c + K3,l v

(6.9)

with operators Ki,l , i = 1, 2, 3 defined in Theorem 6.2. It holds that Al (c, Rc) = AlFFL c for c ≥ 0. Now, we are able to state the minimization problem we want to

6 Radon-based Image Reconstruction for MPI using a continuously rotating FFL

solve min

(c,v)∈C

1 α1 ∥Al (c, v) − ul ∥2L2 + ∥Rc − v∥2L2 + α2 TV (c) , 2∑ 2 l

103

(6.10)

with feasible set C := {(c, v) ∈ D : c ≥ 0, v ≥ 0} , given data ul , l = 1, . . . , L, weighting parameter α1 > 0, and regularization parameter α2 > 0. Note that actively we only penalize the choice of the particle concentration. Nevertheless, if needed an additional regularization term acting on the Radon data v can be included, e.g. directional TV regularization as used in [24]. Theorem 6.3. The minimization problem (6.10) has a solution (c∗ , v∗ ) ∈ C and c∗ ∈ BV (BR , R) . Proof. Rewriting the constrained optimization problem (6.10) using the indicator function δC yields min J (c, v) , (c,v)∈D

J (c, v) :=

α1 1 ∥Al (c, v) − ul ∥2L2 + ∥Rc − v∥2L2 + α2 TV (c) + δC (c, v) . 2∑ 2 l

Note that D is a Hilbertspace with inner product ⟨·, ·⟩D := ⟨·, ·⟩L2 (BR ,R) + ⟨·, ·⟩L2 (Z,R) . Obviously J is proper. The total variation TV (c) is convex and weakly lower semicontinuous according to [1]. Further, Al and R are linear bounded operators and the feasible set C is closed and convex. Thus, J is jointly convex in (c, v) and weakly lower semicontinuous. For existence the only remaining part to show is that J is also coercive. From [1] we know that ∥c∥BV → ∞

=⇒

α1 ∥Rc − v∥2L2 + α2 TV (c) → ∞ 2

(6.11)

as Rc does not annihilate for constant functions. Regard now ∥(c, v)∥2D = ∥c∥2L2 + ∥v∥2L2 → ∞. Then, either ∥c∥L2 or ∥v∥L2 needs to tend to infinity. If ∥v∥L2 → ∞ we directly get that J (c, v) → ∞ as ∥Rc − v∥2L2 → ∞. Thus, we assume that ∥v∥L2 is bounded and ∥c∥L2 → ∞. From (6.8) we get with 1BR denoting the characteristic function √ ∥c∥L2 ≤ ∥c − c 1BR ∥L2 + ∥c 1BR ∥L2 ≤ C TV (c) + πR2 |c|   1 1 = C TV (c) + √ ∥c∥L1 ≤ max C, √ ∥c∥BV πR2 πR2 and thus, ∥c∥BV → ∞ if ∥c∥L2 → ∞. Together with (6.11) we finally obtain that J is coercive yielding existence of a minimizer. ⊔ ⊓

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6.5 Numerical Results Finally, we state numerical results for synthetic data. Thereby, the basis for our data simulation is given by the framework developed by Gael Bringout [4] available at https://github.com/gBringout. Accordingly, the tracer is modeled as 3 O4 ) a solution with 0.5 mol(Fe concentration of magnetite with 30 nm core diamem3 0.6 ter and µ0 T saturation magnetization. Note that since MPI is three dimensional in itself, the concentration is given per volume and for data generation we regard voxel. However, for convenience, hereafter we will stick to our two dimensional formulation. As excitation function we choose Λ (t) = cos (2π fdt)

(6.12)

with drive frequency fd . Further parameters can be found in Table 6.1. Our concentration phantom (cf. Figure 6.4a) is normalized to one and located within a circle around the origin with radius GA , which is the maximum displacement of the     FFL. Thus, we set the FOV to be − GA , GA × − GA , GA . For data generation we divide the FOV into 501 × 501, for image reconstruction into 201 × 201 pixel. In case of sequential line rotation, we gather data for 25 sweeps of the FFL through the FOV and angles equally distributed in [0, π]. More precisely, we regard angles π , j = 1, . . . , 25. Considering simultaneous line rotation, we choose ϕ j = ( j − 1) 25 a total measurement time of 2 f1rot resulting in the same amount of FFL translations through the phantom covering angles again in [0, π]. Table 6.1: Simulation parameters Parameter

Explanation

Value

Unit

µ0

magnetic permeability

TmA−1

kB

Boltzmann constant

4π · 10−7

G

gradient strength

4

T (mµ0 )−1

A

drive peak amplitude

p1 p2

sensitivity of the first receive coil

1.380650424 · 10−23 0.015 T

[0.015/293.29, 0]

T

sensitivity of the second receive coil [0, 0.015/379.71]

JK−1 Tµ0−1 m−1 m−1

fd

drive-field frequency

25

kHz

frot

line rotation frequency

1

kHz

fs

sampling frequency

8

MHz

In the following, we denote

6 Radon-based Image Reconstruction for MPI using a continuously rotating FFL

Kc i,l f (t) :=

K f (t) ,  i,l max K1,l f (t)

105

for i = 1, 2, 3, l = 1, 2.

t

For our reconstructions, we neglect the second term of the forward operator (6.9), which is justified due to Lemma 6.1 and the choice of frequencies. This is emphac sized by Figure 6.3. The left plot shows Kc 1,1 Rc in comparison to K3,1 Rc, whereas e c c the right plot images Kc 2,1 Rc + K3,1 Rc and K3,1 Rc. We find that both additional terms are small compared to Kc 1,1 Rc. Especially the second term seems to have only an influence, when the main part reaches its highest values, while the third term is largest for the zero crossings of Kc 1,1 Rc.

1

0.03 0.02

0.5 0.01

0

0 -0.01

-0.5 -0.02

-1

0

1

2

3

4

5

-0.03

0

1

2

3

(a)

4

5 10-4

10-4

(b)

c Fig. 6.3: Plots showing Kc 1,1 Rc (blue) in comparison to K3,1 Rc (red) together with e c the bound determined in Lemma 6.2 (dark grey) (a) and Kc 2,1 Rc + K3,1 Rc (blue) c in comparison to K3,1 Rc (red) (b) for the phantom in Figure 6.4a. Blocks in the background of the two plots demonstrate periods of the drive field. The total measurement time is 12.5 f1d = 2 f1rot . Figure 6.4a shows the phantom which we want to reconstruct. In order to evaluate the discretized version of (6.9) for a specific time point, the corresponding column in a sinogram filled angle by angle is needed. Thus, to be able to compute data for every sampling point, a sinogram containing a column for each angle the FFL attains during scanning is needed. For sequential line rotation this amounts to the sinogram shown in Figure 6.4b, for simultaneous line rotation to the one in Figure 6.5a. In order to reduce the problem size, we rather aim at reconstructing only the sinogram taking values along the dashed line in Figure 6.5a resulting in Figure 6.5b. Thereby, we get an additional error but because of the shape of the convolution kernel of the

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main part Kc 1,l Rc, which converges to the dirac delta for particle diameters tending to infinity [17], and using a regularization method for reconstruction this should be no further problem. 10-3

1

3.5 0.8

3 2.5

0.6

2 0.4

1.5 1

0.2

0.5 0

0

(a)

(b)

Fig. 6.4: Phantom (a) and corresponding sinogram filled angle by angle (b).

10-3

10-3

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

(a)

(b)

Fig. 6.5: (a) Sinogram for the phantom shown in Figure 6.4a with columns for each angle attained by the FFL during scanning. (b) Adapted version filled following the dashed line in (a). We discretize (6.10) by applying standard methods. We define discretized versions of the Radon transform Rseq and Rsim such that Rseq c yields the sinogram in Figure 6.4b and Rsim c the one in Figure 6.5b. Further, we scale data ul and discretized forward operators Ki,l by dividing by the maximum absolute data value u∗ bl := u

ul , u∗

b i,l := Ki,l . K u∗

For reconstruction we regard the versions of (6.10) specified in Table 6.2. To solve the resulting problem, we use CVX, a package for specifying and solving convex

6 Radon-based Image Reconstruction for MPI using a continuously rotating FFL

107

Table 6.2: Reconstruction methods Method

Regarded minimization problem

M1

1 2 b α1 ∑ ∥K1,l v − ubl ∥22 + 2 ∥Rseq c − v∥22 + α2 ∥|∇c|2 ∥1 c≥0,v≥0 2 l=1

M2 M3

min

1 2 b α1 ∑ ∥K1,l v − ubl ∥22 + 2 ∥Rsim c − v∥22 + α2 ∥|∇c|2 ∥1 c≥0,v≥0 2 l=1  1 2 b α b 3,l v − u bl ∥22 + 1 ∥Rsim c − v∥22 + α2 ∥|∇c|2 ∥1 min ∥ K1,l + K ∑ c≥0,v≥0 2 2 l=1 min

programs ([11], [10]), together with the MOSEK solver [21]. For our weighting  parameters we choose α1 ∈ {1, 2, 4} · 104 and α2 ∈ 0.15.5−0.05i , i = 0, . . . , 49 such that (α1 , α2 ) maximizes the structural similarity (SSIM) of the reconstructed particle concentration with the groundtruth. However, there might be better parameter choices. Note that st , i.e. the distance of the FFL to the origin, is not sampled equidistantly due to the choice of the excitation function (6.12) and because we sample for equidistant time points (cf. [18]). While in [18] they need to process and regrid the signal to be able to use Wiener deconvolution, this is not necessary for our methods. We start with a result for sequential line rotation in Figure 6.6. We obtain good reconstructions for both, the phantom as well as the Radon data. Nevertheless, the vertices of the square are not resolved a hundred percent properly. This is likely due to missing information in the data and because we used an isotropic version of the TV penalty term, which favors rounded edges (cf. [6]). Next, we regard results for

1

10-3 3.5

0.8

3 2.5

0.6 2 0.4

1.5 1

0.2 0.5

Fig. 6.6: Reconstruction of phantom and sinogram for sequential line rotation via M1 . (α1 = 2 · 104 , α2 = 0.14.65 , SSIM (c) = 0.9576) the setting of simultaneous line rotation. Similar to [5], we get a slightly rotated version of the phantom when ignoring the different sampling pattern, see Figure 6.7. Lastly, we compare reconstructions incorporating K3,l R with those neglecting this

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Fig. 6.7: Phantom reconstruction for simultaneous line rotation using model M1 , i.e. ignoring the specific sampling pattern for simultaneous line rotation. Contours of the groundtruth are depicted in white. (α1 = 4 · 104 , α2 = 0.14 , SSIM (c) = 0.7996) term. According to Figure 6.3b we expect only slight differences. Corresponding deliverables with and without added gaussian noise (standard deviation: ca. 0.8% of u∗ ) can be found in Figure 6.8. Our reconstructions show that incorporation of the third term in the model yields slightly higher values for structural similarity. For noisy data the size of K3,l R gets close to the range of added noise and thus, it is reasonable that the gain in the SSIM value is lower compared to the noise-free setting.

6.6 Conclusion In this work, we regarded the relation of MPI and Radon data for a continuously rotating FFL. Because of the time derivative in the signal equation and the additional time-dependencies for this setting, we get two additive terms in the forward model. We derived bounds for these two terms supporting the assumption of [5] and [18] that for sufficiently slow FFL rotation compared to the translation speed, the results from the sequential line rotation setting can be applied. This is also emphasized by our numerical results. Incorporation of the additional terms for image reconstruction does not seem to be necessary on a first view. However, including the third term in our reconstruction approach is simple because both terms together still form a convolution with the Radon data of the particle concentration. Further, its bound can be computed beforehand of reconstruction and thus, its magnitude can be compared to the total signal. In our case, incorporation yielded slightly better structural similarity values. Future research will comprise steps towards a more realistic set of assumptions, like time-dependent particle concentrations, magnetic field imperfections, and that the direct feedthrough of the excitation signal needs to be removed. Moreover, it would be interesting to extend the results of this article to different modeling approaches distinct from the Langevin model.

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Acknowledgment The authors acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 ”Modeling, Simulation and Optimization of Fluid Dynamic Applications”.

References 1. R. Acar and C.R. Vogel. Analysis of bounded variation penalty methods for ill-posed problems. Inverse problems 10(6), 1217, 1994. 2. C. Bathke, T. Kluth, C. Brandt, and P. Maaß. Improved image reconstruction in magnetic particle imaging using structural a priori information. International Journal on Magnetic Particle Imaging 3(1), 2017. 3. K. Bente, M. Weber, M. Graeser, T.F. Sattel, M. Erbe, and T.M. Buzug. Electronic field free line rotation and relaxation deconvolution in magnetic particle imaging. IEEE transactions on medical imaging 34(2), 644–651, 2014. 4. G. Bringout. Field free line magnetic particle imaging: characterization and imaging device up-scaling. PhD thesis, Dissertation, L¨ubeck, Universit¨at zu L¨ubeck, 2016. 5. G. Bringout, W. Erb, and J. Frikel. A new 3d model for magnetic particle imaging using realistic magnetic field topologies for algebraic reconstruction. Inverse Problems 36(12), 124002, 2020. 6. M. Burger and S. Osher. A guide to the TV zoo. In: Level set and PDE based reconstruction methods in imaging, Springer, 1–70, 2013. 7. M. Erbe. Field free line magnetic particle imaging. Springer Science & Business Media, 2014. 8. B. Gleich and J. Weizenecker. Tomographic imaging using the nonlinear response of magnetic particles. Nature 435(7046), 1214–1217, 2005. 9. P.W. Goodwill, J.J. Konkle, B. Zheng, E.U. Saritas, and S.M. Conolly. Projection x-space magnetic particle imaging. IEEE transactions on medical imaging 31(5), 1076–1085, 2012. 10. M. Grant and S. Boyd. Graph implementations for nonsmooth convex programs. In: Recent Advances in Learning and Control, V. Blondel, S. Boyd, and H. Kimura (eds.), Springer, 95– 110, 2008. 11. M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming, version 2.2. Build 1148. http://cvxr.com/cvx, January 2020. ¨ Sarıtas¸, and H.E. G¨uven. Comparison of system12. S. Ilbey, C.B. Top, A. G¨ung¨or, T. C ¸ ukur, E.U. matrix-based and projection-based reconstructions for field free line magnetic particle imaging. International Journal on Magnetic Particle Imaging 3(1):1–8, 2017. 13. B. Kaltenbacher, T. Thi Ngoc Nguyen, A. Wald, and T. Schuster. Parameter identification for the Landau–Lifshitz–Gilbert equation in magnetic particle imaging. In: Time-dependent Problems in Imaging and Parameter Identification, Springer, 377–412, 2021. 14. T. Kluth. Mathematical models for magnetic particle imaging. Inverse Problems 34(8), 083001, 2018. 15. T. Kluth, B. Jin, and G. Li. On the degree of ill-posedness of multi-dimensional magnetic particle imaging. Inverse Problems 34(9), 095006, 2018. 16. T. Kluth, P. Szwargulski, and T. Knopp. Towards accurate modeling of the multidimensional magnetic particle imaging physics. New Journal of Physics 21(10), 103032, 2019. 17. T. Knopp and T.M. Buzug. Magnetic particle imaging: an introduction to imaging principles and scanner instrumentation. Springer Science & Business Media, 2012. 18. T. Knopp, M. Erbe, T.F. Sattel, S. Biederer, and T.M. Buzug. A Fourier slice theorem for magnetic particle imaging using a field-free line. Inverse Problems 27(9), 095004, 2011. 19. T. Knopp, N. Gdaniec, and M. M¨oddel. Magnetic particle imaging: from proof of principle to preclinical applications. Physics in Medicine & Biology 62(14), R124, 2017.

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´ K. Brandt, C. Cooley, and L. Wald. OS-MPI: 20. E. Mattingly, E. Mason, K. Herb, M. Sliwiak, an open-source magnetic particle imaging project. International Journal on Magnetic Particle Imaging 6(2 Suppl1), 2020. 21. Mosek ApS. The MOSEK optimization toolbox for MATLAB manual. Version 9.1 https: //docs.mosek.com/9.1/toolbox/index.html, 2020. 22. F. Natterer. The Mathematics of Computerized Tomography, 1986. 23. M. Storath, C. Brandt, M. Hofmann, T. Knopp, J. Salamon, A. Weber, and A. Weinmann. Edge preserving and noise reducing reconstruction for magnetic particle imaging. IEEE transactions on medical imaging 36(1), 74–85, 2016. 24. R. Tovey, M. Benning, C. Brune, M.J. Lagerwerf, S.M. Collins, R.K. Leary, P.A. Midgley, and C.-B. Sch¨onlieb. Directional sinogram inpainting for limited angle tomography. Inverse Problems 35(2), 024004, 2019. 25. P. Vogel, J. Markert, M.A. R¨uckert, S. Herz, B. Keßler, K. Dremel, D. Althoff, M. Weber, T.M. Buzug, T.A. Bley, et al. Magnetic particle imaging meets computed tomography: First simultaneous imaging. Scientific Reports 9(1), 1–9, 2019. 26. J. Weizenecker. The Fokker-Planck equation for coupled Brown-N´eel-rotation. Physics in Medicine & Biology 63(3), 035004, 2018. 27. J. Weizenecker, B. Gleich, and J. Borgert. Magnetic particle imaging using a field free line. Journal of Physics D: Applied Physics 41(10), 105009, 2008. 28. L. Zdun and C. Brandt. Fast MPI reconstruction with non-smooth priors by stochastic optimization and data-driven splitting. Physics in Medicine & Biology 66(17), 175004, 2021.

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Fig. 6.8: Results for simultaneous line rotation using different reconstruction approaches.

Chapter 7

Numerical Simulation of an idealized coupled Ocean-Atmosphere Climate Model Kamal Sharma, Peter Korn

Abstract We present numerical simulations for an idealized coupled oceanatmosphere climate model. Our climate model [2] belongs to the class of intermediate coupled models which are much simpler than the coupled general circulation models of the ocean-atmosphere system but still allow to study the fundamental aspects of ocean-atmosphere interactions. Our model couples an atmosphere system, described by the compressible two-dimensional (2D) Navier-Stokes equations and an advection-diffusion equation for temperature, to an ocean system, given by 2D incompressible Navier-Stokes equations and an advection-diffusion equation for temperature. The finite element method (FEM) is used to discretize the system of PDEs representing the climate model on a 2D periodic domain and the discrete model is solved using Firedrake [7], which is an efficient automated FEM library. The numerical simulation results are visualized using an open-source software called Paraview [1]. To ensure the accuracy of simulation results of the coupled model, we have carried out detailed numerical investigation of its atmosphere and ocean components separately and tested our codes against some benchmark problems available in the literature. Our final goal is to incorporate stochasticity into the coupled oceanatmosphere model following Hasselmann’s paradigm [4] and use the model to study key features of climate phenomena such as El-Ni˜no Southern Oscillation (ENSO) [3]. Therefore, the numerical simulation of a deterministic climate model (presented here), is an important initial step before simulating its stochastic counterpart.

Kamal Sharma Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] Peter Korn Max Planck Institute for Meteorology, Hamburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Iske, T. Rung (eds.), Modeling, Simulation and Optimization of Fluid Dynamic Applications, Lecture Notes in Computational Science and Engineering 148, https://doi.org/10.1007/978-3-031-45158-4_7

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7.1 Introduction Our goal is to understand and study key features of climatic events, which occur as a result of interactions between the ocean and the atmosphere. One such well known event is the El-Ni˜no Southern Oscillation (ENSO) which takes place in the tropical eastern Pacific ocean as a result of strong ocean-atmosphere interactions [3]. These climatic events are studied with the help of mathematical models consisting of PDEs which simulate the behavior of the atmosphere, the ocean and their interaction. These mathematical models are known as coupled ocean-atmosphere models and they can be divided into three main categories ((1) conceptual models and simple models, (2) intermediate models, and (3) coupled general circulation models (GCMs)) depending on their dynamical complexity [6]. The model that we present in this paper belongs to the intermediate models category. Intermediate models are sophisticated enough (in comparison to conceptual and simple models) to produce realistic solutions but simple enough to be able to easily diagnose in comparison to general circulation models. Our climate model [2, 5] tries to capture very few fundamental processes that can occur in the ocean and the atmosphere through the study of interactions between quantities like the velocity and the temperature in a two-dimensional framework. Therefore, we call it, an idealized coupled ocean-atmosphere climate model. Throughout this paper, we will use the terms like the climate model or the coupled ocean-atmosphere model to refer to our idealized model. The complete set of equations for the climate model in non-dimensionalized form is given by, Atmosphere: ∂ ua 1 1 1 + (ua · ∇) ua + a ua⊥ + a ∇θ a = a △ua , ∂t Ro Ro Re ∂θa 1 a a a o a + (u · ∇) θ = γ (θ − θ ) + a ∆ θ , ∂t Pe Ocean: 1 ∂ uo 1 1 + (uo · ∇) uo + o uo⊥ + o ∇po = σ (uo − ua ) + o △uo , ∂t Ro Ro Re ∂θo 1 + (uo · ∇) θ o = o ∆ θ o , ∂t Pe ∇ · uo = 0, Initial conditions: ua (t0 ) = ua0 , θ a (t0 ) = θ0a , uo (t0 ) = uo0 , θ o (t0 ) = θ0o . (7.1) The vector variable u and the scalar variables θ and p (with super-index for the atmosphere and ocean components) denote the velocity, temperature, and pressure fields respectively. These variables are the function of 2D space variable, x = (x, y) and time, t. We use the notation u⊥ := (u1 , u2 )⊥ = (−u2 , u1 ) to represent the Cori-

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olis force where u1 and u2 are the velocity components in x and y directions respectively. Initial conditions for the velocity and temperature fields are specified at time, t = 0. The climate model equations are defined on a 2D unit-square domain (denoted by Ω ) with periodic boundary conditions in both x and y directions. The term ua := ua − uaavg , where uaavg denotes an average of atmosphere velocity field over the domain. The numbers Re, Pe, and Ro denote the Reynolds, P´eclet and the Rossby numbers respectively. The constants γ and σ are called the coupling coefficients and their values are always set to be less than zero. The magnitude of these constants signifies the strength of interaction between the atmosphere and ocean components. The atmosphere is modelled by 2D compressible Navier-Stokes equations coupled to a 2D advection-diffusion equation for temperature, θ a . The atmosphere velocity field ua , transports the atmosphere temperature and the gradient in the atmosphere temperature field, in turn, changes the atmosphere velocity. The atmosphere temperature is coupled to the ocean temperature through the coupling term, γ (θ a − θ o ) in the advection-diffusion equation. This term can be thought of as playing the role of transferring heat from the ocean to the atmosphere. The ocean is modelled by 2D incompressible Navier-Stokes equations and an equation for the ocean temperature θ o , that is passively advected by the ocean velocity field, uo . The atmosphere and ocean velocities are coupled through a coupling term, σ (uo − ua ). This term plays the role of shear force acting on the ocean system whose value depends on the difference between the local ocean velocity and the local deviation of the atmosphere velocity away from its mean velocity.

Atmosphere

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z x

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Fig. 7.1: 3D and 2D domain representations for a coupled ocean-atmosphere model The 2D domain of our climate model is shown in figure 7.1c. The figure 7.1 shows how our model’s domain differs from another 2D domain (figure 7.1b) which might look like a more natural choice for representing a coupled ocean-atmosphere

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system. Figure 7.1b shows one of the possible ways to represent a coupled oceanatmosphere system in 2D. In order to study the ocean-atmosphere interaction in a domain (figure 7.1b) like that, the equations for the atmosphere and ocean have to be solved on two different sub-domains with coupling conditions specified as additional boundary conditions at the interface. We, however, take a different approach in representing the coupled ocean-atmosphere system on a 2D domain (figure 7.1c). In our approach, we solve both the ocean and atmosphere systems on the same domain with the coupling conditions specified as additional terms in the system equations. One way to visualize such 2D domain is to start with a 3D domain (figure 7.1a) and take the average of all the desirable variables (such as ocean velocity, atmosphere velocity etc.) in the z direction up to the point of ocean-atmosphere interface. We will then obtain a 2D domain on which both the ocean and atmosphere variables are defined at each point. We seek to solve our climate model on such a 2D domain.

Incompressible Navier-Stokes equations

Advectiondiffusion equation

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Fig. 7.2: Breakdown of a coupled ocean-atmosphere model into simpler models We want to numerically solve the equations (7.1) and carry out simulations to understand the dynamics of our climate model and thus gain further insights into the underlying physical processes. Our climate model essentially consists of 2D Navier-Stokes equations and advection diffusion equations, in a rotating frame of reference, which are modified to include the coupling between the ocean and the atmosphere. Figure 7.2 shows how we can arrive at the coupled model starting from three fundamental models, namely the incompressible Navier-Stokes equations, the advection-diffusion equation, and the compressible Navier-Stokes equations. Therefore, before solving the equations for the climate model, we broke it down into simpler models/equations and did their numerical investigation first. We used FEM to

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discretize the equations and performed numerical simulations using Firedrake software. We solved several benchmark test problems for the three fundamental models and compared the simulation results with the analytical/experimental results available in the literature to ensure the accuracy of our numerical codes. We moved on to the numerical analysis of relatively complex models (i.e. the ocean model and the atmosphere model) only after understanding the dynamics of the fundamental models and establishing the accuracy of their numerical codes. In this paper, we present, the numerical investigation of three models, namely the idealized ocean model, the idealized atmosphere model, and the coupled ocean atmosphere model. In section 7.2, we introduce an idealized ocean model and present its discretization along with some simulation results. Section 7.3 addresses the simulation of an idealized atmosphere model. Numerical investigation of the coupled ocean-atmosphere model is presented in section 7.4 along with some simulation results for two test problems. We end our paper by some concluding remarks and talking briefly, in section 7.5, about our approach for integrating stochasticity into the climate model.

7.2 Idealized ocean model We can decouple the climate model by setting the values of coupling coefficients σ and γ to zero and consider only its ocean component. This leads to the following mathematical model for the ocean, ∂u 1 1 1 + (u · ∇) u + u⊥ + ∇p = △u, ∂t Ro Ro Re ∂θo 1 + (u · ∇) θ = ∆ θ , ∂t Pe ∇ · u = 0, u (t0 ) = u0 , θ (t0 ) = θ0 .

(7.2)

We call this an idealized ocean model. This model consists of 2D incompressible Navier-Stokes equations and an equation for the ocean temperature variable θ , that is passively advected by the ocean velocity field u. Note that the equations (7.2) are in non-dimensionalized form. We are interested in simulating events which occur on large-scales in the ocean and therefore, the variables in equations (7.2) are non-dimensionalized with respect to their characteristic values for large-scale flows. Table 7.1 shows typical values of scales for large-scale flows in the ocean and the atmosphere. We have used these values (given in table 7.1) to non-dimensionlize the ocean model variables. The ocean temperature is non-dimensionalized with respect to a reference temperature of, T = 300 K. We solve the ocean model equations on a 2D domain, Ω = [0, 1]2 with periodic boundary conditions in both x and y directions. The finite element weak formulation for this model reads as; find u ∈ V1 ⊂ H 1 (Ω )2 , p ∈ V2 ⊂ L2 (Ω ), and θ ∈ V3 ⊂

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H 1 (Ω ) such that, ∂u 1 1 · v dx + ((u · ∇) u) · v dx + u⊥ · v dx − Ro Ω Ro Ω ∂t Ω Z Z 1 ∇u · ∇v dx + ∇ · u q dx = 0, + Re Ω Ω Z Z Z ∂θ 1 φ dx + u · ∇θ φ dx + ∇θ · ∇φ dx = 0 Pe Ω Ω ∂t Ω

Z

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∇ · v p dx (7.3)

for all v ∈ V1 , q ∈ V2 and φ ∈ V3 where V1 ,V2 and V3 are compatible finite element spaces. Here, H 1 (Ω )2 and L2 (Ω ) denote the usual Sobolev space and Lebesgue space, respectively. Note that, in equations (7.3), there are no boundary terms. This is because of the application of periodic boundary conditions. We use the CrankNicolson time-stepping scheme to obtain a fully discrete problem; given un , θ n (discretized velocity and temperature fields) at time tn , find un+1 , pn+1 and θ n+1 at time tn+1 = tn + ∆t, which satisfy 1 un+1 − un · v dx + ((un+1 · ∇)un+1 + (un · ∇)un ) · v dx ∆t 2 Ω Ω Z Z 1 1 (un+1 + un )⊥ · v dx − pn+1 ∇ · v dx + 2Ro Ω Ro Ω Z Z 1 + (∇un+1 + ∇un ) · ∇v dx + ∇ · un+1 q dx = 0, 2Re Ω Ω Z Z θ n+1 − θ n 1 φ dx + (un · ∇θ n + un+1 · ∇θ n+1 )φ dx ∆t 2 Ω Ω Z 1 + (∇θ n+1 + ∇θ n ) · ∇φ dx = 0 2Pe Ω Z

Z

(7.4)

∀ v ∈ V1 , q ∈ V2 and φ ∈ V3 . We are interested in finding out, how the velocity and temperature fields evolve over time for different initial conditions. Therefore, we solve many test problems for the ocean model using Firedrake for different initial conditions and different values for Re, Pe, and Ro, and analyzed the simulation results. Here, we present simulation Table 7.1: Scales of large-scale flows in the ocean and the atmosphere Variable Scaling (x, y) t (u1 , u2 )

L T U

Meaning

Atmosphere value Horizontal length 106 m Time-scale 1 day (105 s) Horizontal veloc- 10 ms−1 ity

Ocean value 105 m 10 days (106 s) 0.1 ms−1

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results for one of the test problems. u0 = [1, 0] and θ0 = 1+0.03· f (x, y), where f (x, y) = Consider the initial conditions, p 0.5 + 0.5 · cos(π · min(4 (x − 0.5)2 + (y − 0.5)2 , 1)) is a bell-function. Figure 7.3 shows a plot for the temperature and velocity fields at time, t = 0 on the domain [0, 1]2 . Temperature field is color-coded while the velocity field is represented by arrows whose direction and size gives a measure of velocity vectors. The discrete

Fig. 7.3: Initial velocity and temperature fields for the ocean model test problem equations (7.4) are coded using Firedrake and the simulation is run up to time, t = 0.2 with non-dimensionalized parameters set to, Re = 103 , Ro = 0.1, Pe = 102 . Figure 7.4 shows the evolution of velocity and temperature fields over time. We can observe that the temperature field is being transported by the velocity field and it dissipates over time. The term (u · ∇)θ in equation (7.2) is responsible for the advection. The term ∆ θ causes the temperature to dissipate over time and the value of P´eclet number decides how fast the temperature dissipate over time. The Coriolis 1 ⊥ u in the momentum equation, acts perpendicular to the velocity direction. force Ro We can observe its effect on the velocity field over time from the simulation results.

7.3 Idealized atmosphere model The mathematical model for the atmosphere component is shown below.

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(a) Velocity and temperature fields at t = 0.05

(b) Velocity and temperature fields at t = 0.1

(c) Velocity and temperature fields at t = 0.15

(d) Velocity and temperature fields at t = 0.20

Fig. 7.4: Simulation results for the ocean model test problem ∂u 1 1 1 + (u · ∇) u + u⊥ + ∇θ = △u, ∂t Ro Ro Re ∂θ 1 + (u · ∇) θ = ∆ θ , ∂t Pe u(t0 ) = u0 , θ (t0 ) = θ0 .

(7.5)

This is an idealized atmosphere model in a non-dimensionalized form where the variables are non-dimensionalized with respect to their characteristic values for large-scale flows (refer table 7.1). Note that, unlike the ocean model (equations (7.2)) where the temperature is passively advected by velocity field, here (equations (7.5)), the atmosphere velocity field u transports the temperature that provides the gradient term for the momentum equation. Therefore, the atmosphere velocity

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and temperature fields are inherently coupled. The finite element weak formulation for the equations (7.5) is then to find u ∈ V1 ⊂ H 1 (Ω )2 and θ ∈ V2 ⊂ H 1 (Ω ) such that ∂u 1 1 · v dx + ((u · ∇) u) · v dx + u⊥ · v dx + Ro Ω Ro Ω ∂t Ω Z 1 + ∇u · ∇v dx = 0, Re Ω Z Z Z ∂θ 1 φ dx + u · ∇θ φ dx + ∇θ · ∇φ dx = 0, Pe Ω Ω ∂t Ω

Z

Z

Z

Z Ω

∇θ · v dx (7.6)

∀ v ∈ V1 and φ ∈ V2 where V1 and V2 are compatible finite element spaces. The fully discrete problem is to find un+1 and θ n+1 at time, tn+1 = tn + ∆t given θ n at time tn , which satisfy

un ,

1 un+1 − un · v dx + ((un+1 · ∇)un+1 + (un · ∇)un ) · v dx ∆t 2 Ω Ω Z Z 1 1 n+1 n ⊥ (u + u ) · v dx + (∇θ n + ∇θ n+1 ) · v dx + 2Ro Ω 2Ro Ω Z 1 + (∇un+1 + ∇un ) · ∇v dx = 0, 2Re Ω Z Z θ n+1 − θ n 1 φ dx + (un · ∇θ n + un+1 · ∇θ n+1 )φ dx ∆t 2 Ω Ω Z 1 + (∇θ n+1 + ∇θ n ) · ∇φ dx = 0, 2Pe Ω

Z

Z

(7.7)

∀ v ∈ V1 and φ ∈ V2 . We solve the discrete equations (7.7) on a 2D domain, Ω = [0, 1]2 with periodic boundary conditions using Firedrake. Figure 7.6 shows the evolution of temperature and velocity fields over time for the initial conditions (figure 7.5), θ0 = 3000 + 50 · f (x, y) and u0 = [1, 0], where f (x, y) is the bell function defined earlier. We want to point out that, in the atmosphere model equations (7.5), the atmosphere temperature field is non-dimensionalized with respect to a reference temperature value of T = 0.1 K instead of a typical temperature value of 300 K. This is because during the non-dimensionalization process of the atmosphere equations, we forced the denominator of the gradient term ∇θ to be equal to the Rossby number (Re). As a result of that we are restricted in our choice of reference temperature value for the atmosphere. Therefore, to simulate atmospheric flows with initial temperature values around 300 K, we take the non-dimensionalized temperature variable θ0 , to be closer to 3000. We run the simulation for these initial conditions from time, t = 0 to t = 0.2 with the non-dimensionalized parameters set to, Re = 10, Ro = 1, and Pe = 10. Unlike the ocean model where the velocity is mainly driven by Coriolis force, here, the

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Fig. 7.5: Initial velocity and temperature fields for the atmosphere model test problem

velocity is also driven by temperature gradients. We chose the initial temperature field in such a manner that the value of atmospheric temperature decreases radially from the center of the domain to some radial distance and then remain constant afterwards. This kind of temperature distribution forces the atmosphere to flow radially outwards from the domain center. The outward flow of atmosphere can be seen in the numerical results (figure 7.6a). As the atmosphere flow outwards it carries the high temperature fluid along with it (because of the advection term u · ∇θ ) and spreads it in the domain. The term ∆ u in the advection-diffusion equation (7.5) then dissipates the temperature field, thus reducing the magnitude of temperature gradient in the domain. As a result, the magnitude of atmospheric velocity reduces considerably.

7.4 Climate model Most of the ideas used to discretize the idealized ocean and atmosphere models can still be applied for the discretization of climate model equations (7.1). The finite element weak formulation for the climate model reads as follows; find (uo , ua ) ∈ V1 ⊂ H 1 (Ω )2 , po ∈ V2 ⊂ L2 (Ω ), (θ o , θ a ) ∈ V3 ⊂ H 1 (Ω ) such that

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(a) Velocity and temperature fields at t = 0.05

(b) Velocity and temperature fields at t = 0.10

(c) Velocity and temperature fields at t = 0.15

(d) Velocity and temperature fields at t = 0.20

Fig. 7.6: Simulation results for an idealized atmosphere model ∂ uo o 1 1 · v dx + ((uo · ∇) uo ) · vo dx + o uo⊥ · vo dx − o ∇ · vo po dx Ro Ω Ro Ω Ω ∂t Ω Z Z Z 1 + o ∇uo · ∇vo dx − σ (uo − ua ) · vo dx + ∇ · uo q dx = 0, Re Ω Ω Ω Z Z Z ∂θo 1 o o o ψ dx + u · ∇θ ψ dx + o ∇θ · ∇ψ dx = 0, Pe Ω Ω ∂t Ω Z Z Z Z ∂ ua a 1 1 a a a a⊥ a · v dx + ((u · ∇) u ) · v dx + a u · v dx + a ∇θ a · va dx Ro Ω Ro Ω Ω ∂t Ω Z 1 + a ∇ua · ∇va dx = 0, Re Ω Z Z Z Z ∂θa 1 φ dx + ua · ∇θ a φ dx + a ∇θ a · ∇φ dx − γ(θ a − θ o )φ dx = 0. Pe Ω Ω ∂t Ω Ω (7.8) Z

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for all (vo , va ) ∈ V1 , q ∈ V2 and (ψ, φ ) ∈ V3 where V1 ,V2 and V3 are compatible finite element spaces. It can be observed that most of the terms in (7.8) are similar to the weak formulation of idealized ocean and atmosphere models. Therefore, we only need to make slight adjustments to the discrete form of idealized ocean and atmosphere models in order to obtain the discrete climate model. There are two new terms in the climate model which were absent in the idealized ocean and atmosphere models. These are the coupling terms and we want to study their effect on the dynamics of the coupled ocean-atmosphere system. We created two test problems, whose solution might involve the interaction of ocean and atmosphere, and then solved them using the numerical code for our climate model. These test problems mimic some physical scenarios that we might encounter in the ocean and atmosphere. The idea is to see how numerical simulation results compare with our observations for these physical scenarios. The initial conditions for the first test problem (we call it, test problem 1) are as

Fig. 7.7: Initial velocity field of the atmosphere for test problem 1 follows: uo = [0, 0], ua = [sin(2πy), 0], θ o = θ a = 3000. These initial conditions (figure 7.7) can be interpreted physically, as a wind gust over the ocean surface with the atmosphere and ocean in thermal equilibrium. For this physical scenario, one might expect that over time the ocean gains momentum and move in the wind direction while the wind slows down due to the viscous forces. To simulate this problem using our model we set the following values for different parameters in the numerical code: σ = −1.0, γ = −1.0, Rea = 102 , Pea = 102 , Roa = 1.0, Reo = 103 , Peo = 104 , Roo = 1.0. The simulation is then run for a 2D unit-square periodic domain from time, t = 0 to t = 0.2. As expected, in the numerical solution (fig. 7.8) we observe that the ocean picks up momentum and starts to move with the ocean while on the other hand, the atmospheric velocity drops in magnitude. We can also observe that the ocean velocity field looks similar to the atmosphere velocity field. This is due to the term σ (uo − ua ) in the ocean momentum equation which forces the ocean velocity to match local deviations of atmosphere velocity from its mean.

7 Numerical Simulation of an idealized coupled Ocean-Atmosphere Climate Model

(a) atmosphere velocity field at t = 0.05

(b) ocean velocity field at t = 0.05

(c) atmosphere velocity field at t = 0.10

(d) ocean velocity field at t = 0.10

(e) atmosphere velocity field at t = 0.20

(f) ocean velocity field at t = 0.20

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Fig. 7.8: Evolution of ocean and atmosphere velocity fields over time for test problem 1 (Note: the ocean and atmosphere velocities are scaled differently)

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The second test problem (we call it, test problem 2) that we solve has the following initial conditions: uo = [0, 0], ua = [0, 0], θ a = 3000, θ o = 3000 + 50 · f (x, y) where f (x, y) is the bell-function defined earlier. These initial conditions correspond to a physical situation where the ocean and the atmosphere are at rest with same temperatures but at some point there is a sudden rise in ocean temperature locally maybe due to some chemical processes. For this physical scenario, one might expect that over time the temperature of the atmosphere rises because of heat transfer from the ocean and maybe the air starts to move if there are temperature gradients in the atmosphere. We solve this test problem using the numerical code for our climate model and set the following parameters for simulation: σ = −1.0, γ = −1.0, Rea = 10, Pea = 102 , Roa = 1.0, Reo = 102 , Peo = 103 , Roo = 1.0. We run the simulation for a 2D unit-square domain from time, t = 0 to t = 0.2. The results from the numerical simulation are presented in figures 7.10 and 7.11.

Fig. 7.9: Initial temperature field of the ocean for test case 2

7 Numerical Simulation of an idealized coupled Ocean-Atmosphere Climate Model

(a) Atmosphere temperature field at t = 0.05

(b) ocean temperature field at t = 0.05

(c) Atmosphere temperature field at t = 0.10

(d) ocean temperature field at t = 0.10

(e) Atmosphere temperature field at t = 0.20

(f) ocean temperature field at t = 0.20

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Fig. 7.10: Evolution of ocean and atmosphere temperature fields over time for test problem 2 (Note: the ocean and atmosphere temperatures are scaled differently)

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(a) Atmosphere velocity field at t = 0.05

(b) ocean velocity field at t = 0.05

(c) Atmosphere velocity field at t = 0.10

(d) ocean velocity field at t = 0.10

(e) Atmosphere velocity field at t = 0.20

(f) ocean velocity field at t = 0.20

Fig. 7.11: Evolution of ocean and atmosphere velocity fields over time for test problem 2 (Note: the ocean and atmosphere velocities are scaled differently)

7 Numerical Simulation of an idealized coupled Ocean-Atmosphere Climate Model

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Figure 7.10 shows how the temperature fields of the ocean and atmosphere changes with time starting from initial conditions (figure 7.9). We can see that over time (as expected) the atmospheric temperature rises and its distribution is quite similar to the temperature distribution in the ocean. The presence of temperature gradients in the atmosphere forces the air to move from high temperature regions to low temperature regions (figure 7.11. We have seen earlier (in test problem 1) that a flow of air in the atmosphere forces the ocean to move along with it. The simulation results of test problem 1 also shows that the ocean velocity distribution roughly matches the velocity distribution of the atmosphere. But, we observe something different in the simulation results of test problem 2 (see figure 7.11). The magnitude of the ocean velocity increases over time but its distribution is not similar to the atmosphere velocity distribution. In the atmosphere, the flow is in radially outward direction but in the ocean, the flow is rotational. We believe that, although the coupling term σ (uo − ua ) is forcing the ocean to flow radially outwards, the incompressiblity condition ∇ · uo = 0, on the other hand, is opposing this flow tendency. This is because radially outward flows violate the incompressiblity condition and therefore the ocean can not flow outwards. The ocean, as a result, ends up flowing in circles due to the action of Coriolis forces alone. The circular flow slows down the heat dissipation process in the ocean in comparison to the atmosphere where the flow is radially outwards (figure 7.10). The simulation results for the two test problems show that our climate model is capable of modeling some physical phenomena which involve ocean-atmosphere interactions. Although, our numerical approach has produced consistent and stable solutions for various test problems, it is necessary that the climate model has some desirable mathematical properties like the local well-posedness. This is all the more important because of the presence of Navier-Stokes-like equations in the model whose well-posedness can be very difficult to prove. It has been proven that our coupled ocean-atmosphere model (7.1) does possess local well-posedness property [2]. We refer the reader to [2] for more details.

7.5 Conclusions and future work At present, we have a numerical code for solving an idealized climate model which gives consistent and stable solutions for various test problems. We can use the numerical simulations to study fundamental ocean-atmosphere interactions involving quantities such as the velocity and temperature. Our next step is to incorporate stochasticity into this climate model by following Hasselmann’s paradigm [4] and numerically solve the resulting system of stochastic PDEs. We will follow the approach presented in [2] to incorporate stochasticity into our climate model. This will be the subject of our future work.

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Acknowledgement The authors acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 ”Modeling, Simulation and Optimization of Fluid Dynamic Applications”.

References 1. J. Ahrens, B. Geveci, and C. Law. Paraview: An end-user tool for large data visualization. The visualization handbook 717(8), 2005. 2. D. Crisan, D.D. Holm, and P. Korn. Hasselmann’s Paradigm for Stochastic Climate Modelling based on Stochastic Lie Transport. arXiv preprint arXiv:2205.04560, 2022. 3. H.A. Dijkstra. Nonlinear physical oceanography: a dynamical systems approach to the large scale ocean circulation and El Nino, volume 28, Springer Science & Business Media, 2005. 4. K. Hasselmann. Stochastic climate models Part I. Theory. Tellus 28(6), 473–485, 1976. 5. P. Korn. A regularity-aware algorithm for variational data assimilation of an idealized coupled Atmosphere-Ocean model. Journal of Scientific Computing 79(2), 748–786, 2019. 6. J.P. McCreary Jr. and D.L.T. Anderson. An overview of coupled Ocean-Atmosphere models of El Ni˜no and the Southern oscillation. Journal of Geophysical Research: Oceans 96(S01):3125– 3150, 1991. 7. F. Rathgeber, D.A. Ham, L. Mitchell, M. Lange, F. Luporini, A.T.T. McRae, G.-T. Bercea, G.R. Markall, and P.H.J. Kelly. Firedrake: automating the finite element method by composing abstractions. ACM Transactions on Mathematical Software (TOMS) 43(3), 1–27, 2016.

Chapter 8

Application of p-Laplacian relaxed Steepest Descent to Shape Optimizations in Two-Phase Flows Peter Marvin M¨uller, Martin Siebenborn, Thomas Rung

Abstract The paper is concerned with the minimal drag problem in shape optimization of merchant ships exposed to turbulent two-phase flows. Attention is directed to the solution of Reynolds Averaged Navier-Stokes equations using a Finite Volume method. Central aspects are the use of a p-Laplacian relaxed steepest descent direction and the introduction of crucial technical constraints to the optimization procedure, i.e. the center of buoyancy and the displacement of the underwater hull. The example included refers to the frequently investigated Kriso container ship (KCS).

8.1 Introduction In this paper we formulate the minimal drag problem applied to the fluid dynamic shape optimization of merchant ships exposed to turbulent two-phase flows. Such optimization problems are additionally restricted by geometric constraints on the displacement and the center of gravity location. To this end, we propose an algorithm for handling these constraints based on first-order descent methods for this optimization problem as well as for the related sub-problems. The shape optimization problem can be seen as an optimal control problem where the state is described by a set of partial differential equations (PDEs) which depend on a control. The control, however, is given by the domain where again the Peter Marvin M¨uller Institute for Fluid Dynamics and Ship Theory, Hamburg University of Technology e-mail: [email protected] Martin Siebenborn Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] Thomas Rung Institute for Fluid Dynamics and Ship Theory, Hamburg University of Technology e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Iske, T. Rung (eds.), Modeling, Simulation and Optimization of Fluid Dynamic Applications, Lecture Notes in Computational Science and Engineering 148, https://doi.org/10.1007/978-3-031-45158-4_8

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state is defined on. In addition the domain often has to fulfill additional geometric properties which yield a finite number of additional constraints. For the constraint problem we apply the method of Lagrange multipliers where the derivative of the objective with respect to the control is expressed by primal (physical) and adjoint state variables. It is well known that the adjoint approach is efficient for handling PDE constraint optimization problems featuring a large number of degrees of freedom (control variables) and can be realized with first order descent methods [7]. Utilizing second-order methods, one could obtain an expression for updating the Lagrange multipliers in compliance with additional geometric constraints. For example the method of mappings investigated in [14] allows to consider geometric constraints that depend on the state as the variables (primal and adjoint) are determined on the transformed domain. A drawback of this approach, is that the whole optimality system has to be solved simultaneously, in line with -for example- oneshot methods [16], which violates the sequential framework of many engineering CFD procedures [6]. When solving the optimality system sequentially, the general approach is (1) compute the primal (physical) state, (2) compute the adjoint state variables or Lagrange multipliers and finally (3) determine a descent direction and shape deformation field, respectively. The deformation field is obtained from the shape derivative of the objective function [17, 4]. In [13] a first-order approach for the shape optimization problem with geometric constraints was investigated for geometric constraints that do not depend on the state and thus decouple from the shape optimization problem. This allows exclusively consider the constraints when computing the shape deformation field by applying Newton’s method with the Schur complement method for solving the related saddle point problem. The fluid dynamic problems considered in the present study, however, consider geometric constraints that also depend on the state. Strictly speaking, this does not allow to decouple the geometric constraints from the shape optimization problem. Nevertheless, we will state that a decoupled strategy can be pursued for small step sizes if the shape derivative is computed in the reference domain. When considering the minimal drag problem of a free floating vessel it is necessary to conserve the displacement and center of buoyancy of the hull. This is different to the typical geometric constraints that are given in an aerodynamic shape optimization problem, e.g. volume of a wing or area of a wing section [11]. The displacement has to be maintained in order to guarantee that the optimized ship has the ability to transport the required payload. Preserving the center of buoyancy serves two purposes: Firstly it supports maintaining the hydrostatic floating position and secondly it prevents the ship hull from being moved out of the computational domain. Note that this is not sufficient to also account for changes of the floating position induced by the fluid dynamics. This would also require to consider the rigid body dynamics, which balances the inertia and fluid dynamic forces and moments of the rigid vessel, and is ignored herein. The notation of this work employs J ′ to indicate the derivative of a shape-based objective function J. The gradient of a function with respect to cartesian spatial coordinates is denoted by the nabla operator ∇(·) and the Jacobian refers to D(·).

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The operator A∗ denotes the adjoint operator to A and ⟨·, ·⟩ is used to represent a dual pairing.

8.2 Mathematical Model We consider two and three dimensional flow domains Ω ⊂ Rd (d = 2, 3) which feature a boundary Γ and obstacles Ωobs with Lipschitz boundary Γobs embedded in the flow domain. We aim at minimizing the drag of the obstacle Ωobs by deformation of a reference domain which at the same time serves as the initial configuration. The domain is parameterized by a deformation field u : Rd 7→ Rd . We follow the approach in, e.g., [17, 4] where the domain is transformed by the perturbation of the identity. Therewith the perturbed domain is defined by Ω˜ = (id + u)(Ω ) := {x + u : x ∈ Ω }

(8.1)

with u ∈ W 1,∞ (Ω , Rd ). In the following we use the abbreviation T (x) := x+u(x) and T is a injective mapping with weakly differentiable inverse for sufficiently small u. With this we approximate the shape derivative of a shape function J(Ω ) by the Fr´echet derivative of the mapping W 1,∞ (Ω , Rd ) ∋ u 7→ J((id + u)(Ω )), viz. J(Ω˜ ) = J(Ω ) + J ′ (Ω )u + o(∥u∥) for ∥u∥ → 0

(8.2)

where u 7→ J ′ (Ω )u is linear regarding u. To outline the central idea of this paper we first consider a generic constraint shape optimization problem min Ω

J (Ω , y) s.t. e(Ω , y) = 0

and gi (Ω , y) = 0,

i = 1, . . . , m.

(8.3)

where e(Ω , y) denotes the PDE constraint that describes the state y and gi (Ω , y), i = 1, . . . , m are a finite number of additional geometric constraints on the domain Ω . It is worth mentioning that the geometric constraints are restricting the shape of the domain Ω rather than contributing to the characterization of the state y. More precisely the state is fully described by the underlying boundary value problem e(Ω , y) = 0 for a specific domain Ω . The change of the geometry, however, is restricted by the state and the solution of the PDE, respectively. For the hydrodynamic problem at hand, this is described in greater detail in Section 8.3. Assuming that the state constraint e has a unique solution on Ω and thus the control-to-state map Ω 7→ y(Ω ) exists, we obtain the reduced objective function J(Ω ) := J (Ω , y(Ω )). Furthermore we assume that the shape function J(Ω ), as well as e(Ω , y) and gi (Ω , y), i = 1, . . . , m are continuously Fr´echet differentiable. Upon this we define the augmented Lagrange function L(Ω , y, z, λ ) = J(Ω ) + ⟨z, e(Ω , y)⟩ + λ T g(Ω , y) (8.4)

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with the Lagrange multipliers z and λ = (λ1 , . . . , λm )T ∈ Rm . In the following z is also referred to as the adjoint state. Algorithm 1 outlines the general shape optimizaAlgorithm 1 Shape optimization procedure 1: repeat 2: Compute y at u = 0 with e(Ω , y) = 0. 3: Compute the adjoint state z at u = 0 that fulfills

 0 = Jy (Ω , y) δy + ey (Ω , y)∗ z, δy + λ T gy (Ω , y) δy

∀δy

(8.5)

Find a descent direction V at u = 0 by solving the minimization problem

4:

min

V ∈W 1,∞ , ∥DV ∥≤1

J ′ (Ω )V =

min

V ∈W 1,∞ , ∥DV ∥≤1

 Ju (Ω , y) + eu (Ω , y)∗ z + (λ )T gu (Ω , y), V (8.6)

with an iterative scheme while successively updating λ ← λ + τ⟨gu (Ω , y),V ⟩ with a suitable step size τ > 0. 5: Choose a sufficient step size ε > 0 6: Update the shape by applying the transformation Ω ← (id + εV )(Ω ) 7: until converged

tion procedure which is based on an augmented Lagrange method of multipliers. Usually the procedure would contain two nested loops, where the shape optimization problem by itself is solved several times with constant values of λ , and the update of the multiplier λ is performed after each shape optimization loop. To reduce the related efforts in practical applications, the update of λ is performed within the sub-optimization problem in (8.6), and the previous value of λ is used (as an approximation) in (8.5) for the current shape optimization step. This allows to solve the shape optimization problem only once, but does not guarantee the convergence of the algorithm. Indeed, all numerical experiments discussed in Section 8.4 show stable reductions of the objective functional while the geometric constraints are all met within a prescribed tolerance at each iteration. The identification of a deformation field V in accordance with Line 4 of algorithm 1 is itself demanding. To this end, we follow the approach suggested in [3, 12] and determine a descent direction by finding a minimizer of the p-Laplace relaxed problem min

V ∈W 1,p

1 p

Z Ω

 p (∇V : ∇V ) 2 dx + Ju (Ω , y) + eu (Ω , y)∗ z + λ T gu (Ω , y), V . (8.7)

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Relation (8.7) approaches the limiting problem (8.6) for p → ∞, which characterizes the steepest descent direction in W 1,∞ -topology and therefore adheres to the firstorder optimality condition p−2

Z

(∇V : ∇V ) 2 ∇V : ∇U dx Ω

 + Ju (Ω , y) + eu (Ω , y)∗ z + λ T gu (Ω , y), U ≥ 0 ∀U .

(8.8)

8.3 Computational Model

Γwall

v∞ Ω

ΓobsD

Γin Ωobs

ΓobsN

c=1

Γout

c=0

x2 x1 Γwall

Fig. 8.1: Illustration of the flow and obstacle domains and the boundaries. The distribution of the fluid phase is indicated by the air volume concentration c, i.e. c = 1[0] for air[water] filled regions. As outlined in Figure 8.1, the flow domain is multiple-connected with one or several interior boundaries in addition to a single outer boundary, and is occupied by two immiscible incompressible fluids, i.e. air and water. The outer boundary is subdivided into the disjoint subsets inlet Γin , outlet Γout and lateral as well as horizontal walls Γwall . The boundary Γobs of the obstacle Ωobs is fractioned into a nondeformable part ΓobsD and a deformable part ΓobsN . The optimization aims at minimal resistance of the obstacle Ωobs by deforming the boundary ΓobsN . Because the flow domain Ω is the difference of the overall holdall domain and Ωobs finding a optimal shape of Ωobs is equivalent to finding the optimal shape of the domain Ω . On the one hand, the state is given by the velocity v : R+ × Ω → Rd , the total pressure p : R+ × Ω → R and the indicator function/volume concentration c : R+ × Ω → [0, 1] to distinguish between the two immiscible fluid phases air and water. On the other hand the additional state variables kinetic turbulent energy k : R+ × Ω →

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R+ and dissipation rate of kinetic turbulent energy ω : R+ × Ω → R are introduced for turbulence modeling. Here we consider the Wilcox k-ω turbulence model [20]. The total pressure p consists of the pressure p and the hydrostatic pressure −ρg · r with the acceleration due to gravity g and the position r. For the case sketched in Figure 8.1 the gravitation is pointing in negative x2 -direction and thus with the basis vector of unit length r = e2 the total pressure reads p = p − ρgx2 . For the turbulent two phase flow we consider the following RANS equations on [0, T ] × Ω   ∂ (ρv) k T + div ρv ⊗ v) − (µ + ρ )(∇v + ∇v ) + ∇p − f = 0, ∂t ω div(v) = 0, ∂c + (v · ∇)c = 0,  ∂t  ∂ (ρk) k + div ρvk) − (µ + σk ρ )∇k − P + β ∗ ρkω = 0, ∂t ω   ∂ (ρω) ω k ρ + div ρvω − (µ + σω ρ )∇ω − γ P + β ρω 2 − σd ∇k · ∇ω = 0 ∂t ω k ω (8.9) where the turbulent production refers to P = ρ ωk (∇v + ∇vT ) : ∇v and f : R+ × Ω → Rd is a general source term that does not depend on the state. The parameters µ, ρ > 0 are the molecular viscosity and density of the respective fluids. They are composed from the bulk properties for air and water phase, i.e. ρair , ρwater and µair , µwater , which are considered constant and the linear algebraic equation of state, viz. ρ = ρair c + ρwater (1 − c) and µ = µair c + µwater (1 − c), holds. This finally yields a solenoidal velocity field with div(v) = 0. The system is closed by the following set of initial and boundary conditions ∂k ∂ω = 0, =0 ∂n ∂n ρk 3 v = v∞ , c = c∞ , k = ∥v∞ ∥22 α 2 , ω = 2 µνt+ k ∂k ∂ω ∂c =0 = 0, =0 (µ + ρ )(∇v + ∇vT ) · n = pn, ω ∂n ∂n ∂n 3 ρk and v(0) = v0 , c(0) = c0 , k(0) = ∥v0 ∥2 α 2 , ω(0) = 2 µνt+ v = 0,

∂c = 0, ∂n

onΓobs and Γwall , on Γin , on Γout in Ω .

(8.10) The parameters σk , σω , σd , γ, β , β ∗ , νt+ are real valued positive constants of the turbulence model and α ∈ (0, 1]. The force vector F acting on the boundary of the obstacle Γobs is given by   Z
k T F(Γobs ) = (µ + ρ ) ∇v + ∇v · n − pn ds . (8.11) ω Γobs

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The resistance or drag is associated with the component of the force in (8.11) in direction of the approaching flow which we assume to be aligned with the x1 -direction. Hence, the resistance of the obstacle is given by the projection −F(Γobs ) · e1 where e1 is the basis vector of unit length in x1 -direction. For the formulation of the shape derivative it is favorable to consider the volume formulation of the objective function. Introducing a smooth extension η : Ω → Rd with η|Γobs ≡ −e1 and η|Γ \Γobs ≡ 0, one obtains the equivalent volume formulation through integration by parts of (8.11) [2, Section 5.1]  Z 
k J(Ω ) = (ρ(v · ∇)v − f ) · η + (µ + ρ ) ∇v + ∇vT : ∇η − p div(η) dx . ω Ω (8.12) We focus upon the steady state resistance and thus assume the flow to be stationary and all time derivatives in (8.9) vanish in the converged state. In practice this means that the average over a sufficient pseudo-time/iteration period of the state variables is used to suppress minor remaining variations of the flow and the objective functional. The geometric constraints for preserving the water displacement of Ωobs and the center of buoyancy the obstacle are give by gi (Ω , c) = gd+1 (Ω , c) =

Z ZΩ Ω

(1 − c)xi dx, (1 − c) dx.

i = 1, . . . , d

and (8.13)

The appearance of the concentration c in (8.13) secures that the displacement of the underwater hull of the vessel (water wetted part) is preserved rather than the volume of the whole hull. The formulation of the geometric constraints here differs substantially from previous work, e.g. [14] and [13], as it depends on the geometry and the solution of a PDE. Because the derivation as well as implementation of the adjoint problem corresponding to the primal problem (8.9) - (8.10) holds several challenging aspects we follow common practice and neglecting the turbulence model for the adjoint system. This simplification is also known as frozen turbulence assumption [5, 15, 19] where the state variables k and ω are treated as constants when computing the derivative w.r.t. the state as well as the shape and the deformation field, respectively. Hence, the state variable is considered y = (v, p, c) in the following. Hereon we define the augmented Lagrange function

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L(Ω , (v, p, c), (w, q, h), γ, λ ) = J(Ω ) "  Z   k div ρv ⊗ v − (µ + ρ )(∇v + ∇vT ) + ∇p − f · w + ω Ω # − div(v)q + div(vc)h dx +

Z

(8.14)

γ · v dx

Γobs d

+ ∑ λi

Z Ω

i=1

(1 − c) xi dx + λd+1

Z Ω

(1 − c) dx,

where the multiplier γ corresponds to the Dirichlet boundary conditions of the velocity v = 0 that hold on Γobs , and λ = (λ1 , . . . , λd , λd+1 ) is associated with the center of buoyancy and displacement constraint. The adjoint state is characterized by the derivative of (8.14) w.r.t. the state y = (v, p, c) which leads to the variational form 0=

k )(∇w + ∇wT ) : ∇δv − ρv · (∇w + ∇wT ) · δv ω Ω − div(δv ) q − δ p div(w) + div(δv c)h + div(vδc ) h i + δc ∆ρ (v ⊗ v) : ∇w + δc ∆ µ (∇w + ∇wT ) : ∇v dx

Z h

+

(µ + ρ

Z Γobs d

γ · δv dx

+ ∑ λi

Z

i=1

Ω

−δc xi dx + λd+1

Z Ω

(8.15)

−δc dx

∀δy = (δv , δ p , δc ) where ∆ρ = ρair − ρwater and ∆ µ = µair − µwater . The boundary integrals vanish if the boundary conditions w = −η

and (µ + ρ


k ) ∇w + ∇wT · n = (q − ch)n ω

on Γ \ Γout on Γout .

(8.16)

hold and by choosing γ := −(µ + ρ


k ) ∇w + ∇wT · n + (q − ch)n. ω

(8.17)

In order to derive the directional derivative of the reduced objective J ′ (Ω )V we formally apply C´ea’s method. For a detailed description see [1, Section 4.6]. In general the shape derivative of a objective function J(Ω ) has a volume and an equivalent surface formulation. For computational reasons it is favorable to consider the surface formulation even though it requires higher regularity of the solutions (v, p, c)

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139

and (w, q, h) of the primal and adjoint problem (8.9) - (8.10) and (8.15), respectively. Utilizing [1, Theorem 4.2 and 4.3] and assuming that (v, p, c) and (w, q, h) have sufficient regularity we obtain   Z k ∂w ∂v ′ J (Ω )V = −(µ + ρ ) · V · n dx ω ∂n ∂n Γobs (8.18) Z d + ∑ (λi − τgi (Ω , c)) (1 − c)xi V · n dx. Γobs

i=1

As mentioned in Section 8.1 the deformation field is obtained from the directional shape derivative by solving the minimization problem (8.7) 1 1,p d V ∈W (Ω ,R ) p min

Z Ω

p

(DV : DV ) 2 dx + J ′ (Ω )V.

(8.19)

To ensure that the outer boundary remains unchanged the Dirichlet condition u = 0 holds almost everywhere on Γin ∪ Γout ∪ Γwall . In addition parts of the obstacle may be fixed and thus u = 0 also holds a.e. on ΓobsD and natural boundary conditions hold on ΓobsN where the boundary is deformed. To computing the shape deformation field V characterized by the minimization problem (8.19) we suggest the procedure sketched in Algorithm 2. For the sake Algorithm 2 Picard Iteration for Augmented p-Laplacian Problem 1: λ ← 0, p ← 2, u ← 0 2: while p < pmax do 3: k←0 4: repeat 5: Obtain a preliminary V˜ k by solving the linearized problem

Z Ω

6: 7:

τ > 0.

(∇V k−1 : ∇V k−1 )

p−2 2

∇V˜ k : ∇U dx + J ′ (Ω )U

for all U

Relax update V k ← V k + ω(V˜ k −V k−1 ) with ω ∈ (0, 2) Update multiplier λ k ← λ k−1 + τ⟨gu (Ω , y),V k ⟩ with a suitable step size

8: k ← k+1 k − λ k−1 ∥2 + |λ k − λ k−1 |2 ≤ tol 9: until Rk = ∥V k −V k−1 ∥2L2 + ∥λbc v v 2 bc 10: p ← p + pinc 11: end while

of briefness we omit the precise solution method in Line 5 of Algorithm 2 as the solution strategy does not depend on the discretization. Nevertheless, in Section 8.4 we consider the finite volume method wherefore a formulation can be found by partial integration [12].

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8.4 Numerical Results Results presented in this paper are obtained from the finite volume procedure FreSCo+ [18] for the KCS ship in model scale [10] at Reynolds- and Froude numbers of Re = 1.43 · 107 and Fn = 0.26. Also the shape deformation field obtained from the p-Laplacian relaxed problem in (8.8) is approximated with a Picard iteration and finite volume discretization. Figure 8.2 shows the initial configuration with the hull of the KCS and free surface elevation. We investigate two different cases to obtain the shape deformation

Fig. 8.2: Initial hull shape and elevation of the free surface. field from (8.19) which differ in the boundary conditions along the hull. Firstly, we consider the whole hull to be free for deformation, and secondly the air-wetted part of the hull remains fixed and only the underwater part ΓobsN of the hull is deformed. In both studies the deck as well as the transom and a part of the propeller shaft remain fixed. For compatibility the step size for both cases is ε = 1/100 which corresponds to a maximum displacement is approximately 1/1000 of the length of the ship. As stated in [3, 8, 12] the values for p should be large in order to obtain a sufficient approximation for a descent direction in W 1,∞ . However, due to the nonlinearity of (8.19) the numerical computation for large values of p is demanding [9] and we consider pmax = 2.6 as an for both test cases. For Algorithm 2 to converge for p > 2 it requires a good initial guess u0 . Therefore we consider the iteration over a sequence in p = {2, 2.3, 2.6} to compute the initial guess for (8.19) with p = pmax [12, 13]. To review Algorithm 2 we exemplary look at the first iteration of the shape optimization procedure in Algorithm 1 for the first case. Figure 8.4 shows the residuals of the procedure in Algorithm 2. The graphs display the individual contributions to the residual Rk from Line 9 in Algorithm 2 for the tolerance tol = 10−9 . The procedure is stable with the penalty factor τ = 10 whereat the multipliers converge faster

8 Application of p-Laplacian relaxed Steepest Descent

141

ΓobsD

ΓobsN 1st test case.

2nd test case.

Fig. 8.3: Boundary type layout for first test case (left) and second test case (right). The fixed part ΓobsD is colored in light gray and the deformed part ΓobsN in blue. The first test case assigns the complete hull to ΓobsN . The second test case only assigns the underwater hull to ΓobsN , whereas air-wetted hull parts above the water line belong to ΓobsD . 10−5

p = 2.0

p = 2.3

p = 2.6

10−7

Residual

10−9

10−11 ∥V k −V k−1 ∥2L2

10−13

k − λ k−1 ∥2 , ∥λbc 2 bc |λvk − λvk−1 |2 Rk

10−15 100

101

102

103

Iteration k

Fig. 8.4: Residual plots of the p-Laplacian problem for the sequence of p = {2, 2.3, 2.6}. Displayed graphs refer to the residuals of the deformation field V , the Lagrangian multipliers λbc = (λ1 , . . . , λd ) and λv = λd+1 as well as the total residual R.

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than V k → V . The multiplier λ and thus the choice of τ however heavily depends on the computed flow. For both cases the normalized values of the objective function −F(Γobs ) · e1 are shown in Figure 8.5. It can be observed that the functional values decline faster

J(Ω )/J(Ω0 )

1

0.98

all deforming submerged only

0.96 0

5

10

15

20

# Shape

Fig. 8.5: Evolution of the normalized drag force objective function obtained with p = 2.6 when deforming the whole hull (red line, test case 1) and only the underwater hull (blue line, test case 2). for the first test case where the deformation is not limited to the submerged part (solid line). A possible explanation might be the larger maximum value of the displacement in the first case. The geometric constraints in the second case are more restrictive and therefore might result in smaller values of the displacement fields. One could choose a larger step size for the second case but the strategy for choosing a step size which is suitable might be computationally demanding. However, this is concomitant with large deformations at the intersection of the hull and the deck, particularly in the bow regime, cf. Fig. 8.6. As outlined by the magnification in Fig. 8.6, we observe locally vanishing cell volumes after 22 iterations in this regime and the simulations terminate. The issue only occurs for the first test case, where the whole vessel can deform. Using the second approach, the deformation is confined to the water-wetted surface and one could perform further iterations. Figure 8.7 compares the body plans of the initial (black) and the two modified designs of the 22nd iteration. The two strategies predict virtually the same underwater hull deformations. However, differences occur when the free surface is approached, and more pronounced deformations are experienced in the first case, where the whole vessel can deform. Moreover, differences also occur in the bow regime, where the submerged only design (blue) predicts a stronger displacement in the upper part, as indicated by the magnification of the section lines close to the bow in Fig. 8.7.

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143

Fig. 8.6: Illustration of the experienced grid deterioration in the bow region of the KCS container vessel after 22 design iterations in combination with the first test case, where both the water and air wetted hull sections are deformed (cf. Fig. 8.3).

8.5 Summary We presented an algorithmic approach for fluid dynamic shape optimization of floating ships exposed to turbulent two-phase flows under geometric constraints. The main goal was to consider geometric constraints for displacement and the center of buoyancy in order to exclude trivial or undesirable optimal solutions in conjunction with shape updates which approximate the steepest descent direction in a Banach space. The presented algorithm is based on the augmented Lagrange method of multipliers for the geometric constraints and the PDE constraints are treated utilizing the corresponding adjoint operator. Numerical experiments were carried out based on the geometry of the KRISO Container Ship in model scale at realistic test conditions [10]. Results show that the suggested approach leads to deformation fields that fulfill the geometric constraints up to a predetermined tolerance. However, the attainable drag reductions are limited by the degeneration of the computational grid and the discretization of the domain still becomes unfeasible coherent with the successive shape updates. Because the domain of definition of the deformation field is not the holdall domain, but the flow domain (i.e. without the obstacle), it is globally not an injection and geometry overlaps can mathematically not be avoided. Future research may thus consider a discretization of the entire domain including the interior of the obstacle. Moreover, the algorithm may be applied to free-floating vessels sub-

0.7

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0.6

0.8 0.7

0.5

0.6

0.4

z

0.5 0.4 0.3

0.3

0.2 0.1

0.2

0 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

y

0.1 0 -0.6

Fig. 8.7: Body plan of the initial hull (dashed black line) and the optimized hulls of the 22nd design candidate predicted by the all deforming (red line) and only underwater hull deforming (blue line) approaches.

-0.4

-0.2

0

0.2

jected to rigid-body dynamics, which eventually gives a technically more realistic optimization problem. y

Acknowledgements The authors acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 “Modeling, Simulation and Optimization of Fluid Dynamic Applications”. The computations were performed with resources provided by the North-German Super-computing Alliance (HLRN).

Replication of results The geometry of the KCS is available at http://www.simman2008.dk/KCS/ kcs_geometry.htm. A proprietary software is used for mesh generation. Computations are carried out with the in-house finite volume code FreSCo+ .

8 Application of p-Laplacian relaxed Steepest Descent

145

References 1. G. Allaire, C. Dapogny, and F. Jouve. Chapter 1 - shape and topology optimization. In: Geometric Partial Differential Equations - Part II, volume 22 of Handbook of Numerical Analysis, Elsevier, 1–132, 2021. 2. C. Brandenburg, F. Lindemann, M. Ulbrich, and S. Ulbrich. A continuous adjoint approach to shape optimization for Navier-Stokes flow. In: Optimal Control of Coupled Systems of Partial Differential Equations. Intl. Series of Numerical Mathematics. Birkh¨auser, Basel, 2009. 3. K. Deckelnick, P.J. Herbert, and M. Hinze. A novel W 1,∞ y approach to shape optimisation with Lipschitz domains. ESAIM: COCV, 28, 2022. 4. M.C. Delfour and J.-P. Zol´esio. Shapes and Geometries. SIAM, 2011. 5. R.P. Dwight and J. Brezillon. Effect of approximations of the discrete adjoint on gradient-based optimization. AIAA 44(12), 2012. 6. J.H. Ferziger and M. Peri´c. Computational Methods for Fluid Dynamics. Springer, Berlin, 4th edition, 2020. 7. M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich. Optimization with PDE Constraints. Volume 1, Springer, 2009. 8. H. Ishii and P. Loreti. Limits of solutions of p-Laplace equations as p goes to infinity and related variational problems. SIAM Journal on Mathematical Analysis 37(2), 411 – 437, 2005. 9. S. Loisel. Efficient algorithms for solving the p-Laplacian in polynomial time. Numerische Mathematik 146(2), 369–400, 2020. 10. Maritime and Ocean Engineering Research Institute. Kriso container ship geometry. http: //www.simman2008.dk/KCS/kcs_geometry.htm. 11. B. Mohammadi and O. Pironneau. Applied Shape Optimization for Fluids. Volume 2, Oxford University Press, 2010. 12. P.M. M¨uller, N. K¨uhl, M. Siebenborn, K. Deckelnick, M. Hinze, and T. Rung. A novel pharmonic descent approach applied to fluid dynamic shape optimization. Struct. Multidic. Optim. 64, 2021. 13. P.M. M¨uller, J. Pinz´on, T. Rung, and M. Siebenborn. A scalable algorithm for shape optimization with geometric constraints in Banach spaces. https://arxiv.org/abs/2205. 01912, 2022. 14. S. Onyshkevych and M. Siebenborn. Mesh quality preserving shape optimization using nonlinear extension operators. Journal of Optimization Theory and Applications 189, 291–316, 2020. 15. C. Othmer. A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows. Numerical Methods in Fluids 58(8), 861–877, 2008. ¨ 16. E. Ozkaya and N.R. Gauger. Single-step one-shot aerodynamic shape optimization. In: Optimal Control of Coupled Systems of Partial Differential Equations, Birkh¨auser, Basel, 191–204, 2009. 17. J. Sokolovski and J.-P. Zol´esio. Introduction to Shape Optimization. Springer-Verlag, 1992. 18. A. St¨uck and T. Rung. Adjoint rans with filtered shape derivatives for hydrodynamic optimisation. Computers & Fluids 47(1), 22–32, 2011. 19. A. St¨uck and T. Rung. Adjoint complement to viscous finite-volume pressure-correction methods. Journal of Computational Physics 248, 402–419, 2013. 20. D.C. Wilcox et al. Turbulence modeling for CFD. Vol. 2, DCW industries La Canada, 1998.

Chapter 9

Towards Computing High-Order p-Harmonic Descent Directions and Their Limits in Shape Optimization Henrik Wyschka, Martin Siebenborn

Abstract We present an extension of an algorithm for the classical scalar p-Laplace Dirichlet problem to the vector-valued p-Laplacian with mixed boundary conditions in order to solve problems occurring in shape optimization using a p-harmonic approach. The main advantage of the proposed method is that no iteration over the order p is required and thus allows the efficient computation of solutions for higher orders. We show that the required number of Newton iterations remains polynomial with respect to the number of grid points and validate the results by numerical experiments considering the deformation of shapes. Further, we discuss challenges arising when considering the limit of these problems from an analytical and numerical perspective, especially with respect to a change of sign in the source term.

9.1 Introduction Shape optimization constrained by partial differential equations is a vivid field of research with high relevance for industrial grade applications. Mathematically, we consider the problem min J(vs , Ω ) Ω ∈A

s.t. E (vs , Ω ) = 0

where J denotes the objective or shape functional depending on a state variable vs and a Lipschitz domain Ω ⊂ Rd , which is to be minimized over the set of admissible shapes A . Further, the state and the domain have to fulfill a PDE constraint E . A Henrik Wyschka Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] Martin Siebenborn Department of Mathematics, Universit¨at Hamburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. Iske, T. Rung (eds.), Modeling, Simulation and Optimization of Fluid Dynamic Applications, Lecture Notes in Computational Science and Engineering 148, https://doi.org/10.1007/978-3-031-45158-4_9

147

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common solution technique for this type of problem is to formulate it as a sequence of deformations to the initial shape [18]. Each of these transformations has the form Ω˜ = (id + tv)(Ω )

(9.1)

for a step-size 0 < t ≤ 1 and a descent vector field v : Rd → Rd in the sense J ′ (Ω ) v < 0 with shape derivative denoted by J ′ . Hence, the set A is implicitly defined by all shapes reachable via such transformations to the initial shape. From the analytical derivation of this procedure it is required that the v are at least W 1,∞ (Rd , Rd ) inherently ensuring that all admissible shapes remain Lipschitz. However, in practice this condition is often neglected and so-called Hilbert space methods [1] are used. The most prominent example of those consists of finding v ∈ W k,2 (Ω , Rd ) for some k ∈ N. This results in smooth shapes, which are relevant for some applications, but are not necessarily optimal. Further, the mesh quality often deteriorates and it would require a high differentiability order k to obtain a Lipschitz transformation from the Sobolev embedding. Recent development suggests using a p-harmonic approach [3] to determine descent directions. This technique can be obtained by considering the steepest descent in the space of Lipschitz transformations arg min

v∈W 1,∞ (Ω ,Rd )

J ′ (Ω ) v.

(9.2)

∥∇v∥L∞ ≤1

and then potentially relax the problem. The descent field is obtained by the solution of a minimization problem for 2 ≤ p < ∞ reading arg min

v∈W 1,p (Ω ,Rd )

1 p

Z Ω

∥∇v∥2p dx + J ′ (Ω ) v.

(9.3)

Since the classical Hilbert space setting is recovered for the linear case p = 2, this approach can also be understood as a regularization of such methods. It is demonstrated that this approach is superior in terms of representation of sharp corners as well as the overall mesh quality and yields improving results for increasing p [12]. On the downside, the stated numerical examples also made clear that it is challenging to compute solutions for p > 5 due to serious difficulties in numerical accuracy and the need to iterate over increasing p with the presented solution technique. We now assume that the shape derivative J ′ (Ω ) can be expressed via integrals over the domain and the boundary. From the perspective of shape calculus [4, 18] this is a reasonable choice to cover relevant applications. An exemplary computation including geometric constraints can be found in [16]. For the finite setting we recover the general formulation of a problem for the p-Laplacian ∆ p v = ∇ · (∥∇v∥2p−2 ∇v) reading

9 High-Order p-Harmonic Descent Directions and Their Limits

arg min J p (u) = arg min u∈U

p

u∈U

p

1 p

Z

∥∇(u + g)∥2p dx − | {z } Ω

Z Γ

149

hu dΓ −

Z Ω

f u dx

(9.4)

p

=: ∥u+g∥X p (Ω )

over the set U p := {u ∈ W 1,p (Ω , Rd ) : u = 0 a.e. on ∂ Ω \ Γ } with v := u + g for an arbitrary extension of g to the whole domain in {v ∈ W 1,p (Ω , Rd ) : v = g a.e. on ∂ Ω \ Γ }. The corresponding Euler-Lagrange equation is given by  −∆ p v = f in Ω ,  . ∥∇v∥2p−2 ∂η v = h on Γ ,  v=g on ∂ Ω \ Γ Consequently, in order to obtain an efficient shape optimization algorithm, it is necessary to find an efficient routine to solve this problem for a preferably high order p. On the other hand, it is desirable to directly solve the non-relaxed limit case for p = ∞ to obtain analytically valid and possibly superior results. The remainder is structured as follows: In section 9.2 we construct an algorithm for the vector-valued p-Laplace problem by extending a scalar algorithm and integrating support for Neumann boundary conditions. We present numerical results obtained with this algorithm in section 9.3. After that, we discuss a version of the presented algorithm for p = ∞ before we draw conclusions in section 9.5.

9.2 High-Order p-Harmonic Descent In [11] an algorithm to solve scalar p-Laplace problems with Dirichlet boundary conditions is presented in order to show they are solvable in polynomial time. The approach relies on interior-point methods and the theory of self-concordant barriers, including estimates in terms of the barrier parameter ν given by Nesterov [13]. Besides the computational complexity, one of the main advantages is that the solution to the linear Laplacian is a sufficient initial guess for any p and no iteration over p is required. We construct an extension of the algorithm for vector-valued functions featuring mixed Dirichlet and Neumann boundary conditions in order to apply it to shape deformation problems. However, we will only introduce minor changes to the original proof and show that the polynomial estimate holds with mixed boundary conditions in the scalar setting. First, we recall some basic notation for finite elements. Let hΩ be a parameter and ThΩ a triangulation of Ω with n nodes and m elements. For linear elements, the affine coordinate mapping from the reference element to a physical element Ki can be expressed by x = Ji r + bi . The triangulation is called quasi-uniform if there exists parameter ρΩ such that hΩ ≤ ∥Ji ∥2 ≤ ρΩ hΩ for all elements. Further let VhΩ denote the space of vector-valued, piece-wise linear Lagrange elements over ThΩ .

150

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′

For a general function u : Rd → Rd , the coefficient vector u ∈ Rnd is given here by blocks of length d ′ for each node k. In each of those blocks, all entries are associated with the corresponding classical scalar nodal basis function Φk . This allows us to extend the notion of discrete derivative matrices to vector-valued functions by ′ ( j,r) D( j,r) ∈ Rm×nd . The entries are given by Di,d ′ (k−1)+r = ∂∂x j Φk (x(i) ) consisting of derivatives of the basis functions evaluated at the element midpoints x(i) . Note that, only the columns that correspond to the respective image dimension r in the block of node k are non-zero. By this construction, the multiplication to a coefficient vector returns the discrete derivative in direction x j of the r-th image dimension at each element midpoint x(i) . Subsequently, the matrices are sparse with entries only in every d ′ -th column and in these columns only if the node k is in the support of the element i. The vector of weights is given by ω (l) , where l denotes the number of local quadrature points. This also means for triangular elements using the mid-point rule ω (1) = ω is the vector of element volumes. The discretization of the p-Laplacian term from the problem (9.4) is then given by m

d′

d

∥u + g∥Xp p (Ω ) = ∑ ωi

∑ ∑ [D( j,r) (u + g)]2i

i=1

! 2p .

j=1 r=1 ′

′

Further, we define basis matrices E ∈ Rmld ×nd yielding the function value for all image dimensions on the l local quadrature points of all elements on multiplication to a coefficient vector. This discrete operator will later simplify the proof by allowing us apply similar techniques as those for D( j,r) also for the occurring mass matrices M = [E (l) ]⊺W (l) E (l) where W (l) = diag(ω (l) ). Note that all definitions hold similarly for boundary elements and will be denoted (l)

(l) (l)

by a bar, e.g. M = [E ]⊺W E

′

′

∈ Rnd ×nd .

With these connections established, we can now state the discretized and reformulated problem for the vector-valued p-Laplacian with mixed Dirichlet and Neumann boundary data in the following lemma. Lemma 9.1. The problem (9.4) of minimizing J p (u) over the given finite element space VhΩ with 1 ≤ p < ∞ satisfying the additional upper bound ωi ∥∇(u+g)|Ki ∥2p ≤ R is equivalent to the classical convex problem   −M f − Mh (9.5) arg min ⟨c, x⟩ with c = ω x∈Q p

p

with the constrained search set given by   ! 2p   d d′ ∧ ωi si ≤ R . Q p = (u, s) ∈ Rn × Rm : si ≥ ∑ ∑ [D( j,r) (u + g)]2i   j=1 r=1

(9.6)

9 High-Order p-Harmonic Descent Directions and Their Limits

151

The obtained problem is now a classical convex optimization problem by the minimization of a scalar product over a constrained set. An algorithm is obtained by constructing a ν-self-concordant barrier for Q p , computing its first and second derivative and then applying an interior-point method [13]. Lemma 9.2. A 4m-self-concordant barrier for Q p is given by the function F(u, s) = − ∑ log zi − ∑ log τi i

d

zi =

d′

( j,r) u + D( j,r) g)i ]2 ∑ ∑ [(D | {z }

2/p si −

j=1 r=1

where

i

=:y( j,r)

and τi = R − ωi si .

Remark 9.1. By construction the barrier function F is twice differentiable with the first derivative reading   F ′ F = u where Fs d

Fu = 2 ∑

d′

∑ [D( j,r) ]⊺

j=1 r=1

y( j,r) z

and Fs = −

2 1 2/p−1 ω s + . pz τ

The second derivative is given by   F Fus F ′′ = uu where ⊺ Fus Fss d

Fuu = 2 ∑

d′

∑ [D( j,r) ]⊺ Z −1 D( j,r)

j=1 r=1

+4

d

d′

d

d′

∑ ∑ ∑ ∑ (Y ( j1 ,r1 ) D( j1 ,r1 ) )⊺ Z −2 (Y ( j2 ,r2 ) D( j2 ,r2 ) ),

j1 =1 r1 =1 j2 =1 r2 =1

Fus = −

4 p

2 Fss = − p

d

d′

∑ ∑ (Y ( j,r) D( j,r) )⊺ Z −2 S2/p−1 ,

j=1 r=1



 2 4 − 1 Z −1 S2/p−2 + 2 Z −2 S4/p−2 +W 2 T −2 , p p

S = diag(s), W = diag(ω), Y = diag(y), Z = diag(z), T = diag(τ). Remark 9.2. Note that in the construction the 2p -th power of the norm has been moved. Thus the additional constrained si ≥ 0 would be required, as stated in the original version. However, zi → ∞ as si ↘ ∥∇(u + g)|Ki ∥22 or si ↗ −∥∇(u + g)|Ki ∥22 and thus leave us with a correct barrier on the intended set as well additional separated set. Therefore, we drop that condition here and in practice we ensure it by the choice of the initial value with si ≥ 0 and keep track via the line-search in the

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adaptive path-following. This not only simplifies the notation, but also reduces the computational effort. For completeness, we state the proof for the computational complexity in the scalar case with additional Neumann boundary conditions. Note that the obtained bound on the iterations differs from the one in the original version [11, Theo. 1]. This stems from a change we have to introduce to the Hessian and subsequent computations as well as using a different estimate in terms of the barrier parameter, which features ˆ ∗x∗ . However, the estimate on the required iterations for a naive ∥c∥x∗∗ instead of ∥x∥ F F interior-point method with fixed step size [13, Sec. 4.2] is only of theoretical interest. Even though the bound is not sharp, the required iterations are not reasonable for practical applications. Therefore, variations with modified step lengths [14] or an adaptive step size control [11] are used. Although estimates are worse in these settings, we will see in section 9.3 that the latter performs significantly better in practice. Consequently, we will not show theoretical results for the vector-valued setting. Theorem 9.1. Let 1 ≤ p < ∞. Assume that Ω ⊂ Rd is a Lipschitz polytype of width L and that ThΩ is a quasi-uniform triangulation of Ω , parametrized by 0 < hΩ < 1 and with quasi-uniformity parameter 1 ≤ ρΩ < ∞. Further assume g ∈ W 1,p (Ω ), f ∈ Lq (Ω ) and h ∈ Lq (Γ ) with conjugated exponents 1p + 1q = 1 are piece-wise lin-

ear on ThΩ and let VhΩ ⊂ W01,p (Ω ) be the piece-wise linear finite element space on ThΩ whose trace vanishes. Fix a quadrature Q with positive weights such that the integration is exact, R ≥ R∗ := 2(1 + ∥g∥Xp p (Ω ) ) sufficiently large and let ε > 0 be an accuracy.

In exact arithmetic, a naive interior-point method consisting of auxiliary and main path-following using the barrier function from lemma 9.2 to minimize J p (u) over u ∈ VhΩ , starting from xˆ = (0, s) ˆ with sˆi = 1 + (∑ j ∑r [D( j,r) g]2i ) p/2 , converges to the global minimizer in VhΩ in at most q N ≤ 14.4 |Ω |d!h−d Ω

[K ∗ + log(ε −1 h−1−7.5d R5 (1 + ∥g∥X p (Ω ) )(∥ f ∥Lq (Ω ) + ∥h∥Lq (Γ ) + 1))]. Ω

√ iterations. This results in a computational complexity denoted by O( n log(n)). The constant K ∗ = K ∗ (Ω , ρΩ , Q) depends on the domain Ω , the quasi-uniformity parameter ρΩ of the triangulation and the quadrature Q. At convergence, u satisfies J p (u) ≤

min

v∈VhΩ

1 ∥v+g∥ p ≤R p Xp

J p (v) + ε.

9 High-Order p-Harmonic Descent Directions and Their Limits

153

Proof. In order to compute the final estimate, we need to find a bound for ∥c∥2 . Let f ∈ Lq (Ω ) and h ∈ Lq (Γ ) piece-wise linear such that the quadrature on l points is ˆ 2 in the exact. We can obtain a bound on a similar way as the bound for ∥F ′ (x)∥ original paper. Start with

 

−M f − Mh

≤ ∥M f ∥2 + ∥Mh∥2 + 1 ∥ω∥2 . ∥c∥2 = ω

p p 2 In accordance with the original proof we can bound the first term by ∥M f ∥2 = ∥[E (l) ]⊺W (l) E (l) f ∥2 ≤ ∥[E (l) ]⊺W (l) ∥2 ∥E (l) f ∥2 (l)

with the same idea of using ∥[E (l) ]⊺W (l) ∥22 ≤ ωmax ρ([E (l) ]⊺W (l) E (l) ) and then bounding the spectral radius ρ. This is even easier here, because w⊺ E ⊺W Ew =

Z Ω

2 w2 dx = ∥w∥2L2 (Ω ) ≤ [KΩ′ ]2 h2d Ω ∥w∥2

can be estimated without further inequalities. The remainder can now be bound correspondingly with the equivalence of p-norms in finite dimensions ∥E f ∥2 ≤ ≤

(l) [ωmin ]−1/2

ml

∑

i=1

!1 (l) ωi [E (l) f ]2i

(l) [ωmin ]−1/2 (ml)1/2

≤(

ml

∑

i=1

2

(l) ωi [E (l) f ]qi

!1 q

ωmin −1/2 ) (ml)1/2 ∥ f ∥Lq (Ω ) CQ

where we used that the weights on the reference elements are fixed positive and thus there exists a constant CQ > 0 that only depends on the quadrature Q such that (l) CQ−1 ωmin ≤ ωmin ≤ ωmin . Combing the results, we get    −1/2 1/2 ∥M f ∥2 ≤ KΩ′ hdΩ ωmin CQ l 1/2 m1/2 ∥ f ∥Lq (Ω ) ≤ KQ KΩ′′ ∥ f ∥Lq (Ω ) .

A bound for the middle term can be obtained in the same way, just in (d − 1) dimensions. Using m

|Γ | = ∑ ω i ≥ mω min ≥ m i=1

this reads

hd−1 Ω ⇔ m ≤ |Γ |(d − 1)!hd−1 Ω (d − 1)!

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   −1/2 1/2 q ∥Mh∥2 ≤ KΩ′ hd−1 ω (C ml) ∥h∥ Q L (Γ ) Ω min

≤ KQ KΩ′′ ∥h∥Lq (Γ ) .
√ (ρhΩ )d 2 1/2 ≤ √mω Subsequently, using ∥ω∥2 = ∑m and p−1 ≤ 1 we max ≤ m d! i=1 ωi are able to obtain a bound in the form ∗ ∗ ∥c∥2 ≤ CΩ CQ (∥ f ∥Lq (Ω ) + ∥h∥Lq (Γ ) + 1).

Together with the previous results from the original proof we can now determine the relevant bounds as:  −1
−1 ∗ ∗ ∥c∥∗x∗ ≤ λmin ∥c∥2 ≤ c′Ω R−2 h2d CΩ CQ (∥ f ∥Lq (Ω ) + ∥h∥Lq (Γ ) + 1) Ω F

∥F ′ (x)∥ ˆ ∗x∗

F

∗ ∗ 2 −2d = [c′Ω ]−1CΩ CQ R hΩ (∥ f ∥Lq (Ω ) + ∥h∥Lq (Γ ) + 1)  −1   −1 ∗ −1−1.5d ≤ λmin ∥F ′ (x)∥ ˆ 2 ≤ c′Ω R−2 h2d CΩ hΩ R(1 + ∥g∥X p (Ω ) ) Ω ∗ 3 −1−3.5d = [c′Ω ]−1CΩ R hΩ (1 + ∥g∥X p (Ω ) )

Plugging these into the bound on the required Newton iterations in terms of the barrier parameter ν by Nesterov [13, Sec. 4.2.5] √ N ≤ 7.2 ν [2 ln(ν) + ln(∥F ′ (x)∥ ˆ ∗x∗ ) + ln(∥c∥∗x∗ ) + ln(1/ε)], F

F

we compute a bound in terms of number of grid points as √ N ≤ 7.2 4m[2 log(4m) + log(1/ε) ∗ 3 −1−3.5d + log([c′Ω ]−1CΩ R hΩ (1 + ∥g∥X p (Ω ) ))

∗ ∗ 2 −2d + log([c′Ω ]−1CΩ CQ R hΩ (∥ f ∥Lq (Ω ) + ∥h∥Lq (Γ ) + 1))] q ′ −2 ∗ 2 ∗ 2 2 ≤ 14.4 |Ω |d!h−d Ω [log([cΩ ] [CΩ ] CQ 16|Ω | (d!) )]

+ log(ε −1 h−1−7.5d R5 (1 + ∥g∥X p (Ω ) )(∥ f ∥Lq (Ω ) + ∥h∥Lq (Γ ) + 1))] Ω q ∗ = 14.4 |Ω |d!h−d Ω [K (Ω , ρΩ , Q)

+ log(ε −1 h−1−7.5d R5 (1 + ∥g∥X p (Ω ) )(∥ f ∥Lq (Ω ) + ∥h∥Lq (Γ ) + 1))]. Ω

Using h−d Ω = O(n), we obtain the corresponding complexity estimate by inspection. Finally, the convergence error is given by substituting J p into the general formula associated with the iteration bound in terms of the barrier parameter. □ Remark 9.3. To find the global minimum, R has to be chosen sufficiently large, so that the solution is contained in the search set Q p . For 2 ≤ p < ∞ such an upper bound exists and is given by

9 High-Order p-Harmonic Descent Directions and Their Limits

155

R = 2 + 4∥g∥Xp p (Ω ) + 8(p − 1) Lq ∥ f ∥qLq (Ω ) +CTq (Lq + 1)∥h∥qLq (Γ ) h

i

where CT denotes the Sobolev trace constant [5], which only depends on p and Ω . While there even exists an uniform upper bound for it [6], usually neither of them is computable. Therefore, in the actual implementation we choose a heuristic approach. Start by dropping the term resulting from the Neumann boundary condition and if the values during the iteration come close to the bound, restart with an increased version. Proof. We follow the original proof for the pure Dirichlet case, showing an upper bound for a minimizing sequence uk using H¨older’s inequality, the modified Friedrichs inequality for ∥·∥X p (Ω ) and the Sobolev trace theorem for the new term. Start by assuming ∥u∥X p (Ω ) ≥ ∥g∥X p (Ω ) , since otherwise the bound is trivial, and compute 1 ∥u + g∥Xp p (Ω ) − f u dx − hu dΓ p Ω Γ 1 p ≥ (∥u∥X p (Ω ) − ∥g∥X p (Ω ) ) − ∥ f ∥Lq (Ω ) ∥u∥L p (Ω ) − ∥h∥Lq (Γ ) ∥u∥L p (Γ ) p 1 1 1 ≥ ∥u∥Xp p (Ω ) − ∥g∥Xp p (Ω ) − ∥ f ∥Lq (Ω ) Lp− p ∥u∥Xp p (Ω ) p p Z

J(u) =

Z

1

− ∥h∥Lq (Γ )CT (Lp− p + 1)∥u∥Xp p (Ω ) . Now we can modify the application of Young’s inequality for two forcing terms 1 1 1 1 1 to a1 = 4 p Lp− p ∥ f ∥Lq (Ω ) , a2 = 4 p CT (Lp− p + 1)∥h∥Lq (Γ ) and b1,2 = 4− p ∥u∥X p (Ω ) and thus get J(u) − J(0) ≥

2 1 qp − qp q 1 ∥u∥Xp p (Ω ) − ∥g∥Xp p (Ω ) − |{z} 4 p L ∥ f ∥qLq (Ω ) |{z} 2p p q ≤4

for p≥2

≤1

1 1 q − 4 p CTq (Lp− p + 1)∥h∥qLq (Γ ) . | {z } q q

− ≤(Lq p p +1)

Therefore, if h i ˜ ∥u∥Xp p (Ω ) > 4∥g∥Xp p (Ω ) + 8(p − 1) Lq ∥ f ∥qLq (Ω ) +CTq (Lq + 1)∥h∥qLq (Γ ) =: R, then J(u) − J(0) ≥ 0. By contradiction any minimizing sequence uk must fulfill ∥uk ∥Xp p (Ω ) ≤ R˜ for some k large enough. Thus, it is in the set Q p and the final bound ˜ is obtained by R := 2(R+1) to ensure the construction condition for the initial value. □

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9.3 Numerical Results In this section, we present results for numerical experiments with the scheme presented in section 9.2. The data was calculated with an implementation in julia based on the finite element library MinFEM [17] and visualized with Paraview. We choose julia since it allows straightforward computations using matrix or vector based operations, enables easy adaptation of the code and provides great accessibility to accuracy parameters. The code and the examples presented here are available in an online repository [19] and as the package PLaplace in the julia registry.

(a) ∥v∥2

(b) ∥v − v∗ ∥2

(c) vr − v∗r

Fig. 9.1: Solution and error for validation by method of manufactured solutions for v∗ = 12 ∥x∥2 · [1, 1]⊺ on Ω = [0, 1]2 . We start by validating the algorithm and the implementation. For that purpose, we use the method of manufactured solutions [15] to approximate the analytical solution v∗ = 12 ∥x∥2 · [1, 1]⊺ given for the problem    p−2 1  −∆ p v = −p 2 2 ∥x∥2p−2 · in Ω   1   . 1  v = 12 ∥x∥22 · on ∂ Ω   1 We only test the vector-valued setting, since it contains the scalar case, which has not been validated so far, per component. Figure 9.1 shows the obtained solution and the error on the unit square discretized by a regular mesh with 40000 nodes. The error in the two components is identical and in the order of the intended accuracy ε = 10−6 . The largest errors naturally occurs in the bottom left corner, where a function value close to 0 has to be approximated. Thus we consider the results obtained by our implementation as valid.

9 High-Order p-Harmonic Descent Directions and Their Limits Γ

157

η

Ω

∂Ω \ Γ

Fig. 9.2: Sketch of domain for exemplary problem.

From now on, we focus on the additional Neumann boundary conditions. Therefore, we consider the domain Ω as the unit square, where left and upper boundary are free for deformation and denoted by Γ . A sketch of this domain can be found in Figure 9.2. On the free boundary, we will work with combinations of the function ˆ h(x) = sin(2πx1 ) − sin(2πx2 ). This is a sine wave cycle on each part of the boundary, where the one on the upper boundary is inverted such that the two positive parts are next to the upper left corner. As we will see later, this construction leads towards a distance function on the boundary of the limit solution instead of multiple hats.

h= n= p= 2 p= 3 p= 5 p= 8 p = 15 p = 25

ˆ h(x) 2500 10000 40000 95 102 116 95 104 114 92 103 113 118 213 198 191 253 296 233 308 464

ˆ ·η h(x) 2500 10000 40000 94 101 115 93 101 114 92 102 107 86 99 107 111 126 148 152 204 181

ˆ · [1, 1]⊺ h(x) 2500 10000 40000 95 104 116 96 107 115 98 129 116 186 204 213 228 278 326 280 361 481

Table 9.1: Required Newton iterations for solving problems for different boundary ˆ source terms with h(x) = sin(2πx1 ) − sin(2πx2 ), number of grid points and PDE parameters p. For this setting, we can observe the number of required Newton iterations in the path-following with adaptive step-size control for various PDE parameter p and refinements the grid. Table 9.1 shows these values for the scalar setting and two difˆ ferent prolongations of h(x) to a vector-valued setting. Note that the vector-valued ˆ problems can be a significantly different problem. For example applying h(x) to the outer normal vector results in a problem, where per component only one of the free edges features a sine wave and the other one is homogeneous. This is component

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wise a simpler problem than the regular scalar one, which is despite the connection of the components visible in the required iterations. The first observation is that much higher orders p are possible when the boundary features a free part. For the pure Dirichlet setting the numerical maximal value was around p = 5, here the source term reduces the stiffness of the problem such that even p = 25 is possible. In general, we see for all settings the expected behavior of increasing iterations for increasing p and n with some exceptions due to the adaptive stepping. Further, the overall iterations stay comparably small to the number of grid points and thus significantly better than the theoretical estimate. In the scalar Dirichlet case this behavior for the adaptive stepping was already known, but it was unclear if it transfers to the vector-valued setting, especially since the analytical problem is inherently more difficult due to nature of the Frobenius norm in the operator. For the intended application to shape optimization this is a crucial observation, since the computation of many deformation fields is required and thus determines the overall runtime. ˆ · η to calculate the deformation of the Now we will use results obtained for h(x) square by perturbation of identity (9.1). A selected sequence of transformed domains are shown in Figure 9.3 with a reduced number of grid points for improved visibility. For comparison we take the domain for p = 2. This features strong bends near the two fixed endpoints of the free boundary, where the mesh is highly deteriorated. Increasing the order to p = 5, the magnitude of the deformation increases significantly and the bends reduce, however the mesh quality is still poor. When changing to p = 15, the magnitude changes only slightly as well as the bend. However, the mesh at the bends is not deteriorated anymore and the elements in the interior are deformed more uniformly.

9.4 Descent in W 1,∞ In this section, we consider directly solving the steepest descent problem (9.2), commonly associated with the variational problem for the ∞-Laplacian and discuss the challenges arising. The first important property of the ∞-Laplacian is that the solutions are in general non-unique. However, by the approach of regularization with the p-Laplacian, we want the limit of those so-called p-Extensions, which is known as the absolutely minimizing Lipschitz extension (AMLE) due to early work by Aronsson [2]. Here, the term “absolutely minimizing” denotes functions v ∈ C(Ω ) with Lipschitz constant Lv (V ) = Lv (∂V ) for all V ⋐ Ω . While the minimization formulation for the problem is common, there is no variational formulation with test functions under the integral [9]. Further, only for zero forcing a reformulation to an Euler-Lagrange equation is possible. With this approach it was shown that unique solutions in the above sense exist for boundary extension problems [8]. This remains true for homogeneous Dirichlet problems with non-negative source terms [7]. Here, the unique limit solution can be split into two

9 High-Order p-Harmonic Descent Directions and Their Limits

159

(a) Initial mesh

(b) p = 2

(c) p = 5

(d) p = 15

Fig. 9.3: Deformation of a square domain with h(x) = (sin(2πx1 ) − sin(2πx2 )) · η prescribed on the left and upper boundary for increasing p.

regions. In the support of the source term, the solution is given by the distance to the boundary dist(x, ∂ Ω ). The other region is then given by the unique solution of the ∞-Laplacian without forcing and the distance as Dirichlet boundary condition. In one dimension, the situation is simplified and analytical solutions can be computed even for source terms with changing sign. Figure 9.4 shows such solutions and corresponding solutions of p-harmonic relaxations for different source terms. We can observe that the limit solution always has slope 1. Its magnitude does not depend on the magnitude of the source term, but only on the length of the interval between two sign changes. However, the p-harmonic solutions depend on it, meaning that the approximation quality depends on the magnitude of the source term as well. Another interesting observation can be made in the lower right plot. The solu-

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Fig. 9.4: Convergence of numerical solutions for the p-Laplacian with increasing order and the analytical limit p = ∞ in [0, 1] for g = 0 and different source terms f .

tions are able to eliminate certain areas with different signs. While the limit solution is identical to the one with constant source term, the p-harmonic approximations are different. Especially one can still see an impact of the sign change close to the boundary and the kink at tip occurs later. Further, this one dimensional source term is similar to a normalized version of the Neumann boundary condition or boundary source term used in Figure 9.3. Thus, we would expect a solution with slope 1 everywhere on the boundary for the limit deformation. Additionally to the algorithm for the finite setting, which we modified in section 9.2, an algorithm for the limit problem is proposed in [11]. There, the limit of problem (9.4) is formulated as arg min J∞ (u) = arg min sup ∥∇(u + g)∥ − u∈U ∞ u∈U ∞ x∈Ω {z } | =:∥u+g∥X ∞ (Ω )

Z Γ

hu dΓ −

Z Ω

f u dx.

(9.7)

9 High-Order p-Harmonic Descent Directions and Their Limits

(a) Expected limit.

(b) Inner norm ∥·∥2 .

161

(c) Inner norm ∥·∥∞ .

Fig. 9.5: Deformation of a square domain with h(x) = (sin(2πx1 ) − sin(2πx2 )) · η prescribed on the left and upper boundary for different different algorithmic realizations of the ∞-Laplacian.

First, note that we leave the interpretation of the interior norm in the first term open for now. While there is no actual derivation of the formulation given, we can state some arguments to consider it. Interpreting the primary term in equation (9.4) as ∥u + g∥Xp p (Ω ) one would obtain this primary term as ∥u + g∥X ∞ (Ω ) in the same way classical limits for L p -norms are constructed. By the theory of Lipschitz extensions [8], one can understand the ∞-Laplacian as the minimization of the sup norm of the gradient, given further justification to the idea. We will not give the construction of the algorithm here again, since it is similar to the finite setting and the extensions are done in the same manner. However, it is interesting that the reformulated discrete problem (9.5) essentially becomes a problem for the 1-Laplacian over the subspace of constant s meaning that the Lipschitz constant on each element is bounded uniformly and the bound is minimized. On the Neumann boundary this approach would result in v|Γ = 0 and thus we add the constraint s ≥ 1 in order to obtain the desired slope on the boundary. From the sequence of solutions for finite p and the observations in 1D combined with the theoretical definition of the AMLE, we expect the limit artificially shown in Figure 9.5a. Figure 9.5b shows the result for the choice of the inner norm ∥·∥2 in equation (9.7) as proposed in [11]. It overshoots the intended tip slightly, yielding

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an improvement to the solution for p = 15 from Figure 9.3. But it is not able to resolve the intended free boundaries, leading to worse results than the finite setting. However, the grid quality at those boundaries remains good and instead the center becomes significantly worse. By the method of manufactured solutions, we can also show that it is not able to yield the unique solution for a reference problems with the 4/3 4/3 well-known viscosity solution v = x1 − x2 [10]. Another approach is choosing the supremum norm also for the inner norm, which we computed in Figure 9.5c. While this choice yields the correct outer boundary, it still does not deform the interior mesh uniformly.

9.5 Conclusion We added support for Neumann boundary conditions to a present algorithm for the scalar p-Laplacian and proved that the theoretical estimate on the required Newton steps remains polynomial. Further, we constructed the extension of the algorithm to vector-valued problems and performed numerical experiments including validation and shape deformations. The results demonstrate that the extension is indeed applicable to problems occurring in p-harmonic shape optimization and yields solutions for higher-order p > 5 without iterating over p. Those provide further improvements in terms of preserved mesh quality and obtained boundary shape. However, we saw that results obtained for the ∞-Laplacian are significantly different and even highorder solutions do not yield a sufficient approximation, especially since they depend on the magnitude of the source term. First experiments with a modified algorithm to solve the ∞-Laplacian problem directly did not yield the desired results, but small changes could already achieve improvements, allowing the idea to be considered further. For future research it remains to construct a proper algorithm for the limit problem. On the other hand, the results for finite p may be applied in shape optimization problems potentially including 3D settings. This includes modifications to the implementation for high-performance computer architecture and analysis considering scalability.

Acknowledgment The authors acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group GRK 2583 “Modeling, Simulation and Optimization of Fluid Dynamic Applications”.

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