Asymptotic Bounds to Outputs of the Exact Weak Solution of the Three-Dimensional Helmholtz Equation Shahin Ghomeshi A Thesis in The Department of Mechanical and Industrial Engineering Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy (Mechanical Engineering) at Concordia University Montreal, Quebec, Canada March 2010 ©Shahin Ghomeshi, 2010 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 OttawaONK1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-80163-5 Our file Notre reference ISBN: 978-0-494-80163-5 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non­ support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. 1*1 Canada ABSTRACT Asymptotic Bounds to Outputs of the Exact Weak Solution of the Three-Dimensional Helmholtz Equation Shahin Ghomeshi, Ph.D. Concordia University, 2010 In engineering practice, the design is based on certain design quantities or "outputs of interest" which are functionals of field variables such as displacement, velocity field, or pressure. In order to gain confidence in the numerical approximation of "outputs," a method of obtaining sharp, rigorous upper and lower bounds to outputs of the exact solution have been developed for symmetric and coercive problems (the Poisson equation and the elasticity equation), for non-symmetric coercive problems (advection-diffusion-reaction equation), and more recently for certain constrained problems (Stokes equation). In this thesis we develop the method for the Helmholtz equation. The common approach relies on decomposing the global problem into independent local elemental sub-problems by relaxing the continuity along the edges of a partition­ ing of the entire domain, using approximate Lagrange multipliers. The method exploits the Lagrangian saddle point property by recasting the output problem as a constrained minimization problem. The upper and lower computed bounds then hold for all levels of refinement and are shown to approach the exact solution at the same rate as its underlying finite element approach. The certificate of precision can then determine the best as well as the worst case scenario in an engineering design problem. This thesis addresses bounds to outputs of interest for the complex Helmholtz equation. The Helmholtz equation is in gen- iii IV eral non-coercive for high wave numbers and therefore, the previous approaches that relied on duality theory of convex minimization do not directly apply. Only in the asymptotic regime does the Helmholtz equation become coercive, and reliable (guaranteed) bounds can thus be obtained. Therefore, in order to achieve good bounds, several new ingredients have been introduced. The bounds procedure is firstly formulated with appropriate extension to complex-valued equations. Secondly, in the computation of the inter-subdomain continuity multipliers we follow the FETI-H approach and regularize the system matrix with a complex term to make the system non-singular. Finally, in order to obtain sharper output bounds in the presence of pollution errors, higher order nodal spectral element method is employed which has several computational advantages over the traditional finite element approach. We performed verification of our results and demonstrate the bounding properties for the Helmholtz problem. Acknowledgements First and foremost, I would like to thank my advisor, Professor Marius Paraschivoiu, for his continual supply of ideas, as well as for his encouragement and patience during the course of this research. Secondly, I am grateful to Professor Frederic Magoules from Ecole Centrale Paris for his feedback and insight into my work, which was not only very helpful, but also very encouraging. I would also like thank my committee members at Concordia university for their useful feedback and advice on the thesis. These include Professors Georgios Vatistas, and Ramin Sedaghati from the mechanical engineering department and Professor Tien Bui from the computer science department. I am also extremely thankful for the warm atmosphere of the CFD lab where many of my colleagues there provided me with very fruitful discussions. Finally, I could not have overcome the many frustrations and disappointments experienced during the course of my studies without the support of my family and friends. I would especially like to thank my parents Eshrat and Behrang, whose constant love and encouragement gave me the strength and encouragement when times seemed most difficult. v Table of Contents Chapter 1 Introduction 1 1.1 Acoustic Wave Propagation Problem 2 1.1.1 Model Problems in Acoustics 3 1.1.1 Interor Problems 4 1.1.1 Exterior Problems 5 1.1.2 Numerical Challenges 7 1.2 Error Estimation 7 1.3 Objectives and Scope of Thesis 13 Chapter 2 Numerical Methods for the Helmholtz Equation 15 2.1 Model Problem 15 2.1.1 Strong Formulation 15 2.1.2 Preliminaries from Functional Analysis 16 2.1.3 Notations and Weak Formulation 19 2.1.4 Linear Functional Outputs 20 2.2 Discretization Methods 21 vi Table of Contents vn 2.2.1 Nodal Points 24 2.2.2 Basis Functions 25 2.3 Some Key Concepts 28 2.3.1 Positive Definite Forms 29 2.3.2 inf — sup Condition 30 2.3.3 V-Coercive Forms 32 2.3.4 Variational Methods 33 2.3.4 Convergence Properties 35 2.3.5 An Observation 37 2.4 Pollution Effect 38 2.4.1 Piecewise linear case 38 2.4.2 Higher-Order Elements-Zip FEM 41 Chapter 3 Domain Decomposition Methods 43 3.1 The FETI Procedure - Poisson Example 45 3.1.1 Discretization 47 3.1.2 The FETI PCPG Iterative Procedure 50 3.2 Domain Decomposition for Helmholtz 51 3.2.1 Discretization 53 3.2.2 Iterative Method Used 56 3.2.2 Some Comments on Preconditioning 60 Table of Contents viii Chapter 4 Exact Bounds Method for the Poisson Equation 61 4.1 Bounds on Energy 62 4.2 Bounds on Quantitative Outputs 69 4.2.2 Lagrange Multiplier Approximation 71 4.2.3 Local Dual Sub-Problems 71 4.2.3 Sub-problem Approximation 73 4.2.4 Output Bounds 75 4.3 Implementation 76 4.3.1 Interpolation of Hybrid Fluxes 76 4.4 Discrete forms and Sub-problems Computation 78 4.4.1 Numerical Examples 80 4.4.1 Constructed Exact Solution 80 4.4.1 Uniformly Forced Domain 82 4.5 Some Results on Stokes' Problem 85 4.5.1 Model Problem 85 4.5.2 Output Functional 87 4.5.3 Numerical Example of Bound Calculation for Stokes 87 Chapter 5 Bounds for the Helmholtz Equation 91 5.1 Problem Statement 92 5.1.1 Governing Equations 92 Table of Contents IX 5.1.2 Approximation Spaces 92 5.2 Error Bound Formulation 94 5.2.1 Output Functional 94 5.2.2 Error Formulation 94 5.2.3 Lagrange Multiplier Approximation 95 5.2.3 Modified Lagrangian 95 5.2.4 Local Problems 99 5.3 Localized Lagrangian 102 5.4 An Equivalent Lagrangian 103 5.5 Elemental Sub-problems 105 5.6 Bounding Property 107 5.7 Sub-problem Computations 109 5.7.1 Discrete Forms and Approximations 110 5.8 Computation of Bounds 112 5.9 Convergence Properties 113 Chapter 6 Numerical Validation and Results 117 6.1 Discussion of Results 118 6.1.1 For p = 2, q = 3 118 6.1.1 Case I: k = 1 118 6.1.1 Case II: k = 3 118 6.1.1 Case III: k = 5 119 Table of Contents x 6.1.2 For p = 4, q = 5 120 6.1.2 Case I: k = 3 120 6.1.2 Case II: k = 5 121 Chapter 7 Conclusions and Future Work 129 7.1 Conclusions 129 7.2 Future Work 131 Bibliography 134 List of Figures Figure 1.1 Computational domain £7 for a vibro-acoustic problem 5 Figure 1.2 Radiation problem for a vibrating body D 6 Figure 3.1 Representation of the checkerboard partitioning of the mesh 55 Figure 4.1 Conforming nature of the upper bound 63 Figure 4.2 Lower bounding property 68 Figure 4.3 Hybrid flux interpolation on the faces of the subdomains 79 Figure 4.4 Output bounds obtained for constructed solution 83 Figure 4.5 Output bounds obtained for constant forcing, p = 1, q = 2 84 Figure 4.6 Output bounds obtained for constant forcing, p = 2, q = 3 84 Figure 4.7 Geometry for the Stokes problem 86 Figure 4.8 Bounds for the Stokes output 88 Figure 6.1 Convergence study for k = 1 123 xi List of Figures xn Figure 6.2 Bounding property of the bounds for k = 1 123 Figure 6.3 Convergence study for k = 3 124 Figure 6.4 Upper and lower bounds for k = 3 124 Figure 6.5 Convergence study for k = 5 125 Figure 6.6 Upper and lower bounds for k — 5 125 Figure 6.7 Convergence study for k — 3 using higher-order elements 126 Figure 6.8 Upper and lower bounds for k = 3 using higher-order elements.