Time domain reciprocal absorbing boundaries

Copyright by Cuong Tan Nguyen 2017 The Dissertation Committee for Cuong Tan Nguyen certifies that this is the approved version of the following dissertation: Time-domain Reciprocal Absorbing Boundaries Committee: John L Tassoulas, Supervisor Lance Manuel Mark E Mear Krishnaswamy Ravi-Chandar Spyros A Kinnas Time-domain Reciprocal Absorbing Boundaries by Cuong Tan Nguyen, DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN December 2017 Dedicated to our parents, my beloved wife Na Huynh and son Austin Nguyen Acknowledgments First of all, I would like to express my sincere gratitude and deep appreciation to my doctoral advisor, Professor John L Tassoulas, for his innovative ideas on this research, for his great source of knowledge, inspiration and encouragement I am thankful to Professors Lance Manuel, Mark E Mear, K RaviChandar, Spyros A Kinnas for serving on my doctoral committee, for their valuable suggestions and for their inspiring courses I wish to thank my friends at MUSE Lab, Heedong Goh, Babak Poursartip, Arash Fathi, Seungbum Koo, Ying-Chuan Chen, Nan-You Lu and Sedef Kocakaplan They are definitely a part of my memorable time at UT Austin I would also like to thank CAEE staffs, Leslie McCroddan and Velma Vela for their great supports I thank the Vietnam Education Foundation for granting a Fellowship during my first two years at UT Austin Futhermore, I thank the Cockrell School of Engineering for giving me the chance to serve as the teaching assistant of several classes related to Computer methods and Mechanics Most of all, I would like to thank my parents, parents-in-law and express my deep appreciation to my beloved wife Na Huynh and son Austin Nguyen Without their endless love, supports, encouragement and sympathy, this Dissertation would not have been possible v Time-domain Reciprocal Absorbing Boundaries Publication No Cuong Tan Nguyen, Ph.D The University of Texas at Austin, 2017 Supervisor: John L Tassoulas Accurate and efficient computational treatments of wave propagation in unbounded media rely on special absorbing boundaries In most of the problems, the region of interest is only a small part, the “near field”, of an extensive, arguably, infinite domain On the boundary of the near field, absorbing boundaries are introduced in order to mimic the absorption of waves into the exterior domain They represent to various degrees of approximation the “far field” as viewed from the near field which normally contains irregularities and nonlinearities In general, absorbing boundaries for the time-domain analysis of wave propagation in unbounded media can be classified into two categories, global and local ones The aim of this disseration is to develop novel “global” absorbing boundaries by means of reciprocity theorems The study deals with elastodynamics in anti-plane shear and plane strain, acoustic waves in waveguides The domains under consideration include layered strata, layered half-spaces and the full-space vi Table of Contents Acknowledgments v Abstract vi List of Tables xiii List of Figures xiv Chapter Introduction 1.1 Absorbing boundaries 1.2 Dissertation outline Chapter 2.1 2.2 2.3 2.4 Reciprocal absorbing boundary condition for the time-domain numerical analysis of wave motion in unbounded layered media Introduction Notation and definitions 2.2.1 Elastodynamic state 2.2.2 Convolution Time-domain reciprocity theorems for elastodynamics Derivation of reciprocal absorbing boundary condition 2.4.1 Convolution quadrature 2.4.2 Time-stepping algorithm 2.4.3 Material Damping 2.4.4 Reciprocal absorbing boundary condition (RABC) 2.4.5 Initial values 2.4.5.1 Semi-infinite beam on simple supports 2.4.5.2 Semi-infinite beam on flexible supports 2.4.5.3 Semi-infinite rod 1 vii 5 10 10 12 12 17 17 18 19 19 23 23 25 29 2.5 2.6 2.7 2.8 2.4.5.4 Stratum in plane strain 2.4.5.5 Stratum in antiplane shear Numerical examples 2.5.1 Semi-infinite beam on simple supports 2.5.1.1 Convergence 2.5.1.2 Stability 2.5.1.3 Accuracy 2.5.2 Semi-infinite beam on flexible supports 2.5.3 Semi-infinite rod 2.5.4 Semi-infinite layer in anti-plane shear 2.5.5 Semi-infinite layer in plane strain 2.5.6 Stratum under Ricker wavelet load Conclusions Appendix 2.7.1 Beam element 2.7.2 Rod element 2.7.3 Rectangular element in antiplane shear and plane strain 2.7.4 Matrices for the consistent transimitting boundary 2.7.4.1 Anti-plane shear 2.7.4.2 Plane strain Error Analysis 2.8.1 Problem setting 2.8.2 Analysis 2.8.2.1 Stability 2.8.2.2 Convergence Application of reciprocal absorbing boundary condition to transient analysis of acoustic wave propagation 3.1 Introduction 3.2 Reciprocity theorems for acoustics 3.2.1 The linear wave equation for velocity potential 3.2.2 Acoustic reciprocity theorems 29 30 30 30 31 32 32 34 35 38 39 40 42 45 45 46 47 47 47 48 48 48 51 51 54 Chapter viii 58 59 62 62 63 3.2.2.1 Reciprocity in the frequency domain 3.2.2.2 Reciprocity in the time domain 3.3 Reciprocal absorbing boundary condition 3.3.1 Finite elements for acoustic wave propagation 3.3.1.1 The semi-discrete form 3.3.1.2 Time-stepping algorithm 3.3.2 Derivation of reciprocal absorbing boundary condition 3.3.2.1 The original semi-infinite problem 3.3.2.2 Convolution quadrature 3.3.2.3 The construction of reciprocal absorbing boundary condition 3.4 Numerical examples 3.4.1 One-dimensional problem 3.4.1.1 Accuracy 3.4.1.2 Convergence 3.4.1.3 Stability 3.4.2 Two-dimensional problem 3.5 Conclusions Reciprocal absorbing boundary condition with perfectly matched discrete layers for SH waves in a layered half-space 4.1 Introduction 4.2 Reciprocal absorbing boundary condition 4.2.1 Reciprocity theorems 4.2.2 Derivation of reciprocal absorbing boundary condition 4.2.2.1 Convolution quadrature 4.2.2.2 Time-stepping algorithm 4.2.2.3 Material damping 4.2.2.4 Reciprocal absorbing boundary condition 4.3 Perfectly Matched Discrete Layers 4.3.1 Finite element discretization 4.3.2 Time-stepping 63 64 66 66 66 68 69 69 70 71 74 75 76 76 77 78 83 Chapter ix 91 92 94 94 97 97 97 99 99 102 106 108 4.4 Reciprocal absorbing boundary condition with perfectly matched discrete layers 109 4.5 Numerical Examples 110 4.5.1 A two-layer stratum truncated by RABC 110 4.5.2 A homogeneous layer truncated by PMDLs 111 4.5.3 A homogeneous half-space truncated by RABC and PMDLs113 4.5.4 A layered half-space truncated by RABC & PMDLs 115 4.6 Conclusions 121 Chapter 5.1 5.2 5.3 5.4 5.5 Reciprocal absorbing boundary condition with perfectly matched discrete layers for SV-P waves in a layered half-space Introduction Reciprocal absorbing boundary condition for a stratum 5.2.1 Derivations 5.2.1.1 Derivation on the basis of the displacement-based reciprocity theorem 5.2.1.2 Derivation based on the dynamic stiffness matrix 5.2.1.3 Derivation based on the acceleration unit-impulse response matrix 5.2.2 Convolution quadrature and time-stepping scheme 5.2.3 The construction of reciprocal absorbing boundary condition (RABC) 5.2.4 Computing initial values Perfectly matched discrete layers for SV-P waves 5.3.1 Finite element discretization 5.3.2 Time-stepping Reciprocal absorbing boundary condition with perfectly matched discrete layers Numerical examples 5.5.1 RABC for a two-layer stratum 5.5.1.1 Accuracy 5.5.1.2 Convergence 5.5.2 A homogeneous layer x 122 123 125 125 126 130 133 134 135 136 138 141 146 147 149 149 149 150 153 1.6 1.2 0.8 0.4 -0.4 -0.8 -1.2 12 16 20 24 Figure 6.15: Vertical harmonic response: f = 0.1 Hz, a0 = 0.1π, ∆t = 0.01 s and the snapshot of w 0.6 0.4 0.2 -0.2 -0.4 10 12 Figure 6.16: Vertical harmonic response: f = 0.5 Hz, a0 = 0.5π, ∆t = 0.01 s and the snapshot of w 0.15 0.1 0.05 -0.05 -0.1 0.4 0.8 1.2 1.6 2.4 Figure 6.17: Vertical harmonic response: f = 2.5 Hz, a0 = 1.25π, ∆t = 0.005 s and the snapshot of w 204 0.6 0.4 0.2 -0.2 -0.4 10 12 Figure 6.18: Radial harmonic response: f = 0.2 Hz, a0 = 0.2π, ∆t = 0.01 s and the snapshot of ur 0.2 0.15 0.1 0.05 -0.05 -0.1 -0.15 Figure 6.19: Radial harmonic response: f = Hz, a0 = π, ∆t = 0.01 s and the snapshot of ur 0.04 0.03 0.02 0.01 -0.01 -0.02 -0.03 0.2 0.4 0.6 0.8 1.2 Figure 6.20: Radial harmonic response: f = Hz, a0 = 1.25π, ∆t = 0.0025 s and the snapshot of ur 205 0.6 0.4 0.2 -0.2 -0.4 10 12 Figure 6.21: Torsional harmonic response: f = 0.2 Hz, a0 = 0.2π, ∆t = 0.01 s and the snapshot of uθ 0.3 0.2 0.1 -0.1 -0.2 Figure 6.22: Torsional harmonic response: f = Hz, a0 = π, ∆t = 0.01 s and the snapshot of uθ 0.06 0.04 0.02 -0.02 -0.04 0.2 0.4 0.6 0.8 1.2 Figure 6.23: Torsional harmonic response: f = Hz, a0 = 1.25π, ∆t = 0.0025 s and the snapshot of uθ 206 0.7 0.5 0.3 0.1 -0.1 -0.3 -0.5 10 12 Figure 6.24: Horizontal harmonic response: f = 0.2 Hz, a0 = 0.2π, ∆t = 0.01 s and the snapshot of u 0.2 0.1 -0.1 -0.2 Figure 6.25: Horizontal harmonic response: f = Hz, a0 = π, ∆t = 0.01 s and the snapshot of u 0.04 0.03 0.02 0.01 -0.01 -0.02 -0.03 -0.04 0.2 0.4 0.6 0.8 1.2 Figure 6.26: Horizontal harmonic response: f = Hz, a0 = 1.25π, ∆t = 0.0025 s and the snapshot of u 207 6.6 Conclusions We have presented an application of Reciprocal Absorbing Boundary Condition with Perfectly Matched Discrete Layers to the problem of wave motions due to excitations of a circular disk in full space Specifically, four cases were considered, including axial (out-of-plane), radial, torsional and lateral (in-plane) vibrations Numerical examples shown in this paper demonstrate that coupling of the aforementioned absorbing boundaries yields accurate solutions in both transient and harmonic analysis The combination proposed herein is amenable to extension in three dimensions, a topic of future research on Reciprocal Absorbing Boundaries 208 Bibliography [1] Jan D Achenbach Reciprocity and related topics in elastodynamics Applied Mechanics Reviews, 59(1):13–32, 2006 [2] Jan Drewes Achenbach 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and Environmental Engineering Permanent address: H01/17, K137, Le Van Hien, My Khe Ngu Hanh Son, Da Nang, Vietnam This dissertation was typeset with LATEX† by the author † A LT EX is a document preparation system developed by Leslie Lamport as a special version of Donald Knuth’s TEX Program 218 ... dissertation: Time- domain Reciprocal Absorbing Boundaries Committee: John L Tassoulas, Supervisor Lance Manuel Mark E Mear Krishnaswamy Ravi-Chandar Spyros A Kinnas Time- domain Reciprocal Absorbing Boundaries. .. Introduction 1.1 Absorbing boundaries 1.2 Dissertation outline Chapter 2.1 2.2 2.3 2.4 Reciprocal absorbing boundary condition for the time- domain numerical... 6.6 Transient analysis of full-space unbounded domains by reciprocal absorbing boundaries 172 Introduction 172 Reciprocal absorbing boundary condition (RABC) 176