Luận án TS: Biên hấp thụ đối ứng miền thời gian cho truyền sóng không giới hạn

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 appre- ciation to my doctoral advisor, Professor John L. Tassoulas, for his innovative ideas on this research, for his great source of knowledge, inspiration and en- couragement. I am thankful to Professors Lance Manuel, Mark E. Ravi- Chandar, 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 assis- tant of several classes related to Computer methods and Mechanics. Most of all, I would like to thank my parents, parents-in-law and ex- press 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. 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, ab- sorbing 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 irregu- larities 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” absorb- ing boundaries by means of reciprocity theorems. The study deals with elasto- dynamics 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 1.

Reciprocal absorbing boundary condition for the time-domain numerical analysis of wave motion in unbounded layered media 5 2.2 Notation and definitions .3 Time-domain reciprocity theorems for elastodynamics .4 Derivation of reciprocal absorbing boundary condition .2 Time-stepping algorithm .4 Reciprocal absorbing boundary condition (RABC) .1 Semi-infinite beam on simple supports .2 Semi-infinite beam on flexible supports .3 Semi-infinite rod .4 Stratum in plane strain .5 Stratum in antiplane shear .1 Semi-infinite beam on simple supports .2 Semi-infinite beam on flexible supports .3 Semi-infinite rod .4 Semi-infinite layer in anti-plane shear .5 Semi-infinite layer in plane strain .6 Stratum under Ricker wavelet load .3 Rectangular element in antiplane shear and plane strain 47 2.4 Matrices for the consistent transimitting boundary .1 Anti-plane shear. Application of reciprocal absorbing boundary con- dition to transient analysis of acoustic wave propa- gation 58 3.2 Reciprocity theorems for acoustics .1 The linear wave equation for velocity potential .2 Acoustic reciprocity theorems .1 Reciprocity in the frequency domain .2 Reciprocity in the time domain .3 Reciprocal absorbing boundary condition .1 Finite elements for acoustic wave propagation .1 The semi-discrete form .2 Time-stepping algorithm .2 Derivation of reciprocal absorbing boundary condition .1 The original semi-infinite problem .3 The construction of reciprocal absorbing bound- ary condition .1 One-dimensional problem .2 Two-dimensional problem. Reciprocal absorbing boundary condition with perfectly matched discrete layers for SH waves in a layered half-space 91 4.2 Reciprocal absorbing boundary condition .2 Derivation of reciprocal absorbing boundary condition .2 Time-stepping algorithm .4 Reciprocal absorbing boundary condition .3 Perfectly Matched Discrete Layers .1 Finite element discretization .4 Reciprocal absorbing boundary condition with perfectly matched discrete layers .1 A two-layer stratum truncated by RABC .2 A homogeneous layer truncated by PMDLs .3 A homogeneous half-space truncated by RABC and PMDLs113 4.4 A layered half-space truncated by RABC & PMDLs. Reciprocal absorbing boundary condition with perfectly matched discrete layers for SV-P waves in a layered half-space 122 5.2 Reciprocal absorbing boundary condition for a stratum .1 Derivation on the basis of the displacement-based reciprocity theorem .2 Derivation based on the dynamic stiffness matrix 130 5.3 Derivation based on the acceleration unit-impulse response matrix .2 Convolution quadrature and time-stepping scheme .3 The construction of reciprocal absorbing boundary con- dition (RABC) .4 Computing initial values .3 Perfectly matched discrete layers for SV-P waves .1 Finite element discretization .4 Reciprocal absorbing boundary condition with perfectly matched discrete layers .1 RABC for a two-layer stratum .3 A homogeneous half-space .4 A layered half-space.

Transient analysis of full-space unbounded domains by reciprocal absorbing boundaries 172 6.2 Reciprocal absorbing boundary condition (RABC) .1 Acceleration based reciprocity theorem .2 The construction of Reciprocal Absorbing Boundary Con- dition .3 Perfectly matched discrete layers (PMDLs) .1 Semidiscrete equation of motion .1 Anti-plane shear: .4 Reciprocal absorbing boundary condition with perfectly matched discrete layers .2 Steady-state phase for harmonic forces: .2 Steady-state phase for harmonic forces: .2 Steady-state phase for harmonic forces: .2 Steady-state phase for harmonic forces:. 208 Bibliography 209 xi Vita 218 xii List of Tables 2.1 Parameters for time and space discretization.1 Material properties of the layered half-space.1 Parameters of the stratum.1 Vertical response: Relative error in L2 norm at sample points between RABCs-PMDLs and extended mesh solution.2 Radial response: Relative error in L2 norm at sample points between RABCs-PMDLs and extended-mesh solution.3 Torsional response: Relative error in L2 norm at sample points between RABCs-PMDLs and extended mesh solution.4 Horizontal response: Relative error in L2 norm at sample points between RABC-PMDLs and extended mesh solution. 200 xiii List of Figures 2.1 A soil-structure system.2 Reciprocal absorbing boundary condition.3 Two regions to which the reciprocity theorem is applied.4 Semi-infinite beam on simple supports.5 Semi-infinite beam on flexible supports.6 Semi-infinite rod.7 Convergence test for a semi-infinite beam on simple supports (left) and a semi-infinite rod (right) subjected to a step force.8 Stability test for a semi-infinite beam on simple supports sub- jected to a step force at node 1.9 Transient response of a semi-infinite beam on simple supports subjected to a step force at node 1.10 Profiles of symmetrical triangular pulse, Ricker wavelet, or rect- angular pulse forces at node 1.11 Transient response of a semi-infinite beam on simple supports subjected to a symmetrical triangular pulse force at at node 1.12 Transient response of a semi-infinite beam on simple supports subjected to a Ricker wavelet force at node 1.13 Transient response of a semi-infinite beam on simple supports subjected to a rectangular pulse force at node 1.14 Transient response of a semi-infinite beam on flexible supports subjected to a step force at node 1.15 Transient response of a semi-infinite beam on flexible supports subjected to a step moment at node 1.16 Response of a semi-infinite rod subjected to an axially step force at node 1.17 Response of a semi-infinite rod subjected to an axially step force at node 1.18 A homogeneous layer in antiplane shear.19 Displacement at nodes 1 and 2 due to force at node 1.20 Displacement at nodes 1 and 2 due to force at node 2.21 A homogeneous layer in plane strain.22 Displacement in x and y directions due to a horizontal force.23 Displacement in x and y directions due to a vertical force.24 Stratum under a Ricker wavelet load.25 Stratum under a Ricker wavelet load: the computational do- main and the extended mesh.26 Transient response of a stratum under a Ricker wavelet load (ν = 0.33), due to Pu (top) and Pv (bottom).27 Transient response of a stratum under a Ricker wavelet load (ν = 0.4), due to Pu (top) and Pv (bottom).28 Transient response of a stratum under a Ricker wavelet load (ν = 0.45), due to Pu (top) and Pv (bottom).1 A semi-infinite waveguide: (a) the origional semi-infinite prob- lem, (b) the sketch of problem with truncated domain.2 The construction of RABC: (a) the column of elements in RABC, (b) state I: the domain starts from the interior boundary Γ0I , (c) state E: the domain starts from the exterior boundary Γ0E .3 The profile of Ricker wavelet source at x = 0.4 Responses at node 1 (Γ0I ): x1 = 0 and node 2 (Γ0E ): x2 = l for two source cases: (left) Heaviside step source and (right) Ricker wavelet source.5 Convergence test for one-dimensional problem when the normal source f (t) is a Heaviside step function.6 Stability test for one-dimensional problem when the normal source f (t) is a Heaviside step function.7 A circular cavity embedded in an infinite channel.8 Finite element mesh for the channel problem.9 Transient response of point A.10 Transient response of point B.11 Transient response of point C.12 Wave front taken at t = 5 s.13 Wave front taken at t = 9 s.14 Wave front taken at t = 15 s.15 Wave front taken at t = 19 s.16 Wave front taken at t = 24 s.17 Wave front taken at t = 30 s.18 Wave front taken at t = 40 s.19 Wave front taken at t = 50 s.20 Wave fronts taken at t = 5 s and t = 9 s.21 Wave fronts taken at t = 15 s and t = 24 s.22 Wave fronts taken at t = 30 s and t = 40 s.1 Reciprocal absorbing boundary condition.2 Splitting the half-space into a layer and the lower half-space.3 RABC with PMDLs.4 A two-layer stratum.5 Transient response of the two-layer stratum subjected to a step force at node 1.6 Transient response of the two-layer stratum subjected to a step force at node 2.7 A homogeneous layer truncated by PMDLs.8 The profile of Ricker wavelet force at point A.9 The transient response of points A & B.10 The transient response of points C & D.11 Snapshots of wave front at t = 1.12 Snapshots of wave front at t = 2.13 Snapshots of wave front at t = 3.14 Snapshots of wave front at t = 4.15 RABC with PMDLs for a homogeneous half-space.16 Snapshots of wave front at t = 1.17 Snapshots of wave front at t = 2.18 Snapshots of wave front at t = 3.19 Snapshots of wave front at t = 4.20 RABC with PMDLs for a layered half-space.21 Snapshots of wave front at t = 2.22 Snapshots of wave front at t = 3.23 Snapshots of wave front at t = 4.1 Reciprocal absorbing boundary condition, ΓI = Γ0I ∪ ΓN I ∪ ΓI ∪ D 0 N ∞ D ΓI , ΓE = ΓE ∪ ΓE ∪ ΓE ∪ ΓE .3 Splitting the half-space into a layer and the lower half-space.4 Stretching PMDL elements: (a) regular element, (b) PMDLx edge element, (c) PMDLz edge element, (d) PMDLxz corner element.5 Sketch of Gaussian points for regular and PMDL elements in the problem of a half-space.6 RABC with PMDLs.7 A two-layer stratum.8 Transient responses of node 1 (left) and node 2 (right).9 Convergence test for the two-layer stratum under Heaviside step-force H(t).11 The profile of Ricker wavelet force.