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ZAHID, MD. ANWAR. Efficient Absorbing Boundary Conditions for Modeling Wave Propagation in

Unbounded Domains. (Under the direction of Murthy N. Guddati)

Many engineering problems (e.g. soil-structure interaction, medical imaging and nondestructive evaluation) encounter the phenomena of wave propagation. Among these problems some involve domains of infinite extent. Standard numerical methods such as finite element and finite difference methods cannot handle the unbounded domain as they are designed for the analysis of bounded domains. In order to solve an unbounded-domain problem, the domain is truncated around a region of interest, and absorbing boundary conditions (ABCs) are applied on the truncation boundary. These ABCs are expected to absorb outgoing waves and mimic the effect of the truncated exterior.

Continued-fraction absorbing boundary conditions (CFABCs) are a class of highly efficient ABCs for modeling acoustic wave absorption into unbounded domains. The current versions of CFABCs are applicable only to non-dispersive scalar wave equation and are not effective for dispersive or elastic wave propagation problems. This dissertation contains extensions of CFABCs to dispersive and elastic wave propagation problems.

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propagation problems. Elastic wave propagation is inherently complex because of the strong coupling of pressure and shear waves that propagate at different speeds. It turns out that straightforward extension of acoustic CFABC tends to be unstable for elastic wave propagation problems. Modifying the CFABC by altering the parameters to complex numbers appears to rectify the stability problem. This stabilized CFABC, named “complex CFABC”, is not as efficient as the original CFABC, but is superior to existing ABCs for elastic media. The complex CFABC necessitates modification of the implementation including careful operator splitting to achieve efficient explicit computational procedure. These modifications result in an effective and stable complex CFABC for elastic wave propagation, which is illustrated with the help of numerical examples.

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in Unbounded Domains

by

MD. ANWAR ZAHID

A dissertation

submitted to the Graduate Faculty of

North Carolina State University

in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy

CIVIL ENGINEERING

Raleigh

2005

APPROVED BY:

Dr. C. C. David Tung

Dr. M. Shamimur Rahman

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To my FATHER

Who always inspired me for my PhD, Whom I lost during my PhD

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ACKNOWLEDGEMENTS

First and foremost, I want to thank my parents for their unconditional love and support. Their blessing, inspiration and teaching helped me in every aspect of life. Specially, I remember those days in my childhood, when my father relocated for enrolling me in the best school of our hometown and chose to commute long distances everyday for his work. Both of them never hesitated to take any kind of suffering for the education and career of their children.

I would like to express my sincere appreciation and profound gratitude to my advisor Dr. Murthy N. Guddati for his invaluable guidance, endless patience, constant encouragement and finally for his sincere friendship. He is somebody whom I like to be in touch through out the rest of my life.

Deepest appreciations are due to Dr. Tung, Dr. Rahman and Dr. Haider for serving on my PhD committee.

I am thankful to Bin, Humayoun, Keng-Wit and other group members for their support and help.

I also like to thank my sisters Laboni and Lubna and my sisters in law Mili, Nomi, Seema for always inspiring me to finish the work. Also I like to express my thanks to Iftekhar Bhai.

Thanks are also extended to the wonderful Bangladeshi community members here in North Carolina.

Did I forget to thank my family? No way. Finally, I like to thank my wonderful wife Samina, our son Ethan and our daughter Raaiqa for their unconditional support, endless inspiration, true sacrifice and

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Table of Contents

LIST OF FIGURES...VII

1. INTRODUCTION... 1

1.1 Background and Objectives... 1

1.2 Organization... 4

References ... 4

2. PADDED CONTINUED FRACTION ABSORBING BOUNDARY CONDITIONS FOR DISPERSIVE WAVES ... 7

2.1 Introduction ... 7

2.2 Continued Fraction Absorbing Boundary Conditions (CFABC)...10

2.2.1 CFABC for Straight Computational Boundaries...10

2.2.2 CFABC for Corner Regions...14

2.3 CFABC for Dispersive Wave Equation...14

2.3.1 Characteristics of the Solution for Dispersive Wave Equation...15

2.3.2 Padded CFABCs for Propagating and Evanescent Waves...16

2.4 Finite Element Discretization...18

2.5. Time Stepping...20

2.5.1 Combined Crack-Nicholson / Newmark Time Stepping...21

2.5.2 Extended Central Difference Method with Splitting...23

2.6 Numerical Examples...26

2.6.1 Explosion in a Waveguide...27

2.6.2 Explosion in Full Space...28

2.7 Concluding Remarks...32

References ...36

3. COMPLEX CONTINUED FRACTION ABSORBING BOUNDARY CONDITIONS FOR TRANSIENT ELASTODYNAMICS OF CONVEX POLYGONAL DOMAINS ...40

3.1 Introduction ...40

3.2 Preliminaries...44

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3.4.1 Finite Element Discretization...55

3.5 Time Stepping...56

3.6 Selection of complex CFABC parameters...58

3.7 Numerical Examples...60

3.7.1 Explosion in Full Space...60

3.7.2 Half Space Problem...61

3.7.3 Semi Infinite Wave Guide Problem...62

3.7 Summary and Concluding Remarks ...67

References ...69

4 CONCLUDING REMARKS...72

4.1 CFABCs for Dispersive Waves...72

4.2 CFABCs for Elastic Waves...73

4.3 Closure ...73

Appendix A: Contribution Matrices for Different Elements in Dispersive Case ...73

A.1 Edge Absorber Contribution in Padded CFABC...74

A.2 Corner Absorber Contribution in Padded CFABC...76

Appendix B: Contribution Matrices for Different Elements in Elastic Case ...79

B.1 Edge Absorber Contribution in Elastic CFABC...79

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List of Figures

Figure 2.1 Replacement of full-space with half space augmented with ABC...11

Figure 2.2 Infinite one-point integrated absorbing layers to mimic right half space ...11

Figure 2.3 Truncated mesh with Dirichlet boundary condition on the right...12

Figure 2.4 Corner absorbers (midpoint integration is used for the corner absorbers)...14

Figure 2.5 Schematic of finite element discretization of CFABC with padding...19

Figure 2.6 Waveguide problem...28

Figure 2.7 Performance of CFABC for waveguide problem with low dispersion...29

Figure 2.8 Performance of CFABC for waveguide problem with high dispersion...30

Figure 2.9 Schematic of the full-space problem with CFABC (no padding) ...32

Figure 2.10 CFABC solution of the full-space problem with three absorbing layers ...33

Figure 2.11 CFABC solution of the full-space problem with six absorbing layers...34

Figure 2.12 Schematic of the full-space problem with padded CFABC ...34

Figure 2.13 CFABC solution of the full-space problem with three absorbing layers and one padding layer ...35

Figure 3.1 Replacement of full-space with half space augmented with ABC...47

Figure 3.2 Infinite one-point integrated absorbing layers to mimic right half space ...48

Figure 3.3 Truncated mesh with Dirichlet boundary condition on the right...49

Figure 3.4 A schematic of continued fraction absorbing boundary conditions: one point integration is used for the edge absorber in the direction perpendicular to the boundary and midpoint integration is used for the corner absorbers ...53

Figure 3.5 Schematic of finite element discretization of CFABC ...55

Figure 3.6 Problem schematic: polygonal domain with oblique corners ...61

Figure 3.7 Performance of complex CFABC for polygonal domain with oblique corners...62

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Figure 3.9 Performance of complex CFABC for the rectangular half space problem ...63

Figure 3.10 Waveguide problem ...64

Figure 3.11 Performance of complex CFABC for waveguide problem with three absorbers...65

Figure 3.12 Performance of complex CFABC for waveguide problem with five absorbers...66

Figure 3.13 Wave guide problem: Effect of midpoint integration ...68

Figure 3.14 Comparison of CFABC solution with the reference solution for the wave guide problem. Time history of vertical displacement at the top rightmost point in the domain ...68

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1. INTRODUCTION

1.1 Background and Objectives

Unbounded domains are common in engineering problems, especially in those associated with wave propagation. In most of the problems, the goal of the modeling is to obtain the field variable in a small bounded part of the domain, henceforth called the interior or the computational domain. Often, the bounded domain of interest is modeled using domain based methods such as finite elements or finite differences and special boundary conditions are applied on the boundary of the interior. These boundary conditions are expected to mimic the absorption of waves into the unbounded exterior, and hence referred to as transmitting, non-reflecting, or absorbing boundary conditions (ABCs). Due to the importance of the problem, several researchers have attempted to devise effective absorbing boundary conditions.

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ABCs is the rational approximation of the exact impedance of the exterior. The exact impedance (or the associated dispersion relationship) involves square-root of the differential operators, translating into pseudo differential operators involving expensive convolution operations. Rational approximation of the square-root function facilitates the conversion of the pseudo differential operator into differential operators, making the boundary condition amenable for numerical computation. The use of this idea for unbounded domain modeling is pioneered independently by Engquist and Majda [3, 4] and Lindman[5]. They made use of rational /continued fraction approximations to develop a series of ABCs of increasing accuracy. In spite of their theoretical potential for high accuracy, only low order versions are used, mainly due to the fact that higher order ABCs contain higher order derivatives and cannot be implemented into the standard finite element and finite difference setting. The multi-direction absorbers developed by Higdon [6] are equivalent to Engquist-Majda boundary conditions and have the same limitations. The situation has changed in the past decade, with several researchers presenting practical approaches (based on auxiliary variables) to implement high-order local ABCs (see [7] for a review). These ABCs are found to be highly effective in modeling wave absorption into unbounded domains [8]. Most of these boundary conditions are developed for straight computational boundaries, with some of them extended to orthogonal corner regions.

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extensions and generalizations of the PML were subsequently given (e.g. [11-16]), and research in the optimization of the PML parameters remains active till the present. Despite the attractiveness of PML, it is observed by Hagstrom [8] that, due to the discretization and truncation errors, the performance of PML is inferior to the local ABCs, with the latter’s drawback being the limited applicability.

Considering the accuracy of local ABCs and broader applicability of PMLs, it would be desirable to obtain a boundary condition combining the advantages of the two methods. While the two boundary conditions are based on completely different ideas, there is a surprising link between them which is utilized to develop new ABCs combining the strengths of local ABCs and PML. This link is explored by Guddati [17], who observed similarities in the performance of the continued fraction ABCs (CFABCs) of [18] and PML. Later, CFABCs are linked to optimal finite difference discretization of PML [19] indicating that CFABCs are expected to perform better than general PML. While the work reported in [19] is the optimal local ABC, it is limited to the context of staggered grid finite difference methods. Recently, Guddati and Lim [20] developed an extremely simple link between CFABCs and PML, which translates into simple and efficient implementation in C0 finite element setting. Moreover, they extend

CFABCs to polygonal computational domains. While their CFABC is highly efficient, it is limited to acoustic waves represented by standard non-dispersive scalar wave equation. It is desirable if a similar CFABC can be developed for more general and complex wave propagation problems.

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1.2 Organization

The dissertation is divided into four chapters. This introductory chapter is followed by two completely independent chapters that constitute two separate papers. The second chapter presents the development of padded CFABC for dispersive media, while the third chapter presents the development of complex CFABC for elastic media. Chapter 4 contains some closing remarks on both the developments in this dissertation along with some recommendations for future research.

References

[1] K. L. Shlager and J. B. Schneider, Selective Survey of the Finite-Difference Time Domain Literature, Ieee Antennas and Propagation Magazine 37 (1995) 39-57.

[2] A. Bayliss and E. Turkel, Radiation boundary conditions for wave-like equations, Communications on Pure and Applied Mathematics 33 (1980) 707-725.

[3] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves, Mathematics of Computation 31 (1977) 629--651.

[4] B. Engquist and A. Majda, Radiation Boundary-Conditions for Acoustic and Elastic Wave Calculations, Communications on Pure and Applied Mathematics 32 (1979) 313-357.

[5] E. L. Lindman, Free-Space Boundary-Conditions for Time-Dependent Wave-Equation, Journal of Computational Physics 18 (1975) 66-78.

[6] R. L. Higdon, Absorbing Boundary-Conditions for Difference Approximations to the Multidimensional Wave-Equation, Mathematics of Computation 47 (1986) 437-459.

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[8] T. Hagstrom, New results on absorbing layers and radiation boundary conditions, in: Topics in Computational Wave Propagation. M. Ainsworth, et al. Eds., (Springer-Verlag, New York, 2003) pp. 1-42.

[9] M. Israeli and S. A. Orszag, Approximation of radiation boundary conditions, Journal of Computational Physics 41 (1981) 115-135.

[10] J. P. Berenger, A Perfectly Matched Layer for the Absorption of Electromagnetic-Waves, Journal of Computational Physics 114 (1994) 185-200.

[11] W. C. Chew and W. H. Weedon, A 3d Perfectly Matched Medium from Modified Maxwells Equations with Stretched Coordinates, Microwave and Optical Technology Letters 7 (1994) 599-604.

[12] W. C. Chew and Q. H. Liu, Perfectly matched layers for elastodynamics: A new absorbing boundary condition, Journal of Computational Acoustics 4 (1996) 341-359.

[13] J. P. Berenger, Improved PML for the FDTD solution of wave-structure interaction problems, IEEE Antennas and Propagation 45 (1997) 466-473.

[14] S. A. Cummer, A simple, nearly perfectly matched layer for general electromagnetic media, IEEE Microwave and Wireless Components Letters 13 (2003) 128-130.

[15] J. Y. Fang and Z. H. Wu, Generalized perfectly matched layer for the absorption of propagating and evanescent waves in lossless and lossy media, IEEE Transactions on Microwave Theory and Techniques 44 (1996) 2216-2222.

[16] J.-L. Vay, Assymetric perfectly matched layer for the absorption of waves, Journal of Computational Physics 183 (2002) 367-399.

[17] M. N. Guddati, Comparison of continued-fraction absorbing boundary conditions with perfectly matched layers, in: 14th ASCE Engineering Mechanics Conference (EM 2000), Austin, TX, (2000)

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[19] S. Asvadurov, V. Druskin, M. N. Guddati, and L. Knizhnerman, On optimal finite-difference approximation of PML, Siam Journal on Numerical Analysis 41 (2003) 287-305.

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2. Padded Continued Fraction Absorbing Boundary Conditions

for Dispersive Waves

Abstract

Continued fraction absorbing boundary conditions (CFABCs) are new arbitrarily high order local

absorbing boundary conditions that are highly effective in simulating wave propagation in

unbounded domains. The current versions of CFABCs are developed for the non-dispersive

acoustic wave equation with convex polygonal computational domains. In this chapter, the

CFABCs are modified through augmentation of special padding elements, and are effective for

absorbing evanescent waves occurring in dispersive wave problems while retaining their

absorption properties for propagating waves. The padded CFABCs for dispersive wave equations

result in fourth order evolution equations, which are solved using an efficient combination of

Crank Nicholson method, Newmark time-stepping scheme, and operator splitting ideas. The

effectiveness of the padded CFABCs and their implementation is illustrated through numerical

examples with varying levels of dispersion. (Accepted in Computer Methods in Applied Mechanics and

Engineering)

2.1 Introduction

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waves by eliminating the inward propagation operator. Material-based ABCs, on the other hand, are realized by surrounding the computational domain with a lossy material that dampens the outgoing waves. Differential-equation-based ABCs can be further classified into two sub categories: exact (global) ABCs and approximate (local) ABCs. Computation with global absorbing boundaries involves obtaining the exterior Green’s function and coupling it with the interior, which involves expensive convolution operations [2]. Global ABCs are useful for small-scale wave propagation problems and lead to highly accurate results. In spite of many innovations, they still tend to be computationally expensive for large scale problems. With large-scale problems in mind, we focus our discussion on local ABCs and material based ABCs. In particular, we develop in this chapter an effective ABC for large-scale dispersive wave propagation problems. Furthermore, we limit our discussion to polygonal domains with straight computational boundaries, and note that the use of circular and curved boundaries (see e.g. [3]) could be an efficient alternative to polygonal boundaries.

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mentioned researchers focused on non-dispersive waves, except for Higdon [9], who proposed a sequence of ABCs for the dispersive (Klein-Gordon) wave equation, which is recently implemented in a modified approach by Givoli, Neta and coworkers in finite difference [14, 15] as well as finite element settings [16]. Givoli and Neta also developed similar ABC for dispersive shallow water equations [17].

Material ABCs involve adding a layer with artificial damping right next to the boundary, so that the incident waves decay thus minimizing reflections. The most successful material ABC is the perfectly matched layer (PML) which was originally developed for electromagnetics by Berenger [18], and triggered explosive development of material-based ABCs; we mention few of the developments that are most relevant to the current development. Chew and Liu [19] noted that PML is equivalent to stretching the domain into complex space, a useful analogy that aided further development of PML. Due to the physical and geometrical basis of PML, unlike differential-equation based ABCs, they are easily extendible to corner regions. The original PML is effective mainly for propagating waves. Extensions to evanescent waves are proposed by Fang and Wu [20] and Berenger [21-23]. PML is also extended to dispersive wave problems by several researchers [24-27]. Recently, the PML is compared with rational absorbers by Hagstrom [12], who observed that due to the discretization and truncation errors, PML is less effective than the rational absorbers, but emphasized that PML has better flexibility with respect to treating corner regions.

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standard finite element and finite difference methods. The CFABC, the original version and later derivations, are developed and tested for non-dispersive acoustic wave equation.

In this chapter, we extend the CFABCs for dispersive wave equations and test their effectiveness. In particular we extend CFABC to simultaneously absorb propagating waves as well as evanescent waves, which could be predominant when the dispersion is significant. The resulting CFABC is named padded CFABC as it involves a highly efficient way of inserting a padding region that is aimed at absorbing evanescent waves. The padded CFABC, when discretized using finite element method, results in a fourth order evolution equation, which is different from the standard second order equation arising in wave propagation problems. We solve the fourth order equation efficiently by carefully combining Crank-Nicholson method, Newmark time-stepping scheme and operator splitting ideas. The effectiveness of the proposed ABC and its implementation is illustrated using several numerical examples.

The outline of the rest of the chapter is as follows. The basic ideas behind the latest derivation of CFABCs are summarized in Section 2. In Section 3, CFABCs are extended to dispersive wave equations (Klein-Gordon equation). Finite element discretization is discussed in Section 4, while section 5 focuses on the time integration of the resulting evolution equations. Numerical examples are presented in Section 6, and the chapter is concluded with some closing remarks in Section 7.

2.2 Continued Fraction Absorbing Boundary Conditions (CFABC)

2.2.1 CFABC for Straight Computational Boundaries

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(a) The first step in the derivation is to discretize the right half-space using an infinite number of finite element layers (note that the discretization is performed only in the direction perpendicular to the boundary, see Figure 2.2). The displacement is assumed to vary linearly within each layer. Such a discretization results in errors, triggering the need for rather thin finite element layers.

Figure 2.1 Replacement of full-space with half space augmented with ABC

Figure 2.2 Infinite one-point integrated absorbing layers to mimic right half space

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elimination of the discretization error is achieved by a rather simple procedure: using midpoint integration rule to compute the contribution matrices. It turns out that the finite element approximation error in the half-space impedance is exactly countered by the error in one-point integration (see [31, 32] for details). A consequence of one-point integration is that the thickness of the finite element layers can be arbitrary; they can even be imaginary or complex.

(c) In order to make the problems computationally tractable, the number of layers needs to be truncated (Figure 2.3). Such a truncation introduces error, which can be measured in terms of the reflection at the interface between the left half-space and the discretized right half-space. The reflection error can be analyzed easily for any wavemode of the form aeikx, with horizontal

wavenumber k and is given by [31]:

2

1

2 / 2 / n

n

j

j j

k i L R

k i L =

 − 

= 

+

 

, (1)

where Ljare the lengths of the one-point integrated finite element layers and n is the number of layers.

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(d) By examining (1), we note that the reflection coefficient is equal to unity for real k, making it

completely ineffective for propagating waves. However, armed with flexibility of choosing arbitrary Lj, one can choose Lj to be not only real, but also imaginary or complex. Choosing imaginary or complex element lengths would reduce the reflection coefficient for propagating waves, which constitutes the final step of the new CFABC. In specific, for the CFABC developed for non-dispersive acoustic wave equation, the element lengths are chosen to be purely imaginary and inversely proportional to the frequency, i.e. Lj =2icj ω, where cj are the

chosen (reference) phase velocities and ωis the frequency. With such a choice, the reflection coefficient in (1) becomes:

2

1 n

p j

j p j

c c R

c c =

 − 

= 

+

 

(2)

It is clear from the above equation the CFABC are effective in absorbing propagating waves (when cp is real).

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2.2.2 CFABC for Corner Regions

In light of the new derivation of CFABC in terms of mesh extension into imaginary/complex space, CFABC’s treatment of corners of computational boundaries becomes rather intuitive. For any (convex) corner, the absorbers can be devised by taking the tensor product of the two imaginary meshes associated with the two adjoining edges. Consequently, the resulting elements would be parallelograms in shape (for general non-orthogonal corners), have imaginary lengths in both directions, and the element matrices are computed by 1 1× integration rule (see Figure 2.4). Further details can be found in [31].

Figure 2.4 Corner absorbers (midpoint integration is used for the corner absorbers)

2.3 CFABC for Dispersive Wave Equation

This chapter focuses on the extension of the CFABC to dispersive wave propagation problems. In particular, we consider the Klein-Gordon equation, which is a dispersive equation of the form:

(

)

2 2

0

xx zz tt

c u u u f u

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where c is the wave velocity andf is the dispersion parameter. Klein-Gordon equation is considered here due to its simplicity, but it is expected that the proposed ideas can be easily extended to more complex dispersive wave equations arising in several contexts of engineering and physics (e.g. flexural waves, shallow water equations, meteorology and quantum mechanics).

It turns out that the extension of the CFABC to dispersive wave equation requires special care. Due to the presence of the dispersive term

( )

2

f u in (3), spatial discretization of CFABC results in fourth order evolution equation in time, as opposed to third order equation for the non-dispersive case. Furthermore, while the non-dispersive case involves predominantly propagating waves, there is a significant presence of evanescent waves in the dispersive case, necessitating modification of the original CFABC. Before proceeding to the development of CFABC for dispersive waves, we look at the specific characteristics of dispersive waves that would govern the design of the CFABC.

2.3.1 Characteristics of the Solution for Dispersive Wave Equation

Unlike in non-dispersive media, wave propagation in dispersive media involves both propagating and evanescent waves, which are generated even in the absence of physical boundaries or heterogeneities. In this section, we take a closer look at these wave modes. Due to the linearity of the equation, the solution of (3) can be written in terms of Fourier modes:

( )

i kx lz t

u

=

e

+ −ω , (4)

where k is the horizontal wavenumber, l is the vertical wavenumber and ω is the frequency. The dispersion relation is obtained by substituting (4) in (3):

(

)

2 2 2 2 2

c k l f

ω = + + . (5)

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2 2 2 cos

f k

c θ

ω

= − . (6)

It can be seen that the waves are propagating (have real wavenumber) for frequencies greater than the dispersion parameter, but are evanescent for lower frequencies. The propagating waves are characterized by their phase velocity, cp =ω/k, which takes the form:

2

1 cos

p

c c

f

θ ω

= −

 

 

 

 

, (7)

indicating that ccp < ∞. Similarly, evanescent waves can be characterized by their decay rate

2 2 cos

1 f ik

c f θ

ω

σ = − = − , (8)

which clearly varies between 0 and f c/ .

By examining the reflection coefficient in (1), we note that the CFABCs completely reflect all the evanescent waves (since R=1 when the wavenumbers are imaginary). We thus need to modify the

CFABCs so that they can absorb propagating as well as evanescent waves. Such a modification is presented in the following subsection. We note that the proposed modification is valid not only for dispersive wave problems, but also for non-dispersive systems where evanescent waves are significant, such as in media with interfaces.

2.3.2 Padded CFABCs for Propagating and Evanescent Waves

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lengths in the direction perpendicular to the boundary. The combination of this efficiently discretized buffer (padding) region, laced with regular CFABC is called the padded CFABC, and would be effective in simultaneously absorbing evanescent and propagating waves. Essentially, padded CFABC is similar to regular CFABC, but with some inner layers having real lengths that are frequency independent. Thus the parameters of padded CFABC are the number of padding layers

( )

m , thickness of the padding layers

(

Lj,j=1Lm

)

, the number of absorbing layers

( )

n , and the reference phase velocities associated with

these absorbing layers

(

cj,j=1Ln

)

.

In order to devise effective padded CFABC, it is important to analyze the approximation error in the half-space impedance. Similar to propagating waves, the error in the treatment of evanescent waves could be characterized with the help of the reflection coefficient. Emphasizing that the absorbing layers have no effect on the evanescent waves (R=1 if there are no padding layers), we have the reflection coefficient for evanescent waves as:

2

1

2 2

m

j

j j

L R

L

σ σ

=

 − 

= 

+

 

, (9)

where σ is the decay rate of the evanescent wave and Lj is the thickness of the jth padding layer.

Similarly, we note that the reflection coefficient for propagating waves is unaltered by the padding layers, and is the same as (2). Based on the expressions for the reflection coefficients (2) and (9), one can carefully optimize for the lengths of layers in order to minimize the error. Such optimization procedure is beyond the scope of this chapter, but we make a few observations that could be helpful in choosing the parameters of the padded CFABC.

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velocities are chosen such that cj =c/ cos

( )

θj , where 0≤θj <π/ 2. Unlike the case of non-dispersive

wave propagation, in the presence of dispersion, θj does not have any immediate physical significance.

j

θ can be chosen to cover its entire range by choosing θj =

(

j−1

)

π/ 2n. For example, in most of the

numerical examples presented in this chapter, we choose to use three absorbing layers

(

n=3

)

, with

0 , 30 , 60

j

θ = o o o.

Selection of the lengths of padding layers: Referring to the previous subsection, we note that the decay coefficients for evanescent waves range from 0 and f c/ . Based on this, the lengths of the padding elements could be chosen between 2 /c f and ∞. A geometric mesh advocated in [33] could possibly be used for the padding region, a strategy that is not explored here. For the numerical examples presented here, we required only one padding layer whose length is obtained in an ad-hoc manner.

2.4 Finite Element Discretization

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Figure 2.5 Schematic of finite element discretization of CFABC with padding

In the subsequent discussion, superscript j stands for the element number and the subscripts

, ,

i e c stand for interior / padding, edge and corner regions respectively. The contribution from the edge

absorbing elements takes the form (see (A.11) in the appendix):

j j j j

e e + e e

C V R W , (10)

where the matrices j e

C and Rej are given by (A.12) in the appendix, V= ∂ ∂U t andW=

Udt, where U

is the displacement vector. The contribution from the corner absorbing elements takes the form (see (A.20)in the appendix):

j j j j c c+ c c

K U S X , (11)

where j c

K , j

c

S are given by (A.18) and (A.19), and X=

Wdt. Note that the second term in (11) is

unique to the dispersive wave equation. The contribution for the interior / padding elements results in standard mass and stiffness terms as:

j j j j

i i + i i

M A K U , (12)

where j i

M and j i

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Assembling the contribution from the interior finite elements, the padding and absorbing elements results in the global system of evolution equations:

+ + + + =

MA CV KU RW SX F, (13)

where F is the load vector resulting from the interior loading, and A V U W X, , , , are respectively acceleration, velocity, displacement, the integral of Uand integral of W. The matrix Khas contribution

from the interior / padding elements and corner absorbers

[

K=Ki+Kc

]

, M has contribution from the

interior / padding regions, C and Rhave contribution from the edge absorbers, and S has contribution

from the corner absorbers.

2.5. Time Stepping

The presence of the W and X terms in (13) makes the evolution equation fourth order in time,

as opposed to second order equations in standard dynamics and third order equations resulting from CFABC for non-dispersive wave equations [31]. While it may appear that special effort is necessary to devise time-stepping schemes for (13), the schemes developed for non-dispersive CFABC in [31] could be easily extended to the dispersive case as explained in this section.

Before proceeding to the solution of (13), we summarize the time-stepping scheme for non-dispersive CFABC developed in [31]:

a. Crank-Nicholson method is utilized to link the edge absorber to the interior. Since the corner absorber is effectively an absorber to the edge absorber, trapezoidal rule is applied twice to link the corner variables to the interior. Applying the Crank-Nicholson method transforms the third order evolution equation into a standard second order equation.

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i.e., the coupled Crank-Nicholson / Newmark method has the same stability condition as that associated with solving the interior problem with Newmark method.

c. When constant-average acceleration method (CAA) is used for solving the second order system, the proposed time-stepping scheme is equivalent to the extended constant average acceleration (ECAA) method proposed in [29] which could be used for implicit solution of low-frequency problems. In order to obtain more efficient computational method for medium and high-frequency wave propagation problems, an (almost) explicit time stepping procedure based on central difference method is proposed and is named the extended central difference method (ECDM). The computation is not fully explicit because of the imaginary element lengths and one-point integration, which are crucial for the accuracy of the method Fortunately, careful selection of spatial integration rules renders the computation almost explicit, making it attractive for large-scale computations.

In the remainder of the section, we extend the time-stepping scheme to the fourth order evolution equation (13). The general time-stepping scheme is presented first by combining Crank-Nicholson and Newmark methods. Subsequently, we design an efficient explicit time-stepping scheme for padded CFABC.

2.5.1 Combined Crack-Nicholson / Newmark Time Stepping

The first step of the proposed time stepping scheme is to rewrite spatially discretized equation in (13) at t=tn+1, while separating the contribution from different zones:

1 1 1 1 1 1 1

n n n n n n n

i i e e i i c c e e c c

+ + + + + + + + + + X + = +

M A C V K U K U R W S F . (14)

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(

)

1 1

2

n n n n

e e i i

t

+ = ++ +

U U V V , (15)

where ∆t is the time-step size. The above equation immediately translates to the integration formulas for

e

W and Ve:

(

)

(

)

1 1 1 1 , . 2 2

n n n n

e e i i

n n n n

e e i i

t t + + + + ∆ = + + ∆ = + +

W W U U

V V A A

(16)

Since the corner absorber can be viewed as an absorber to the edge absorber, we apply trapezoidal rule twice to link the corner displacement to the interior variables, i.e.

(

)

(

)

1 1 2 1 . 2 4

n n n n

c c e e

n n n n

c e i i

t t t + + + ∆ = + + ∆ = + ∆ + +

U U V V

U V A A

(17)

In dispersive case, X (the integral of W) is the extra term that arises in the equation, which can be

treated by similarly applying trapezoidal rule twice:

(

)

(

)

1 1 2 1 . 2 4 n e

n n n n

c c e e

n n n

c i i

t t t + + + ∆ = + + ∆ = + ∆ + + X X X W W W U U (18)

Substituting (16), (17) and (18) in (14) results in:

1 1 1

n n n

i i

+ + + = +

MA KU F , (19)

where, 2 2 2 1 1 2 , 2 4 , 2 4 2 4 . 2 4

i e c

i e c

n n n n n n n

e e i c c e i

n n n n n

e e i c c e i

t t t t t t t t t t + + ∆ ∆ = + + ∆ ∆ = + +   ∆ ∆   = − + + ∆ +       ∆ ∆   − + + ∆ +    

M M C K

K K R S

F F C V A K U V A

R W U S X W U

(20)

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extensive numerical experiments, even for the dispersive case, it appears that the above coupling does not impose any additional stability limitations. All the stability conditions, if any, arise from the time stepping procedure employed for solving (19).

The interior displacement and velocity are related to the acceleration according to the Newmark method:

(

)

(

)

2 1 1 1 1 , ,

1 2 2

2 1

n n n n n

i i i i i

n n n n

i i i i

t t t β β γ γ + + + +         ∆ = + ∆ + − + = + ∆ − +

U U V A A

V V A A

(21)

where β and γ are the parameters of the method. Substituting the above expressions in (19) results in

1 1

n n

i

eff eff

+ = +

M A F , (22)

where,

2

1 1 2 1

2

eff

n n n n n

i i i

eff t t t β β + +          = + ∆ = − + ∆ + ∆ −

M M K

F F K U V A (23)

When constant-average acceleration method

(

CAA, β = /1 4,γ =1/ 2

)

is used, the proposed time-stepping is unconditionally stable. Furthermore, due to the second order accuracy of Crack-Nicholson and CAA methods, the combination is also second order accurate.

2.5.2 Extended Central Difference Method with Splitting

The ECAA is an unconditionally stable method imposing no restriction on the time step size, but requires expensive implicit computation. In order to reduce the computational cost, we utilize central difference method to solve (22) with β =0 ,γ =1/ 2. Unfortunately, straightforward substitution of

0 , 1/ 2

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deficiencies. Firstly, we split the stiffness operator into dispersive part and the non-dispersive part, and treat the dispersive part, which is diagonal, implicitly. The second modification involves special lumping that renders the coefficient matrix block diagonal with small blocks. We describe the procedure in the following paragraphs.

Noting that the dispersion term has contributions to the matricesK R Si, e, c, we split the matrices as follows: d nd d nd , , e e e i = i + i

= +

K K K

R R R

(24)

where the superscripts nd and d stand for non-dispersive and dispersive parts. The matrix Sc results

only from dispersion term and requires no splitting. Equation (19) is then rewritten in a split form:

d nd

1 1,d 1,nd 1

n n n n

i i i

+ + + + + = +

MA K U K U F , (25)

where n1,d

i +

U and n 1,nd

i +

U are the discretized field variables associated with dispersive and non-dispersive

terms respectively. The terms in (25) are given by:

d nd nd nd d d 2 2 d nd 2 1 1 2

,nd ,nd ,d ,d ,d

, 2 4 , 2 4 , 2 2 4 .

2 2 4

e e

e e

i e c

i c

i

n n n n n n n

e e i c c e i

n n n n n n n

e i e i c c e i

t t

t t

t

t t

t

t t t

t + + ∆ ∆ = + + ∆ ∆ = + + ∆ = +   ∆ ∆   = − + + ∆ +       ∆ ∆ ∆     − + + + ∆ +      

M M C K

K K R S

K K R

F F C V A K U V A

R W U R W U S X W U (26)

We now use explicit stepping

(

β=0

)

for the non-dispersive part and implicit stepping

(

β =1/ 4

)

for the dispersion term:

2

1,nd ,nd

2

1,d ,d 1

, . 2 4 n i

n n n

i i i

n n n n n

i i i i i

t t t t + + + +   ∆ = + ∆ + ∆ = + ∆ + A

U U V

U U V A A

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Such a modification in the time-stepping alters the final linear equation (22) into

1 1

n n

i

eff eff

+ = +

M A F , (28)

where,

d

d nd

2

2 2

1 1 ,d ,nd

4

4 2

eff

n n n n n n n n

i i i i i i

eff

t

t t

t t

+ +    

   

   

∆ = +

∆ ∆

= − + ∆ + + ∆ +

M M K

F F K U V A K U V A

(29)

It can be easily shown that the dispersion term, since it is treated implicitly, introduces no additional stability constraint, resulting in efficient computation.

By close examination of M K, d in (26), we note that these matrices are not diagonal, making

eff

M non-diagonal, thus precluding explicit solution of (28). The matrices Miand Kid can be

diagonalized easily through nodal point integration (as they are mass-type matrices). On the other hand, the matrices, Ce,Kc and

d

e

R cannot be evaluated using nodal point integration because midpoint

integration is an essential part of CFABC. Furthermore, Ce and Kc are associated with terms involving

spatial derivatives, which prevents complete diagonalization. Fortunately, Ce and

d

e

R can be converted

into a block-diagonal form, by utilizing nodal point integration in the direction of the boundary and retaining the important midpoint integration in the perpendicular direction. Such an integration decouples the nodes in the direction of the boundary, while keeping the nodes coupled in the direction perpendicular to the boundary. Note that typically there are only three or four nodes perpendicular to the boundary, making the block size very small and keeping the computational cost very low. The computation of Kc is not altered in any way, and the corner nodes remain completely coupled. Again,

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be used for the padding region. However, this adds no additional complexity since the padding region can be topologically embedded into the absorbing region for the sake of computation, i.e., the nodes in the padding as well as the absorber are coupled in the direction perpendicular to the boundary, and uncoupled along the boundary.

Another important observation made during numerical experiments is that, in order to attain stability, the integration rule used along the boundary for interior, padding and absorbing regions must be consistent. Since nodal point integration rule is used along the boundary, for the sake of simplicity, we utilized nodal point integration (in the direction of the boundary) for all the terms and regions. This translates to evaluating all the matrices using (a) nodal point integration for the interior, (b) nodal point integration along the boundary and midpoint integration in the direction perpendicular to the boundary, for edge padding and absorbing layers, and (c) center point integration used for the corner padding and absorbing elements.

2.6 Numerical Examples

In this section, numerical examples are presented to illustrate the performance of the CFABCs with or without padding for the dispersive waves. Due to its computational efficiency we use the explicit computation with splitting for all the numerical examples. For all the problems, a Gaussian explosion used in [31] is utilized to generate the dispersive waves, i.e. the forcing function is given by:

( )

(

)

( )

2 2 2

2

3 2 2 2

2 1 , if 2 ,

,

0, otherwise o o

f t t

o o o

r

f t t e t t r a

f r t a

π

π − −

 

 

− − − ≤ ≤

= . (30)

In the above, fo is the central frequency of excitation, to =1 fo, r is the distance from the center of the

(37)

parameters are chosen as a=5h and fo =c 20h, where h is the element size. The motivation for such a choice is to ensure that the spatial discretization error is not visible in the results. Two levels of dispersion is considered for the numerical examples: low dispersion with the dispersion parameter f = fo 5, and high dispersion with f =1.5fo. The wave velocity for all the examples is taken as c=2000.

2.6.1 Explosion in a Waveguide

Here we simulate the propagation of explosion in an unbounded waveguide, with Dirichlet conditions applied at the top and bottom boundaries. The height of the waveguide

( )

H is taken as 6 units. The explosion is applied at a distance of H 4 from the top boundary. The domain is truncated on

the left and right at a distance of H from the center of the explosion. Due to the symmetry of the problem, only the right half is analyzed. The resulting 6 6× square domain is discretized using 40 40×

square elements with edge length h=0.15. The time step size is taken as the critical value, i.e.

2 t h c

∆ = . Since the boundary is placed far away from the load, we expect the evanescent waves to decay, indicating that padding layers may not be necessary. Based on this, we start our analysis with applying three-layer CFABC. According to the discussion in Section 2.3.2, the phase velocities are chosen as cj =2000, 2309, 4000, corresponding to θj =0 , 30 , 60o o o. The schematic of the problem is shown in

(38)

Figure 2.6 Waveguide problem.

(Only the right half of the problem is analyzed, due to symmetry considerations.)

Wave motion in a waveguide is complex in nature as the energy is trapped within its boundaries. Due to the complexity of computing the exact solution, we obtained the reference solution by analyzing a larger computational domain of size 72 6× , laced again with three-layer CFABC. We verified that the validity of the reference solution by ensuring that there is no visible difference by further increasing the size of the computational domain. The reference solutions corresponding to the low and high dispersion cases are plotted in the left columns of Figures 2.7 and 2.8 respectively. By comparing the reference solution with the solution obtained by using 6 6× domain with three-layer CFABC, it is clear that just

three absorbing layers were sufficient to properly simulate the unbounded exterior indicating high efficiency of CFABC for dispersive waves.

2.6.2 Explosion in Full Space

(39)
(40)
(41)

domain is chosen such that the explosion is at a distance of

(

3.75,3.75

)

from the bottom left corner of the domain. Similar to the previous problem, the interior is discretized with elements with edge

lengthh=0.15. The time step size is taken as the stability limit

(

∆ =t h 2c

)

. We analyzed the problem

for both low and high dispersion. It appears that the CFABC used for the waveguide problem suffices for the low-dispersion case (results are not presented here), while high dispersion case requires further refinement. We thus focus only on the high dispersion case for the rest of the section.

The computational domain is laced with three absorbing layers all around, with reference phase velocities cj =2000, 2309, 4000 (a schematic is shown in Figure 2.9). The snapshots of the solution at

different time instances are shown in Figure 2.10. While the CFABC was effective for earlier times, the circular wave front becomes distorted in the third and fourth snapshots, indicating that the three-layer CFABC is introducing errors. With the goal of increasing the accuracy, we increased the number of CFABC layers to six with the phase velocities corresponding to θj =0 ,10 , 30 , 40 , 60 80o o o o o, o. The resulting snapshots, shown in Figure 2.11, indicate that increasing the number of absorbing layers is not sufficient for reducing the error. This appears reasonable in that since the computational boundary is very close to the source, there are significant evanescent waves at the boundaries which are not treated properly by the absorbing layers.

In order to treat the evanescent waves properly, we laced the computational domain with a padded CFABC, with just one padding layer added to the three previously used absorbing layers. The padding thickness is chosen as 12, which is 80 times the size of the interior finite element size. The

(42)

from padded CFABC analysis are shown in Figure 2.13, with all the wave fronts circular in shape, clearly illustrating the effectiveness of padded CFABC in accurately absorbing dispersive waves with minimal computational cost.

Figure 2.9 Schematic of the full-space problem with CFABC (no padding)

2.7 Concluding Remarks

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computation accurate. For the numerical examples considered in this chapter, padded CFABC required just one padding layer and three absorbing layers, in contrast to more than ten PML layers that are typically used for effective absorption [20, 23, 24].

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Figure 2.11 CFABC solution of the full-space problem with six absorbing layers

Figure 2.12 Schematic of the full-space problem with padded CFABC.

(45)

Figure 2.13 CFABC solution of the full-space problem with three absorbing layers and one padding layer

(46)

While the present chapter focuses on Klein-Gordon equation, due to the generality of the underlying ideas, padded CFABC would be applicable for more complex dispersive wave systems. Padded CFABC are also applicable for any systems where evanescent waves are significant, such as non-dispersive systems with boundaries and interfaces. An essential next step to the current development is devising strategies for automatically choosing the parameters of CFABC, which is a subject of ongoing investigations. We also note that the midpoint integration idea is very general and applicable to any problem with governing equation is second order in space [32]. Implementation of these ideas for elastic and poro-elastic ABCs will be reported in future publications.

References

[1] K. L. Shlager and J. B. Schneider, Selective Survey of the Finite-Difference Time-Domain Literature, IEEE Antennas and Propagation Magazine 37 (1995) 39-57.

[2] J. P. Wolf and C. Song, Finite Element Modelling of Unbounded Media. Willey, Chichester, 1996.

[3] D. Givoli, Numerical Methods for Problems in Infinite Domains. Elsevier, Amsterdam, 1992. [4] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of

waves, Mathematics of Computation 31 (1977) 629--651.

[5] B. Engquist and A. Majda, Radiation Boundary Conditions for Acoustic and Elastic Wave Calculations, Communications on Pure and Applied Mathematics 32 (1979) 313-357.

[6] E. L. Lindman, Free Space Boundary Conditions for Time-dependent Wave Equation, Journal of Computational Physics 18 (1975) 66-78.

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[8] R. L. Higdon, Radiation Boundary Conditions for Elastic Wave Propagation, SIAM Journal on Numerical Analysis 27 (1990) 831-869.

[9] R. L. Higdon, Radiation Boundary-Conditions for Dispersive Waves, SIAM Journal on Numerical Analysis 31 (1994) 64-100.

[10] D. Givoli, High order local non-reflecting boundary conditions: a review, Wave Motion 39 (2004) 319-326.

[11] F. Collino, High order absorbing boundary conditions for wave propagation models., in: Proceedings of 2nd international conference on mathematical and numerical aspects of wave propagation phenomenon, (1993)

[12] T. Hagstrom, New results on absorbing layers and radiation boundary conditions, in: Topics in Computational Wave Propagation. M. Ainsworth, et al. Eds., (Springer-Verlag, New York, 2003) pp. 1-42.

[13] T. Hagstrom and T. Warburton, A new auxiliary variable formulation of high-order local radiation boundary conditions: corner compatibility conditions and extensions to first-order systems, Wave Motion 39 (2004) 327-338.

[14] D. Givoli and B. Neta, High-order non-reflecting boundary scheme for time-dependent waves, Journal of Computational Physics 186 (2003) 24-46.

[15] D. Givoli and B. Neta, High order non-reflecting boundary conditions for dispersive waves, Wave Motion 37 (2003) 257-271.

[16] D. Givoli, B. Neta, and I. Patlashenko, Finite element analysis of time-dependent semi-infinite wave-guides with high-order boundary treatment, International Journal of Numerical Methods in Engineering 58 (2003) 1955-1983.

[17] D. Givoli and B. Neta, High-order nonreflecting boundary conditions for the dispersive shallow water equations, Journal of Computational and Applied Mathematics 158 (2003) 49-60.

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[19] W. C. Chew and Q. H. Liu, Perfectly matched layers for elastodynamics: A new absorbing boundary condition, Journal of Computational Acoustics 4 (1996) 341-359.

[20] J. Y. Fang and Z. H. Wu, Generalized Perfectly Matched Layer - an Extension of Berengers Perfectly Matched Layer Boundary-Condition, IEEE Microwave and Guided Wave Letters 5 (1995) 451-453.

[21] J. P. Berenger, Improved PML for the FDTD solution of wave-structure interaction problems, IEEE Transactions on Antennas and Propagation 45 (1997) 466-473.

[22] J. P. Berenger, Application of the CFSPML to the absorption of evanescent waves in waveguides, IEEE Microwave and Wireless Components Letters 12 (2002) 218-220.

[23] J. P. Berenger, An effective PML for the absorption of evanescent waves in waveguides, IEEE Microwave and Guided Wave Letters 8 (1998) 188-190.

[24] T. Uno, Y. W. He, and S. Adachi, Perfectly matched layer absorbing boundary condition for dispersive medium, IEEE Microwave and Guided Wave Letters 7 (1997) 264-266.

[25] M. Fujii and P. Russer, A nonlinear and dispersive APML ABC for the FD-TD methods, IEEE Microwave and Wireless Components Letters 12 (2002) 444-446.

[26] G. X. Fan and Q. H. Liu, An FDTD algorithm with perfectly matched layers for general dispersive media, IEEE Transactions on Antennas and Propagation 48 (2000) 637-646.

[27] I. M. Navon, B. Neta, and M. Y. Hussaini, A perfectly matched layer approach to the linearized shallow water equations models, Monthly Weather Review 132 (2004) 1369-1378.

[28] M. N. Guddati, Comparison of continued-fraction absorbing boundary conditions with perfectly matched layers, in: 14th ASCE Engineering Mechanics Conference (EM 2000), Austin, TX, (2000)

[29] M. N. Guddati and J. L. Tassoulas, Continued-fraction absorbing boundary conditions for the wave equation, Journal of Computational Acoustics 8 (2000) 139-156.

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[31] M. N. Guddati and K-W. Lim, Continued fraction absorbing boundary conditions for convex polygonal domains, International Journal of Numerical Methods in Engineering (2004) (in review).

[32] M. N. Guddati, Arbitrarily Wide Angle Wave Equations for complex media, Computer Methods in Applied Mechanics and Engineering (2004) (to appear).

[33] D. Ingerman, V. Druskin, and L. Knizhnerman, Optimal finite difference grids and rational approximations of the square root I. Elliptic problems, Communications on Pure and Applied Mathematics 53 (2000) 1039-1066.

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3. Complex Continued Fraction Absorbing Boundary Conditions for

Transient Elastodynamics of Convex Polygonal Domains

Abstract

This chapter presents a new absorbing boundary condition (ABC) that is effective in modeling wave

absorption into elastic unbounded domains. The proposed boundary condition is an extension of the

scalar continued fraction ABCs (CFABCs) which combine the accuracy of rational-approximation

based ABCs and flexibility of the perfectly matched layers. The main hurdles involved in the

extension were that the straightforward extension of the acoustic CFABC results in instabilities and,

even if they were stable, could not be implemented explicitly. The instability problem is resolved by

choosing complex CFABC parameters, while the explicit computation is achieved by a new finite

element implementation utilizing special time-integration procedures. The resulting elastic CFABCs,

similar to their acoustic counterparts, are applicable to polygonal computational domains. This

chapter contains the details of the derivation and implementation of the elastic CFABCs, as well as

several numerical examples illustrating their effectiveness.

3.1 Introduction

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introduction of a radiation condition in any unbounded direction: waves should radiate outwards from a source toward an unbounded direction without any spurious wave motion in the reverse direction. Irregularities in the geometry of the domain or the physical material often require a numerical solution of the problem, thus requiring the use of a bounded domain, along with an artificial boundary condition named as absorbing boundary condition (ABC). These ABCs are expected to absorb outgoing waves and mimic the effect of the truncated unbounded part. Since the 1970s many researchers have proposed several ABCs, which are classified into two broad classes: differential-equation-based and material-based. Differential-equation-based ABCs are obtained by factoring the wave equation into outward and inward propagating operators and permitting only outgoing waves by eliminating the inward propagation operator. Material-based ABCs, on the other hand, are realized by surrounding the computational domain with a fictitious material that dampens the outgoing waves. Differential-equation-based ABCs can be further classified into two sub categories: exact (global) ABCs and approximate (local) ABCs. Computation with global absorbing boundaries involves obtaining the exterior Green’s function and coupling it with the interior, which involves expensive convolution operations [1]. Global ABCs are useful for small-scale wave propagation problems and lead to highly accurate results. In spite of many innovations, they still tend to be computationally expensive for large scale problems. With large-scale problems in mind, we focus our discussion on local ABCs and material based ABCs. In particular, we develop in this chapter an effective ABC for large-scale elastic wave propagation problems. Furthermore, we limit our discussion to polygonal domains with straight computational boundaries, and note that the use of circular and curved boundaries (see e.g. [2]) could be an efficient alternative to polygonal boundaries.

(52)

problem from a different direction and devised multi-directional absorbers [6], which can be proven to be equivalent to rational absorbers. Despite the possibility of high accuracy, only less accurate low-order versions of rational absorbers were used, mainly because the higher order ABCs contain higher order derivatives and were difficult to implement in the standard finite element and finite difference settings. In the past decade this situation has improved, with many researchers presenting practical approaches (based on auxiliary variables) to implement high-order local ABCs (see [7] for a review). One important point is that local ABCs require special treatment at corner regions; while several treatments are developed to treat corners [8, 9], they are rather complex involving cumbersome implementation (when compared to material ABCs). Most of the above mentioned developments focused on scalar wave equation. In contrast, elastic wave propagation usually has three components of waves, namely P, SV and SH waves. Depending on the boundary conditions and material heterogeneity there may also be Rayleigh and Stoneley waves. The out of plane SH waves are similar to the scalar wave but the in-plane P and SV waves are strongly coupled, making the elastic case significantly more difficult to analyze than the scalar case. Clayton and Engquist [10] were the first to extend the idea of differential equation based ABC for the elastic case. Noting the complexity of dispersion relationship for the elastic ABCs, they use the scalar case to provide a hint to the paraxial approximations and fit the coefficients by matching to the full elastic wave equation; their development is limited to low-order ABCs that are not highly accurate. Later Higdon [11] extended his multi-directional absorption idea to the elastic case. While Higdon’s ABCs are theoretically as accurate as their scalar counterparts, the higher order versions do not appear to have been implemented. Randall [12, 13] extended Lindman’s ABCs to elastic waves by considering the displacement-potential formulation, and is not applicable to more general and versatile displacement formulation.

Figure

Figure 2.1 Replacement of full-space with half space augmented with ABC
Figure 2.3 Truncated mesh with Dirichlet boundary condition on the right
Figure 2.4 Corner absorbers (midpoint integration is used for the corner absorbers)
Figure 2.5 Schematic of finite element discretization of CFABC with padding
+7

References

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