CONTINUED-FRACTION ABSORBING BOUNDARY CONDITIONS FOR THE WAVE EQUATION

Transcription

1 Journal of Computational Acoustics, Vol. 8, No. 1 (2) c IMACS CONTINUED-FRACTION ABSORBING BOUNDARY CONDITIONS FOR THE WAVE EQUATION MURTHY N. GUDDATI Department of Civil Engineering, North Carolina State University, Campus Box 798, Raleigh, NC , USA JOHN L. TASSOULAS Department of Civil Engineering, The University of Texas at Austin, Austin, TX 78712, USA Received 25 June 1999 Revised 1 October 1999 Absorbing boundary conditions are generally required for numerical modeling of wave phenomena in unbounded domains. Local absorbing boundary conditions are generally preferred for transient analysis because of their computational efficiency. However, their accuracy is severely limited because the more accurate high-order boundary conditions cannot be implemented easily. In this paper, a new arbitrarily high-order absorbing boundary condition based on continued fraction approximation is presented. Unlike the existing boundary conditions, this one does not contain high-order derivatives, thus making it amenable to implementation in conventional C finite element and finite difference methods. The superior numerical properties and implementation aspects of this boundary condition are discussed. Numerical examples are presented to illustrate the performance of these new high-order boundary condition. 1. Introduction Numerical modeling of wave propagation in unbounded domains requires special treatment as the standard methods for bounded domain analysis are not adequate. The customary way to analyze unbounded domain problems is to truncate the unbounded domain around the region of interest and apply the so-called absorbing boundary condition (ABC) on the truncation boundary. This truncated problem can then be solved using standard numerical methods such as the finite element method (FEM) and finite difference method (FDM). The critical aspect in maintaining the accuracy of this procedure is to obtain an ABC that accurately simulates the effect of the truncated exterior. The past 3 years have seen significant research efforts towards obtaining accurate and efficient ABCs for both time-harmonic and transient analyses. 1 Most of these boundary conditions can be classified into two broad categories: global and local. The global boundary conditions attempt to simulate the effect of the exterior in an exact sense and are fully coupled in space and, for transient analysis, in time. On the other hand, the local boundary conditions are approximations of the exterior Green s function and are local in both space and time, making them attractive for transient analysis. More recently, a third type of 139

2 14 M. N. Guddati & J. L. Tassoulas boundary conditions based on perfectly matched layers (PML, introduced by Berenger 2 )is attracting significant attention. This paper focuses on local absorbing boundary conditions, and a comparision with global boundary conditions and perfectly matched layers will be the subject of future work. The main idea behind the local boundary conditions is to perfectly absorb waves impinging on the truncation boundary. The first local absorbing boundary condition was proposed by Lysmer and Kuhlemeyer. 3 Their boundary conditions are viscous dampers that absorb waves travelling in any prescribed direction. They produce spurious reflections for waves travelling in other directions. Lindman 4 and Engquist and Majda 5,6 proposed a series of local boundary conditions with increasing order of accuracy. Their boundary conditions are based on approximating the one-way wave equation in the Fourier domain by rational approximations. Despite the theoretical merit of the Engquist-Majda boundary conditions, their practical success is limited by the difficulties posed in implementing the high-order derivatives occurring in the high-order boundary conditions. Lindman s approach is better suited for implementation in explicit finite differences, but its stability properties are not very clear. Furthermore, their implementation for finite elements and implicit finite differences is not clear. There have been several other kinds of local boundary conditions proposed (e.g., Bayliss and Turkel, 7 Higdon, 8 and others). Although the philosophy behind these derivations might be different, all these boundary conditions share the same drawback as the Engquist-Majda boundary conditions: their high-order versions involve high-order derivatives, making it very difficult to implement them in conventional finite-element and finite-difference settings. To overcome this limitation, research effort has been directed towards implementation issues of the high-order boundary conditions. The implementation proposed by Kallivokas and Bielak 9 fits well into the context of the finite element method, but is limited to secondorder boundary conditions. The multi-stage boundary conditions developed by Safjan 1 do not have this limitation, but require that several initial-boundary value problems be solved. In this paper, we surpass the obstacle of high-order derivatives by proposing a series of high-order local boundary conditions that contain only second-order derivatives (this method was previously outlined in Guddati 11 and Guddati and Tassoulas. 12 ) This is accomplished by introducing auxiliary layers of degrees of freedom next to the boundary. The augmented coefficient matrices arising from the proposed boundary conditions are sparse, symmetric and positive definite. For the first time, these continued fraction based boundary conditions present a way of using arbitrarily high-order ABCs in standard finite element and finite difference settings. The outline of the rest of the paper is as follows. In Sec. 2, we present the basic idea behind the continued fraction boundary condition and draw similarities with existing ABCs. Section 3 contains the key idea of reducing the order of the boundary condition by viewing the continued fraction iteration in a matrix form. The implementation details are given in Sec. 4. Section 5 contains some numerical examples illustrating the effectiveness of high-order boundary conditions. Finally, the paper is summarized in Sec. 6 with some recommendations for future research.

3 Continued-Fraction Absorbing Boundary Conditions for the Wave Equation Absorbing Boundary Conditions Based on Continued Fractions We start our presentation by examining the wave equation in two spatial dimensions: G(u xx + u zz )+ρu tt =, (2.1) where the subscripts denote differentiation with respect to the corresponding variable. The solution for the wave equation can be expressed in terms of the Fourier modes of the form e (ikx+ilz)/ G+iωt/ ρ. (2.2) In the above, k and l are scaled wave numbers corresponding to x and z, andωis the scaled frequency. We then have the following duality between the space-time and the Fourier (wave number-frequency) domains: ik G x ; il G z ; iω ρ (2.3) t Using this duality, the wave equation can be transformed into the Fourier domain resulting in a relationship between the wave numbers and the frequency: k 2 + l 2 = ω 2. (2.4) We limit our discussion to absorbing boundary conditions for straight computational boundaries. Without any loss of generality, we can assume that the outward normal is in the positive x direction. An ideal boundary condition would annihilate all the reflections from the boundary, indicating that the waves should have non-negative phase velocity in the x direction. This implies that the exact solution contains modes with non-positive k/ω. Thisis easily accomplished by writing the boundary condition in the form of the following one-way wave equation: (ik + iω 1 σ 2 ) u =, (2.5) where σ =(il)/(iω). For traveling waves, σ canbeviewedassin(θ), where θ is the angle of incidence with respect to the direction normal to the boundary. Although the one-way wave equation represents the dispersion relationship for exact absorbing boundary condition in the Fourier domain, it cannot be implemented in space and time coordinates. This is because the inverse transform of the square-root term results in a pseudo-differential operator that cannot be implemented using numerical methods. To obtain manageable boundary conditions consisting of differential operators, a rational approximation can be used for 1 σ 2. Then the boundary condition takes the form ik + iωf n =, (2.6) where f n is an nth order rational approximation of 1 σ 2. Engquist and Majda 6 proposed a series of approximations based on truncated continued fraction expansions of 1 σ 2 as follows: f 1 =1, (2.7)

4 142 M. N. Guddati & J. L. Tassoulas f 2 =1 σ2 2, (2.8) f n+1 =1 σ2. (2.9) 1+f n We generalize their approach and propose a modified continued fraction expansion as follows: f 1 =cos(θ 1 ), (2.1) f 2 =cos(θ 2 ) cos2 (θ 2 ) (1 σ 2 ), cos(θ 2 )+f 1 (2.11) f n+1 =cos(θ n+1 ) cos2 (θ n+1 ) (1 σ 2 ). (2.12) cos(θ n+1 )+f n In the above, θ n are angles that are chosen to accelerate the convergence of the iteration based on the directionality of the wave propagation. It can also be shown that the iterative procedure in Eqs. (2.1) (2.12) is a contraction mapping (just like Eqs. (2.7) (2.9)) for σ 1 and thus models the absorption of traveling waves progressively better. It is clear that if θ n = for every iteration, this procedure is identical to the expansion used by Engquist and Majda (Eqs. (2.7) (2.9)) Reflection Coefficients The ratio of reflected and incident wave amplitudes provides a measure of the effectiveness of the absorbing boundary conditions. The reflection coefficients for travelling waves for any approximation of the dispersion relationship is given by 6 : R n = f n cos(θ) f n +cos(θ), (2.13) where θ is the angle of incidence. Notice that if f n is exact (f n = 1 σ 2 =cos(θ)), the reflection coefficient is zero. It can be easily shown that the reflection coefficient for the nth-order approximation in Eqs. (2.1) (2.12) is R n = n k=1 cos(θ k ) cos(θ) cos(θ k )+cos(θ). (2.14) This reflection coefficient is identical to the one for the multi-directional transmitting boundary proposed by Higdon 8 : n ( cos(θ k ) x ) u =. (2.15) t k=1 Indeed, it can be shown that the nth-order continued fraction boundary condition is exactly the same as the nth-order multi-directional boundary conditions with the same choice of angles.

5 Continued-Fraction Absorbing Boundary Conditions for the Wave Equation 143 Although the continued fraction form (Eqs. (2.1) (2.12)) appears to be much more complicated than the multi-directional form (Eq. (2.15)), we illustrate in the next section that the former is much more amenable to numerical implementation. 3. Mixed Form of Absorbing Boundary Conditions To eliminate the high-order spatial and temporal derivatives, we present a mixed form by defining auxiliary variables. The implementation is restricted to even-order approximations with θ 2k 1 = θ 2k. Then the iterative process for even-order approximations takes the form: In the above, a n and b n are given by f 2 = a 1, (3.1) b 2 n+1 f 2n+2 = a n+1. a n+1 + f 2n (3.2) a n = cos(θ 2n) 2 b n = cos(θ 2n) σ2 2cos(θ 2n ), (3.3) + 1 σ2 2cos(θ 2n ). (3.4) To satisfy the approximate local boundary condition (ik + iωf 2n )u =, it is sufficient to satisfy the following matrix equation: ik iω + a n b n b n a n + f 2n 2 [ ] u = u 1 [ ], (3.5) where u 1 is an auxiliary variable, whose definition is included in the above matrix equation (the second equation). The elimination of u 1 through factorization of Eq. (3.5) results in the desired approximate boundary condition (ik + iωf 2n )u =. This idea of viewing the continued fraction iteration in matrix form is the key to the ease of implementation of continued fraction boundary conditions. The procedure described above can be used recursively to define additional auxiliary variables u 1, u 2,..., u n 1, and to obtain the following matrix form: ik iω + a n b n b n a n + a n 1 b n 1 b n b2 b 2 a 2 + a 1 u u 1 u 2. u n 1 =. (3.6).

6 144 M. N. Guddati & J. L. Tassoulas Using Eqs. (3.3) and (3.4) and noting that σ =(il)/(iω), the above equation can be rewritten as: iku u u u 1 + iω(a + B).. (il)2 iω B u 1. =.. (3.7) u n 1 u n 1 In the above, the matrices A and B are given by: c n c n A = 1 c n c n + c n 1 c n 1. 2 c..... n 1, (3.8) c2 c 2 c 2 + c 1 where, 1/c n 1/c n B = 1 1/c n (1/c n )+(1/c n 1 ) 1/c n /c..... n 1, (3.9) /c2 1/c 2 (1/c 2 )+(1/c 1 ) c n =cos(θ 2n ). (3.1) Using the duality relationships (Eq. (2.3)), and scaling by G, the above equation can be transformed into space and time as follows: G u x u = u ρg(a + B) u 1 t.. G G 2 ρ B u 1 z 2 dt, (3.11). u n 1 u n 1 where the coefficient matrices A and B are given by Eqs. (3.8) and (3.9). Notice that Eq. (3.11) gives a relationship between traction ( G u/ x) and displacement (u), that can be used directly for the boundary term in the variational formulation of the interior. This is the continued fraction absorbing boundary condition. The major advantage of this boundary condition over the existing local boundary conditions is that we see only second-order spatial derivatives and first-order time derivatives. This property makes this boundary condition very amenable to numerical implementation. Also, it can be seen that the matrices A and B are symmetric and positive definite, resulting in a symmetric and positive-definite discretized boundary matrix in a finite element setting. The implementation details and advantages are discussed in the following section.

7 Continued-Fraction Absorbing Boundary Conditions for the Wave Equation Finite Element Discretization The discretization of u 1,..., u n 1 is chosen to be identical to the discretization of u on the boundary, that is, in the z direction as shown in Fig. 1. We apply Galerkin approximation to the boundary condition. Under these circumstances, the application of the above boundary condition can be viewed as adding additional nodes and elements to the existing truncated problem. As shown in Fig. 1, we add a layer of elements for every two levels of approximation. The contribution of these absorbing elements to the coefficient matrices is symmetric and positive definite. It should also be noted that the connectivity is local and preserves the sparsity of the coefficient matrices. Thus, the computational problem in Fig. 1 can be viewed as a standard finite element problem and all the computational techniques developed for standard analysis such as domain-decomposition techniques apply directly to our computational problem. The interior problem, when augmented with the boundary conditions, results in the following integro-differential equation in time: MU tt + CU t + KU + R t Udt = F. (4.1) From Eq. (3.11), we notice that the matrices C and R have contributions from the boundary conditions. Due to the existence of the time-integral term, this discrete evolution equation differs from those of standard wave equations and an appropriate time stepping scheme should be devised. The constant-average acceleration method 13 is extended in a consistent manner to handle this situation as follows: v n+1 = v n + a n + a n+1 t, (4.2) 2 u n+1 = u n + v n + v n+1 t, (4.3) 2 w n+1 = w n + u n + u n+1 t, (4.4) 2 where, a is the second derivative, v is the first derivative and w is the integral of u with respect to time. This scheme appears to work well for the numerical examples presented in this paper. It may be possible to generalize the scheme to induce numerical dissipation. However, this possibility is not explored here. 5. Numerical Examples With the easy-to-implement Continued Fraction Boundary Conditions (CFRAC BCs) developed in the previous section, it is now possible to examine the accuracy properties of higher order boundary conditions. In this section, the CFRAC BCs are used to solve two example problems. The first one involves modeling propagation of individual wave modes in a half-space or a layer. The second one is a more complicated wave scattering problem.

8 146 M. N. Guddati & J. L. Tassoulas Finite Elements (Interior) z x n-2 2n n-2 n-1 Order of approximation Number of additional layers Layers of absorbing elements Finite Element Node Auxiliary Node Fig. 1. Implementation of higher order absorbing boundary conditions: each line of auxiliary nodes represents discretization of an auxiliary variable. A layer of absorbing elements between two lines of auxiliary nodes is not a physical layer, but a mathematical artifact. With the help of these examples, it is concluded that the accuracy is significantly improved by using higher order CFRAC BCs Propagation of Wave Modes in a Half-space Wave propagation in an acoustic half-space (x > and z R) can be represented by several wave modes in one-dimensional half-space (x >). These wave modes are standard Fourier modes in the z direction, and satisfy the following dispersive wave equation in x and t: u xx + u tt + l 2 u =, (5.1) where l is the wave number associated with z. Notice that the coefficients G and ρ (Eq. (2.1)) are chosen to be unity. The CFRAC boundary condition can be applied to the above equation by replacing 2 / z 2 by l 2 in Eq. (3.11). It is our goal to simulate the effect of the one-dimensional half-space (x >) governed by Eq. (5.1). More specifically, we need the value of u/ x at x = for an applied displacement u at x =. The applied displacement has the form of a squared sine pulse: u x= = sin 2 (Ωt) for Ωt π and u x= = otherwise. We choose Ω = 2 to minimize the evanescent waves, that is, the solution contains predominantly travelling waves. The exact solution for u/ x is obtained by using the frequency domain solution (k = ω 2 l 2 ) in conjunction with Fourier transforms. The half-space is then replaced by various high-order CFRAC BCs (order = 2, 4, 6 and 8). The time step size is chosen to be.25 so as to minimize the error due to time stepping. The results from the CFRAC BCs are compared with the exact solution in Fig. 2. It is clearly evident that the second-order and fourth-order BCs are considerably erroneous. On the other hand, the sixth- and eighth-order BCs produce remarkably accurate results Cylindrical Cavity in an Infinite Channel In this problem, a cylindrical cavity of radius (R = 5) is embedded in an infinite channel of width = 1. The channel is filled with acoustic fluid with wave velocity c = 4.A squared sine

9 Continued-Fraction Absorbing Boundary Conditions for the Wave Equation Reaction Force EXACT Second Order Fourth Order Sixth Order Eighth Order Time Fig. 2. Performance of high-order absorbing boundary conditions. Fig. 3. Analysis of wave propagation in an acoustic channel around a cavity. radial velocity pulse is applied on the cavity: v =sin 2 (t)for t πand v = otherwise. The problem setting is shown in Fig. 3. The aim of the analysis is to obtain the velocity potential history on the cavity wall, for example at Point-A. It is also of interest to examine the wave propagation phenomenon around the cavity (near-field), which is arbitrarily chosen as a 5 1 rectangular region surrounding the cavity. Due to the symmetry of the problem, only one quarter of the near-field will be examined as shown in Fig. 3.

10 148 M. N. Guddati & J. L. Tassoulas Channel Boundary (Normal Velocity = ) Artificial Boundary Conditions (ABC) Symmetric Boundary Conditions (Normal Velocity = ) Symmetric Boundary Conditions (Normal Velocity = ) Fig. 4. Finite element mesh for the channel problem. For the purposes of numerical simulation, a velocity potential formulation is employed. The near-field (one quarter) is discretized using 1211 nodes and 1125 bilinear finite elements (Fig. 4). Homogeneous Neumann boundary conditions are applied at the axes of symmetry and at the channel boundary. Applied velocity (nonhomogenous Neumann) boundary condition is applied on the cavity wall. The unbounded domain on the left of the truncation boundary is replaced by an absorbing boundary condition. The time-step size is chosen as.25 units. To obtain the exact solution, the problem is solved by applying the robust characteristics method (Guddati, 11 Guddati and Tassoulas 14 ) on the artificial boundary. The mesh and time-step size are refined to examine the convergence of the solution. It is noticed that the original mesh and time-step size result in exact solutions for all practical purposes. The problem is then solved with the local boundary conditions applied at the artificial boundary. Four analyses are performed by using various boundary conditions: (1) viscous damper, (2) second-order Engquist-Majda BC, (3) fourth-order CFRAC BC with θ 2 =,θ 4 =45,and (4) sixth-order CFRAC BC with θ 2 =,θ 4 =3,θ 6 =6. The velocity potential histories at Point-A obtained from different analyses are compared with the exact history in Figs. 5 and 6. Notice that the top part of the figure gives the long term response, and the bottom part gives the short term response. It is of interest to examine

11 Continued-Fraction Absorbing Boundary Conditions for the Wave Equation The variation of the velocity potential at Point-A Exact Viscous Damper Second Order E-M BC Velocity Potential Time 1.2 The variation of the velocity potential at Point-A Exact Viscous Damper Second Order E-M BC Velocity Potential Time Fig. 5. Existing boundary conditions: velocity potential history at Point-A. how accurate different approaches are in the short term, and how well they capture the oscillatory behavior in the long term. Figure 5 contains the results for the viscous damper and the second-order Engquist-Majda BC. It is clearly seen that the viscous damper gives a highly dissipative response in the long term, and highly erroneous results in the short term. The results from the second-order Engquist-Majda BC are less erroneous in the short term, but still overly dissipative in the long term. The results from the CFRAC BCs are given

12 15 M. N. Guddati & J. L. Tassoulas 1.2 The variation of the velocity potential at Point-A Exact Fourth Order CFRAC (angles =, 45) Sixth Order CFRAC (angles =, 3, 6) Velocity Potential Time 1.2 The variation of the velocity potential at Point-A Exact Fourth Order CFRAC (angles =, 45) Sixth Order CFRAC (angles =, 3, 6) Velocity Potential Time Fig. 6. Continued fraction boundary conditions: velocity potential history at Point-A. in Fig. 6. Notice that the fourth-order CFRAC BC is very accurate in the short term and captures the oscillatory behavior in the long term, although with some error. The sixth-order CFRAC BC results in fairly accurate results: almost negligible error in the short term and reduced amplitude and phase errors in the long term. The snapshots of the velocity potential at different times for the exact solution are given in Fig. 7. Note that each snapshot represents a change in the wave front. At the first snapshot

13 Continued-Fraction Absorbing Boundary Conditions for the Wave Equation 151 Time = 5. Time = Time = 16. Time = Fig. 7. Exact solution: snapshots of velocity potential in the near-field. (t = 5), the wave front is contained in the near field and is fully visible and circular. At time t = 9, the wave front expands beyond the near field, but does not encounter any physical boundaries, and is still circular. At time t = 16, the wave front reaches the channel boundary and starts reflecting. At time t = 24, the reflected wave scatters off the cavity as can be

14 152 M. N. Guddati & J. L. Tassoulas Time = 5. Time = Time = 16. Time = Fig. 8. First-order viscous damper: snapshots of velocity potential in the near-field. clearly seen in the figure. The ability to capture all these phenomena accurately is arguably a good test of the performance of absorbing boundary conditions. The snapshots for the viscous damper are given in Fig. 8. It can be clearly seen that, at t = 9, the wave front is not properly absorbed by the damper, giving erroneous results

15 Continued-Fraction Absorbing Boundary Conditions for the Wave Equation 153 Time = 5. Time = Time = 16. Time = Fig. 9. Second-order Engquist-Majda boundary condition: snapshots of velocity potential in the near-field. at subsequent times. The results from the second-order Engquist-Majda BCs (Fig. 9) are better in that the results seem fairly similar until time t = 16, but the scattered wave front at t = 24 is completely erroneous. On the other hand, the results from both the fourth- and sixth-order CFRAC BCs (Figs. 1 and 11) are almost identical to the exact results.

16 154 M. N. Guddati & J. L. Tassoulas Time = 5. Time = Time = 16. Time = Fig. 1. Fourth-order continued-fraction boundary condition: snapshots of velocity potential in the near-field. 6. Conclusions We have presented a continued fraction local absorbing boundary condition that can be implemented very elegantly in a C finite element setting. It is believed that this is the first arbitrarily high-order local boundary condition that is amenable to numerical

17 Continued-Fraction Absorbing Boundary Conditions for the Wave Equation 155 Time = 5. Time = Time = 16. Time = Fig. 11. Sixth-order continued-fraction boundary condition: snapshots of velocity potential in the near-field. implementation in a standard finite element setting. The proposed technique is based on a continued fraction approximation of the dispersion relationship. The key element of this boundary condition is that the continued fraction approximation is viewed in a matrix form to arrive at a mixed form of boundary condition. The boundary conditions can be considered as a generalization of the Engquist-Majda boundary conditions, and are equivalent

18 156 M. N. Guddati & J. L. Tassoulas to the multi-directional wave absorbers. The attractive feature of the continued fraction boundary condition is that the order can simply be increased by adding layers of nodes and elements that can be viewed topologically as a finite element mesh. The augmentation of the continued fraction boundary condition with the interior finite element mesh preserves sparsity, symmetry and positive definiteness of the coefficient matrix. To treat the time integral term resulting from the boundary condition, the constant-average acceleration is consistently extended. Numerical examples illustrate the superior performance of these local boundary conditions. The continued fraction boundary condition can be extended and improved in several aspects. Time-stepping schemes with flexibility of induced numerical damping should be explored. Our method, in its present form, is limited to straight interaction boundaries. It should be extended to curved boundaries, and if possible, to boundaries with corners. The possibility of extending the method to elastic wave equation should also be explored. References 1. D. Givoli, Numerical Methods for Problems in Infinite Domains (Elsevier, Amsterdam, 1992). 2. J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comp. Physics 114 (1994), J. Lysmer and R. L. Kuhlemeyer, Finite dynamic model for infinite media, ASCE J. of Engr. Mech. 95 (1969), E. L. Lindman, Free-space boundary conditions for the time dependent wave equations, J. Comp. Physics 18 (1975), B. Engquist and A. Majda, Absorbing boundary conditions for the numerical simulation of waves,, Math. Comp. 31 (1977), B. Engquist and A. Majda, Radiation boundary conditions for acoustic and elastic wave calculations, Comm. Pure Appl. Math. 32 (1979), A. Bayliss, and E. Turkel, Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math. 33 (198), R. L. Higdon, Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comp. 47 (1986), L. F. Kallivokas and J. Bielak, Time-domain analysis of transient structural acoustics problems based on the finite element method and a novel absorbing boundary element, J. Acoust. Soc. Am. 94 (1993), A. Safjan, Highly accurate non-reflecting boundary conditions for finite element simulation of transient acoustics problems, TICAM Report 95-11, Texas Inst. for Comp. and Appl. Math., The University of Texas at Austin (1995). 11. M. N. Guddati, Efficient methods for modeling transient wave propagation in unbounded domains, Ph.D. Thesis, The University of Texas at Austin, M. N. Guddati and J. L.Tassoulas, Transient analysis of wave propagation in unbounded media: space-time methods and continued fraction implementations, in IUTAM Symposium on Computational Methods for Unbounded Media, eds. T. Geers (Kluwer Academic Publishers, 1998), pp K.-J. Bathe, Finite Element Procedures (Prentice-Hall, Englewood Cliffs, 1996). 14. M. N. Guddati and J. L. Tassoulas, Characteristics methods for transient analysis of unbounded media, Comp. Meths. Appl. Mech. Eng. 164 (1998),