Continued Fraction Absorbing Boundary Conditions for Transient Elastic Wave Propagation Modeling

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1 Continued Fraction Absorbing Boundary Conditions for Transient Elastic Wave Propagation Modeling Md Anwar Zahid and Murthy N. Guddati 1 Summary This paper presents a novel absorbing boundary condition (ABC) that can accurately simulate elastic wave absorption into unbounded exteriors that are often encountered in problems related to seismology and soil-structure interaction. By linking the material-damping ABCs (the perfectly matched layers) with differential-equation-based ABCs, the paper draws on their respective advantages and presents an ABC named the continued fraction ABC (CFABC). The CFABC is highly efficient and appears to be the first ABC applicable for polygonal computational domains. Representative numerical examples are presented to illustrate the effectiveness of the proposed ABCs. Introduction It has long been recognized that proper modeling of unbounded domains is vital for accurate computation of earthquake response of soil-structure, fluidstructure and other systems involving coupled media. Similar situation occurs in computational seismology, e.g. in simulation of an earthquake response of a basin. Standard numerical methods such as finite element and finite difference methods cannot handle the unbounded domains as they are designed for the analysis of bounded domains. The standard procedure is to truncate the domain around a region of interest and apply so-called absorbing boundary conditions (ABCs) to mimic the wave absorption properties of the truncated exterior. Development of an accurate ABC for transient elastic wave propagation problems is the subect of this paper. 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 subcategories: global ABCs and local ABCs. While global ABCs may be highly accurate, they tend to be prohibitively expensive for large-scale wave 1 North Carolina State University, Raleigh, NC ; mnguddat@ncsu.edu

2 propagation problems. In contrast, local ABCs are more economical. However, there does not appear to be an accurate local ABCs for elastic wave propagation problems that is amenable for easy numerical implementation. In contrast to differential-equation based ABCs, Material ABCs are physically motivated; they involve adding a layer with artificial damping right next to the boundary, which makes the incident waves decay, thus reducing the reflections. The most successful material ABC is the perfectly matched layer (PML) which was originally developed for electromagnetics by Berenger [2], and triggered explosive development of material-based ABCs. Various versions of elastic PML are now available (see [1] for a review). The PML was compared with local ABCs by Hagstrom who observed that, due to the discretization and truncation errors, PML is less effective than the local ABCs, but emphasized that PML has a better flexibility with respect to treating corner regions [3]. It is thus desirable if the advantages of PML and local ABCs are combined. Guddati and Lim [4] developed such an ABC. They developed a simple link between PML and the local ABCs, leading to a new local ABC, named the continued fraction ABC (CFABC). The CFABC is as flexible as the PML, while retaining the accuracy of local ABCs. The development of CFABC in [4] is limited to acoustic wave equation. In this paper, we extend the CFABC for elastic wave equations and arrive at a new displacement based formulation that is effective for absorbing various types of waves occurring in elastic media. Presented below are (a) an outline of the concepts behind CFABC, (b) its extension to elastic wave propagation, and (c) numerical examples illustrating the effectiveness of elastic CFABC. Basic Idea of Continued Fraction ABCs The CFABC is applicable where the artificial (truncation) boundary is polygonal. It is a composition of edge and corner CFABCs. We first describe the derivation of the edge CFABC, followed by the extension to corner CFABC. Edge CFABC: Without any loss of generality, we explain the derivation of CFABC for a vertical computational boundary. Essentially, our goal is to replace a full space with a left half-space and an ABC simulating the effect of the right half-space (Figure 1a). The procedure entails several steps as described below. (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 Figure 1b) 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.

3 (a) (b) Figure 1: (a) The obective of edge CFABC; (b) Infinite midpoint-integrated layers to mimic right half space; (c) Truncation of the number of layers. (b) In the second step, we make an important observation that the artificial reflection due to discretization can be completely eliminated, thus facilitating the use of finite element layers of arbitrary thickness. The 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 midpoint integration (see [4] for details). A consequence of midpoint integration is that the thickness of the finite element layers can be arbitrary. (c) In order to make the problems computationally tractable, the number of layers needs to be truncated (Figure 1c). 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 ikx analyzed easily for any wave mode of the form ae, with the wavenumber in the direction perpendicular to the boundary k, and is given by [4]: (c)

4 2 n k 2/ i L R =, (1) = 1 k+ 2/ i L where L are the lengths of the one-point integrated finite element layers and n is the number of layers. (d) By examining equation (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 L, one can choose L 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 CFABC. Corner CFABC: At any corner, the CFABC is obtained by taking the tensor product of the two imaginary meshes associated with the two adoining edges. Consequently, the resulting elements would be parallelograms in shape and the element matrices are computed by 1 1 integration rule (see Figure 2). Edge Absorbers Corner Absorbers Edge Absorbers Figure 2: A schematic of edge and corner CFABCs. In summary, the CFABC can be viewed as a finite element mesh as extended to the exterior using rectangular elements in the direction perpendicular to the edge and parallelogram elements at the corners. The simple but important differences are that (a) the elements are of imaginary length in the direction(s) perpendicular to the boundary, and (b) the contribution from these elements are evaluated using midpoint integration rule in the direction perpendicular to the boundary.

5 Complex CFABC for Elastic Wave Propagation It can be shown that the above procedure should work for elastic wave propagation problems as wee (see [5] for a proof). Unfortunately, this extension turned out to be not trivial. The straightforward extension of CFABC to elastic waves has two significant shortcomings. Firstly, the absorbing boundary condition, while stable for homogeneous full and half space problems, tends to be unstable for layered media. Secondly, original CFABC is not amenable to explicit computation, making itself expensive for large-scale wave propagation problems. We overcome these limitations by modifying the original CFABC in two ways: (a) using complex element lengths that render the ABC stable for layered media, giving rise to complex CFABC; (b) Using operator splitting that renders the computational almost explicit. Additionally, the complex CFABC is implemented in a manner different from earlier versions of CFABC. The new implementation demands less storage and, as opposed to earlier CFABC implementations, does not increase the order of the differential equation. The complex CFABC with the new implementation fits well into standard finite element settings and is shown to provide effective absorption of various types of waves occurring in elastic media. The reader is referred to [1] for further details. Numerical Examples Two representative numerical examples are presented here: the first one illustrating the effectiveness of the CFABC for polygonal computational domains and the second one illustrating the applicability of the CFABC for layered media. Example 1 (Explosion in full space): In this example, a vertical Gaussian explosion is simulated in an elastic full-space. The domain is truncated using a polygonal boundary around the load and the CFABC is applied all around the computational domain (see the schematic in figure 3a). As seen in the figure, (a) Figure 3: (a) Problem schematic; (b) A snap shot of the wave fronts. (b)

6 both the vertically propagating pressure wave-front as well as the horizontally propagating shear wave-front are well absorbed by the CFABC. It is important to note that only 3 layers of CFABC are used here, clearly illustrating the efficiency of CFABC. Example 2 (Pulse load on an elastic Layer): In this example we simulate the propagation of a vertical pulse load in an unbounded horizontal layer resting on a rigid rock. Note that only the right half is analyzed due to symmetry. Thus, symmetric boundary conditions are applied on the left. The domain is truncated on the right and the CFABC is applied on the truncation domain. Figure 4 shows the comparison of the CFABC results with results obtained from very large mesh. Clearly, CFABC results in accurate response predictions. Free Free Symmetric BC CFABC Symmetric BC CFABC Rigid rock (fixed) (a) Figure 4: (a) the solution obtained from large mesh; (b) CFABC solution. References Rigid rock (fixed) 1. M. A. Zahid (2005), Efficient absorbing boundary conditions for modeling wave propagation in unbounded domains, Ph.D. dissertation, NC State Univ. 2. T. Hagstrom (2003), New results on absorbing layers and radiation boundary conditions, Topics in Computational Wave Propagation. Springer, pp J. P. Berenger, A Perfectly Matched Layer for the Absorption of Electromagnetic-Waves, J. of Computational Physics 114 (1994) M. N. Guddati and K. W. Lim (2005), Continued fraction absorbing boundary conditions for convex polygonal domains, Int. J. Num. Meth. Engr. (in revision). 5. M. N. Guddati (2005), Arbitrarily Wide Angle Wave Equations for complex media, Comp. Meth. Appl. Mech. Engr. (in press). (b)