Boundary Integral Equations and Boundary Elements Methods in Elastodynamics

doi:10.1016/S0065-2687(06)48003-1

Advances in Geophysics

Volume 48, 2007, Pages 157-189

https://doi.org/10.1016/S0065-2687(06)48003-1 Get rights and content

We review the application of boundary integral equation (BIE) methods to elastic wave propagation problems. BIE methods express the wavefield as an integral equation defined over the boundary of the domain studied. They can be grouped into two families. “Direct” BIE relate the wavefield (generally the displacements and tractions) to the values of the wavefield at the boundary of the domain, while “Indirect” BIE rely on an intermediate unknown, which is usually a distribution of fictitious sources along the boundary. We present the mathematical bases of the methods and discuss and review their various numerical implementations. We illustrate some of the applications for seismic wave propagation.

Introduction

In several areas of physics, like solid mechanics, electromagnetism, fluid dynamics, or thermodynamics, the resolution of a problem can be reduced to an integral equation defined over the boundary of the domain studied. This expresses the fact that the totality of the information pertaining to the domain is carried by the boundary. Apparently this remarkable fact was discovered by G. Green although some attribute the finding to C.F. Gauss. In any event, the Boundary Integral Equations (BIE) formulate the solution of a problem in terms of values at the domain's boundary. BIE methods arise from potential theory and can be grouped into two families. “Direct” BIE relate the physical variables (the wavefield of displacements and tractions in elastodynamics) in a given domain to the values taken by these variables at the boundary. “Indirect” BIE, on the other hand, rely on an intermediate unknown, which is usually a distribution of fictitious (or apparent) sources along the boundary.

The resolution of the direct or indirect BIE must be done numerically. The first step consists in discretizing the boundary into a set of points or into a set of elements. In the latter case, the BIE method is often referred to as a Boundary Element Method (BEM) while the indirect BIE may be named as indirect BEM (IBEM). In elastodynamics, which is our domain of interest here, the second step is the evaluation of the medium Green's functions. In doing so, one needs to avoid the inherent singularities of having to evaluate the wavefield at the source location itself. To this effect, various formulations of the BIE have been developed which are un-conditionally stable. The third and final step is to solve the resulting linear system of equations.

Compared to domain methods, like finite-differences or finite-elements, BIE methods have an interesting conceptual advantage which is the reduction of one space dimension for both discretization and handling of the unknowns. For instance, two-dimensional (2-D) problems are reduced to integral representations along lines (thus, this is equivalent to 1-D problems). This advantage is enhanced because the domain effectively considered can be much larger than in domain methods which require to make explicit such size. Moreover, BIE methods (in frequency) do not need absorbing boundaries. Indeed, we have to mention that BIE are more accurate than domain methods. They match more easily the boundary conditions and do not suffer from grid dispersion. They easily fulfill radiation conditions at infinity and can handle complex boundary geometries for which domain methods would require complicated grid setting. On the other hand, the benefits are counter-balanced by the global character of boundary approaches, each equation involves the full wavefield (at least in the considered sub domain), while in domain methods, only the local field values are related by the governing equations. In the BIE the field at each point of the boundary is linked to all boundary field values and, as a consequence, the linear system that needs to be solved is dense. For this reason, sometimes they are more demanding in computer time and memory space, because they require solving non-sparse linear systems.

As the BIE methods directly express the physics of the problem, they also provide potentially valuable insight into the problems they tackle. They cannot be easily designed to handle non-linear problems. They are relatively well adapted to study acoustic or seismic wave propagation in the shallow Earth which can often be described as consisting of relatively homogeneous layers separated by sharp boundaries.

After describing the direct and indirect BIE methods, we will review and discuss some salient numerical aspects of these techniques. We will also present and discuss other boundary methods, in which approximate expressions of the wavefield, such as plane wave decompositions or basis function expansions are used to set up boundary equations from which relatively fast, approximate solutions can be obtained.

Excellent reviews of the available literature on BEM in elastodynamics up to the early 90s are those by Manolis and Beskos (1988), Brebbia and Domínguez (1992), and Domínguez (1993). These researchers focused on various structural engineering applications. Herein the emphasis is laid on seismological aspects.

Section snippets

The Direct BIE

The formulation of the direct BIE in elastodynamics can be traced back to the pioneering work of Somigliana (1886), more than a century ago. It stems from the theory of elastic potentials and states that field values of a problem of linear elasticity in a given domain are fully determined from the displacements and tractions along the domain's boundary. It can be seen as an application of the Maxwell–Betti reciprocity theorem. Somigliana's work is based upon the classical work by Stokes (1849)

The Indirect BIE

The indirect formulation of the elastodynamic problem expresses the wavefield as an integral over the boundary of elementary source radiations: $u_{i} (x, t) = \int_{0}^{t} d τ \int_{S} ϕ_{j} (ξ, τ) G_{i j} (x, t, ξ, τ) d S (ξ)$ where $ϕ_{j} (ξ, τ)$ denotes a force density distribution applied at time τ at point ξ of the surface S. In this equation we have again assumed that the volumic body sources are null. If present, their contribution simply needs to be added to the integral term.

An example of this formulation is given in Vai et al. (1999);

Approximate Representations

Early applications of BIE methods in elastodynamics have dealt with the inherent singularities of the integrand kernels, as the Green's functions have to be evaluated at the very location of the sources. The pioneering works by Wong and Jennings (1975), and Sills (1978) are good examples of these efforts. However, in several practical instances a simplified approach was needed. The chosen strategy was to avoid these singularities and reliable methods have been developed where the sources are

Discretization and Inversion

The discretization of the boundaries leads to the discretization of the BIE and the application of the boundary conditions transforms the BIE into a system of linear equations which, in general, is not symmetric. This fact is sometimes undesirable, particularly for large size matrices. The resolution of this system is usually done implicitly, and in the frequency domain. The discretization of the boundaries is generally done by approximating the surfaces by a set of elements over which the

Time Domain Implementation

In order to overcome memory limitations, which arise from solving very large linear systems, BIE methods may be implemented in the time domain. The BIE then gives the expression of the displacement field u at time t as a function of the values of u and t at times $τ < t$ . Its discretization leads to an implicit formulation which can be solved for each time step (Mansur and Brebbia, 1982, Mansur and Brebbia, 1985, Rice and Sadd, 1984, Banerjee).

It is, however, often more interesting to render this

Other Boundary Methods

One of the most successful boundary method in elastodynamics is the Aki–Larner method. This method, developed by Aki and Larner (1970), represents the harmonic wave field diffracted by a surface or interface as a superposition of plane waves with unknown coefficients propagating away from the diffracting boundary. Inhomogeneous waves are allowed. This discrete summation results from an assumed spatial periodicity of the medium. The equations, in displacement and stress, expressing the boundary

Hybrid Methods

In many ways, boundary and domain methods can be viewed as complementary techniques, and their combination may lead to more efficient solutions than relying on one technique only. Formulations combining BE and FE have been proposed by Mossessian and Dravinski (1987), Regan and Harkrider (1989), Fujiwara (1996b), Zhang et al. (1998), and Fu (2002). This combination is particularly relevant for media containing both irregular boundaries and volume heterogeneities (Fu, 2002). Figure 16, shows the

Domains of Application

BIE methods have been extensively applied to model or simulate seismic wave propagation in the shallow earth. They have helped understand and quantify the effect of irregular topography, like hills, mountains, or valleys on earthquake ground motion (Bouchon, 1973, Bouchon, 1985, Wong and Jennings, 1975, Sánchez-Sesma, 1978, Sánchez-Sesma, 1983, Sills, 1978, Sánchez-Sesma and Rosenblueth, 1979, Bard, 1982, Wong, 1982, Sánchez-Sesma, Géli, Kawase, 1988, Gaffet and Bouchon, 1989, Axilrod and

Concluding Remarks

This rapid passage over the BIE in elastodynamics essentially covers certain developments in the Earth sciences and seismology of the last three decades. There is a number of problems that may benefit from advances in other areas.

Variational, domain decomposition and subdomain hybrid formulations hold the promise to give the power to deal with more complex configurations…

Acknowledgements

The critical comments of V. Maupin and R.-S. Wu helped to sharpen the focus of this work. Thanks are given to M. Campillo, O. Coutant for their comments and suggestions. The multifarious help of E. Flores was crucial to finish this work. Our research benefited from various sources of support along the years. We want to thank the French CNRS, and the Mexican CONACYT and DGAPA-UNAM.

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Boundary Integral Equations and Boundary Elements Methods in Elastodynamics

Introduction

Section snippets

The Direct BIE

The Indirect BIE

Approximate Representations

Discretization and Inversion

Time Domain Implementation

Other Boundary Methods

Hybrid Methods

Domains of Application

Concluding Remarks

Acknowledgements

Wave Motion

Wave motion

Appl. Math. Mod.

Soil Dyn. Earthquake Engrg.

Surface motion of a layered medium having an irregular interface due to incident plane SH waves

J. Geophys. Res.

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