The Discrete Wave Number Formulation of Boundary Integral Equations and Boundary Element Methods: A Review with Applications to the Simulation of Seismic Wave Propagation in Complex Geological Structures

1996 ◽  
pp. 3-20
Author(s):  
Michel Bouchon
Keyword(s):  
Wave Propagation ◽  
Boundary Element ◽  
Integral Equations ◽  
Seismic Wave ◽  
Boundary Integral Equations ◽  
Seismic Wave Propagation ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Wave Number ◽  
Geological Structures

Modeling of Wave Propagation in the Unsaturated Soils Using Boundary Element Method

2017 ◽  
Vol 743 ◽  
pp. 158-161
Author(s):  
Andrey Petrov ◽  
Sergey Aizikovich ◽  
Leonid A. Igumnov
Keyword(s):  
Integral Equation ◽  
Wave Propagation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equation ◽  
Boundary Integral Equations ◽  
Water Pressure ◽  
Boundary Integral ◽  
Element Method

Problems of wave propagation in poroelastic bodies and media are considered. The behavior of the poroelastic medium is described by Biot theory for partially saturated material. Mathematical model is written in term of five basic functions – elastic skeleton displacements, pore water pressure and pore air pressure. Boundary element method (BEM) is used with step method of numerical inversion of Laplace transform to obtain the solution. Research is based on direct boundary integral equation of three-dimensional isotropic linear theory of poroelasticity. Green’s matrices and, based on it, boundary integral equations are written for basic differential equations in partial derivatives. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. To approximate the boundary consider its decomposition to a set of quadrangular and triangular 8-node biquadratic elements, where triangular elements are treated as singular quadrangular. Every element is mapped to a reference one. Interpolation nodes for boundary unknowns are a subset of geometrical boundary-element grid nodes. Local approximation follows the Goldshteyn’s generalized displacement-stress matched model: generalized boundary displacements are approximated by bilinear elements whereas generalized tractions are approximated by constant. Integrals in discretized boundary integral equations are calculated using Gaussian quadrature in combination with singularity decreasing and eliminating algorithms.

scholarly journals A Preconditioned 3-D Multi-Region Fast Multipole Solver for Seismic Wave Propagation in Complex Geometries

2012 ◽  
Vol 11 (2) ◽  
pp. 594-609 ◽  
Author(s):  
S. Chaillat ◽  
J.F. Semblat ◽  
M. Bonnet
Keyword(s):  
Wave Propagation ◽  
Seismic Wave ◽  
Seismic Waves ◽  
Plane Waves ◽  
Seismic Wave Propagation ◽  
Boundary Element Methods ◽  
Complex Geometries ◽  
Fast Multipole ◽  
Geological Structures ◽  
Actual Configuration

AbstractThe analysis of seismic wave propagation and amplification in complex geological structures requires efficient numerical methods. In this article, following up on recent studies devoted to the formulation, implementation and evaluation of 3-D single- and multi-region elastodynamic fast multipole boundary element methods (FM-BEMs), a simple preconditioning strategy is proposed. Its efficiency is demonstrated on both the single- and multi-region versions using benchmark examples (scattering of plane waves by canyons and basins). Finally, the preconditioned FM-BEM is applied to the scattering of plane seismic waves in an actual configuration (alpine basin of Grenoble, France), for which the high velocity contrast is seen to significantly affect the overall efficiency of the multi-region FM-BEM.

scholarly journals Energetic BEM for the numerical analysis of 2D Dirichlet damped wave propagation exterior problems

2017 ◽  
Vol 8 (1) ◽  
pp. 103-127
Author(s):  
A. Aimi ◽  
M. Diligenti ◽  
C. Guardasoni
Keyword(s):  
Wave Propagation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Hyperbolic Partial Differential Equations ◽  
Weak Formulation ◽  
Exterior Problems ◽  
Element Method

Abstract Time-dependent problems modeled by hyperbolic partial differential equations can be reformulated in terms of boundary integral equations and solved via the boundary element method. In this context, the analysis of damping phenomena that occur in many physics and engineering problems is a novelty. Starting from a recently developed energetic space-time weak formulation for 1D damped wave propagation problems rewritten in terms of boundary integral equations, we develop here an extension of the so-called energetic boundary element method for the 2D case. Several numerical benchmarks, whose numerical results confirm accuracy and stability of the proposed technique, already proved for the numerical treatment of undamped wave propagation problems in several dimensions and for the 1D damped case, are illustrated and discussed.

scholarly journals A numerical study of energetic BEM-FEM applied to wave propagation in 2D multidomains

2014 ◽  
Vol 96 (110) ◽  
pp. 5-22 ◽  
Author(s):  
A. Aimi ◽  
L. Desiderio ◽  
M. Diligenti ◽  
C. Guardasoni
Keyword(s):  
Wave Propagation ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Numerical Study ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Weak Formulation ◽  
Multilayered Media ◽  
The Stability ◽  
Coupling Algorithm

Starting from a recently developed energetic space-time weak formulation of boundary integral equations related to wave propagation problems defined on single and multidomains, a coupling algorithm is presented, which allows a flexible use of finite and boundary element methods as local discretization techniques, in order to efficiently treat unbounded multilayered media. Partial differential equations associated to boundary integral equations will be weakly reformulated by the energetic approach and a particular emphasis will be given to theoretical and experimental analysis of the stability of the proposed method.

Regularization Techniques Applied to Boundary Element Methods

1994 ◽  
Vol 47 (10) ◽  
pp. 457-499 ◽  
Author(s):  
Masataka Tanaka ◽  
Vladimir Sladek ◽  
Jan Sladek
Keyword(s):  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Scalar Potential ◽  
Fundamental Solutions ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Element Formulation ◽  
Hypersingular Integrals ◽  
Regularization Techniques

This review article deals with the regularization of the boundary element formulations for solution of boundary value problems of continuum mechanics. These formulations may be singular owing to the use of two-point singular fundamental solutions. When the physical interpretation is irrelevant for this topic of computational mechanics, we consider various mechanical problems simultaneously within particular sections selected according to the main topic. In spite of such a structure of the paper, applications of the regularization techniques to many mechanical problems are described. There are distinguished two main groups of regularization techniques according to their application to singular formulations either before or after the discretization. Further subclassification of each group is made with respect to basic principles employed in individual regularization techniques. This paper summarizes the substances of the regularization procedures which are illustrated on the boundary element formulation for a scalar potential field. We discuss the regularizations of both the strongly singular and hypersingular integrals, occurring in the boundary integral equations, as well as those of nearly singular and nearly hypersingular integrals arising when the source point is near the integration element (as compared to its size) but not on this element. The possible dimensional inconsistency (or scale dependence of results) of the regularization after discretization is pointed out. Finally, we discuss the numerical approximations in various boundary element formulations, as well as the implementations of solutions of some problems for which derivative boundary integral equations are required.

scholarly journals Quotient-space boundary element methods for scattering at complex screens

2021 ◽  
Author(s):  
Xavier Claeys ◽  
Lorenzo Giacomel ◽  
Ralf Hiptmair ◽  
Carolina Urzúa-Torres
Keyword(s):  
Boundary Element ◽  
Integral Equations ◽  
Quotient Space ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Link Type ◽  
Trace Spaces ◽  
Space Boundary ◽  
Single Trace

AbstractA complex screen is an arrangement of panels that may not be even locally orientable because of junction lines. A comprehensive trace space framework for first-kind variational boundary integral equations on complex screens has been established in Claeys and Hiptmair (Integr Equ Oper Theory 77:167–197, 2013. https://doi.org/10.1007/s00020-013-2085-x) for the Helmholtz equation, and in Claeys and Hiptmair (Integr Equ Oper Theory 84:33–68, 2016. https://doi.org/10.1007/s00020-015-2242-5) for Maxwell’s equations in frequency domain. The gist is a quotient space perspective that allows to make sense of jumps of traces as factor spaces of multi-trace spaces modulo single-trace spaces without relying on orientation. This paves the way for formulating first-kind boundary integral equations in weak form posed on energy trace spaces. In this article we extend that idea to the Galerkin boundary element (BE) discretization of first-kind boundary integral equations. Instead of trying to approximate jumps directly, the new quotient space boundary element method employs a Galerkin BE approach in multi-trace boundary element spaces. This spawns discrete boundary integral equations with large null spaces comprised of single-trace functions. Yet, since the right-hand-sides of the linear systems of equations are consistent, Krylov subspace iterative solvers like GMRES are not affected by the presence of a kernel and still converge to a solution. This is strikingly confirmed by numerical tests.

scholarly journals Stress Analysis of Three-Dimensional Media Containing Localized Zone by FEM-SGBEM Coupling

2011 ◽  
Vol 2011 ◽  
pp. 1-27
Author(s):  
Jaroon Rungamornrat ◽  
Sakravee Sripirom
Keyword(s):  
Stress Analysis ◽  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Numerical Technique ◽  
Computational Cost ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
Element Method

This paper presents an efficient numerical technique for stress analysis of three-dimensional infinite media containing cracks and localized complex regions. To enhance the computational efficiency of the boundary element methods generally found inefficient to treat nonlinearities and non-homogeneous data present within a domain and the finite element method (FEM) potentially demanding substantial computational cost in the modeling of an unbounded medium containing cracks, a coupling procedure exploiting positive features of both the FEM and a symmetric Galerkin boundary element method (SGBEM) is proposed. The former is utilized to model a finite, small part of the domain containing a complex region whereas the latter is employed to treat the remaining unbounded part possibly containing cracks. Use of boundary integral equations to form the key governing equation for the unbounded region offers essential benefits including the reduction of the spatial dimension and the corresponding discretization effort without the domain truncation. In addition, all involved boundary integral equations contain only weakly singular kernels thus allowing continuous interpolation functions to be utilized in the approximation and also easing the numerical integration. Nonlinearities and other complex behaviors within the localized regions are efficiently modeled by utilizing vast features of the FEM. A selected set of results is then reported to demonstrate the accuracy and capability of the technique.

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