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.

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 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.

Numerical Implementation

1994 ◽  
Author(s):  
Nhan Phan-Thien ◽  
Sangtae Kim
Keyword(s):  
Boundary Element ◽  
Numerical Integration ◽  
Integral Equations ◽  
Large Scale ◽  
Boundary Integral Equations ◽  
Numerical Technique ◽  
Boundary Integral ◽  
Practical Implementation ◽  
Computing Environment ◽  
The Individual

Analytical solutions to a set of boundary integral equations are rare, even with simple geometries and boundary conditions. To make any reasonable progress, a numerical technique must be used. There are basically four issues that must be discussed in any numerical scheme dealing with integral equations. The first and most basic one is how numerical integration can be effected, together with an effective way of dealing with singular kernels of the type encountered in elastostatics. Numerical integration is usually termed numerical quadrature, meaning mathematical formulae for numerical integration. The second issue is the boundary discretization: when integration over the whole boundary is replaced by a sum of the integrations over the individual patches on the boundary. Each patch would be a finite element, or in our case, a boundary element on the surface. Obviously a high-order integration scheme can be devised for the whole domain, thus eliminating the need for boundary discretization. Such a scheme would be problem dependent and therefore would not be very useful to us. The third issue has to do with the fact that we are constrained by the very nature of the numerical approximation process to search for solutions within a certain subspace of L2, say the space of piecewise constant functions in which the unknowns are considered to be constant over a boundary element. It is the order of this subspace, together with the order and the nature of the interpolation of the geometry, that gives rise to the names of various boundary element schemes. Finally, one is faced with the task of solving a set of linear algebraic equations, which is usually dense (the system matrix is fully populated) and potentially ill-conditioned. A direct solver such as Gauss elimination may be very efficient for small- to medium-sized problems but will become stuck in a large-scale simulation, where the only feasible solution strategy is an iterative method. In fact, iterative solution strategies lead naturally to a parallel algorithm under a suitable parallel computing environment. This chapter will review various issues involved in the practical implementation of the CDL-BIEM on a serial computer and on a distributed computing environment.

Removing Non-Uniqueness in Symmetric Galerkin Boundary Element Method for Elastostatic Neumann Problems and its Application to Half-Space Problems

2020 ◽  
Vol 36 (6) ◽  
pp. 749-761
Author(s):  
Y. -Y. Ko
Keyword(s):  
Boundary Element Method ◽  
Rigid Body ◽  
Boundary Element ◽  
Integral Equations ◽  
Half Space ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Neumann Problems ◽  
Galerkin Boundary Element Method ◽  
Element Method

ABSTRACTWhen the Symmetric Galerkin boundary element method (SGBEM) based on full-space elastostatic fundamental solutions is used to solve Neumann problems, the displacement solution cannot be uniquely determined because of the inevitable rigid-body-motion terms involved. Several methods that have been used to remove the non-uniqueness, including additional point support, eigen decomposition, regularization of a singular system and modified boundary integral equations, were introduced to amend SGBEM, and were verified to eliminate the rigid body motions in the solutions of full-space exterior Neumann problems. Because half-space problems are common in geotechnical engineering practice and they are usually Neumann problems, typical half-space problems were also analyzed using the amended SGBEM with a truncated free surface mesh. However, various levels of errors showed for all the methods of removing non-uniqueness investigated. Among them, the modified boundary integral equations based on the Fredholm’s theory is relatively preferable for its accurate results inside and near the loaded area, especially where the deformation varies significantly.

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.

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