The consistent boundary element method for potential and elasticity: Part II — machine-precision numerical evaluations for 2D problems

In linear poroelasticity so far only collocation boundary element methods have been available. However, in some applications, e.g., when coupling with finite elements is desired, a symmetric formulation is preferable. Choosing a Galerkin approach which involves the second boundary integral equation, such a formulation is possible. Here, a previously presented integration by part technique for the regularization of the first boundary integral equation is extended to the second boundary integral equation as well. While the weakly singular representation of the double layer operator has been presented before, the emphasis lies here on the so called hyper‐singular boundary integral operator. Due to the regularization, this operator can be evaluated numerically and, hence, be used within a numerical scheme for the first time. Different numerical studies will be presented to show the behavior of the established symmetric Galerkin boundary element method, also comparing it with collocation boundary element methods. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

To solve boundary integral equations for potential problems using collocation Boundary Element Method (BEM) on smooth curved 3D geometries, an analytical singularity extraction technique is employed. By adopting the isoparametric approach, curved geometries that are represented by mapped rectangles or triangles from the parametric domain are considered. The singularity extraction on the governing singular integrals can be performed either as an operation of subtraction or division, each having some advantage.A particular series expansion of a singular kernel about a source point is investigated. The series in the parametric coordinates consists of functions of a type Rpxqyr, where R is a square root of a quadratic bivariate homogeneous polynomial with coefficients corresponding to the first fundamental form of a smooth surface, and p,q,r are integers, satisfying p≤−1 and q,r≥0. By extracting more terms from the series expansion of the singular kernel, the smoothness of the regularized kernel at the source point can be increased. Analytical formulae for integrals of such terms are obtained from antiderivatives of Rpxqyr, using recurrence formulae, and by evaluating them at the edges of rectangular or triangular parametric domains.Numerical tests demonstrate that the singularity extraction technique can be a useful prerequisite for a numerical quadrature scheme to obtain high order of convergence for the governing singular integrals in 3D collocation BEM with respect to the number of quadrature nodes or with respect to size of the discretization mesh cells.

In this paper, we extend the SABO technique (Semi-Analytical method for Barrier Options), based on collocation Boundary Element Method (BEM), to the pricing of Barrier Options with payoff dependent on more than one asset. The efficiency and accuracy already revealed in the case of a single asset is confirmed by the presented numerical results.

Abstract: In the boundary element method, treatment of all the possible singular integrals is very important for the correctness and accuracy of the solutions. Generally, the directional derivative of a weakly singular integral is computed by an integral in the sense of Cauchy principle value if the directional derivative of the weakly singular integral kernel is strongly singular or in the sense of Hadamard finite part integral if the the directional derivative of the weakly singular integral kernel is hypersingular. We will try to discover how the strongly singular and hypersingular integrals are generated and propose a method to avoid the appearance of such kind of strongly singular and hypersingular integrals. Here, using some examples, we will show that with an ’exact derivation’ the directional derivative of a weakly singular integral will still be a weakly singular integral or a normal integral. That is none strongly or hypersingular integrals are generated in such a process. And therefore, Cauchy principle value and Hadamard finite part integral are not indispensable. The result will help us to form a new approach to overcome the so-called strongly and hypersingular integrals occurring in the boundary element method.

Evaluation of surface electric fields, which is well described by Poisson equation, is of great importance on many occasions in power system. This paper proposes a boundary element method for long slender conductors. Examples show the method is convergent and shows great potential for the evaluation of electric field generated by slender conductors.

Solutions of 2D and 3D non-homogeneous potential problems by using a boundary element-collocation method

Integration of singular integrals for the Galerkin-type boundary element method in 3D elasticity

In this paper, an iterative algorithm based on the conjugate gradient method (CGM) in combination with the boundary element method (BEM) for obtaining stable approximate solutions to the Cauchy problem in linear elasticity is analysed. An efficient stopping criterion for the CGM proposed by Nemirovskii in 1986 is employed and in addition the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data.

In electrocardiographic imaging, epicardial potentials are reconstructed computationally from electrocardiographic measurements. The reconstruction is typically done with the help of the boundary element method (BEM), using the point collocation weighting and constant or linear basis functions. In this paper, we evaluated the performance of constant and linear point collocation and Galerkin BEMs in the epicardial potential problem. The integral equations and discretizations were formulated in terms of the single- and double-layer operators. All inner element integrals were calculated analytically. The computational methods were validated against analytical solutions in a simplified geometry. On the basis of the validation, no method was optimal in all testing scenarios. In the forward computation of the epicardial potential, the linear Galerkin (LG) method produced the smallest errors. The LG method also produced the smallest discretization error on the epicardial surface. In the inverse computation of epicardial potential, the electrode-specific transfer matrix performed better than the full transfer matrix. The Tikhonov 2 regularization outperformed the Tikhonov 0. In the optimal modeling conditions, the best BEM technique depended on electrode positions and chosen error measure. When large modeling errors such as omission of the lungs were present, the choice of the basis and weighting functions was not significant.

Finite Element Method (FEM1) is pervasively used in most of 3D product design analysis, in which Computer Aided Design (CAD) models need to be converted in to mesh models first and then enriched with some material features and boundary conditions data, etc. The interaction between CAD models and FEM models is intensive. Boundary Element Method (BEM) has been expected to be advantageous in large-scale problems in recent years owing to its reduction of the dimensionality and its reduced complexity in mesh generation. However, the BEM application has so far been limited to relatively small problems due to the memory and computational complexity for matrix buildup are O(N2). The fast multipole BEM (FMBEM) combined with BEM and fast multipole method (FMM) can overcome the defect of the traditional BEM, and provides an effective method to solve the large-scale problem. Combining GPU parallel computing with FMBEM can further improve its efficiency significantly. Based on the three-dimensional elastic mechanics problems, the parallelisms of the multipole moment (ME), multipole moment to multipole moment (M2M) translation, multipole moment to local expansion (M2L) translation, local expansion to local expansion (L2L) translation and near-field direct calculation were analyzed respectively according to the characteristics of the FMM, and the parallel strategies under CUDA were presented in this paper. Three main major parts are included herein: (1) FMBEM theory in 3D elastostatics, (2) the parallel FMBEM algorithm using CUDA, and (3) comparison the GPU parallel FMBEM with BEM, FEM and FMBEM respectively by engineering examples. Numerical example results show the 3D elastostatics GPU FMBEM not only can speed up the boundary element calculation process, but also save memory which can be effective to solve the large-scale engineering problems.

The integrals required in the computation of influence coefficient matrices of the boundary element method (BEM) depend on the distance r(x,x′) from the collocation point or field point x to the source or load point x′. As a consequence, a distinction must be made between the case where the collocation point does not belong to the integration domain (proper integrals) and the case where the collocation point does belong to the integration domain (improper integrals). Moreover, situations arise where x comes close to x′ and the integrals, albeit of a regular character, behave almost as improper, this case being referred to as nearly singular integration. Analytical integration captures best the singular or nearly singular kernel behavior, but this technique can only be carried out in very simple situations as, for instance, boundary integrals over straight elements. In the present paper a set of useful analytical integration formulas for the 2D BEM with curved elements is deduced, employing a symbolic computational algebra system. © 1997 John Wiley & Sons, Ltd.

The boundary element method (BEM) is a numerical method for solving boundary-value or initial-value problems formulated by use of boundary integral equations. In the BEM, only the boundaries - that is, surfaces for three-dimensional problems or curves for two dimensional (2D) problems - of a problem domain need to be discretized. However the boundary element method (BEM) has been limited to solving problems with a few thousand degrees of freedom (DOFs) on a personal computer. This is because the conventional BEM, in general, produces dense and nonsymmetric matrices. The main idea of the fast multipole (FM) BEM is to employ iterative solvers to solve the BEM system of equations. Using this method we can solve models with more than one million equations on a laptop computer. In this paper, the governing equations for elasticity problems are reviewed first. Numerical examples are provided to demonstrate the accuracy and efficiencies of fast multipole method (FMM) for solving 2D elasticity problems.

A Monte Carlo solution method for linear elasticity

Poroelasticity is concerned with heterogeneous media consisting of an elastic solid skeleton saturated by a diffusing pore fluid. Its range of application covers a variety of important real-life problems, such as design of nuclear reactor cores, exploitation of oil or gas deposits, simulation of living bone behaviour for orthopedical surgery, control of filtration leakage from reservoirs, and manufacturing process design for composite materials. Poroelasticity is now the subject of a fairly abundant literature, stemming from Terzaghi’s concept of effective stress and Biot’s linear consolidation theory. From a computational mechanics point of view, poroelastic analysis has been conducted using either the finite element method or the traditional (collocation) boundary element method [1–3].

An adaptive cell-based domain integration method for treatment of domain integrals in 3D boundary element method for potential and elasticity problems