Boundary Element Equations Research Articles

A boundary element method for dynamic plate bending problems

In this paper the fundamental solution for Kirchhoff plate in the Laplace transform domain is deduced and the boundary element formulation is established to study plate bending problem. By use of the moment integral equation, the dual boundary element equation is developed to solve the cracked plate problem subjected to dynamic loads. Durbin’s Laplace transform inversion method is used and the dynamic stress intensity factors are determined by equivalent stress technique. An infinite plate containing an isolated crack or two cracks, or one set of parallel cracks under Heaviside load on the crack surfaces is studied. The numerical results obtained demonstrate the efficiency and accuracy of the proposed formulation.

BEM for crack-inclusion problems of plane thermopiezoelectric solids

The problem of interactions between an inclusion and multiple cracks in a thermopiezoelectric solid is considered by boundary element method (BEM) in this paper. First of all, a BEM for the crack–inclusion problem is developed by way of potential variational principle, the concept of dislocation, and Green's function. In the BE model, the continuity condition of the interface between inclusion and matrix is satisfied, a priori, by the Green's function, and not involved in the boundary element equations. This is then followed by expressing the stress and electric displacement (SED) and elastic displacements and electric potential (EDEP) in terms of polynomials of complex variables ξt and ξk in the transformed ξ-plane in order to simulate SED intensity factors by the BEM. The least-squares method incorporating the BE formulation can, then, be used to calculate SED intensity factors directly. Numerical results for a piezoelectric plate with one inclusion and a crack are presented to illustrate the application of the proposed formulation. Copyright © 2000 John Wiley & Sons, Ltd.

A boundary element method for analysis of contaminant transport in porous media I: homogeneous porous media

A boundary element method is developed for the analysis of contaminant migration in porous media. The technique involves, firstly, taking the Laplace transform with respect to time then followed by a co-ordinate transform and a mathematical transform of the well-known advection-dispersion equation. The series of transforms reduce the equation into the modified Helmholz equation and this greatly facilitates the formulation of the boundary integral equation and a system of approximating algebraic boundary element equations. The algebraic equations are solved simultaneously in the transform space before being inverted numerically to obtain the concentration of the contaminant in real time and space. The application of this technique is demonstrated by some illustrative examples.

A distributed algorithm for multi-region problem in BEM

The situation of multi-region problem may often appear when boundary element method (BEM) is applied in practical problems especially in VLSI-CAD. It is difficult to deal with this problem if traditional methods are used. Particularly, when the problem to be solved contains a lot of materials, the advantages of using BEM such as simplicity, convenience and rapidity will be weakened due to the complexity of solving complex boundary element equation. In this paper a distributed algorithm for multi-region problem in BEM is presented. This algorithm has been implemented in a distributed system consisting of 3 workstations to extract VLSI layout parameters. The results show that the calculation time of this distributed algorithm is less than that of the traditional methods. The results also demonstrate that this algorithm can speed up the computation and has the features of parallelism and high efficiency.

A symmetric Galerkin multi-zone boundary element method for cohesive crack growth

In the present paper, a fully symmetric Galerkin multi-zone boundary element method (GMZBEM) is proposed for the cohesive crack growth. For interface problems, the system matrix of each subdomain formulated by GMZBEM is not completely symmetric. Based on regarding the coefficient matrix as the sum of a symmetric part and a non-symmetric one, this problem is solved by adopting a proper assembly. According to this approach, the cohesive crack growth is simulated by marching one of craze-tips step-by-step and by using a special iterative algorithm. During the fracture process, the crack is simulated by means of the cohesive crack model, and the materials in this paper follow linear-elastic and isotropic models. The iterative enforcement of the cohesive crack involves the solution from boundary element equations at each iteration, but these equations in each subdomain need be factored only once for the entire problem unless the interfaces are modified. Numerical results obtained by using the present method are compared with the results from previous methods, which shows that GMZBEM is accurate and effective.

The Conditioning of Boundary Element Equations on Locally Refined Meshes and Preconditioning by Diagonal Scaling

Consider a boundary integral operator on a bounded,d-dimensional surface in $\mbox{\smallBbb R}^{d+1}$. Suppose that the operator is a pseudodifferential operator of order $2m,$ $m\in\mbox{\smallBbb R},$ and that the associated bilinear form is symmetric and positive-definite. (The surface may be open or closed, and m may be positive or negative.) Let B denote the stiffness matrix arising from a Galerkin boundary element method with standard nodal basis functions. If local mesh refinement is used, then the partition may contain elements of very widely differing sizes, and consequently B may be very badly conditioned. In fact, if the elements are nondegenerate and 2|m| < d, then the $\ell_2$ condition number of B satisfies {\rm cond}({\mbox {\bf {\em B}}})\leq C N^{2|m|/d} \left(\frac{h_{\rm max}}{h_{\rm min}}\right)^{d-2m}, where hmax and h min are the sizes of the largest and smallest elements in the partition, and N is the number of degrees of freedom. However, if B is preconditioned using a simple diagonal scaling, then the condition number is reduced to ${\cal O}(N^{2|m|/d})$. That is, diagonal scaling restores the condition number of the linear system to the same order of growth as that for a uniform partition. The growth in the critical case 2|m|=d is worse by only a logarithmic factor.

Adsorption of Molecules on a Charged Polarizable Surface in an Electrolyte

We present a method for the calculation of the binding and rotational energies of neutral (H2S) and charged (HS−) molecules impinging upon a charged and polarizable (Cu 〈100〉) surface in the presence of an electrolyte. A molecular surface is constructed surrounding the H2S and HS−molecules forming boundary elements. A coupled Schrödinger–Poisson–Boltzmann iterative procedure treats the electronic structure of the molecules at the 6-31G**/MP2 level of theory and includes solvation effects through the single and double layers of charge induced by the electronic distribution. The molecule, together with its charged layers, forms a molecular single and double layer, an object which then interacts with a polarizable Gouy–Chapman plane within the electrolyte. The induced charge at the molecular surface resulting from this external electric field is obtained by solving a second set of boundary element equations. The induced polarization of the solid surface created by the impinging molecular ion is treated by a modified method of images. Repulsive interactions between the atoms of the molecule and those of the surface are obtained using a rigid-ion Hartree–Fock method. Binding energies of the molecule to the surface are determined as a function of the real surface charge imposed and also the ionic strength of the solution. It is found that surface charges can completely (180°) reorient these molecules and that the counterions in the solution can completely screen binding effects of even large surface charges. Solid surface polarization can be significant in low dielectric constant solvents and is also reduced by counterions.

A Generalized SOR Method for Dense Linear Systems of Boundary Element Equations

In this paper an iterative scheme of first degree is developed for the purpose of solving linear systems of boundary element equations of the form $Hx = c$ where H is a dense square nonsingular matrix. The iterative scheme considered is $$ (D + (\Omega H)_{sl})x^{(k + 1)} = (D - (\Omega H)_{u})x^{(k)} + \Omega c, $$ \noindent where $(\Omega H)_u$ and $(\Omega H)_{s1}$ are defined as the upper triangular and strictly lower triangular terms of $\Omega H$, respectively. The parameter matrix $\Omega$ is selected to minimize the Frobenius norm $\|D - (\Omega H)_u\|_F$. Mathematical arguments and numerical experiments are presented to show that minimizing $\|D - (\Omega H)_u\|_F$ provides for faster convergence. Numerical tests are performed for systems of boundary element equations generated by three-dimensional potential and elastostatic problems. Computation times are determined and compared against those for Gaussian elimination and Gauss--Seidel iteration.

A Generalized SOR Method for Dense Linear Systems of Boundary Element Equations

In this paper an iterative scheme of first degree is developed for the purpose of solving linear systems of boundary element equations of the form $Hx = c$ where $H$ is a dense square nonsingular m...

Electromagnetic force analysis of moving conducting slab by using 3D IBIEM

This paper presents a new scheme to calculate the magnetic forces between an arbitrary current distribution and a conducting slab in steady motion. The IBIEM is adopted for the electromagnetic force analysis which includes a moving conducting slab. The 3D moving effect is reflected in the Green's function of diffusion type. This IBIEM excludes spurious oscillations that occur in finite element solutions. Especially, in the case of the conducting slab, the coefficient matrix of the boundary element equations is sparse. Therefore, computer memory and computing time can be greatly saved. Numerical results based on the IBIEM agreed well with their analytic solutions.

The preconditioned GMRES method for systems of coupled FEM-BEM equations.

We analyze the generalized minimal residual method (GMRES) as a solver for coupled finite element and boundary element equations. To accelerate the convergence of GMRES we apply a hierarchical basis block preconditioner for piecewise linear finite elements and piecewise constant boundary elements. It is shown that the number of iterations which is necessary to reach a given accuracy grows only poly-logarithmically with the number of unknowns.

Multigrid Schur Complement Preconditioning in BE‐DD Methods

AbstractWe consider Domain Decomposition (DD) Boundary Element (BE) equations and analyze a multigrid preconditioner, derived from the hypersingular operator, for the Schur complement. The robustness of the preconditioner with respect to the number of subdomains is being investigated.

On a Class of Preconditioning Methods for Dense Linear Systems from Boundary Elements

In this paper we discuss several sparse-matrix-based preconditioners suitable for preconditioning dense linear systems of boundary element equations. All preconditioners involve only O(n) nonzeros. We provide a framework for constructing operator-splitting-based preconditioners and use it to analyze a class of sparse preconditioners. For singular integral equations, a more efficient preconditioner is proposed that has a band-2 structure.

ON THE USE OF FFT ALGORITHM FOR THE CIRCUMFERENTIAL CO-ORDINATE IN BOUNDARY ELEMENT FORMULATION OF AXISYMMETRIC PROBLEMS

A new numerical method is proposed for the boundary element analysis of axisymmetric bodies. The method is based on complex Fourier series expansion of boundary quantities in circumferential direction, which reduces the boundary element equation to an integral equation in (r–z) plane involving the Fourier coefficients of boundary quantities, where r and z are the co-ordinates of the (r, θ, z) cylindrical co-ordinate system. The kernels appearing in these integral equations can be computed effectively by discrete Fourier transform formulas together with the fast Fourier transform (FFT) algorithm, and the integral equations in (r–z) plane can be solved by Gaussian quadrature, which establishes the Fourier coefficients associated with boundary quantities. The Fourier transform solution can then be inverted into (r, θ, z) space by using again discrete Fourier transform formulas together with FFT algorithm. In the study, first we present the formulation of the proposed method which is outlined above. Then, the method is assessed by using three sample problems. A good agreement is observed in the comparisons of the predictions of the method with those available in the literature. It is further found that the proposed method provides considerable saving in computer time compared to existing methods of literature. © 1997 by John Wiley & Sons, Ltd.

A preconditioning strategy for the solution of linear boundary element systems using the GMRES method

In this paper a novel preconditioning strategy is presented that is designed to improve the convergence rates of the Generalized Minimal Residual (GMRES) method when applied to dense linear systems of boundary element equations of the form Hx = c. The GMRES method is applied to the preconditioned system (D+ L) −1ΩHx = (D+ L) −1Ωc , where D = diag( H), L is the strictly lower triangular part of ΩH and Ω is a sparsely populated upper triangular matrix. The coefficients in Ω are determined via the minimization of the square of the Frobenius norm map; U+ D−Dmap; F , where U is the strictly upper triangular part of ΩH and D= diag(ΩH) . Several proofs are given to demonstrate that minimizing map; U+ D−Dmap; F 2 provides for improved conditioning and consequently faster convergence rates. Numerical experiments are performed on systems of boundary element equations generated by three-dimensional potential and elastostatic problems. Computation times are determined and compared against those for Jacobi preconditioned GMRES, preconditioned Gauss-Seidel and Gaussian elimination. Moreover, condition numbers are noted and up to 100-fold reductions are observed for the systems tested.

Three-dimensional eddy current calculation by an adaptive three-component boundary element algorithm

An adaptive technique of calculating three-dimensional eddy currents by using the three-component boundary element algorithm is presented. The three-component algorithm is a minimum-order boundary element method in the calculation of three-dimensional eddy currents, which employs only three scalar unknowns on each mesh node, so that about 40% computer memory and 50% computational time can be saved relative to the conventional four-component algorithm that usually employs a scalar and a three-component vector as unknowns. The error estimate of the three-component algorithm and the implementation of the adaptive technique are described. The error estimate is based on the residue of the boundary element equation at the non-collocation mesh nodes. The adaptive technique is able to save computer memory and increase the accuracy of the solution greatly.

H2S and HS-Adsorption on a Charged Cu Surface in an Electrolyte: Effect of Ionic Strength

ABSTRACTWe present a method for the calculation of the binding and rotational energies of neutral (H2S) and charged (HS-) molecules impinging upon a charged (Cu &lt;100&gt;) surface in the presence of an electrolyte. A molecular surface is constructed surrounding the H2S and HS-molecules forming boundary elements. A coupled Schrödinger-Poisson-Boltzmann iterative procedure treats the electronic structure of the molecules at the 6–31G**/MP2 level of theory and includes solvation effects through the single and double layers of charge induced by the electronic distribution. The molecule, together with its charged layers, forms a Molecular Single and Double Layer (MSDL), an object which then interacts with a Gouy-Chapman plane within the electrolyte. The additional induced charge at the molecular surface resulting from this electric field is obtained by solving a second set of boundary element equations. Repulsive interactions between the atoms of the molecule and those of the surface are obtained using a rigid-ion Hartree-Fock method. Binding energies of the molecule to the surface are determined as a function of the real surface charge imposed and also the ionic strength of the solution. It is found that surface charges can completely (180°) reorient these molecules and that the counterions in the solution can completely screen binding effects of even large surface charges.Work supported by the United States Department of Energy under contract #DE-AC04–94AL85000.

Efficient parallel solvers for boundary element equations using data decomposition

The boundary element method (BEM) may be computationally expensive if complex problems are to be solved, thus there is the need for implementing it on fast computer architectures, especially parallel computers. One main aspect is the efficient parallel implementation for solvers for the linear system of equations which is dense and non-symmetric if a standard collocation method is used. In this paper, we discuss the numerical and parallel efficiency of various iterative solvers. Their parallel implementation on a distributed memory system (MIMD computer) is described. The parallelization is performed by data partitioning which leads to a very high efficiency. As a model problem, we assume linear elasticity but the parallelization strategy is well suited for other types of problems. The efficiency of our implementation is shown by test examples.

Boundary element analysis by Krylov subspace method on MIMD parallel multiprocessor system

The parallel boundary element solution on a distributed-memory multiprocessor computer is presented. The proposed computational strategy does not require the formulation of the system of boundary element equations explicitly. According to the number of processors of the MIMD (multiple instruction stream-multiple data stream) parallel computer, the total boundary element nodes are divided into the same number of groups and distributed to each processor which stores relevant information and generates corresponding boundary element equations. The Krylov subspace method is employed to solve in parallel the system of boundary element equations that are distributed in the processors. The results of boundary element analysis, i.e. strain and stress of internal points are also given by each processor in parallel.

Application of the boundary element method to rubber-like elasticity

A boundary element method solution for the problem of finite deformation of rubber-like materials is presented. The solution is obtained by solving the nonlinear boundary element equations using an incremental-iterative procedure. The procedure is then implemented in a computer code that is tested through three numerical examples. In the first example, the plane strain problem of a thick-walled cylindrical pressure vessel is considered. In the second and third examples, which represent the plane stress case, a square rubber sheet with and without a central hole is subjected to finite plane deformation.