Grid redistribution based on measurable error indicators for the direct boundary element method

The Laplace problem subject to the Dirichlet or Neumann boundary condition in the direct and indirect boundary element methods (BEM) sometimes both may result in a singular or ill-conditioned system (some special situations) for the interior problem. In this paper, the direct and indirect BEMs are revisited to examine the uniqueness of the solution by introducing the Fichera’s idea and the self-regularized technique. In order to construct the complete range of the integral operator in the BEM lacking a constant term in the case of a degenerate scale, the Fichera’s method is provided by adding the constraint and a slack variable to circumvent the problem of degenerate scale. We also revisit the Fredholm alternative theorem by using the singular value decomposition (SVD) in the discrete system and explain why the direct BEM and the indirect BEM are not indeed equivalent in the solution space. According to the relation between the SVD structure and Fichera’s technique, a self-regularized method is proposed in the matrix level to deal with non-unique solutions of the Neumann and Dirichlet problems which contain rigid body mode and degenerate scale, respectively, at the same time. The singularity and proportional influence matrices of 3 by 3 are studied by using the property of the symmetric circulant matrix. Finally, several examples are demonstrated to illustrate the validity and the effectiveness of the self-regularized method.

A direct (potential based) boundary element method (BEM) developed in our company (Romanian Research and Design Institute for Shipbuilding) is briefly presented. It may be applied for the calculation of the potential flow paste the ship hull, also around the lifting bodies like: hydrofoil, nozzle, rudder, propeller. The lifting bodies hydrodynamic calculation has some modelling particularities, e.g.: the wake shape determination, and numerical difficulties, e.g.: the BEM integrals calculation. Therefore the BEM application to such kind of bodies substantially differs from standard BEM application.

Time-domain boundary element method in viscoelasticity with application to a spherical cavity

The objective of this study is to examine the applicability of the boundary element method to analysing a plate on elastic foundation. The fundamental solution of the problem is presented as a Fourier‐Bessel integral. For the computation of the values of the fundamental solution an algorithm was developed in which the Fourier‐Bessel integral was decomposed into an alternative convergent sequence. Equations based on the direct and indirect boundary element method were derived for a plate situated on a one‐ or two‐parametric elastic foundation. According to the theory presented, computer programs based on the direct and indirect boundary element method were developed. These programs can be used for examining the behaviour of a smooth‐boundary plate on a one‐ or two‐parametric elastic foundation. The computer programs were tested by several examples. The results obtained by using a small number of boundary elements compared favourably to the results obtained by a fine finite element mesh. The study shows that the boundary element method is applicable to the analysis of a plate on elastic foundation.

The boundary element method has evolved rapidly within the past decade and is now recognized as a reliable and efficient alternative to finite element and finite difference procedures, especially for problems encountered in Potential Theory and Elasto-statics. The technique consists in transforming the partial differential equations which govern the behavior in the domain over to a set of integral equations relating quantities associated with the boundary, and applying numerical procedures to generate approximate solutions for the boundary variables. This paper describes a general procedure, based on the weighted residual approach, for transforming the differential equations to integral expressions that form the basis for the Direct Boundary Element Method, and shows that boundary element and finite element methods can be interpreted as variants of the fundamental weak formulation for the problem.

Vector and multithread computation of silencer performance prediction on a dual-processor PC workstation

The possibilities of the algorithm for applying the boundary element method to solving boundary value problems are discussed on the example of the two-dimensional Poisson differential equation. The algorithm does not change significantly when the type of boundary conditions changes: the Dirichlet problem, the Neumann problem, or a mixed boundary value problem. The idea of the algorithm is taken from the work of John T. Katsikadelis [1]. The algorithm is described in detail in the next sequence of actions. 1) The boundary- value problem for a two-dimensional finite domain is formulated. The desired function in the domain, its values, and its normal derivative on the boundary contour are connected by means of the second Green formula. 2) We pass from the boundary value problem for the Poisson equation to the boundary value problem for the Laplace equation. This simplifies the process of constructing an integral equation. We obtain the integral equation on the boundary contour using the boundary conditions. 3) In the integral equation, we divide the boundary contour into a finite number of boundary elements. The desired function and its normal derivative are considered constant values on each boundary element. We compose a system of linear algebraic equations considering these values. 4) We modify the system of linear algebraic equations taking into account the boundary conditions. After that, we solve it using the Gauss method. The computer program has been developed according to the developed algorithm. We used it in the learning process. The software implementation of the algorithm takes into account the capabilities of modern computer technology and modern needs of the educational process. The work of the program is shown in the test case. Further modification of the described algorithm is possible

A numerical procedure for the calculation of the magnetic field of coils with open iron cores is presented. Considering that the field to be calculated is an open boundary problem with axisymmetric geometry, the boundary element method with double scalar magnetic potentials is used. The axisymmetric fundamental solution of 3-D Laplace's equation and its normal derivative are introduced. The method has advantages for the solution of axisymmetric unbounded magnetostatic problems. >

In this paper, we describe Laplace domain boundary element method (BEM) for transient dynamic problems of three-dimensional finite homogeneous anisotropic linearly elastic solids. The employed boundary integral equations for displacements are regularized using the static traction fundamental solution. Modified integral expressions for the dynamic parts of anisotropic fundamental solutions and their first derivatives are obtained. Boundary elements with mixed approximation of geometry and field variables with the standard nodal collocation procedure are used for spatial discretization. In order to obtain time-domain solutions, the classic Durbin’s method is applied for numerical inversion of Laplace transform. Problem of alleviating Gibbs oscillations is addressed. Dynamic boundary element analysis of the model problem involving trigonal material is performed to test presented formulation. Obtained results are compared with finite element solutions.

Time domain elastodynamic boundary element methods are prone to numerical instabilities. Under suitable conditions, these instabilities can swamp the transient response of a system. We show evidence of these instabilities in both the direct and indirect boundary element methods. We summarize the literature on the evidence and causes of these instabilities, and refer to improved algorithms and alternative formulations which are less prone to numerical instabilities. Finally, we make suggestions as to where research should concentrate so that these methods can reach their full potential.

The non-linear buckling of composite laminates, triggered by geometric imperfections, is here analysed adopting a boundary element methodology. The non-linear theory for thin anisotropic plates couples in-plane forces causing buckling with the consequent bending deformation. The adopted formulation for in-plane forces in terms of the stress function is mathematically identical to that for the bending problem, thus boundary integral equations and fundamental solutions of the same form are used. Differential equations governing increments of the stress function and the deflection are obtained; the resulting integral equations include irreducible domain integrals depending on powers or products of the second derivatives of the stress function and the deflection. The latter are calculated through complementary integral equations obtained from the original ones by differentiating the field variables in the domain. The solution process accounts for the domain integrals through an iterative scheme; thus there is no need for modelling any field variables in the plate domain although domain meshing is necessary for performing numerical integrations. The boundary is meshed into quadratic discontinuous elements while the domain is divided into triangular cells with linear discontinuous variation of the relevant field variables for integration purposes. The analysis is implemented through a suit of C codes and applied to a rectangular symmetrically laminated plate. Its predictions are compared with published results obtained by other methods. The accuracy and limitations of the formulation are discussed and alternative approaches for expanding its scope pointed out.

3D mixed boundary elements for elastostatic deformation field analysis

Revisit of a degenerate scale: A semi-circular disc

Chapter Three - The BEM for Potential Problems in Two Dimensions

Boundary element analysis of mode I and mixed mode (I and II) crack problems of 2-D gradient elasticity