Symmetric Complex Boundary Element Scheme for 2-D Stokes Mixed Boundary Value Problem

For 2-D Stokes mixed boundary value problems we construct a boundary<br />integral equation which couples a conventional boundary integral equation<br />for the velocity with a hypersingular boundary integral equation for the<br />traction. Expressing terms in the equation by complex variables, we obtain a<br />complex boundary integral equation and realize symmetrization of boundary<br />element scheme by Galerkin method. Applying a boundary limit method, we<br />obtain exact calculation formulae for calculation of hypersingular boundary<br />integrals. It is shown that all divergent terms in hypersingular integrals<br />cancel each other out.

Solving the complete-electrode direct model of ERT using the boundary element method and the method of fundamental solutions

This paper discusses solving the forward problem for electrical resistance tomography (ERT). The mathematical model is governed by Laplace's equation with the most general boundary conditions forming the so-called complete electrode model (CEM). We examine this problem in simply-connected and multiply - connected domains (rigid inclusion, cavity and composite bi-material). This direct problem is solved numerically using the boundary element method (BEM) and the method of fundamental solutions (MFS). The resulting BEM and MFS solutions are compared in terms of accuracy, convergence and stability. Anticipating the findings, we report that the BEM provides a convergent and stable solution, whilst the MFS places some restrictions on the number and location of the source points.

A Study of Speed of the Boundary Element Method as applied to the Realtime Computational Simulation of Biological Organs

In this work, possibility of simulating biological organs in realtime using the Boundary Element Method (BEM) is investigated. Biological organs are assumed to follow linear elastostatic material behavior, and constant boundary element is the element type used. First, a Graphics Processing Unit (GPU) is used to speed up the BEM computations to achieve the realtime performance. Next, instead of the GPU, a computer cluster is used. Results indicate that BEM is fast enough to provide for realtime graphics if biological organs are assumed to follow linear elastostatic material behavior. Although the present work does not conduct any simulation using nonlinear material models, results from using the linear elastostatic material model imply that it would be difficult to obtain realtime performance if highly nonlinear material models that properly characterize biological organs are used. Although the use of BEM for the simulation of biological organs is not new, the results presented in the present study are not found elsewhere in the literature.

Determination of a space-dependent force function in the one-dimensional wave equation

<p class="p1">The determination of an unknown spacewice dependent force function acting on a vibrating string from over-specied Cauchy boundary data is investigated numerically using the boundary element method (BEM) combined with a regularized method of separating variables. This linear inverse problem is ill-posed since small errors in the input data cause large errors in the output force solution. Consequently, when the input data is contaminated with noise we use the Tikhonov regularization method in order to obtain a stable solution. The choice of the regularization parameter is based on the L-curve method. Numerical results show that the solution is accurate for exact data and stable for noisy data.</p>

A DRBEM For Time-Dependent Infiltration from Periodic Irrigation Channels in a Homogeneous Soil

<p><span style="font-size: 10.000000pt; font-family: 'CMR10';">In this paper, a problem involving time-dependent water flow in a homogeneous soil is considered. The problem involves water infiltration from periodic identical trapezoidal channels. A governing equation of the problem is the Richard’s equation, which can be studied more conveniently by transforming the equation to a Helmholtz equation using the Kirchhoff transformation with dimensionless variables and a Laplace’s transform. A dual reciprocity boundary element method (DRBEM) is employed to solve the Helmholtz equation numerically. Results obtained are found to be physically reasonable. </span></p>

Suction Potential and Water Absorption from Periodic Channels in Different Types of Homogeneous Soils

<p><span style="font-size: 10.000000pt; font-family: 'CMR10';">In this paper, problems involving infiltration from periodic identical trapezoidal channels into homogeneous soils with root water uptake are considered. The governing equation of infiltration through soil is transformed to a modified Helmholtz equation using Kirchoff transformation with dimensionless variables. A DRBEM with a predictor-corrector scheme is employed to obtain numerical solutions to the modified Helmholtz equation. The proposed method is used to solve infiltration problems in three different types of soil. Results are shown to be physically meaningful. </span></p>

Recent developments in the evaluation of the 3D fundamental solution and its derivatives for transversely isotropic elastic materials

<p class="p1">Explicit closed-form real-variable expressions of a fundamental solution and its derivatives for three-dimensional problems in transversely linear elastic isotropic solids are presented. The expressions of the fundamental solution in displacements <span class="s1">U</span><span class="s2">ik </span>and its derivatives, originated by a unit point force, are valid for any combination of material properties and for any orientation of the radius vector between the source and field points. An ex- pression of <span class="s1">U</span><span class="s2">ik </span>in terms of the Stroh eigenvalues on the oblique plane normal to the radius vector is used as starting point. Working from this expression of <span class="s1">U</span><span class="s2">ik</span>, a new approach (based on the application of the rotational symmetry of the material) for deducing the first and second order derivative kernels, <span class="s1">U</span><span class="s2">ik,j </span>and <span class="s1">U</span><span class="s2">ik,jl </span>respectively, has been developed. The expressions of the fundamental solution and its derivatives do not suffer from the difficulties of some previous expressions, obtained by other authors in different ways, with complex valued functions appearing for some combinations of material parameters and/or with division by zero for the radius vector at the rotational symmetry axis. The expressions of <span class="s1">U</span><span class="s2">ik</span>, <span class="s1">U</span><span class="s2">ik,j </span>and <span class="s1">U</span><span class="s2">ik,jl </span>are presented in a form suitable for an efficient computational implementation in BEM codes.</p>

Recovering boundary data in planar heat conduction using a boundary integral equation method

We consider a Cauchy problem for the heat equation, where the temperature field is to be reconstructed from the temperature and heat flux given on a part of the boundary of the solution domain. We employ a Landweber type method proposed in ~\cite{Bast}, where a sequence of mixed well-posed problems are solved at each iteration step to obtain a stable approximation to the original Cauchy problem. We develop an efficient boundary integral equation method for the numerical solution of these mixed problems, based on the method of Rothe. Numerical experiments are presented both with exact and noisy data, showing the efficiency and stability of the proposed procedure and approximations.

Boundary integral equations for plane orthotropic bodies and exterior regions

Assuming linear displacements and constant strains and stresses at infinity, we re-formulate the equations of the direct boundary element method for plane problems of elasticity. We consider a body made of orthotropic material. The reformulated equations make it possible to attack plane problems on exterior regions without replacing the region by a bounded one.

A Technique to Remove Second Order Singularity Application in the Boundary Element Method for Elastoplasticity Plane Stress Analysis

This paper presents a technique to establish the strain incremental formulation in the boundary element method applied to elastoplasticity problem. In this technique, the application of second order singularity problem is avoided, and only first order singularity problem is sufficient. The proposed technique is applied to analyse a timber beam structure at the plastic stage. The solution is compared with existing strain formulation method proposed by established publication. The result gives an improved solution compared with the existing method. The proposed technique is a simplified formulation where there is no second order singularity involved in the formulation.