Hossein Hosseinzadeh

Solution of Poisson’s equation by analytical boundary element integration

The solution of Poisson’s equation is essential for many branches of science and engineering such as fluid-mechanics, geosciences, and electrostatics. Solution of two-dimensional Poisson’s equations is carried out by BEM based on Galerkin Vector Method in which the integrals that appear in the boundary element method are expressed by analytical integration. In this paper, the Galerkin vector method is developed for more general case than presented in the previous works. The integrals are computed for constant and linear elements in BEM. By employing analytical integration in BEM computation, the numerical schemes and coordinate transformations can be avoided. The presented method can also be used for the multiple domain case. The results of the analytical integration are employed in BEM code and the obtained analytical expression will be applied to several examples where the exact solution exists. The produced results are in good agreement with the exact solution.

A simple and accurate scheme based on complex space C to calculate boundary integrals of 2D boundary elements method

Abstract In this work a semi-analytical algorithm is presented to calculate the boundary integrals of higher order which appear in boundary elements method (BEM). In fact treating singularity and near singularity of the boundary integrals using complex space C is the main aim of this paper. The integrals are computed for linear, quadratic, cubic and other higher order elements when the geometry of the boundary elements is curved. The main advantages of the new algorithm are its applicability, simplicity and high accuracy which enable the conventional higher order BEM to solve Poisson’s problems, accurately. The potentials at the interior points very close to boundary can be evaluated by the scheme developed in this report. Some test problems are given and numerical simulations are presented. Numerical results demonstrate that the new algorithm proposed in the current paper can effectively handle singular and near singular boundary integrals of BEM, specially for solving partial differential equations which arise in thin body problems.

Obtaining the upper bound of discretization error and critical boundary integrals of circular arc boundary element method

Abstract The boundary element method (BEM) is a popular method of solving linear partial differential equations, and it can be applied in many areas of engineering and science including fluid mechanics, acoustics, electromagnetics. The circular arc element (CAE) method is a scheme to discretize the boundary of problems arising in the BEM. In Dehghan and Hosseinzadeh (2011) [10] the order of the convergence of CAE discretization was obtained for the 2D Laplace equation. The current work extends the formulation developed in Dehghan and Hosseinzadeh (2011) [10] to convert CAE to a robust discretization method. In the present paper the upper bound of the CAE’s discretization error is determined theoretically for the 2D Laplace equation. Also we present a new method based on the complex space C to obtain CAE’s boundary integrals without facing singularity and near singularity. Since there is no efficient approach to treat the near singular integrals of CAE in the BEM literature, the new scheme presented in this paper enhances the CAE discretization significantly. Several test problems are given and the numerical simulations are obtained which confirm the theoretical results.

The use of continuous boundary elements in the boundary elements method for domains with non-smooth boundaries via finite difference approach

A numerical method is presented in this article to deal with the drawback of boundary elements method (BEM) at corner points. The use of continuous elements instead of the discontinuous ones has been recommended in the BEM literature widely because of the simplicity and accuracy. However the continuous elements lead to certain difficulties for problems where their domains contain corners. In this paper the finite difference method (FDM) has been applied to obtain some constraints for boundary points near the corners to deal with this drawback. Because of its simplicity and capability, the new scheme is applicable on BEM problems for all geometries, all governing equations and general boundary conditions, easily. Since the Dirichlet boundary condition is more critical than the other ones, we will focus on it in the numerical implementation. The numerical results show that the new treatment improves the accuracy of BEM significantly.

On uniqueness of numerical solution of boundary integral equations with 3-times monotone radial kernels

The uniqueness of solution of boundary integral equations (BIEs) is studied here when geometry of boundary and unknown functions are assumed piecewise constant. In fact we will show BIEs with 3-times monotone radial kernels have unique piecewise constant solution. In this paper nonnegative radial function F ź 3 is introduced which has important contribution in proving the uniqueness. It can be found from the paper if ź 3 is sufficiently small then eigenvalues of the boundary integral operator are bigger than F ź 3 / 2 . Note that there is a smart relation between ź 3 and boundary discretization which is reported in the paper, clearly. In this article an appropriate constant c 0 is found which ensures uniqueness of solution of BIE with logarithmic kernel ln ( c 0 r ) as fundamental solution of Laplace equation. As a result, an upper bound for condition number of constant Galerkin BEMs system matrix is obtained when the size of boundary cells decreases. The upper bound found depends on three important issues: geometry of boundary, size of boundary cells and the kernel function. Also non-singular BIEs are proposed which can be used in boundary elements method (BEM) instead of singular ones to solve partial differential equations (PDEs). Then singular boundary integrals are vanished from BEM when the non-singular BIEs are used. Finally some numerical examples are presented which confirm the analytical results.