AN INDIRECT BOUNDARY ELEMENT METHOD FOR PLATE BENDING ANALYSIS

An indirect Boundary Element Method is employed for the static analysis of homogeneous isotropic and linear elastic Kirchhoff plates of an arbitrary geometry. The objectives of this paper consists of a construction and a study of the resulting boundary integral equations as well as a development of stable powerful algorithms for their numerical approximation. These equations involve integrals with high-order kernel singularities. The treatment of singular and hypersingular integrals and a construction of solutions in the neighborhood of the irregular points on the boundary are discussed. Numerical examples illustrate the procedure and demonstrate its advantages. © 1997 by John Wiley & Sons, Ltd.     

On the Cauchy principal value of the surface integral in the boundary integral equation of 3D elasticity

A simple demonstration of the existence of the Cauchy principal value (CPV) of the strongly singular surface integral in the Somigliana Identity at a non-smooth boundary point is presented. First a regularization of the strongly singular integral by analytical integration of the singular term in the radial direction in pre-image planes of smooth surface patches is carried out. Then it is shown that the sum of the angular integrals of the characteristic of the tractions of the Kelvin fundamental solution is zero, a formula for the transformation of angles between the tangent plane of a suface patch and the pre-image plane at smooth mapping of the surface patch being derived for this purpose.     

A new method for the determination of the flexural rigidity of an orthotropic plate

In this paper a new method for the determination of flexural rigidities in orthotropic plate bending problems is presented. Boundary integral equations are established for the curvatures and the deflections inside the domain. By a simple discretization of the boundary and the inside plate, the elimination of curvatures is possible. If the fundamental solution of isotropic plates is chosen, then a linear system of n equations with three unknowns is obtained. These equations are provided by the knowledge of the deflections inside the plates, and the unknowns are the flexural rigidities. By using the least square method, the computation of these rigidities becomes easy.     

Electrostatic BEM for MEMS with thin conducting plates and shells

Micro-electro-mechanical systems (MEMS) sometimes use beam or plate shaped conductors that can be very thin-with h/L≈O(10⁻²−10⁻³) (in terms of the thickness h and length L of the side of a square pate). Conventional Boundary Element Method (BEM) analysis of the electric field in a region exterior to such thin conductors can become difficult to carry out accurately and efficiently—especially since MEMS analysis requires computation of charge densities (and then surface tractions) separately on the top and bottom surfaces of such plates. A new boundary integral equation (BIE) is derived in this work that, when used together with the standard BIE with weakly singular kernels, results in a powerful technique for the BEM analysis of such problems. This new approach, in fact, works best for very thin plates. This thin plate BEM is derived and discussed in this work. Numerical results, from several BEM based methods, are presented and compared for the model problem of a parallel plate capacitor.     

Galerkin formulation and singularity subtraction for spectral solutions of boundary integral equations

A new spectral Galerkin formulation is presented for the solution of boundary integral equations. The formulation is carried out with an exact singularity subtraction procedure based on analytical integrations, which provides a fast and precise way to evaluate the coefficient matrices. The new Galerkin formulation is based on the exact geometry of the problem boundaries and leads to a non-element method that is completely free of mesh generation. The numerical behaviour of the method is very similar to the collocation method; for Dirichlet problems, however, it leads to a symmetric coefficient matrix and therefore requires half the solution time of the collocation method. © 1998 John Wiley & Sons, Ltd.     

The boundary element solution of the Laplace and biharmonic equations subjected to noisy boundary data

This study investigates the numerical solution of the Laplace and biharmonic equations subjected to noisy boundary data. Since both equations are linear, they are numerically discretized using the Boundary Element Method (BEM), which does not use any solution domain discretization, to reduce the problem to solving a system of linear algebraic equations for the unspecified boundary values. It is shown that when noisy, lower-order derivatives are prescribed on the boundary, then a direct approach, e.g. Gaussian elimination, for solving the resulting discretized system of linear equations produces an unstable, i.e. unbounded and highly oscillatory, numerical solution for the unspecified higher-order boundary derivatives data. In order to overcome this difficulty, and produce a stable solution of the resulting system of linear equations, the singular value decomposition approach (SVD), truncated at an optimal level given by the L-curve method, is employed. © 1998 John Wiley & Sons, Ltd.     

HYPERSINGULAR RESIDUALS—A NEW APPROACH FOR ERROR ESTIMATION IN THE BOUNDARY ELEMENT METHOD

This paper presents a new approach for a posteriori ‘pointwise’ error estimation in the boundary element method. The estimator relies upon evaluation of the residual of hypersingular integral equations, and is therefore intrinsic to the boundary integral equation approach. A methodology is developed for approximating the error on the boundary as well as in the interior of the domain. Extensive computational experiments have been performed for the two-dimensional Laplace equation and the numerical results indicate that the error estimates successfully track the form of the exact error curve. Moreover, a reasonable estimate of the magnitude of the actual error is also predicted.