A System of Boundary Integral Equations with Hypersingular Kernels for the Free Edge Plate

Abstract

We propose a system of boundary integral equations for the free edge plate. This BIE system is particularly well suited for an infinite plate with holes contained in a bounded subset. First, we prove existence and uniqueness in a weighted Sobolev space, of the solution of the problem of the infinite plate with bounded free edges. Then, we derive a system of boundary integral equations. We write this system variationally and we prove that it has a solution unique up to the traces of polynomials of degree one. Left to itself, the corresponding bilinear form would give rise to hypersingular kernels. Therefore, using a method developped by J.C. Nedelec for potential or 3D-elasticity problems, we transform it to obtain an expression with only weakly singular kernels. This expression is well suited for numerical computations, allowing in particular, the use of C 1 boundary elements.