A Spectral Boundary Element Method

Abstract

The Boundary Element Method (BEM) is not local and generates a full and nonsymmetric matrix, therefore the matrix solution time could easily grow beyond acceptable limits. Furthermore, in many cases a decoupling between geometry description and solution approximationwould be advantageous and desirable. This work aims at offering the BEM a new approximation capability, which could reduce the number of degrees of freedom (dofs) needed, and at providing the desired decoupling. The method proposed here takes advantage of the less restrictive requirements in BEM compared with FEM for the field approximation. The base of the new approximation becomes what has been called “Spectral Element” (SE): a super-entity gathering many elements, which are used only for geometry description and numerical integration. Spectral shape functions are generated using smooth positive definite compact supported Radial Basis Functions (RBF). The method was applied to linear elasticity in 3D and numerical tests were performed for different typical cases in order to highlight advantages, disadvantages and limits of the proposed methodology. 1 Approximation field in the BEM 1.1 Some general properties of the “finite element approximation” Both BEM and FEM, in order to reduce the infinite dimensional space of the solution to a finite one, use the so called finite transform. If x represents the vector of space coordinates and u the approximation field, a set of “generalised degrees of freedom” in the form of a vector α must be chosen so that: u(x) = A(x)α (1) Boundary Elements XXVII 165 © 2005 WIT Press WIT Transactions on Modelling and Simulation, Vol 39, www.witpress.com, ISSN 1743-355X (on-line) In equation (1) A(x) represents the coordinate function matrix, which contains functions fixed in space. The finite elements appear when the continuum under analysis is ideally cut into “small” pieces, the elements of the mesh. In order to ensure that the values of the field in few specific points (nodes) belonging to an element govern the whole behaviour of u inside it, equation (1) must be imposed simultaneously in each of the n nodes of an element:   u1(x) u2(x) .. un(x)   = U = Cα =   A(x1) A(x2) .. A(xn)  α (2) Now if C is square, i.e. if the number of nodes n is equal to the number of α dofs, and is not singular, the following equations hold: u(x) = A(x)α = A(x)C−1U = Φ(x)U where Φ = A(C)−1 (3) Generally the Φ matrix is called shape matrix. It interpolates the dof of the element, represented by the nodal values, in between the nodes. In the FEM four general properties are needed for the functions in Φ, called shape functions, each of which belongs to a different node: 1. Completeness condition: in order to be able to represent a constant field, (first term in a Taylor series development of any field) the sum of all functions must be one in every point of the element: