Efficient analytical integration of symmetric Galerkin boundary integrals over curved elements; elasticity formulation

Abstract

Abstract Substantial improvements are reported in the computational efficiency of Galerkin boundary element analysis (BEA) 1employing curved continuous boundary elements for elasticity problems. A direct analytical treatment of the singular and hypersingular double integrations, employing a limit to the boundary definition is adapted to gain computational economy. Symbolic manipulation has been employed to derive the analytical expressions for the singular contributions to these double integrals. The analytical regularization procedure separates the potentially singular Galerkin integrands into an essentially singular but simple part, plus a regular remainder that can be integrated numerically. The exact finite contributions from the simplified singular terms are then computed analytically. It is shown that the key to containing the explosive growth in the length of the formulae associated with such a hybrid analytical/numerical integration scheme is the strategic timing of when to take the limit to the boundary. This regularization also isolates the contribution from the curvature of the boundary element, thus facilitating enhanced computational efficiency in problems with many straight elements. Although the evaluation of the integrals of the Kelvin fundamental solution essentially follows the procedures described earlier for the scalar Laplace equation, the analytical treatment, is now more involved, in terms of the complexity and number of terms present in each of these second rank tensor kernel functions. The complete analytical formulation is explicitly provided in a series of tables. Example problems indicate that, with these techniques, Galerkin symmetric BEA can be more efficient than collocation for moderate or large scale problems.