Rapid re-analysis in BEM elastostatic calculations

The real time solution and updating of stress contours for small elasticity problems allows the rapid and interactive evaluation of multiple design proposals and the convergence on an optimum solution in a significantly reduced time. The increased level of interactivity allows the user to see the potentially complex interactions between multiple geometric features. A number of strategies are proposed and compared with current techniques. Boundary element integrals are accelerated by the use of Look-Up Tables (LUT's) containing pre-computed integrals. These can be employed in the matrix assembly and the internal point calculations for both flat and arc elements. The effect of problem size on propagated error has been investigated and the effect of interpolation has been considered. An error analysis has been performed and a suitable scheme for LUT refinement proposed. The approach has been benchmarked against regular Gauss-Legendre quadrature. The large storage requirement of the LUT's has been considered and the use of a least squares surface fit to the LUT's has been investigated and the computational cost compared with Gauss-Legendre quadrature. Iterative re-solution of the problem is accelerated by the use of an approximate LU preconditioner that is updated as required by a secondary thread of execution. The proposed preconditioner has been benchmarked against other preconditioners and reduces solve times to typically less than 30% of the direct solve time. The proposed techniques achieve the aim of providing a real-time update of contours for small problems on a PC.