Techniques to accelerate BEM computation to provide virtual reality update of stress solutions

We address the problem of rapid re-analysis of small problems in elasticity, in which the aim is to enable updated stress contours to be displayed in real time as a design geometry is dynamically modified. For the small to medium sized problems that are the focus of the work, it cannot be assumed that the solution phase dominates, and so the evaluation of boundary integrals is considered as well as the equation solution. Two strategies are employed for acceleration of boundary element integrals: the use of look-up tables (LUTs) containing pre-computed integrals and the use of approximate analytical expressions derived from surface fits. These may be used in the matrix assembly and internal point calculations. This paper concentrates on the extension of the LUT technique presented earlier [Trevelyan J, Wang P. Interactive re-analysis in mechanical design evolution. Part 2: rapid evaluation of boundary element integrals. Comput Struct 2001;79:939–51 [1]] to consider circular arc elements as well as extending the method to the derivative kernels to allow the use of LUTs with the stress boundary integral equation. Further details are provided on suitable LUT refinement and the approach is benchmarked against conventional Gauss–Legendre integration. The surface fit approach is presented as an alternative to LUTs that does not incur the considerable memory cost associated with LUTs. The equation solution is cast in a re-solution framework, in which we use a GMRES algorithm. Convergence is greatly accelerated by using an approximate complete LU preconditioner updated periodically using multi-threading. The resulting system achieves the aim of providing real-time update of contours for small to medium sized problems on a PC. This development is expected to allow a qualitative change in the way engineers might use computer aided engineering tools, in which design ideas may rapidly be assessed immediately as a change is made.