Techniques to accelerate boundary element contributions in elasticity

Abstract

The problem of rapid re-analysis of small problems in elasticity is investigated. The aim is to enable updated stress contours to be displayed in real-time as a design geometry is dynamically modified. The focus of this work is small to medium sized problems; as a result 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 precomputed integrals and the use of approximate analytical expressions derived from surface fits. These may be used in the matrix assembly and internal point calculations. LUTs are derived for both flat and circular arc elements for both the displacement and stress boundary integral equation. Details are provided on suitable LUT refinements and the approach is benchmarked against conventional Gauss-Legendre quadrature. The surface fit approach is presented as an alternative to LUTs that does not incur the considerable memory cost associated with LUTs. This approach has been limited to flat elements. The equation solution is cast in a re-solution framework, in which we use a GM-RES iterative solver. Convergence is greatly accelerated by using an approximate but complete LU preconditioner updated periodically using multi-threading. Consideration of the period of update is investigated with reference to the spread of eigenvalues in the preconditioned system. The resulting system achieves the aim of providing real time update of contours for small to medium size 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.