Three-dimensional modeling of fracture by Fast Multipole Symmetric Galerkin Boundary Element Method. Application to multi-fractured media

The solution of three dimensional elastostatic problem using the Symmetric Galerkin Boundary Element Method (SGBEM) gives rise to fully populated (albeit symmetric) matrix equations, entailing long solution times for large models. By integrating the Fast Multipole Method (FMM) and the iterative solver GMRES (Generalized Minimal Residual) into SGBEM, the performance of the combined method is drastically improved. Consequently, some models involving up to 105 unknowns can be considered. However, the only stocked matrix in FM-SGBEM that gathers all contributions of near particles (Knear) has never been fully used as a preconditioner by GMRES. For this reason, the solution phase still takes a lot of iterations to reach the desired convergence. Flexible GMRES, on the other hand, provides an inner-outer scheme which has proved to be able to boost the computation since the matrix of near interactions Knear can be efficiently exploited. This article presents first the general features of FM-SGBEM then the implementation of Flexible GMRES into FM-SGBEM. Some numerical experiments that validate the performance of Flexible GMRES on test problems concerning cracks in the unbounded domain is reported.