Fast Hierarchical Boundary Element Method for Large Scale 3-D Elastic Problems

This chapter reviews recent developments in the strategies for the fast solution of boundary element systems of equations for large scale 3D elastic problems. Both isotropic and anisotropic materials as well as cracked and uncracked solids are considered. The focus is on the combined use the hierarchical representation of the boundary element collocation matrix and iterative solution procedures. The hierarchical representation of the collocation matrix is built starting from the generation of the cluster and block trees that take into account the nature of the considered problem, i.e. the possible presence of a crack. Low rank blocks are generated through adaptive cross approximation (ACA) algorithms and the final hierarchical matrix is further coarsened through suitable procedures also used for the generation of a coarse preconditioner, which is built taking full advantage of the hierarchical format. The final system is solved using a GMRES iterative solver. Applications show that the technique allows considerable savings in terms of storage memory, assembly time and solution time without accuracy penalties. Such features make the method appealing for large scale applications.