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

Steklov Spectral Geometry for Extrinsic Shape Analysis

We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.Comment: Additional experiments adde

Fast Solution of 3D Elastodynamic Boundary Element Problems by Hierarchical Matrices

In this paper a fast solver for three-dimensional elastodynamic BEM problems formulated in the Laplace transform domain is presented, implemented and tested. The technique is based on the use of hierarchical matrices for the representation of the collocation matrix for each value of the Laplace parameter of interest and uses a preconditioned GMRES for the solution of the algebraic system of equations. The preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. An original strategy for speeding up the overall analysis is presented and tested. The reported numerical results demonstrate the effectiveness of the technique