235 research outputs found
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
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