A Black-Box Algorithm for Fast Matrix Assembly in Isogeometric Analysis

Dr. Clemens Hofreither

May 30, 2017, 3:30 p.m. S2 059

We present a fast algorithm for assembling stiffness matrices in
Isogeometric Analysis with tensor product spline spaces. The procedure
exploits the facts that (a) such matrices have block-banded structure,
and (b) they often have low Kronecker rank. Combined, these two
properties allow us to reorder the nonzero entries of the stiffness
matrix into a relatively small, dense matrix or tensor of low rank. A
suitable black-box low-rank approximation algorithm is then applied to
this matrix or tensor. This allows us to approximate the nonzero entries
of the stiffness matrix while explicitly computing only relatively few
of them, leading to a fast assembly procedure.

The algorithm does not require any further knowledge of the used spline
spaces, the geometry transform, or the partial differential equation,
and thus is black-box in nature. Existing assembling routines can be
reused with minor modifications.

Numerical examples demonstrate significant speedups over a standard
Gauss quadrature assembler for several geometries in two and three
dimensions. The runtime scales sublinearly with the number of degrees of
freedom in a large pre-asymptotic regime.