Assembling mass and stiffness matrices for isogeometric solving

Dr. Angelos Mantzaflaris

May 22, 2012, 3:30 p.m. S2 059

We present techniques for the assembly of matrices needed in isogeometric
solving, such as the Gram, sti ness, mass or advection matrices. Computing
such matrices is dominating the computational time of isogeometric methods
for boundary value problems. Typically, the entries of these matrices are inte-
grals of products of shape functions and their derivatives. These integrals over
elements in the physical domain are transformed to integrals over the support
of the basis functions, resulting in integrants involving the (inverse of the)
Jacobian of the geometry map.
First, we review quadrature approaches for numerical integration in the
context of isogeometric methods. A problem with quadrature rules is that
they require evaluation of the basis functions over a usually large number of
quadrature points. Recent works try to reduce the number of evaluations by
deriving specialized quadrature rules that are optimal for the underlying spline
space.
Second, we explore a quadrature-free method for the assembly of the afore-
mentioned matrices. The proposed method consists in an initial approximation
of the rational part that appears in the integrant and a fast look-up operation
for the resulting integrals, involving only products of B-Splines. This strategy
aims at overcoming the evaluations bottleneck that appears in quadrature-
based approaches.