# 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, stiness, 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.