# Assembling mass and stiffness matrices for isogeometric solving

## Dr. Angelos Mantzaflaris

**May 22, 2012, 1:30 p.m. S2 059**

We present techniques for the assembly of matrices needed in isogeometric solving, such as the Gram, stiffness, 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 integrals 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 aforementioned 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.