Sum-factorization techniques in Isogeometric Analysis

TL;DR

This paper introduces sum-factorization algorithms for isogeometric analysis that significantly reduce the computational complexity of assembling matrices, especially at higher spline degrees, by applying a global or macro-element approach instead of element-wise methods.

Contribution

The paper presents novel sum-factorization algorithms for matrix assembly in isogeometric analysis that improve computational efficiency by changing the application scope from element-wise to global or macro-element levels.

Findings

Complexity grows as p^{d+2} with the new algorithms.

Algorithms outperform standard element-wise approaches at high spline degrees.

Global/macro-element approach reduces computational cost significantly.

Abstract

The fast assembling of stiffness and mass matrices is a key issue in isogeometric analysis, particularly if the spline degree is increased. We present two algorithms based on the idea of sum factorization, one for matrix assembling and one for matrix-free methods, and study the behavior of their computational complexity in terms of the spline order $p$. Opposed to the standard approach, these algorithms do not apply the idea element-wise, but globally or on macro-elements. If this approach is applied to Gauss quadrature, the computational complexity grows as $p^{d+2}$ instead of $p^{2d+1}$ as previously achieved.