Fast formation of isogeometric Galerkin matrices by weighted quadrature

• Published in: Computer methods in applied mechanics and engineering 2017-04, Vol.316, p.606-622
• Main Authors: Calabrò, F., Sangalli, G., Tani, M.
• Format: Article
• Language: English
• Subjects: [formula omitted]-refinement; Algorithms; Basis functions; Computational efficiency; Computing time; Cost analysis; Floating point arithmetic; Floating structures; Galerkin method; Geometry; Isogeometric analysis; Loops; Parameterization; Splines; Weight; Weighted quadrature
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2016.09.013

In this paper we propose an algorithm for the formation of matrices of isogeometric Galerkin methods. The algorithm is based on three ideas. The first is that we perform the external loop over the rows of the matrix. The second is that we calculate the row entries by weighted quadrature. The third is that we exploit the (local) tensor product structure of the basis functions. While all ingredients have a fundamental role for computational efficiency, the major conceptual change of paradigm with respect to the standard implementation is the idea of using weighted quadrature: the test function is incorporated in the integration weight while the trial function, the geometry parametrization and the PDEs coefficients form the integrand function. This approach is very effective in reducing the computational cost, while maintaining the optimal order of approximation of the method. Analysis of the cost is confirmed by numerical testing, where we show that, for p large enough, the time required by the floating point operations is less than the time spent in unavoidable memory operations (the sparse matrix allocation and memory write). The proposed algorithm allows significant time saving when assembling isogeometric Galerkin matrices for all the degrees of the test spline space and paves the way for a use of high-degree k-refinement in isogeometric analysis. •We discuss the formation of isogeometric Galerkin matrices by row-loop and weighted quadrature.•We give an estimate of the computational cost.•We perform tests showing that the proposed algorithm over-performs other approaches.