Fast formation and assembly of isogeometric Galerkin matrices for trimmed patches

TL;DR

This paper introduces two efficient methods for assembling isogeometric Galerkin matrices on trimmed patches, demonstrating significant speed-ups over traditional finite element methods and analyzing their accuracy and scalability.

Contribution

It compares hybrid Gauss-point-based and discontinuous weighted quadrature approaches for matrix assembly on trimmed patches, highlighting their advantages and limitations.

Findings

Significant speed-ups over standard finite element assembly.

Discontinuous weighted quadrature scales well with basis degree.

Hybrid Gauss approach is suitable for moderate degrees.

Abstract

This work explores the application of the fast assembly and formation strategy from [8, 17] to trimmed bi-variate parameter spaces. Two concepts for the treatment of basis functions cut by the trimming curve are investigated: one employs a hybrid Gauss-point-based approach, and the other computes discontinuous weighted quadrature rules. The concepts' accuracy and efficiency are examined for the formation of mass matrices and their application to L2-projection. Significant speed-ups compared to standard element by element finite element formation are observed. There is no clear preference between the concepts proposed. While the discontinuous weighted scheme scales favorably with the degree of the basis, it also requires additional effort for computing the quadrature weights. The hybrid Gauss approach does not have this overhead, which is determined by the complexity of the trimming curve. Hence, it is well-suited for moderate degrees, whereas discontinuous-weightedquadrature has potential for high degrees, in particular, if the related weights are computed in parallel.