Recursive Singular Value Decomposition Compression of Refined Isogeometric Analysis Matrices as a Tool to Speedup Iterative Solvers Performance

Abstract

The isogeometric analysis (IGA) uses higher-order and continuity basis functions as compared to the traditional finite element method. IGA has many applications in simulations of time-dependent problems. These simulations are often performed using an explicit time-integration scheme, which requires the solution of a system of linear equations with the mass matrix, constructed with high-order and continuity basis functions. The iterative solvers are most commonly applied for large problems simulated over complex geometry. This paper focuses on recursive decomposition of the mass matrix using the Singular Value Decomposition algorithm (SVD). We build a recursive tree, where submatrices are expressed as multi-columns multiplied by multi-rows. When we keep the mass matrix compressed in such a way, the multiplication of a matrix by a vector, as performed by an iterative solver, can be performed in O(Nr) instead of $O(N^2)$ computational cost, where N is the number of rows of input matrix, r is the number of singular values bigger than given value. Next, we focus on refined isogeometric analysis (rIGA). We introduce the C0 separators into IGA submatrices and analyze the SVD recursive compression and computational cost of an iterative solver when increasing the patch size and the order of B-spline basis functions.