Abstract
In this paper, we consider the computational cost of a multi-frontal direct solver used for the factorization of matrices resulting from a discretization of isogeometric analysis with T-splines, and analysis-suitable T-splines. We start from model projection or model heat transfer problems discretized over two-dimensional meshes, either uniformly refined or refined towards a point or an edge. These grids preserve several symmetries and they are the building blocks of more complicated grids constructed during adaptive isotropic refinement procedures. A large class of computational problems construct meshes refined towards point or edge singularities. We propose an ordering that permutes the matrix in a way that the computational cost of a multi-frontal solver executed on adaptive grids is linear. We show that analysis-suitable T-splines with our ordering, besides having other well-known advantages, also significantly reduce the computational cost of factorization with the multi-frontal direct solver. Namely, the factorization with N T-splines of order p over meshes refined to a point has a linear O(Np4) cost, and the factorization with T-splines on meshes refined to an edge has O(N2pp2) cost. We compare the execution time of the multi-frontal solver with our ordering to the Approximate Minimum Degree (AMD) and Cuthill–McKee orderings available in Octave.
Highlights
Higher-order and continuity basis functions, such as B-splines, Non-Uniform Rational B-splines (NURBS) [1], and T-splines [2], are used in computer-aided design (CAD)/computer-aided engineering (CAE) systems for modeling of the geometry of engineering design
This paper focuses on the T-spline basis functions, which can be defined on adaptive grids
We propose ordering algorithms resulting in a permutation of the sparse matrix such that the computational cost of multi-frontal solver algorithm factorization is linear for grids refined towards point or edge singularities when using analysis-suitable T-splines
Summary
Higher-order and continuity basis functions, such as B-splines, Non-Uniform Rational B-splines (NURBS) [1], and T-splines [2], are used in computer-aided design (CAD)/computer-aided engineering (CAE) systems for modeling of the geometry of engineering design. Multi-frontal solvers construct orderings and elimination trees based on a matrix’s sparsity pattern multi-frontal solvers applied to isogeometric analysis with T-spline basis functions on adaptive resulting grids. To find the computational cost of the elimination process, we can look at the spans of B-spline basis functions interface [25], such as MUMPS solver [11,12,13], Scalapack [26], PaStiX [27], and SuperLU [28].
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