Matrix-free weighted quadrature for a computationally efficient isogeometric k-method

Abstract

The k-method is the isogeometric method based on splines (or NURBS, etc.) with maximum regularity. When implemented following the paradigms of classical finite element methods, the computational resources required by the k−method are prohibitive even for moderate degree. In order to address this issue, we propose a matrix-free strategy combined with weighted quadrature, which is an ad-hoc strategy to compute the integrals of the Galerkin system. Matrix-free weighted quadrature (MF-WQ) speeds up matrix operations, and, perhaps even more important, greatly reduces memory consumption. Our strategy also requires an efficient preconditioner for the linear system iterative solver. In this work we deal with an elliptic model problem, and adopt a preconditioner based on the Fast Diagonalization method, an old idea to solve Sylvester-like equations. Our numerical tests show that the isogeometric solver based on MF-WQ is faster than standard approaches (where the main cost is the matrix formation by standard Gaussian quadrature) even for low degree. But the main achievement is that, with MF-WQ, the k-method gets orders of magnitude faster by increasing the degree, given a target accuracy. Therefore, we are able to show the superiority, in terms of computational efficiency, of the high-degree k-method with respect to low-degree isogeometric discretizations. What we present here is applicable to more complex and realistic differential problems, but its effectiveness will depend on the preconditioner stage, which is as always problem-dependent. This situation is typical of modern high-order methods: the overall performance is mainly related to the quality of the preconditioner.Matrix-free isogeometric analysis 2 of degree p goes up to C p−1 . We denote k-method the isogeometric method where splines of highest possible regularity are employed. In this setting, higher accuracy is achieved by k-refinement, that is, raising the degree p and refining the mesh, with C p−1 regularity at the new inserted knots, see [18].The k-method leads to several advantages, such as higher accuracy per degree-of-freedom and improved spectral accuracy, see for example [18,12,5,6,19,14]. At the same time, the k-method brings significant challenges at the computational level. Indeed, using standard finite element routines, its computational cost grows too fast with respect to the degree p, making degree raising unfavourable from the viewpoint of computational efficiency. As a consequence, in complex isogeometric simulations quadratic C 1 NURBS are preferred, see [4,27].The computational cost of the k-method is well understood in literature. The standard formation of the Poisson (Galerkin) stiffness matrix, by Gauss quadrature and element assembly, takes O(N p 9 ) floating point operations (flops) in 3D, where N is the number of degrees of freedom (see, e.g., [8] for the details). Computational limitations appear also in the solution phase of the linear system: the large support overlap associated with B-splines makes the associated Galerkin matrices ...

Space–time least–squares isogeometric method and efficient solver for parabolic problems

“…We emphasize that it is possible, though beyond the scope of this paper, to take the matrix-free paradigm one step further by using the approach developed in [36]. Using this approach, where even the factors of A as in (30) are not needed, would significantly improve the overall iterative solver in terms of memory and computational cost (both for the setup and for the matrix-vector computations).…”

“…The k-method, based on splines of degree p and C p−1 regularity, delivers higher accuracy per degree-of-freedom, comparing to C 0 or discontinuous hp-FEM [8,13,17]. However, the k-method also requires ad-hoc algorithms, otherwise when the polynomial degree p increases, the computational cost per degree-of-freedom increases dramatically, both in the formation of the matrix and in the solution of the linear system [11,36].…”

Sum factorization techniques in Isogeometric Analysis

“…The cost of applying A D , excluding the evaluation of u(x)F(x), is as in (21), (22) and (23). Relabeling and accumulating onto v has cost R M .…”

The surrogate matrix methodology: Low-cost assembly for isogeometric analysis

“…• For our new surrogate methods, it would be most efficient if the matrix entries which require quadrature were to be computed based on individual basis function interactions. This leaves out standard element-by-element assembly strategies, but would work well with the control point pairwise method proposed in [32] or, ideally, with a row-byrow approach; e.g., [11,25,38]. Nevertheless, one should still expect to see significant speed-ups with surrogate methods in standard element-by-element codes, at least for moderate polynomial orders.…”