BPX preconditioners for isogeometric analysis using analysis-suitable T-splines

Cho, Durkbin; Vázquez, Rafael

doi:10.1093/imanum/dry032

Cited by 12 publications

(17 citation statements)

References 44 publications

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“…On the other hand, handling trimming or complex geometries (e.g., singular or highly distorted) is a specific difficulty of the isogeometric method, and it is under study in the community. The same is true for locally-refinable spaces, see the recent paper [9] and the references therein.…”

Section: Discussionmentioning

confidence: 70%

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

Sangalli

Tani

2018

Computer Methods in Applied Mechanics and Engineering

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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 ...

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Section: Discussionmentioning

confidence: 70%

“…. , 9,10. We only consider MF-WQ, with the Krylov method, the preconditioner and the stopping criterion as above.…”

Section: Numerical Experimentsmentioning

confidence: 99%

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

Sangalli

Tani

2018

Computer Methods in Applied Mechanics and Engineering

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“…We also define, for an arbitrary hyperrectangular element in the index domain q Q = Π d i=1 (a i , b i ), the bisection operator in the j-th direction (compare with [157, Definition 2.5] and [62,Section 4…”

Section: T-meshes Refined By Bisectionmentioning

confidence: 99%

“…We also mention that [62] has recently introduced a local multilevel preconditioner for the stiffness matrix of symmetric problems which leads to uniformly bounded condition numbers for T-splines on admissible T-meshes. An important consequence is that the corresponding PCG solver is uniformly contractive.…”

Section: Remark 31mentioning

confidence: 99%

Mathematical Foundations of Isogeometric Analysis

et al. 2020

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isogeometric analysis, with special focus on the mathematical theory. This includes an overview of available spline technologies for the local resolution of possible singularities as well as the state-of-the-art formulation of convergence and quasi-optimality of adaptive algorithms for both the finite element method (FEM) and the boundary element method (BEM) in the frame of isogeometric analysis (IGA).

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“…Balancing Domain Decomposition by Constraints (BDDC) preconditioners for Galerkin IGA have been studied in [10,13,14,43]. In addition to our previous works, we also mention [15,18,19] on BPX preconditioners, [31,32,39] on IGA multigrid, [35,36,37,38] on IGA Discontinuous Galerkin methods, and [40,48,49,42] on other IGA solvers. A recent comparison between spectral elements and IGA discretizations and solvers can be found in [33].…”

Section: Introductionmentioning

confidence: 99%

Overlapping Additive Schwarz preconditioners for isogeometric collocation discretizations of linear elasticity

Cho

Pavarino

Scacchi

2021

Computers & Mathematics with Applications

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Overlapping Additive Schwarz (OAS) preconditioners are here constructed for isogeometric collocation discretizations of the system of linear elasticity in both two and three space dimensions. Isogeometric collocation methods are recent variants of isogeometric analysis based on the numerical approximation of the strong form of partial differential equations at appropriate collocation points. Numerical results in two and three dimensions show that two-level OAS preconditioners are scalable in the number of subdomains N , quasi-optimal with respect to the mesh size h and optimal with respect to the spline polynomial degree p. Moreover, two-level OAS preconditioners are more robust than one-level OAS and non-preconditioned GMRES solvers when the material tends to the incompressible limit, as well as in the presence of strong deformation of the NURBS geometry.

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BPX preconditioners for isogeometric analysis using analysis-suitable T-splines

Cited by 12 publications

References 44 publications

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

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

Mathematical Foundations of Isogeometric Analysis

Overlapping Additive Schwarz preconditioners for isogeometric collocation discretizations of linear elasticity

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