Adaptive isogeometric analysis by local h-refinement with T-splines

Dörfel, Michael R.; Jüttler, Bert; Simeon, Bernd

doi:10.1016/j.cma.2008.07.012

Cited by 338 publications

(196 citation statements)

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“…A shared memory implementation for CPUs would also form a perfect example to illustrate the estimates derived in the paper. As future work, we consider to extend the results to T-splines, which allow for local adaptivity [19].…”

Section: Discussionmentioning

confidence: 99%

Computational cost estimates for parallel shared memory isogeometric multi-frontal solvers

Woźniak

Kuźnik

Paszyński

et al. 2014

Computers & Mathematics with Applications

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In this paper we present computational cost estimates for parallel shared memory isogeometric multi-frontal solver. The estimates show that the ideal isogeometric shared memory parallel direct solver scales as O(p 2 log(N/p)) for one dimensional problems, O(Np 2 ) for two dimensional problems, and O(N 4/3 p 2 ) for three dimensional problems, where N is the number of degrees of freedom, and p is the polynomial order of approximation. The computational costs of the shared memory parallel isogeometric direct solver are compared with those corresponding to the sequential isogeometric direct solver, being the latest equal to O(Np 2 ) for the one dimensional case, O(N 1.5 p 3 ) for the two dimensional case, and O(N 2 p 3 ) for the three dimensional case. The shared memory version significantly reduces both the scalability in terms of N and p. Theoretical estimates are compared with numerical experiments performed with linear, quadratic, cubic, quartic, and quintic B-splines, in one and two spatial dimensions.Keywords: isogeometric finite element method, multi-frontal direct solver, computational cost, NVIDIA CUDA GPU Preprint submitted to Computers & Mathematics with ApplicationsMarch 27, 20141. Introduction Classical higher order finite element methods (FEM) [17,18] maintain only C 0 -continuity at the element interfaces, while isogeometric analysis (IGA) utilizes B-splines as basis functions, and thus, it delivers C k global continuity [14]. The higher continuity obtained across elements allows IGA to attain optimal convergence rates for any polynomial order, while using fewer degrees of freedom [3,1]. Nevertheless, this reduced count in the number of degrees of freedom may not immediately correlate with a computational cost reduction, since solution time per degree of freedom augments as the continuity is increased [10,13]. In spite of the increased cost of highercontinuous spaces, they have proven very popular and useful. For example, higher-continuous spaces have allowed the solution of higher-order partial di↵erential equations with elegance [7,28,29,51,16,15] as well as several non-linear problems of engineering interest [31,6,30,5,20,11,9,4]. Thus, e cient multi-frontal solvers for higher-continuous spaces are important.The multi-frontal solver is one of the state-of-the art algorithm for solving linear systems of equations [22,26]. It is a generalization of the frontal solver algorithm proposed in [33,21]. The multi-frontal algorithm constructs an assembly tree based on the analysis of the connectivity data or the geometry of the computational mesh. Finite elements are joint into pairs and fully assembled unknowns are eliminated within frontal matrices associated to multiple branches of the tree. The process is repeated until the root of the assembly tree is reached. Finally, the common interface problem is solved and partial backward substitutions are recursively called on the assembly tree.There exist parallel versions of the multi-frontal direct solver algorithm targeting distributed-memory, share...

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Section: Discussionmentioning

confidence: 99%

Computational cost estimates for parallel shared memory isogeometric multi-frontal solvers

Woźniak

Kuźnik

Paszyński

et al. 2014

Computers & Mathematics with Applications

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“…, the corresponding refinement procedure may cause a propagation of the refinement beyond the regions marked by the error estimator [3]. In particular, the linear independence of the T-spline blending functions can be guaranteed only by considering a restricted subset of Tsplines [1,13,16].…”

Section: Neverthelessmentioning

confidence: 99%

THB-splines: The truncated basis for hierarchical splines

Giannelli

Jüttler

Speleers

2012

Computer Aided Geometric Design

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The construction of classical hierarchical B-splines can be suitably modified in order to define locally supported basis functions that form a partition of unity. We will show that this property can be obtained by reducing the support of basis functions defined on coarse grids, according to finer levels in the hierarchy of splines. This truncation not only decreases the overlapping of supports related to basis functions arising from different hierarchical levels, but it also improves the numerical properties of the corresponding hierarchical basis -which is denoted as truncated hierarchical B-spline (THB-spline) basis. Several computed examples will illustrate the adaptive approximation behavior obtained by using a refinement algorithm based on THB-splines.

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“…Sederberg et al [24] defined the notion of T-splines that allow us to reduce the number of those control points. In [5] Dörfel et al used T-splines for local h-refinement in isogeometric analysis.…”

Section: Patchmentioning

confidence: 99%