PetIGA-MF: A multi-field high-performance toolbox for structure-preserving B-splines spaces

We propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers. Starting from a highly continuous Isogeometric Analysis (IGA) discretization, we introduce C 0 -separators to reduce the interconnection between degrees of freedom in the mesh. By doing so, both the solution time and best approximation errors are simultaneously improved. We call the resulting method "refined Isogeometric Analysis (rIGA)". To illustrate the impact of the continuity reduction, we analyze the number of Floating Point Operations (FLOPs), computational times, and memory required to solve the linear system obtained by discretizing the Laplace problem with structured meshes and uniform polynomial orders. Theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between p 2 and p 3 , with p being the polynomial order of the discretization. Numerical results indicate that our proposed rIGA method delivers a speed-up factor proportional to p 2 . In a 2D mesh with four million elements and p = 5, the linear system resulting from rIGA is solved 22 times faster than the one from highly continuous IGA. In a 3D mesh with one million elements and p = 3, the linear rIGA system is solved 15 times faster than the IGA one.

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“…PetIGA has been used to model many engineering applications since its inception [3,5,8,9,42,[45][46][47][48][49].…”

Section: Implementation Detailsmentioning

confidence: 99%

The value of continuity: Refined isogeometric analysis and fast direct solvers

García

Pardo

Dalcín

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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“…Isogeometric analysis based on NURBS (Non-Uniform Rational B-Splines) has been described in a number of papers (e.g. [12,24,25,41]) and the efficient implementation of the method in open source software has been discussed in [26,29,54,57]. Isogeometric analysis employs piecewise polynomial curves composed of linear combinations of Bspline basis functions.…”

Section: Problem Statementmentioning

confidence: 99%

Dispersion-optimized quadrature rules for isogeometric analysis: Modified inner products, their dispersion properties, and optimally blended schemes

Puzyrev

Deng

Calo

2017

Computer Methods in Applied Mechanics and Engineering

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This paper introduces optimally-blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations. To quantify the approximation errors when we modify the inner products, we generalize the Pythagorean eigenvalue theorem of Strang and Fix. The proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and nonuniform meshes as well as for different polynomial orders and continuity of the basis. The optimally-blended schemes improve the convergence rate of the method by two orders with respect to the fully-integrated Galerkin method. The proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems.

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“…The strategy was implemented in PetIGA-MF [18], a toolbox built on top of PetIGA [19] extending it to support flexible and parallel multi-field discretizations based on structure-preserving B-spline spaces, in the sense of [2]. For examples of the Navier-Stokes-Cahn-Hilliard model using PetIGA-MF see [44,45] .…”

Section: Performance Resultsmentioning

confidence: 99%

“…In that study, the spectral equivalence of the block factorization strategy (thus, the invariance in the number of iterations of the iterative solver with mesh refinement) was shown possible, even though we used non-scalable direct methods to solve for the blocks. The question of a scalable choice for the blocks was only possible to address with our parallel implementation of divergence-conforming spaces [18] based on the high-performance isogeometric framework , 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 named PetIGA [19], built on top of PETSc [20], that releases a whole set of solvers and preconditioners. Here we investigate an approach built from state-of-the-art and black-box preconditioning strategies that furnish an h-independent preconditioner for the Stokes problem discretized with divergence-conforming B-spline spaces, with good performance for both sequential and parallel setups.…”

Section: Introductionmentioning

confidence: 99%

A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem

Côrtes

Dalcín

Sarmiento

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. AbstractThe recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity-pressure pairs for viscous incompressible flows that are at the same time inf − sup stable and pointwise divergence-free. When applied to the discretized Stokes problem, these spaces generate a symmetric and indefinite saddle-point linear system. The iterative method of choice to solve such system is the Generalized Minimum Residual Method. This method lacks robustness, and one remedy is to use preconditioners. For linear systems of saddle-point type, a large family of preconditioners can be obtained by using a block factorization of the system. In this paper, we show how the nesting of "black-box" solvers and preconditioners can be put together in a block triangular strategy to build a scalable block preconditioner for the Stokes system discretized by divergence-conforming B-splines. Besides the known cavity flow problem, we used for benchmark flows defined on complex geometries: an eccentric annulus and hollow torus of an eccentric annular cross-section.

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PetIGA-MF: A multi-field high-performance toolbox for structure-preserving B-splines spaces

Cited by 38 publications

References 31 publications

The value of continuity: Refined isogeometric analysis and fast direct solvers

The value of continuity: Refined isogeometric analysis and fast direct solvers

Dispersion-optimized quadrature rules for isogeometric analysis: Modified inner products, their dispersion properties, and optimally blended schemes

A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem

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