Micropolar Fluids Using B-spline Divergence Conforming Spaces

Sarmiento, A. F.; García, Daniel; Dalcín, Lisandro; Collier, Nathan; Calo, Victor M.

doi:10.1016/j.procs.2014.05.089

Cited by 6 publications

(7 citation statements)

References 11 publications

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“…The numerator is a quadratic polynomial with coefficients (28) which proves (32). Inserting (28) into (35) and solving for ω n and ω n+1 then gives (33) and completes the proof.…”

Section: Gaussian Quadrature Formulaementioning

confidence: 80%

“…In the B-spline literature [11,17,19], the knot sequence is usually written with knots' multiplicities. However, in the isogeometric analysis literature, see e.g., [9,33], the knot vector is usually split into a vector carrying the partition of the interval and a vector containing continuity information (knot multiplicity). As in this paper the multiplicity is always four at every knot, we follow the latter notation and, throughout the paper, write X n without multiplicity, i.e., x k < x k+1 , k = 0, .…”

Section: Quintic Splines With Uniform Knot Sequencesmentioning

confidence: 99%

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Gaussian quadrature rules for C1 quintic splines with uniform knot vectors

Bartoň

Ait‐Haddou²,

Calo

2017

Journal of Computational and Applied Mathematics

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We provide explicit quadrature rules for spaces of C 1 quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids numerical solvers. Each rule is optimal, that is, requires the minimal number of nodes, for a given function space. For each of n subintervals, generically, only two nodes are required which reduces the evaluation cost by 2/3 when compared to the classical Gaussian quadrature for polynomials over each knot span. Numerical experiments show fast convergence, as n grows, to the "two-third" quadrature rule of Hughes et al. [23] for infinite domains.

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“…The numerator is a quadratic polynomial with coefficients (28) which proves (32). Inserting (28) into (35) and solving for ω n and ω n+1 then gives (33) and completes the proof.…”

Section: Gaussian Quadrature Formulaementioning

confidence: 80%

Section: Quintic Splines With Uniform Knot Sequencesmentioning

confidence: 99%

Gaussian quadrature rules for C1 quintic splines with uniform knot vectors

Bartoň

Ait‐Haddou²,

Calo

2017

Journal of Computational and Applied Mathematics

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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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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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“…In the B-spline literature [6,12,14], the knot sequence is usually written with knots' multiplicities. However, in the isogeometric analysis literature, see e.g., [5,23], the knot vector is usually split into a vector carrying the partition of the interval and a vector containing continuity information (knot multiplicity). As in this paper the multiplicity is always four at every knot, we follow the latter notation and, throughout the paper, write X n without multiplicity, i.e., x k < x k+1 , k = 0, .…”

Section: Quintic Splines With Uniform Knot Sequencesmentioning

confidence: 99%

“…By Corollary 1, there are exactly two nodes inside (x k−1 , x k ). Due to Lemma 2.3, only the roots of the quadratic factor in (23) contribute to the computation of the nodes and hence solving Q k (α k ) = 0 with coefficients from (27) gives α k and β k . Combining these with (15) results in (29).…”

Section: Gaussian Quadrature Formulaementioning

confidence: 99%

Gaussian quadrature rules for $C^1$ quintic splines

Bartoň,

Ait-Haddou,

Calo

2015

Preprint

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We provide explicit expressions for quadrature rules on the space of C 1 quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal, that is, it requires the minimal number of nodes. For each of n subintervals, generically, only two nodes are required which reduces the evaluation cost by 2/3 when compared to the classical Gaussian quadrature for polynomials. Numerical experiments show fast convergence, as n grows, to the "two-third" quadrature rule of Hughes et al.[16] for infinite domains.

show abstract

Micropolar Fluids Using B-spline Divergence Conforming Spaces

Cited by 6 publications

References 11 publications

Gaussian quadrature rules for C1 quintic splines with uniform knot vectors

Gaussian quadrature rules for C1 quintic splines with uniform knot vectors

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

Gaussian quadrature rules for $C^1$ quintic splines

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