Reduced integration at superconvergent points in isogeometric analysis

Fahrendorf, Frederik; Lorenzis, Laura De; Gómez, Héctor

doi:10.1016/j.cma.2017.08.028

Cited by 44 publications

(21 citation statements)

References 22 publications

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“…Second, all coefficients a (i ) j 1 ,j 2 are specified by the second set of interpolation conditions in (14). Third, thanks to the fulfillment of conditions (14) and (15) and the fact that the function g i is given by (10), we get…”

Section: The Vertex Subspace W V (I)mentioning

confidence: 99%

“…, j 2 = 0, 1, 2, 3, and g , respectively, across the interface Γ (i ) , cf. [26,Section 7], the first set of interpolation conditions in (14) uniquely determine all coefficients a Γ (i ) j 1 ,j 2 . Moreover, as the functions g Γ (i ) i ensure C 2 -smoothness across the interfaces Γ (i ) , we also obtain…”

Section: The Vertex Subspace W V (I)mentioning

confidence: 99%

“…The two methods [15] and [32] select different specific subsets of the superconvergent points to achieve that the number of collocation points coincides with the number of degrees of freedom. While the points 1 in [15] are chosen in an alternating way, which reduces for the case of odd spline degree p the convergence order also to O(h p ) in the L 2 (L ∞ ) norm, the points in [32] are selected in a clustered way to maintain the convergence behavior of [1].In [1,2,14,32,38], the concept of isogeometric collocation has been mostly used for solving the Poisson's equation. Further problems and PDEs of interest in the framework of isogeometric collocation are amongst others linear and nonlinear elasticity, elastostatistics and elastodynamics [1, 3, 13-15, 19, 36, 38], plate/shell problems [15,30] and beam problems [5,31].As already mentioned before, most of the existing isogeometric collocation methods have in common that they are mainly restricted to the case of one-patch domains.…”

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confidence: 99%

“…In [1,2,14,32,38], the concept of isogeometric collocation has been mostly used for solving the Poisson's equation. Further problems and PDEs of interest in the framework of isogeometric collocation are amongst others linear and nonlinear elasticity, elastostatistics and elastodynamics [1, 3, 13-15, 19, 36, 38], plate/shell problems [15,30] and beam problems [5,31].…”

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Isogeometric collocation on planar multi-patch domains

Kapl

Vitrih

2020

Computer Methods in Applied Mechanics and Engineering

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We present an isogeometric framework based on collocation to construct a C 2 -smooth approximation of the solution of the Poisson's equation over planar bilinearly parameterized multi-patch domains. The construction of the used globally C 2 -smooth discretization space for the partial differential equation is simple and works uniformly for all possible multipatch configurations. The basis of the C 2 -smooth space can be described as the span of three different types of locally supported functions corresponding to the single patches, edges and vertices of the multi-patch domain. For the selection of the collocation points, which is important for the stability and convergence of the collocation problem, two different choices are numerically investigated. The first approach employs the tensor-product Greville abscissae as collocation points, and shows for the multi-patch case the same convergence behavior as for the one-patch case [2], which is suboptimal in particular for odd spline degree. The second approach generalizes the concept of superconvergent points from the one-patch case (cf. [1,15,32]) to the multi-patch case. Again, these points possess better convergence properties than Greville abscissae in case of odd spline degree.While in case of a one-patch domain, the higher continuity of the functions can be easily guaranteed by using spline functions with the desired smoothness within the patch, the construction of C s -smooth (s ≥ 1) isogeometric spline spaces over multi-patch domains is challenging, and is the task of current research, see e.g. [6-9, 20-23, 28, 29, 33, 34, 40, 41] for s = 1 and [24][25][26][27]39] for s = 2. The design of C s -smooth multi-patch spline spaces is linked to the concept of geometric continuity of multi-patch surfaces, cf. [17,35], due to the fact that an isogeometric function is C s -smooth on a multi-patch domain if and only if its associated multi-patch graph surface is G s -smooth, cf. [16,28].So far, the problem of isogeometric collocation has been mostly explored on one-patch domains, see e.g. [1, 2, 5, 13-15, 30-32, 36, 38]. For analyzing the convergence behavior of a particular approach, the errors are generally computed with respect to L ∞ , W 1,∞ and W 2,∞ norm, or equivalently with respect to L 2 , H 1 and H 2 norm, respectively. The study of isogeometric collocation methods has started in [2] by using the Greville and Demko abscissae as collocation points. In comparison to the Galerkin approach, both choices show a suboptimal convergence behavior with respect to L 2 (L ∞ ) norm, namely of orders O(h p ) and O(h p−1 ) under h-refinement for even and odd spline degree p, respectively, and additionally a suboptimal convergence order O(h p−1 ) with respect to H 1 (W 1,∞ ) norm but just in case of odd spline degree p.In [1], an isogeometric collocation method has been presented, which is based on the computation and on the use of specific collocation points called superconvergent points. The proposed technique possesses only for even spline degree p in case of the L 2 ...

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Section: The Vertex Subspace W V (I)mentioning

confidence: 99%

Section: The Vertex Subspace W V (I)mentioning

confidence: 99%

mentioning

confidence: 99%

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Isogeometric collocation on planar multi-patch domains

Kapl

Vitrih

2020

Computer Methods in Applied Mechanics and Engineering

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“…Moreover, the high smoothness degree that B-splines or NURBS typically possess across the elements makes Gauss integration rules suboptimal [21,22]. The development of more efficient integration schemes is currently an open problem, although significant progress has been made in [23][24][25][26][27][28]. IGA-C represents an interesting solution for this problem since the need for numerical quadrature is completely removed due to the discretization of the strong form of the governing equations.…”

Section: Introductionmentioning

confidence: 99%

Effects of parameterization and knot placement techniques on primal and mixed isogeometric collocation formulations of spatial shear-deformable beams with varying curvature and torsion

Marino

Hosseini

Hashemian

et al. 2020

Computers & Mathematics with Applications

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We present a displacement-based and a mixed isogeometric collocation (IGA-C) formulation for free-form, three-dimensional, shear-deformable beams with high and rapidly-varying curvature and torsion. When such complex shapes are concerned, the approach used to build the IGA geometric model becomes relevant. Although IGA-C has been so far successfully applied to a wide range of problems, the effects that different parameterization and knot placement techniques may have on the accuracy of collocation-based formulations is still an unexplored field. To fill this gap, primal and mixed formulations are used combining two parameterization methods (chord-length and equally spaced) with two knot placement techniques (uniformly spaced and De Boor). With respect to the space-varying Frenet local frame, we derive the strong form of the governing equations in a compact form through the definition of two matrix operators conveniently used to perform first and second order derivatives of the vector fields involved in the formulations. This approach is very efficient and easy to implement within a collocation-based scheme. Several challenging numerical experiments allow to test the different considered parameterizations and knot placement techniques, revealing in particular that with the primal formulation an equally spaced parameterization is definitively the most recommended choice and it should always be used with an approximation degree of, at least, p = 6, although some caution must be adopted when very high

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