Explicit higher-order accurate isogeometric collocation methods for structural dynamics

Evans, John A.; Hiemstra, René R.; Hughes, Thomas J.R.; Reali, Alessandro

doi:10.1016/j.cma.2018.04.008

Cited by 65 publications

(28 citation statements)

References 34 publications

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“…Moreover, we obtain that the function φ v (i) I ;j 1 ,j 2 is well defined and C 2 -smooth on Ω, where the latter property is the direct consequence of satisfying the homogeneous linear system (11). Fourth, the C 4 -smoothness of the function φ v (i) I ;j 1 ,j 2 at vertex v (i) follows directly from the fulfillment of the interpolation conditions (13).…”

Section: The Vertex Subspace W V (I)mentioning

confidence: 70%

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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: 70%

“…by interpolating the conditions (13). The only two differences in the construction of the functions are that the lower index of the indices i is not taken modulo ν i , and that for the patches Ω (i 0 ) and Ω (i −1 ) the functions g…”

Section: The Vertex Subspace W V (I)mentioning

confidence: 99%

Isogeometric collocation on planar multi-patch domains

Kapl

Vitrih

2020

Computer Methods in Applied Mechanics and Engineering

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“…The RK method is summarized as

{\overset{boldu}{true}}_{t_{s}} = {boldM}^{- 1} boldf ((), {boldu}_{t_{s}}, {\overset{boldu}{true}}_{t_{s}}, t_{s})

{\overset{boldu}{true}}_{t_{s} + normalΔ t false/ 2} = {boldu}_{t_{s}} + \frac{1}{2} normalΔ t {\overset{boldu}{true}}_{t_{s}}

{\overset{\overset{boldu}{true}}{true}}_{t_{s} + normalΔ t false/ 2} = {\overset{boldu}{true}}_{t_{s}} + \frac{1}{2} normalΔ t {\overset{boldu}{true}}_{t_{s}}

{\overset{\overset{boldu}{true}}{true}}_{t_{s} + normalΔ t false/ 2} = {boldM}^{- 1} boldf ((), {\overset{boldu}{true}}_{t_{s} + normalΔ t false/ 2}, {\overset{\overset{boldu}{true}}{true}}_{t_{s} + normalΔ t false/ 2}, t_{s} + \frac{normalΔ t}{2})

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…”

Section: A Brief Review Of the Various Methods Used In This Studyunclassified

“…Step‐by‐step direct time integration methods are essential tools for effective analyses of structural dynamics . Linear and nonlinear transient analyses of large and complex structural systems can be effectively done by using proper direct time integration methods.…”

Section: Introductionmentioning

confidence: 99%

“…Step-by-step direct time integration methods are essential tools for effective analyses of structural dynamics. [1][2][3] Linear and nonlinear transient analyses of large and complex structural systems can be effectively done by using proper direct time integration methods. For this reason, the development of accurate and efficient time integration methods has long been a major interest in structural dynamics.…”

Section: Introductionmentioning

confidence: 99%

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An accurate two‐stage explicit time integration scheme for structural dynamics and various dynamic problems

Kim

2019

Numerical Meth Engineering

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Summary In this article, a two‐stage explicit time integration method is presented for the numerical analysis of various dynamic problems described by second‐order ordinary differential equations in time. The newly proposed method have negligible damping and period errors in the important low‐frequency range and can introduce moderate numerical damping into the spurious high‐frequency range. The computational effort per step required in the new method is the same as the effort needed in the existing two‐stage explicit methods. The proposed method can provide fourth‐order accurate solutions for velocity independent problems and second‐order accurate solutions for velocity dependent problems, which is not possible in the existing methods. The proposed new method is expected to be an excellent tool for numerical analyses of various dynamic problems including structural dynamics.

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Global Collocation Using Macroelements

Provatidis

2019

Precursors of Isogeometric Analysis

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Explicit higher-order accurate isogeometric collocation methods for structural dynamics

Cited by 65 publications

References 34 publications

Isogeometric collocation on planar multi-patch domains

Isogeometric collocation on planar multi-patch domains

An accurate two‐stage explicit time integration scheme for structural dynamics and various dynamic problems

Global Collocation Using Macroelements

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