Isogeometric collocation methods for the Reissner–Mindlin plate problem

Kiendl, Josef; Auricchio, Ferdinando; Veiga, L. Beirão da; Lovadina, Carlo; Reali, Alessandro

doi:10.1016/j.cma.2014.09.011

Cited by 81 publications

(38 citation statements)

References 31 publications

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“…Therefore, the optimal convergence in H 1 (and in H 2 for the Kirchhoff plate) delivered by the non-symmetric Nitsche method is of primary importance, while its reduced L 2 accuracy is acceptable. We note that isogeometric collocation [67] is another recent analysis technology with optimal accuracy in the derivatives, but reduced convergence rates in the L 2 error that has been successfully applied for structural analysis [6,42,64]. The conservativity and superior accuracy for the diffusive flux at immersed boundaries, e.g., in terms of boundary stresses or moments, is a significant added benefit of the non-symmetric Nitsche method.…”

Section: Weak Pointsmentioning

confidence: 99%

The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements

Schillinger

Harari

Hsu

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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We explore the use of the non-symmetric Nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh. In contrast to symmetric Nitsche methods, it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters. We first review the available mathematical background, recollecting relevant aspects of the method from a numerical analysis viewpoint. We then compare accuracy and convergence of symmetric and non-symmetric Nitsche methods for a Laplace problem, a Kirchhoff plate, and in 3D elasticity. Our numerical experiments confirm that the non-symmetric method leads to reduced accuracy in the L2" role="presentation"> error, but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux, bending moments or stress. Based on our numerical evidence, the non-symmetric Nitsche method is a viable alternative for problems with diffusion-type operators, in particular when the accuracy of derivative quantities is of primary interest. KeywordsImmersed finite element methods, Weak boundary and coupling conditions, Non-symmetric Nitsche method AbstractWe explore the use of the non-symmetric Nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh. In contrast to symmetric Nitsche methods, it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters. We first review the available mathematical background, recollecting relevant aspects of the method from a numerical analysis viewpoint. We then compare accuracy and convergence of symmetric and non-symmetric Nitsche methods for a Laplace problem, a Kirchhoff plate, and in 3D elasticity. Our numerical experiments confirm that the non-symmetric method leads to reduced accuracy in the L 2 error, but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux, bending moments or stress. Based on our numerical evidence, the non-symmetric Nitsche method is a viable alternative for problems with diffusion-type operators, in particular when the accuracy of derivative quantities is of primary interest.

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Section: Weak Pointsmentioning

confidence: 99%

The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements

Schillinger

Harari

Hsu

et al. 2016

Computer Methods in Applied Mechanics and Engineering

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show abstract

“…An IGA-C approach for Bernoulli-Euler beams and Kirchhoff plates has been proposed in [29], where accuracy, stability, and robustness of the method are shown. A further development of the method is presented in [30], where isogeometric collocation methods for Reissner-Mindlin plates are proposed for the first time. IGA-C has been used to successfully solve a new single-variable formulation for shear-deformable beams recently introduced in [31].…”

Section: Introductionmentioning

confidence: 99%

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Marino

2016

Computer Methods in Applied Mechanics and Engineering

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Highlights• Isogeometric collocation is extended to geometrically exact beams.• Consistent linearization of the strong form of the governing equations is derived.• Incremental rotations are parametrized through Eulerian rotation vectors.• Numerical tests show efficiency and high accuracy. AbstractWe extend the isogeometric collocation method to the geometrically nonlinear beams. An exact kinematic formulation, able to represent three-dimensional displacements and rotations without any restriction in magnitude, is presented without the introduction of the moving frame concept. A displacement-based formulation is adopted. Full linearization of the strong form of the governing equations is derived consistently with the underlying geometric structure of the configuration manifold. Incremental rotations are parametrized through Eulerian rotation vectors and configuration updates are performed by means of the exponential map. Numerical tests demonstrate that the proposed combination of isogeometric collocation method with the chosen rotations parametrization results in an efficient computational scheme able to model complex problems with high accuracy.

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“…The formulations of bending moment M y , twist moment M xy and shear force Q y can be found in Kiendl et al (2015), Beirão da Veiga et al (2015). The material parameters and thickness are E = 10.92 × 10 6 , ν = 0.3 and t = 0.1 m. The convergence rate of Nitsche method in L 2 -norm is studied here with the comparison to that from the isogeometric analysis on single patch, using different level of refinement.…”

Section: Clamped Square Plate Under a Suitable Body Loadmentioning

confidence: 99%

Nitsche method for isogeometric analysis of Reissner–Mindlin plate with non-conforming multi-patches

Zhao

Wang

2015

Computer Aided Geometric Design

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Isogeometric collocation methods for the Reissner–Mindlin plate problem

Cited by 81 publications

References 31 publications

The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements

The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

Nitsche method for isogeometric analysis of Reissner–Mindlin plate with non-conforming multi-patches

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