Galerkin formulations of isogeometric shell analysis: Alleviating locking with Greville quadratures and higher-order elements

Zou, Z.; Hughes, Thomas J.R.; Scott, Michael A.; Sauer, Roger A.; Savitha, Eshwar J.

doi:10.1016/j.cma.2021.113757

Cited by 50 publications

(22 citation statements)

References 69 publications

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“…where W n is the numerical solution at time level t n . The coefficient matrices in the updating formula (32) for the C2 scheme are given by…”

Section: C2 and Upc2 Schemesmentioning

confidence: 99%

“…Since its introduction, IGA has gained increasing attention in engineering and applied sciences communities for simulating challenging PDEs in domains with complex geometries. Recently, IGA and its variations (e.g., isogeometric collocation or Galerkin) have been well developed and widely used for problems with higher order derivatives such as various plate and shell models; see for example [23,[28][29][30][31][32][33] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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Stable finite difference methods for Kirchhoff-Love plates on overlapping grids

Tang

2021

Preprint

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In this work, we propose and develop efficient and accurate numerical methods for solving the Kirchhoff-Love plate model in domains with complex geometries. The algorithms proposed here employ curvilinear finitedifference methods for spatial discretization of the governing PDEs on general composite overlapping grids. The coupling of different components of the composite overlapping grid is through numerical interpolations. However, interpolations introduce perturbation to the finite-difference discretization, which causes numerical instability for time-stepping schemes used to advance the resulted semi-discrete system. To address the instability, motivated by an upwind scheme for solving the wave equation, we propose to add a fourthorder hyper-dissipation to the spatially discretized system to stabilize its time integration; this additional dissipation term captures the essential upwinding effect of the original upwind scheme. The investigation of strategies for incorporating the upwind dissipation term into several time-stepping schemes (both explicit and implicit) leads to the development of four novel algorithms. For each algorithm, formulas for determining a stable time step and a sufficient dissipation coefficient on curvilinear grids are derived by performing a local Fourier analysis. Quadratic eigenvalue problems for a simplified model plate in 1D domain are considered to reveal the weak instability due to the presence of interpolating equations in the spatial discretization. This model problem is further investigated for the stabilization effects of the proposed algorithms. Carefully designed numerical experiments are carried out to validate the accuracy and stability of the proposed algorithms, followed by two benchmark problems to demonstrate the capability and efficiency of our approach for solving realistic applications. Results that concern the performance of the proposed algorithms are also presented.

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“…where W n is the numerical solution at time level t n . The coefficient matrices in the updating formula (32) for the C2 scheme are given by…”

Section: C2 and Upc2 Schemesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stable finite difference methods for Kirchhoff-Love plates on overlapping grids

Tang

2021

Preprint

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“…The scientific attention on the developments of discretization methods for Reissner-Mindlin plates is still very animated [2][3][4], with a recent focus on polygonal elements [5][6][7][8]. Indeed, polygonal meshes may be particularly appealing for a number of applications, e.g.…”

Section: Introductionmentioning

confidence: 99%

“…The main assumptions with regards to the calculation of the consistent and stabilization terms of the stiffness matrix are discussed in Sect. 4. Numerical results are presented in Sect.…”

Section: Introductionmentioning

confidence: 99%

First-order VEM for Reissner–Mindlin plates

et al. 2021

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In this paper, a first-order virtual element method for Reissner–Mindlin plates is presented. A standard displacement-based variational formulation is employed, assuming transverse displacement and rotations as independent variables. In the framework of the first-order virtual element, a piecewise linear approximation is assumed for both displacement and rotations on the boundary of the element. The consistent term of the stiffness matrix is determined assuming uncoupled polynomial approximations for the generalized strains, with different polynomial degrees for bending and shear parts. In order to mitigate shear locking in the thin-plate limit while keeping the element formulation as simple as possible, a selective scheme for the stabilization term of the stiffness matrix is introduced, to indirectly enrich the approximation of the transverse displacement with respect to that of the rotations. Element performance is tested on various numerical examples involving both thin and thick plates and different polygonal meshes.

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“…Since then, rotation-free IGA shells have been steadily advanced, for example to PHT-splines (Nguyen-Thanh et al, 2011), anisotropic materials (Nagy et al, 2013), damage (Deng et al, 2015), biological materials (Tepole et al, 2015), fracture (Kiendl et al, 2016), liquid shells , elasto-plasticity (Ambati et al, 2018), phase separation (Zimmermann et al, 2019), thermo-mechanical coupling (Vu-Bac et al, 2019), multi-patch constraints (e.g. see the recent review in Paul et al Paul et al (2020)), and reduced quadrature (Zou et al, 2021). Balobanov et al Balobanov et al (2019) present a general strain gradient theory and its corresponding isogeometric finite element formulation for Kirchhoff-Love shells.…”

Section: Introductionmentioning

confidence: 99%

A general isogeometric finite element formulation for rotation-free shells with embedded fibers and in-plane bending

Duong¹,

Itskov²,

Sauer³

2021

Preprint

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This paper presents a general, nonlinear finite element formulation for rotationfree shells with embedded fibers that captures anisotropy in stretching, shearing, twisting and bending -both in-plane and out-of-plane. These capabilities allow for the simulation of large sheets of heterogeneous and fibrous materials either with or without matrix, such as textiles, composites, and pantographic structures. The work is a computational extension of our earlier theoretical work (Duong et al., 2021) that extends existing Kirchhoff-Love shell theory to incorporate the in-plane bending resistance of initially straight or curved fibers. The formulation requires only displacement degrees-of-freedom to capture all mentioned modes of deformation. To this end, isogeometric shape functions are used in order to satisfy the required C 1 -continuity for bending across element boundaries. The proposed formulation can admit a wide range of material models, such as surface hyperelasticity that does not require any explicit thickness integration. To deal with possible material instability due to fiber compression, a stabilization scheme is added. Several benchmark examples are used to demonstrate the robustness and accuracy of the proposed computational formulation.

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Galerkin formulations of isogeometric shell analysis: Alleviating locking with Greville quadratures and higher-order elements

Cited by 50 publications

References 69 publications

Stable finite difference methods for Kirchhoff-Love plates on overlapping grids

Stable finite difference methods for Kirchhoff-Love plates on overlapping grids

First-order VEM for Reissner–Mindlin plates

A general isogeometric finite element formulation for rotation-free shells with embedded fibers and in-plane bending

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