A lowest-order free-stabilization Virtual Element Method for the Laplacian eigenvalue problem

Meng, Jian; Wang, Xue; Bu, Linlin; Mei, Liquan

doi:10.1016/j.cam.2021.114013

Cited by 10 publications

(2 citation statements)

References 31 publications

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“…In conventional VEM, the stabilization term is needed because the strain order is too low to accurately describe the energy within the element. By increasing the order of the strain polynomial, the stabilization-free virtual element method (SFVEM) [33] has been recently proposed and has been successfully used in different fields [34][35][36]. However, these applications are usually limited to two-dimensional linear problems.…”

Section: Introductionmentioning

confidence: 99%

Stabilization-free virtual element method for 3D hyperelastic problems

Xu,

Peng,

Wriggers

2024

Comput Mech

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In this work, we present a first-order stabilization-free virtual element method (SFVEM) for three-dimensional hyperelastic problems. Different from the conventional virtual element method, which necessitates additional stabilization terms in the bilinear formulation, the method developed in this work operates without the need for any stabilization. Consequently, it proves highly suitable for the computation of nonlinear problems. The stabilization-free virtual element method has been applied in two-dimensional hyperelasticity and three-dimensional elasticity problems. In this work, the format will be applied to three-dimensional hyperelasticity problems for the first time. Similar to the techniques used in the two-dimensional stabilization-free virtual element method, the new virtual element space is modified to allow the computation of the higher-order $$L_2$$ L 2 projection of the gradient. This paper reviews the calculation process of the traditional $$\mathcal {H}_1$$ H 1 projection operator; and describes in detail how to calculate the high-order $$L_2$$ L 2 projection operator for three-dimensional problems. Based on this high-order $$L_2$$ L 2 projection operator, this paper extends the method to more complex three-dimensional nonlinear problems. Some benchmark problems illustrate the capability of the stabilization-free VEM for three-dimensional hyperelastic problems.

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Section: Introductionmentioning

confidence: 99%

Stabilization-free virtual element method for 3D hyperelastic problems

Xu,

Peng,

Wriggers

2024

Comput Mech

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“…The well‐posedness was proven in References 45,48 and the discrete problem can be solved without a stabilizing bilinear form. According to these developments, the stabilization‐free virtual element method has been extended to the Laplacian eigenvalue problem, 49 linear plane elasticity in References 47,50, and 3D elasticity in Reference 51. All the above contributions are considered linear problems.…”

Section: Introductionmentioning

confidence: 99%

Stabilization‐free virtual element method for 2D elastoplastic problems

Xu,

Wang,

Wriggers

2024

Numerical Meth Engineering

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In this paper, a novel first‐ and second‐order stabilization‐free virtual element method is proposed for two‐dimensional elastoplastic problems. In contrast to traditional virtual element methods, the improved method does not require any stabilization, making the solution of nonlinear problems more reliable. The main idea is to modify the virtual element space to allow the computation of the higher‐order projection operator, ensuring that the strain and stress represent the element energy accurately. Considering the flexibility of the stabilization‐free virtual element method, the elastoplastic mechanical problems can be solved by radial return methods known from the traditional finite element framework. plasticity with hardening is considered for modeling the nonlinear response. Several numerical examples are provided to illustrate the capability and accuracy of the stabilization‐free virtual element method.

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When rational functions meet virtual elements: the lightning virtual element method

Trezzi,

Zerbinati

2024

Calcolo

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We propose a lightning Virtual Element Method that eliminates the stabilisation term by actually computing the virtual component of the local VEM basis functions using a lightning approximation. In particular, the lightning VEM approximates the virtual part of the basis functions using rational functions with poles clustered exponentially close to the corners of each element of the polygonal tessellation. This results in two great advantages. First, the mathematical analysis of a priori error estimates is much easier and essentially identical to the one for any other non-conforming Galerkin discretisation. Second, the fact that the lightning VEM truly computes the basis functions allows the user to access the point-wise value of the numerical solution without needing any reconstruction techniques. The cost of the local construction of the VEM basis is the implementation price that one has to pay for the advantages of the lightning VEM method, but the embarrassingly parallelizable nature of this operation will ultimately result in a cost-efficient scheme almost comparable to standard VEM and FEM.

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A lowest-order free-stabilization Virtual Element Method for the Laplacian eigenvalue problem

Cited by 10 publications

References 31 publications

Stabilization-free virtual element method for 3D hyperelastic problems

Stabilization-free virtual element method for 3D hyperelastic problems

Stabilization‐free virtual element method for 2D elastoplastic problems

When rational functions meet virtual elements: the lightning virtual element method

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