Lowest order stabilization free Virtual Element Method for the Poisson equation

Berrone, Stefano; Borio, Andrea; Marcon, Francesca

doi:10.48550/arxiv.2103.16896

Cited by 6 publications

(11 citation statements)

References 38 publications

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“…where N dof E indicates the number of degrees of freedom on element E. The stabilization term can be avoided resorting to a slightly different VEM formulation [19].…”

Section: The Virtual Element Methodsmentioning

confidence: 99%

A new quality preserving polygonal mesh refinement algorithm for Virtual Element Methods

Berrone¹,

D'Auria²

2021

Preprint

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Mesh adaptivity is a useful tool for efficient solution to partial differential equations in very complex geometries. In the present paper we discuss the use of polygonal mesh refinement in order to tackle two common issues: first, adaptively refine a provided good quality polygonal mesh preserving quality, second, improve the quality of a coarse poor quality polygonal mesh during the refinement process on very complex domains. For finite element methods and triangular meshes, convergence of a posteriori mesh refinement algorithms and optimality properties have been widely investigated, whereas convergence and optimality are still open problems for polygonal adaptive methods. In this article, we propose a new refinement method for convex cells with the aim of introducing some properties useful to tackle convergence and optimality for adaptive methods. The key issues in refining convex general polygons are: a refinement dependent only on the marked cells for refinement at each refinement step; a partial quality improvement, or, at least, a non degenerate quality of the mesh during the refinement iterations; a bound of the number of unknowns of the discrete problem with respect to the number of the cells in the mesh. Although these properties are quite common for refinement algorithms of triangular meshes, these issues are still open problems for polygonal meshes.

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“…where N dof E indicates the number of degrees of freedom on element E. The stabilization term can be avoided resorting to a slightly different VEM formulation [19].…”

Section: The Virtual Element Methodsmentioning

confidence: 99%

A new quality preserving polygonal mesh refinement algorithm for Virtual Element Methods

Berrone¹,

D'Auria²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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“…The second approach proposed is basically an extension to plane elasticity of the strategy presented in [18] for the Poisson equation, similarly to what very recently done in [19].…”

Section: Remarkmentioning

confidence: 99%

“…The key operator in the present VEM formulation is the compatibility matrix C defined in (18), projecting the symmetric part of the displacement gradient field onto the space P k−1 of polynomials of degree up to k − 1 of the approximate strain field. The computation of C requires the computation of the symmetric and invertible matrix G, defined in (16), and of A, defined in (19).…”

Section: Virtual Element Formulationmentioning

confidence: 99%

“…Matrix A 2 in (45) contains the integral of the unknown displacement shape functions N u and cannot be computed as it is. According to the proposed strategy, this term is computed by means of a suitable projection of the gradient of N u onto the gradient of known polynomial functions N 1 of order 1 [18,19].…”

Section: Self-stabilized Elements Without Additional Internal Degrees...mentioning

confidence: 99%

“…Even though these new DOFs can be easily condensed out, since they are internal to the element and not shared with neighboring elements, their presence implies a small additional computational effort. To avoid this, an alternative approach, based on a projection of the unknown virtual displacement field, based on what very recently proposed in [18,19], has also been considered, leading to a self-stabilized element, not requiring additional DOFs. It should be noted that while this work was in progress, an approach substantially identical to the one considered here, though not based on a Hu-Washizu formulation of the VEM, has been published in [20], leading to the same stabilized stiffness matrix in the case k = 1.…”

Section: Introductionmentioning

confidence: 99%

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A Hu–Washizu variational approach to self-stabilized virtual elements: 2D linear elastostatics

et al. 2023

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An original, variational formulation of the Virtual Element Method (VEM) is proposed, based on a Hu–Washizu mixed variational statement for 2D linear elastostatics. The proposed variational framework appears to be ideal for the formulation of VEs, whereby compatibility is enforced in a weak sense and the strain model can be prescribed a priori, independently of the unknown displacement model. It is shown how the ensuing freedom in the definition of the strain model can be conveniently exploited for the formulation of self-stabilized and possibly locking-free low order VEs. The superior performances of the VEs formulated within this framework has been verified by application to several numerical tests.

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Stress‐hybrid virtual element method on quadrilateral meshes for compressible and nearly‐incompressible linear elasticity

Chen,

Sukumar

2023

Numerical Meth Engineering

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In this article, we propose a robust low‐order stabilization‐free virtual element method on quadrilateral meshes for linear elasticity that is based on the stress‐hybrid principle. We refer to this approach as the stress‐hybrid virtual element method (SH‐VEM). In this method, the Hellinger–Reissner variational principle is adopted, wherein both the equilibrium equations and the strain‐displacement relations are variationally enforced. We consider small‐strain deformations of linear elastic solids in the compressible and near‐incompressible regimes over quadrilateral (convex and nonconvex) meshes. Within an element, the displacement field is approximated as a linear combination of canonical shape functions that are virtual. The stress field, similar to the stress‐hybrid finite element method of Pian and Sumihara, is represented using a linear combination of symmetric tensor polynomials. A 5‐parameter expansion of the stress field is used in each element, with stress transformation equations applied on distorted quadrilaterals. In the variational statement of the strain‐displacement relations, the divergence theorem is invoked to express the stress coefficients in terms of the nodal displacements. This results in a formulation with solely the nodal displacements as unknowns. Numerical results are presented for several benchmark problems from linear elasticity. We show that SH‐VEM is free of volumetric and shear locking, and it converges optimally in the norm and energy seminorm of the displacement field, and in the norm of the hydrostatic stress.

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Lowest order stabilization free Virtual Element Method for the Poisson equation

Cited by 6 publications

References 38 publications

A new quality preserving polygonal mesh refinement algorithm for Virtual Element Methods

A new quality preserving polygonal mesh refinement algorithm for Virtual Element Methods

A Hu–Washizu variational approach to self-stabilized virtual elements: 2D linear elastostatics

Stress‐hybrid virtual element method on quadrilateral meshes for compressible and nearly‐incompressible linear elasticity

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