A low-order locking-free virtual element for linear elasticity problems

Tang, Xuanhe; Liu, Zhibin; Zhang, Baiju; Feng, Meiqiang

doi:10.1016/j.camwa.2020.04.032

Cited by 15 publications

(12 citation statements)

References 34 publications

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“…Most of the materials in this section are stated without proof. Related proofs can be founded in our recent work [56]. We just apply the results to Biot's consolidation model in poroelasticity.…”

Section: A Low Order Schemementioning

confidence: 98%

“…) are the degrees of freedom given bỹ︀ D V 1-̃︀ D V 3. Similar to higher order case, we established the following lemma in [56].…”

Section: A Low Order Schemementioning

confidence: 99%

“…Therefore, we develop another low-order virtual element method for the Biot's consolidation model. Such an element is inspired by the works of Fortin [35], and Bernardi and Raugel [20], and has been analyzed in our previous paper [56] for linear elasticity problem. The main idea is to introduce extra degrees of freedom related to the normal component of v on each edge, so that the related Fortin operator can be easily constructed, which is an important operator in deriving robust error estimates.…”

Section: Introductionmentioning

confidence: 99%

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On the locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity

et al. 2021

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We propose and analyze two locking-free three-field virtual element methods for Biot's consolidation model in poroelasticity. One is a high-order scheme, and the other is a low-order scheme. For time discretization, we use the backward Euler scheme. The proposed methods are well-posed, and optimal error estimates of all the unknowns are obtained for fully discrete solutions. The generic constants in the estimates are uniformly bounded as the Lamé coefficient λ tends to infinity, and as the constrained specific storage coefficient is arbitrarily small. Therefore the methods are free of both Poisson locking and pressure oscillations. Numerical results illustrate the good performance of the methods and confirm our theoretical predictions.

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Section: A Low Order Schemementioning

confidence: 98%

“…) are the degrees of freedom given bỹ︀ D V 1-̃︀ D V 3. Similar to higher order case, we established the following lemma in [56].…”

Section: A Low Order Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity

et al. 2021

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“…No additional degrees of freedom are introduced (displacement-based formulation). In the existing VEM approaches, the volumetric locking is alleviated by the B-bar formulation [1,2], mixed formulation [3], enhanced strain formulation [4], hybrid formulation [5], nonconforming formulations [6][7][8], or addition of degrees of freedom related to the normal components of the displacement field on the element's edges to satisfy the inf-sup condition [9].…”

Section: Introductionmentioning

confidence: 99%

A node-based uniform strain virtual element method for compressible and nearly incompressible elasticity

Ortiz-Bernardin¹,

Silva-Valenzuela²,

Salinas³

et al. 2022

Preprint

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We propose a displacement-based approach to solve problems in compressible and nearly incompressible plane elasticity that combines a nodal integration technique and the virtual element method (VEM), wherein the strain is averaged around nodes from the strain of surrounding virtual elements. For the strain averaging procedure, a nodal averaging operator is constructed using a generalization to virtual elements of the node-based uniform strain approach for finite elements.We refer to these new elements as node-based uniform strain virtual elements (NVEM). A salient feature of the NVEM is that the stresses and strains become nodal variables just like displacements, which can be exploited in nonlinear simulations thereby providing room for further development of this novel approach. Through several benchmark problems in plane elasticity, we demonstrate that the NVEM is accurate and optimally convergent, and devoid of volumetric locking in the nearly incompressible limit.

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“…The order requirement is to ensure the so-called discrete inf-sup condition and the optimal convergence. For the lowest-order case, a Bernardi-Raugel type VEM can be found in [29], where the uniform convergence is achieved by adding extra degrees of freedom so that the inf-sup condition can be satisfied easily. Nonconforming VEMs for the linear elasticity problems are first introduced in [31] for the pure displacement/traction formulation in two or three dimensions.…”

Section: Introductionmentioning

confidence: 99%

A lowest-order locking-free nonconforming virtual element method based on the reduced integration technique for linear elasticity problems

Yu¹

2021

Preprint

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We develop a lowest-order nonconforming virtual element method for planar linear elasticity, which can be viewed as an extension of the idea in Falk (1991) to the virtual element method (VEM), with the family of polygonal meshes satisfying a very general geometric assumption. The method is shown to be uniformly convergent for the nearly incompressible case with optimal rates of convergence. The crucial step is to establish the discrete Korn's inequality, yielding the coercivity of the discrete bilinear form. We also provide a unified locking-free scheme both for the conforming and nonconforming VEMs in the lowest order case. Numerical results validate the feasibility and effectiveness of the proposed numerical algorithms.

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A low-order locking-free virtual element for linear elasticity problems

Cited by 15 publications

References 34 publications

On the locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity

On the locking-free three-field virtual element methods for Biot’s consolidation model in poroelasticity

A node-based uniform strain virtual element method for compressible and nearly incompressible elasticity

A lowest-order locking-free nonconforming virtual element method based on the reduced integration technique for linear elasticity problems

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