Stress-hybrid virtual element method on six-noded triangular meshes for compressible and nearly-incompressible linear elasticity
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| Format: | Preprint |
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2024
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| author | Chen, Alvin Bishop, Joseph E. Sukumar, N. |
| author_facet | Chen, Alvin Bishop, Joseph E. Sukumar, N. |
| contents | In this paper, we present a first-order Stress-Hybrid Virtual Element Method (SH-VEM) on six-noded triangular meshes for linear plane elasticity. We adopt the Hellinger--Reissner variational principle to construct a weak equilibrium condition and a stress based projection operator. On applying the divergence theorem to the weak strain-displacement relations, the stress projection operator is expressed in terms of the nodal displacements, which leads to a displacement-based formulation. This stress-hybrid approach assumes a globally continuous displacement field while the stress field is discontinuous across each element. The stress field is initially represented by divergence-free tensor polynomials based on Airy stress functions. However, for flexibility in choosing basis functions, we also present a formulation that uses a penalty term to enforce the element equilibrium conditions. This method is referred to as the Penalty Stress-Hybrid Virtual Element Method (PSH-VEM). Numerical results are presented for PSH-VEM and SH-VEM, and we compare their convergence to the composite triangle FEM and B-bar VEM on benchmark problems in linear elasticity. The SH-VEM converges optimally in the $L^2$ norm of the displacement, energy seminorm, and the $L^2$ norm of hydrostatic stress. Furthermore, the results reveal that PSH-VEM converges in most cases at a faster rate than the expected optimal rate, but it requires the selection of a suitably chosen penalty parameter. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06280 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stress-hybrid virtual element method on six-noded triangular meshes for compressible and nearly-incompressible linear elasticity Chen, Alvin Bishop, Joseph E. Sukumar, N. Numerical Analysis In this paper, we present a first-order Stress-Hybrid Virtual Element Method (SH-VEM) on six-noded triangular meshes for linear plane elasticity. We adopt the Hellinger--Reissner variational principle to construct a weak equilibrium condition and a stress based projection operator. On applying the divergence theorem to the weak strain-displacement relations, the stress projection operator is expressed in terms of the nodal displacements, which leads to a displacement-based formulation. This stress-hybrid approach assumes a globally continuous displacement field while the stress field is discontinuous across each element. The stress field is initially represented by divergence-free tensor polynomials based on Airy stress functions. However, for flexibility in choosing basis functions, we also present a formulation that uses a penalty term to enforce the element equilibrium conditions. This method is referred to as the Penalty Stress-Hybrid Virtual Element Method (PSH-VEM). Numerical results are presented for PSH-VEM and SH-VEM, and we compare their convergence to the composite triangle FEM and B-bar VEM on benchmark problems in linear elasticity. The SH-VEM converges optimally in the $L^2$ norm of the displacement, energy seminorm, and the $L^2$ norm of hydrostatic stress. Furthermore, the results reveal that PSH-VEM converges in most cases at a faster rate than the expected optimal rate, but it requires the selection of a suitably chosen penalty parameter. |
| title | Stress-hybrid virtual element method on six-noded triangular meshes for compressible and nearly-incompressible linear elasticity |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2401.06280 |