Efficient virtual element formulations for compressible and incompressible finite deformations

Wriggers, Peter; Reddy, B. D.; Rust, Wilhelm; Hudobivnik, Blaž

doi:10.1007/s00466-017-1405-4

Cited by 148 publications

(131 citation statements)

References 36 publications

Supporting

Mentioning

124

Contrasting

Order By: Relevance

“…For simplicity we will only deal with elements with a curved edge, since the case of polygonal elements has already been treated in [15], [25]. Moreover, always for simplicity, we fix our attention on the dofi-dofi stabilization (30). Under our assumptions on the mesh the validity of (27), that is, estimates from below, can be easily proved with the techniques used in [15], [25].…”

Section: Interpolation Estimatesmentioning

confidence: 99%

The Virtual Element Method with curved edges

2019

View full text Add to dashboard Cite

In this paper we initiate the investigation of Virtual Elements with curved faces. We consider the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain or with a curved interface. While an approximation of the domain with polygons leads, for degree of accuracy k≥2, to a sub-optimal rate of convergence, we show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence.

show abstract

Section: Interpolation Estimatesmentioning

confidence: 99%

The Virtual Element Method with curved edges

2019

View full text Add to dashboard Cite

show abstract

“…Because of the flexibility and robustness in selecting element shapes, the VEM was employed to solve various partial differential equations of physical problems, such as elliptic, hyperbolic, and parabolic problems . For engineering applications, elastostatic problems were investigated for small and finite strain conditions . Computational results demonstrated that the VEM provided a more stable solution than the finite element method (FEM) for soft materials under a large deformation, and a locking‐free behavior was observed for both nearly incompressible and incompressible materials .…”

Section: Introductionmentioning

confidence: 99%

“…[3][4][5][6] For engineering applications, elastostatic problems were investigated for small and finite strain conditions. [7][8][9][10] Computational results demonstrated that the VEM provided a more stable solution than the finite element method (FEM) for soft materials under a large deformation, and a locking-free behavior was observed for both nearly incompressible and incompressible materials. 9,10 Elastodynamic examples with nonconvex meshes were solved, and computational results illustrated that the VEM was able to consistently handle general nonconvex elements.…”

Section: Introductionmentioning

confidence: 99%

“…[7][8][9][10] Computational results demonstrated that the VEM provided a more stable solution than the finite element method (FEM) for soft materials under a large deformation, and a locking-free behavior was observed for both nearly incompressible and incompressible materials. 9,10 Elastodynamic examples with nonconvex meshes were solved, and computational results illustrated that the VEM was able to consistently handle general nonconvex elements. 11 Furthermore, the VEM was also utilized to investigate fracture problems, [12][13][14] contact mechanics, 15 and topology optimization.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical recipes for elastodynamic virtual element methods with explicit time integration

Park

Chi

Paulino

2019

Numerical Meth Engineering

View full text Add to dashboard Cite

Summary We present a general framework to solve elastodynamic problems by means of the virtual element method (VEM) with explicit time integration. In particular, the VEM is extended to analyze nearly incompressible solids using the B‐bar method. We show that, to establish a B‐bar formulation in the VEM setting, one simply needs to modify the stability term to stabilize only the deviatoric part of the stiffness matrix, which requires no additional computational effort. Convergence of the numerical solution is addressed in relation to stability, mass lumping scheme, element size, and distortion of arbitrary elements, either convex or nonconvex. For the estimation of the critical time step, two approaches are presented, ie, the maximum eigenvalue of a system of mass and stiffness matrices and an effective element length. Computational results demonstrate that small edges on convex polygonal elements do not significantly affect the critical time step, whereas convergence of the VEM solution is observed regardless of the stability term and the element shape in both two and three dimensions. This extensive investigation provides numerical recipes for elastodynamic VEMs with explicit time integration and related problems.

show abstract

“…The technique is well known for its formal elegance and flexibility, which allows using meshes composed of general polygons/polyhedra and allowing for the presence of hanging nodes and nonconforming grids . The method, originally proposed in 2012 in its displacement‐based formulation and presented for the Laplace operator in a two‐dimensional context, is rapidly developing and has been already applied to numerous physical problems as well as extended to three‐dimensional cases and mixed formulations …”

Section: Introductionmentioning

confidence: 99%

An equilibrium‐based stress recovery procedure for the VEM

Artioli

Miranda

Lovadina

et al. 2018

Numerical Meth Engineering

View full text Add to dashboard Cite

Summary Within the framework of the displacement‐based virtual element method (VEM), namely, for plane elasticity, an important topic is the development of optimal techniques for the evaluation of the stress field. In fact, in the classical VEM formulation, the same projection operator used to approximate the strain field (and then evaluate the stiffness matrix) is employed to recover, via constitutive law, the stress field. Considering a first‐order formulation, strains are locally mapped onto constant functions, and stresses are piecewise constant. However, the virtual displacements might engender more complex strain fields for polygons, which are not triangles. This leads to an undesirable loss of information with respect to the underlying virtual stress field. The recovery by compatibility in patches, originally proposed for finite element schemes, is here extended to VEM, aiming at mitigating such an effect. Stresses are recovered by minimizing the complementary energy of patches of elements over an assumed set of equilibrated stress modes. The procedure is simple, efficient, and can be readily implemented in existing codes. Numerical tests confirm the good performance of the proposed technique in terms of accuracy and indicate an increase of convergence rate with respect to the classical approach in many cases.

show abstract

Efficient virtual element formulations for compressible and incompressible finite deformations

Cited by 148 publications

References 36 publications

The Virtual Element Method with curved edges

The Virtual Element Method with curved edges

Numerical recipes for elastodynamic virtual element methods with explicit time integration

An equilibrium‐based stress recovery procedure for the VEM

Contact Info

Product

Resources

About