Mesh generation of curvilinear polygons for the high-order virtual element method (VEM)

We present a unifying viewpoint on hybrid high-order and virtual element methods on general polytopal meshes in dimension $2$ or $3$, in terms of both formulation and analysis. We focus on a model Poisson problem. To build our bridge (i) we transcribe the (conforming) virtual element method into the hybrid high-order framework and (ii) we prove $H^m$ approximation properties for the local polynomial projector in terms of which the local virtual element discrete bilinear form is defined. This allows us to perform a unified analysis of virtual element/hybrid high-order methods, that differs from standard virtual element analyses by the fact that the approximation properties of the underlying virtual space are not explicitly used. As a complement to our unified analysis we also study interpolation in local virtual spaces, shedding light on the differences between the conforming and nonconforming cases.

Convergence analysis of virtual element method for the electric interface model on polygonal meshes with small edges

In this paper, we propose a positivity‐preserving conservative scheme based on the virtual element method (VEM) to solve convection–diffusion problems on general meshes. As an extension of finite element methods to general polygonal elements, the VEM has many advantages such as substantial mathematical foundations, simplicity in implementation. However, it is neither positivity‐preserving nor locally conservative. The purpose of this article is to develop a new scheme, which has the same accuracy as the VEM and preserves the positivity of the numerical solution and local conservation on primary grids. The first step is to calculate the cell‐vertex values by the lowest‐order VEM. Then, the nonlinear two‐point flux approximations are utilized to obtain the nonnegativity of cell‐centered values and the local conservation property. The new scheme inherits both advantages of the VEM and the nonlinear two‐point flux approximations. Numerical results show that the new scheme can reach the optimal convergence order of the virtual element theory, that is, the second‐order accuracy for the solution and the first‐order accuracy for its gradient. Moreover, the obtained cell‐centered values are nonnegative, which demonstrates the positivity‐preserving property of our new scheme.

We present a virtual element method (VEM)-based topology optimization framework using polyhedral elements, which allows for convenient handling of non-Cartesian design domains in three dimensions. We take full advantage of the VEM properties by creating a unified approach in which the VEM is employed in both the structural and the optimization phases. In the structural problem, the VEM is adopted to solve the three-dimensional elasticity equation. Compared to the finite element method, the VEM does not require numerical integration (when linear elements are used) and is less sensitive to degenerated elements (e.g., ones with skinny faces or small edges). In the optimization problem, we introduce a continuous approximation of material densities using the VEM basis functions. When compared to the standard element-wise constant approximation, the continuous approximation enriches the geometrical representation of structural topologies. Through two numerical examples with exact solutions, we verify the convergence and accuracy of both the VEM approximations of the displacement and material density fields. We also present several design examples involving non-Cartesian domains, demonstrating the main features of the proposed VEM-based topology optimization framework. The source code for a MATLAB implementation of the proposed work, named PolyTop3D, is available in the (electronic) Supplementary Material accompanying this publication.

A virtual element method (VEM) with the first-order optimal convergence order is developed for solving two-dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background unfitted mesh. A novel virtual space is introduced on a virtual triangulation of the polygonal mesh satisfying a maximum angle condition, which shares exactly the same degrees of freedom as the usual [Formula: see text]-conforming virtual space. This new virtual space serves as the key to prove that the optimal error bounds of the VEM are independent of high aspect ratio of the possible anisotropic polygonal mesh near the interface.

Machine learning based refinement strategies for polyhedral grids with applications to virtual element and polyhedral discontinuous Galerkin methods

Virtual element methods (VEM) are the latest evolution of the Mimetic Finite Difference Method and can be considered to be more close to the Finite Element approach. They combine the ductility of mimetic finite differences for dealing with rather weird element geometries with the simplicity of implementation of Finite Elements. Moreover, they make it possible to construct quite easily high-order and high-regularity approximations (and in this respect they represent a significant improvement with respect to both FE and MFD methods). In the present paper we show that, on the other hand, they can also be used to construct DG-type approximations, although numerical tests should be done to compare the behavior of DG-VEM versus DG-FEM.

The virtual element method (VEM) allows discretization of the problem domain with polygons in 2D. The polygons can have an arbitrary number of sides and can be concave or convex. These features, among others, are attractive for meshing complex geometries. VEM applied to linear elasticity problems is now well established. Nonlinear problems involving plasticity and hyperelasticity have also been explored by researchers using VEM. Clearly, techniques for extending the method to nonlinear problems are attractive. In this work, a novel first‐order consistent VEM is applied within a static co‐rotational framework. To the author's knowledge, this has not appeared before in the literature with virtual elements. The formulation allows for large displacements and large rotations in a small strain setting. For some problems avoiding the complexity of finite strains, and alternative stress measures, is warranted. Furthermore, small strain plasticity is easily incorporated. The basic method, VEM specific implementation details for co‐rotation, and representative benchmark problems are illustrated. Consequently, this research demonstrates that the co‐rotational VEM formulation successfully solves certain classes of nonlinear solid mechanics problems. The work concludes with a discussion of results for the current formulation and future research directions.

The Virtual Element Method (VEM) is an extension of the Finite Element Method (FEM) to handle polytopal meshes. After giving a short introduction of the VEM for a two dimensional Laplacian problem, we show the differences between an implementation of a VEM and a FEM code highlighting which are the main issues associated with the VEM framework. Furthermore, this paper will show one of the possible ways to face such issues: a library developed to “deal and play” with the VEM discretisation. This library deals with the VEM, since there are several partial differential equations in two/three dimensions coming from both academic and engineering problems. Then, one can “play” with the VEM, since has been designed so that one can plug-in new features such as new polytopes quadrature rules, new solvers and new virtual element spaces in a smart way.

We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [A. H. Schatz, An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comput. 28 (1974) 959–962] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214]), we use here, in a systematic way, the [Formula: see text]-projection operators as designed in [B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013) 376–391]. In particular, the present method does not reduce to the original Virtual Element Method of [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] for simpler problems as the classical Laplace operator (apart from the lowest-order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] to the case of variable coefficients produces, in general, sub-optimal results.

Stabilization techniques in combination with reduced integration are often employed in numerical methods to address problems of locking while reducing computational costs. In finite elements, concepts such as enhanced assumed strain (EAS) methods are usually employed in combination with stabilization techniques to tackle certain locking phenomena, such as volumetric or shear locking. This combination has been shown to improve the performance of low‐order finite element formulations while addressing the downsides of reduced integration. In recent years, the virtual element method (VEM) has emerged as a numerical approach capable of handling arbitrary polygonal and polyhedral meshes, offering flexibility in mesh generation and refinement. This flexibility makes VEM particularly suitable for a wide range of applications involving distorted, non‐convex, and irregular meshes. Similar to reduced integration, VEM formulations require stabilization to avoid rank deficiencies and ensure numerical consistency. Different stabilization techniques have been employed to overcome this issue. In this contribution, a stabilization technique based on reduced integration is proposed for VEM. The formulation is validated using two numerical examples.

This work presents a stabilization-free virtual element method (VEM) for phase field fracture. The distinctive feature of the virtual element method is its ability to utilize elements of general shape. However, the existence of additional stabilization term in the traditional virtual element method has some drawbacks when solving complex phase field fracture models. Different from the conventional virtual element method, the approach employed in this work eliminates the need for additional stabilization terms, making it more suitable for the phase field modeling of fracture. In this work, the anisotropic phase field fracture model is considered. In order to improve the calculation efficiency, the non-matching mesh ability of VEM and adaptive technique are employed. Since the virtual element method is automatically applicable to elements with general shape, it is easy to handle an arbitrary number of nodes and thus also hanging nodes resulting from the non-matching meshes used to adapt the meshes. Several representative benchmarks show the accuracy and efficiency of the proposed method.

Since its introduction, the virtual element method (VEM) was shown to be able to deal with a large variety of polygons, while achieving good convergence rates. The regularity assumptions proposed in the VEM literature to guarantee the convergence on a theoretical basis are therefore quite general. They have been deduced in analogy to the similar conditions developed in the finite element method (FEM) analysis. In this work, we experimentally show that the VEM still converges, with almost optimal rates and low errors in the L2, H1 and $L^{\infty }$ norms, even if we significantly break the regularity assumptions that are used in the literature. These results suggest that the regularity assumptions proposed so far might be overestimated. We also exhibit examples on which the VEM sub-optimally converges or diverges. Finally, we introduce a mesh quality indicator that experimentally correlates the entity of the violation of the regularity assumptions and the performance of the VEM solution, thus predicting if a mesh is potentially critical for VEM.

We study virtual element methods (VEMs) for solving the obstacle problem, which is a representative elliptic variational inequality of the first kind. VEMs can be regarded as a generalization of standard finite element methods with the addition of some suitable nonpolynomial functions, and the degrees of freedom are carefully chosen so that the stiffness matrix can be computed without actually computing the nonpolynomial functions. With this special design, VEMS can easily deal with complicated element geometries. In this paper we establish a priori error estimates of VEMs for the obstacle problem. We prove that the lowest-order ($k=1$) VEM achieves the optimal convergence order, and suboptimal order is obtained for the VEM with $k=2$. Two numerical examples are reported to show that VEM can work on very general polygonal elements, and the convergence orders in the $H^1$ norm agree well with the theoretical prediction.

In this article, we propose new gradient recovery schemes for the BEM‐based Finite Element Method (BEM‐based FEM) and Virtual Element Method (VEM). Supporting general polytopal meshes, the BEM‐based FEM and VEM are highly flexible and efficient tools for the numerical solution of boundary value problems in two and three dimensions. We construct the recovered gradient from the gradient of the finite element approximation via local averaging. For the BEM‐based FEM, we show that, under certain requirements on the mesh, superconvergence of the recovered gradient is achieved, which means that it converges to the true gradient at a higher rate than the untreated gradient. Moreover, we propose a simple and very efficient a posteriori error estimator, which measures the difference between the unprocessed and recovered gradient as an error indicator. Since the BEM‐based FEM and VEM are specifically suited for adaptive refinement, the resulting adaptive algorithms perform very well in numerical examples.