Virtual Element and Discontinuous Galerkin Methods

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

We introduce a new class of Discontinuous Galerkin (DG) methods for solving nonlinear conservation laws on unstructured Voronoi meshes that use a local Virtual Element basis defined within each polygonal control volume. The basis functions are evaluated as an $L_2$ projection of the virtual basis which remains unknown, along the lines of the Virtual Element Method (VEM). Contrary to the VEM approach, the new basis functions lead to a nonconforming representation of the solution with discontinuous data across the element boundaries, as employed in DG discretizations. The discretization in time is carried out following the ADER (Arbitrary order DERivative Riemann problem) methodology, which yields one-step fully discrete schemes that make use of a coupled space-time representation of the numerical solution. The space-time basis functions are constructed as a tensor product of the novel local virtual basis in space and a one-dimensional Lagrange nodal basis in time. The resulting space-time stiffness matrix is stabilized by an extension of the dof–dof stabilization technique adopted in the VEM framework, hence allowing an element-local space-time Galerkin finite element predictor to be evaluated. The new VEM-DG algorithms are rigorously validated against a series of benchmarks in the context of compressible Euler and Navier–Stokes equations.

• Weak formulation of configurational forces (CFs) in the VEM is derived. • Automated mesh refinement based on CFs improves VEM approximations. • Accurate crack tip loading analyses are demonstrated. • Internal moments in virtual elements must not be disregarded in fracture analyses. • Efficient and accurate crack growth simulations with non-convex crack tip elements. The Virtual Element Method (VEM) constitutes a generalization of the Finite Element Method, allowing for spatial discretizations with arbitrary, even non-convex polygonal elements. This offers significant advantages, particularly for problems involving moving discontinuities, local stress concentrations, or multiple phases with distinctly different dimensions. The theory of configurational forces (CF) represents a very flexible basis to quantify driving forces for virtual defect displacements within a thermodynamical framework, and at the same time provides an indicator for the local mesh quality in numerical calculations. In combination, the VEM and the CF theory thus constitute an effective basis for numerical simulations of crack growth within the context of a classical approach with strong discontinuities and sharp crack tips. Challenges to cope with are attributable to the lacking knowledge of shape functions within virtual elements, the numerical integration on arbitrary polygonal elements, and the requirement of an automated generation of polygonal meshes. The weak form of the energy-momentum balance is derived from a variational approach, uniquely separating nodal CFs acting at free surfaces from those driving the defect. Furthermore, internal degrees of freedom, which are inherent to higher-order virtual elements, have to be interpreted within the context of CFs. Unphysical, so-called spurious, CFs are exploited for an improved mesh design at crack tips, eventually generating accurate results of crack tip loading and crack path simulation.

SummaryIn this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C0 Lagrange finite element methods. In the VEM, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of ‐continuous VEMs on polygonal and polyhedral meshes are described, which reveals that the variational approach adopted in the VEM affords a generalized and robust means to stabilize underintegrated finite elements. We focus on the heat conduction (Poisson) equation and present a virtual element approach for the isoparametric four‐node quadrilateral and eight‐node hexahedral elements. In addition, we show quantitative comparisons of the consistency and stabilization matrices in the VEM with those in the hourglass control method of Belytschko and coworkers. Numerical examples in two and three dimensions are presented for different stabilization parameters, which reveals that the method satisfies the patch test and delivers optimal rates of convergence in the L2 norm and the H1 seminorm for Poisson problems on quadrilateral, hexahedral, and arbitrary polygonal meshes. Copyright © 2015 John Wiley & Sons, Ltd.

The virtual element method (VEM) was developed not too long ago, starting with the paper (Beirão-da-Veiga et al. in SIAM J Numer Anal 51:794–812, 2013) related to elasticity in solid mechanics. The virtual element method allows to revisit the construction of different elements, however has so far not applied to space-time formulations for one-dimensional structural elements like strings, trusses and beams. Here we study several VEM elements suitable for space-time formulations that are build upon the Hamilton’s principle. It will be shown that these elements can be easily incorporated in classical finite element codes since they have the same number of unknowns. Furthermore, we show that the property of VEM to deal with non-conforming meshes is of special interest for holistic space time formulation: VEM formulations allow locally varying time discretizations (time increments) in a natural and efficient way.

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.

Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

Meshless method based on strong form has the advantages of high computational speed and convenient application, but it suffers low accuracy and instability. In this paper, we propose a meshless virtual element method (VEM) based on strong-form to promote accuracy and instability. Virtual element is built around each node for establishing local function, which used to calculate the derivative of shape function. The optimal selection of point-cloud with adaptive algorithm is proposed to balance the stability and accuracy of interpolation. Numerical simulation demonstrates that VEM has improvement in stability compared with other conventional strong-form method, just needing additional small calculation. and surpass finite element method (FEM) in accuracy and the ability for catching high gradient change.

The thesis deals with theoretical and applicative aspects of some innovative numerical techniques for the simulation of the flow in Discrete Fracture Networks (DFN). In particular, the recently developed Virtual Element Method (VEM) is considered. A VEM-SUPG stabilized formulation for advection-diffusion problems is defined and studied theoretically and numerically, as well as a residual a posteriori error estimate which does not include any term depending on the VEM stabilization form. Regarding DFN flow simulations, an approach based on Virtual Elements and standard domain decomposition techniques such as Mortar methods is introduced and studied, also in combination with the use of orthogonal polynomials to avoid numerical instabilities that arise when computing polynomial projections on very badly shaped elements. Finally, we consider a constrained optimization formulation of the problem of computing the flow in DFNs and we develop a residual based a posteriori error estimate that contains non standard terms related to the geometrical non-conformity of the mesh on each fracture to the intersections between them.

Abstract. Additively-manufactured metallic lattice materials are a class of architectured solids that is becoming increasingly popular due to their unique cellular structure, which can be engineered to meet specific design requirements. Understanding and modelling the damage in these innovative materials is a significant challenge that must be addressed for their effective use in aerospace applications. The Virtual Element Method (VEM) is a numerical technique recently introduced as a generalisation of the FEM capable of handling meshes comprising an assemblage of generic polytopes. This advantage in creating domain discretisation has already been used to model the behaviour of materials with complex microstructures. This work employs a numerical framework based on a nonlinear VEM formulation combined with a continuum damage model to study the fracture behaviour of two-dimensional metallic lattice material under static loading. VEM's effectiveness in modelling lattice failure behaviour is assessed through several numerical tests. The influence of micro-architecture on the material's failure behaviour and macroscopic mechanical performance is discussed.

Dealing with complex shell surfaces using the finite element method, we are often limited to simple geometric meshes such as triangles and quadrangles and have to refine the meshes to meet the calculation accuracy requirements, significantly increasing the calculation cost. The virtual element method (VEM), a new numerical method with high mesh flexibility, has been widely applied to many physical and mechanical problems. To the best of our knowledge, this method has yet to be studied for membrane shell models so far. In this paper, for the first time, we provide an enriched conforming VEM discrete scheme for the two‐dimensional three‐component (2D‐3C) generalized membrane shell (GMS) model proposed by Ciarlet et al. Furthermore, we prove the VEM discrete solution's existence and uniqueness for the GMS. Numerical experiments demonstrate that this discrete scheme is convergent and stable and can accurately represent the membrane stress state of conical, cylindrical, hyperbolical, and spherical shells clamped along a portion of their lateral faces. At the same time, we show the diversity of the grid subdivision. Thus, we develop the VEM for the GMS model successfully. In the future, we will continue to study the VEM for other shell models.

Numerical algorithm of concave polygon elements in solid mechanics: Theoretical analysis and performance evaluation of S-FEM and VEM

This study investigates the Pennes bioheat transfer model within the framework of the Virtual Element Method, presenting for the first time its application as a computational approach to analyze the thermal behavior of biological tissues. The virtual element method, conceived as a natural extension of the finite element approach, is particularly advantageous for handling any type of complex meshes within the framework of Galerkin approximation. To ensure the robustness of this discretization technique, we establish the theoretical foundations of the model, including the derivation of optimal convergence rates, error estimates in the ‐norm, and proofs of existence and uniqueness of the discrete solution. A numerical investigation of the proposed scheme is conducted by applying it to the generalized bioheat transfer equation, discretized in time using Euler's method. A comparative analysis against the classical finite element method is carried out to rigorously evaluate the effectiveness of the proposed virtual element scheme. The results underscore the computational advantages of the virtual element method, demonstrating its ability to preserve accuracy and stability over complex mesh structures while providing deeper insights into the thermo‐regulatory mechanisms of biological tissues.

Conforming virtual element approximations of the two-dimensional Stokes problem

We present a systematic description and comparison of the Finite Element Method (FEM) with the relatively new Virtual Element Method (VEM) for solving boundary value problems in linear elasticity, including primal and mixed formulations. The description highlights the common base and the essential difference between FEM and VEM: discretisation of the same primal (Galerkin) and mixed weak formulations and assembly of element-wise quantities, but different approaches to element shape functions. The mathematical formulations are complemented with detailed description of the computer implementation of all methods, including all versions of VEM, which will benefit readers willing to develop their own computational framework. Numerical solutions of several boundary value problems are also presented in order to discuss the weaker and stronger sides of the methods.