Virtual Element Method Research Articles

On a space-time implementation of the wave equation using virtual elements

AbstractThe 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.

A Higher Order Nonconforming Virtual Element Method for the Cahn–Hilliard Equation

AbstractIn this paper we develop a fully nonconforming virtual element method of arbitrary approximation order for the two dimensional Cahn–Hilliard equation. We carry out the error analysis for the semidiscrete (continuous-in-time) scheme and verify the theoretical convergence result via numerical experiments. We present a fully discrete scheme which uses a convex splitting Runge–Kutta method to discretize in the temporal variable alongside the virtual element spatial discretization.

Mesh optimization for the virtual element method: How small can an agglomerated mesh become?

We present an optimization procedure for generic polygonal or polyhedral meshes, tailored for the Virtual Element Method (VEM). Once the local quality of the mesh elements is analyzed through a quality indicator specific to the VEM, groups of elements are agglomerated to optimize the global mesh quality. A user-set parameter regulates the percentage of mesh elements, and consequently of faces, edges, and vertices, to be removed. This significantly reduces the total number of degrees of freedom associated with a discrete problem defined over the mesh with the VEM, particularly for high-order formulations. We show how the VEM convergence rate is preserved in the optimized meshes, and the approximation errors are comparable with those obtained with the original ones. We observe that the optimization has a regularization effect over low-quality meshes, removing the most pathological elements. In such cases, these “badly-shaped” elements yield a system matrix with very large condition number, which may cause the VEM to diverge, while the optimized meshes lead to convergence. We conclude by showing how the optimization of a real CAD model can be used effectively in the simulation of a time-dependent problem.

Particle Virtual Element Method (PVEM): an agglomeration technique for mesh optimization in explicit Lagrangian free-surface fluid modelling

Explicit solvers are commonly used for simulating fast dynamic and highly nonlinear engineering problems. However, these solvers are only conditionally stable, requiring very small time-step increments determined by the characteristic length of the smallest, and often most distorted, element in the mesh. In the Lagrangian description of fluid motion, the computational mesh quickly deteriorates. To circumvent this problem, the Particle Finite Element Method (PFEM) creates a new mesh (e.g., through a Delaunay tessellation, based on node positions) when the current one becomes overly distorted. A fast and efficient remeshing technique is therefore of pivotal importance for an effective PFEM implementation in explicit dynamics. Unfortunately, the 3D Delaunay tessellation does not guarantee well-shaped elements, often generating zero- or near-zero-volume elements (slivers), which drastically reduce the stable time-step size. Available mesh optimization techniques have limited applicability due to their high computational cost when runtime remeshing is required. An innovative possibility to overcome this problem is the use of the Virtual Element Method (VEM), a variant of the finite element method that can make use of polyhedral elements of arbitrary shapes and number of nodes. This paper presents the formulation of a 3D first-order Particle Virtual Element Method (PVEM) for weakly compressible flows. Starting from a tetrahedral mesh, poorly shaped elements, such as slivers, are agglomerated to form polyhedral Virtual Elements (VEs) with a controlled characteristic length. This approach ensures full control over the minimum time-step size in explicit dynamics simulations, maintaining stability throughout the entire analysis.

A unified local projection-based stabilized virtual element method for the coupled Stokes-Darcy problem

A unified local projection-based stabilized virtual element method for the coupled Stokes-Darcy problem

A simple, effective and high-precision boundary meshfree method for solving 2D anisotropic heat conduction problems with complex boundaries

A simple, effective and high-precision boundary meshfree method called virtual boundary meshfree Galerkin method (VBMGM) is used to tackle 2D anisotropic heat conduction problems with complex boundaries. Temperature and heat flux are expressed by virtual boundary element method. The virtual source function is constructed through the utilization of radial basis function interpolation. Calculation model diagram and discrete model diagram of real boundaries, and schematic diagram of VBMGM are demonstrated. Using Galekin method and considering boundary conditions, the integral equation and the discrete formula of VBMGM are given in detail. The benefits of the Galerkin, meshfree, and boundary element methods are all presented in VBMGM. Seven numerical examples of general anisotropic heat conduction problems (including three numerical examples with complex boundaries and four numerical examples with mixed boundary conditions) are computed and contrasted with precise solutions and different numerical methods. The computation time of each example is given. The number of degrees of freedom used in many examples is half or less than that of the numerical method being compared. The suggested method has been demonstrated to be effective and high-precision for solving the 2D anisotropic heat conduction problems with complex boundaries.

Arbitrary Order Virtual Element Methods for High‐Order Phase‐Field Modeling of Dynamic Fracture

Arbitrary Order Virtual Element Methods for High‐Order Phase‐Field Modeling of Dynamic Fracture

Unconditional error analysis of weighted implicit-explicit virtual element method for nonlinear neutral delay-reaction–diffusion equation

In this paper, we develop a virtual element method in space for the nonlinear neutral delay-reaction–diffusion equation, while a weighted implicit-explicit scheme is utilized in time. The nonlinear term is adopted by using the Newton linearized method. The calculation efficiency is improved using an implicit scheme to analyze linear terms and an explicit scheme to deal with nonlinear terms. Then, the time–space error splitting technique and G-stability are creatively combined to rigorously analyze the unconditionally optimal convergence results of the numerical scheme without any restrictions on the mesh ratio. Finally, numerical examples on a set of polygonal meshes are provided to confirm the theoretical results. In particular, when the weighted parameter is taken θ=12 (or θ=1), the method degenerates into the Crank–Nicolson (or second-order backward differential formula (BDF2)) scheme.

Two mixed virtual element formulations for parabolic integro-differential equations with nonsmooth initial data

This article presents and examines two distinctive approaches to the mixed virtual element method (VEM) applied to parabolic integro-differential equations (PIDEs) with non-smooth initial data. In the first part of the paper, we introduce and analyze a mixed virtual element scheme for PIDE that eliminates the need for the resolvent operator. Through the introduction of a novel projection involving a memory term, coupled with the application of energy arguments and the repeated use of an integral operator, this study establishes optimal L2-error estimates for the two unknowns p and σ. Furthermore, optimal error estimates are derived for the standard mixed formulation with a resolvent kernel. The paper offers a comprehensive analysis of the VEM, encompassing both formulations.

Unconditional error analysis of the linearized transformed [formula omitted] virtual element method for nonlinear coupled time-fractional Schrödinger equations

This paper constructs a linearized transformed L1 virtual element method for the generalized nonlinear coupled time-fractional Schrödinger equations. The solutions to such problems typically exhibit singular behavior at the beginning. To avoid this pitfall, we introduce an identical s-fractional differential system derived from a smoothing transformation of variables t=s1/α, 0<α<1. By utilizing the discrete complementary convolution kernels, we prove the boundedness and error estimates of the solution of time-discrete system. Moreover, the unconditionally optimal error bounds of the proposed fully discrete scheme are derived in L2-norm without restriction on the grid ratio. Finally, numerical tests on a set of polygonal meshes are presented to verify the theoretical results.

VEMcomp: a Virtual Elements MATLAB package for bulk-surface PDEs in 2D and 3D

AbstractWe present a Virtual Element MATLAB solver for elliptic and parabolic, linear and semilinear Partial Differential Equations (PDEs) in two and three space dimensions, which is coined VEMcomp. Such PDEs are widely applicable to describing problems in material sciences, engineering, cellular and developmental biology, among many other applications. The library covers linear and nonlinear models posed on different simple and complex geometries, involving time-dependent bulk, surface, and bulk-surface PDEs. The solver employs the Virtual Element Method (VEM) of lowest polynomial order $${k=1}$$ k = 1 on general polygonal and polyhedral meshes, including the Finite Element Method (FEM) of order $${k=1}$$ k = 1 as a special case when the considered mesh is simplicial. VEMcomp has three main purposes. First, VEMcomp generates polygonal and polyhedral meshes optimized for fast matrix assembly. Triangular and tetrahedral meshes are encompassed as special cases. For surface PDEs, VEMcomp is compatible with the well-known Matlab package DistMesh for mesh generation. Second, given a mesh for the considered geometry, possibly generated with an external package, VEMcomp computes all the matrices (mass and stiffness) required by the VEM or FEM method. Third, for multiple classes of stationary and time-dependent bulk, surface and bulk-surface PDEs, VEMcomp solves the considered PDE problem with the VEM or FEM in space and IMEX Euler in time, through a user-friendly interface. As an optional post-processing, VEMcomp comes with its own functions for plotting the numerical solutions and evaluating the error when possible. An extensive set of examples illustrates the usage of the library.

Step-by-step solving virtual element schemes based on scalar auxiliary variable with relaxation for Allen–Cahn-type gradient flows

In this paper, we consider integrating the scalar auxiliary variable time discretization with the virtual element method spatial discretization to obtain energy-stable schemes for Allen–Cahn-type gradient flow problems. In order to optimize CPU time during calculations, we propose two step-by-step solving SAV algorithms by introducing a novel auxiliary variable to replace the original one. Then, linear, decoupled, and unconditionally energy-stable numerical schemes are constructed. However, due to truncation errors, the auxiliary variable is not equivalent to the continuous case in the original definition. Therefore, we propose a novel relaxation technique to preserve the original energy dissipation rule. It not only retains all the advantages of the above algorithms but also improves accuracy and consistency. Finally, a series of numerical experiments are conducted to demonstrate the effectiveness of our method.

A virtual element scheme for the time-fractional parabolic PDEs over distorted polygonal meshes

We extend the virtual element method to the two-dimensional time-fractional parabolic PDE, characterized by a fractional derivative of order α∈(0,1) in time. To illustrate the working of this fractional virtual element scheme, a numerical investigation of the following time-fractional problem over distorted polygonal meshes is conducted. cDtαu(z,t)−Δu=f(z,t)inz∈Ω,t∈(0,T],where Ω is a spatial domain, α is fractional order, and t is time variable. Our methodology is based on the fundamental technical component, fractional version of the Grunwald–Letnikov approximation. We prove the method’s well-posedness, that is the approximate solution’s existence and uniqueness. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the L2-norm and H1-norm over distorted mesh configuration is validated by numerical results, underlining the practical effectiveness of the proposed method.

A linearized BDF2 virtual element method for the unsteady Brinkman–Forchheimer equations with variable time step

This paper is devoted to a lowest order virtual element method for the unsteady incompressible Brinkman–Forchheimer equations on polygonal mesh, while a linearized variable-time-step second order backward differentiation formula is adopted in time. We utilize discrete orthogonal convolution and discrete complementary convolution kernels to obtain error estimates for the solutions of time-discrete system. By virtue of the temporal–spatial error splitting approach, and under the current mildest adjacent time-step ratio condition: 0<rk≔τk/τk−1≤rmax≈4.8645, we have established the L∞ boundedness of the fully discrete velocity solution and its L2-norm optimal error estimate. Numerical experiments are conducted on various polygonal meshes, validating the accuracy of the theoretical analysis.

Novel coupled hydromechanical model considering multiple flow mechanisms for simulating underground hydrogen storage in depleted low-permeability gas reservoir

Depleted low-permeability gas reservoirs are the primary storage media for underground hydrogen storage (UHS). An accurate assessment of the geomechanics and complex flow mechanisms in a depleted low-permeability gas reservoir is crucial for predicting the injection and production capabilities of UHS facilities. This paper proposes a novel hydromechanical multicomponent model that couples non-Darcy flow, relative permeability hysteresis (RPH), and geomechanics to evaluate the hydrogen storage capacity of UHS facilities in a depleted low-permeability gas reservoir. The coupled model was solved using a fully coupled strategy, employing the finite volume method to solve non-Darcy flow equations and the virtual element method to solve geomechanical problems. The results indicate that the high-velocity non-Darcy (hvnD) and geomechanical effects reduce the working gas volume by 12.82% and 10.56%, respectively. Additionally, the low-velocity non-Darcy (lvnD) effect causes the third stress/strain distribution to exhibit a “focusing” phenomenon, resulting in a 13.19% reduction in the working gas volume. This paper also describes a case where residual gas is captured by mechanisms such as “water lock” and “capillary capture” through the RPH effect. Considering the RPH effect, the gas-water transition zone expanded by 30%, and the peak volume of gas-containing pores in the water zone increased by 1.48 times. Ignoring the RPH effect led to a 19.73% overestimation of the utilization rate of the gas-containing pore space. Hence, this study achieves safe and successful seasonal hydrogen storage and provides theoretical support for designing multicycle operational plans.

Orthogonal polynomial bases in the mixed virtual element method

AbstractThe use of orthonormal polynomial bases has been found to be efficient in preventing ill‐conditioning of the system matrix in the primal formulation of Virtual Element Methods (VEM) for high values of polynomial degree and in presence of badly‐shaped polygons. However, we show that using the natural extension of a orthogonal polynomial basis built for the primal formulation is not sufficient to cure ill‐conditioning in the mixed case. Thus, in the present work, we introduce an orthogonal vector‐polynomial basis which is built ad hoc for being used in the mixed formulation of VEM and which leads to very high‐quality solution in each tested case. Furthermore, a numerical experiment related to simulations in Discrete Fracture Networks (DFN), which are often characterised by very badly‐shaped elements, is proposed to validate our procedures.

Numerical analysis of the stochastic Stefan problem

The gradient discretisation method (GDM) – a generic framework encompassing many numerical methods – is studied for a general stochastic Stefan problem with multiplicative noise. The convergence of the numerical solutions is proved by compactness method using discrete functional analysis tools, Skorokhod theorem and the martingale representation theorem. The generic convergence results established in the GDM framework are applicable to a range of different numerical methods, including for example mass-lumped finite elements, but also some finite volume methods, mimetic methods, lowest-order virtual element methods, etc. Theoretical results are complemented by numerical tests based on two methods that fit in GDM framework.

Virtual element method for solving a viscoelastic contact problem with long memory

In this paper, we use the virtual element method to solve a history-dependent hemivariational inequality arising in contact problems. The contact problem concerns the deformation of a viscoelastic body with long memory, subjected to a contact condition with non-monotone normal compliance and unilateral constraints. A fully discrete scheme based on the trapezoidal rule for the discretization of the time integration and the virtual element method for the spatial discretization are analyzed. We provide a unified priori error analysis for both internal and external approximations. For the linear virtual element method, we obtain the optimal order error estimate. Finally, three numerical examples are reported, providing numerical evidence of the theoretically predicted optimal convergence orders.

Meshing theory and contact analysis of double enveloping hourglass worm drive with planar generatrix

The meshing and limit equations of worm drive usually have strong nonlinearities such as multiple solutions, solution nonexistence and equation singularity. Meanwhile, the tooth surfaces of worm drive are in nonconforming line contact, which often requires mesh refinement of contact region for the loaded contact analysis. These two challenges cause that the modeling of worm drive heavily relies on manual adjustment and the loaded contact analysis of worm drive is still rare especially when edge contact and assembly error are concerned. Focusing on the double enveloping hourglass worm (DEHW) drive with planar generatrix, this work presents procedures to solve meshing and limit equations with global convergence. The instantaneous contact line, meshing limit line, curvature interference limit line and tooth surface grid discretization are adaptively generated, without manual trial or adjustment. On the basis of adaptive mesh refinement of tooth surface, the mortar virtual element method is adopted for loaded contact analysis of DEHW drive with edge contact and center distance error. Under sliding friction, the discrete system of governing equalities and inequalities is solved by semi-smooth Newton algorithm after constraint condensation. Numerical results for meshing theory and loaded contact analysis of DEHW drive with planar generatrix are discussed.

The stabilized nonconforming virtual element method for the Darcy–Stokes problem

A stabilized nonconforming virtual element method is designed in order to solve the Darcy–Stokes problem, which preserves a divergence-free approximation to the velocity. The same degrees of freedom as the usual Crouzeix–Raviart-type virtual element is used, but a different virtual element space is obtained by modifying the conforming Stokes virtual element with the H1-projection operator. The proposed stabilized scheme contains two jump penalty terms over edges. One is the penalty for jumps of velocity approximation and the other one is the penalty for jumps of its normal component. We analyze this method’s well-posedness and prove its uniform convergence in a discrete energy norm. Finally, we verify the validity of this stabilized scheme by some numerical experiments.