Virtual Element Method Research Articles

A Robust and Mass Conservative Virtual Element Method for Linear Three-field Poroelasticity

A Robust and Mass Conservative Virtual Element Method for Linear Three-field Poroelasticity

On the Coupling of the Curved Virtual Element Method with the One-Equation Boundary Element Method for 2D Exterior Helmholtz Problems

We consider the Helmholtz equation defined in unbounded domains, external to 2D bounded ones, endowed with a Dirichlet condition on the boundary and the Sommerfeld radiation condition at infinity. To solve it, we reduce the infinite region, in which the solution is defined, to a bounded computational one, delimited by a curved smooth artificial boundary and we impose on this latter a non reflecting condition of boundary integral type. Then, we apply the curved virtual element method in the finite computational domain, combined with the one-equation boundary element method on the artificial boundary. We present the theoretical analysis of the proposed approach and we provide an optimal convergence error estimate in the energy norm. The numerical tests confirm the theoretical results and show the effectiveness of the new proposed approach.

CVEM-BEM Coupling with Decoupled Orders for 2D Exterior Poisson Problems

For the solution of 2D exterior Dirichlet Poisson problems, we propose the coupling of a Curved Virtual Element Method (CVEM) with a Boundary Element Method (BEM), by using decoupled approximation orders. We provide optimal convergence error estimates, in the energy and in the weaker textit{L}^text {2}-norm, in which the CVEM and BEM contributions to the error are separated. This allows for taking advantage of the high order flexibility of the CVEM to retrieve an accurate discrete solution by using a low order BEM. The numerical results confirm the a priori estimates and show the effectiveness of the proposed approach.

Sharper Error Estimates for Virtual Elements and a Bubble-Enriched Version

In the present contribution we develop a sharper error analysis for the Virtual Element Method, applied to a model elliptic problem, that separates the element boundary and element interior contributions to the error. As a consequence we are able to propose a variant of the scheme that allows one to take advantage of polygons with many edges (such as those composing Voronoi meshes or generated by agglomeration procedures) in order to yield a more accurate discrete solution. The theoretical results are supported by numerical experiments.

Virtual element method for the Sobolev equations

The virtual element method for the Sobolev equations is proposed in this paper, where the semi‐discrete scheme and the fully discrete scheme are both discussed. With the help of the energy projection operator defined by the discrete bilinear form, the corresponding optimal error estimates in the norm and semi‐norm for both the semi‐discrete solution and the fully discrete solution are deduced. Finally, three numerical examples are carried out to verify the theoretical results.

The nonconforming virtual element method for semilinear elliptic problems

The nonconforming virtual element method (VEM) for the semilinear elliptic problem is developed in this paper. The nonlinear right-hand side is approximated by using the L2 projection. The optimal convergence of the nonconforming VEM in the broken H1 norm is proved. Finally, some numerical experiments are carried out to support the theoretical results.

BDDC Preconditioners for Divergence Free Virtual Element Discretizations of the Stokes Equations

The virtual element method (VEM) is a new family of numerical methods for the approximation of partial differential equations, where the geometry of the polytopal mesh elements can be very general. The aim of this article is to extend the balancing domain decomposition by constraints preconditioner to the solution of the saddle-point linear system arising from a VEM discretization of the two-dimensional Stokes equations. Under suitable hypotesis on the choice of the primal unknowns, the preconditioned linear system results symmetric and positive definite, thus the preconditioned conjugate gradient method can be used for its solution. We provide a theoretical convergence analysis estimating the condition number of the preconditioned linear system. Several numerical experiments validate the theoretical estimates, showing the scalability and quasi-optimality of the method proposed. Moreover, the solver exhibits a robust behavior with respect to the shape of the polygonal mesh elements. We also show that a faster convergence could be achieved with an easy to implement coarse space, slightly larger than the minimal one covered by the theory.

A numerical implementation for the high-order 2D virtual element method in MATLAB

We present a numerical implementation for the Virtual Element Method that incorporates high order spaces. We include all the required computations in order to assemble the stiffness and mass matrices, and right hand side. Convergence of the method is verified for different polygonal partitions. An specific mesh-free application that allows to approximate harmonic functions is discussed, which is based on high-order projections onto polynomial spaces of degree k; this approach only requires to solve a k(k − 1)/2 linear system, reducing significantly the number of operations compared to usual finite or virtual element methods, and can be modified for different virtual spaces and elliptic partial differential equations.

VEMPIC

The comprehensive visual modeling of fluid motion has historically been a challenging task, due in no small part to the difficulties inherent in geometries that are non-manifold, open, or thin. Modern geometric cut-cell mesh generators have been shown to produce, both robustly and quickly, workable volumetric elements in the presence of these problematic geometries, and the resulting volumetric representation would seem to offer an ideal infrastructure with which to perform fluid simulations. However, cut-cell mesh elements are general polyhedra that often contain holes and are non-convex; it is therefore difficult to construct the explicit function spaces required to employ standard functional discretizations, such as the Finite Element Method. The Virtual Element Method (VEM) has recently emerged as a functional discretization that successfully operates with complex polyhedral elements through a weak formulation of its function spaces. We present a novel cut-cell fluid simulation framework that exactly represents boundary geometry during the simulation. Our approach enables, for the first time, detailed fluid simulation with "in-the-wild" obstacles, including ones that contain non-manifold parts, self-intersections, and extremely thin features. Our key technical contribution is the generalization of the Particle-In-Cell fluid simulation methodology to arbitrary polyhedra using VEM. Coupled with a robust cut-cell generation scheme, this produces a fluid simulation algorithm that can operate on previously infeasible geometries without requiring any additional mesh modification or repair.

A Virtual Element-Based Postural Optimization Method for Improved Ergonomics During Human-Robot Collaboration

Human-robot collaboration is becoming increasingly popular in the manufacturing industry, opening the door to a large range of applications by combining the complementary skills of the human worker and the robot. Collaborative robots are also a solution to decrease the operator workload and indirectly reduce the risk of occupational injuries such as musculoskeletal disorders (MSDs). The latter represents one of the major causes of absenteeism at work. Thanks to the development of human tracking devices, it is possible to monitor the operator, analyze the postures, and assess the associated MSD risk. In this paper, we present a novel ergonomics optimization framework that performs postural optimization based on the virtual element method. A feedback interface is developed whereby the user is informed about non-ergonomic postures and an improved body pose is proposed. The workpiece position controller module acts on the cobot end-effector and indirectly on the co-manipulated part in such a way that the operator’s posture is improved. The framework was validated by a user study performed on a human-robot collaboration task whereby the subject polishes a part hold by the robot. The conducted study of the user’s perception and REBA scores showed promising results. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper is motivated by the problem of non-ergonomic posture of workers in hybrid workcells. The proposed approach makes use of virtual elements (springs and dampers) to build a mechanical model of the human body posture and perform postural optimization. The obtained body joint angles are fed into two modules of the framework. First, a graphical interface displays the current pose of the user and proposes to him the improved posture. Second, a controller adapts the pose of the workpiece hold by the collaborative robot. This is realized by computing a displacement vector between the wrist current and optimized positions. The use of such a framework was demonstrated on a collaborative polishing task whereby the robot adjusts the position of the workpiece. After a user study test with 10 participants, joint data were collected and the REBA scores of different subtasks were measured and compared. The results from these preliminary experiments showed that the proposed approach improves the human body postures and offers a promising solution to enhance ergonomics by the robot assistance in case of robotic workcells. The conducted survey also shows an overall positive subject’s perception of the system.

Virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media

In this paper, we propose a conforming virtual element method for the Helmholtz transmission eigenvalue problem of anisotropic media. By using [Formula: see text]-coercivity theory, the spectral approximation theory of compact operator and the projection and interpolation error estimates, we prove the spectral convergence of the discrete scheme and the optimal a priori error estimates for the discrete eigenvalues and eigenfunctions. The virtual element method has great flexibility in handling polygonal meshes, which motivates us to construct a fully computable a posteriori error estimator for the virtual element method. Then the upper bound of the approximation error is derived from the residual equation and the inf-sup condition. In turn, the related lower bound is established by using the bubble function strategy. Finally, we provide numerical examples to verify the theoretical results, including the optimal convergence of the virtual element scheme on uniformly refined meshes and the efficiency of the estimator on adaptively refined meshes.

Virtual elements on agglomerated finite elements to increase the critical time step in elastodynamic simulations

AbstractIn this article, we use the first‐order virtual element method (VEM) to investigate the effect of shape quality of polyhedra in the estimation of the critical time step for explicit three‐dimensional elastodynamic finite element (FE) simulations. Low‐quality FEs are common when meshing realistic complex components, and while tetrahedral meshing technology is generally robust, meshing algorithms cannot guarantee high‐quality meshes for arbitrary geometries or for non‐water‐tight computer‐aided design models. For reliable simulations on such meshes, we consider FE meshes with tetrahedral and prismatic elements that have badly shaped elements—tetrahedra with dihedral angles close to and , and slender prisms with triangular faces that have short edges—and agglomerate such “bad” elements with neighboring elements to form a larger polyhedral virtual element. On each element, the element‐eigenvalue inequality is used to estimate the critical time step. For a suite of illustrative FE meshes with being a mesh‐coordinate parameter that leads to poor mesh quality, we show that adopting VEM on the agglomerated polyhedra yield critical time steps that are insensitive as . The significant reduction in solution time on meshes with agglomerated virtual elements vis‐à‐vis tetrahedral meshes is demonstrated through explicit dynamics simulations on a tapered beam.

Computed tomography (CT) Image-based analysis of concrete microstructure using virtual element method

The analysis of actual concrete microstructure is challenging because the microstructure has various length scales associated with small voids and various sizes of aggregate particles. To efficiently and effectively analyze complex microstructure like concrete, the image-based virtual element method (VEM) is presented. Concrete microstructure is reconstructed using the complementarity of X-ray and neutron computed tomography (CT) images with the morphological filtering. Based on the reconstructed microstructure, a VEM mesh is generated through integrating microstructural and homogeneous meshes. A microstructural mesh is directly obtained from the reconstructed microstructure, while a homogeneous mesh is employed to control the level of accuracy in the computational simulation. Then, the mesh refinement in the image-based VEM is easily performed through increasing the number of elements in a homogeneous mesh. Computational results converge to the reference solution of strain error under mesh refinement while capturing local stress and strain concentration.

A priori and a posteriori error analysis of the lowest-order NCVEM for second-order linear indefinite elliptic problems

The nonconforming virtual element method (NCVEM) for the approximation of the weak solution to a general linear second-order non-selfadjoint indefinite elliptic PDE in a polygonal domain Omega is analyzed under reduced elliptic regularity. The main tool in the a priori error analysis is the connection between the nonconforming virtual element space and the Sobolev space H^1_0(Omega ) by a right-inverse J of the interpolation operator I_h. The stability of the discrete solution allows for the proof of existence of a unique discrete solution, of a discrete inf-sup estimate and, consequently, for optimal error estimates in the H^1 and L^2 norms. The explicit residual-based a posteriori error estimate for the NCVEM is reliable and efficient up to the oscillation terms. Numerical experiments on different types of polygonal meshes illustrate the robustness of an error estimator and support the improved convergence rate of an adaptive mesh-refinement in comparison to the uniform mesh-refinement.

Some Estimates for Virtual Element Methods in Three Dimensions

Abstract Some estimates on virtual element methods (VEMs), including inverse inequalities, norm equivalence, and interpolation error estimates, are developed for star-shaped polyhedral meshes whose faces admit virtual regular and quasi-uniform triangulations. This mesh regularity covers the usual ones used for theoretical analysis of VEMs, and the proofs are carried out in a straightforward way.

High-order interpolatory Serendipity Virtual Element Method for semilinear parabolic problems

We present an efficient method for the numerical approximation of a general class of two dimensional semilinear parabolic problems on polygonal meshes. The proposed approach takes advantage of the properties of the serendipity version of the Virtual Element Method, which not only reduces the number of degrees of freedom compared to the original Virtual Element Method, but also allows for the introduction of an approximation of the nonlinear term that is computable from the degrees of freedom of the discrete solution with a low computational cost, thus significantly improving the efficiency of the method. An error analysis for the semi-discrete formulation is carried out, and an optimal estimate for the error in the L_2-norm is obtained. The accuracy and efficiency of the proposed method when combined with a second order Strang operator splitting time discretization is illustrated in our numerical experiments, with approximations up to order 6.

A free‐cutting mesh strategy for optimal shape synthesis in magnetics

The authors propose an innovative technique for dealing with optimal shape design problems that exploits the flexibility of the virtual element method in generating meshes composed of general polygonal and polyhedral elements. Virtual element method and finite element method can coexist on the same discretized domain; therefore, the possibility of dealing with hanging nodes and gluing sub-domain meshes is ensured. Accordingly, the shape synthesis of a magnetic pole is considered as the case study. It is shown that the proposed technique is effective in handling the shape variations dictated by an algorithm of evolutionary optimisation.

Conforming virtual element approximations of the two-dimensional Stokes problem

The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present two different conforming virtual element formulations for the numerical approximation of the Stokes problem that work on polygonal meshes. The velocity vector field is approximated in the virtual element spaces of the two formulations, while the pressure variable is approximated through discontinuous polynomials. Both formulations are inf-sup stable and convergent with optimal convergence rates in the L2 and energy norm. We assess the effectiveness of these numerical approximations by investigating their behavior on a representative benchmark problem. The observed convergence rates are in accordance with the theoretical expectations and a weak form of the zero-divergence constraint is satisfied at the machine precision level.

A 2-dimentional contact analysis using second-order virtual element method

A 2-dimentional contact analysis using second-order virtual element method

Bend 3d mixed virtual element method for Darcy problems

In this study, we propose a virtual element scheme to solve the Darcy problem in three physical dimensions. The main novelty is that curved elements are naturally handled without any degradation of the solution accuracy. Indeed, in presence of curved boundaries, or internal interfaces, the geometrical error introduced by planar approximations may dominate the convergence rate limiting the benefit of high-order approximations. We consider the Darcy problem in its mixed form to directly obtain accurate and mass conservative fluxes without any post-processing. An important step to derive the proposed scheme is the integration over curved polyhedrons, here presented and discussed. Finally, we show the theoretical analysis of the scheme as well as several numerical examples to support our findings.