Bricks for the mixed high-order virtual element method: Projectors and differential operators

Robust and scalable adaptive BDDC preconditioners for virtual element discretizations of elliptic partial differential equations in mixed form

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

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.

• 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.

In this work we introduce and analyze a mixed virtual element method for the two-dimensional nonlinear Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions. For the continuous formulation we consider a dual-mixed approach in which the main unknowns are given by the gradient of the velocity and the pseudostress, whereas the velocity itself and the pressure are computed via simple postprocessing formulae. In addition, because of analysis reasons we add a redundant term arising from the constitutive equation relating the pseudostress and the velocity, so that the well-posedness of the resulting augmented formulation is established by using known results from nonlinear functional analysis. Then, we introduce the main features of the mixed virtual element method, which employs an explicit piecewise polynomial subspace and a virtual element subspace for approximating the aforementioned main unknowns, respectively. In turn, the associated computable discrete nonlinear operator is defined in terms of the $$\mathbb {L}^2$$ -orthogonal projector onto a suitable space of polynomials, which allows the explicit integration of the terms involving deviatoric tensors that appear in the original setting. Next, we show the well-posedness of the discrete scheme and derive the associated a priori error estimates for the virtual element solution as well as for the fully computable projection of it. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken $$\mathbb {H}(\mathbf {div})$$ -norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.

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.

We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

A comparison of differential quadrature methods for the solution of partial differential equations

The material in this thesis is the result of a year's experience in the solution of problems on the Caltech Electric Analog Computer. Although much work has been done elsewhere, the solution of partial differential equations is a relatively new field for the Caltech Computer. It is natural that such an undertaking should initiate points of view and techniques that differ from those of other investigators., This thesis contains the development of certain ideas that have been useful to the author in the solution of problems on the computer. In Part I, finite difference methods are treated with reference to problems with one space variable. Techniques are developed for the representation of differential operators by means of electrical networks and the question of unequal lumping is discussed. The solutions of the fourth order differential equations of a beam are treated in Part II. Such practical considerations as the effects of parasitic impedances and cell size are investigated. Solutions are presented for the normal modes of a cantilever beam, the transient vibration of a cantilever beam, and the coupled modes of vibration of an airplane wing. Problems involving the scalar Laplacian operator are treated in Part III. A general asymmetric network is developed that is useful for problems with irregular boundaries and for problems where it is desired to have variable cell size. These techniques are illustrated with respect to a cavity resonator problem and an electromagnetic radiation problem. Elastic plate problems are treated in Part IV. The analogy for the elastic plate is an extension of the beam analogy to two dimensions. Here the difficult problems are those relating to the representation of boundary conditions, particularly of boundary conditions along an irregular edge. Some conclusions regarding the construction of a network analyzer designed specifically for the solution of partial differential equations are given in Part V. The chief conclusion is that such a computer must contain a much larger number of electrical elements than are at present available in the Caltech Computer.

Recovery-based error estimators for the VEM and BEM-based FEM

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.

Several research papers experimentally validated the derived theoretical solution of the partial differential equation (PDE) of Transient Land Drainage (TLD) using laboratory sand tanks, Hele-Shaw models, resistance networks, and field data. In this research, the main objective is to use an Arduino microcontroller linked with the LabVIEW software to derive experimentally-by using an RC analog circuit connected to the Arduino microcontroller - a solution to Kirchhoff’s partial differential equation describing the unsteady electrical flow through the RC circuit. The RC circuit electrically simulates a non-dimensional virtual aquifer for which an experimental Fourier sine series solution in terms of the analog components is to be derived and used for comparison with the theoretical Fourier sine series solution of the transient partial differential equation of land drainage. The derived Fourier sine series solution of the partial differential equation governing the unsteady electrical flow through the RC analog is written in a non-dimensional form to enable its general application to the case study of this research. Isochrones of the water table drawdown based on those derived from the analog solution are similar in trend and behavior to those of the theoretical solution, indicating that both Fourier solutions are physically and mathematically identical.

A fully-discrete virtual element method for the nonstationary Boussinesq equations in stream-function form

The problem of checking existence of infinitely differentiable solutions of linear partial differential equations with zero boundary conditions is considered. The coefficients of the equations are assumed to be polynomials over Z in independent variables. It is proved that this problem is algorithmically undecidable. This result extends results of our earlier studies of analytic solutions. The proof relies on the result obtained by Denef and Lipshitz concerning the relationship between a certain subset of nonhomogeneous differential equations of the considered form (but without boundary conditions) and Diophantine equations.

A variationally consistent numerical approach based on the Virtual Element Method (VEM) is presented for the analysis of 2D elastoplasticity problems. The mixed Hu-Washizu functional of elasticity is extended to incorporate the energy contributions specific to the finite-step elastoplastic problem. It is demonstrated how the governing equations of the discretized elastoplastic problem - including the loading-unloading conditions - emerge naturally as the stationarity conditions of the VEM-discretized functional. Spurious hourglass modes are prevented by formulating a self-stabilized version of Virtual Elements (VEs) that exploits the possibility offered by the mixed approach to define strain and displacement approximations of the same order. The insensitivity of VEs to element distortion and the possibility to use polygonal elements with any shape and number of edges is tested with the analysis of several benchmarks from the literature. It is shown how accurate solutions can be obtained also in the case of non-convex quadrilateral or pentagonal elements. Additionally, the role of internal moment degrees of freedom in preventing elastoplastic locking at the plastic failure limit is elucidated.