Polygonal methods for PDEs: theory and applications

Online Conference, May 17-19, 2021


Polygonal methods for PDEs: theory and applications

Online Conference, May 17-19, 2021


The development and analysis of numerical methods for the approximation of the solution to Partial Differential Equations (PDEs) on polygonal and polyhedral meshes have undergone anexplosive interest in recent years among the scientific community. Indeed, polygonal and polyhedral meshes offer a very flexible framework to handle, for instance, hanging nodes, different cellshapes within the same mesh and non-matching interfaces, resulting thus in an improved geometric flexibility to correctly represent complicated geometries, interfaces and heterogeneous media.




Monday, May 17   Tuesday, May 18   Wednesday, May 19
09:00-09:45 Brezzi   09:00-09:45 Wriggers  
09:45-10:15 Georgoulis   09:45-10:15 Verani   09:30-10:15 Houston
10:15-10:45 Park   10:15-10:45 Russo   10:15-10:45 Lovadina
BREAK(10:45-11:15)   BREAK(10:45-11:15)   BREAK(10:45-11:15)
11:15-11:45 Artioli   11:15-11:45 Formaggia   11:15-11:45 Pingaro
11:45-12:05 Prada   11:45-12:05 Mascotto   11:45-12:05 D’Auria
12:05-12:25 Busetto   12:05-12:25 Aldakheel   12:05-12:25 Vacca
12:25-12:45 Dassi   12:25-12:45 Hudobivnik   12:25-12:45 Scacchi
12:45-13:05 Mazzieri   12:45-13:05 Böhm   12:45-13:05 Visinoni
LUNCH (13:05-14:30)   LUNCH (13:05-14:30)   LUNCH (13:05-14:30)
14:30-15:00 Di Pietro   14:30-15:15 Ern   14:30-15:00 Manzini
15:00-15:30 Mora   15:15-15:35 Patruno   15:00-15:20 Desiderio
15:30-15:50 Gardini   15:35-15:55 Marcon   15:20-15:40 Botti
BREAK (15:50-16:00 )   15:55-16:15 Manuzzi      
16:00-16:45 Sukumar   16:15-16:35 Frittelli      

Manuzzi - Tuesday, May 18 - 15:55-16:15

Refinement of polygonal grids using Convolutional Neural Networks with applications to polygonal Discontinuous Galerkin and Virtual Element methods

We propose new strategies to handle polygonal grids refinement based on Convolutional Neural Networks (CNNs). We show that CNNs can be successfully employed to identify correctly the “shape” of a polygonal element so as to design suitable refinement criteria to be possibly employed within adaptive refinement strategies. We propose two refinement strategies that exploit the use of CNNs to classify elements’ shape, at a low computational cost. We test the proposed idea considering two families of finite element methods that support arbitrarily shaped polygonal elements, namely Polygonal Discontinuous Galerkin (PolyDG) methods and Virtual Element Methods (VEMs). We demonstrate that the proposed algorithms can greatly improve the performance of the discretization schemes both in terms of accuracy and quality of the underlying grids. Moreover, since the training phase is performed off-line and is problem independent the overall computational costs are kept low.

Verani - Tuesday, May 18 - 09:00-09:45

Anisotropic a posteriori error estimate for the virtual element method

We present an anisotropic a posteriori error estimate for the adaptive conforming virtual element approximation of a paradigmatic two-dimensional elliptic problem. Hinging upon the reliability of the a posteriori error estimate we design an adaptive polygonal anisotropic algorithm based on the classical paradigm: SOLVE-ESTIMATE-MARK-REFINE. Numerical tests assess the superiority of the proposed algorithm in comparison with standard polygonal isotropic mesh refinement schemes.

Mazzieri - Monday, May 17 - 12:45-13:05

A polytopal discontinuous Galerkin method for wave propagation problem in coupled poroelastic-acoustic media

We introduce and analyze a finite element discontinuous Galerkin method on polytopal meshes for the numerical discretization of acoustic waves propagation through poroelastic materials. Wave propagation is modeled by the acoustics equations in the acoustic domain and the low-frequency Biot’s equations in the poroelastic one. The coupling is realized by means of (physically consistent) transmission conditions, imposed on the interface between the domains, modeling different pore configurations. For the space discretization, we introduce and analyze a high-order discontinuous Galerkin method on polytopal meshes, which is then coupled with Newmark-β time integration schemes. Stability analysis and error estimates for the energy norm are presented. We show a wide set of numerical results and examples of physical interest to investigate the capability of the proposed method in different scenarios.

Mascotto - Tuesday, May 18 - 11:45-12:05

Enriching virtul elements with singular functions

Solutions to elliptic PDEs on polygonal and polyhedral domains are singular at the corners of the domain. The singular behaviour is known a priori: the solution can be split as a sum of a smooth term, plus series of singular terms that belong to the kernel of the differential operator appearing in the PDE. The standard virtual element method (VEM) is a generalization of the finite element method (FEM) to polygonal/polyhedral meshes and is based on ``polynomial-like'' approximation spaces consisting of solutions to local problems mimicking the target PDE. In this talk, we present an enrichment strategy of standard virtual element (VE) spaces by means of singular functions. Notably, we modify the definition of the local spaces tuning the boundary conditions of local problems. By doing this, local VE spaces contain as many as desired singular functions appearing at the corners of the domain. This procedure results in an effective and minor modification of already existing VE codes, as well as in a natural extension of existing theoretical results.

Park - Monday, May 17 - 10:15-10:45

B-bar Virtual Element Methods for both nearly incompressible and compressible materials

For the analysis of both nearly incompressible and compressible materials, a B-bar formulation of the virtual element method (VEM) is presented. In B-bar VEM, the material stiffness is decomposed into dilatational and deviatoric parts; only the deviatoric part is utilized for stabilization of the element stiffness matrix, which successfully eliminates locking behavior for nearly incompressible material. The accuracy and convergence of the B-bar VEM are demonstrated for both 2D and 3D examples with various element shapes, i.e., convex and non-convex.

Patruno - Tuesday, May 18 - 15:15-15:35

An enhanced VEM formulation with enriched strain representation for plane elasticity

An enhanced Virtual Element Method (VEM) formulation is proposed for plane elasticity, based on the enrichment of the strain representation within the element without altering the displacement representation at its boundary. The aim is to fully exploit polygonal elements characterized by a high number of edges which, in the classical VEM formulation, are characterized by numerous displacement degrees of freedom despite a relatively poor representation of the internal strain field. In addition, an energy norm in the projection operator definition is introduced and the approach is framed within a generalization of the classic VEM formulation. Depending on the edge numerosity, various strain field representations are proposed and tested numerically. Results show a considerable increase of accuracy with respect to standard VEM while keeping the optimal convergence rate and, in many cases, avoiding the need for stabilization.

Prada - Monday, May 17 - 11:45-12:05

Stabilization of the nonconforming virtual element method

We address the issue of designing robust stabilization terms for the nonconforming virtual element method. To this end, we transfer the problem of defining the stabilizing bilinear form from the elemental nonconforming virtual element space, whose functions are not known in closed form, to the dual space spanned by the known functionals providing the degrees of freedom. By this approach, we manage to construct different bilinear forms yielding optimal or quasi-optimal stability bounds and error estimates, under weaker assumptions on the tessellation than the ones usually considered in this framework. In particular, we prove optimality under geometrical assumptions allowing a mesh to have a very large number of arbitrarily small edges per element. Finally, we numerically assess the performance of the VEM for several different stabilizations fitting with our new framework on a set of representative test cases.

Lovadina - Wednesday, May 19 - 09:45-10:15

Mixed Virtual Element Methods for elasticity problems

The Virtual Element Method (VEM) is an emerging methodology for the approximation of partial differential equation problems. The initial motivation of VEM is the need to construct an accurate conforming Galerkin scheme with the capability to deal with highly general polygonal/polyhedral meshes, including hanging vertexes and non-convex shapes. We present Virtual Element schemes, both 2D and 3D, based on the Hellinger-Reissner variational principle. As it is well-known, imposing both the symmetry of the stress tensor and the continuity of the tractions at the inter-element is typically a great source of troubles in the framework of classical Galerkin schemes. We exploit the great flexibility of VEM to present alternative methods, which provide symmetric stresses, continuous tractions and are reasonably cheap with respect to the obtained accuracy. VEMs reach this goal by abandoning the local polynomial approximation concept, a feature originally used to design conforming Galerkin schemes on general polytopal meshes. In this talk, we detail the ideas which led to the design of our schemes, we state the theoretical results, and we present several numerical tests to assess the actual computational performance of the our approach. Finally, we discuss some possible future extensions.

Dassi - Monday, May 17 - 12:25-12:45

Bend Mixed Virtual Element Method

We propose an extension of the mixed Virtual Element Method (VEM) for bi-dimensional polygonal meshes characterized by curved boundaries or interfaces starting from the idea described in the pioneering work by Beirao et al. 2019. This extension is a key aspect in the resolution of partial differential equations. Indeed, a ``bad'' approximation of the curved domain introduces a geometrical error on the numerical solution that can corrupt the expected convergence order of the numerical scheme itself. We show some numerical experiments that underline this aspect and show that the proposed approach does overcome this issue and results in a robust method with respect to element distortion. This work represents the first step of the wider project called ``Bend Vem 3D'' founded by Gruppo Nazionale per il Calcolo Scientifico whose main objective is to analyze and solve mixed problem characterized by curved boundaries in 3d.

Marcon - Tuesday, May 18 - 15:35-15:55

Lowest order stabilization free Virtual Element Method

In this work, we introduce and analyse the first order Enlarged Enhancement Virtual Element Method (E^2VEM) for the 2D Poisson problem. The method has the interesting property of allowing the definition of bilinear forms that do not require a stabilization term. It is based on the use of higher order polynomial projections in the discrete bilinear form with respect to the standard one [1] and on a modification of the VEM space to allow the computation of such projections, maintaining the same set of degrees of freedom. We provide a proof of well-posedness and optimal order a priori error estimates. Numerical tests on convex and non-convex polygonal meshes confirm the theoretical convergence rates. References [1] L. Beirão da Veiga, F. Brezzi, L. D. Marini, and A. Russo. Virtual element methods for general second order elliptic problems on polygonal meshes. Mathematical Models and Methods in Applied Sciences, 26(04):729–750, 2015. [2] Stefano Berrone, Andrea Borio, and Francesca Marcon. Lowest order stabilization free virtual element method for the poisson equation. 2021, arXiv:2103.16896.

Georgoulis - Monday, May 17 - 09:45-10:15

A posteriori error estimates for discontinuous Galerkin methods on polygonal and polyhedral meshes

We present a new a posteriori error analysis for hp-version interior penalty disconinuous Galerkin (dG) methods for linear elliptic problems. The a posteriori error bounds are proven for meshes consisting of extremely general polygonal/polyhedral element shapes. In particular, arbitrary number of very small faces are allowed on each polygonal/polyhedral element, as long as certain mild shape regularity assumptions are satisfied. The case of simplicial and/or box-type elements is included in the analysis as a special case. As such the present analysis generalizes the known a posteriori error analysis results for hp-dG methods to admit arbitrary number of irregular hanging nodes per element. The proof hinges on a new recovery strategy in conjunction with a generalized Helmholtz decomposition formula. The resulting a posteriori error bound involves jumps on the tangential derivatives along elemental faces. Local lower bounds are also proven, indicating the optimality of the proposed approach. A series of numerical experiments is also presented, highlighting the good performance of the a posteriori error bounds. The presented results are based on joint work with Andrea Cangiani (SISSA, Trieste) and Zhaonan Dong (INRIA, Paris).

Vacca - Wednesday, May 19 - 12:05-12:25

Vorticity-stabilized Virtual Elements for the Oseen Equation

In the present talk we extend the divergence-free VEM of [Beir~ao da Veiga, Lovadina, Vacca, 2018] to the Oseen problem, including a suitable stabilization procedure that guarantees robustness in the convection dominated case without disrupting the divergence-free property. The stabilization includes local SUPG-like terms of the vorticity equation, internal jump terms for the velocity gradients, and an additional VEM stabilization. The theoretical convergence results underline the robustness of the scheme in different regimes, including the convection dominated case. Furthermore, as in the non-stabilized case, the influence of the pressure on the velocity error is moderate, as it appears only through higher order terms.

Scacchi - Wednesday, May 19 - 12:25-12:45

Parallel block preconditioners for three-dimensional virtual element discretizations of elliptic equations in mixed form

In recent years, several research groups worldwide have focused on the development of numerical methods for the approximation of partial differential equations (PDEs) on polygonal or polyhedral grids. Among the different methodologies proposed, the Virtual Element Method (VEM) represents a generalization of the Finite Element Method that can easily handle general polytopal meshes. In this talk, we first present a new VEM discretization for the solution of the mixed formulation of three-dimensional elliptic equations. Then, we focus on the parallel solution of the linear system arising from such discretization, considering both direct and iterative parallel solvers. In the latter case, we develop two block preconditioners, one based on the approximate Schur complement and one on a regularization technique. The numerical tests performed on a Linux cluster show that the proposed VEM discretization recovers the expected theoretical convergence rates and we analyze the performance of the direct and iterative parallel solvers taken into account. This is a joint work with Franco Dassi, from the University of Milano-Bicocca.

Ern - Tuesday, May 18 - 14:30-15:15

Hybrid high-order methods for wave propagation on unfitted meshes

We design and analyze an unfitted hybrid high-order (HHO) method for the acoustic wave equation. The wave propagates in a domain where a curved interface separates subdomains with different material properties. The key feature of the space discretization method is that the interface can cut more or less arbitrarily through the mesh cells. We address both the second-order formulation in time of the wave equation and its reformulation as a first-order system. For explicit time-stepping schemes, we study the CFL condition and observe that the unfitted approach combined with local cell agglomeration leads to a comparable condition as when using fitted meshes for planar interfaces.

Di Pietro - Monday, May 17 - 14:30-15:00

Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra

In this work, merging ideas from compatible discretisations and polyhedral methods, we construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these sequences are directly amenable to computer implementation. Besides proving exactness, we show that the usual three-dimensional sequence of trimmed Finite Element spaces forms, through appropriate interpolation operators, a commutative diagram with our sequence, which ensures suitable approximation properties. The discrete de Rham (DDR) sequence is then used to design a stable arbitrary-order approximation of a magnetostatics problem.

Visinoni - Wednesday, May 19 - 12:45-13:05

Hybridization of VEM for Hellinger-Reissner elasticity problems

In this talk, we present the hybridization procedure proposed in [Arnold, Brezzi, 1985, M2AN] to the Virtual Element Method (VEM) for linear elasticity problems based on the Hellinger-Reissner principle. This procedure consists of using Lagrange multipliers to impose the required continuity constraints across the inter-elements, rather than enforcing them directly in the approximation spaces, and employing a static condensation technique to reduce the computational costs and obtain a symmetric and positive definite linear system. Moreover, exploiting the available information derived from multipliers, we also show how to design a better approximation of the displacement field using a straightforward post-processing procedure. Some numerical tests are provided to show the validity and the potential of the proposed method. This work is in collaboration with F. Dassi and C. Lovadina.

Busetto - Monday, May 17 - 12:05-12:25

A Virtual Element Approximation for Immiscible Incompressible Two-phase Flow in Porous Media

In the present contribution we investigate the potentialities of the Virtual Element Method (VEM) in the framework of the two-phase flow of immiscible incompressible fluids in porous media. This phenomenon is mathematically described by a system of time-dependent coupled nonlinear partial differential equations. Due to the nature of the equations the numerical simulation of this problem is still a challenging task, but extremely important from an applicative point of view. Indeed, these equations are of interest in many scientific and industrial fields including petroleum and chemical engineering, hydrogeology and nuclear disposal safety. In this work we discretize the governing equations in time and in space combining an iterative Implicit-Pressure-Implicit-Saturation method with a primal conforming virtual element discretization. We analyse the performance of the resulting fully discrete scheme and we show its potentialities in terms of simplified construction of high-order approximations and mesh flexibility. This latter feature is very attractive for the numerical modelling of realistic flow processes since porous media are typically characterized by geometric complexity. We present the results obtained testing the method on a problem having known analytical solution and on some well known benchmark problems that are of interest for applications.

Formaggia - Tuesday, May 18 - 11:15-11:45

Block preconditioners for Darcy problem in fractured media

We will report some recent results on block preconditioners for a hybrid dimensional numerical model of flow in fractured porous media. We will discuss a formulation based on the mimetic finite difference for the flow in the rock coupled with a description of flow in an immersed fracture network. For the latter, a primal formulation is used. The final system shows a double saddle point structure. We show an analysis of the spectral properties of the system and some block preconditioning techniques that have proven robust for this class of problems.

Mora - Monday, May 17 - 15:00-15:30

Virtual Elements for the Transmission Eigenvalue Problem

The aim of this talk is to analyze a C^1 Virtual Element Method (VEM) on polygonal meshes for solving a quadratic and non-selfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. Optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are obtained. Numerical experiments will be provided to verify the theoretical error estimates.

D’Auria - Wednesday, May 19 - 11:45-12:05

Polygonal and Polyhedral Mesh Refinement

Refinement strategies for polythopal meshes including techniques that can be considered a generalizations of refinement strategies for triangular or tetrahedral meshes is a topic still under investigation. In this talk we will present several refinement strategies viable for polytopal elements suitable for VEM discretizations. In particular we will analyse the behaviour of the error and the quality of the mesh produced. The proposed methods can be extended to consider information provided by anisotropic error estimate. The proposed refinement strategies are tested on simple configurations in order to investigate the error decay and in the framework of the fractured media, where a natural algorithm for producing minimal conforming polygonal meshes is introduced.

Manzini - Wednesday, May 19 - 14:30-15:00

The virtual element method for magnetohydrodynamics applications.

We apply the framework of the Virtual Element Method (VEM) to a model in Magneto-HydroDynamics (MHD), that incorporates a coupling between electromagnetics and fluid flow, to construct novel discretizations for simulating realistic phenomenon in MHD. First, we study two chains of spaces approximating the electromagnetic and fluid flow components of the model. Then, we show that this VEM approximation will yield divergence free discrete magnetic fields that is an important property in any simulation in MHD. We present a linearization strategy to solve the VEM approximation which respects the divergence free condition on the magnetic field. This linearization will require that, at each non-linear iteration, a linear system be solved. We study these linear systems and show that they represent well-posed saddle point problems. We conclude by presenting numerical experiments exploring the performance of the VEM applied to the subsystem describing the electromagnetics, and in particular the magnetic reconnection problem.

Botti - Wednesday, May 19 - 15:20-15:40

Nonconforming polyhedral discretization methods for multiple-network poroelasticity

We develop and analyze nonconforming monolithic discretizations of multiple-network poroelasticity problems, modeling seepage through deformable fissured media. The proposed schemes are based on Hybrid High-Order and discontinuous Galerkin methods and are designed to support general polygonal and polyhedral elements. This is a veritable advantage in geological modelling, where the need for general elements arises due to the presence of fractures and the onset of degenerate elements to account for compaction or erosion. We use as a starting point a mixed weak formulation where an additional total pressure variable is considered, ensuring the robustness in the entire range of geophysical parameters. The resulting methods have several appealing features: they supports general meshes and arbitrary approximation order; they can be applied to any number of pore networks; and they are well-behaved for quasi-incompressible porous matrices. The stability and convergence results rely on a novel abstract framework for the analysis of nonconforming fully coupled discretizations that rests on mild regularity assumptions. The theoretical results are validated on a complete set of numerical tests.

Sukumar - Monday, May 17 - 16:00-16:45

Scaled boundary cubature scheme for numerical integration over planar regions with affine and curved boundaries

In this talk, I will introduce the scaled boundary cubature (SBC) scheme for accurate and efficient integration of functions over polygons and two-dimensional regions bounded by parametric curves. Over two-dimensional domains, the SBC method reduces integration over a region bounded by m curves to integration over m regions (referred to as curved triangular regions), where each region is bounded by two line segments and a curve. With proper (counterclockwise) orientation of the boundary curves, the scheme is applicable to convex and nonconvex domains. Additionally, for star-convex domains, a tensor-product cubature rule with positive weights and integration points in the interior of the domain is obtained. If the integrand is homogeneous, we show that this new method reduces to the homogeneous numerical integration scheme; however, the SBC scheme is more versatile since it is equally applicable to both homogeneous and non-homogeneous functions. I will also introduce methods for smoothing integrands with point singularities and near-singularities. When these methods are used, highly efficient integration of weakly singular functions is realized. I will present the application of the SBC method to several benchmark problems to demonstrate its broad applicability and superior performance when compared to existing methods for integration. This is joint-work with Eric B. Chin at Lawrence Livermore National Laboratory.

Böhm - Tuesday, May 18 - 12:45-13:05

Multiphysics response of polycrystalline materials: Computational homogenization via the Virtual Element Method

Multiphysics response of polycrystalline materials: Computational homogenization via the Virtual Element Method Christoph Böhm, Blaž Hudobivnik, Michele Marino, Peter Wriggers Institute of Continuum Mechanics Leibniz Universität Hannover An der Universität 1, 30826 Hannover, Germany boehm@ikm.uni-hannover.de Throughout the last years, problems, coupled to electric(-magnetically) behaviour, gaining continuously interests. Such a behaviour, applied to ferro-electrics/ferro-magnetics, exhibits an electro-mechanical/magneto- mechanical response at the micro-structure of the considered materials. A transition of resulting microscopic quantities to macroscopic length-scales is accomplished by computational homogenization techniques. However, when modelling within the micro-structural framework of ferro-electrics/ferro-magnetics, the observed struc- ture is a crystalline aggregate [1]. Such heterogeneous environments have demands to a flexibility of meshes regarding discretization techniques in classical finite element simulations. Consequently, the discretization ends up in partial highly refined meshes in order to maintain a certain accuracy of the solution. Contrary, the Vir- tual Element Method (VEM) is able to handle elements by imposing nodal degrees of freedom as point-wise introduced values at the vertices of the particular virtual element [2]. This ability gives the freedom to treat elements of arbitrary shapes, i.e. arbitrary number of nodes and possibly non-convex shapes. Hence, the virtual element method is able to fit the underlying grain structure perfectly. Moreover, VEM showed abilities to be more accurate (in terms of a computational error) regarding computational homogenization of microscopic pure mechanical as well as coupled electro-mechanical/magneto-mechanical frameworks, when compared to classical finite element approaches at the same number of degrees of freedom [3, 4]. The codes for the generation of the element residual and the element tangent matrix were obtained by utilizing the automatic differentiation based software tool AceGen [5]. By utilizing virtual elements within such heterogeneous environments, the ability arises to either decrease the amount of computational costs or to increase the accuracy (by i.e. an increase of the number of degrees of freedom) at the same amount of computational costs. The authors acknowledge financial support to this work by the DFG Collaborative Research Centre CRC 1153 "Process chain for the production of hybrid high-performance components through tailored forming", project no. 252662854, as well as by the Italian Ministry of Education, University and Research (MIUR) in the framework of the Rita Levi Montalcini Program. References [1] Schröder, J., Labusch, M. and Keip, M.-A. Algorithmic two-scale transition for magneto-electro-mechanically coupled problems: FE2-scheme: localization and homogenizaton. Computer Methods in Applied Mechanics and Engineering, 302, 253–280, 2016. [2] Beirão da Veiga, L., Brezzi, F., Marini, L. D. and Russo, A. The hitchhiker’s guide to the virtual element method. Mathematical models and methods in applied sciences, 24(08), 1541–1573, 2014. [3] Marino, M., Hudobivnik, B. and Wriggers, P. Computational homogenization of polycrystalline materials with the Virtual Element Method. Computer Methods in Applied Mechanics and Engineering, 355, 349–372, 2019. [4] Böhm, C., Hudobivnik, B., Marino, M. and Wriggers, P. Electro-magneto-mechanically response of poly- crystalline materials: Computational Homogenization via the Virtual Element Method. Computer Methods in Applied Mechanics and Engineering, 380, 2021. [5] Korelc, J. and Wriggers, P. Automation of Finite Element Methods. Springer, 2016.

Sacco - Wednesday, May 19 - 10:15-10:45

A Virtual Element approach for cohesive fracture nucleation and evolution

Numerical methods for modeling fracture nucleation and evolution are mostly based on the finite element methods (FEM). As the fracture curve is not known a priori, significant remeshing is often required to refine the mesh around the crack tip and to introduce and open interfaces between elements along the fracture curve. The remeshing technique allows to obtain valuable and accurate results in fracture mechanics problems [1,2]. Nevertheless, the remeshing procedure is computationally almost expensive, as new elements and nodes have to be generated with a continuous burdensome updating of the discretization. An interesting and effective finite element technique for studying the fracture evolution, known as extended finite element method (XFEM), has been proposed for brittle and cohesive fracture in [3,4]. The XFEM technique allows to obtain very accurate results introducing a minimal remeshing of the discretization. In fact, it is based on the idea of splitting the element crossed by the crack in two elements. This technique received great success and several modifications and improvements have been proposed in literature during the last 20 years. In this paper, a technique for studying the crack evolution is presented, conjugating the idea of splitting the cracked element in two elements with the development of the Virtual Element Method (VEM). In fact, the VEM, proposed initially in [5], is much more flexible than standard FEM, allowing to use elements with arbitrary shape and with how many edges and nodes are necessary, without limitation. The use of VEM in fracture mechanics has been presented in [6]. Herein, a quadrilateral twelve node virtual element characterized by linear interpolation of the displacement field on the boundary is introduced. An accurate evaluation of the strain field around the crack is performed adopting a complementary energy minimization approach in the single element. Then, a nonlocal measure of the strain at the crack tip, accounting also of the displacement jump of the open crack, is determined; this quantity is assumed to govern the crack evolution. Once the crack direction is determined the virtual element containing the tip on its boundary is split either in two 6-nodes elements or in a 3-nodes and a 9 nodes elements. The split elements are then joined by a cohesive interface. Several applications are performed in order to assess the ability of the proposed procedure in reproducing the crack evolution, comparing the obtained results with the experimental and numerical evidence available in literature. References [1] P.A. Wawrzynek, A.R. Ingraffea, An interactive approach to local remeshing around a propagating crack, Finite Elements in Analysis and Design, 5(1), 1989, pp. 87-96. [2] P. Bocca, A. Carpinteri, and S. Valente, Mixed mode fracture of concrete, International Journal of Solids and Structures, 27(9), 1991 pp. 1139-1153. [3] N. Moës, J. Dolbow, and T. Belytschko, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering, 46(1), 1999, pp. 131-150. [4] N. Moës and T. Belytschko, Extended finite element method for cohesive crack growth, Engineering Fracture Mechanics, 69(7), 2002, pp. 813-833. [5] L. Beirão da Veiga , F. Brezzi , A. Cangiani, G. Manzini, L.D. Marini, A. Russo, Basic principles of virtual element methods. Math Models Methods Appl Sci, 2013, 23(1), pp. 199-214. [6] E. Artioli, S. Marfia, E. Sacco, VEM-based tracking algorithm for cohesive/frictional 2D fracture, Computer Methods in Applied Mechanics and Engineering, 365, 2020, 112956.

Brezzi - Monday, May 17 - 09:00-09:45

Virtual Elements and non-polynomial approximations

After a brief sketch of the basic ideas behind Virtual Element Methods, the talk will discuss the interest and pros-and-cons of approximations using non-polynomial functions, and analyze their application on more conventional grids (as triangles, quadrilaterals, and the like) compared with classical approaches using polynomials or rational functions. The attention will be pointed, mostly, on situations where the use of traditional approaches is more delicate (as incompressibility conditions, higher order continuity requirements, and the like).

Pingaro - Wednesday, May 19 - 11:15-11:45

Fast Homogenization Procedure for particle composite materials using Virtual Element Methods

The study of mechanical behaviour of particle composite and polycrystalline materials with inter-granular phases grown of interest in the last decades. They are modern composite materials extremely relevant for a wide range of engineering applications. Their complex microstructure is often characterized by stochastically disordered distributions, having a direct impact on the overall mechanical behaviour. A computational homogenization procedure in conjunction with a statistical approach have been successfully adopted for the definition of the Representative Volume Element (RVE) size, that in random media is an unknown of the problem, and of the related equivalent elastic moduli. Drawback of such a statistical approach to homogenization is the high computational cost, which prevents the possibility to perform series of parametric analyses. To overcome this problem an integrated framework that automates all the steps to perform has been developed in the so-called Fast Statistical Homogenization Procedure (FSHP). The approach, combined with the Virtual Element Method (VEM) used as a valuable tool to keep computational costs down, allow us to efficiently solve the series (hundreds) of simulations and to rapidly converge to the RVE size detection. Numerical examples of different materials like porous and cermet-like linear elastic composites will be presented.

Houston - Wednesday, May 19 - 09:00-09:45

Fast Solution Techniques for High-Order Discontinuous Galerkin Methods on Polytopic Grids

In this talk we consider the efficient implemention of high-order/hp-version discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polytopic meshes. In particular, we consider two key aspects: fast evaluation of integrals over general polytopic elements, based on employing properties of homogeneous functions, as well as the design and analaysis of a class of two-level non-overlapping additive Schwarz preconditioners for the solution of the underlying linear system of equations. Here, the preconditioner is based on a coarse space and a non-overlapping partition of the computational domain where local solvers are applied in parallel. In particular, the coarse space can potentially be chosen to be non-embedded with respect to the finer space; indeed it can be obtained from the fine grid by employing agglomeration and edge coarsening techniques. We investigate the dependence of the condition number of the preconditioned system with respect to the diffusion coefficient and the discretization parameters, i.e., the mesh size and the polynomial degree of the fine and coarse spaces. Numerical examples are presented which confirm the theoretical bounds.

Gardini - Monday, May 17 - 15:30-15:50

Approximation of PDE eigenvalue problems involving parameter dependent matrices

We discuss the solution of eigenvalue problem associated with partial differential equations that can be written as generalized algebraic eigenvalue problem with matrices depending on scalar parameters. Parameter dependent matrices occur frequently when stabilized formulations are used for the numerical approximation of partial differential equations. With the help of classical numerical examples, we show that the presence of one (or both) parameters can produce unexpected results. As application, we consider the Virtual Element Method and discuss the dependence of the convergence behavior on the choice of the stabilization parameters. The optimal strategy for the choice of the parameters is still the object of ongoing research; nevertheless, the quick and easy recipe will be that the parameter related to the stiffness matrix should be large enough, while the parameter related to the mass matrix should be small enough and possibly equal to zero.

Russo - Tuesday, May 18 - 10:15-10:45

The Virtual Element Method for curved polygons

In this talk I will present some recent (and not so recent) results concerning the Virtual Element Method for polygons with curved edges.

Frittelli - Tuesday, May 18 - 16:15-16:35

Virtual Element Method for bulk-surface PDEs and an application to battery modeling

We consider a coupled bulk-surface PDE in d=2 or d=3 space dimensions. The model consists of a PDE in the bulk (a compact set Ω in R^d) that is coupled to another PDE on the surface Γ = ∂Ω through general nonlinear boundary conditions. In this talk we focus on the spatial discretisation of such bulk-surface PDEs, for which we propose a novel method, based on coupling a virtual element method [Beirao da Veiga et al., 2013] in Ω to (i) a surface finite element method on Γ [Dziuk & Elliott, 2013] when d=2 and (ii) a surface virtual element method on Γ [Frittelli & Sgura, 2018] when d=3. The proposed method, which we coin the Bulk-Surface Virtual Element Method (BSVEM) includes, as a special case, the bulk-surface finite element method (BSFEM) on simplicial meshes [Madzvamuse & Chung, 2016]. The method exhibits second-order convergence in space, provided the exact solution is H^2 on the surface Γ and H^(2+(2d-3)/4) in the bulk Ω. Such additional fractional regularity is required only in the simultaneous presence of surface curvature and non-simplicial bulk elements. Two novel techniques introduced in our analysis are (i) an L^2-preserving inverse trace operator for the analysis of boundary conditions and (ii) the Sobolev extension as a replacement of the lifting operator [Elliott & Ranner, 2013] for sufficiently smooth exact solutions. The generality of the polytopal mesh can be exploited to optimize the computational time of matrix assembly. The method takes an optimised matrix-vector form that also simplifies the known special case of BSFEM on triangular meshes [Madzvamuse & Chung, 2016]. Numerical examples illustrate our findings. As an application, we present a bulk-surface PDE system for battery modeling. The cycling instability in the recharge process is controlled by the interaction between the "shape" and the "chemistry" of materials here described by a reaction-diffusion PDE system (RD-PDE) with source-terms accounting for the battery operating conditions.

Desiderio - Wednesday, May 19 - 15:00-15:20

On the coupling of Curved Virtual Element and one-equation Boundary Element Methods for 2D exterior Helmholtz problems

We consider the Helmholtz equation defined in unbounded domains, external to 2D bounded ones having curved shape, endowed with a Dirichlet condition on the boundary and a radiation condition at infinity. For its solution, we reduce the infinite region to a bounded computational one, delimited by a curved smooth artificial boundary and we impose on it a non-reflecting boundary condition of boundary integral type. Then, we apply the combined use of the Curved Virtual Element Method and the one-equation Boundary Element Method. We present a theoretical analysis and provide an optimal convergence error estimates in the energy norm. The numerical tests confirm the theoretical results and show the effectiveness of the new proposed approach.

Artioli - Monday, May 17 - 11:15-11:45

Curvilinear virtual elements for solid mechanics applications

The talk will address the development of innovative curvilinear virtual element methods for 2D solid mechanics applications. The proposed virtual elements are suitable to exact representation of the curved computational domain, retaining all the typical features of a standard conforming VEM. Applications will range from material nonlinearity, to asymptotic homogenization of doubly periodic and random fibre-reinforced composite materials, and to two-dimensional contact mechanics, targeting the main advantages of the proposed formulation in comparison to standard straight-edge finite element techniques.

Wriggers - Tuesday, May 18 - 09:00-09:45

Application of virtual elements for simulations in solid mechanics

Virtual elements (VEM) were developed during the last decade and applied to various problems in solid mechanics. The method includes elements that can have arbitrary shape including non convex polyhedra. This flexibility with respect to the geometry can be explored and utilized within engineering applications for specific problems.

This lecture will cover several applications of the virtual element method in the area of solids mechanics which are related to

  • contact and interfaces  
  • fracture using classical and phase field approaches
  • homogenization of polycrystal microstructural response
  • discrete flexible element methods and
  • design of specific virtual elements for legacy FEM codes

For these problem classes we will discuss the pros and cons of virtual elements for efficient, reliable and robust solutions in the engineering world.

Aldakheel - Tuesday, May 18 - 12:05-12:25

Virtual elements in fracture mechanics

Hudobivnik - Tuesday, May 18 - 12:25-12:45

NURBS based geometries: A mapping approach for virtual serendipity elements

A NURBS-based second order serendipity virtual element method for general arbitrary element shapes is presented. In comparison with the already existing serendipity VEM, a general mapping scheme is developed allowing to deviate from the assumption of straight edges of virtual elements. A number of numerical examples illustrates the robustness and accuracy of the new mapping methodology.
The results are very promising and underline the advantages of the formulations for dealing with arbitrary geometries.


F. Brezzi (IMATI-CNR, Pavia)
A. Ern (University of Paris-Est)
P. Houston (University of Nottingham)
S. Sukumar (University of California, Davis)
P. Wriggers (University of Hannover)


P.F. Antonietti (Politecnico di Milano) L. Beirao da Veiga (Università di Milano-Bicocca) S. Berrone (Politecnico di Torino)