The virtual element method for linear elastodynamics models. Design, analysis, and implementation

We design the conforming virtual element method for the numerical simulation of two dimensional time-dependent elastodynamics problems. We investigate the performance of the method both theoretically and numerically. We prove the stability and the convergence of the semi-discrete approximation in the energy norm and derive optimal error estimates. We also show the convergence in the $L^2$ norm. The performance of the virtual element method is assessed on a set of different computational meshes, including non-convex cells up to order four in the h-refinement setting. Exponential convergence is also experimentally seen in the p-refinement setting.

[1]  Ohannes A. Karakashian,et al.  On multistep-Galerkin discretizations of semilinear hyperbolic and parabolic equations , 1980 .

[2]  Lourenço Beirão da Veiga,et al.  A virtual element method for the acoustic vibration problem , 2016, Numerische Mathematik.

[3]  Alfio Quarteroni,et al.  Generalized Galerkin approximations of elastic waves with absorbing boundary conditions , 1998 .

[4]  Heng Chi,et al.  On nonconvex meshes for elastodynamics using virtual element methods with explicit time integration , 2019, Computer Methods in Applied Mechanics and Engineering.

[5]  高等学校計算数学学報編輯委員会編,et al.  高等学校計算数学学報 = Numerical mathematics , 1979 .

[6]  Alessandro Russo,et al.  Mixed Virtual Element Methods for general second order elliptic problems on polygonal meshes , 2014, 1506.07328.

[7]  Edip Baysal,et al.  Forward modeling by a Fourier method , 1982 .

[8]  Paola F. Antonietti,et al.  The conforming virtual element method for polyharmonic problems , 2018, Comput. Math. Appl..

[9]  Franco Brezzi,et al.  Virtual Element Methods for plate bending problems , 2013 .

[10]  Gianmarco Manzini,et al.  Mimetic finite difference method , 2014, J. Comput. Phys..

[11]  Tim Warburton,et al.  An explicit construction of interpolation nodes on the simplex , 2007 .

[12]  Mrinal K. Sen,et al.  The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion , 2008 .

[13]  Gianmarco Manzini,et al.  Conforming and nonconforming virtual element methods for elliptic problems , 2015, 1507.03543.

[14]  Ahmed Alsaedi,et al.  Equivalent projectors for virtual element methods , 2013, Comput. Math. Appl..

[15]  Z. Alterman,et al.  Propagation of elastic waves in layered media by finite difference methods , 1968 .

[16]  Peter Wriggers,et al.  A computational framework for brittle crack-propagation based on efficient virtual element method , 2019, Finite Elements in Analysis and Design.

[18]  Kyoungsoo Park,et al.  Numerical recipes for elastodynamic virtual element methods with explicit time integration , 2019, International Journal for Numerical Methods in Engineering.

[19]  S. Natarajan,et al.  Implementation of the virtual element method for coupled thermo-elasticity in Abaqus , 2018, Numerical Algorithms.

[20]  Dibyendu Adak,et al.  Virtual element method for semilinear sine-Gordon equation over polygonal mesh using product approximation technique , 2019, Math. Comput. Simul..

[21]  Lorenzo Mascotto,et al.  Ill‐conditioning in the virtual element method: Stabilizations and bases , 2017, 1705.10581.

[22]  Minfu Feng,et al.  Virtual element method for two-dimensional linear elasticity problem in mixed weakly symmetric formulation , 2018, Appl. Math. Comput..

[23]  Shaochun Chen,et al.  The nonconforming virtual element method for elasticity problems , 2019, J. Comput. Phys..

[24]  G. Vacca,et al.  The p- and hp-versions of the virtual element method for elliptic eigenvalue problems , 2018, Comput. Math. Appl..

[25]  Stefano Berrone,et al.  The virtual element method for discrete fracture network simulations , 2014 .

[26]  Paola F. Antonietti,et al.  High-order Discontinuous Galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes , 2018, Computer Methods in Applied Mechanics and Engineering.

[27]  Emmanuil H. Georgoulis,et al.  A posteriori error estimates for the virtual element method , 2016, Numerische Mathematik.

[28]  Peter Wriggers,et al.  Phase-field modeling of brittle fracture using an efficient virtual element scheme , 2018, Computer Methods in Applied Mechanics and Engineering.

[29]  D. Adak,et al.  Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes , 2018, Numerical Methods for Partial Differential Equations.

[30]  Stefano Berrone,et al.  Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method , 2017 .

[31]  A. Quarteroni,et al.  hp -Version Discontinuous Galerkin Approximations of the Elastodynamics Equation , 2017 .

[32]  Lourenço Beirão da Veiga,et al.  A Stream Virtual Element Formulation of the Stokes Problem on Polygonal Meshes , 2014, SIAM J. Numer. Anal..

[33]  H. Brezis.  Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert , 1973 .

[34]  Stefano Berrone,et al.  A Posteriori Error Estimate for a PDE-Constrained Optimization Formulation for the Flow in DFNs , 2016, SIAM J. Numer. Anal..

[35]  Alfio Quarteroni,et al.  Dispersion-dissipation analysis of 3-D continuous and discontinuous spectral element methods for the elastodynamics equation , 2017 .

[36]  Stefano Berrone,et al.  The Virtual Element Method for Underground Flow Simulations in Fractured Media , 2016 .

[37]  Martin Galis,et al.  The Finite-Difference Modelling of Earthquake Motions: Waves and Ruptures , 2014 .

[38]  Gianmarco Manzini,et al.  Convergence Analysis of the mimetic Finite Difference Method for Elliptic Problems with Staggered Discretizations of Diffusion Coefficients , 2016, SIAM J. Numer. Anal..

[39]  M. Putti,et al.  Post processing of solution and flux for the nodal mimetic finite difference method , 2015 .

[40]  B. Reddy,et al.  A virtual element method for transversely isotropic elasticity , 2018, Computational Mechanics.

[41]  Felipe Lepe,et al.  A Virtual Element Method for the Steklov Eigenvalue Problem Allowing Small Edges , 2015, Journal of Scientific Computing.

[42]  Glaucio H. Paulino,et al.  On the Virtual Element Method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes , 2014 .

[43]  Shaochun Chen,et al.  The nonconforming virtual element method for plate bending problems , 2016 .

[44]  D. Komatitsch,et al.  Introduction to the spectral element method for three-dimensional seismic wave propagation , 1999 .

[45]  Blanca Ayuso de Dios,et al.  Stability Analysis of Discontinuous Galerkin Approximations to the Elastodynamics Problem , 2013, J. Sci. Comput..

[46]  K. Lipnikov,et al.  The nonconforming virtual element method , 2014, 1405.3741.

[47]  Peter Wriggers,et al.  A virtual element method for contact , 2016 .

[48]  Ilaria Perugia,et al.  A Plane Wave Virtual Element Method for the Helmholtz Problem , 2015, 1505.04965.

[49]  Gianmarco Manzini,et al.  A posteriori error estimation and adaptivity in hp virtual elements , 2018, Numerische Mathematik.

[50]  Mark A. Taylor,et al.  An Algorithm for Computing Fekete Points in the Triangle , 2000, SIAM J. Numer. Anal..

[51]  Gianmarco Manzini,et al.  Arbitrary-Order Nodal Mimetic Discretizations of Elliptic Problems on Polygonal Meshes , 2011, SIAM J. Numer. Anal..

[52]  F. Brezzi,et al.  Basic principles of Virtual Element Methods , 2013 .

[53]  P. Wriggers,et al.  Finite and Virtual Element Formulations for Large Strain Anisotropic Material with Inextensive Fibers , 2018 .

[54]  P. F. Antonietti,et al.  The fully nonconforming virtual element method for biharmonic problems , 2016, 1611.08736.

[55]  L. Beirao da Veiga,et al.  Mixed Virtual Element Methods for general second order elliptic problems on polygonal meshes , 2014 .

[56]  L. Beirao da Veiga,et al.  Basic principles of hp virtual elements on quasiuniform meshes , 2015, 1508.02242.

[57]  Jan S. Hesthaven,et al.  From Electrostatics to Almost Optimal Nodal Sets for Polynomial Interpolation in a Simplex , 1998 .

[58]  Stefano Berrone,et al.  A hybrid mortar virtual element method for discrete fracture network simulations , 2016, J. Comput. Phys..

[59]  L. Beirao da Veiga,et al.  Serendipity Nodal VEM spaces , 2015, 1510.08477.

[60]  Lourenço Beirão da Veiga,et al.  Virtual Elements for Linear Elasticity Problems , 2013, SIAM J. Numer. Anal..

[61]  Stefano de Miranda,et al.  A family of virtual element methods for plane elasticity problems based on the Hellinger–Reissner principle , 2017, Computer Methods in Applied Mechanics and Engineering.

[62]  Peter Wriggers,et al.  Serendipity virtual element formulation for nonlinear elasticity , 2019, Computers & Structures.

[63]  A. Quarteroni,et al.  Non-conforming high order approximations of the elastodynamics equation , 2012 .

[64]  M. Dumbser,et al.  An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two-dimensional isotropic case with external source terms , 2006 .

[65]  L. Beirao da Veiga,et al.  A Virtual Element Method for elastic and inelastic problems on polytope meshes , 2015, 1503.02042.

[66]  S. de Miranda,et al.  A stress/displacement Virtual Element method for plane elasticity problems , 2017, 1702.01702.

[67]  Giuseppe Vacca,et al.  Virtual Element Methods for hyperbolic problems on polygonal meshes , 2016, Comput. Math. Appl..

[68]  Arthur Frankel,et al.  Three-dimensional simulations of ground motions in the San Bernardino Valley, California, for hypothetical earthquakes on the San Andreas fault , 1993, Bulletin of the Seismological Society of America.

[69]  L. B. D. Veiga,et al.  A virtual element method with arbitrary regularity , 2014 .

[70]  L. Beirao da Veiga,et al.  H(div) and H(curl)-conforming VEM , 2014, 1407.6822.

[71]  Gianmarco Manzini,et al.  Hourglass stabilization and the virtual element method , 2015 .

[72]  Stefano Berrone,et al.  A globally conforming method for solving flow in discrete fracture networks using the Virtual Element Method , 2016 .

[73]  Peter Wriggers,et al.  A low order 3D virtual element formulation for finite elasto–plastic deformations , 2017, Computational Mechanics.

[74]  Lourenco Beirao da Veiga,et al.  Stability Analysis for the Virtual Element Method , 2016, 1607.05988.

[75]  Ahmad Shahba,et al.  Stabilized tetrahedral elements for crystal plasticity finite element analysis overcoming volumetric locking , 2016 .

[76]  J. M. Thomas,et al.  Introduction à l'analyse numérique des équations aux dérivées partielles , 1983 .

[77]  Gianmarco Manzini,et al.  Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes , 2017, J. Comput. Phys..

[78]  Dibyendu Adak,et al.  Virtual element method for semilinear hyperbolic problems on polygonal meshes , 2018, Int. J. Comput. Math..

[79]  Gianmarco Manzini.  The Mimetic Finite Difference Method , 2013 .

[80]  Raducanu Razvan,et al.  MATHEMATICAL MODELS and METHODS in APPLIED SCIENCES , 2012 .

[81]  P. Wriggers,et al.  A virtual element method for frictional contact including large deformations , 2019, Engineering Computations.

[82]  L. Mascotto,et al.  Exploring high-order three dimensional virtual elements: Bases and stabilizations , 2017, Comput. Math. Appl..

[83]  Peter Wriggers,et al.  Computational homogenization of polycrystalline materials with the Virtual Element Method , 2019, Computer Methods in Applied Mechanics and Engineering.

[84]  R. Courant,et al.  Über die partiellen Differenzengleichungen der mathematischen Physik , 1928 .

[85]  David Mora,et al.  A priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations , 2017, IMA Journal of Numerical Analysis.

[86]  Annalisa Buffa,et al.  Mimetic finite differences for elliptic problems , 2009 .

[87]  Glaucio H. Paulino,et al.  Some basic formulations of the virtual element method (VEM) for finite deformations , 2017 .

[88]  Gianmarco Manzini,et al.  The Conforming Virtual Element Method for the convection-diffusion-reaction equation with variable coeffcients. , 2014 .

[89]  Stefano Berrone,et al.  Towards effective flow simulations in realistic discrete fracture networks , 2016, J. Comput. Phys..

[90]  A. Russo,et al.  New perspectives on polygonal and polyhedral finite element methods , 2014 .

[91]  M. Dumbser,et al.  An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes — II. The three-dimensional isotropic case , 2006 .

[92]  G. Burton.  Sobolev Spaces , 2013 .

[93]  Glaucio H. Paulino,et al.  Bridging art and engineering using Escher-based virtual elements , 2015 .

[94]  Gianmarco Manzini,et al.  Virtual Element Methods for Elliptic Problems on Polygonal Meshes , 2017 .

[95]  Stefano Berrone,et al.  A Parallel Solver for Large Scale DFN Flow Simulations , 2015, SIAM J. Sci. Comput..

[96]  Gianmarco Manzini,et al.  A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation , 2014, J. Comput. Phys..

[97]  Peter Wriggers,et al.  Virtual element formulation for isotropic damage , 2018 .

[98]  Stéphane Bordas,et al.  Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods , 2015 .

[99]  D. Kosloff,et al.  Solution of the equations of dynamic elasticity by a Chebychev spectral method , 1990 .

[100]  E. A. S. Neto,et al.  F‐bar‐based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. Part I: formulation and benchmarking , 2005 .

[101]  Peter Wriggers,et al.  A Virtual Element Method for Crack Propagation , 2018, PAMM.

[102]  Elena Zampieri,et al.  Numerical approximation of elastic waves equations by implicit spectral methods , 1997 .

[103]  Heiner Igel,et al.  Wave propagation in three-dimensional spherical sections by the Chebyshev spectral method , 1999 .

[104]  Gianmarco Manzini,et al.  Residual a posteriori error estimation for the Virtual Element Method for elliptic problems , 2015 .

[105]  L. Beirao da Veiga,et al.  Divergence free Virtual Elements for the Stokes problem on polygonal meshes , 2015, 1510.01655.

[106]  Gianmarco Manzini,et al.  The NonConforming Virtual Element Method for the Stokes Equations , 2016, SIAM J. Numer. Anal..

[107]  Stefano Berrone,et al.  The virtual element method for Discrete Fracture Network flow and transport simulations , 2016 .

[108]  Kim B. Olsen,et al.  Three-dimensional simulation of earthquakes on the Los Angeles fault system , 1996, Bulletin of the Seismological Society of America.

[109]  Filánder A. Sequeira,et al.  A mixed virtual element method for a pseudostress-based formulation of linear elasticity , 2019, Applied Numerical Mathematics.

[110]  Glaucio H. Paulino,et al.  A simple and effective gradient recovery scheme and a posteriori error estimator for the Virtual Element Method (VEM) , 2019, Computer Methods in Applied Mechanics and Engineering.

[111]  Ezio Faccioli,et al.  2d and 3D elastic wave propagation by a pseudo-spectral domain decomposition method , 1997 .

[112]  Peter Wriggers,et al.  A Virtual Element Method for 2D linear elastic fracture analysis , 2018, Computer Methods in Applied Mechanics and Engineering.

[113]  Peter Wriggers,et al.  VIRTUAL ELEMENT FORMULATION FOR PHASE-FIELD MODELING OF DUCTILE FRACTURE , 2019, International Journal for Multiscale Computational Engineering.

[114]  Franco Brezzi,et al.  The Hitchhiker's Guide to the Virtual Element Method , 2014 .

[115]  G. Seriani,et al.  Modelling Waves in Anisotropic Media by a Spectral Element Method , 2001 .

[116]  Paola F. Antonietti,et al.  A high-order discontinuous Galerkin approach to the elasto-acoustic problem , 2018, Computer Methods in Applied Mechanics and Engineering.

[117]  Ezio Faccioli,et al.  Spectral-domain decomposition methods for the solution of acoustic and elastic wave equations , 1996 .