A Multiscale Method for the Heterogeneous Signorini Problem

In this paper, we develop a multiscale method for solving the Signorini problem with a heterogeneous field. The Signorini problem is encountered in many applications, such as hydrostatics, thermics, and solid mechanics. It is well-known that numerically solving this problem requires a fine computational mesh, which can lead to a large number of degrees of freedom. The aim of this work is to develop a new hybrid multiscale method based on the framework of the generalized multiscale finite element method (GMsFEM). The construction of multiscale basis functions requires local spectral decomposition. Additional multiscale basis functions related to the contact boundary are required so that our method can handle the unilateral condition of Signorini type naturally. A complete analysis of the proposed method is provided and a result of the spectral convergence is shown. Numerical results are provided to validate our theoretical findings.

[1]  Martin Stynes,et al.  Numerical Treatment of Partial Differential Equations , 2007 .

[2]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[3]  Yalchin Efendiev,et al.  Domain Decomposition Preconditioners for Multiscale Flows in High Contrast Media: Reduced Dimension Coarse Spaces , 2010, Multiscale Model. Simul..

[4]  Barbara I. Wohlmuth,et al.  A Reduced Basis Method for Parametrized Variational Inequalities , 2012, SIAM J. Numer. Anal..

[5]  Faker Ben Belgacem,et al.  Hybrid finite element methods for the Signorini problem , 2003, Math. Comput..

[6]  Michael Ulbrich,et al.  Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces , 2011, MOS-SIAM Series on Optimization.

[7]  Yalchin Efendiev,et al.  Generalized Multiscale Finite-Element Method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media , 2014, J. Comput. Phys..

[8]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[9]  Karen Veroy,et al.  A slack approach to reduced-basis approximation and error estimation for variational inequalities , 2016 .

[10]  R. Glowinski,et al.  Numerical Analysis of Variational Inequalities , 1981 .

[11]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[12]  Yalchin Efendiev,et al.  Mixed Generalized Multiscale Finite Element Methods and Applications , 2014, Multiscale Model. Simul..

[13]  Yalchin Efendiev,et al.  Generalized multiscale finite element methods (GMsFEM) , 2013, J. Comput. Phys..

[14]  Rolf Stenberg,et al.  Mixed and Stabilized Finite Element Methods for the Obstacle Problem , 2017, SIAM J. Numer. Anal..

[15]  Wing Tat Leung,et al.  A Generalized Multiscale Finite Element Method (GMsFEM) for perforated domain flows with Robin boundary conditions , 2019, J. Comput. Appl. Math..

[16]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[17]  Yalchin Efendiev,et al.  Generalized Multiscale Finite Element Methods for Wave Propagation in Heterogeneous Media , 2013, Multiscale Model. Simul..

[18]  Moritz Kassmann,et al.  On weighted Poincaré inequalities , 2012 .

[19]  Yalchin Efendiev,et al.  Generalized multiscale finite element methods for problems in perforated heterogeneous domains , 2015, 1501.03536.

[20]  René Pinnau,et al.  Model Reduction of Contact Problems in Elasticity: Proper Orthogonal Decomposition for Variational Inequalities , 2014 .

[21]  V. Girault,et al.  A Local Regularization Operator for Triangular and Quadrilateral Finite Elements , 1998 .

[22]  J. Oden,et al.  Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods , 1987 .

[23]  Yalchin Efendiev,et al.  Generalized Multiscale Finite Element Methods. Oversampling Strategies , 2013, 1304.4888.

[24]  Gianluigi Rozza,et al.  Reduced Basis Method for Parametrized Elliptic Optimal Control Problems , 2013, SIAM J. Sci. Comput..

[25]  Jinchao Xu,et al.  Some Estimates for a Weighted L 2 Projection , 1991 .

[26]  Kazufumi Ito,et al.  The Primal-Dual Active Set Strategy as a Semismooth Newton Method , 2002, SIAM J. Optim..

[27]  David Ryckelynck,et al.  Hybrid hyper‐reduced modeling for contact mechanics problems , 2018 .

[28]  Guanglian Li,et al.  On the Convergence Rates of GMsFEMs for Heterogeneous Elliptic Problems Without Oversampling Techniques , 2018, Multiscale Model. Simul..

[29]  Charbel Farhat,et al.  Projection‐based model reduction for contact problems , 2015, 1503.01000.

[30]  Jaroslav Haslinger,et al.  Numerical methods for unilateral problems in solid mechanics , 1996 .

[31]  Jessika Eichel,et al.  Partial Differential Equations Second Edition , 2016 .

[32]  A. Ern,et al.  A reduced basis method for parametrized variational inequalities applied to contact mechanics , 2019, International Journal for Numerical Methods in Engineering.

[33]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .