Abstract
We introduce and analyze a new mixed finite element method with reduced symmetry for the standard linear model in viscoelasticity. Following a previous approach employed for linear elastodynamics, the present problem is formulated as a second-order hyperbolic partial differential equation in which, after using the motion equation to eliminate the displacement unknown, the stress tensor remains as the main variable to be found. The resulting variational formulation is shown to be well-posed, and a class of \(\text {H}(\text {div})\)-conforming semi-discrete schemes is proved to be convergent. Then, we use the Newmark trapezoidal rule to obtain an associated fully discrete scheme, whose main convergence results are also established. Finally, numerical examples illustrating the performance of the method are reported.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, D.N., Brezzi, F., Douglas, J.: PEERS: a new mixed finite element method for plane elasticity. Japan J. Appl. Math. 1(2), 347–367 (1984)
Arnold, D.N., Falk, R.S., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comput. 76(260), 1699–1723 (2007)
Arnold, D.N., Lee, J.J.: Mixed methods for elastodynamics with weak symmetry. SIAM J. Numer. Anal. 52(6), 274–2769 (2014)
Bécache, E., Ezziani, A., Joly, P.: A mixed finite element approach for viscoelastic wave propagation. Comput. Geosci. 8, 255–299 (2005)
Bécache, E., Joly, P., Tsogka, C.: A new family of mixed finite elements for the linear elastodynamic problem. SIAM J. Numer. Anal. 39(6), 2109–2132 (2002)
Boffi, D., Brezzi, F., Fortin, M.: Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8(1), 95–121 (2009)
Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013)
Cockburn, B., Gopalakrishnan, J., Guzmán, J.: A new elasticity element made for enforcing weak stress symmetry. Math. Comput. 79, 1331–1349 (2010)
Evans, L.C.: Partial Differential Equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI (2010)
Fabrizio, M., Morro, A.: Mathematical Problems in Linear Viscoelasticity. SIAM, Philadelphia (1992)
García, C., Gatica, G.N., Meddahi, S.: A new mixed finite element method for elastodynamics with weak symmetry. J. Sci. Comput. 72(3), 1049–1079 (2017)
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms, vol. 5. Springer Science & Business Media (2012)
Gurtin, M.E., Sternberg, E.: On the linear theory of viscoelasticity. Arch. Rational Mech. Anal. 11, 291–356 (1962)
Gopalakrishnan, J., Guzmán, J.: A second elasticity element using the matrix bubble. IMA J. Numer. Anal. 32, 352–372 (2012)
Idesman, A., Niekamp, R., Stein, E.: Finite elements in space and time for generalized viscoelastic Maxwell model. Comput. Mech. 27, 49–60 (2001)
Janovsky, V., Shaw, S., Warby, M.K., Whiteman, J.R.: Numerical methods for treating problems of viscoelastic isotropic solid deformation. J. Comput. Appl. Math. 63(1–3), 91–107 (1995)
Lee, J.J.: Analysis of mixed finite element methods for the standard linear solid model in viscoelasticity. Calcolo 54(2), 587–607 (2017)
Marques, S.P., Creus, G.J.: Computational Viscoelasticity. Springer Science & Business Media, Berlin (2012)
Renardy, M., Rogers, R.: An Introduction to Partial Differential Equations. Texts in Applied Mathematics, 13. Springer, New York (2004)
Rivière, B., Shaw, S., Wheeler, M., Whiteman, J.R.: Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity. Numer. Math. 95, 347–376 (2003)
Rivière, B., Shaw, S., Whiteman, J.R.: Discontinuous Galerkin finite element methods for dynamic linear solid viscoelasticity problems. Numer. Methods Partial Diff. Equ. 23(5), 1149–1166 (2007)
Rognes, M., Winther, R.: Mixed finite element methods for linear viscoelasticity using weak symmetry. Math. Models Methods Appl. Sci. 20, 955–985 (2010)
Roubíček, T.: Nonlinear Partial Differential Equations with Applications. Second edition. International Series of Numerical Mathematics, 153. Birkhäuser/Springer Basel AG, Basel, (2013)
Salençon, J.: Viscoelastic Modeling for Structural Analysis. Wiley (2019)
Shaw, S., Warby, M.K., Whiteman, J.R., Dawson, C., Wheeler, M.F.: Numerical techniques for the treatment of quasistatic viscoelastic stress problems in linear isotropic solids. Comput. Methods Appl. Mech. Eng. 118, 211–237 (1994)
Shaw, S., Whiteman, J.R.: Numerical solution of linear quasistatic hereditary viscoelasticity problems. Siam J. Numer. Anal. 38, 80–97 (2000)
Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numer. Math. 53, 513–538 (1988)
Acknowledgements
This research was partially supported by Spain’s Ministry of Economy Project MTM2017-87162-P; by CONICYT-Chile through the project AFB170001 of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal; and by Centro de Investigación en Ingeniería Matemática (CI\(^2\)MA), Universidad de Concepción.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gatica, G.N., Márquez, A. & Meddahi, S. A mixed finite element method with reduced symmetry for the standard model in linear viscoelasticity. Calcolo 58, 11 (2021). https://doi.org/10.1007/s10092-021-00401-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10092-021-00401-0