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Erschienen in: Calcolo 2/2017

30.11.2016

Analysis of mixed finite element methods for the standard linear solid model in viscoelasticity

verfasst von: Jeonghun J. Lee

Erschienen in: Calcolo | Ausgabe 2/2017

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Abstract

We propose mixed finite element methods for the standard linear solid model in viscoelasticity and prove a priori error estimates. In our mixed formulation the governing equations of the problem become a symmetric hyperbolic system, so we can use standard techniques for a priori error estimates and time discretization. Numerical results illustrating our theoretical analysis are included.

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Literatur
1.
Zurück zum Zitat Arnold, D.N., Brezzi, F., Douglas Jr., J.: PEERS: a new mixed finite element for plane elasticity. Jpn. J. Appl. Math. 1, 347–367 (1984)MathSciNetCrossRefMATH Arnold, D.N., Brezzi, F., Douglas Jr., J.: PEERS: a new mixed finite element for plane elasticity. Jpn. J. Appl. Math. 1, 347–367 (1984)MathSciNetCrossRefMATH
2.
Zurück zum Zitat Arnold, D.N., Falk, R.S., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comp. 76(260), 1699–1723 (2007). (electronic)MathSciNetCrossRefMATH Arnold, D.N., Falk, R.S., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comp. 76(260), 1699–1723 (2007). (electronic)MathSciNetCrossRefMATH
3.
4.
Zurück zum Zitat 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). (electronic)MathSciNetCrossRefMATH 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). (electronic)MathSciNetCrossRefMATH
5.
Zurück zum Zitat Bécache, E., Ezziani, A., Joly, P.: A mixed finite element approach for viscoelastic wave propagation. Comput. Geosci. 8(3), 255–299 (2004)MathSciNetCrossRefMATH Bécache, E., Ezziani, A., Joly, P.: A mixed finite element approach for viscoelastic wave propagation. Comput. Geosci. 8(3), 255–299 (2004)MathSciNetCrossRefMATH
6.
Zurück zum Zitat Boffi, D., Brezzi, F., Fortin, M.: Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8(1), 95–121 (2009)MathSciNetMATH Boffi, D., Brezzi, F., Fortin, M.: Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8(1), 95–121 (2009)MathSciNetMATH
7.
Zurück zum Zitat Borcherdt, R.D.: Viscoelastic waves in layered media. Cambridge University Press, Cambridge (2009)CrossRefMATH Borcherdt, R.D.: Viscoelastic waves in layered media. Cambridge University Press, Cambridge (2009)CrossRefMATH
8.
Zurück zum Zitat Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods, volume 15 of Springer Series in computational Mathematics. Springer (1991) Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods, volume 15 of Springer Series in computational Mathematics. Springer (1991)
9.
Zurück zum Zitat Brezzi, F., Douglas Jr., J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217–235 (1985)MathSciNetCrossRefMATH Brezzi, F., Douglas Jr., J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217–235 (1985)MathSciNetCrossRefMATH
10.
Zurück zum Zitat Bulíček, M., Málek, J., Rajagopal., K.R.: On Kelvin-Voigt model and its generalizations. Evol. Equ. Control Theory 1(1), 17–42 (2012)MathSciNetCrossRefMATH Bulíček, M., Málek, J., Rajagopal., K.R.: On Kelvin-Voigt model and its generalizations. Evol. Equ. Control Theory 1(1), 17–42 (2012)MathSciNetCrossRefMATH
11.
Zurück zum Zitat Cockburn, B., Gopalakrishnan, J., Guzmán, J.: A new elasticity element made for enforcing weak stress symmetry. Math. Comp. 79(271), 1331–1349 (2010)MathSciNetCrossRefMATH Cockburn, B., Gopalakrishnan, J., Guzmán, J.: A new elasticity element made for enforcing weak stress symmetry. Math. Comp. 79(271), 1331–1349 (2010)MathSciNetCrossRefMATH
12.
Zurück zum Zitat Evans, L.C.: Partial differential equations, vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1998) Evans, L.C.: Partial differential equations, vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (1998)
13.
Zurück zum Zitat Falk, R.S.: Finite elements for linear elasticity. In: Boffi, D., Gastaldi, L. (eds.) Mixed finite elements: compatibility conditions, vol. 1939. Springer, Berlin Heidelberg (2008) Falk, R.S.: Finite elements for linear elasticity. In: Boffi, D., Gastaldi, L. (eds.) Mixed finite elements: compatibility conditions, vol. 1939. Springer, Berlin Heidelberg (2008)
14.
Zurück zum Zitat Farhloul, M., Fortin, M.: Dual hybrid methods for the elasticity and the Stokes problems: a unified approach. Numer. Math. 76(4), 419–440 (1997)MathSciNetCrossRefMATH Farhloul, M., Fortin, M.: Dual hybrid methods for the elasticity and the Stokes problems: a unified approach. Numer. Math. 76(4), 419–440 (1997)MathSciNetCrossRefMATH
15.
Zurück zum Zitat Fernández, J.R., Hild, P.: A priori and a posteriori error analyses in the study of viscoelastic problems. J. Comput. Appl. Math. 225(2), 569–580 (2009)MathSciNetCrossRefMATH Fernández, J.R., Hild, P.: A priori and a posteriori error analyses in the study of viscoelastic problems. J. Comput. Appl. Math. 225(2), 569–580 (2009)MathSciNetCrossRefMATH
16.
Zurück zum Zitat Gopalakrishnan, J., Guzmán, J.: A second elasticity element using the matrix bubble. IMA J. Numer. Anal. 32(1), 352–372 (2012)MathSciNetCrossRefMATH Gopalakrishnan, J., Guzmán, J.: A second elasticity element using the matrix bubble. IMA J. Numer. Anal. 32(1), 352–372 (2012)MathSciNetCrossRefMATH
17.
Zurück zum Zitat Gurtin, M.E., Sternberg, E.: On the linear theory of viscoelasticity. Arch. Ration. Mech. Anal. 11, 291–356 (1962) Gurtin, M.E., Sternberg, E.: On the linear theory of viscoelasticity. Arch. Ration. Mech. Anal. 11, 291–356 (1962)
18.
Zurück zum Zitat Gurtin, M.E.: An introduction to continuum mechanics, vol. 158 of Mathematics in Science and Engineering. Academic, New York (1981) Gurtin, M.E.: An introduction to continuum mechanics, vol. 158 of Mathematics in Science and Engineering. Academic, New York (1981)
19.
Zurück zum Zitat Guzmán, J.: A unified analysis of several mixed methods for elasticity with weak stress symmetry. J. Sci. Comput. 44(2), 156–169 (2010)MathSciNetCrossRefMATH Guzmán, J.: A unified analysis of several mixed methods for elasticity with weak stress symmetry. J. Sci. Comput. 44(2), 156–169 (2010)MathSciNetCrossRefMATH
20.
Zurück zum Zitat Juntunen, M., Lee, J.: Optimal second order rectangular elasticity elements with weakly symmetric stress. BIT 54(2), 425–445 (2014)MathSciNetCrossRefMATH Juntunen, M., Lee, J.: Optimal second order rectangular elasticity elements with weakly symmetric stress. BIT 54(2), 425–445 (2014)MathSciNetCrossRefMATH
21.
Zurück zum Zitat Karamanou, M., Shaw, S., Warby, M.K., Whiteman, J.R.: Models, algorithms and error estimation for computational viscoelasticity. Comput. Methods Appl. Mech. Eng. 194(2–5), 245–265 (2005)MathSciNetCrossRefMATH Karamanou, M., Shaw, S., Warby, M.K., Whiteman, J.R.: Models, algorithms and error estimation for computational viscoelasticity. Comput. Methods Appl. Mech. Eng. 194(2–5), 245–265 (2005)MathSciNetCrossRefMATH
22.
Zurück zum Zitat Lee, J.J.: Unified analysis of mixed methods for elasticity with weakly symmetric stress. Adv. Comput. Math. 42(2), 361–376 (2016) Lee, J.J.: Unified analysis of mixed methods for elasticity with weakly symmetric stress. Adv. Comput. Math. 42(2), 361–376 (2016)
23.
Zurück zum Zitat Logg, A., Mardal, K.-A., Wells, G.N. (eds.): Automated solution of differential equations by the finite element method, volume 84 of Lecture Notes in Computational Science and Engineering. The FEniCS book. Springer, Heidelberg (2012) Logg, A., Mardal, K.-A., Wells, G.N. (eds.): Automated solution of differential equations by the finite element method, volume 84 of Lecture Notes in Computational Science and Engineering. The FEniCS book. Springer, Heidelberg (2012)
24.
Zurück zum Zitat Pazy, A.: Semigroups of linear operators and applications to partial differential equations, vol. 44 of Applied Mathematical Sciences. Springer, New York (1983) Pazy, A.: Semigroups of linear operators and applications to partial differential equations, vol. 44 of Applied Mathematical Sciences. Springer, New York (1983)
25.
Zurück zum Zitat Provenzano, P.P., Lakes, R.S., Corr, D.T., Vanderby, R.: Application of nonlinear viscoelastic models to describe ligament behavior. Biomech. Model. Mechanobiol. 1(1), 45–57 (2002)CrossRef Provenzano, P.P., Lakes, R.S., Corr, D.T., Vanderby, R.: Application of nonlinear viscoelastic models to describe ligament behavior. Biomech. Model. Mechanobiol. 1(1), 45–57 (2002)CrossRef
26.
Zurück zum Zitat Provenzano, P., Lakes, R., Keenan, T., Vanderby Jr., R.: Nonlinear ligament viscoelasticity. Ann. Biomed. Eng. 29(10), 908–914 (2001)CrossRef Provenzano, P., Lakes, R., Keenan, T., Vanderby Jr., R.: Nonlinear ligament viscoelasticity. Ann. Biomed. Eng. 29(10), 908–914 (2001)CrossRef
27.
Zurück zum Zitat Rajagopal, K.R.: A note on a reappraisal and generalization of the Kelvin-Voigt model. Mech. Res. Commun. 36(2), 232–235 (2009)CrossRefMATH Rajagopal, K.R.: A note on a reappraisal and generalization of the Kelvin-Voigt model. Mech. Res. Commun. 36(2), 232–235 (2009)CrossRefMATH
28.
Zurück zum Zitat Rivière, B., Shaw, S., Wheeler, M.F., Whiteman, J.R.: Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity. Numer. Math. 95(2), 347–376 (2003)MathSciNetCrossRefMATH Rivière, B., Shaw, S., Wheeler, M.F., Whiteman, J.R.: Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity. Numer. Math. 95(2), 347–376 (2003)MathSciNetCrossRefMATH
29.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
30.
Zurück zum Zitat Rognes, M., Winther, R.: Mixed finite element methods for linear viscoelasticity using weak symmetry. Math. Models Methods Appl. Sci. 20(6), 955–985 (2010)MathSciNetCrossRefMATH Rognes, M., Winther, R.: Mixed finite element methods for linear viscoelasticity using weak symmetry. Math. Models Methods Appl. Sci. 20(6), 955–985 (2010)MathSciNetCrossRefMATH
31.
Zurück zum Zitat 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(3–4), 211–237 (1994)MathSciNetCrossRefMATH 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(3–4), 211–237 (1994)MathSciNetCrossRefMATH
32.
Zurück zum Zitat Shaw, S., Whiteman, J.R.: Numerical solution of linear quasistatic hereditary viscoelasticity problems. SIAM J. Numer. Anal. 38(1), 80–97 (2000). (electronic)MathSciNetCrossRefMATH Shaw, S., Whiteman, J.R.: Numerical solution of linear quasistatic hereditary viscoelasticity problems. SIAM J. Numer. Anal. 38(1), 80–97 (2000). (electronic)MathSciNetCrossRefMATH
33.
Zurück zum Zitat Stenberg, R.: On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math. 48(4), 447–462 (1986)MathSciNetCrossRefMATH Stenberg, R.: On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math. 48(4), 447–462 (1986)MathSciNetCrossRefMATH
35.
Zurück zum Zitat Tschoegl, N.W.: The phenomenological theory of linear viscoelastic behavior. Springer, Berlin (1989)CrossRefMATH Tschoegl, N.W.: The phenomenological theory of linear viscoelastic behavior. Springer, Berlin (1989)CrossRefMATH
Metadaten
Titel
Analysis of mixed finite element methods for the standard linear solid model in viscoelasticity
verfasst von
Jeonghun J. Lee
Publikationsdatum
30.11.2016
Verlag
Springer Milan
Erschienen in
Calcolo / Ausgabe 2/2017
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-016-0200-5

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