nach oben

Journal of Scientific Computing

Erschienen in:

17.02.2017

A New Mixed Finite Element Method for Elastodynamics with Weak Symmetry

verfasst von: Carlos García, Gabriel N. Gatica, Salim Meddahi

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2017

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We provide a new mixed finite element analysis for linear elastodynamics with reduced symmetry. The problem is formulated as a second order system in time by imposing only the Cauchy stress tensor and the rotation as primary and secondary variables, respectively. We prove that the resulting variational formulation is well-posed and provide a convergence analysis for a class of \({\mathrm {H}}(\mathop {{\mathrm {div}}}\nolimits )\)-conforming semi-discrete schemes. In addition, we use the Newmark trapezoidal rule to obtain a fully discrete version of the problem and carry out the corresponding convergence analysis. Finally, numerical tests illustrating the performance of the fully discrete scheme are presented.

Vorheriger Artikel A Numerical Strategy for Freestream Preservation of the High Order Weighted Essentially Non-oscillatory Schemes on Stationary Curvilinear Grids

Nächster Artikel Regularization of Singularities in the Weighted Summation of Dirac-Delta Functions for the Spectral Solution of Hyperbolic Conservation Laws

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Arnold, D.N., Brezzi, F., Douglas, J.: PEERS: A new mixed finite element method for plane elasticity. Japan J. Appl. Math. 1, 347–367 (1984)MathSciNetCrossRefMATH

Arnold, D.N., Falk, R.S., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comp. 76, 1699–1723 (2007)MathSciNetCrossRefMATH

Arnold, D.N., Lee, J.J.: Mixed methods for elastodynamics with weak symmetry. SIAM J. Numer. Anal. 52, 274–2769 (2014)MathSciNetCrossRefMATH

Bécache, E., Joly, P., Tsogka, C.: A new family of mixed finite elements for the linear elastodynamic problem. SIAM J. Numer. Anal. 39, 2109–2132 (2002)MathSciNetCrossRefMATH

Boffi, D., Brezzi, F., Fortin, M.: Reduced symmetry elements in linear elasticity. Comm. Pure Appl. Anal. 8, 1–28 (2009)MathSciNetMATH

Boulaajine, L., Farhloul, M., Paquet, L.: A priori error estimation for the dual mixed finite element method of the elastodynamic problem in a polygonal domain II. J. Comput. Appl. Math. 235(5), 1288–1310 (2011)MathSciNetCrossRefMATH

Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013)MATH

Cockburn, B., Gopalakrishnan, J., Guzmán, J.: A new elasticity element made for enforcing weak stress symmetry. Math. Comp. 79, 1331–1349 (2010)MathSciNetCrossRefMATH

Douglas Jr., J., Gupta, C.P.: Superconvergence for a mixed finite element method for elastic wave propagation in a plane domain. Numer. Math. 49, 189–202 (1986)MathSciNetCrossRefMATH

10.

Evans, L.C.: Partial Differential Equations. Second edition. Graduate Studies in Mathematics, vol 19. American Mathematical Society, Providence, RI (2010)

11.

Farhloul, M., Fortin, M.: Dual hybrid methods for the elasticity and the Stokes problems: a unified approach. Numer. Math. 76, 419–440 (1997)MathSciNetCrossRefMATH

12.

Gatica, G.N.: Analysis of a new augmented mixed finite element method for linear elasticity allowing \(\mathbb{RT}_0-{\mathbb{P}}_1-{\mathbb{P}}_0\) approximations. M2AN. Math. Model. Numer. Anal. 40, 1–28 (2006)MathSciNetCrossRef

13.

Gatica, G.N.: A simple introduction to the mixed finite element method. Theory and Applications. Springer Briefs in Mathematics. Springer, Cham (2014)CrossRefMATH

14.

Gopalakrishnan, J., Guzmán, J.: A second elasticity element using the matrix bubble. IMA J. Numer. Anal. 32, 352–372 (2012)MathSciNetCrossRefMATH

15.

Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numerica 11, 237–339 (2002)MathSciNetCrossRefMATH

16.

Logg, A., Mardal, K.-A., Wells, G.N., et al.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)CrossRefMATH

17.

Makridakis, ChG: On mixed finite element methods for linear elastodynamics. Numer. Math. 61, 235–260 (1992)MathSciNetCrossRefMATH

18.

Meddahi, S., Mora, D., Rodríguez, R.: Finite element spectral analysis for the mixed formulation of the elasticity equations. SIAM J. Numer. Anal. 51, 1041–1063 (2013)MathSciNetCrossRefMATH

19.

Renardy, M., Rogers, R.: An introduction to Partial Differential Equations. Texts in Applied Mathematics, vol. 13. Springer, New York (2004)MATH

20.

Roubíček, T.: Nonlinear Partial Differential Equations with Applications. 2nd edition. International Series of Numerical Mathematics, vol. 153. Birkhuser/Springer Basel AG, Basel (2013)

21.

Stenberg, R.: A family of mixed finite elements for the elasticity problem. Numer. Math. 53, 513–538 (1988)MathSciNetCrossRefMATH

Titel: A New Mixed Finite Element Method for Elastodynamics with Weak Symmetry
verfasst von: Carlos García
Gabriel N. Gatica
Salim Meddahi
Publikationsdatum: 17.02.2017
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 3/2017
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0384-0

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 3/2017

Fast Iterative Method with a Second-Order Implicit Difference Scheme for Time-Space Fractional Convection–Diffusion Equation

Improved Approximation of Phase-Space Densities on Triangulated Domains Using Discrete Flow Mapping with p-Refinement

A Numerical Strategy for Freestream Preservation of the High Order Weighted Essentially Non-oscillatory Schemes on Stationary Curvilinear Grids

A Parallel Finite Element Method for 3D Two-Phase Moving Contact Line Problems in Complex Domains

High-Order Accurate Adaptive Kernel Compression Time-Stepping Schemes for Fractional Differential Equations

Regularization of Singularities in the Weighted Summation of Dirac-Delta Functions for the Spectral Solution of Hyperbolic Conservation Laws