Skip to main content
SpringerLink
Log in
Book cover

Advanced Finite Element Technologies pp 131–175Cite as

Least-Squares Mixed Finite Element Formulations for Isotropic and Anisotropic Elasticity at Small and Large Strains

  • Chapter
  • First Online:

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 566))

Abstract

The performance of least-squares finite element formulations for geometrically linear and nonlinear problems is investigated in this work. We consider different elastic material behaviors as, e.g., quasi-incompressibility and transverse isotropy. Basis for the provided element formulations is a first-order system of differential equations consisting of the residual forms of the balance of momentum, a constitutive relation, and a (redundant) residual enforcing a stronger control of the balance of moment of momentum. The sum of the squared \(L^2(\mathcal{B})\)-norms of the residuals leads to a functional, which is the basis for the related minimization problem. As unknown fields the displacements (approximated in \(W^{1,p}(\mathcal{B})\)) and the stresses (approximated in \(W^{q}({\text {div}},\mathcal{B})\)) are chosen. Here, the choice of the polynomial orders of the interpolation functions for the displacements and stresses is not restricted by the so-called LBB condition; they can be chosen independently. Numerical examples for the proposed formulations are presented and compared to standard and mixed Galerkin formulations.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    de Boer introduced the tensor cross product instead of the cofactor.

  2. 2.

    Alternative invariance requirements in terms of first/second Piola–Kirchhoff stress tensors are \( {{\varvec{Q}}}{{\varvec{P}}}({{\varvec{F}}}) = {{\varvec{P}}}({{\varvec{F}}}^+) \forall {{\varvec{Q}}}\in \mathcal{{SO}}(3) \) and \( {{\varvec{S}}}({{\varvec{F}}}) = {{\varvec{S}}}({{\varvec{F}}}^+) \forall {{\varvec{Q}}}\in \mathcal{{SO}}(3) \), respectively.

References

  • Ahrens, J., Geveci, B., & Law, C. (2005). ParaView: An end-user tool for large data visualization, visualization handbook. Champaign: Elsevier. Version 10.1 edition.

    Google Scholar 

  • Antman, S. S. (1995). Nonlinear problems of elasticity. New York: Springer.

    Book  MATH  Google Scholar 

  • Babuška, I. (1973). The finite element method with lagrangian multipliers. Numerische Mathematik, 20(3), 179–192.

    Article  MathSciNet  MATH  Google Scholar 

  • Ball, J. M. (1977a). Convexity conditions and existence theorems in non-linear elasticity. Archive of Rational Mechanics and Analysis, 63, 337–403.

    Article  MATH  Google Scholar 

  • Ball, J. M. (1977b). Constitutive inequalities and existence theorems in nonlinear elastostatics. In R. J. Knops (Ed.), Herriot Watt symposion: Nonlinear analysis and mechanics (Vol. 1, pp. 187–238). London: Pitman.

    Google Scholar 

  • Balzani, D. (2006) Polyconvex anisotropic energies and modeling of damage applied to arterial walls. Ph.D. thesis. University Duisburg-Essen, Verlag Glückauf Essen.

    Google Scholar 

  • Balzani, D., Neff, P., Schröder, J., & Holzapfel, G. A. (2006). A polyconvex framework for soft biological tissues. Adjustment to experimental data. International Journal for Numerical Methods in Engineering, 43(20), 6052–6070.

    MathSciNet  MATH  Google Scholar 

  • Bathe, K. J. (1995). Finite Element Procedures. Prentice: Prentice Hall.

    MATH  Google Scholar 

  • Bathe, K.-J. (2001). The inf-sup condition and its evaluation for mixed finite element methods. Computers and Structures, 79(2), 243–252.

    Article  MathSciNet  Google Scholar 

  • Bertrand, F., Münzenmaier, S., & Starke, G. (2014). First-order system least squares on curved boundaries. SIAM Journal on Scientific Computing, 52, 880–894.

    MATH  Google Scholar 

  • Bochev, P., & Gunzburger, M. (2009). Least-squares finite element methods. New York: Springer.

    MATH  Google Scholar 

  • Boehler, J. P. (1978). Lois de comportement anisotrope des milieux continus. Journal de Mécanique, 17(2), 153–190.

    MathSciNet  MATH  Google Scholar 

  • Boehler, J. P. (1979). A simple derivation of representations for non-polynomial constitutive equations in some cases of anisotropy. Zeitschrift für angewandte Mathematik und Mechanik, 59, 157–167.

    Article  MathSciNet  MATH  Google Scholar 

  • Boehler, J. P. (1987). Introduction to the invariant formulation of anisotropic constitutive equations. In J. P. Boehler (Ed.), Applications of tensor functions in solid mechanics (Vol. 292, pp. 13–30). CISM courses and lectures. Vienna: Springer.

    Google Scholar 

  • Braess, D. (1997). Finite Elemente (2nd ed.). Berlin: Springer.

    Book  MATH  Google Scholar 

  • Brezzi, F. (1974). On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. Revue française d’automatique, informatique, recherche opérationnelle. Analyse numérique, 8(2), 129–151.

    MathSciNet  MATH  Google Scholar 

  • Cai, Z., & Starke, G. (2003). First-order system least squares for the stress-displacement formulation: Linear elasticity. SIAM Journal on Numerical Analysis, 41, 715–730.

    Article  MathSciNet  MATH  Google Scholar 

  • Cai, Z., & Starke, G. (2004). Least-squares methods for linear elasticity. SIAM Journal on Numerical Analysis, 42, 826–842.

    Article  MathSciNet  MATH  Google Scholar 

  • Cai, Z., & Westphal, C. R. (2009). An adaptive mixed least-squares finite element method for viscoelastic fluids of Oldroyd type. Journal of Non-Newtonian Fluid Mechanics, 159, 72–80.

    Article  MATH  Google Scholar 

  • Cai, Z., Manteuffel, T. A., & McCormick, S. F. (1995). First-order system least squares for velocity-vorticity-pressure form of the Stokes equations, with application to linear elasticity. Electronic Transactions on Numerical Analysis, 3, 150–159.

    MathSciNet  MATH  Google Scholar 

  • Cai, Z., Manteuffel, T. A., & McCormick, S. F. (1997). First-order system least squares for the Stokes equation, with application to linear elasticity. SIAM Journal on Numerical Analysis, 34, 1727–1741.

    Article  MathSciNet  MATH  Google Scholar 

  • Cai, Z., Manteuffel, T. A., McCormick, S. F., & Parter, S. V. (1998). First-order system least squares (FOSLS) for planar linear elasticity: Pure traction problem. SIAM Journal on Numerical Analysis, 35, 320–335.

    Article  MathSciNet  MATH  Google Scholar 

  • Cai, Z., Lee, C.-O., Manteuffel, T. A., & McCormick, S. F. (2000a). First-order system least squares for linear elasticity: Numerical results. SIAM Journal on Scientific Computing, 21, 1706–1727.

    Article  MathSciNet  MATH  Google Scholar 

  • Cai, Z., Lee, C.-O., Manteuffel, T. A., & McCormick, S. F. (2000b). First-order system least squares for the Stokes and linear elasticity equations: Further results. SIAM Journal on Scientific Computing, 21, 1728–1739.

    Article  MathSciNet  MATH  Google Scholar 

  • Ciarlet, P. G. (1988). Mathematical elasticity: Three-dimensional elasticity (Vol. I). Amsterdam: Elsevier Science Ltd.

    MATH  Google Scholar 

  • Ciarlet, P. G. (1991). Handbook of numerical analysis, Vol II: Finite element methods, Part 1. Amsterdam: Elsevier Science Ltd.

    MATH  Google Scholar 

  • Dacorogna, B. (1989). Direct methods in the calculus of variations (1st ed., Vol. 78). Applied mathematical sciences. Berlin: Springer.

    Google Scholar 

  • de Boer, R. (1993). Vektor- und Tensorrechnung für Ingenieure. Berlin: Springer.

    MATH  Google Scholar 

  • Deang, J. M., & Gunzburger, M. D. (1998). Issues related to least-squares finite element methods for the Stokes equations. SIAM Journal on Scientific Computing, 20(3), 878–906.

    Article  MathSciNet  MATH  Google Scholar 

  • Eason, E. D. (1976). A review of least-squares methods for solving partial differential equations. International Journal for Numerical Methods in Engineering, 10, 1021–1046.

    Article  MathSciNet  MATH  Google Scholar 

  • Ebbing, V. (2010). Design of polyconvex energy functions for all anisotropy classes. Ph.D. thesis, Institut für Mechanik, Abteilung Bauwissenschaften, Fakultät für Ingenieurwissenschaften, Universität Duisburg-Essen.

    Google Scholar 

  • Ern, A., & Guermond, J.-L. (2013). Theory and practice of finite elements (Vol. 159). New York: Springer Science & Business Media.

    MATH  Google Scholar 

  • Jiang, B.-N. (1998). The least-squares finite element method. Berlin: Springer.

    Book  MATH  Google Scholar 

  • Jiang, B.-N., & Wu, J. (2002). The least-squares finite element method in elasticity. Part I: Plane stress or strain with drilling degrees of freedom. International Journal for Numerical Methods in Engineering, 53, 621–636.

    Article  MathSciNet  MATH  Google Scholar 

  • Korelc, J. (1997). Automatic generation of finite-element code by simultaneous optimization of expressions. Theoretical Computer Science, 187, 231–248.

    Article  MATH  Google Scholar 

  • Korelc, J. (2002). Multi-language and multi-environment generation of nonlinear finite element codes. Engineering with Computers, 18, 312–327.

    Article  Google Scholar 

  • Ladyzhenskaya, O. A. (1969). The mathematical theory of viscous incompressible flow (Vol. 76). New York: Gordon and Breach.

    MATH  Google Scholar 

  • Lynn, P. P., & Arya, S. K. (1973). Use of the least squares criterion in the finite element formulation. International Journal for Numerical Methods in Engineering, 6, 75–88.

    Article  MathSciNet  MATH  Google Scholar 

  • Manteuffel, T. A., McCormick, S. F., Schmidt, J. G., & Westphal, C. R. (2006). First-order system least squares (FOSLS) for geometrically nonlinear elasticity. SIAM Journal on Numerical Analysis, 44, 2057–2081.

    Article  MathSciNet  MATH  Google Scholar 

  • Marsden, J. E., & Hughes, J. R. (1983). Mathematical foundations of elasticity. Prentice: Prentice-Hall.

    MATH  Google Scholar 

  • Morrey, C. B. (1952). Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific Journal of Mathematics, 2, 25–53.

    Article  MathSciNet  MATH  Google Scholar 

  • Morrey, C. B. (1966). Multiple integrals in the calculus of variations. Berlin: Springer.

    MATH  Google Scholar 

  • Müller, B., Starke, G., Schwarz, A., & Schröder, J. (2014). A first-order system least squares method for hyperelasticity. SIAM Journal on Scientific Computing, 36, 795–816.

    Article  MathSciNet  MATH  Google Scholar 

  • Neumann, F. E. (1885). Vorlesungen über die Theorie der Elastizität der festen Körper und des Lichtäthers. Leipzig: Teubner.

    MATH  Google Scholar 

  • Pontaza, J. P. (2003). Least-squares variational principles and the finite element method: theory, form, and model for solid and fluid mechanics. Ph.D. thesis. Texas A&M University.

    Google Scholar 

  • Pontaza, J. P., & Reddy, J. N. (2003). Spectral/\(hp\) least-squares finite element formulation for the Navier-Stokes equation. Journal of Computational Physics, 190, 523–549.

    Article  MathSciNet  MATH  Google Scholar 

  • Raviart, P. A., & Thomas, J. M. (1977). A mixed finite element method for 2nd order elliptic problems. Mathematical aspects of finite element methods. Lecture notes in mathematics. New York: Springer.

    Google Scholar 

  • Schröder, J. (2010). Anisotropic polyconvex energies. In J. Schröder & P. Neff (Eds.), Poly-, quasi- and rank-one convexity in applied mechanics (Vol. 516, pp. 53–106). CISM courses and lectures. Vienna: Springer.

    Google Scholar 

  • Schröder, J., & Neff, P. (2001). On the construction of polyconvex anisotropic free energy functions. In Miehe, C. (Ed.) Proceedings of the IUTAM Symposium Computational Mechanics of Solid Materials at Large Strains (pp. 171–180). Kluwer Academic Publishers.

    Google Scholar 

  • Schröder, J., & Neff, P. (2003). Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions. International Journal of Solids and Structures, 40, 401–445.

    Article  MathSciNet  MATH  Google Scholar 

  • Schröder, J., Neff, P., & Ebbing, V. (2008). Anisotropic polyconvex energies on the basis of crystallographic motivated structural tensors. Journal of the Mechanics and Physics of Solids, 56(12), 3486–3506.

    Article  MathSciNet  MATH  Google Scholar 

  • Schwarz, A. (2009). Least-squares mixed finite elements for solid mechanics. Ph.D. thesis, University Duisburg-Essen.

    Google Scholar 

  • Schwarz, A., & Schröder, J. (2007). Least-squares mixed finite elements with applications to anisotropic elasticity and viscoplasticity. Proceedings of Applied Mathematics and Mechanics, 7, 4040043–4040044.

    Article  Google Scholar 

  • Schwarz, A., Schröder, J., & Starke, G. (2009). Least-squares mixed finite elements for small strain elasto-viscoplasticity. International Journal for Numerical Methods in Engineering, 77, 1351–1370.

    Article  MathSciNet  MATH  Google Scholar 

  • Schwarz, A., Schröder, J., & Starke, G. (2010). A modified least-squares mixed finite element with improved momentum balance. International Journal for Numerical Methods in Engineering, 81, 286–306.

    MathSciNet  MATH  Google Scholar 

  • Schwarz, A., Steeger, K., & Schröder, J. (2014). Weighted overconstrained least-squares mixed finite elements for static and dynamic problems in quasi-incompressible elasticity. Computational Mechanics, 54(1), 603–612.

    Article  MathSciNet  MATH  Google Scholar 

  • Schwarz, A., Steeger, K., & Schröder, J. (2015). Weighted overconstrained least-squares mixed finite elements for hyperelasticity. Proceedings of Applied Mathematics and Mechanics, 15, 227–228.

    Article  MATH  Google Scholar 

  • Smith, G. F. (1970). On a fundamental error in two papers of C.-C. Wang, “On representations for isotropic functions, Parts I and II”. Archive for Rational Mechanics and Analysis, 36, 161–165.

    Article  MathSciNet  MATH  Google Scholar 

  • Smith, G. F. (1971). On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. International Journal of Engineering Science, 9, 899–916.

    Article  MathSciNet  MATH  Google Scholar 

  • Smith, G. F., Smith, M. M., & Rivlin, R. S. (1963). Integrity bases for a symmetric tensor and a vector. The crystal classes. Archive for Rational Mechanics and Analysis, 12, 93–133.

    Article  MathSciNet  MATH  Google Scholar 

  • Spencer, A. J. M. (1965). Isotropic integrity bases for vectors and second-order tensors. Archive for Rational Mechanics and Analysis, 18, 51–82.

    Article  MathSciNet  MATH  Google Scholar 

  • Spencer, A. J. M. (1971). Theory of invariants. In A. C. Eringen (Ed.), Continuum physics (Vol. 1, pp. 239–353). New-York: Academic Press.

    Google Scholar 

  • Tchonkova, M., & Sture, S. (1997). A mixed least squares method for solving problems in linear elasticity: Formulation and initial results. Computational Mechanics, 19, 317–326.

    Article  MathSciNet  MATH  Google Scholar 

  • Tchonkova, M., & Sture, S. (2002). A mixed least squares method for solving problems in linear elasticity: Theoretical study. Computational Mechanics, 29, 332–339.

    Article  MathSciNet  MATH  Google Scholar 

  • Truesdell, C., & Noll, W. (1965). The nonlinear field theories of mechanics. In S. Flügge (Ed.), Handbuch der Physik III/3. Berlin: Springer.

    Google Scholar 

  • Šilhavý, M. (1997). The mechanics and thermodynamics of continuous media. Berlin: Springer.

    MATH  Google Scholar 

  • Wang, C.-C. (1969a). On representations for isotropic functions. Part I. Isotropic functions of symmetric tensors and vectors. Archive for Rational Mechanics and Analysis, 33, 249–267.

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, C.-C. (1969b). On representations for isotropic functions. Part II. Isotropic functions of skew-symmetric tensors, symmetric tensors, and vectors. Archive for Rational Mechanics and Analysis, 33, 268–287.

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, C.-C. (1970a). A new representation theorem for isotropic functions: An answer to professor G. F. Smith’s Criticism of my papers on representations for isotropic functions. Part 1. Scalar-valued isotropic functions. Archive for Rational Mechanics and Analysis, 36, 166–197.

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, C.-C. (1970b). A new representation theorem for isotropic functions: An answer to professor G. F. Smith’s criticism of my papers on representations for isotropic functions. Part 2. Vector-valued isotropic functions, symmetric tensor-valued isotropic functions, and skew-symmetric tensor-valued isotropic functions. Archive for Rational Mechanics and Analysis, 36, 198–223.

    Article  MathSciNet  MATH  Google Scholar 

  • Wang, C.-C. (1971). Corrigendum to my recent papers on “representations for isotropic functions”. Archive for Rational Mechanics and Analysis, 43, 392–395.

    Article  MathSciNet  Google Scholar 

  • Wolfram Research Inc. (2015). Mathematica. Champaign: Wolfram Research, Inc. Version 10.1 edition.

    Google Scholar 

  • Wriggers, P. (2001). Nichtlineare Finite-Element-Methoden. Berlin: Springer.

    Book  Google Scholar 

  • Zheng, Q.-S. (1994). Theory of representations for tensor functions - a unified invariant approach to constitutive equations. Applied Mechanics Reviews, 47, 545–587.

    Article  Google Scholar 

  • Zhu, J. Z., Taylor, R. L., & Zienkiewicz, O. C. (2005). The finite element method: its basis and fundamentals. Oxford: Butterworth Heinemann.

    MATH  Google Scholar 

  • Zienkiewicz, O. C., Owen, D. R., & Lee, K. N. (1974). Least square-finite element for elasto-static problems. Use of ‘reduced’ integration. International Journal for Numerical Methods in Engineering, 8, 341–358.

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the German Research Foundation (DFG) for financial support: research grant SCHR 570/14-1. Furthermore, the authors are grateful to Gerhard Starke and Benjamin Müller with whom they have shared inspiring work on least-squares finite element methods. Special thanks go to Nils Viebahn for many hours of support especially for the implementation and visualization and to Maximilian Igelbüscher, Carina Nisters, and Serdar Serdas for fruitfull discussions.

Author information

Authors and Affiliations

  1. Faculty of Engineering, Department of Civil Engineering, Institute of Mechanics, University of Duisburg-Essen, Universitätsstr. 15, 45141, Essen, Germany

    Jörg Schröder, Alexander Schwarz & Karl Steeger

Authors
  1. Jörg Schröder
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexander Schwarz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Karl Steeger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jörg Schröder .

Editor information

Editors and Affiliations

  1. Institute of Mechanics, University of Duisburg-Essen, Essen, Germany

    Jörg Schröder

  2. University of Hannover, Hannover, Germany

    Peter Wriggers

Rights and permissions

Reprints and permissions

Copyright information

© 2016 CISM International Centre for Mechanical Sciences

About this chapter

Cite this chapter

Schröder, J., Schwarz, A., Steeger, K. (2016). Least-Squares Mixed Finite Element Formulations for Isotropic and Anisotropic Elasticity at Small and Large Strains. In: Schröder, J., Wriggers, P. (eds) Advanced Finite Element Technologies. CISM International Centre for Mechanical Sciences, vol 566. Springer, Cham. https://doi.org/10.1007/978-3-319-31925-4_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-31925-4_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-31923-0

  • Online ISBN: 978-3-319-31925-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation