Optimal Calculation of Solid-Body Deformations with Prescribed Degrees of Freedom over Smooth Boundaries

  • Vitoriano Ruas
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)


One of the reasons why the finite element method became the most used technique in Computational Solid Mechanics is its versatility to deal with bodies having a curved shape. In this case method’s isoparametric version for meshes consisting of curved triangles or tetrahedra has been mostly employed to recover the optimal approximation properties known to hold for standard elements in the case of polygonal or polyhedral domains. However isoparametric finite elements helplessly require the manipulation of rational functions and the use of numerical integration. This can be a brain teaser in many cases, especially if the problem at hand is non linear. We consider a simple alternative to deal with boundary conditions commonly encountered in practical applications, that bypasses these drawbacks, without eroding the quality of the finite-element model. More particularly we mean prescribed displacements or forces in the case of solids. Our technique is based only on polynomial algebra and can do without curved elements. Although it can be applied to countless types of problems in Continuum Mechanics, it is illustrated here in the computation of small deformations of elastic solids.


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This work was partially accomplished while the author was working at PUCRio, Brazil, as a CNPq research grant holder.


  1. Bertrand F, Münzenmaier S, Starke G (2014) First-order system least squares on curved boundaries: Higher-order raviart–thomas elements. SIAM Journal on Numerical Analysis 52(6):3165–3180Google Scholar
  2. Brenner SC, RScott L (2008) The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, vol 15. SpringerGoogle Scholar
  3. Ciarlet PG (1978) The Finite Element Method for Elliptic Problems. North Holland, AmsterdamGoogle Scholar
  4. Nitsche J (1972) On dirichlet problems using subspaces with nearly zero boundary conditions. In: Aziz A (ed) The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic PressGoogle Scholar
  5. Ruas V (2017a) Methods of arbitrary optimal order with tetrahedral finite-element meshes forming polyhedral approximations of curved domains. arXiv Numerical AnalysisGoogle Scholar
  6. Ruas V (2017b) Optimal simplex finite-element approximations of arbitrary order in curved domains circumventing the isoparametric technique. ArXiv e-printsGoogle Scholar
  7. Ruas V (2017c) A simple alternative for accurate finite-element modeling in curved domains. In: Comptes-rendus du Congrès Français de Mécanique, LilleGoogle Scholar
  8. Ruas V, Silva Ramos MA (2017) A hermite fe method for maxwell’s equations. AIP Conference Proceedings 1863(1):370,003Google Scholar
  9. Scott LR (1973) Finite element techniques for curved boundaries. Phd, MITGoogle Scholar
  10. Žénišek A (1978) Curved triangular finite c m-elements. Aplikace Matematiky 23(5):346–377Google Scholar
  11. Zienkiewicz OC (1971) The Finite Element Method in Engineering Science. McGraw-Hill,Google Scholar
  12. Zlámal M (1973) Curved elements in the finite element method. i. SIAM Journal on Numerical Analysis 10(1):229–240Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.UMR 7190, Institut Jean Le Rond d’AlembertSorbonne Université, Centre National de la Recherche ScientifiqueParisFrance

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