hp-FEM for three-dimensional elastic plates

Monique Dauge; Christoph Schwab

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 4, page 597-630
  • ISSN: 0764-583X

Abstract

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In this work, we analyze hierarchic hp-finite element discretizations of the full, three-dimensional plate problem. Based on two-scale asymptotic expansion of the three-dimensional solution, we give specific mesh design principles for the hp-FEM which allow to resolve the three-dimensional boundary layer profiles at robust, exponential rate. We prove that, as the plate half-thickness ε tends to zero, the hp-discretization is consistent with the three-dimensional solution to any power of ε in the energy norm for the degree p = 𝒪 ( log ε ) and with 𝒪 ( p 4 ) degrees of freedom.

How to cite

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Dauge, Monique, and Schwab, Christoph. "hp-FEM for three-dimensional elastic plates." ESAIM: Mathematical Modelling and Numerical Analysis 36.4 (2010): 597-630. <http://eudml.org/doc/194118>.

@article{Dauge2010,
abstract = { In this work, we analyze hierarchic hp-finite element discretizations of the full, three-dimensional plate problem. Based on two-scale asymptotic expansion of the three-dimensional solution, we give specific mesh design principles for the hp-FEM which allow to resolve the three-dimensional boundary layer profiles at robust, exponential rate. We prove that, as the plate half-thickness ε tends to zero, the hp-discretization is consistent with the three-dimensional solution to any power of ε in the energy norm for the degree $p=\{\cal O\}(\left|\{\log \varepsilon\}\right|)$ and with $\{\cal O\}(\{p^4\})$ degrees of freedom. },
author = {Dauge, Monique, Schwab, Christoph},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Plates; hp-finite elements; exponential convergence; asymptotic expansion.; two-scale asymptotic expansion; energy norm},
language = {eng},
month = {3},
number = {4},
pages = {597-630},
publisher = {EDP Sciences},
title = {hp-FEM for three-dimensional elastic plates},
url = {http://eudml.org/doc/194118},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Dauge, Monique
AU - Schwab, Christoph
TI - hp-FEM for three-dimensional elastic plates
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 4
SP - 597
EP - 630
AB - In this work, we analyze hierarchic hp-finite element discretizations of the full, three-dimensional plate problem. Based on two-scale asymptotic expansion of the three-dimensional solution, we give specific mesh design principles for the hp-FEM which allow to resolve the three-dimensional boundary layer profiles at robust, exponential rate. We prove that, as the plate half-thickness ε tends to zero, the hp-discretization is consistent with the three-dimensional solution to any power of ε in the energy norm for the degree $p={\cal O}(\left|{\log \varepsilon}\right|)$ and with ${\cal O}({p^4})$ degrees of freedom.
LA - eng
KW - Plates; hp-finite elements; exponential convergence; asymptotic expansion.; two-scale asymptotic expansion; energy norm
UR - http://eudml.org/doc/194118
ER -

References

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  1. I. Babuska and L. Li, Hierarchic modelling of plates. Comput. & Structures40 (1991) 419-430.  
  2. I. Babuska and L. Li, The problem of plate modelling - theoretical and computational results. Comput. Methods Appl. Mech. Engrg.100 (1992) 249-273.  Zbl0764.73040
  3. P. Bolley, J. Camus and M. Dauge, Régularité Gevrey pour le problème de Dirichlet dans des domaines à singularités coniques. Comm. Partial Differential Equations10 (1985) 391-432.  Zbl0573.35024
  4. P.G. Ciarlet, Mathematical Elasticity II: Theory of Plates. Elsevier Publ., Amsterdam (1997).  Zbl0888.73001
  5. M. Dauge, I. Djurdjevic, E. Faou and A. Rössle, Eigenmodes asymptotic in thin elastic plates. J. Math. Pures Appl.78 (1999) 925-964.  Zbl0966.74027
  6. M. Dauge and I. Gruais, Asymptotics of arbitrary order for a thin elastic clamped plate. I: Optimal error estimates. Asymptot. Anal.13 (1996) 167-197.  Zbl0856.73029
  7. M. Dauge and I. Gruais, Asymptotics of arbitrary order for a thin elastic clamped plate. II: Analysis of the boundary layer terms. Asymptot. Anal.16 (1998) 99-124.  Zbl0941.74031
  8. M. Dauge and I. Gruais, Edge layers in thin elastic plates. Comput. Methods Appl. Mech. Engrg.157 (1998) 335-347.  Zbl0961.74040
  9. M. Dauge, I. Gruais and A. Rössle, The influence of lateral boundary conditions on the asymptotics in thin elastic plates. SIAM J. Math. Anal.31 (1999/00) 305-345 (electronic).  Zbl0958.74034
  10. E. Faou, Développements asymptotiques dans les coques linéairement élastiques. Thèse, Université de Rennes 1 (2000).  
  11. E. Faou, Élasticité linéarisée tridimensionnelle pour une coque mince : résolution en série formelle en puissances de l'épaisseur. C. R. Acad. Sci. Paris Sér. I Math.330 (2000) 415-420.  
  12. R.D. Gregory and F.Y. Wan, Decaying states of plane strain in a semi-infinite strip and boundary conditions for plate theory. J. Elasticity14 (1984) 27-64.  Zbl0536.73047
  13. B. Guo and I. Babuska, Regularity of the solutions for elliptic problems on nonsmooth domains in R3. I. Countably normed spaces on polyhedral domains. Proc. Roy. Soc. Edinburgh Sect. A127 (1997) 77-126.  Zbl0874.35019
  14. B. Guo and I. Babuska, Regularity of the solutions for elliptic problems on nonsmooth domains in R3. II. Regularity in neighbourhoods of edges. Proc. Roy. Soc. Edinburgh Sect. A127 (1997).  Zbl0884.35022
  15. V.A. Kondrat'ev, Boundary-value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc.16 (1967) 227-313.  
  16. J.M. Melenk and C. Schwab, HP FEM for reaction-diffusion equations. I. Robust exponential convergence. SIAM J. Numer. Anal.35 (1998) 1520-1557 (electronic).  Zbl0972.65093
  17. C.B. Morrey and L. Nirenberg, On the analyticity of the solutions of linear elliptic systems of partial differential equations. Comm. Pure Appl. Math.10 (1957) 271-290.  Zbl0082.09402
  18. C. Schwab, Boundary layer resolution in hierarchical models of laminated composites. RAIRO Modél. Math. Anal. Numér.28 (1994) 517-537.  Zbl0817.73038
  19. C. Schwab,p- and hp-finite element methods. Theory and applications in solid and fluid mechanics. The Clarendon Press Oxford University Press, New York (1998).  Zbl0910.73003
  20. C. Schwab and S. Wright, Boundary layer approximation in hierarchical beam and plate models. J. Elasticity38 (1995) 1-40.  Zbl0834.73040
  21. E. Stein and S. Ohnimus, Coupled model- and solution-adaptivity in the finite-element method. Comput. Methods Appl. Mech. Engrg.150 (1997) 327-350. Symposium on Advances in Computational Mechanics, Vol. 2 (Austin, TX, 1997).  Zbl0926.74127

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