Skip to main content
SpringerLink
Log in

Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

We discuss, in this paper, a flux-free method for the computation of strict upper bounds of the energy norm of the error in a Finite Element (FE) computation. The bounds are strict in the sense that they refer to the difference between the displacement computed on the FE mesh and the exact displacement, solution of the continuous equations, rather than to the difference between the displacements computed on two FE meshes, one coarse and one refined. This method is based on the resolution of a series of local problems on patches of elements and does not require the resolution of a previous problem of flux equilibration, as happens with other methods. The paper concentrates more specifically on linear solid mechanics issues, and on the assessment of the energy norm of the error, seen as a necessary tool for the estimation of the error in arbitrary quantities of interest (linear functional outputs). Applications in both 2D and 3D are presented.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Babuška I, Oden JT (2004) Verification and validation in computational engineering and science: basic concepts. Comp Meths Appl Mech Eng 193: 4057–4066

    Article  MATH  Google Scholar 

  2. Ainsworth M, Oden JT (2000) A posteriori error estimation in finite element analysis. Pure and applied mathematics. Wiley-Interscience, New York

    Google Scholar 

  3. Wiberg NE, Díez P (2006) Adaptive modeling and simulation. Comp Meths Appl Mech Eng 195(4–6): 205–480

    Article  MATH  Google Scholar 

  4. Stein E, Stephan EP (2007) Error controlled hp-adaptive FE and FE-BE methods for variational equalities and inequalities including model adaptivity. Comp Mech 39(5): 555–680

    Article  Google Scholar 

  5. Ladevèze P, Pelle JP (2005) Mastering calculations in linear and nonlinear mechanics. Mechanical engineering. Springer, Heidelberg

    Google Scholar 

  6. Zienkiewicz OC, Zhu JZ (1987) A simple error estimator and adaptive procedure for practical engineering analysis. Int J Numer Meth Eng 24: 337–357

    Article  MATH  MathSciNet  Google Scholar 

  7. Babuška I, Rheinboldt WC (1978) Error estimates for adaptive finite element computations. SIAM J Numer Anal 15(4): 736–755

    Article  MATH  MathSciNet  Google Scholar 

  8. Babuška I, Rheinboldt WC (1978) A posteriori error estimates for the finite element method. Int J Numer Meth Eng 12(10): 1597–1615

    Article  MATH  Google Scholar 

  9. Ladevèze P, Leguillon D (1983) Error estimate procedure in the finite element method and applications. SIAM J Numer Anal 20(3): 485–509

    Article  MATH  MathSciNet  Google Scholar 

  10. Demkowicz L, Oden JT, Strouboulis T (1984) Adaptative finite elements for flow problems with moving boundaries. Part I: variational principles and a posteriori error estimates. Comp Meths Appl Mech Eng 46(2): 217–251

    Article  MATH  MathSciNet  Google Scholar 

  11. Zhu JZ (1997) A posteriori error estimation, the relationship between different procedures. Comp Meths Appl Mech Eng 150(1–4): 411–422

    Article  MATH  Google Scholar 

  12. Choi HW, Paraschivoiu M (2004) Adaptive computations of a posteriori finite element output bounds: a comparison of the ‘hybrid-flux’ approach and the ‘flux-free’ approach. Comp Meths Appl Mech Eng 193(36–38): 4001–4033

    Article  MATH  Google Scholar 

  13. Parés N, Díez P, Huerta A (2006) Subdomain-based flux-free a posteriori error estimators. Comp Meths Appl Mech Eng 195(4–6): 297–323

    Article  MATH  Google Scholar 

  14. Carstensen C, Funken SA (1999-2000) Fully reliable localized error control in the FEM. SIAM J Sci Comput 21(4): 1465–1484

    Article  MATH  MathSciNet  Google Scholar 

  15. Machiels L, Maday Y, Patera AT (2000) A ‘flux-free’ nodal Neumann subproblem approach to output bounds for partial differential equations. Comptes-Rendus de l’Académie des Sci Ser I Math 330(3): 249–254

    Article  MATH  MathSciNet  Google Scholar 

  16. Morin P, Nochetto RH, Siebert KG (2003) Local problems on stars: a posteriori error estimators, convergence, and performance. Math Comp 72: 1067–1097

    Article  MATH  MathSciNet  Google Scholar 

  17. Prudhomme S, Nobile F, Chamoin L, Oden JT (2004) Analysis of a subdomain-based error estimator for finite element approximations of elliptic problems. Numer Meths Partial Diff Eqs 20(2): 165–192

    Article  MATH  MathSciNet  Google Scholar 

  18. Moitinho de Almeida JP, Maunder EAW (2000) Recovery of equilibrium on star patches using a partition of unity technique. Int J Numer Meth Eng (accepted). doi:10.1002/nme.2623

  19. Parés N (2005) Error assessment for functional outputs of PDE’s: bounds and goal-oriented adaptivity. Ph.D. thesis, Universitat Politècnica de Catalunya, Barcelona, Spain

  20. Sauer-Budge AM, Bonet J, Huerta A, Peraire J (2004) Computing bounds for linear functionals of exact weak solutions to Poisson’s equation. SIAM J Numer Anal 42(4): 1610–1630

    Article  MATH  MathSciNet  Google Scholar 

  21. Parés N, Bonet J, Huerta A, Peraire J (2006) The computation of bounds for linear-functional outputs of weak solutions to the two-dimensional elasticity equations. Comp Meths Appl Mech Eng 195(4–6): 406–429

    Article  MATH  Google Scholar 

  22. Parés N, Díez P, Huerta A (2008) Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the discontinuous galerkin time discretization. Comp Meths Appl Mech Eng 197(19–20): 1641–1660

    Google Scholar 

  23. Parés N, Díez P, Huerta A (2008) Bounds of functional outputs for parabolic problems. Part II: Bounds of the exact solution. Comp Meths Appl Mech Eng 197(19–20): 1661–1679

    Google Scholar 

  24. Rannacher R, Suttmeier FT (1997) A feed-back approach to error control in finite element methods: application to linear elasticity. Comp Mech 19(5): 434–446

    Article  MATH  MathSciNet  Google Scholar 

  25. Prudhomme S, Oden JT (1999) On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comp Meths Appl Mech Eng 176(1–4): 313–331

    Article  MATH  MathSciNet  Google Scholar 

  26. Sarrate J, Peraire J, Patera AT (1999) Finite element error bounds for nonlinear outputs of the Helmholtz equation. Int J Numer Meth Fluids 11(1): 17–36

    Article  MathSciNet  Google Scholar 

  27. Oden JT, Prudhomme S (2001) Goal-oriented error estimation and adaptivity for the finite element method. Comp Maths Appl 41: 735–756

    Article  MATH  MathSciNet  Google Scholar 

  28. Díez P, Calderón G (2007) Goal-oriented error estimation for transient parabolic problems. Comp Mech 39(5): 631–646

    Article  MATH  Google Scholar 

  29. Rüter M, Stein E (2007) On the duality of finite element discretization error control in computational newtonian and eshelbian mechanics. Comp Mech 39(5): 609–630

    Article  MATH  Google Scholar 

  30. Cast3m. http://www-cast3m.cea.fr. Last accessed: 8 August (2008)

  31. Matlab. http://www.mathworks.com/. Last accessed: 8 August (2008)

  32. Peraire J, Patera AT (1997) Bounds for linear-functional outputs of coercive partial differential equations: local indicators and adaptive refinement. In: Ladevèze P, Oden JT (eds) Workshop on new advances in adaptive computational methods in mechanics. Elsevier Science Ltd, Oxford, pp 199–216

    Google Scholar 

  33. Paraschivoiu M, Peraire J, Patera AT (1997) A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations. Comp Meths Appl Mech Eng 150(1–4): 23–50

    MathSciNet  Google Scholar 

  34. Ladevèze P (2007) Strict upper error bounds on computed outputs of interest in computational structural mechanics. Comp Mech 42(2): 271–286

    Article  Google Scholar 

  35. Chamoin L, Ladevèze P (2007) Bounds on history-dependent or independent local quantities in viscoelasticity problems solved by approximate methods. Int J Numer Meth Eng 71: 1387–1411

    Article  Google Scholar 

Download references

Author information

Author notes

  1. Régis Cottereau

    Present address: Laboratoire MSSMat, CNRS UMR 8579 École Centrale Paris, Châtenay-Malabry, France

Authors and Affiliations

  1. International Center for Numerical Methods in Engineering (CIMNE), E. T. S. d’Enginyers de Camins, Canals i Ports de Barcelona, Universitat Politècnica de Catalunya, Barcelona, Spain

    Régis Cottereau

  2. Laboratori de Càlcul Numèric, Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Barcelona, Spain

    Pedro Díez & Antonio Huerta

Authors
  1. Régis Cottereau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pedro Díez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Antonio Huerta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Régis Cottereau.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cottereau, R., Díez, P. & Huerta, A. Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method. Comput Mech 44, 533–547 (2009). https://doi.org/10.1007/s00466-009-0388-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-009-0388-1

Keywords

Search

Navigation