Advertisement

Reduced Basis Approximation and A Posteriori Error Estimation: Applications to Elasticity Problems in Several Parametric Settings

  • Dinh Bao Phuong Huynh
  • Federico Pichi
  • Gianluigi Rozza
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 15)

Abstract

In this work we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for elasticity problems in affinely parametrized geometries. The essential ingredients of the methodology are: a Galerkin projection onto a low-dimensional space associated with a smooth “parametric manifold”—dimension reduction; an efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations—rapid convergence; an a posteriori error estimation procedures—rigorous and sharp bounds for the functional outputs related with the underlying solution or related quantities of interest, like stress intensity factor; and Offline-Online computational decomposition strategies—minimum marginal cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present several illustrative results for linear elasticity problem in parametrized geometries representing 2D Cartesian or 3D axisymmetric configurations like an arc-cantilever beam, a center crack problem, a composite unit cell or a woven composite beam, a multi-material plate, and a closed vessel. We consider different parametrization for the systems: either physical quantities—to model the materials and loads—and geometrical parameters—to model different geometrical configurations—with isotropic and orthotropic materials working in plane stress and plane strain approximation. We would like to underline the versatility of the methodology in very different problems. As last example we provide a nonlinear setting with increased complexity.

Notes

Acknowledgements

We are sincerely grateful to Prof. A.T. Patera (MIT) and Dr. C.N. Nguyen (MIT) for important suggestions, remarks, insights, and codevelopers of the rbMIT and RBniCS (http://mathlab.sissa.it/rbnics) software libraries used for the numerical tests. We acknowledge the European Research Council consolidator grant H2020 ERC CoG 2015 AROMA-CFD GA 681447 (PI Prof. G. Rozza).

References

  1. 1.
    Almroth, B.O., Stern, P., Brogan, F.A.: Automatic choice of global shape functions in structural analysis. AIAA J. 16, 525–528 (1978)CrossRefGoogle Scholar
  2. 2.
    Benner, P., Gugercin, S., Willcox, K.: A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Benner, P., Cohen, A., Ohlberger, M., Willcox, K.: Model Reduction and Approximation: Theory and Algorithms. Computational Science and Engineering Series. Society for Industrial and Applied Mathematics, Philadelphia (2017)zbMATHCrossRefGoogle Scholar
  4. 4.
    Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds.): Model Reduction of Parametrized Systems. Springer, Berlin (2017)Google Scholar
  5. 5.
    Berger, M.: On Von Kármán equations and the buckling of a thin elastic plate, I the clamped plate. Commun. Pure Appl. Math. 20, 687–719 (1967)zbMATHCrossRefGoogle Scholar
  6. 6.
    Brezzi, F., Rappaz, J., Raviart, P.A.: Finite dimensional approximation of nonlinear problems. part I: branches of non singular solutions. Numer. Math. 36(1), 1–25 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Brezzi, F., Rappaz, J., Raviart, P.A.: Finite dimensional approximation of nonlinear problems. part II: limit points. Numer. Math. 37(1), 1–28 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Brezzi, F., Rappaz, J., Raviart, P.A.: Finite dimensional approximation of nonlinear problems. part III: simple bifurcation points. Numer. Math. 38(1), 1–30 (1982)zbMATHCrossRefGoogle Scholar
  9. 9.
    Canuto, C., Tonn, T., Urban, K.: A posteriori error analysis of the reduced basis method for nonaffine parametrized nonlinear PDEs. SIAM J. Numer. Anal. 47(3), 2001–2022 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Carroll, W.F.: A Primer for Finite Elements in Elastic Structures. Wiley, Hoboken (1998)Google Scholar
  11. 11.
    Chinesta, F., Ladevèze, P.: Separated Representations and PGD-Based Model Reduction: Fundamentals and Applications. CISM International Centre for Mechanical Sciences. Springer, Vienna (2014)Google Scholar
  12. 12.
    Chinesta, F., Huerta, A., Rozza, G., Willcox, K.: Model order reduction: a survey. In: Wiley Encyclopedia of Computational Mechanics. Wiley, Hoboken (2016)Google Scholar
  13. 13.
    Ciarlet, P.G.: Mathematical Elasticity, Volume I: Three-Dimensional Elasticity. Elsevier, Amsterdam (1988)zbMATHGoogle Scholar
  14. 14.
    Ciarlet, P.G.: Mathematical Elasticity: Volume II: Theory of Plates. Studies in Mathematics and its Applications. Elsevier, Amsterdam (1997)Google Scholar
  15. 15.
    Grepl, M.A., Patera, A.T.: A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. Math. Model. Numer. Anal. 39(1), 157–181 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Grepl, M., Maday, Y., Nguyen, N., Patera, A.: Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Math. Model. Numer. Anal. 41(3), 575–605 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Hesthaven, J., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics. Springer, Berlin (2015)Google Scholar
  18. 18.
    Hutchingson, J.: Nonlinear Fracture Mechanics. Monograph, Department of Solid Mechanics, Technical University, Lyngby (1979)Google Scholar
  19. 19.
    Huynh, D.B.P.: Reduced-basis approximation and applications in fracture mechanics. Ph.D. thesis, Singapore-MIT Alliance, National University of Singapore (2007)Google Scholar
  20. 20.
    Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for stress intensity factors. Int. J. Numer. Methods Eng. 72(10), 1219–1259 (2007)zbMATHCrossRefGoogle Scholar
  21. 21.
    Huynh, D.B.P., Rozza, G., Sen, S., Patera, A.T.: A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. CR Acad. Sci. Paris Ser. I 345, 473–478 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Huynh, D.B.P., Nguyen, N.C., Rozza, G., Patera, A.T.: rbMIT software http://augustine.mit.edu/methodology/methodology_rbMIT_System.htm. Copyright MIT, Technology Licensing Office, case 12600, Cambridge, MA (2007–2009)
  23. 23.
    Huynh, D., Knezevic, D.J., Chen, Y., Hesthaven, J.S., Patera, A.T.: A natural-norm Successive Constraint Method for inf-sup lower bounds. Comput. Methods Appl. Mech. Eng. 199(29–32), 1963–1975 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Milani, R., Quarteroni, A., Rozza, G.: Reduced basis method for linear elasticity problems with many parameters. Comput. Methods Appl. Mech. Eng. 197, 4812–4829 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Murakami, Y.: Stress Intensity Factors Handbook. Elsevier, Amsterdam (2001)Google Scholar
  26. 26.
    Nguyen, N., Veroy, K., Patera, A.: Certified real-time solution of parametrized partial differential equations. In: Yip, S. (ed.) Handbook of Materials Modeling: Methods, pp. 1529–1564. Springer, Dordrecht (2005)Google Scholar
  27. 27.
    Noor, A.K.: Recent advances in reduction methods for nonlinear problems. Comput. Struct. 13, 31–44 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Noor, A.K.: On making large nonlinear problems small. Comput. Methods Appl. Mech. Eng. 34, 955–985 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Noor, A.K., Peters, J.M.: Reduced basis technique for nonlinear analysis of structures. AIAA J. 18(4), 455–462 (1980)CrossRefGoogle Scholar
  30. 30.
    Parks, D.M.: A stiffness derivative finite element technique for determination of crack tip stress intensity factors. Int. J. Fract. 10(4), 487–502 (1974)CrossRefGoogle Scholar
  31. 31.
    Patera, A., Rozza, G.: Reduced basis approximation and A posteriori error estimation for Parametrized Partial Differential Equation. MIT Pappalardo Monographs in Mechanical Engineering (Copyright MIT (2007–2010)). http://augustine.mit.edu
  32. 32.
    Pichi, F., Rozza, G.: Reduced basis approaches for parametrized bifurcation problems held by non-linear Von Kármán equations (2018). arXiv:1804.02014Google Scholar
  33. 33.
    Quateroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations, 2nd edn. Springer, Berlin (1997)Google Scholar
  34. 34.
    Quarteroni, A., Rozza, G., Manzoni, A.: Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1(1), 3 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations: An Introduction. UNITEXT. Springer, Berlin (2015)Google Scholar
  36. 36.
    Rozza, G., Huynh, D., Patera, A.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics. Arch. Comput. Meth. Eng. 15, 229–275 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Rozza, G., Nguyen, N.C., Patera, A., Deparis, S.: Reduced basis methods and a posteriori error estimators for heat transfer problems. In: Proceedings of the ASME HT 2009 Summer Conference, Heat Transfer Conference. ASME, New York (2009). Paper No. HT2009-88211, pp. 753–762Google Scholar
  38. 38.
    Sneddon, I., Dautray, R., Benilan, P., Lions, J., Cessenat, M., Gervat, A., Kavenoky, A., Lanchon, H.: Mathematical Analysis and Numerical Methods for Science and Technology. Volume 1: Physical Origins and Classical Methods. Mathematical Analysis and Numerical Methods for Science and Technology. Springer, Berlin (1999)Google Scholar
  39. 39.
    Sneddon, I., Dautray, R., Benilan, P., Lions, J., Cessenat, M., Gervat, A., Kavenoky, A., Lanchon, H.: Mathematical Analysis and Numerical Methods for Science and Technology. Volume 2: Functional and Variational Methods. Mathematical Analysis and Numerical Methods for Science and Technology. Springer, Berlin (2000)Google Scholar
  40. 40.
    Veroy, K.: Reduced-basis methods applied to problems in elasticity: analysis and applications. Ph.D. thesis, Massachusetts Institute of Technology (2003)Google Scholar
  41. 41.
    Young, W., Budynas, R.: Roark’s Formulas for Stress and Strain, 7th edn. McGraw, New York (2001)Google Scholar
  42. 42.
    Zanon, L.: Model order reduction for nonlinear elasticity: applications of the reduced basis method to geometrical nonlinearity and finite deformation. Ph.D. thesis, RWTH Aachen University (2017)Google Scholar
  43. 43.
    Zanon, L., Veroy-Grepl, K.: The reduced basis method for an elastic buckling problem. Proc. Appl. Math. Mech. 13(1), 439–440 (2013)CrossRefGoogle Scholar
  44. 44.
    Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z.: The Finite Element Method: Its Basis and Fundamentals, 6th edn. Butterworth-Heinemann, Oxford (2005)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Dinh Bao Phuong Huynh
    • 1
  • Federico Pichi
    • 2
  • Gianluigi Rozza
    • 2
  1. 1.Akselos SA and Department of Mechanical EngineeringMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.mathLab, Mathematics Area, SISSAInternational School for Advanced StudiesTriesteItaly

Personalised recommendations