Skip to main content
SpringerLink
Log in

On a Constraint-Based Regularization Technique for Configurational r-Adaptivity and 3D Shape Optimization

  • Conference paper
IUTAM Symposium on Progress in the Theory and Numerics of Configurational Mechanics

Part of the book series: IUTAM Bookseries ((IUTAMBOOK,volume 17))

  • 529 Accesses

Abstract

This contribution deals with a numerical regularization technique for con-figurational r-adaptivity and shape optimization based on a fictitious energy con-straint. The notion configurational refers to the fact that both r-adaptivity and shape optimization rely on an optimization of the potential energy with respect to changes of the (discrete) reference configuration. In the case of r-adaptivity, the minimization of the total potential energy optimizes the mesh and thus improves the accuracy of the finite element solution, whereas the maximization of the total potential energy by varying the initial shape increases the stiffness of the structure. In the context of r-adaptivity, the energy constraint sets the distortion of the mesh to a reasonable limit and improves the solvability of the problem. The application of the energy constraint to a node-based shape optimization is a remedy for well-known problems of node-based shape optimization methods with maintaining a smooth and regular boundary

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

Access this chapter

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 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Armero, F., Love, E.: An arbitrary Lagrangian—Eulerian finite element method for finite strain plasticity. Int. J. Numer. Meth. Engng. 57, 2003, 471–508.

    Article  MATH  MathSciNet  Google Scholar 

  2. Askes, H., Kuhl, E., Steinmann, P., An ALE formulation based on spatial and material settings of continuum mechanics. Part 2: Classification and applications. Comput. Methods Appl. Mech. Engrg. 193, 2004, 4223–4245.

    Article  MATH  MathSciNet  Google Scholar 

  3. Bathe, K.J., Sussman, T.D., An algorithm for the construction of optimal finite element meshes in linear elasticity. ASME, AMD, Computer Methods for Nonlinear Solids and Structural Mechanics 54, 1983, 15–36.

    Google Scholar 

  4. Belegundu, A.D., Rajan, S.D., A shape optimization approach based on natural design variables and shape functions. Comput. Methods Appl. Mech. Engrg. 66, 1988, 87–106.

    Article  MATH  Google Scholar 

  5. Belytschko, T., Liu, W.K., Moran, B., Nonlinear Finite Elements for Continua and Structures. Wiley, 2000.

    Google Scholar 

  6. Bendsøe, M.P., Sigmund, O., Topology Optimization. Springer, 2003.

    Google Scholar 

  7. Bletzinger, K.U., Firl, M., Linhard, J., Wüchner, R., Optimal shapes of mechanically motivated surfaces. Comput. Methods Appl. Mech. Engrg., 2008, published online.

    Google Scholar 

  8. Braibant, V., Fleury, C., Shape optimal design using B-splines. Comput. Methods Appl. Mech. Engrg. 44, 1984, 247–267.

    Article  MATH  Google Scholar 

  9. Braibant, V., Fleury, C., An approximation-concepts approach to shape optimal design. Comput. Methods Appl. Mech. Engrg. 53, 1985, 119–148.

    Article  MATH  MathSciNet  Google Scholar 

  10. Braun, M., Configurational forces induced by finite-element discretization. Proc. Estonian Acad. Sci. Phys. Math. 46, 1997, 24–31.

    MATH  MathSciNet  Google Scholar 

  11. Breitfeld, M.G., Shanno, D.F., A globally convergent penalty-barrier algorithm for nonlinear programming and its computational performance. Technical Report RRR 12–94, RUT-COR, 1994.

    Google Scholar 

  12. Breitfeld, M.G., Shanno, D.F., Computational experience with penalty-barrier methods for nonlinear programming. Ann. Operations Res. 62, 1996, 439–463.

    Article  MATH  MathSciNet  Google Scholar 

  13. Carroll, W.E., Barker, R.M., A theorem for optimum finite-element idealizations. Int. J. Solids Structures 9, 1973, 883–895.

    Article  MathSciNet  Google Scholar 

  14. Felippa, C.A., Numerical experiments in finite element grid optimization by direct energy search. Appl. Math. Modelling 1, 1976, 239–244.

    Article  Google Scholar 

  15. Felippa, C.A., Optimization of finite element grids by direct energy search. Appl. Math. Modelling 1, 1976, 93–96.

    Article  MATH  Google Scholar 

  16. Haftka, R.T., Grandhi, R.V., Structural shape optimization — A survey. Comput. Methods Appl. Mech. Engrg. 57, 1986, 91–106.

    Article  MATH  MathSciNet  Google Scholar 

  17. Hughes, T.J.R., The Finite Element Method. Dover Publications, 1987.

    Google Scholar 

  18. Knupp, P.M., Achieving finite element mesh quality via optimization of the Jacobian matrix norm, associated quantities. Part II — A framework for volume mesh optimization and the condition number of the Jacobian matrix. Int. J. Numer. Meth. Engng. 48, 2000, 1165–1185.

    Article  MATH  Google Scholar 

  19. Knupp, P.M., Algebraic mesh quality measures. SIAM J. Sci. Computing 23, 2001, 193–218.

    Article  MATH  MathSciNet  Google Scholar 

  20. Kuhl, E., Askes, H., Steinmann, P., An ALE formulation based on spatial and material settings of continuum mechanics. Part 1: Generic hyperelastic formulation. Comput. Methods Appl. Mech. Engrg. 193, 2004, 4207–4222.

    Article  MATH  MathSciNet  Google Scholar 

  21. Lindby, T., Santos, J.L.T., 2-D and 3-D shape optimization using mesh velocities to integrate analytical sensitivities with associative CAD. Structural Optimization 13, 1997, 213–222.

    Article  Google Scholar 

  22. Materna, D., Barthold, F.J., On variational sensitivity analysis and configurational mechanics. Comput. Mech. 41, 2008, 661–681.

    Article  MATH  MathSciNet  Google Scholar 

  23. McNeice, G.M., Marcal, P.V., Optimization of finite element grids based on minimum potential energy. Trans. ASME, J. Engrg. Ind. 95(1), 1973, 186–190.

    Google Scholar 

  24. Mosler, J., Ortiz, M., On the numerical implementation of varational arbitrary Lagrangian—Eulerian (VALE) formulations. Int. J. Numer. Meth. Engng. 67, 2006, 1272–1289.

    Article  MATH  MathSciNet  Google Scholar 

  25. Mueller, R., Kolling, S., Gross, D., On configurational forces in the context of the finite element method. Int. J. Numer. Meth. Engng. 53, 2002, 1557–1574.

    Article  MATH  MathSciNet  Google Scholar 

  26. Munson, T., Mesh shape-quality optimization using the inverse mean-ratio metric. Math. Program. 110, 2007, 561–590.

    Article  MathSciNet  Google Scholar 

  27. Scherer, M., Denzer, R., Steinmann, P., Energy-based r-adaptivity: A solution strategy and applications to fracture mechanics. Int. J. Frac. 147, 2007, 117–132.

    Article  MATH  Google Scholar 

  28. Steinmann, P., Ackermann, D., Barth, F.J., Application of material forces to hyperelasto-static fracture mechanics. Part II: Computational setting. Int. J. Solids Structures 38, 2001, 5509–5526.

    Article  MATH  Google Scholar 

  29. Sussman, T., Bathe, K.J., The gradient of the finite element variational indicator with respect to nodal point co-ordinates: An explicit calculation and application in fracture mechanics and mesh optimization. Int. J. Numer. Meth. Engng. 21, 1985, 763–774.

    Article  MATH  Google Scholar 

  30. Thoutireddy, P., Ortiz, M., A variational r-adaption and shape-optimization method for finite-deformation elasticity. Int. J. Numer. Meth. Engng. 61, 2004, 1–21.

    Article  MATH  MathSciNet  Google Scholar 

  31. Wriggers, P., Nichtlineare Finite-Element-Methoden. Springer, 2001.

    Google Scholar 

  32. Yao, T.M., Choi, K.K., 3-D shape optimal design and automatic finite element regridding. Int. J. Numer. Meth. Engng. 28, 1989, 369–384.

    Article  MATH  Google Scholar 

  33. Zhang, S., Belegundu, A.D., A systematic approach for generating velocity fields in shape optimization. Structural Optimization 5, 1992, 84–94.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chair of Applied Mechanics, University of Erlangen-Nuremberg, Egerlandstraße 5, Erlangen, D-91058, Germany

    Michael Scherer & Paul Steinmann

  2. Department of Mechanical and Structural Engineering, University of Trento, via Mesiano 77, Trento, I-38100, Italy

    Ralf Denzer

Authors
  1. Michael Scherer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ralf Denzer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Paul Steinmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Scherer .

Editor information

Editors and Affiliations

  1. Chair of Applied Mechanics, University of Erlangen-Nuremberg, Erlangen, Germany

    P. Steinmann

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science + Business Media B.V.

About this paper

Cite this paper

Scherer, M., Denzer, R., Steinmann, P. (2009). On a Constraint-Based Regularization Technique for Configurational r-Adaptivity and 3D Shape Optimization. In: Steinmann, P. (eds) IUTAM Symposium on Progress in the Theory and Numerics of Configurational Mechanics. IUTAM Bookseries, vol 17. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3447-2_2

Download citation

Publish with us

Policies and ethics

Search

Navigation