Structural and Multidisciplinary Optimization

, Volume 48, Issue 4, pp 711–715 | Cite as

A mesh evolution algorithm based on the level set method for geometry and topology optimization

Brief Note

Abstract

We propose an approach for structural optimization which combines the flexibility of the level set method for handling large deformations and topology changes with the accurate description of the geometry provided by an exact mesh of the shape. The key ingredients of our method are efficient algorithms for (i) moving a level set function on an unstructured mesh, (ii) remeshing the surface corresponding to the zero level set and (iii) simultaneously adaptating the volumic mesh which fits to this surfacic mesh.

Keywords

Geometry and topology optimization Level set method Local mesh modifications 

Notes

Acknowledgments

This work has been supported by the RODIN project (FUI AAP 13). G. A. is a member of the DEFI project at INRIA Saclay Ile-de-France.

References

  1. Allaire G, Dapogny C, Frey P (2011) Topology and geometry optimization of elastic structures by exact deformation of simplicial mesh. C R Acad Sci Paris, Ser I 349(17):999–1003MathSciNetCrossRefMATHGoogle Scholar
  2. Allaire G, Jouve F, Toader AM (2004) Structural optimization using shape sensitivity analysis and a level-set method. J Comput Phys 194:363–393MathSciNetCrossRefMATHGoogle Scholar
  3. Chopp D (1993) Computing minimal surfaces via level-set curvature flow. J Comput Phys 106:77–91MathSciNetCrossRefMATHGoogle Scholar
  4. Dapogny C Ph.D. thesis of Université Pierre et Marie Curie (in preparation)Google Scholar
  5. Dapogny C, Frey P (2012) Computation of the signed distance function to a discrete contour on adapted triangulation. Calcolo 49(3):193–219MathSciNetCrossRefMATHGoogle Scholar
  6. Delfour MC, Zolesio J-P (2011) Shapes and Geometries: metrics, analysis, differential calculus, and optimization, 2nd edn. SIAM, PhiladelphiaGoogle Scholar
  7. Frey PJ, George PL (2008) Mesh Generation: application to finite elements, 2nd edn. Wiley, HobokenCrossRefGoogle Scholar
  8. Ha S-H, Cho S (2008) Level set based topological shape optimization of geometrically nonlinear structures using unstructured mesh. Comput Struct 86:844–868CrossRefGoogle Scholar
  9. Osher SJ, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79:12–49MathSciNetCrossRefMATHGoogle Scholar
  10. Persson P-O, Strang G (2004) A Simple mesh generator in MATLAB. SIAM Review 46(2):329–345MathSciNetCrossRefMATHGoogle Scholar
  11. Strain J (1999) Semi-lagrangian methods for level set equations. J Comput Phys 151:498–533MathSciNetCrossRefMATHGoogle Scholar
  12. Xia Q, Shi T, Liu S, Wang MY (2012) A level set solution to the stress-based structural shape and topology optimization. Comput Struct 90–91:55–64CrossRefGoogle Scholar
  13. Yamasaki S, Nomura T, Kawamoto A, Nishiwaki S (2011) A level set-based topology optimization method targeting metallic waveguide design problems. Int J Numer Meth Engng 87:844–868MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Grégoire Allaire
    • 1
  • Charles Dapogny
    • 2
    • 3
  • Pascal Frey
    • 2
  1. 1.Centre de Mathématiques Appliquées (UMR 7641)École PolytechniquePalaiseauFrance
  2. 2.UMR 7598, Laboratoire J.-L. LionsUPMC Univ Paris 06ParisFrance
  3. 3.Renault DREAM-DELT’AGuyancourtFrance

Personalised recommendations