Geodesic
Hybrid level set phase field method for topology optimization of contact problems
Mathematica Bohemica, Tome 140 (2015) no. 4, pp. 419-435

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
The paper deals with the analysis and the numerical solution of the topology optimization of system governed by variational inequalities using the combined level set and phase field rather than the standard level set approach. The standard level set method allows to evolve a given sharp interface but is not able to generate holes unless the topological derivative is used. The phase field method indicates the position of the interface in a blurry way but is flexible in the holes generation. In the paper a two-phase topology optimization problem is formulated in terms of the modified level set function and regularized using the Cahn-Hilliard based interfacial energy term rather than the standard perimeter term. The derivative formulae of the cost functional with respect to the level set function is calculated. Modified reaction-diffusion equation updating the level set function is derived. The necessary optimality condition for this optimization problem is formulated. The finite element and finite difference methods are used to discretize the state and adjoint systems. Numerical examples are provided and discussed.
The paper deals with the analysis and the numerical solution of the topology optimization of system governed by variational inequalities using the combined level set and phase field rather than the standard level set approach. The standard level set method allows to evolve a given sharp interface but is not able to generate holes unless the topological derivative is used. The phase field method indicates the position of the interface in a blurry way but is flexible in the holes generation. In the paper a two-phase topology optimization problem is formulated in terms of the modified level set function and regularized using the Cahn-Hilliard based interfacial energy term rather than the standard perimeter term. The derivative formulae of the cost functional with respect to the level set function is calculated. Modified reaction-diffusion equation updating the level set function is derived. The necessary optimality condition for this optimization problem is formulated. The finite element and finite difference methods are used to discretize the state and adjoint systems. Numerical examples are provided and discussed.
DOI : 10.21136/MB.2015.144460
Classification : 35J86, 49K20, 49Q10, 49Q12, 74N20, 74P10
Keywords: topology optimization; unilateral problem; level set approach; phase field method 
Myśliński, Andrzej; Koniarski, Konrad. Hybrid level set phase field method for topology optimization of contact problems. Mathematica Bohemica, Tome 140 (2015) no. 4, pp. 419-435. doi: 10.21136/MB.2015.144460
@article{10_21136_MB_2015_144460,
     author = {My\'sli\'nski, Andrzej and Koniarski, Konrad},
     title = {Hybrid level set phase field method for topology optimization of contact problems},
     journal = {Mathematica Bohemica},
     pages = {419--435},
     year = {2015},
     volume = {140},
     number = {4},
     doi = {10.21136/MB.2015.144460},
     mrnumber = {3432543},
     zbl = {06537674},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2015.144460/}
}
TY  - JOUR
AU  - Myśliński, Andrzej
AU  - Koniarski, Konrad
TI  - Hybrid level set phase field method for topology optimization of contact problems
JO  - Mathematica Bohemica
PY  - 2015
SP  - 419
EP  - 435
VL  - 140
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2015.144460/
DO  - 10.21136/MB.2015.144460
LA  - en
ID  - 10_21136_MB_2015_144460
ER  - 
%0 Journal Article
%A Myśliński, Andrzej
%A Koniarski, Konrad
%T Hybrid level set phase field method for topology optimization of contact problems
%J Mathematica Bohemica
%D 2015
%P 419-435
%V 140
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2015.144460/
%R 10.21136/MB.2015.144460
%G en
%F 10_21136_MB_2015_144460

[1] Allaire, G., Jouve, F., Toader, A.-M.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004), 363-393. | DOI | MR | Zbl

[2] Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences 147 Springer, New York (2006). | DOI | MR | Zbl

[3] Blank, L., Garcke, H., Farshbaf-Shaker, M. Hassan, Styles, V.: Relating phase field and sharp interface approaches to structural topology optimization. ESAIM Control Optim. Calc. Var. 20 (2014), 1025-1058. | DOI | MR

[4] Blank, L., Garcke, H., Sarbu, L., Srisupattarawanit, T., Styles, V., Voigt, A.: Phase-field approaches to structural topology optimization. Constrained Optimization and Optimal Control for Partial Differential Equations G. Leugering et al. International Series of Numerical Mathematics 160 Birkhäuser, Basel 245-256 (2012). | MR

[5] Bourdin, B., Chambolle, A.: The phase-field method in optimal design. M. P. Bendsø{e} et al. IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials. Status and Perspectives. Proc. of the Conf. Rungstedgaard, Denmark, 2005 Springer, Dordrecht (2006).

[6] Chan, T. F., Vese, L. A.: Active contours without edges. IEEE Trans. Image Process. 10 (2001), 266-277. | DOI | Zbl

[7] Choi, J. S., Yamada, T., Izui, K., Nishiwaki, S., Yoo, J.: Topology optimization using a reaction-diffusion equation. Comput. Methods Appl. Mech. Eng. 200 (2011), 2407-2420. | DOI | MR | Zbl

[8] Deaton, J. D., Grandhi, R. V.: A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct. Multidiscip. Optim. 49 (2014), 1-38. | DOI | MR

[9] Droske, M., Ring, W., Rumpf, M.: Mumford-Shah based registration: a comparison of a level set and a phase field approach. Comput. Vis. Sci. 12 (2009), 101-114. | DOI | MR

[10] Gain, A. L., Paulino, G. H.: Phase-field based topology optimization with polygonal elements: a finite volume approach for the evolution equation. Struct. Multidiscip. Optim. 46 (2012), 327-342. | DOI | MR | Zbl

[11] Haslinger, J., Mäkinen, R. A. E.: Introduction to Shape Optimization. Theory, Approximation, and Computation. Advances in Design and Control 7 SIAM, Philadelphia (2003). | MR | Zbl

[12] Kronbichler, M., Kreiss, G.: A hybrid level-set-phase-field method for two-phase flow with contact lines. Technical Report 2011-026, University of Uppsala, Department of Information Technology, 2011.

[13] Myśliński, A.: Phase field approach to topology optimization of contact problems. Proc. of the 10th World Congress on Structural and Multidisciplinary Optimization R. Haftka ISSMO (2013), Paper 5434, 9 pages.

[14] Myśliński, A.: Shape and topology optimization of elastic contact problems using piecewise constant level set method. Proc. of the 11th International Conf. on Computational Structural Technology B. H. V. Topping Civil-Comp Press Stirlingshire (2012), Paper 233, 12 pages.

[15] Myśliński, A.: Radial basis function level set method for structural optimization. Control Cybern. 39 (2010), 627-645. | MR | Zbl

[16] Myśliński, A.: Level set method for shape and topology optimization of contact problems. Eng. Anal. Bound. Elem. 32 (2008), 986-994 System Modeling and Optimization 2009 IFIP Adv. Inf. Commun. Technol. 312 Elsevier, Oxford (2009), pp. 397-410 A. Korytowski et al. | DOI | MR

[17] Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences 153 Springer, New York (2003). | MR | Zbl

[18] Penzler, P., Rumpf, M., Wirth, B.: A phase-field model for compliance shape optimization in nonlinear elasticity. ESAIM, Control Optim. Calc. Var. 18 (2012), 229-258. | DOI | MR | Zbl

[19] Scherer, M., Denzer, R., Steinmann, P.: A fictitious energy approach for shape optimization. Int. J. Numer. Methods Eng. 82 (2010), 269-302. | DOI | MR | Zbl

[20] Sokołowski, J., Żochowski, A.: On topological derivative in shape optimization. Optimal Shape Design and Modelling T. Lewiński et al. Academic Printing House EXIT Warsaw, Poland (2004), 55-143. | MR

[21] Sokołowski, J., Zolésio, J.-P.: Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer Series in Computational Mathematics 16 Springer, Berlin (1992). | DOI | Zbl

[22] Takezawa, A., Nishiwaki, S., Kitamura, M.: Shape and topology optimization based on the phase field method and sensitivity analysis. J. Comput. Phys. 229 (2010), 2697-2718. | DOI | MR | Zbl

[23] Dijk, N. P. van, Maute, K., Langelaar, M., Keulen, F. van: Level-set methods for structural topology optimization: a review. Struct. Multidiscip. Optim. 48 (2013), 437-472. | DOI | MR

[24] Wallin, M., Ristinmaa, M., Askfelt, H.: Optimal topologies derived from a phase-field method. Struct. Multidiscip. Optim. 45 (2012), 171-183. | DOI | MR | Zbl

[25] Yamada, T., Izui, K., Nishiwaki, S., Takezawa, A.: A topology optimization method based on the level set method incorporating a fictitious interface energy. Comput. Methods Appl. Mech. Eng. 199 (2010), 2876-2891. | DOI | MR | Zbl

Cité par Sources :