| Title:
|
Hybrid level set phase field method for topology optimization of contact problems (English) |
| Author:
|
Myśliński, Andrzej |
| Author:
|
Koniarski, Konrad |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
140 |
| Issue:
|
4 |
| Year:
|
2015 |
| Pages:
|
419-435 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
topology optimization |
| Keyword:
|
unilateral problem |
| Keyword:
|
level set approach |
| Keyword:
|
phase field method |
| MSC:
|
35J86 |
| MSC:
|
49K20 |
| MSC:
|
49Q10 |
| MSC:
|
49Q12 |
| MSC:
|
74N20 |
| MSC:
|
74P10 |
| idZBL:
|
Zbl 06537674 |
| idMR:
|
MR3432543 |
| DOI:
|
10.21136/MB.2015.144460 |
| . |
| Date available:
|
2015-11-17T20:48:35Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144460 |
| . |
| Reference:
|
[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. Zbl 1136.74368, MR 2033390, 10.1016/j.jcp.2003.09.032 |
| Reference:
|
[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). Zbl 1110.35001, MR 2244145, 10.1007/978-0-387-44588-5 |
| Reference:
|
[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. MR 3264233, 10.1051/cocv/2014006 |
| Reference:
|
[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 3060477 |
| Reference:
|
[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). |
| Reference:
|
[6] Chan, T. F., Vese, L. A.: Active contours without edges.IEEE Trans. Image Process. 10 (2001), 266-277. Zbl 1039.68779, 10.1109/83.902291 |
| Reference:
|
[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. Zbl 1230.74151, MR 2803635, 10.1016/j.cma.2011.04.013 |
| Reference:
|
[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. MR 3182450, 10.1007/s00158-013-0956-z |
| Reference:
|
[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. MR 2485788, 10.1007/s00791-008-0084-2 |
| Reference:
|
[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. Zbl 1274.74332, MR 2969787, 10.1007/s00158-012-0781-9 |
| Reference:
|
[11] Haslinger, J., Mäkinen, R. A. E.: Introduction to Shape Optimization. Theory, Approximation, and Computation.Advances in Design and Control 7 SIAM, Philadelphia (2003). Zbl 1020.74001, MR 1969772 |
| Reference:
|
[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. |
| Reference:
|
[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. |
| Reference:
|
[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. |
| Reference:
|
[15] Myśliński, A.: Radial basis function level set method for structural optimization.Control Cybern. 39 (2010), 627-645. Zbl 1283.49052, MR 2791364 |
| Reference:
|
[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. MR 2648854, 10.1016/j.enganabound.2007.12.008 |
| Reference:
|
[17] Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces.Applied Mathematical Sciences 153 Springer, New York (2003). Zbl 1026.76001, MR 1939127 |
| Reference:
|
[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. Zbl 1251.49054, MR 2887934, 10.1051/cocv/2010045 |
| Reference:
|
[19] Scherer, M., Denzer, R., Steinmann, P.: A fictitious energy approach for shape optimization.Int. J. Numer. Methods Eng. 82 (2010), 269-302. Zbl 1188.74045, MR 2656022, 10.1002/nme.2764 |
| Reference:
|
[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 1691940 |
| Reference:
|
[21] Sokołowski, J., Zolésio, J.-P.: Introduction to Shape Optimization. Shape Sensitivity Analysis.Springer Series in Computational Mathematics 16 Springer, Berlin (1992). Zbl 0761.73003, 10.1007/978-3-642-58106-9_1 |
| Reference:
|
[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. Zbl 1185.65109, MR 2586210, 10.1016/j.jcp.2009.12.017 |
| Reference:
|
[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. MR 3107583, 10.1007/s00158-013-0912-y |
| Reference:
|
[24] Wallin, M., Ristinmaa, M., Askfelt, H.: Optimal topologies derived from a phase-field method.Struct. Multidiscip. Optim. 45 (2012), 171-183. Zbl 1274.74408, MR 2872841, 10.1007/s00158-011-0688-x |
| Reference:
|
[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. Zbl 1231.74365, MR 2740765, 10.1016/j.cma.2010.05.013 |
| . |
