Abstract
This paper deals with simultaneous topology and shape optimization of elastic contact problems. The structural optimization problem for an elastic contact problem is formulated. Shape as well as topological derivatives formulae of the cost functional are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization and to calculate a descent direction in numerical algorithm. Level set based numerical algorithm for the solution of this optimization problem is proposed. Numerical examples are provided and discussed.
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Allaire, G., Jouve, F., Toader, A.: Structural optimization using sensitivity analysis and a level set method. Journal of Computational Physics 194(1), 363–393 (2004)
Burger, M., Hackl, B., Ring, W.: Incorporating topological derivatives into level set methods. Journal of Computational Physics 194(1), 344–362 (2004)
Bergonioux, M., Kunisch, K.: Augmented Lagrangian techniques for elliptic state constrained optimal control problems. SIAM Journal on Control and Optimization 35, 1524–1543 (1997)
Chopp, H., Dolbow, J.: A hybrid extended finite element / level set method for modelling phase transformations. International Journal for Numerical Methods in Engineering 54, 1209–1232 (2002)
Delfour, M., Zolesio, J.-P.: Shapes and Geometries: Analysis, Differential Calculus and Optimization. SIAM Publications, Philadelphia (2001)
Fulmański, P., Laurain, A., Scheid, J.F., Sokołowski, J.: A Level Set Method in Shape and Topology Optimization for Variational Inequalities. Int. J. Appl. Math. Comput. Sci. 17, 413–430 (2007)
Garreau, S., Guillaume, Ph., Masmoudi, M.: The topological asymptotic for PDE systems: the elasticity case. SIAM Journal on Control Optimization 39, 1756–1778 (2001)
Gomes, A., Suleman, A.: Application of spectral level set methodology in topology optimization. Structural Multidisciplinary Optimization 31, 430–443 (2006)
de Gourmay, F.: Velocity extension for the level set method and multiple eigenvalue in shape optimization. SIAM Journal on Control and Optimization 45(1), 343–367 (2006)
Haslinger, J., Mäkinen, R.: Introduction to Shape Optimization. Theory, Approximation, and Computation. SIAM Publications, Philadelphia (2003)
He, L., Kao, Ch.Y., Osher, S.: Incorporating topological derivatives into shape derivatives based level set methods. Journal of Computational Physics 225, 891–909 (2007)
Hintermüller, M., Ring, W.: A level set approach for the solution of a state-constrained optimal control problem. Numerische Mathematik 98, 135–166 (2004)
Myśliński, A.: Level set method for shape optimization of contact problems. In: Neittaanmäki, P., Jyväskylä. (eds.) Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering, Finland (2004)
Myśliński, A.: Topology and shape optimization of contact problems using a level set method. In: Herskovits, J., Mazorche, S., Canelas, A. (eds.) Proceedings of VI World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil (2005)
Myśliński, A.: Shape Optimization of Nonlinear Distributed Parameter Systems. Academic Printing House EXIT, Warsaw (2006)
Myśliński, A.: Level Set Method for Optimization of Contact Problems. Engineering Analysis with Boundary Elements 32, 986–994 (2008)
Novotny, A.A., Feijóo, R.A., Padra, C., Tarocco, E.: Topological derivative for linear elastic plate bending problems. Control and Cybernetics 34(1), 339–361 (2005)
Norato, J.A., Bendsoe, M.P., Haber, R., Tortorelli, D.A.: A topological derivative method for topology optimization. Structural Multidisciplinary Optimization 33, 375–386 (2007)
Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2003)
Sokołowski, J., Zolesio, J.-P.: Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer, Berlin (1992)
Sokołowski, J., Żochowski, A.: Optimality conditions for simultaneous topology and shape optimization. SIAM Journal on Control 42(4), 1198–1221 (2003)
Sokołowski, J., Żochowski, A.: On topological derivative in shape optimization. In: Lewiński, T., Sigmund, O., Sokołowski, J., Żochowski, A. (eds.) Optimal Shape Design and Modelling, pp. 55–143. Academic Printing House EXIT, Warsaw (2004)
Sokołowski, J., Żochowski, A.: Modelling of topological derivatives for contact problems. Numerische Mathematik 102(1), 145–179 (2005)
Stadler, G.: Semismooth Newton and augmented Lagrangian methods for a simplified friction problem. SIAM Journal on Optimization 15(1), 39–62 (2004)
Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering 192, 227–246 (2003)
Xia, Q., Wang, M.Y., Wang, S., Chen, S.: Semi-Lagrange method for level set based structural topology and shape optimization. Multidisciplinary Structural Optimization 31, 419–429 (2006)
Wang, S.Y., Lim, K.M., Khao, B.C., Wang, M.Y.: An extended level set method for shape and topology optimization. Journal of Computational Physics 221, 395–421 (2007)
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Myśliński, A. (2009). Level Set Method for Shape and Topology Optimization of Contact Problems. In: Korytowski, A., Malanowski, K., Mitkowski, W., Szymkat, M. (eds) System Modeling and Optimization. CSMO 2007. IFIP Advances in Information and Communication Technology, vol 312. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04802-9_23
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DOI: https://doi.org/10.1007/978-3-642-04802-9_23
Publisher Name: Springer, Berlin, Heidelberg
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