Dynamic programming approach to structural optimization problem - numerical algorithm

• Autorzy: Fulmański P. , Nowakowski A. , Pustelnik J.

Identyfikatory

DOI

http://dx.doi.org/10.7494/OpMath.2014.34.4.699

• Języki publikacji: EN

Abstrakty

EN

In this paper a new shape optimization algorithm is presented. As a model application we consider state problems related to fluid mechanics, namely the Navier-Stokes equations for viscous incompressible fluids. The general approach to the problem is described. Next, transformations to classical optimal control problems are presented. Then, the dynamic programming approach is used and sufficient conditions for the shape optimization problem are given. A new numerical method to find the approximate value function is developed.

Słowa kluczowe

EN

elliptic equations; optimal shape control; structural optimization; stationary Navier-Stokes equations; dynamic programming; sufficient optimality condition; numerical approximation

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2014
• Tom: Vol. 34, no. 4
• Strony: 699--724
• Opis fizyczny: Bibliogr. 23 poz., rys.

Twórcy

autor

Fulmański P.

• University of Łódz Faculty of Mathematics and Computer Science Computer Science Banacha 22, 90-238 Łódz, Poland

autor

Nowakowski A.

• University of Łódz Faculty of Mathematics and Computer Science Computer Science Banacha 22, 90-238 Łódz, Poland

autor

Pustelnik J.

• University of Łódz Faculty of Mathematics and Computer Science Computer Science Banacha 22, 90-238 Łódz, Poland

Bibliografia

• [1] M. Burger, B. Hackl, W. Ring, Incorporating topological derivatives into level set methods, Journal of Computational Physics 194 (2004) 1, 344–362.
• [2] D. Bucur, G. Buttazzo, Variational Methods in Shape Optimization Problems, Birkhäuser, 2005.
• [3] C. Castaing, M. Valadier, Convex analysis and measurable multifunctions, Springer- -Verlag, 1977.
• [4] M. Dambrine, On variations of the shape Hessian and sufficient conditions for the stability of critical shapes, Rev. R. Acad. Cien. Serie A. Mat. RACSAM 96 (2002) 1, 95–121.
• [5] M.C. Delfour, J.P. Zolésio, Shapes and Geometries: Analysis, Differential Calculus and Optimization, Adv. Des. Control, SIAM, Philadelphia, 2001.
• [6] K. Eppler, Second derivatives and sufficient optimality conditions for shape functionals, Control Cybernet. 29 (2000), 485–512.
• [7] K. Eppler, H. Harbrecht, Numerical solution of elliptic shape optimization problems using wavelet-based BEM, Optim. Methods Softw. 18 (2003), 105–123.
• [8] K. Eppler, H. Harbrecht, 2nd order shape optimization using wavelet BEM, Optim. Methods Softw. 21 (2006), 135–153.
• [9] K. Eppler, H. Harbrecht, R. Schneider, On convergence in elliptic shape optimization, SIAM J. Control Optim. 46 (2007) 1, 61–83.
• [10] W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimal Control, New York, Springer-Verlag, 1975.
• [11] P. Fulmanski, A. Laurin, J.F. Scheid, J. Sokolowski, A level set method in shape and topology optimization for variational inequalities, Int. J. Appl. Math. Comput. Sci. 17 (2007), 413–430.
• [12] P. Fulmanski, A. Nowakowski, Dual dynamic approach to shape optimization, Control Cybernet. 35 (2006) 2, 205–218.
• [13] S. Garreau, P. Guillaume, M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case, SIAM J. Control Optim. 39 (2001), 1756–1778.
• [14] J. Haslinger, R. Mäkinen, Introduction to Shape Optimization. Theory, Approximation and Computation, SIAM Publications, Philadelphia, 2003.
• [15] I. Hlavacek, J. Haslinger, J. Necas, J. Lovišek, Solving of variational Inequalities in Mechancs, Mir, Moscow, 1996 [in Russian].
• [16] H. Maurer, J. Oberle, Second order sufficient conditions for optimal control problems with free final time: the Riccati approach, SIAM J. Control Optim. 41 (2002), 380–403.
• [17] A. Nowakowski, Shape optimization of control problems described by wave equations, Control Cybernet. 37 (2008) 4, 1045–1055.
• [18] J. Pustelnik, Approximation of optimal value for Bolza problem, Ph.D. Thesis, 2009 [in Polish].
• [19] J. Sokołowski, J.P. Zolésio, Introduction to Shape Optimiation, Springer-Verlag, 1992.
• [20] J. Sokołowski, A. Zochowski, Optimality conditions for simultaneous topology and shape optimization, SIAM J. Control Optim. 42 (2003) 4, 1198–1221.
• [21] J. Sokołowski, A. Zochowski, On Topological Derivative in Shape Optimization, [in:] T. Lewiski, O. Sigmund, J. Sokołowski, A. Zochowski, Optimal Shape Design and Modelling, Academic Printing House EXIT, Warsaw, Poland, 2004, 55–143.
• [22] J. Sokołowski, A. Zochowski, A Modeling of topological derivatives for contact problems, Numer. Math. 102 (2005) 1, 145–179.
• [23] G. Stadler, Semismooth Newton and augmented Lagrangian methods for a simplified friction problem, SIAM J. Optim. 15 (2004) 1, 39–62.