Radial basis function level set method for structural optimization

• Autorzy: Myśliński A.
• Języki publikacji: EN

Abstrakty

EN

This paper is concerned with simultaneous topology and shape optimization of elastic contact problems. The structural optimization problem for an elastic contact problem consists in finding such topology as well as such shape of the boundary of the domain occupied by the body that the normal contact stress is minimized. Shape and topological derivatives formulae of the cost functional obtained using material derivative and asymptotic expansion methods, respectively, are recalled. These derivatives are employed to formulate the necessary optimality condition and to calculate a descent direction in a numerical algorithm. Level set based method is employed in numerical algorithm for tracking the evolution of the domain boundary on a fixed mesh and finding an optimal domain. In order to increase the efficiency of the level set based numerical algorithm, the radial basis function approach is used to solve the equation governing domain boundary evolution. Numerical examples are provided and discussed.

Słowa kluczowe

EN

contact problem; structural optimization; level set method; radial basis functions

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2010
• Tom: Vol. 39, no 3
• Strony: 627--645
• Opis fizyczny: Bibliogr. 32 poz., rys.

Twórcy

autor

Myśliński A.

• Systems Research Institute of the Polish Academy of Sciences, ul. Newelska 6, 01-447 Warsaw, Poland,

Bibliografia

• ALLAIRE, G., JOUVE, F. and TOADER, A. (2004) Structural Optimization Using Sensitivity Analysis and a Level Set Method. Journal of Computational Physics 194, 363-393.
• BOZTOSUN, I., CHARA, A., ZERROUKAT, M. and DJIDJELI, K. (2002) Thin-Plate Spline Radial Basis Function Scheme for Advection-Diffusion Problems. Electronic Journal of Boundary Elements BETEQ 2001(2), 267-282.
• BURGER, M., HACKL, B. and RING, W. (2004) Incorporating Topological Derivatives into Level Set Methods. Journal of Computational Physics 194(1), 344-362.
• CHOPP, H. and DOLBOW, J. (2002) A Hybrid Extended Finite Element / Level Set Method for Modelling Phase Transformations. International Journal for Numerical Methods in Engineering 54, 1209-1232.
• DELFOUR, M. and ZOLESIO, J.P. (2001) Shape and Geometries: Analysis, Differential Calculus and Optimization. SIAM Publications, Philadelphia.
• DOUAN, Y. (2008) A note on the meshless method using radial basis functions. Computers and Mathematics with Applications 55, 66-75.
• FULMAŃSKI, P., LAURIN, A., SCHEID, J.F. and SOKOŁOWSKI, J. (2007) A Level Set Method in Shape and Topology Optimization for Variational Inequalities. Int. J. Appl. Math. Comput. Sci. 17, 413-430.
• GARREAU, S., GUILLAUME, PH. and MASMOUDI, M. (2001) The Topological Asymptotic for PDE Systems: the Elasticity Case. SIAM Journal on Control Optimization 39, 1756-1778.
• GOMES, A. and SULEMAN, A. (2006) Application of Spectral Level Set Methodology in Topology Optimization. Structural Multidisciplinary Optimization 31, 430-443.
• DE GOURMAY, F. (2006) Velocity Extension for the Level Set Method and Multiple Eigenvalue in Shape Optimization. SIAM Journal on Control and Optimization 45 (1), 343-367.
• HASLINGER, J. and MÄKINEN, R. (2003) Introduction to Shape Optimization. Theory, Approximation, and Computation. SIAM Publications, Philadelphia.
• HLAVAČEK, I., HASLINGER, J., NEČAS, J. and LOVIŠEK, J. (1986) Solving the Variational Inequalities in Mechanics. Mir, Moscow (in Russian).
• HE, L., KAO, CH.Y. and OSHER, S. (2007) Incorporating Topological Derivatives into Shape Derivatives Based Level Set Methods. Journal of Computational Physics 225, 891-909.
• HÜEBER, S., STADLER, G. and WOHLMUTH, B. (2008) A Primal-Dual Active Set Algorithm for Three Dimensional Contact Problems with Coulomb Friction. SIAM J. Sci. Comput. 30 (2), 572-596.
• HINTERMÜLLER, M. and RING, W. (2004) A Level Set Approach for the Solution of a State - Constrained Optimal Control Problems. Numerische Mathematik 98, 135-166.
• ISKE, A. (2003) Radial basis functions: basics, advanced topics and meshfree methods for transport problems. Rend. Sem. Mat. Univ. Pol. Torino. 61 (3), Splines and Radial Functions, 247-285.
• LARSSON, E. and FORENBERG, B. (2003) A Numerical Study of some Radial Basis Function based Solution Methods for Elliptic PDEs. Comput. Math, and Appl. 46 (5), 891-902.
• MYŚLIŃSKI, A. (2004) Level Set Method for Shape Optimization of Contact Problems. In: P. Neittaanmaki, ed., CD ROM Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering, Jyvaskyla, Finland, 24 - 28 July 2004.
• MYŚLIŃSKI, A. (2005) Topology and Shape Optimization of Contact Problems using a Level Set Method. In: J. Herskovits, S. Mazorche, A. Canelas, eds., CD ROM Proceedings of VI World Congresses of Structural and Multidisciplinary Optimization. Rio de Janeiro, Brazil, 30 May - 03 June 2005.
• MYŚLIŃSKI, A. (2006) Shape Optimization of Nonlinear Distributed Parameter Systems. Academic Publishing House EXIT.
• MYŚLIŃSKI, A. (2008) Level Set Method for Optimization of Contact Problems. Engineering Analysis with Boundary Elements 32, 986-994.
• NORATO, J.A., BENDSOE, M.P., HABER, R. and TORTORELLI, D.A. (2007) A Topological Derivative Method for Topology Optimization. Structural Multidisciplinary Optimization 33, 375-386.
• NOVOTNY, A.A., FEIJÓO, R.A., PADRA, C. and TAROCO, E. (2005) Topological Derivative for Linear Elastic Plate Bending Problems. Control and Cybernetics 34 (1), 339-361.
• OSHER, S. and FEDKIW, R. (2003) Level Set Methods and Dynamic Implicit Surfaces. Springer, New York.
• SOKOLOWSKI, J. and ZOLESIO, J.P. (1992) Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer, Berlin.
• SOKOŁOWSKI, J. and ŻOCHOWSKI, A. (2003) Optimality Conditions for Simultaneous Topology and Shape Optimization. SIAM Journal on Control 42 (4), 1198-1221.
• SOKOŁOWSKI, J. and ŻOCHOWSKI, A. (2004) On Topological Derivative in Shape Optimization. In: T. Lewiński, O. Sigmund, J. Sokołowski, A. Żochowski, eds., Optimal Shape Design and Modelling, Academic Publishing House EXIT, Warsaw, 55-143.
• SOKOŁOWSKI, J. and ŻOCHOWSKI, A. (2005) A Modelling of Topological Derivatives for Contact Problems. Numerische Mathematik 102 (1), 145-179.
• STADLER, G. (2004) Semismooth Newton and Augmented Lagrangian Methods for a Simplified Friction Problem. SIAM Journal on Optimization 15 (1), 39-62.
• WANG, M. Y., WANG, X. and QUO, D. (2003) A Level Set Method for Structural Topology Optimization. Computer Methods in Applied Mechanics and Engineering 192, 227-246.
• WANG, S.Y., LIM, K.M., KHAO, B.C. and WANG, M.Y. (2007) An Extended Level Set Method for Shape and Topology Optimization. Journal of Computational Physics 221, 395-421.
• XIA, Q., WANG, M.Y., WANG, S. and CHEN, S. (2006) Semi - Lagrange Method for Level Set Based Structural Topology and Shape Optimization. Multidisciplinary Structural Optimization 31, 419-429.