A new approach for simultaneous shape and topology optimization based on dynamic implicit surface function

• Autorzy: Guo H. , Zhao K. , Wang M. Y.
• Języki publikacji: EN

Abstrakty

EN

In the present paper, a new approach for structural topology optimization based on dynamic implicit surface function (DISF) is proposed. DISF is used to describe the shape/topology of a structure, which is approximated in terms of the nodal values. Then, a relationship is established between the element stiffness and the values of the implicit surface function on its four nodes. In this way and with some non-local treatments of the design sensitivities, not only the shape derivative but also the topological derivative of the optimal design can be incorporated in the numerical algorithm in a unified way. Numerical experiments demonstrate that by employing this approach, the computational efforts associated with DISF (and level set) based algorithms can be diminished. Clear optimal topologies and smooth structural boundaries free from any sign of numerical instability can be obtained simultaneously and efficiently.

Słowa kluczowe

EN

topology optimization; implicit; surface function; topological derivative; level set

PL

optymalizacja topologiczna; pochodna topologiczna

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2005
• Tom: Vol. 34, no 1
• Strony: 255--282
• Opis fizyczny: Bibliogr. 39 poz.

Twórcy

autor

Guo H.

• State Key Laboratory of Structural Analysis for Industrial Equipment Department of Engineering Mechanics, Dalian University of Technology Dalian 116023, China

autor

Zhao K.

• State Key Laboratory of Structural Analysis for Industrial Equipment Department of Engineering Mechanics, Dalian University of Technology Dalian 116023, China

autor

Wang M. Y.

• Department of Automation & Computer-Aided Engineering The Chinese University of Hong Kong Shatin N.T., Hong Kong

Bibliografia

• Allaire, G., Jouve, F. and Toader, A.M. (2002) A level set method for shape optimization. Comptes Rendus de l’Academie des Sciences, Serie I 334, 1125-1130.
• Allaire, G., Jouve, F. and Toader, A.M. (2004) Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics 194, 363-393.
• Allaire, G. and Kohn, R.V. (1993) Optimal design for minimum weight and compliance in plane stress using extremal microstructures. European Journal of Mechanics/A 12, 839-878.
• Ambrosio, L. and Buttazzo, G. (1993) An optimal design problem with perimeter penalization. Calculus of Variations 1 55-69.
• Belytschko, T., Xiao, S.P. and Parimi, C. (2003) Topology optimization with implicit functions and regularization. International Journal for Numerical Methods in Engineering 57, 1177-1196.
• Bendsøe, M.P. (1989) Optimal shape design as a material distribution problem. Structural Optimization 1, 193-202.
• Bendsøe, M.P. (1999) Variable-topology optimization: status and challenges. In: W. Wunderlich, ed., Proceedings of European Conference on Computational Mechanics, M¨unchen, Germany, August 31-September 3. Universitat Munchen, Paper No.137.
• Bendsøe, M.P. and Kikuchi, N. (1988) Generating optimal topologies in optimal design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 71, 197-224.
• Borrvall, T. and Petersson, J. (2001) Topology optimization using regularized intermediate density control. Computer Methods in Applied Mechanics and Engineering 190, 4911-4928.
• Bourdin, B. (2001) Filters in topology optimization International Journal of Numerical Methods in Engineering 50, 2143-2158.
• Burger, M., Hackl, B. and Ring, W. (2004) Incorporating topological derivatives into level set methods. Journal of Computational Physics 194, 344-362.
• Cea, J.S., Garreau, S., Guillaume, P. and Masmoudi, M. (2000) The shape and topological optimizations connection. Computer Methods in Applied Mechanics and Engineering 188, 713-726.
• Cheng, G.D. and Guo, X. (1997) Epsilon-relaxed approach in structural topology optimization. Structural Optimization 13, 258-266.
• Cheng, G.D. and Olhoff, N. (1981) An investigation concerning optimal design of solid elastic plates. International Journal of Solids and Structures 16, 305-323.
• Duysinx, P. and Bendsøe, M.P. (1998) Topology optimization of continuum structures with local stress constraints. International Journal for Numerical Methods in Engineering 43, 1453-1478.
• Eschenauer, H.A., Kobelev, V.V. and Schumacher, A. (1994) Bubble method for topology and shape optimization of structures. Structural Optimization 8, 42-51.
• Eschenauer, H.A. and Olhoff, N. (2001) Topology optimization of continuum structures: A review. Applied Mechanics Review 54, 331-390.
• Garreau, S., Guillaume, P. and Masmoudi, M. (2001) The topological asymptotic for PDE systems: The elasticity case. SIAM Journal on Control and Optimization 39, 1756-1778.
• Guo, X., Zhao, K. and Gu, Y.X. (2004a) Topology optimization with designdependent loads by level set approach. Submitted to Structural and Multidisciplinary Optimization.
• Guo, X., Zhao, K. and Gu, Y.X. (2004b) A new density-stiffness interpolation scheme for topology optimization of continuum structures. Engineering Computations 21, 9-22.
• Haber, R.B., Bendsøe, M.P. and Jog, C.S. (1996) A new approach to variabletopology shape design using a constraint on the perimeter. Structural Optimization 11, 1-12.
• Hintermuller, M. and Ring, W. (2003) A second order shape optimization approach for image segmentation. SIAM Journal on Applied Mathematics 64, 442-467.
• Kohn, R.V. and Strang, G. (1986) Optimal design and relaxation of variational problems. Communications on Pure and Applied Mathematics, 39, 1-25 (PART I), 139-182 (PART II) and 353-377 (PART III).
• Lewinski, T. and Sokolowski, J. (2003) Energy change due to the appearance of cavities in elastic solids. International Journal of Solids and Structures, 40, 1765-1803.
• Lurie, K.A. and Cherkaev, A.V. (1982) Regularization of optimal design problems for bars and plates. Journal of Optimization Theory and Applications 37, 499-522 (PART I), 523-543 (PART II) and 42, 247-282 (PART III).
• Osher, S.J., Fedkiw, R.P. (2003) Level set and dynamic implicit surfaces. Springer.
• Osher, S.J. and Santosa, F. (2001) Level set methods for optimization problems involving geometry and constraints I. Frequence of a two density inhomogeneous drum. Journal of Computational Physics 171, 272-288.
• Petersen, N.L. (2000) Maximization of Eigenvalues using topology optimization. Structural and Multidisciplinary Optimization 20, 2-12.
• Petersson, J. and Sigmund, O. (1998) Slope constrained topology optimization. International Journal for Numerical Methods in Engineering 41, 1417-1434.
• Rozvany, G.I.N., Zhou, M. and Birker, T. (1992) Generalized shape optimization without homogenization. Structural Optimization 4, 250-252.
• de Ruiter, M.J. and van Keulen, F. (2003) The topological derivative in the topology description function approach. Preprint.
• de Ruiter, M.J. and van Keulen, F. (2004) Topology optimization using a topology description function. Structural and Multidisciplinary Optimization 26, 406-416.
• Sethian, J.A. and Wiegmann, A. (2000) Structural boundary design via level set and immersed interface methods. Journal of Computational Physics 163, 489-528.
• Sigmund, O. and Petersson, J. (1998) Numerical instabilities in topology optimization. Structural and Multidisciplinary Optimization 16, 68-75.
• Sokolowski, J. and Zochowski, A. (1999) On the topological derivative in shape optimization. SIAM Journal on Control and Optimization 37, 1251-1272.
• Svanberg, K. (1987) The method of moving asymptotes-a new method for structural optimization. International Journal of Numerical Methods in Engineering 24, 359-373.
• Wang M.Y. and Wang, X.M. (2004) “Color” level sets: A multi-phase method for structural topology optimization with multiple materials. Computer Methods in Applied Mechanics and Engineering 193, 469-496.
• Wang, M.Y., Wang, X.M.and Guo, D.M. (2003) A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering 192, 227-246.
• Wang, X.M., Wang, M.Y. and Guo, D.M. (2004) Structural shape and topology optimization in a level-set framework of region representation. Structural and Multidisciplinary Optimization 27, 1-19.