Abstract
This paper addresses a novel method of topology and shape optimization. The basic idea is the iterative positioning of new holes (so-called “bubbles”) into the present structure of the component. This concept is therefore called the “bubble method”. The iterative positioning of new bubbles is carried out by means of different methods, among others by solving a variational problem. The insertion of a new bubble leads to a change of the class of topology. For these different classes of topology, hierarchically structured shape optimizations that determine the optimal shape of the current bubble, as well as the other variable boundaries, are carried out.
Similar content being viewed by others
References
Allaire, G.; Kohn, R.V. 1993: Optimal design for minimum weight and compliance in plane stress using extremal microstructures.Eur. J. Mech. A/Solids 12, 839–878
Atrek, E. 1989: SHAPE: a program for shape optimization of continuum structures.Proc. 1st Int. Conf. Opti'89, pp. 135–144. Berlin, Heidelberg, New York: Springer
Banichuk, N.V. 1990:Introduction to optimization of structures. Berlin, Heidelberg, New York: Springer
Bendsøe, M.P.; Diaz, A.; Kikuchi, N. 1993: Topology and generalized layout optimization of elastic structures. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.):Topology design of structures, pp. 159–205. Dordrecht: Kluwer
Bendsøe, M.P.; Kikuchi, N. 1988: Generating optimal topologies in structural design using a homogenization method.Comp. Meth. Appl. Mech. Eng. 71, 197–224
Bourgat, J.F. 1973: Numerical experiments of the homogenisation method for operators with periodic coefficients.Lecture Notes in Mathematics 704, pp. 330–356. Berlin, Heidelberg, New York: Springer
Courant, R.; Hilbert, D. 1968:Methods of mathematical physics I. New York: Interscience Publishers
Dems, K. 1991: Fist and second-order shape sensitivity analysis of structures.Struct. Optim. 3, 79–88
Eschenauer, H.A.; Geilen, J.; Wahl, H.J. 1993: SAPOP- an optimization procedure for multicriteria structural design. In: Schittkowski, K.; Hörnlein, H. (eds.)Numerical methods in FE—based structural optimization systems. Int. Series of Num. Math. Basel: Birkhäuser-Verlag
Eschenauer, H.A.; Schnell, W. 1993:Elastizitätstheorie (3rd edition). Mannheim: Bibl. Inst.- Wissenschaftsverlag
Eschenauer, H.A.; Schumacher, A. 1993a: Possibilities of applying various procedures of topology optimization to components subject to mechanical loads.ZAMM 73, T392-T394
Eschenauer, H.A.; Schumacher, A. 1993b: Bubble method: a special strategy for finding best possible initial designs.Proc. ASME Design Technical Conf. -19th Design Automation Conf. (held in Albuquerque, New Mexico), Vol. 65-2, pp. 437–443
Eschenauer, H.A.; Schumacher, A.; Vietor, T. 1993: Decision makings for initial designs made of advanced materials. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology design of structures, pp. 469–480. Dordrecht: Kluwer
Eschenauer, H.A.; Weinert, M. 1992: Optimal layouts of complex shell structures by means of decomposition techniques.Proc. 4th AIAA/USAF/NASA/OAI Symp. on Multidisciplinary Analysis and Optimization (held in Cleveland, OH), pp. 999–1007
Farin, G. 1991:Splines in CAD/CAM. Surveys on mathematics for industry, pp. 39–73. Berlin, Heidelberg, New York: Springer
Fleury, C.; Braibant, V. 1986: Structural optimization: a new dual method using mixed variables.Int. J. Num. Meth. Eng. 23, 405–428
Haug, E.J.; Choi, K.K.; Komkov, V. 1985:Design sensitivity analysis of structural systems. Orlando, Florida: Academic Press
Hashin, Z. 1962: The elastic moduli of heterogeneous materials.ASME J. Appl. Mech. 29, 143–150
Kirsch, U. 1990: On the relationship between optimum structural topologies and geometries.Struct. Optim. 2, 39–45
Kohn, R.; Strang, G. 1986: Optimal design and relaxation of variational problems I–III.Comm. Pure Appl. Math. 39, 113–137, 139–182, 353–377
Korycki, R.; Eschenauer, H.A.; Schumacher, A. 1993: Incorporation of adjoint-method sensitivity analysis with the optimization procedure SAPOP.Proc. 11th Polish Conf. Comp. Meth. Mech. (held in Kielce, Poland)
Lasdon, L.S. 1982: Reduced gradient methods. In: Powell, M.J.D. (ed.)Nonlinear optimization (8th ed.), pp. 243–250. London: Academic Press
Michell, A.G.M. 1904: The limits of economy of materials in frame structures.Phil. Mag. 8, 589–597
Parkinson, A.; Wilson, M. 1986: Development of a hybrid SQP—GRG—algorithm for constrained nonlinear programming.Proc. Design Eng. Tech. Conf. (held in Ohio)
Prager, W. 1974: A note on discretized Michell structures.Comp. Meth. Appl. Mech. Engng. 3, 349–355
Prager, W.; Rozvany, G.I.N. 1977: Optimization of structural geometry. In: Beduarek, A.R.; Cesari, L. (eds.)Dynamical systems, pp. 265–293. New York: Academic Press.
Rozvany, G.I.N.; Zhou, M.; Birker, T; Sigmund, O. 1993: Topolagy optimization using iterative continuum-type optimality criteria (COC) methods for discritized systems. In: Bendsøe, M.P.; Mota Soares, C.A. (eds.)Topology design structures, pp. 273–283. Dordrecht: Kluwer
Rozvany, G.I.N.; Zhou, M.; Gollub, W. 1990: Continuum-type optimality criteria methods with a displacement constraint. Part II.Struct. Optim. 2, 77–104
Rozvany, G.I.N.; Zhou, M.; Rotthaus, M.; Gollub, W.; Spengemann, F. 1989: Continuum-type optimality criteria methods for large finite element systems with a displacement constraint. Part I.Struct. Optim. 1, 47–72
Schmit, L.A.; Mallet, R.H. 1963: Structural synthesis and design parameters.Hierarchy J. Struct. Divi. Proc. ASCE 89, 269–299
Svanberg, K. 1987: The method of moving asymptotes — a new method for structural optimization.Int. J. Num. Meth. Eng. 24, 359–373
Swanson, J.A. 1988:ANSYS user's manual, Vols. I+II. Houston: Swanson Analysis System Inc.
Tartar, L. 1973: Estimation de coefficients homogeneises.Lecture Notes in Mathematics 704, pp. 364–373. Berlin, Heidelberg, New York: Springer
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Eschenauer, H.A., Kobelev, V.V. & Schumacher, A. Bubble method for topology and shape optimization of structures. Structural Optimization 8, 42–51 (1994). https://doi.org/10.1007/BF01742933
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01742933