Topology optimization of elastic continua using restriction. (English) Zbl 1141.74356
Summary: This is a study of restriction methods in topology optimization of linear elastic continua. An ill-posed optimization problem is transformed into a well-posed one by restricting the feasible set, hence the term restriction. A number of restriction methods are presented and compared, of which some have been treated previously in the literature and some are new. Advantages and drawbacks of the methods are discussed from a theoretical as well as a numerical point of view. The problems of minimizing compliance and designing compliant mechanisms constitute the base for these discussions and several numerical examples are presented that illustrate features of the various restriction methods.
MSC:
| 74P15 | Topological methods for optimization problems in solid mechanics |
References:
| [1] | Bendsøc, M. P., Optimization of Structural Topology, Shape and Material (1995), Berlin: Springer-Verlag, Berlin · Zbl 0822.73001 |
| [2] | B. Hassani and E. Hinton (1999),Homogenization and Structural Topology Optimization: Theory, Practice and Software, Springer-Verlag, London Limited. · Zbl 0938.74002 |
| [3] | Rozvany, G. I.N.; Bendsøe, M. P.; Kirsch, U., Layout optimization of structures, Appl. Mech. Rev., 48, 41-119 (1995) |
| [4] | Bendsøe, M. P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng., 71, 197-224 (1988) · Zbl 0671.73065 · doi:10.1016/0045-7825(88)90086-2 |
| [5] | Kohn, R. V.; Strang, G., Optimal design and relaxation of variational problems, Comm. Pure Appl. Math., 39, Part I, 113-137 (1986) · Zbl 0609.49008 · doi:10.1002/cpa.3160390107 |
| [6] | S. Spagnolo (1976), “Convergence in energy for elliptic operators”, in: B. Hubbard (Ed.),Numerical Solutions of Partial Differential Equations III, Academic Press, 469-498. · Zbl 0347.65034 |
| [7] | K.A. Lurie and A.V. Cherkaev (1997), “Effective characteristics of composite materials and the optimal design of structural elements”,Topics in the Mathematical Modelling of Composite Materials, in A. Cherkaev and R. Kohn (Eds.), Birkhäuser, Boston, 175-258. · Zbl 0927.74056 |
| [8] | Allaire, G.; Kohn, R. V., Optimal design for minimum weight and compliance in plane stress using extremal microstructures, Europ. J. Mech. A. Solids, 12, 6, 838-878 (1993) · Zbl 0794.73044 |
| [9] | M.P. Bendsøe, A. Díaz and N. Kikuchi (1993), “Topology and generalized layout optimization of elastic structures”,Topology Design of Structures, in M.P. Bendsøe and C.A. Mota Soares (Eds.), Kluwer Academic Publishers, 159-205. |
| [10] | Allaire, G.; Bonnetier, E.; Francfort, G.; Jouve, F., Shape optimization by the homogenization method, Numer. Math., 76, 27-68 (1997) · Zbl 0889.73051 · doi:10.1007/s002110050253 |
| [11] | Allaire, G.; Kohn, R. V., Optimal bounds on the effective behaviour of a mixture of two well-ordered materials, Q. Appl. Math. LI, 4, 675-699 (1993) · Zbl 0805.73042 |
| [12] | Petersson, J., On stiffness maximization of variable thickness sheet with unilateral contact, Q. Appl. Math., 54, 541-550 (1996) · Zbl 0871.73046 |
| [13] | Bendsøe, M. P.; Guedes, J. P.; Haber, R. B.; Pedersen, P.; Taylor, J. E., An analytical model to predict optimal material properties in the context of optimal structural design, J. Applied Mech., 61, 930-937 (1994) · Zbl 0831.73036 |
| [14] | Yuge, K.; Iwai, N.; Kikuchi, N., Optimization of 2-D structures subjected to nonlinear deformations using the homogenization method, Struct. Optim., 17, 286-299 (1999) · doi:10.1007/BF01207005 |
| [15] | Buhl, T.; Pedersen, C. B.W.; Sigmund, O., Stiffness design of geometrically non-linear structures using topology optimization, Struct. Optim., 19, 2, 93-104 (2000) · doi:10.1007/s001580050089 |
| [16] | Maute, K.; Schwartz, S.; Ramm, E., Adaptive topology optimization of elastoplastic structures, Struct. Optim., 15, 81-91 (1998) · doi:10.1007/BF01278493 |
| [17] | Ananthasuresh, G. K.; Kota, S.; Kikuchi, N., Strategies for systematic synthesis of compliant MEMS, 1994 ASME Winter Annual Meeting, 55-2, 677-686 (1994) |
| [18] | Ananthasuresh, G. K.; Kota, S., Designing compliant mechanisms, Mech. Eng., 117, 11, 93-96 (1995) |
| [19] | Frecker, M. I.; Ananthasuresh, G. K.; Nishiwaki, S.; Kikuchi, N.; Kota, S., Topological synthesis of compliant mechanisms using multi-criteria optimization, Trans. ASME, J. Mech. Des., 119, 238-245 (1997) |
| [20] | Nishiwaki, S.; Frecker, M. I.; Min, S.; Kikuchi, N., Topology optimization of compliant mechanisms using the homogenization method, Int. J. Numer. Methods Eng., 42, 535-559 (1998) · Zbl 0908.73051 · doi:10.1002/(SICI)1097-0207(19980615)42:3<535::AID-NME372>3.0.CO;2-J |
| [21] | Hetrick, J. A.; Kota, S., An energy formulation for parametric size and shape optimization of compliant mechanisms, Trans. ASME, J. Mech. Des., 121, 229-234 (1999) |
| [22] | Larsen, U. D.; Sigmund, O.; Bouwstra, S., Design and fabrication of compliant micromechanisms and structures with negative poisson’s ratio, J. Microelectromech. Syst., 6, 2, 99-106 (1997) · doi:10.1109/84.585787 |
| [23] | Sigmund, O., On the design of compliant mechanisms using topology optimization, Mech. Struct. Mach., 25, 493-524 (1997) · doi:10.1080/08905459708945415 |
| [24] | Claus, B. W.; Pedersen, Thomas Buhl; Sigmund, Ole, Topology synthesis of large-displacement compliant mechanisms, Int. J. Numer. Methods Eng., 50, 2683-2705 (2001) · Zbl 0988.74055 · doi:10.1002/nme.148 |
| [25] | Bruns, T. E.; Tortorelli, D. A., Topology optimization of nonlinear elastic structures and compliant mechanisms, Comput. Methods Appl. Mech. Eng., 190, 3443-3459 (2001) · Zbl 1014.74057 · doi:10.1016/S0045-7825(00)00278-4 |
| [26] | Adams, R. A., Sobolev Spaces (1975), New York: Academic Press, New York · Zbl 0314.46030 |
| [27] | Bendsøe, M. P., Optimal shape design as a material distribution problem, Struct. Optim., 1, 193-202 (1989) · doi:10.1007/BF01650949 |
| [28] | Zhou, M.; Rozvany, G. I.N., The COC algorithm, part II: topological, geometrical and generalized shape optimization, Comput. Methods Appl. Mech. Eng., 89, 309-336 (1991) · doi:10.1016/0045-7825(91)90046-9 |
| [29] | M. Stolpe and K. Svanberg (2001), “An alternative interpolation scheme for minimum compliance topology optimization”, Optimization and Systems Theory, Royal Institute of Technology, Report TRITA/MAT-00-OS13. To appear inStruct. Multidisc. Optim. |
| [30] | Maute, K.; Ramm, E., Adaptive topology optimization, Struct. Optim., 10, 100-112 (1995) · doi:10.1007/BF01743537 |
| [31] | Mlejnek, H. P.; Schirrmacher, R., An engineer’s approach to optimal distribution and shape finding, Comput. Methods Appl. Mech. Eng., 106, 1-26 (1993) · Zbl 0800.73264 · doi:10.1016/0045-7825(93)90182-W |
| [32] | Bendsøe, M. P.; Sigmund, O., Material interpolation schemes in topology optimization, Arch. Appl. Mech., 69, 635-654 (1999) · Zbl 0957.74037 |
| [33] | Jog, C. S.; Haber, R. B., Stability of finite element models for distributed-parameter optimization and topology design, Comput. Methods Appl. Mech. Eng., 130, 203-226 (1996) · Zbl 0861.73072 · doi:10.1016/0045-7825(95)00928-0 |
| [34] | G. Allaire and R.V. Kohn (1993). “Topology optimization and optimal shape design using homogenization”,Topology Design of Structures, in M.P. Bendsøe and C.A. Mota Soares (Eds.), Kluwer Academic Publishers, 207-218. |
| [35] | Rossow, M. P.; Taylor, J. E., A finite element method for the optimal design of variable thickness sheets, AIAA Journal, 11, 1566-1568 (1973) · doi:10.2514/3.50631 |
| [36] | Céa, J.; Malanowski, K., An example of a max-min problem in partial differential equations, SIAM J. Control, 8, 3, 305-316 (1970) · Zbl 0204.46603 · doi:10.1137/0308021 |
| [37] | Svanberg, K., The method of moving asymptotes— a new method for structural optimization, Int. J. Numer. Methods Eng., 24, 359-373 (1987) · Zbl 0602.73091 · doi:10.1002/nme.1620240207 |
| [38] | Petersson, J.; Sigmund, O., Slope constrained topology optimization, Int. J. Numer. Methods Eng., 41, 1417-1434 (1998) · Zbl 0907.73044 · doi:10.1002/(SICI)1097-0207(19980430)41:8<1417::AID-NME344>3.0.CO;2-N |
| [39] | Stolpe, M.; Svanberg, K., On the trajectories of penalization methods for topology optimization, Optimization and Systems Theory, Royal Institute of Technology, Report TRITA/MAT-00-OS02. Struct. Multidisc. Optim., 21, 2, 128-139 (2001) |
| [40] | Díaz, A.; Sigmund, O., Checkerboard patterns in layout optimization, Struct. Optim., 10, 40-45 (1995) · doi:10.1007/BF01743693 |
| [41] | Sigmund, O.; Petersson, J., Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Struct. Optim., 16, 68-75 (1998) · doi:10.1007/BF01214002 |
| [42] | V. Maz’ya (1994). “Approximate approximations”,The Mathematics of Finite Elements and Applications, in J.R. Whiteman (Ed.), John Wiley & Sons Ltd, 77-104. · Zbl 0827.65083 |
| [43] | Temam, R., Mathematical Problems in Plasticity (1985), Paris: BORDAS, Paris · Zbl 0457.73017 |
| [44] | E. Giustu (1984),Minimal Surfaces and Functions of Bounded Variation, Birkhäuser Boston, Inc. · Zbl 0545.49018 |
| [45] | Ambrosio, L.; Butazzo, G., An optimal design problem with perimeter penalization, Calc. Var., 1, 55-69 (1993) · Zbl 0794.49040 · doi:10.1007/BF02163264 |
| [46] | Petersson, J., Some convergence results in perimeter-controlled topology optimization, Comput. Methods Appl. Mech. Eng., 171, 123-140 (1999) · Zbl 0947.74050 · doi:10.1016/S0045-7825(98)00248-5 |
| [47] | Haber, R. B.; Jog, C. S.; Bendsøc, M. P., A new approach to variable-topology shape design using a constraint on perimeter, Struct. Optim., 11, 1-12 (1996) · doi:10.1007/BF01279647 |
| [48] | P. Duysinx (1997), “Layout optimization: A mathematical programming approach”, DCAMM Report 540, Technical University of Denmark. |
| [49] | Beckers, M., Topology optimization using a dual method with discrete variables, Struct. Optim., 17, 14-24 (1999) |
| [50] | Fernandez, P.; Guedes, J. M.; Rodrigues, H., Topology optimization of three-dimensional linear clastic structures with a constraint onperimeter, Comput. Struct., 73, 583-594 (1999) · Zbl 0992.74058 · doi:10.1016/S0045-7949(98)00312-5 |
| [51] | J. Petersson, M. Beckers and P. Duysinx (1999), “Almost isotropic perimeters in topology optimization: Theoretical and numerical aspects”, in C.L. Blocbaum, K.E. Lewis and R.W. Mayne (Eds.), WCSMO-3, Proceedings of theThird World Congress of Structural and Multidisciplinary Optimization, 117-119. |
| [52] | Bendsøc, M. P., On obtaining a solution to optimization problems for solid, elastic plates by restriction of the design space, J. Struct. Mech., 11, 4, 501-522 (1983) |
| [53] | Niordson, F., Optimal design of clastic plates with a constraint on the slope of the thickness function, Int. J. Solids Struct., 19, 141-151 (1983) · Zbl 0502.73074 · doi:10.1016/0020-7683(83)90005-7 |
| [54] | Hlaváček, I.; Bock, I.; Lovišek, J., Optimal control of a variational inequality with applications to structural analysis. II. Local optimization of the stress in a beam. III. Optimal design of an elastic plate, Appl. Math. Optim., 13, 117-136 (1985) · Zbl 0582.73081 · doi:10.1007/BF01442202 |
| [55] | M. Zhou, Y.K. Shyy and H.L. Thomas (1999), “Checkerboard and minimum member size control in topology optimization”, in C.L. Bloebaum, K.E. Lewis and R.W. Mayne (Eds.), WCSMO-3, Proceedings of theThird World Congress of Structural and Multidisciplinary Optimization, 440-442. |
| [56] | O. Sigmund (1994),Design of material structures using topology optimization, Ph.D. thesis, DCAMM, Technical University of Denmark. |
| [57] | O. Sigmund and T. Buhl (2000), “Design of multiphysics actuators using topology optimization”, DCAMM Report 632. Technical University of Denmark. To appear inComput. Methods Appl., Mech. Eng. |
| [58] | Pedersen, N. L., Maximization of eigenvalues using topology optimization, Struct. Multidisc. Optim., 20, 2-11 (2000) · doi:10.1007/s001580050130 |
| [59] | B. Bourdin (1999), “Filters in topology optimization”, DCAMM Report 627, Technical University of Denmark. To appear inInt. J. Numer. Methods Eng. · Zbl 0971.74062 |
| [60] | Borrvall, T.; Petersson, J., Topology optimization using regularized intermediate density control, Comput. Methods Appl. Mech. Eng., 190, 4911-4928 (2001) · Zbl 1022.74035 · doi:10.1016/S0045-7825(00)00356-X |
| [61] | T. Borrvall and J. Petersson (2000), “Large scale topology optimization in 3D using parallel computing”, Department of Mechanical Engineering, Linköping University, LiTH-IKP-R-1130. To appear inComput. Methods Appl. Mech. Eng. · Zbl 1022.74036 |
| [62] | Eschenauer, H. A.; Kobelev, V.; Schumacher, A., Bubble method of topology and shape optimization of structures, Struct. Optim., 8, 42-51 (1994) · doi:10.1007/BF01742933 |
| [63] | Xic, Y. M.; Steven, G. P., A simple evolutionary procedure for structural optimization, Comput. Struct., 49, 885-896 (1993) · doi:10.1016/0045-7949(93)90035-C |
| [64] | Guedes, J. M.; Taylor, J. E., On the prediction of material properties and topology optimization for optimal continuum structures, Struct. Optim., 14, 193-199 (1997) · doi:10.1007/BF01812523 |
| [65] | Rodrigues, H.; Soto, C. A.; Taylor, J. E., A design model to predict optimal two-material composite structures, Struct. Optim., 17, 186-198 (1999) |
| [66] | Ringertz, U. T., On finding the optimal distribution of material properties, Struct. Optim., 5, 265-267 (1993) · doi:10.1007/BF01743590 |
| [67] | M. Torabzadeh-Tari (1999), “Implementation of a topology optimization approach in TRINITAS: a new weight function in the volume constraint,” Department of Mechanical Engineering, Linköping University, LiTH-IKP-Ex-1584. |
| [68] | Kim, Y. Y.; Yoon, G. H., Multi-resolution multi-scale topology optimization—a new paradigm, Int. J. Solids Struct., 37, 5529-5559 (2000) · Zbl 0981.74044 · doi:10.1016/S0020-7683(99)00251-6 |
| [69] | T. Poulsen (2000), “Topology optimization in wavelet space”, DCAMM Report 639, Technical University of Denmark. · Zbl 1112.74464 |
| [70] | Hashin, Z.; Shtrikman, S., A variational approach to the theory of the clastic behaviour of multiphase materials, J. Mech. Phys. Solids, 11, 127-140 (1963) · Zbl 0108.36902 · doi:10.1016/0022-5096(63)90060-7 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
