zbMATH — the first resource for mathematics

Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. (English) Zbl 1273.74398
Summary: A frequent goal of the design of vibrating structures is to avoid resonance of the structure in a given interval for external excitation frequencies. This can be achieved by, e.g., maximizing the fundamental eigenfrequency, an eigenfrequency of higher order, or the gap between two consecutive eigenfrequencies of given order. This problem is often complicated by the fact that the eigenfrequencies in question may be multiple, and this is particularly the case in topology optimization. In the present paper, different approaches are considered and discussed for topology optimization involving simple and multiple eigenfrequencies of linearly elastic structures without damping. The mathematical formulations of these topology optimization problems and several illustrative results are presented.

74P15 Topological methods for optimization problems in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
Full Text: DOI
[1] Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202 · doi:10.1007/BF01650949
[2] Bendsøe MP (2006) Computational challenges for multi-physics topology optimization. In: Mota Soares CA, Martins JAC, Rodrigues HC, Ambrosio JAC (eds) Computational mechanics–solids, structures and coupled problems. Springer, Dordrecht, The Netherlands, pp 1–20
[3] Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224 · Zbl 0671.73065 · doi:10.1016/0045-7825(88)90086-2
[4] Bendsøe MP, Olhoff N (1985) A method of design against vibration resonance of beams and shafts. Optim Control Appl Methods 6:191–200 · Zbl 0569.49009 · doi:10.1002/oca.4660060302
[5] Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654 · Zbl 0957.74037 · doi:10.1007/s004190050248
[6] Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Berlin
[7] Bendsøe MP, Olhoff N, Taylor JE (1983) A variational formulation for multicriteria structural optimization. J Struct Mech 11:523–544 · doi:10.1080/03601218308907456
[8] Bendsøe MP, Olhoff N, Sigmund O (eds) (2006) Topological design optimization of structures, machines and materials–status and perspectives. Proc. IUTAM Symp., Copenhagen, Denmark, October 26–29, 2005. Springer, Dordrecht, The Netherlands
[9] Bratus AS, Seyranian AP (1983a) Bimodal solutions in eigenvalue optimization problems. Prikl Mat Meh 47:546–554
[10] Bratus AS, Seyranian AP (1983b) Bimodal solutions in eigenvalue optimization problems. Appl Math Mech 47:451–457 · Zbl 0569.47004 · doi:10.1016/0021-8928(83)90081-3
[11] Cheng G, Olhoff N (1982) Regularized formulation for optimal design of axisymmetric plates. Int J Solids Struct 18:153–169 · Zbl 0468.73106 · doi:10.1016/0020-7683(82)90023-3
[12] Diaz AR, Kikuchi N (1992) Solutions to shape and topology eigenvalue optimization problems using a homogenization method. Int J Numer Methods Eng 35:1487–1502 · Zbl 0767.73046 · doi:10.1002/nme.1620350707
[13] Diaz AR, Lipton R, Soto CA (1994) A new formulation of the problem of optimum reinforcement of Reissner–Midlin plates. Comp Meth Appl Mechs Eng 123:121–139 · Zbl 1067.74051 · doi:10.1016/0045-7825(94)00777-K
[14] Du J, Olhoff N (2007) Minimization of sound radiation from vibrating bi-material structures using topology optimization. Struct Multidiscipl Optim 33:305–321
[15] Eschenauer H, Olhoff N (2001) Topology optimization of continuum structures: a review. Appl Mech Rev 54(4):331–389 · doi:10.1115/1.1388075
[16] Haftka RT, Gurdal Z, Kamat MP (1990) Elements of structural optimization. Kluwer, Dordrecht
[17] Haug EJ, Choi KK, Komkov V (1986) Design sensitivity analysis of structural systems. Academic, New York · Zbl 0618.73106
[18] Jensen JS, Pedersen NL (2006) On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases. J Sound Vib 289:967–986 · doi:10.1016/j.jsv.2005.03.028
[19] Kosaka I, Swan CC (1999) A symmetry reduction method for continuum structural topology optimization. Comput Struct 70:47–61 · Zbl 0958.74046 · doi:10.1016/S0045-7949(98)00158-8
[20] Krog LA, Olhoff N (1999) Optimum topology and reinforcement design of disk and plate structures with multiple stiffness and eigenfrequency objectives. Comput Struct 72:535–563 · Zbl 1050.74644 · doi:10.1016/S0045-7949(98)00326-5
[21] Lancaster P (1964) On eigenvalues of matrices dependent on a parameter. Numer Math 6:377–387 · Zbl 0133.26201 · doi:10.1007/BF01386087
[22] Ma ZD, Cheng HC, Kikuchi N (1994) Structural design for obtaining desired eigenfrequencies by using the topology and shape optimization method. Comput Syst Eng 5(1):77–89 · doi:10.1016/0956-0521(94)90039-6
[23] Ma ZD, Kikuchi N, Cheng HC (1995) Topological design for vibrating structures. Comput Methods Appl Mech Eng 121:259–280 · Zbl 0849.73045 · doi:10.1016/0045-7825(94)00714-X
[24] Masur EF (1984) Optimal structural design under multiple eigenvalue constraints. Int J Solids Struct 20:211–231 · Zbl 0544.73117 · doi:10.1016/0020-7683(84)90034-9
[25] Masur EF (1985) Some additional comments on optimal structural design under multiple eigenvalue constraints. Int J Solids Struct 21:117–120 · Zbl 0558.73075 · doi:10.1016/0020-7683(85)90028-9
[26] Olhoff N (1976) Optimization of vibrating beams with respect to higher order natural frequencies. J Struct Mech 4:87–122 · doi:10.1080/03601217608907283
[27] Olhoff N (1977) Maximizing higher order eigenfrequencies of beams with constraints on the design geometry. J Struct Mech 5:107–134 · doi:10.1080/03601217708907308
[28] Olhoff N (1989) Multicriterion structural optimization via bound formulation and mathematical programming. Struct Optim 1:11–17 · doi:10.1007/BF01743805
[29] Olhoff N, Du J (2007) Topological design for minimum dynamic compliance of continuum structures subjected to forced vibration. Struct Multidisc Optim (in press)
[30] Olhoff N, Parbery R (1984) Designing vibrating beams and rotating shafts for maximum difference between adjacent natural frequencies. Int J Solids Struct 20:63–75 · Zbl 0529.73079 · doi:10.1016/0020-7683(84)90076-3
[31] Overton ML (1988) On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J Matrix Anal Appl 9(2):256–268 · Zbl 0647.65044 · doi:10.1137/0609021
[32] Pedersen NL (2000) Maximization of eigenvalues using topology optimization. Struct Multidisc Optim 20:2–11 · doi:10.1007/s001580050130
[33] Rozvany G, Zhou M (1991) Applications of the COC method in layout optimization. In: Eschenauer H, Mattheck C, Olhoff N (eds) Proc. Int. Conf. on Engineering Optimization in Design Processes (held in Karlsruhe 1990). Springer, Berlin Heidelberg New York, pp 59–70
[34] Rozvany G, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Optim 4:250–252 · doi:10.1007/BF01742754
[35] Seyranian AP (1987a) Multiple eigenvalues in optimization problems. Prikl Mat Meh 51:349–352
[36] Seyranian AP (1987b) Multiple eigenvalues in optimization problems. Appl Math Mech 51:272–275 · Zbl 0653.49025 · doi:10.1016/0021-8928(87)90076-1
[37] Seyranian AP, Lund E, Olhoff N (1994) Multiple eigenvalues in structural optimization problems. Struct Optim 8(4):207–227 · doi:10.1007/BF01742705
[38] Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25:493–524 · doi:10.1080/08905459708945415
[39] Sigmund O (2001) Microstructural design of elastic band gap structures. In: Cheng GD et al (eds), Proceedings of the 4th World Congress of Structrual and Multidisciplinary Optimization WCSMO4. Liaoning Electronic, Dalian, China, pp 6
[40] Sigmund O, Jensen JS (2003) Systematic design of phononic band-gap materials and structures by topology optimization. Philos Trans R Soc Lond Ser A Math Phys Sci 361:1001–1019 · Zbl 1067.74053 · doi:10.1098/rsta.2003.1177
[41] Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey of procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16:68–75 · doi:10.1007/BF01214002
[42] Svanberg K (1987) The method of moving asymptotes a new method for structural optimization. Int J Numer Methods Eng 24:359–373 · Zbl 0602.73091 · doi:10.1002/nme.1620240207
[43] Taylor JE, Bendsøe MP (1984) An interpretation of min–max structural design problems including a method for relaxing constraints. Int J Solids Struct 20:301–314 · Zbl 0531.73062 · doi:10.1016/0020-7683(84)90041-6
[44] Tcherniak D (2002) Topology optimization of resonating structures using SIMP method. Int J Numer Methods Eng 54:1605–1622 · Zbl 1034.74042 · doi:10.1002/nme.484
[45] Wittrick WH (1962) Rates of change of eigenvalues, with reference to buckling and vibration problems. J R Aeronaut Soc 66:590–591
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.