zbMATH — the first resource for mathematics

A method using successive iteration of analysis and design for large-scale topology optimization considering eigenfrequencies. (English) Zbl 1439.74280
Summary: Repeatedly solving the generalized eigenvalue problems by far dominates the computational cost in large-scale topology optimization involving natural frequency constraints. This study proposes a method for dynamic topology optimization problems considering natural frequencies using successively executed iterations for the structural analysis and design. By using the Rayleigh quotients as approximations of the natural frequencies and achieving sequential approximation of the eigenpairs through inverse iteration-like procedures to improve the eigenvectors along with the topological evolution of the structure, the method avoids solving the time-consuming eigenvalue problem in each design iteration. This makes the method particularly suitable for large-scale frequency-constrained topology optimization problems. The convergence property of the method is analyzed under the assumption of sufficiently small design changes between two successive design iterations. Numerical examples regarding frequency and frequency gap constraints show that this method is able to realize concurrent convergence of the eigenvalue analysis and design optimization, and is more efficient than the conventional double-loop approach.
74P15 Topological methods for optimization problems in solid mechanics
PETSc; top.m
Full Text: DOI
[1] Bendsøe, M. P., Optimal shape design as a material distribution problem, Struct. Optim., 1, 193-202 (1989)
[2] Zhou, M.; Rozvany, G., The COC algorithm part II: topological geometrical and generalized shape optimization, Comput. Methods Appl. Mech. Engrg., 89, 309-336 (1991)
[3] Sigmund, O., A 99 line topology optimization code written in matlab, Struct. Multidiscip. Optim., 21, 120-127 (2001)
[4] Allaire, G.; Jouve, F.; Toader, A.-M., Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194, 363-393 (2004) · Zbl 1136.74368
[5] Wang, M. Y.; Wang, X.; Guo, D., A level set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg., 192, 227-246 (2003) · Zbl 1083.74573
[6] Huang, X.; Xie, Y. M., Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method, Finite Elem. Anal. Des., 43, 1039-1049 (2007)
[7] Diaz, A. R.; Kikuchi, N., Solutions to shape and topology eigenvalue optimization problems using a homogenization method, Internat. J. Numer. Methods Engrg., 35, 1487-1502 (1992) · Zbl 0767.73046
[8] Ma, Z.-D.; Kikuchi, N.; Cheng, H.-C., Topological design for vibrating structures, Comput. Methods Appl. Mech. Engrg., 121, 259-280 (1995) · Zbl 0849.73045
[9] Pedersen, N. L., Maximization of eigenvalues using topology optimization, Struct. Multidiscip. Optim., 20, 2-11 (2000)
[10] Xu, B.; Han, Y. S.; Zhao, L.; Xie, Y. M., Topology optimization of continuum structures for natural frequencies considering casting constraints, Eng. Optim., 51, 941-960 (2019)
[11] Yoon, G. H., Maximizing the fundamental eigenfrequency of geometrically nonlinear structures by topology optimization based on element connectivity parameterization, Comput. Struct., 88, 120-133 (2010)
[12] Zhou, P.; Ou, G.; Du, J., Topology optimization of continua considering mass and inertia characteristics, Struct. Multidiscip. Optim., 1-14 (2019)
[13] An, H.; Chen, S.; Huang, H., Multi-objective optimal design of hybrid composite laminates for minimum cost and maximum fundamental frequency and frequency gaps, Compos. Struct., 209, 268-276 (2019)
[14] Du, J.; Olhoff, N., Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps, Struct. Multidiscip. Optim., 34, 91-110 (2007) · Zbl 1273.74398
[15] Leader, M. K.; Chin, T. W.; Kennedy, G. J., High-resolution topology optimization with stress and natural frequency constraints, AIAA J., 1-17 (2019)
[16] Jensen, J. S., Topology optimization of dynamics problems with Padé approximants, Internat. J. Numer. Methods Engrg., 72, 1605-1630 (2007) · Zbl 1194.74270
[17] Jog, C., Topology design of structures subjected to periodic loading, J. Sound Vib., 253, 687-709 (2002)
[18] Min, S.; Kikuchi, N.; Park, Y.; Kim, S.; Chang, S., Optimal topology design of structures under dynamic loads, Struct. Optim., 17, 208-218 (1999)
[19] Takezawa, A.; Daifuku, M.; Nakano, Y.; Nakagawa, K.; Yamamoto, T.; Kitamura, M., Topology optimization of damping material for reducing resonance response based on complex dynamic compliance, J. Sound Vib., 365, 230-243 (2016)
[20] Kang, Z.; Zhang, X.; Jiang, S.; Cheng, G., On topology optimization of damping layer in shell structures under harmonic excitations, Struct. Multidiscip. Optim., 46, 51-67 (2012) · Zbl 1274.74348
[21] Larsen, A. A.; Laksafoss, B.; Jensen, J. S.; Sigmund, O., Topological material layout in plates for vibration suppression and wave propagation control, Struct. Multidiscip. Optim., 37, 585-594 (2009) · Zbl 1274.74359
[22] Tcherniak, D., Topology optimization of resonating structures using SIMP method, Internat. J. Numer. Methods Engrg., 54, 1605-1622 (2002) · Zbl 1034.74042
[23] Yoon, G. H., Structural topology optimization for frequency response problem using model reduction schemes, Comput. Methods Appl. Mech. Engrg., 199, 1744-1763 (2010) · Zbl 1231.74366
[24] Yoon, G. H.; Choi, H.; So, H., Development and optimization of a resonance-based mechanical dynamic absorber structure for multiple frequencies, J. Low Freq. Noise Vib. Act. Control, Article 1461348419855533 pp. (2019)
[25] Zhao, J.; Wang, C., Topology optimization for minimizing the maximum dynamic response in the time domain using aggregation functional method, Comput. Struct., 190, 41-60 (2017)
[26] Zhu, J.; He, F.; Liu, T.; Zhang, W.; Liu, Q.; Yang, C., Structural topology optimization under harmonic base acceleration excitations, Struct. Multidiscip. Optim., 57, 1061-1078 (2018)
[27] Zhu, J.; Zhang, W.; Xia, L., Topology optimization in aircraft and aerospace structures design, Arch. Comput. Methods Eng., 23, 595-622 (2016) · Zbl 1360.74128
[28] Gao, J.; Luo, Z.; Li, H.; Li, P.; Gao, L., Dynamic multiscale topology optimization for multi-regional micro-structured cellular composites, Compos. Struct., 211, 401-417 (2019)
[29] Li, H.; Luo, Z.; Xiao, M.; Gao, L.; Gao, J., A new multiscale topology optimization method for multiphase composite structures of frequency response with level sets, Comput. Methods Appl. Mech. Engrg., 356, 116-144 (2019)
[30] Liang, X.; Du, J., Concurrent multi-scale and multi-material topological optimization of vibro-acoustic structures, Comput. Methods Appl. Mech. Engrg., 349, 117-148 (2019)
[31] Liu, Q.; Ruan, D.; Huang, X., Topology optimization of viscoelastic materials on damping and frequency of macrostructures, Comput. Methods Appl. Mech. Engrg., 337, 305-323 (2018)
[32] Zhao, J.; Yoon, H.; Youn, B. D., An efficient concurrent topology optimization approach for frequency response problems, Comput. Methods Appl. Mech. Engrg., 347, 700-734 (2019)
[33] Du, J.; Olhoff, N., Minimization of sound radiation from vibrating bi-material structures using topology optimization, Struct. Multidiscip. Optim., 33, 305-321 (2007)
[34] Du, J.; Olhoff, N., Topological design of vibrating structures with respect to optimum sound pressure characteristics in a surrounding acoustic medium, Struct. Multidiscip. Optim., 42, 43-54 (2010) · Zbl 1274.74263
[35] Dühring, M. B.; Jensen, J. S.; Sigmund, O.; Dühring, M. B.; Jensen, J. S., Acoustic design by topology optimization, J. Sound Vib., 317, 557-575 (2008)
[36] Luo, J.; Gea, H. C., Optimal stiffener design for interior sound reduction using a topology optimization based approach, J. Vib. Acoust., 125, 267-273 (2003)
[37] Olhoff, N.; Niu, B., Discrete material optimization of vibrating laminated composite plates for minimum sound emission, Int. J. Solids Struct., 47, 2097-2114 (2010) · Zbl 1194.74254
[38] He, J.; Kang, Z., Achieving directional propagation of elastic waves via topology optimization, Ultrasonics, 82, 1-10 (2018)
[39] Sigmund, O.; Jensen, J. S., Systematic design of phononic band-gap materials and structures by topology optimization, Philos. Trans. Math. Phys. Eng. Sci., 361, 1001-1019 (2003) · Zbl 1067.74053
[40] Diaz, A. R.; Haddow, A. G.; Ma, L., Design of band-gap grid structures, Struct. Multidiscip. Optim., 29, 418-431 (2005)
[41] Halkje̊r, S.; Sigmund, O.; Jensen, J. S., Maximizing band gaps in plate structures, Struct. Multidiscip. Optim., 32, 263-275 (2006)
[42] Li, Y. F.; Huang, X.; Meng, F.; Zhou, S., Evolutionary topological design for phononic band gap crystals, Struct. Multidiscip. Optim., 54, 595-617 (2016)
[43] Aage, N.; Andreassen, E.; Lazarov, B. S.; Sigmund, O., Giga-voxel computational morphogenesis for structural design, Nature, 550, 84-86 (2017)
[44] Duysinx, P.; Bruyneel, M.; Fleury, C., Solution of topology optimization problems with sequential convex programming (2003)
[45] Aage, N.; Lazarov, B. S., Parallel framework for topology optimization using the method of moving asymptotes, Struct. Multidiscip. Optim., 47, 493-505 (2013) · Zbl 1274.74302
[46] Ferrari, F.; Lazarov, B. S.; Sigmund, O., Eigenvalue topology optimization via efficient multilevel solution of the frequency response, Internat. J. Numer. Methods Engrg. (2018)
[47] Perdon, A.; Gambolati, G., Extreme eigenvalues of large sparse matrices by Rayleigh quotient and modified conjugate gradients, Appl. Mech. Eng., 56, 251-264 (1986) · Zbl 0579.65028
[48] Amir, O.; Sigmund, O., On reducing computational effort in topology optimization: how far can we go?, Struct. Multidiscip. Optim., 44, 25-29 (2011) · Zbl 1274.74306
[49] Limkilde, A.; Evgrafov, A.; Gravesen, J., On reducing computational effort in topology optimization: we can go at least this far!, Struct. Multidiscip. Optim., 58, 2481-2492 (2018)
[50] Cheng, G.; Xu, L.; Jiang, L., A sequential approximate programming strategy for reliability-based structural optimization, Comput. Struct., 84, 1353-1367 (2006)
[51] Haftka, R. T., Simultaneous analysis and design, AIAA J., 23, 1099-1103 (1985) · Zbl 0587.73135
[52] Stolpe, M.; Svanberg, K., An alternative interpolation scheme for minimum compliance topology optimization, Struct. Multidiscip. Optim., 22, 116-124 (2001)
[53] Sigmund, O.; Maute, K., Topology optimization approaches, Struct. Multidiscip. Optim., 48, 1031-1055 (2013)
[54] Svanberg, K., The method of moving asymptotes—a new method for structural optimization, Internat. J. Numer. Methods Engrg., 24, 359-373 (1987) · Zbl 0602.73091
[55] Rohn, J.; Deif, A., On the range of eigenvalues of an interval matrix, Computing, 47, 373-377 (1992) · Zbl 0755.65039
[56] Chen, S. H.; Qiu, Z.; Liu, Z., Perturbation method for computing eigenvalue bounds in structural vibration systems with interval parameters, Commun. Numer. Methods. Eng., 10, 121-134 (1994) · Zbl 0790.73079
[57] Deif, A., Rigorous perturbation bounds for eigenvalues and eigenvectors of a matrix, J. Comput. Appl. Math., 57, 403-412 (1995) · Zbl 0823.15017
[58] Bathe, K.-J.; Ramaswamy, S., An accelerated subspace iteration method, Comput. Methods Appl. Mech. Engrg., 23, 313-331 (1980) · Zbl 0438.73069
[59] Bertolini, A.; Lam, Y., Accelerated reduction of subspace upper bound by multiple inverse iteration, Comput. Syst. Eng., 6, 67-72 (1995)
[60] Jensen, J. S.; Pedersen, N. L., On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases, J. Sound Vib., 289, 967-986 (2006)
[61] Hu, J.; Zhang, X.; Kang, Z., Layout design of piezoelectric patches in structural linear quadratic regulator optimal control using topology optimization, J. Intell. Mater. Syst. Struct., 29, 2277-2294 (2018)
[62] Aage, N.; Andreassen, E.; Lazarov, B. S., Topology optimization using PETSc: An easy-to-use, fully parallel, open source topology optimization framework, Struct. Multidiscip. Optim., 51, 565-572 (2015)
[63] Balay, S.; Brown, J.; Buschelman, K.; Eijkhout, V.; Gropp, W. D.; Kaushik, D.; Knepley, M. G.; McInnes, L. C.; Smith, B. F.; Zhang, H., PETSc users manual Tech. Rep (2013), ANL-95/11-Revision 3.4. Argonne National Laboratory
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.