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Bi-directional evolutionary topology optimization of geometrically nonlinear continuum structures with stress constraints. (English) Zbl 07193152
Summary: This paper proposes a design method to maximize the stiffness of geometrically nonlinear continuum structures subject to volume fraction and maximum von Mises stress constraints. An extended bi-directional evolutionary structural optimization (BESO) method is adopted in this paper. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low density elements. The maximum von Mises stress is approximated by the \(p\)-norm global stress. By introducing one Lagrange multiplier, the objective of the traditional stiffness design is augmented with \(p\)-norm stress. The stiffness and \(p\)-norm stress are considered simultaneously by the Lagrange multiplier method. A heuristic method for determining the Lagrange multiplier is proposed in order to effectively constrain the structural maximum von Mises stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topology variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several benchmark design problems.

MSC:
74 Mechanics of deformable solids
76 Fluid mechanics
Software:
top.m
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References:
[1] Bendsøe, M. P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng., 71, 2, 197-224 (1988) · Zbl 0671.73065
[2] Bendsøe, M. P.; Sigmund, O., Topology Optimization: Theory, Methods and Applications (2003), Springer: Springer Berlin · Zbl 1059.74001
[3] Deaton, J. D.; Grandhi, R. V., A survey of structural and multidisciplinary continuum topology optimization: post 2000, Struct. Multidiscip. Optim., 49, 1, 1-38 (2014)
[4] Huang, X.; Xie, Y. M., Evolutionary Topology Optimization of Continuum structures: Methods and Applications (2010), Wiley: Wiley Chichester · Zbl 1279.90001
[5] Zhu, J.; Zhang, W.; Xia, L., Topology optimization in aircraft and aerospace structures design, Arch. Comput. Methods Eng., 23, 4, 595-622 (2016) · Zbl 1360.74128
[6] Li, Z.; Shi, T.; Xia, Q., Eliminate localized eigenmodes in level set based topology optimization for the maximization of the first eigenfrequency of vibration, Adv. Eng. Softw., 107, 59-70 (2017)
[7] Wei, P.; Wang, M. Y., Piecewise constant level set method for structural topology optimization, Int. J. Numer. Methods Eng., 78, 4, 379-402 (2009) · Zbl 1183.74222
[8] Xia, L.; Xia, Q.; Huang, X.; Xie, Y. M., Bi-directional evolutionary structural optimization on advanced structures and materials: a comprehensive review, Arch. Comput. Methods Eng., 25, 2, 437-478 (2018) · Zbl 1392.74074
[9] Duysinx, P.; Bendsøe, M. P., Topology optimization of continuum structures with local stress constraints, Int. J. Numer. Methods Eng., 43, 8, 1453-1478 (1998) · Zbl 0924.73158
[10] Jeong, S. H.; Park, S. H.; Choi, D. H.; Yoon, G. H., Topology optimization considering static failure theories for ductile and brittle materials, Comput. Struct., 110, 116-132 (2012)
[11] Luo, Y.; Kang, Z., Topology optimization of continuum structures with drucker-prager yield stress constraints, Comput. Struct., 90, 65-75 (2012)
[12] Moon, S. J.; Yoon, G. H., A newly developed qp-relaxation method for element connectivity parameterization to achieve stress-based topology optimization for geometrically nonlinear structures, Comput. Methods Appl. Mech. Eng., 265, 226-241 (2013) · Zbl 1286.74076
[13] Takezawa, A.; Yoon, G. H.; Jeong, S. H.; Kobashi, M.; Kitamura, M., Structural topology optimization with strength and heat conduction constraints, Comput. Methods Appl. Mech. Eng., 276, 341-361 (2014) · Zbl 1423.74762
[14] Le, C.; Norato, J.; Bruns, T.; Ha, C.; Tortorelli, D., Stress-based topology optimization for continua, Struct. Multidiscip. Optim., 41, 4, 605-620 (2010)
[15] Svärd, H., Interior value extrapolation: a new method for stress evaluation during topology optimization, Struct. Multidiscip. Optim., 51, 3, 613-629 (2015)
[16] Bruggi, M., Topology optimization with mixed finite elements on regular grids, Comput. Methods Appl. Mech. Eng., 305, 133-153 (2016) · Zbl 1425.74372
[17] Cheng, G.; Guo, X., Epsilon-relaxed approach in structural topology optimization, Struct. Multidiscip. Optim., 13, 4, 258-266 (1997)
[18] Bruggi, M., On an alternative approach to stress constraints relaxation in topology optimization, Struct. Multidiscip. Optim., 36, 125-141 (2008) · Zbl 1273.74397
[19] Bruggi, M.; Duysinx, P., Topology optimization for minimum weight with compliance and stress constraints, Struct. Multidiscip. Optim., 46, 3, 369-384 (2012) · Zbl 1274.74219
[20] Yang, R.; Chen, C., Stress-based topology optimization, Struct. Multidiscip. Optim., 12, 2, 98-105 (1996)
[21] Xia, L.; Zhang, L.; Xia, Q.; Shi, T. L., Stress-based topology optimization using bi-directional evolutionary structural optimization method, Comput. Methods Appl. Mech. Eng., 333, 356-370 (2018)
[22] Zhao, F.; Xia, L.; Lai, W. X.; Xia, Q.; Shi, T. L., Evolutionary topology optimization of continuum structures with stress constraints, Struct. Multidiscip. Optim., 59, 647-658 (2019)
[23] Xie, Y. M.; Steven, G. P., A simple evolutionar‘y procedure for structural optimization, Comput. Struct., 49, 885-896 (1993)
[24] Huang, X.; Xie, Y. M., Convergent and mesh-independent solutions for bi-directional evolutionary structural optimization method, Finite Elem. Anal. Des., 43, 1039-1049 (2007)
[25] Fritzen, F.; Xia, L.; Leuschner, M.; Breitkopf, P., Topology optimization of multiscale elastoviscoplastic structures, Int. J. Numer. Methods Eng., 106, 6, 430-453 (2016) · Zbl 1352.74239
[26] Xia, L.; Fritzen, F.; Breitkopf, P., Evolutionary topology optimization of elastoplastic structures, Struct. Multidiscip. Optim., 55, 2, 569-581 (2017)
[27] Xia, L.; Da, D.; Yvonnet, J., Topology optimization for maximizing the fracture resistance of quasi-brittle composites, Comput. Methods Appl. Mech. Eng., 332, 234-254 (2018)
[28] Bruns, T.; Tortorelli, D., An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms, Int. J. Numer. Methods Eng., 57, 10, 1413-1430 (2003) · Zbl 1062.74589
[29] Buhl, T.; Pedersen, C.; Sigmund, O., Stiffness design of geometrically nonlinear structures using topology optimization, Struct. Multidiscip. Optim., 19, 2, 93-104 (2000)
[30] Gea, H.; Luo, J., Topology optimization of structures with geometrical nonlinearities, Comput. Struct., 79, 20-21, 1977-1985 (2001)
[31] Luo, Y.; Wang, M.; Kang, Z., Topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique, Comput. Methods Appl. Mech. Eng., 286, 422-441 (2015) · Zbl 1423.74754
[32] Pedersen, C.; Buhl, T.; Sigmund, O., Topology synthesis of large-displacement compliant mechanisms, Int. J. Numer. Methods Eng., 50, 12, 2683-2705 (2001) · Zbl 0988.74055
[33] Wang, F.; Lazarov, B.; Sigmund, O.; Jensen, J., Interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems, Comput. Methods Appl. Mech. Eng., 276, 453-472 (2014) · Zbl 1423.74768
[34] Yoon, G.; Kim, Y., Element connectivity parameterization for topology optimization of geometrically nonlinear structures, Int. J. Solids Struct., 42, 7, 1983-2009 (2005) · Zbl 1111.74035
[35] Bendsøe, M.; Guedes, J.; Plaxton, S.; Taylor, J., Optimization of structure and material properties for solids composed of softening material, Int. J. Solids Struct., 33, 12, 1799-1813 (1996) · Zbl 0919.73114
[36] Maute, K.; Schwarz, S.; Ramm, E., Adaptive topology optimization of elastoplastic structures, Struct. Optim., 15, 2, 81-91 (1998)
[37] Schwarz, S.; Maute, K.; Ramm, E., Topology and shape optimization for elastoplastic structural response, Comput. Methods Appl. Mech. Eng., 190, 15-17, 2135-2155 (2001) · Zbl 1067.74052
[38] Yoon, G.; Kim, Y., Topology optimization of material-nonlinear continuum structures by the element connectivity parameterization, Int. J. Numer. Methods Eng., 69, 10, 2196-2218 (2007) · Zbl 1194.74278
[39] Yuge, K.; Kikuchi, N., Optimization of a frame structure subjected to a plastic deformation, Struct. Optim., 10, 3-4, 197-208 (1995)
[40] Yuge, K.; Iwai, N.; Kikuchi, N., Optimization of 2-D structures subjected to nonlinear deformations using the homogenization method, Struct. Optim., 17, 4, 286-299 (1999)
[41] Huang, X.; Xie, Y. M., Topology optimization of nonlinear structures under displacement loading, Eng. Struct., 30, 7, 2057-2068 (2008)
[42] Huang, X.; Xie, Y. M.; Lu, G., Topology optimization of energy-absorbing structures, Int. J. Crash Worthiness, 12, 6, 663-675 (2007)
[43] Jung, D.; Gea, H., Topology optimization of nonlinear structures, Finite Elem. Anal. Des., 40, 11, 1417-1427 (2004)
[44] Buhl, T.; Pedersen, C. B.W.; Sigmund, O., Stiffness design of geometrically nonlinear structures using topology optimization, Struct. Multidiscip. Optim., 19, 93-104 (2000)
[45] Burns, T. E.; Tortorelli, D. A., An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms, Int. J. Num. Methods Eng., 57, 1413-1430 (2003) · Zbl 1062.74589
[46] Crisfield, M. A., Non-linear Finite Element Analysis of Solids and Structures (1991), Wiley: Wiley New York · Zbl 0809.73005
[47] Duysinx, P.; Sigmund, O., New development in handling stress constraints in optimal material distribution, (Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. A Collection of Technical Papers (Held in St. Louis, Missouri), 3 (1998)), 1501-1509
[48] Huang, X.; Xie, Y. M., Evolutionary topology optimization of continuum structures with an additional displacement constraint, Struct. Multidiscip. Optim., 40, 409-416 (2010) · Zbl 1274.74343
[49] Zuo, Z. H.; Xie, Y. M.; Huang, X., Evolutionary topology optimization of structures with multiple displacement and frequency constraints, Adv. Struct. Eng., 15, 2, 359-372 (2012)
[50] Sigmund, O., A 99 line topology optimization code written in MATLAB, Struct. Multidiscip. Optim., 21, 2, 120-127 (2001)
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.