×

Topology optimization of compliant mechanisms considering stress constraints, manufacturing uncertainty and geometric nonlinearity. (English) Zbl 1442.74165

Summary: This paper proposes and investigates two formulations to topology optimization of compliant mechanisms considering stress constraints, manufacturing uncertainty and geometric nonlinearity. The first formulation extends the maximum output displacement robust approach with stress constraints to incorporate the effects of geometric nonlinear behavior during the optimization process. The second formulation relies on the concept of path-generating mechanisms, where not only the final configuration is important, but also the load-displacement equilibrium path. A novel path-generating formulation is thus proposed, not only to achieve the prescribed equilibrium path, but also to take stress constraints and manufacturing uncertainty into account during the optimization process. Although both formulations have different goals, the same main techniques are employed: density approach to topology optimization, augmented Lagrangian method to handle the large number of stress constraints, three-field robust approach to handle the manufacturing uncertainty, and the energy interpolation scheme to handle convergence issues due to large deformation in void regions. Several numerical examples are addressed to demonstrate applicability of the proposed approaches. The optimized results are post-processed with body-fitted finite element meshes. Obtained results demonstrate that: (1) the proposed nonlinear analysis based maximum output displacement approach is able to provide solutions with good performance in situations of large displacements, with stress and manufacturing requirements satisfied; (2) the linear analysis based maximum output displacement approach provides optimized topologies that show large stress constraint violations and rapidly varying stress behavior under uniform boundary variation, when these are post-processed with full nonlinear analysis; (3) the proposed path-generating formulation is able to provide solutions that follow the prescribed control points, including stress robustness.

MSC:

74P15 Topological methods for optimization problems in solid mechanics

Software:

ALGENCAN; Gmsh

References:

[1] Bendsøe, M. P.; Sigmund, O., Topology Optimization: Theory, Methods and Applications (2003), Springer: Springer Berlin · Zbl 1059.74001
[2] Sigmund, O.; Maute, K., Topology optimization approaches, Struct. Multidiscip. Optim., 48, 6, 1031-1055 (2013)
[3] Sigmund, O., On the design of compliant mechanisms using topology optimization, Mech. Struct. Mach., 25, 4, 493-524 (1997)
[4] Wang, F.; Lazarov, B. S.; Sigmund, O.; Jensen, J. S., Interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems, Comput. Methods Appl. Mech. Engrg., 276, 453-472 (2014) · Zbl 1423.74768
[5] Buhl, T.; Pedersen, C.; Sigmund, O., Stiffness design of geometrically nonlinear structures using topology optimization, Struct. Multidiscip. Optim., 19, 2, 93-104 (2000)
[6] Pedersen, C. B.W.; Buhl, T.; Sigmund, O., Topology synthesis of large-displacement compliant mechanisms, Internat. J. Numer. Methods Engrg., 50, 12, 2683-2705 (2001) · Zbl 0988.74055
[7] Bruns, T. E.; Tortorelli, D. A., An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms, Internat. J. Numer. Methods Engrg., 57, 10, 1413-1430 (2003) · Zbl 1062.74589
[8] Yoon, G. H.; Kim, Y. Y., Element connectivity parameterization for topology optimization of geometrically nonlinear structures, Int. J. Solids Struct., 42, 7, 1983-2009 (2005) · Zbl 1111.74035
[9] Klarbring, A.; Strömberg, N., Topology optimization of hyperelastic bodies including non-zero prescribed displacements, Struct. Multidiscip. Optim., 47, 1, 37-48 (2013) · Zbl 1274.74351
[10] 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. Engrg., 265, 226-241 (2013) · Zbl 1286.74076
[11] Luo, Y.; Wang, M. Y.; Kang, Z., Topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique, Comput. Methods Appl. Mech. Engrg., 286, 422-441 (2015) · Zbl 1423.74754
[12] Wallin, M.; Ivarsson, N.; Tortorelli, D., Stiffness optimization of non-linear elastic structures, Comput. Methods Appl. Mech. Engrg., 330, 292-307 (2018) · Zbl 1439.74053
[13] Li, Y.; Zhu, J.; Wang, F.; Zhang, W.; Sigmund, O., Shape preserving design of geometrically nonlinear structures using topology optimization, Struct. Multidiscip. Optim., 59, 4, 1033-1051 (2019)
[14] Lahuerta, R. D.; Simões, E. T.; Campello, E. M.B.; Pimenta, P. M.; Silva, E. C.N., Towards the stabilization of the low density elements in topology optimization with large deformation, Comput. Mech., 52, 4, 779-797 (2013) · Zbl 1311.74099
[15] Duysinx, P.; Bendsøe, M. P., Topology optimization of continuum structures with local stress constraints, Internat. J. Numer. Methods Engrg., 43, 8, 1453-1478 (1998) · Zbl 0924.73158
[16] Pereira, J. T.; Fancello, E. A.; Barcellos, C. S., Topology optimization of continuum structures with material failure constraints, Struct. Multidiscip. Optim., 26, 1-2, 50-66 (2004) · Zbl 1243.74157
[17] Le, C.; Norato, J.; Bruns, T.; Ha, C.; Tortorelli, D., Stress-based topology optimization for continua, Struct. Multidiscip. Optim., 41, 4, 605-620 (2010)
[18] Amstutz, S.; Novotny, A. A., Topological optimization of structures subject to von mises stress constraints, Struct. Multidiscip. Optim., 41, 3, 407-420 (2010) · Zbl 1274.74053
[19] Guo, X.; Zhang, W. S.; Wang, M. Y.; Wei, P., Stress-related topology optimization via level set approach, Comput. Methods Appl. Mech. Engrg., 200, 47, 3439-3452 (2011) · Zbl 1230.74152
[20] 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
[21] Holmberg, E.; Torstenfelt, B.; Klarbring, A., Stress constrained topology optimization, Struct. Multidiscip. Optim., 48, 1, 33-47 (2013) · Zbl 1274.74341
[22] Emmendoerfer, H.; Fancello, E. A., A level set approach for topology optimization with local stress constraints, Internat. J. Numer. Methods Engrg., 99, 2, 129-156 (2014) · Zbl 1352.74238
[23] Collet, M.; Noël, L.; Bruggi, M.; Duysinx, P., Topology optimization for microstructural design under stress constraints, Struct. Multidiscip. Optim., 58, 6, 2677-2695 (2018)
[24] Salazar de Troya, M. A.; Tortorelli, D. A., Adaptive mesh refinement in stress-constrained topology optimization, Struct. Multidiscip. Optim., 58, 6, 2369-2386 (2018)
[25] Picelli, R.; Townsend, S.; Brampton, C.; Norato, J.; Kim, H., Stress-based shape and topology optimization with the level set method, Comput. Methods Appl. Mech. Engrg., 329, 1-23 (2018) · Zbl 1439.74294
[26] De Leon, D. M.; Alexandersen, J.; O. Fonseca, J. S.; Sigmund, O., Stress-constrained topology optimization for compliant mechanism design, Struct. Multidiscip. Optim., 52, 5, 929-943 (2015)
[27] Lopes, C. G.; Novotny, A. A., Topology design of compliant mechanisms with stress constraints based on the topological derivative concept, Struct. Multidiscip. Optim., 54, 4, 737-746 (2016)
[28] Chu, S.; Gao, L.; Xiao, M.; Luo, Z.; Li, H., Stress-based multi-material topology optimization of compliant mechanisms, Internat. J. Numer. Methods Engrg., 113, 7, 1021-1044 (2018) · Zbl 07874740
[29] de Assis Pereira, A.; Cardoso, E. L., On the influence of local and global stress constraint and filtering radius on the design of hinge-free compliant mechanisms, Struct. Multidiscip. Optim., 58, 2, 641-655 (2018)
[30] Conlan-Smith, C.; James, K. A., A stress-based topology optimization method for heterogeneous structures, Struct. Multidiscip. Optim. (2019)
[31] Cheng, G. D.; Guo, X., \( \varepsilon \)-Relaxed approach in structural topology optimization, Struct. Optim., 13, 4, 258-266 (1997)
[32] Bruggi, M., On an alternative approach to stress constraints relaxation in topology optimization, Struct. Multidiscip. Optim., 36, 2, 125-141 (2008) · Zbl 1273.74397
[33] Duysinx, P.; Sigmund, O., New developments in handling stress constraints in optimal material distribution, (7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization (1998)), 1501-1509
[34] da Silva, G. A.; Beck, A. T.; Sigmund, O., Stress-constrained topology optimization considering uniform manufacturing uncertainties, Comput. Methods Appl. Mech. Engrg., 344, 512-537 (2019) · Zbl 1440.74290
[35] da Silva, G. A.; Beck, A. T.; Sigmund, O., Topology optimization of compliant mechanisms with stress constraints and manufacturing error robustness, Comput. Methods Appl. Mech. Engrg., 354, 397-421 (2019) · Zbl 1441.74156
[36] Sigmund, O., Manufacturing tolerant topology optimization, Acta Mech. Sinica, 25, 2, 227-239 (2009) · Zbl 1270.74165
[37] Wang, F.; Lazarov, B. S.; Sigmund, O., On projection methods, convergence and robust formulations in topology optimization, Struct. Multidiscip. Optim., 43, 6, 767-784 (2011) · Zbl 1274.74409
[38] Chen, S.; Chen, W., A new level-set based approach to shape and topology optimization under geometric uncertainty, Struct. Multidiscip. Optim., 44, 1, 1-18 (2011) · Zbl 1274.49056
[39] Wang, F.; Jensen, J. S.; Sigmund, O., Robust topology optimization of photonic crystal waveguides with tailored dispersion properties, J. Opt. Soc. Amer. B, 28, 3, 387-397 (2011)
[40] Schevenels, M.; Lazarov, B.; Sigmund, O., Robust topology optimization accounting for spatially varying manufacturing errors, Comput. Methods Appl. Mech. Engrg., 200, 49, 3613-3627 (2011) · Zbl 1239.74080
[41] Lazarov, B. S.; Schevenels, M.; Sigmund, O., Robust design of large-displacement compliant mechanisms, Mech. Sci., 2, 2, 175-182 (2011)
[42] Lazarov, B. S.; Schevenels, M.; Sigmund, O., Topology optimization with geometric uncertainties by perturbation techniques, Internat. J. Numer. Methods Engrg., 90, 11, 1321-1336 (2012) · Zbl 1242.74075
[43] Lazarov, B. S.; Schevenels, M.; Sigmund, O., Topology optimization considering material and geometric uncertainties using stochastic collocation methods, Struct. Multidiscip. Optim., 46, 4, 597-612 (2012) · Zbl 1274.74360
[44] Guo, X.; Zhang, W.; Zhang, L., Robust structural topology optimization considering boundary uncertainties, Comput. Methods Appl. Mech. Engrg., 253, 356-368 (2013) · Zbl 1297.74089
[45] Christiansen, R. E.; Lazarov, B. S.; Jensen, J. S.; Sigmund, O., Creating geometrically robust designs for highly sensitive problems using topology optimization, Struct. Multidiscip. Optim., 52, 4, 737-754 (2015)
[46] Jansen, M.; Lombaert, G.; Schevenels, M., Robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis, Comput. Methods Appl. Mech. Engrg., 285, 452-467 (2015) · Zbl 1423.74749
[47] Zhang, W.; Kang, Z., Robust shape and topology optimization considering geometric uncertainties with stochastic level set perturbation, Internat. J. Numer. Methods Engrg., 110, 1, 31-56 (2017) · Zbl 1378.74054
[48] Ruiz, D.; Sigmund, O., Optimal design of robust piezoelectric microgrippers undergoing large displacements, Struct. Multidiscip. Optim., 57, 1, 71-82 (2018)
[49] Bathe, K.-J., Finite Element Procedures (1996), Prentice Hall: Prentice Hall Upper Sadle River, New Jersey
[50] Bendsøe, M. P., Optimal shape design as a material distribution problem, Struct. Optim., 1, 4, 193-202 (1989)
[51] Sigmund, O., Morphology-based black and white filters for topology optimization, Struct. Multidiscip. Optim., 33, 4-5, 401-424 (2007)
[52] Ciarlet, P. G., Mathematical Elasticity. Volume I: Three-Dimensional Elasticity (1988), North-Holland Publishing Company: North-Holland Publishing Company Amsterdam · Zbl 0648.73014
[53] Simo, J. C.; Hughes, T. J.R., Computational Inelasticity (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0934.74003
[54] Bonet, J.; Wood, R. D., Nonlinear Continuum Mechanics for Finite Element Analysis (2008), Cambridge University Press: Cambridge University Press New York · Zbl 1142.74002
[55] Coda, H. B.; Greco, M., A simple fem formulation for large deflection 2d frame analysis based on position description, Comput. Methods Appl. Mech. Engrg., 193, 33, 3541-3557 (2004) · Zbl 1068.74072
[56] Coda, H. B.; Paccola, R. R., An alternative positional fem formulation for geometrically non-linear analysis of shells: Curved triangular isoparametric elements, Comput. Mech., 40, 1, 185-200 (2007) · Zbl 1178.74163
[57] Coda, H. B., A solid-like fem for geometrically non-linear 3d frames, Comput. Methods Appl. Mech. Engrg., 198, 47, 3712-3722 (2009) · Zbl 1230.74174
[58] Fancello, E. A.; Pereira, J. T., Structural topology optimization considering material failure constraints and multiple load conditions, Lat. Am. J. Solids Struct., 1, 1, 3-24 (2003)
[59] Birgin, E. G.; Martínez, J. M., Practical Augmented Lagrangian Methods for Constrained Optimization (2014), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA · Zbl 1339.90312
[60] Kreisselmeier, G.; Steinhauser, R., Systematic control design by optimizing a vector performance index, IFAC Proc. Vol., 12, 7, 113-117 (1979) · Zbl 0443.49028
[61] Olhoff, N., Multicriterion structural optimization via bound formulation and mathematical programming, Struct. Optim., 1, 1, 11-17 (1989)
[62] da Silva, G. A.; Beck, A. T.; Cardoso, E. L., Topology optimization of continuum structures with stress constraints and uncertainties in loading, Internat. J. Numer. Methods Engrg., 113, 1, 153-178 (2018) · Zbl 07867260
[63] Clausen, A.; Andreassen, E., On filter boundary conditions in topology optimization, Struct. Multidiscip. Optim., 56, 5, 1147-1155 (2017)
[64] Geuzaine, C.; Remacle, J.-F., Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79, 11, 1309-1331 (2009) · Zbl 1176.74181
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.