zbMATH — the first resource for mathematics

A surrogate assisted adaptive framework for robust topology optimization. (English) Zbl 1440.74283
Summary: This paper presents a novel approach for robust topology optimization (RTO). The proposed approach integrates the hybrid polynomial correlated function expansion (H-PCFE) with the deterministic topology optimization (DTO) algorithm, where H-PCFE is used for computing the moments associated with the objective function in robust design optimization (RDO). It is argued that re-training the H-PCFE model at every iteration will make the overall algorithm computationally expensive and time-consuming. To alleviate this issue, an adaptive algorithm is proposed that automatically decides whether to retrain the H-PCFE model or to use the available H-PCFE model. Since RTO involves a large number of design variables, we also argue that, from a computational point of view, gradient-based optimization algorithms are better suited over gradient-free optimization techniques. However, the bottleneck of using gradient-based optimization algorithms is associated with the computation of the gradients. To address this issue, we propose a procedure for computing the gradients of the objective function in RTO. The proposed approach is highly flexible and can be integrated with the already available DTO codes from literature. To illustrate the performance of the proposed approach, three well-known benchmark problems have been solved; two of them being on the 2D MBB beam and one more on a 3D cantilever beam. As compared to the results of the DTO, RTO yields conservative results which are visible from the additional limbs appearing in the optimized topology of the structure. Furthermore, to justify the use of H-PCFE as a surrogate within the proposed framework, a comparative assessment of the proposed approach against the popular surrogate models has also been presented. It is observed that H-PCFE outperforms other popular surrogate models such as Kriging, radial basis function and polynomial chaos expansion, in both accuracy and efficiency.
74P15 Topological methods for optimization problems in solid mechanics
top88.m; top.m
Full Text: DOI
[1] Bendsøe, M. P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Engrg., 71, 2, 197-224 (1988) · Zbl 0671.73065
[2] Bendsøe, M. P., Optimal shape design as a material distribution problem, Struct. Optim., 1, 4, 193-202 (1989)
[3] Zhou, M.; Rozvany, G. I.N., The COC algorithm, Part II: Topological, geometrical and generalized shape optimization, Comput. Methods Appl. Mech. Engrg., 89, 1-3, 309-336 (1991)
[4] Mlejnek, H. P., Some aspects of the genesis of structures, Struct. Optim., 5, 1-2, 64-69 (1992)
[5] Allaire, G.; Jouve, F.; Toader, A. M., Une méthode de lignes de niveaux pour l’optimisation de forme, C. R. Math., 334, 12, 1125-1130 (2002)
[6] Allaire, G.; Jouve, F.; Toader, A. M., Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194, 1, 363-393 (2004) · Zbl 1136.74368
[7] Wang, M.; Wang, X.; Guo, D., A level set method for structural topology optimization, Comput. Methods Appl. Mech. Engrg., 192, 227-246 (2003) · Zbl 1083.74573
[8] Ghasemi, H.; Park, H. S.; Rabczuk, T., A level-set based IGA formulation for topology optimization of flexoelectric materials, Comput. Methods Appl. Mech. Engrg., 313, 239-258 (2017)
[9] Ghasemi, H.; Park, H. S.; Rabczuk, T., A multi-material level set-based topology optimization of flexoelectric composites, Comput. Methods Appl. Mech. Engrg., 332, 47-62 (2018)
[10] Sokolowski, J.; Zochowski, A., On the topological derivative in shape optimization, SIAM J. Control Optim., 37, 4, 1251-1272 (1999) · Zbl 0940.49026
[11] Bourdin, B.; Chambolle, A., Design-dependent loads in topology optimization, ESAIM Control Optim. Calc. Var., 9, 19-48 (2003) · Zbl 1066.49029
[12] Xie, Y. M.; Steven, G. P., A simple evolutionary procedure for structural optimization, Comput. Struct., 49, 5, 885-896 (1993)
[13] Yamada, T.; Izui, K.; Nishiwaki, S.; Takezawa, A., A topology optimization method based on the level set method incorporating a fictitious interface energy, Comput. Methods Appl. Mech. Engrg., 199, 45-48, 2876-2891 (2010) · Zbl 1231.74365
[14] Maute, K.; Ramm, E., General shape optimization - an integrated model for topology and shape optimization, Struct. Multidiscip. Optim., 964 (1995)
[15] Eschenauer, H. A.; Kobelev, V. V.; Schumacher, A., Bubble method for topology and shape optimization of structures, Struct. Optim., 8, 1, 42-51 (1994)
[16] Kim, D.-H.; Lee, S. B.; Kwank, B. M.; Kim, H.-G.; Lowther, D., Smooth boundary topology optimization for electrostatic problems through the combination of shape and topological design sensitivities, IEEE Trans. Magn., 44, 6, 1002-1005 (2008)
[17] Le, C.; Bruns, T.; Tortorelli, D., A gradient-based, parameter-free approach to shape optimization, Comput. Methods Appl. Mech. Engrg., 200, 9-12, 985-996 (2011) · Zbl 1225.74065
[18] Arnout, S.; Firl, M.; Bletzinger, K.-U., Parameter free shape and thickness optimisation considering stress response, Struct. Multidiscip. Optim., 45, 6, 801-814 (2012) · Zbl 1274.74444
[19] Liu, K.; Tovar, A., An efficient 3D topology optimization code written in Matlab, Struct. Multidiscip. Optim., 50 (2014)
[20] Sigmund, O., On the design of compliant mechanisms using topology optimization*, Mech. Struct. Mach., 25, 4, 493-524 (1997)
[21] Petersson, J.; Sigmund, O., Slope constrained topology optimization, Internat. J. Numer. Methods Engrg., 41, 8, 1417-1434 (1998) · Zbl 0907.73044
[22] Ambrosio, L.; Buttazzo, G., An optimal design problem with perimeter penalization, Calc. Var. Partial Differential Equations, 1, 1, 55-69 (1993) · Zbl 0794.49040
[23] Bruns, T. E.; Tortorelli, D. A., Topology optimization of non-linear elastic structures and compliant mechanisms, Comput. Methods Appl. Mech. Engrg., 190, 26-27, 3443-3459 (2001) · Zbl 1014.74057
[24] Bourdin, B., Filters in topology optimization, Int. J. Numer. Methods Eng., 50, 2143-2158 (2001) · Zbl 0971.74062
[25] Guest, J. K.; Prévost, J. H.; Belytschko, T., Achieving minimum length scale in topology optimization using nodal design variables and projection functions, Internat. J. Numer. Methods Engrg., 61, 2, 238-254 (2004) · Zbl 1079.74599
[26] Sigmund, O., Morphology-based black and white filters for topology optimization, Struct. Multidiscip. Optim., 33, 4-5, 401-424 (2007)
[27] Xu, S.; Cai, Y.; Cheng, G., Volume preserving nonlinear density filter based on heaviside functions, Struct. Multidiscip. Optim., 41, 4, 495-505 (2000) · Zbl 1274.74419
[28] Stolpe, M.; Svanberg, K., An alternative interpolation scheme for minimum compliance topology optimization, Struct. Multidiscip. Optim., 22, 2, 116-124 (2001)
[29] Burger, M.; Osher, S. J., A survey on level set methods for inverse problems and optimal design, European J. Appl. Math., 16, 2, 263-301 (2005) · Zbl 1091.49001
[30] Norato, J.; Haber, R.; Tortorelli, D.; Bendsøe, M. P., A geometry projection method for shape optimization, Internat. J. Numer. Methods Engrg., 60, 14, 2289-2312 (2004) · Zbl 1075.74702
[31] de Ruiter, M.; van Keulen, F., Topology optimization using a topology description function, Struct. Multidiscip. Optim., 26, 6, 406-416 (2004)
[32] Wang, S.; Wang, M. Y., Radial basis functions and level set method for structural topology optimization, Internat. J. Numer. Methods Engrg., 65, 12, 2060-2090 (2006) · Zbl 1174.74331
[33] Luo, Z.; Tong, L.; Wang, M. Y.; Wang, S., Shape and topology optimization of compliant mechanisms using a parameterization level set method, J. Comput. Phys., 227, 1, 680-705 (2007) · Zbl 1127.65043
[34] Gomes, A. A.; Suleman, A., Application of spectral level set methodology in topology optimization, Struct. Multidiscip. Optim., 31, 6, 430-443 (2006) · Zbl 1245.74072
[35] Huang, S. P.; Quek, S. T.; Phoon, K. K., Convergence study of the truncated Karhunen-Loeve expansion for simulation of stochastic processes, Internat. J. Numer. Methods Engrg., 52, 1029-1043 (2001) · Zbl 0994.65004
[36] Chakraborty, S.; Chowdhury, R., Uncertainty propagation using hybrid HDMR for stochastic field problems, (International Conference on Structural Engineering and Mechanics (2013))
[37] Schuëller, G.; Jensen, H., Computational methods in optimization considering uncertainties An overview, Comput. Methods Appl. Mech. Engrg., 198, 1, 2-13 (2008) · Zbl 1194.74258
[38] Zheng, Y.; Gao, L.; Xiao, M.; Li, H.; Luo, Z., Robust topology optimization considering load uncertainty based on a semi-analytical method, Int. J. Adv. Manuf. Technol., 94, 9-12, 3537-3551 (2018)
[40] Dubourg, V.; Sudret, B.; Bourinet, J.-M., Reliability-based design optimization using kriging surrogates and subset simulation, Struct. Multidiscip. Optim., 44, 5, 673-690 (2011)
[41] Ditlevsen, O.; Madsen, H. O., Structural Reliability Methods (2007), Wiley-Blackwell
[42] Tu, J.; Choi, K. K.; Park, Y. H., Design potential method for robust system parameter design, AIAA J., 39, 4, 667-677 (2001)
[43] Nguyen, T. H.; Song, J.; Paulino, G. H., Single-loop system reliability-based topology optimization considering statistical dependence between limit-states, Struct. Multidiscip. Optim., 44, 5, 593-611 (2011) · Zbl 1274.74373
[44] Royset, J.; Der Kiureghian, A.; Polak, E., Reliability-based optimal structural design by the decoupling approach, Reliab. Eng. Syst. Saf., 73, 3, 213-221 (2001)
[45] Lógó, J., New type of optimal topologies by iterative method, Mech. Based Des. Struct. Mach., 33, 2, 149-171 (2005)
[46] Lógó, J.; Ghaemi, M.; Rad, M. M., Optimal topologies in case of probabilistic loading: the influence of load correlation, Mech. Based Des. Struct. Mach., 37, 3, 327-348 (2009)
[47] 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
[48] Chatterjee, T.; Chakraborty, S.; Chowdhury, R., A bi-level approximation tool for the computation of FRFs in stochastic dynamic systems, Mech. Syst. Signal Process., 70-71, 484-505 (2016)
[49] Chakraborty, S.; Chowdhury, R., Moment independent sensitivity analysis: H-PCFEBased approach, J. Comput. Civ. Eng., 31, 1 (2016), 06016001:1-06016001:11
[50] Chakraborty, S.; Chowdhury, R., Hybrid framework for the estimation of rare failure event probability, J. Eng. Mech.-ASCE, 143, 5 (2017)
[51] Chakraborty, S.; Chowdhury, R., An efficient algorithm for building locally refined hp adaptive H-PCFE: Application to uncertainty quantification, J. Comput. Phys., 351 (2017)
[52] Chakraborty, S.; Majumder, D., Hybrid reliability analysis framework (HRAF) for reliability analysis of tunnels, J. Comput. Civ. Eng., 32, 4 (2018)
[53] Majumder, D.; Chakraborty, S.; Chowdhury, R., Probabilistic analysis of tunnels: A hybrid polynomial correlated function expansion based approach, Tunnelling Underground Space Technol., 70 (2017)
[54] Chakraborty, S.; Chowdhury, R., Towards h-p adaptive’ generalised ANOVA, Comput. Methods Appl. Mech. Engrg., 320, 558-581 (2017)
[55] Chakraborty, S.; Chowdhury, R., Modelling uncertainty in incompressible flow simulation using Galerkin based generalized ANOVA, Comput. Phys. Comm., 208, 73-91 (2016) · Zbl 1375.76039
[56] Bilionis, I.; Zabaras, N.; Konomi, B. A.; Lin, G., Multi-output separable Gaussian process: Towards an efficient, fully Bayesian paradigm for uncertainty quantification, J. Comput. Phys., 241, 212-239 (2013) · Zbl 1349.76760
[57] Mukhopadhyay, T.; Chakraborty, S.; Dey, S.; Adhikari, S.; Chowdhury, R., A critical assessment of kriging model variants for high-fidelity uncertainty quantification in dynamics of composite shells, Arch. Comput. Methods Eng., 24, 3, 495-518 (2017) · Zbl 1376.62133
[58] Biswas, S.; Chakraborty, S.; Chandra, S.; Ghosh, I., Kriging-Based Approach for Estimation of Vehicular Speed and Passenger Car Units on an Urban Arterial, J. Transp. Eng. Part A: Syst., 143, 3, 04016013 (2017)
[59] Bendsøe, M. P., Optimization of Structural Topology, Shape, and Material, 271 (1995), Springer · Zbl 0822.73001
[60] Chakraborty, S.; Chowdhury, R., Polynomial correlated function expansion, (Modeling and Simulation Techniques in Structural Engineering (2017), IGI Global), 348-373
[61] Chakraborty, S.; Chatterjee, T.; Chowdhury, R.; Adhikari, S., A surrogate based multi-fidelity approach for robust design optimization, Appl. Math. Model., 47, 726-744 (2017) · Zbl 1446.90009
[62] Kaymaz, I., Application of Kriging method to structural reliability problems, Struct. Saf., 27, 2, 133-151 (2005)
[63] Chakraborty, S.; Chowdhury, R., Multivariate function approximations using D-MORPH algorithm, Appl. Math. Model., 39, 23-24, 7155-7180 (2015) · Zbl 1443.41018
[64] Rao, C. R.; Mitra, S. K., Generalized inverse of a matrix and its applications, (Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (1971))
[65] Li, G.; Rabitz, H., D-MORPH regression: application to modeling with unknown parameters more than observation data, J. Math. Chem., 48, 4, 1010-1035 (2010) · Zbl 1303.62032
[66] Li, G.; Rey-de Castro, R.; Rabitz, H., D-MORPH regression for modeling with fewer unknown parameters than observation data, J. Math. Chem., 50, 7, 1747-1764 (2012) · Zbl 1314.62154
[67] Sobol, I. M., Uniformly distributed sequences with an additional uniform property, USSR Comput. Math. Math. Phys., 16, 236-242 (1976) · Zbl 0391.10033
[68] Blatman, G.; Sudret, B., Adaptive sparse polynomial chaos expansion based on least angle regression, J. Comput. Phys., 230, 6, 2345-2367 (2011) · Zbl 1210.65019
[69] Sudret, B., Global sensitivity analysis using polynomial chaos expansions, Reliab. Eng. Syst. Saf., 93, 7, 964-979 (2008)
[70] Tripathy, R.; Bilionis, I.; Gonzalez, M., Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation, J. Comput. Phys., 321, 191-223 (2016) · Zbl 1349.65049
[71] Bilionis, I.; Zabaras, N.; Konomi, B. A.; Lin, G., Multi-output separable Gaussian process: Towards an efficient, fully Bayesian paradigm for uncertainty quantification, J. Comput. Phys., 241, 212-239 (2013) · Zbl 1349.76760
[72] Bilionis, I.; Zabaras, N., Multi-output local Gaussian process regression: Applications to uncertainty quantification, J. Comput. Phys., 231, 17, 5718-5746 (2012) · Zbl 1277.60066
[73] Sigmund, O., A 99 line topology optimization code written in Matlab, Struct. Multidiscip. Optim., 21, 2, 120-127 (2001)
[74] Andreassen, E.; Clausen, A.; Schevenels, M.; Lazarov, B. S.; Sigmund, O., Efficient topology optimization in MATLAB using 88 lines of code, Struct. Multidiscip. Optim., 43 (2011) · Zbl 1274.74310
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.