Journal of Applied and Computational Mechanics

Proportional Topology Optimization under ‎Reliability-based Constraints

Document Type : Research Paper

Authors

1 Graduate Program in Mechanical Engineering, Federal University of Rio Grande do Sul, Av. Sarmento Leite, 425, Sala 202, 2º Andar, 90050-170, Porto Alegre, RS. Brazil‎

2 Graduate Program in Civil Engineering, Federal University of Rio Grande do Sul, Av. Osvaldo Aranha, 99, 3º Andar, 90035-190, Porto Alegre, RS. Brazil‎

3 Graduate Program in Mechanical Engineering, Federal University of Rio Grande do Sul,‎ Av. Sarmento Leite, 425, Sala 202, 2º Andar, 90050-170, Porto Alegre, RS. Brazil‎

Abstract

Topology optimization is a methodology widely used in the design phase that has gained space in engineering. On the other hand, uncertainty is present in material properties, loads, and boundary conditions in practically any design. The main goal for this paper lies in the coupling of the two subjects to account for uncertainties in the topology optimization. The Proportional Topology Optimization method renders the possibility of treating the stress constraints in a unified way. This allows topologies that at the same time preserve structural reliability and optimize costs. The Proportional Topology Optimization method under the reliability constraint is presented for isostatic and hyperstatic beam examples with stress and displacement LSF.

Keywords

Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 

[1] Bekdas, G., Nigdeli, S.M., Kayabekir, A.E., Yang, X., Optimization in Civil Engineering and Metaheuristic Algorithms: A Review of State-of-the-Art Developments, In: Computational Intelligence, Optimization and Inverse Problems with Applications in Engineering, 2019. https://doi.org/10.1007/978-3-319-96433-1_6.
[2] Kongwat, S., Hasegawa H., Optimization on mechanical structure for material nonlinearity based on proportional topology method, Journal of Advanced Simulation in Science and Engineering, 6 (2), 2019, 354-366. https://doi.org/10.15748/jasse.6.354.
[3] Wang, Y., Xu, H., Pasini, D., Multiscale isogeometric topology optimization for lattice materials, Computer Methods in Applied Mechanics and Engineering, 316, 2017, 568-585. https://doi.org/10.1016/j.cma.2016.08.015.
[4] Zhang, J., Sato, Y., Yanagimoto, J., Homogenization-based topology optimization integrated with elastically isotropic lattices for additive manufacturing of ultralight and ultra-stiff structures, CIRP Annals – Manufacturing Technology, 70, 2021, 111-114. https://doi.org/10.1016/j.cirp.2021.04.019.
[5] Zhang, J., Yanagimoto, J., Topology optimization of micro lattice dome with enhanced stiffness and energy absorption for additive manufacturing, Composite Structures, 255, 2021, 112889. https://doi.org/10.1016/j.compstruct.2020.112889.
[6] Gan, N., Wang, Q., Topology optimization design of improved response surface method for time-variant reliability, Advances in Engineering Software, 146, 2020, 102828. https://doi.org/10.1016/j.advengsoft.2020.102828.
[7] da Silva, G.A., Cardoso, E.L., Beck, A.T., Comparison of robust, reliability-based and non-probabilistic topology optimization under uncertain loads and stress constraints, Probabilistic Engineering Mechanics, 59, 2020, 103039. https://doi.org/10.1016/j.probengmech.2020.103039.
[8] Hasofer, A.M., Lind, D., An exact and invariant first-order reliability format, Journal of Engineering Mechanics, 100, 1974, 111-121.
[9] Breitung, K., Asymptotic Approximations for Multinormal Integrals, Journal of Engineering Mechanics, 110(3), 1984, 357–366.
[10] Ang, A.H.S., Tang, W.H., Probability Concepts in Engineering: Emphasis on Applications to Civil and Environmental Engineering: Wiley; 2nd Edition, 2006.
[11] Rostami, S. A. L., Kolahdooz, A., Zhang, J., Robust topology optimization under material and loading uncertainties using an evolutionary structural extended finite element method, Engineering Analysis with Boundary Elements, 133, 2021, 61–70. https://doi.org/10.1016/j.enganabound.2021.08.023.
[12] Biyikli, E., To, A.C., Proportional Topology Optimization: A New Non-Sensitivity Method for Solving Stress Constrained and Minimum Compliance Problems and Its Implementation in MATLAB, PLoS ONE, 10(12), 2015. https://doi.org/10.1371/journal.pone.0145041.
[13] Li, Q., Steven, G.P., Xie, Y. M., On the equivalence between stress criterion and stiffness criterion in evolutionary structural optimization, Structural Optimization, 18, 1999, 67-73.
[14] Kharmanda, G., Olhoff, N., Mohamed, A., Lemaire, M., Reliability-based topology optimization, Structural and Multidisciplinary Optimization, 26, 2004, 295-307. https://doi.org/10.1007/s00158-003-0322-7.
[15] Jung, H., Cho, S., Reliability-based topology optimization of geometrically nonlinear structures with loading and material uncertainties, Finite Element in Analysis and Design, 43, 2004, 311-331. https://doi:10.1016/j.finel.2004.06.002.
[16] Tootkaboni, M., Asadbourne, A., Guest, J.K., Topology optimization of continuum structures under uncertainty – A polynomial Chaos approach, Computer Methods in Applied Mechanics and Engineering, 201-204, 2012, 263-276. https://doi.org/10.1016/j.cma.2011.09.009.
[17] Ghasemi, H., Rafiee, R., Zhuang, X., Muthu, J., Rabczuk, T., Uncertainties propagation in metamodel-based probabilistic optimization of CNT/polymer composite structure using stochastic multi-scale modeling, Computational Materials Science, 85, 2014, 295-305. http://dx.doi.org/10.1016/j.commatsci.2014.01.020.
[18] Ghasemi, H., Park, H. S., Rabczuk, T., A multi-material level set-based topology optimization of flexoelectric composites, Computer Methods in Applied Mechanics and Engineering, 332, 2018, 47-62. https://doi.org/10.1016/j.cma.2017.12.005.
[19] Guo, X., Zhang, W., Zhang, L., Robust structural topology optimization considering boundary uncertainties, Computer Methods in Applied Mechanics and Engineering, 253, 2013, 356-368. http://dx.doi.org/10.1016/j.cma.2012.09.005.
[20] Luo, Y., Zhou, M., Wang, M.Y., Deng, Z., Reliability-based topology optimization for continuum structures with local failure constraints, Computer and Structures, 143, 2014, 73-84. http://dx.doi.org/10.1016/j.compstruc.2014.07.009.
[21] Cheng, G., Guo, X., e-relaxed approach in structural topology optimization, Engineering Optimization, 13, 1997, 258–266.
[22] da Silva, G.A., Cardoso, E.L., Stress-based topology optimization of continuum structures under uncertainties, Computer Methods in Applied Mechanics and Engineering, 313, 2017, 647-672. http://dx.doi.org/10.1016/j.cma.2016.09.049.
[23] António, C.C., Hoffbauer, L.N., Reliability-based design optimization and uncertainty quantification for optimal conditions of composite structures with nonlinear behavior, Engineering Structures, 153, 2017, 479-490. http://dx.doi.org/10.1016/j.engstruct.2017.10.041.
[24] Chun, J., Song, J., Paulino, G.H., System reliability-based design and topology of structures under constraints on first-passage probability, Structural Safety, 76, 2019, 81-94. https://doi.org/10.1016/j.strusafe.2018.06.006.
[25] Keshavarzzadeh, V., Kirby, R.M., Narayan A., Stress-based topology optimization under uncertainty via simulation-based Gaussian process, Computer Methods in Applied Mechanics and Engineering, 365, 2020, 112992. https://doi.org/10.1016/j.cma.2020.112992.
[26] Liu, B., Jiang, C., Li, G., Huang, X., Topology optimization of structures considering local material uncertainties in additive manufacturing, Computer Methods in Applied Mechanics and Engineering, 360, 2020, 112786. https://doi.org/10.1016/j.cma.2019.112786.
[27] Gomez, F., Spencer, B.F., Carrion, J., Topology optimization of buildings subjected to stochastic base excitation, Engineering Structures, 223, 2020, 111111. https://doi.org/10.1016/j.engstruct.2020.111111.
[28] Rao, S.S., Engineering Optimization: Theory and Practice, John Wiley and Sons, 4th Edition, 2009. https://doi.org/10.1002/9780470549124.
[29] Bendsøe, M.P., Optimal shape design as a material distribution problem, Structural Optimization, 1989, 193-202. https://doi.org/10.1007/BF01650949
[30] Le, C., Norato, J., Brums, T., Ha, C., Tortorelli, D., Stress-based topology optimization for continua, Structural Multidisciplinary Optimization, 41, 2010, 605-620. https://doi.org/10.1007/s00158-009-0440-y.
[31] Huang, X., Xie, Y.M., Evolutionary topology optimization of continuum structures: methods and applications, Wiley, 2010.
[32] Kuhinek, D., Zorić, I., Hrženjak, P., Measurement Uncertainty in Testing of Uniaxial Compressive Strength and Deformability of Rock Samples, Measurement Science Review, 11, 2011, 112-117. https://doi.org/10.2478/v10048-011-0021-2.
Journal of Applied and Computational Mechanics
Volume 8, Issue 1
January 2022
Pages 319-330