Reliability-based topology optimization under shape uncertainty modeled in Eulerian description
Tóm tắt
This paper presents a reliability-based topology optimization method under geometrical uncertainties. First, we briefly introduce the concept of topology optimization. Then, we explain how shape uncertainty is modeled in Eulerian description, using an advection equation and a Karhunen-Loève expansion. Based on the shape uncertainty modeling, we formulate a reliability measure for the shape uncertainty, briefly introducing the inverse reliability method. Two optimization problems, a minimum mean compliance problem and an optimum design problem for a compliant mechanism, are then formulated using the proposed shape uncertainty modeling. The design sensitivity analysis for the reliability analysis and optimization procedure, performed using the adjoint variable method, is then explained. A two-level optimization algorithm is constructed next, in which the inner iteration is used for reliability analysis and the outer is used for updating design variables. Finally, three numerical examples are provided to demonstrate the validity and the utility of the proposed method.
Tài liệu tham khảo
Alban A, Darji HA, Imamura A, Nakayama MK (2017) Efficient Monte Carlo methods for estimating failure probabilities. Reliab Eng Syst Saf 165:376–394
Alexandersen J, Aage N, Andreasen CS, Sigmund O (2014) Topology optimisation for natural convection problems. Int J Numer Methods Fluids 76(10):699–721
Allaire G (2002) Shape optimization by the homogenization method. Springer
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9-10):635–654
Betz W, Papaioannou I, Straub D (2014) Numerical methods for the discretization of random fields by means of the Karhunen–Loève expansion. Comput Methods Appl Mech Eng 271:109–129
Bichon BJ, McFarland JM, Mahadevan S (2011) Efficient surrogate models for reliability analysis of systems with multiple failure modes. Reliab Eng Syst Saf 96(10):1386–1395
Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158
Chen S, Chen W (2011) A new level-set based approach to shape and topology optimization under geometric uncertainty. Struct Multidiscip Optim 44(1):1–18
Chen S, Chen W, Lee S (2010) Level set based robust shape and topology optimization under random field uncertainties. Struct Multidiscip Optim 41(4):507–524
Chen X, Hasselman TK, Neill DJ et al (1997) Reliability based structural design optimization for practical applications. In: Proceedings of the 38th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, pp 2724–2732
Ezzati G, Mammadov M, Kulkarni S (2015) A new reliability analysis method based on the conjugate gradient direction. Struct Multidiscip Optim 51(1):89–98
Furuta K, Izui K, Yaji K, Yamada T, Nishiwaki S (2017) Level set-based topology optimization for the design of a peltier effect thermoelectric actuator. Struct Multidiscip Optim 55(5):1671–1683
Guest JK, Igusa T (2008) Structural optimization under uncertain loads and nodal locations. Comput Methods Appl Mech Eng 198(1):116–124
Guest JK, Prévost J H, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254
Guo X, Zhang W, Zhang L (2013) Robust structural topology optimization considering boundary uncertainties. Comput Methods Appl Mech Eng 253:356–368
Iga A, Nishiwaki S, Izui K, Yoshimura M (2009) Topology optimization for thermal conductors with heat convection and conduction including design-dependent effects. Int J Heat Mass Transfer 52:2721–2732
Ito M, Kim NH, Kogiso N (2018) Conservative reliability index for epistemic uncertainty in reliability-based design optimization. Struct Multidiscip Optim 57(5):1919–1935
Jansen M, Lombaert G, Diehl M, Lazarov BS, Sigmund O, Schevenels M (2013) Robust topology optimization accounting for misplacement of material. Struct Multidiscip Optim 47(3):317–333
Jansen M, Lombaert G, Schevenels M (2015) Robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis. Comput Methods Appl Mech Eng 285:452–467
Kang Z, Liu P (2018) Reliability-based topology optimization against geometric imperfections with random threshold model. International Journal for Numerical Methods in Engineering. https://doi.org/10.1002/nme.5797
Kawamoto A, Matsumori T, Yamasaki S, Nomura T, Kondoh T, Nishiwaki S (2011) Heaviside projection based topology optimization by a PDE-filtered scalar function. Struct Multidiscip Optim 44(1):19–24
Keshtegar B, Lee I (2016) Relaxed performance measure approach for reliability-based design optimization. Struct Multidiscip Optim 54(6):1439–1454
Kharmanda G, Olhoff N, Mohamed A, Lemaire M (2004) Reliability-based topology optimization. Struct Multidiscip Optim 26(5):295–307
Kiureghian AD, Stefano MD (1991) Efficient algorithm for second-order reliability analysis. J Eng Mech 117(12):2904–2923
Kogiso N, Ahn W, Nishiwaki S, Izui K, Yoshimura M (2008) Robust topology optimization for compliant mechanisms considering uncertainty of applied loads. J Adv Mech Des Syst Manuf 2(1):96–107
Kogiso N, Yang YS, Kim BJ, Lee JO (2012) Modified single-loop-single-vector method for efficient reliability-based design optimization. J Adv Mech Des Syst Manuf 6(7):1206–1221
Lang C, Sharma A, Doostan A, Maute K (2015) Heaviside enriched extended stochastic FEM for problems with uncertain material interfaces. Comput Mech 56(5):753–767
Lazarov BS, Sigmund O (2009) Sensitivity filters in topology optimisation as a solution to Helmholtz type differential equation. In: 8th world congress on structural and multidisciplinary optimization
Lazarov BS, Schevenels M, Sigmund O (2012) Topology optimization with geometric uncertainties by perturbation techniques. Int J Numer Methods Eng 90(11):1321–1336
Lee JO, Yang YS, Ruy WS (2002) A comparative study on reliability-index and target-performance-based probabilistic structural design optimization. Comput Struct 80(3):257–269
Ma ZD, Kikuchi N, Cheng HC (1995) Topological design for vibrating structures. Comput Methods Appl Mech Eng 121(1):259–280
Matsumori T, Kondoh T, Kawamoto A, Nomura T (2013) Topology optimization for fluid–thermal interaction problems under constant input power. Struct Multidiscip Optim 47(4):571–581
Meng Z, Li G, Wang BP, Hao P (2015) A hybrid chaos control approach of the performance measure functions for reliability-based design optimization. Comput Struct 146:32–43
Nishiwaki S, Frecker MI, Min S, Kikuchi N (1998) Topology optimization of compliant mechanisms using the homogenization method. Int J Numer Methods Eng 42:535–559
Nouy A, Clement A (2010) eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces. Int J Numer Methods Eng 83(10):1312–1344
Rackwitz R, Flessler B (1978) Structural reliability under combined random load sequences. Comput Struct 9(5):489–494
Rashki M, Miri M, Moghaddam MA (2014) A simulation-based method for reliability based design optimization problems with highly nonlinear constraints. Autom Constr 47:24–36
Schevenels M, Lazarov BS, Sigmund O (2011) Robust topology optimization accounting for spatially varying manufacturing errors. Comput Methods Appl Mech Eng 200(49):3613–3627
Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4-5):401–424
Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mech Sinica 25(2):227–239
Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization 16(1):68–75
Svanberg K (1987) The method of moving asymptotes–a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373
Tu J, Choi KK, Park YH (1999) A new study on reliability-based design optimization. Trans Am Soc Mech Eng J Mech Des 121(4):557–564
Wang MY, Wang X (2004) Color level sets: a multi-phase method for structural topology optimization with multiple materials. Comput Methods Appl Mech Eng 193(6):469–496
Xiao NC, Zuo MJ, Zhou C (2018) A new adaptive sequential sampling method to construct surrogate models for efficient reliability analysis. Reliab Eng Syst Saf 169:330–338
Yamada T, Izui K, Nishiwaki S (2011) A level set-based topology optimization method for maximizing thermal diffusivity in problems including design-dependent effects. J Mech Des 133(3):031,011
Yin L, Ananthasuresh G (2001) Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme. Struct Multidiscip Optim 23(1):49–62
Youn BD, Choi KK (2004) An investigation of nonlinearity of reliability-based design optimization approaches. J Mech Des 126:403–411
Youn BD, Choi KK, Park YH (2003) Hybrid analysis method for reliability-based design optimization. J Mech Des 125(2):221–232
Zhang W, Kang Z (2017) Robust shape and topology optimization considering geometric uncertainties with stochastic level set perturbation. Int J Numer Methods Eng 110(1):31–56
Zuo W, Saitou K (2017) Multi-material topology optimization using ordered SIMP interpolation. Struct Multidiscip Optim 55(2):477–491