Topology optimization of continuum structures with stress constraints and uncertainties in loading. (English) Zbl 07867260
Summary: Topology optimization using stress constraints and considering uncertainties is a serious challenge, since a reliability problem has to be solved for each stress constraint, for each element in the mesh. In this paper, an alternative way of solving this problem is used, where uncertainty quantification is performed through the first-order perturbation approach, with proper validation by Monte Carlo simulation. Uncertainties are considered in the loading magnitude and direction. The minimum volume problem subjected to local stress constraints is formulated as a robust problem, where the stress constraints are written as a weighted average between their expected value and standard deviation. The augmented Lagrangian method is used for handling the large set of local stress constraints, whereas a gradient-based algorithm is used for handling the bounding constraints. It is shown that even in the presence of small uncertainties in loading direction, different topologies are obtained when compared to a deterministic approach. The effect of correlation between uncertainties in loading magnitude and direction on optimal topologies is also studied, where the main observed result is loss of symmetry in optimal topologies.
{Copyright © 2017 John Wiley & Sons, Ltd.}
{Copyright © 2017 John Wiley & Sons, Ltd.}
MSC:
| 74Pxx | Optimization problems in solid mechanics |
| 74Sxx | Numerical and other methods in solid mechanics |
| 90Cxx | Mathematical programming |
Software:
ALGENCANReferences:
| [1] | GuestJK, IgusaT. Structural optimization under uncertain loads and nodal locations. Comput Meth Appl Mech Eng. 2008;198(1):116‐124. https://dx.doi.org/10.1016/j.cma.2008.04.009; Computational Methods in Optimization Considering Uncertainties. · Zbl 1194.74238 · doi:10.1016/j.cma.2008.04.009 |
| [2] | SilvaM, TortorelliDA, NoratoJA, HaC, BaeHR. Component and system reliability‐based topology optimization using a single‐loop method. Struct Multidiscipl Optim. 2010;41(1):87‐106. https://doi.org/10.1007/s00158-009-0401-5 · Zbl 1274.74392 · doi:10.1007/s00158-009-0401-5 |
| [3] | DunningPD, KimA, MullineuxG. Introducing loading uncertainty in topology optimization. AIAA J. 2011;49(4):760‐768. https://doi.org/10.2514/1.J050670 · doi:10.2514/1.J050670 |
| [4] | SchevenelsM, LazarovBS, SigmundO. Robust topology optimization accounting for spatially varying manufacturing errors. Comput Meth Appl Mech Eng. 2011;200(4952):3613‐3627. https://dx.doi.org/10.1016/j.cma.2011.08.006 · Zbl 1239.74080 · doi:10.1016/j.cma.2011.08.006 |
| [5] | LazarovBS, SchevenelsM, SigmundO. Topology optimization with geometric uncertainties by perturbation techniques. Int J Numer Methods Eng. 2012;90(11):1321‐1336. https://doi.org/10.1002/nme.3361 · Zbl 1242.74075 · doi:10.1002/nme.3361 |
| [6] | TootkaboniM, AsadpoureA, GuestJK. Topology optimization of continuum structures under uncertainty – a polynomial chaos approach. Comput Meth Appl Mech Eng. 2012;201204:263‐275. https://dx.doi.org/10.1016/j.cma.2011.09.009 · Zbl 1239.74081 · doi:10.1016/j.cma.2011.09.009 |
| [7] | DunningPD, KimA. Robust topology optimization: minimization of expected and variance of compliance. AIAA J. 2013;51(11):2656‐2664. https://doi.org/10.2514/1.J052183 · doi:10.2514/1.J052183 |
| [8] | LuoY, ZhouM, WangMY, DengZ. Reliability based topology optimization for continuum structures with local failure constraints. Comput Struct. 2014;143:73‐84. https://dx.doi.org/10.1016/j.compstruc.2014.07.009 · doi:10.1016/j.compstruc.2014.07.009 |
| [9] | ZhaoJ, WangC. Robust topology optimization under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices. Comput Meth Appl Mech Eng. 2014;273:204‐218. https://doi.org/10.1016/j.cma.2014.01.018 · Zbl 1296.74085 · doi:10.1016/j.cma.2014.01.018 |
| [10] | GuoX, ZhaoX, ZhangW, YanJ, SunG. Multi‐scale robust design and optimization considering load uncertainties. Comput Meth Appl Mech Eng. 2015;283:994‐1009. https://doi.org/10.1016/j.cma.2014.10.014 · Zbl 1423.74747 · doi:10.1016/j.cma.2014.10.014 |
| [11] | HolmbergE, ThoreCJ, KlarbringA. Worst‐case topology optimization of self‐weight loaded structures using semi‐definite programming. Struct Multidiscipl Optim. 2015;52(5):915‐928. https://doi.org/10.1007/s00158-015-1285-1 · doi:10.1007/s00158-015-1285-1 |
| [12] | HolmbergE, ThoreCJ, KlarbringA. Game theory approach to robust topology optimization with uncertain loading. Struct Multidiscipl Optim. 2016:1‐15. https://doi.org/10.1007/s00158-016-1548-5 · doi:10.1007/s00158-016-1548-5 |
| [13] | JalalpourM, TootkaboniM. An efficient approach to reliability‐based topology optimization for continua under material uncertainty. Struct Multidiscipl Optim. 2016;53(4):759‐772. https://doi.org/10.1007/s00158-015-1360-7 · doi:10.1007/s00158-015-1360-7 |
| [14] | da SilvaGA, CardosoEL. Topology optimization of continuum structures subjected to uncertainties in material properties. Int J Numer Methods Eng. 2016;106(3):192‐212. nme.5126. · Zbl 1352.74235 |
| [15] | da SilvaGA, CardosoEL. Stress‐based topology optimization of continuum structures under uncertainties. Comput Meth Appl Mech Eng. 2017;313:647‐672. https://dx.doi.org/10.1016/j.cma.2016.09.049 · Zbl 1439.74264 · doi:10.1016/j.cma.2016.09.049 |
| [16] | KeshavarzzadehV, FernandezF, TortorelliDA. Topology optimization under uncertainty via non‐intrusive polynomial chaos expansion. Comput Meth Appl Mech Eng. 2017;318:120‐147. https://doi.org/10.1016/j.cma.2017.01.019 · Zbl 1439.74281 · doi:10.1016/j.cma.2017.01.019 |
| [17] | ThoreCJ, HolmbergE, KlarbringA. A general framework for robust topology optimization under load‐uncertainty including stress constraints. Comput Meth Appl Mech Eng. 2017;319:1‐18. https://doi.org/10.1016/j.cma.2017.02.015 · Zbl 1439.74296 · doi:10.1016/j.cma.2017.02.015 |
| [18] | BeckAT, GomesWJS, LopezRH, MiguelLFF. A comparison between robust and risk‐based optimization under uncertainty. Struct Multidiscipl Optim. 2015;52(3):479‐492. https://doi.org/10.1007/s00158-015-1253-9 · doi:10.1007/s00158-015-1253-9 |
| [19] | AgarwalH. Reliability based design optimization: formulations and methodologies. PhD Thesis. Notre Dame, Indiana: Graduate School of the University of Notre Dame; 2004. |
| [20] | LopezRH, BeckAT. Reliability‐based design optimization strategies based on form: a review. J Braz Soc Mech Sci & Eng. 201212;34:506‐514. https://doi.org/10.1590/S1678-58782012000400012 · doi:10.1590/S1678-58782012000400012 |
| [21] | BeyerHG, SendhoffB. Robust optimization – a comprehensive survey. Comput Meth Appl Mech Eng. 2007;196(3334):3190‐3218. https://dx.doi.org/10.1016/j.cma.2007.03.003 · Zbl 1173.74376 · doi:10.1016/j.cma.2007.03.003 |
| [22] | ParkGJ, LeeTH, LeeKH, HwangKH. Robust design: an overview. AIAA. 2006;44(1):181‐191. https://doi.org/10.2514/1.13639 · doi:10.2514/1.13639 |
| [23] | DuysinxP, BendseMP. Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng. 1998;43(8):1453‐1478. https://doi.org/10.1002/(SICI)1097-0207(19981230)43:81453::AID-NME4803.0.CO;2-2 · Zbl 0924.73158 · doi:10.1002/(SICI)1097-0207(19981230)43:81453::AID-NME4803.0.CO;2-2 |
| [24] | FancelloEA, PereiraJT. Structural topology optimization considering material failure constraints and multiple load conditions. Lat Am J Solids Stru. 2003;1(1):3‐24. |
| [25] | PereiraJT, FancelloEA, BarcellosCS. Topology optimization of continuum structures with material failure constraints. Struct Multidiscipl Optim. 2004;26(1):50‐66. https://doi.org/10.1007/s00158-003-0301-z · Zbl 1243.74157 · doi:10.1007/s00158-003-0301-z |
| [26] | FancelloEA. Topology optimization for minimum mass design considering local failure constraints and contact boundary conditions. Struct Multidiscipl Optim. 2006;32(3):229‐240. https://doi.org/10.1007/s00158-006-0019-9 · Zbl 1245.74071 · doi:10.1007/s00158-006-0019-9 |
| [27] | BruggiM. On an alternative approach to stress constraints relaxation in topology optimization. Struct Multidiscipl Optim. 2008;36(2):125‐141. https://doi.org/10.1007/s00158-007-0203-6 · Zbl 1273.74397 · doi:10.1007/s00158-007-0203-6 |
| [28] | BruggiM, VeniniP. A mixed fem approach to stress‐constrained topology optimization. Int J Numer Methods Eng. 2008;73(12):1693‐1714. https://doi.org/10.1002/nme.2138 · Zbl 1159.74397 · doi:10.1002/nme.2138 |
| [29] | LeC, NoratoJ, BrunsT, HaC, TortorelliD. Stress‐based topology optimization for continua. Struct Multidiscipl Optim. 2010;41(4):605‐620. https://doi.org/10.1007/s00158-009-0440-y · doi:10.1007/s00158-009-0440-y |
| [30] | ParísJ, NavarrinaF, ColominasI, CasteleiroM. Block aggregation of stress constraints in topology optimization of structures. Adv Eng Software. 2010;41(3):433‐441. https://dx.doi.org/10.1016/j.advengsoft.2009.03.006; Advances in optimum engineering design. · Zbl 1303.74033 · doi:10.1016/j.advengsoft.2009.03.006 |
| [31] | BruggiM, DuysinxP. Topology optimization for minimum weight with compliance and stress constraints. Struct Multidiscipl Optim. 2012;46(3):369‐384. https://doi.org/10.1007/s00158-012-0759-7 · Zbl 1274.74219 · doi:10.1007/s00158-012-0759-7 |
| [32] | HolmbergE, TorstenfeltB, KlarbringA. Stress constrained topology optimization. Struct Multidiscipl Optim. 2013;48(1):33‐47. https://doi.org/10.1007/s00158-012-0880-7 · Zbl 1274.74341 · doi:10.1007/s00158-012-0880-7 |
| [33] | EmmendoerferH, FancelloEA. A level set approach for topology optimization with local stress constraints. Int J Numer Methods Eng. 2014;99(2):129‐156. https://doi.org/10.1002/nme.4676 · Zbl 1352.74238 · doi:10.1002/nme.4676 |
| [34] | SvärdH. Interior value extrapolation: a new method for stress evaluation during topology optimization. Struct Multidiscipl Optim. 2015;51(3):613‐629. https://doi.org/10.1007/s00158-014-1171-2 · doi:10.1007/s00158-014-1171-2 |
| [35] | EmmendoerferH, FancelloEA. Topology optimization with local stress constraint based on level set evolution via reactiondiffusion. Comput Meth Appl Mech Eng. 2016;305:62‐88. https://dx.doi.org/10.1016/j.cma.2016.02.024 · Zbl 1425.74373 · doi:10.1016/j.cma.2016.02.024 |
| [36] | LeeK, AhnK, YooJ. A novel p‐norm correction method for lightweight topology optimization under maximum stress constraints. Comput Struct. 2016;171:18‐30. http://doi.org/10.1016/j.compstruc.2016.04.005 · doi:10.1016/j.compstruc.2016.04.005 |
| [37] | BendsøeMP, SigmundO. Topology Optimization: Theory, Methods and Applications. Berlin: Springer; 2003. https://doi.org/10.1007/978-3-662-05086-6 · doi:10.1007/978-3-662-05086-6 |
| [38] | SvanbergK. The method of moving asymptotesa new method for structural optimization. Int J Numer Methods Eng. 1987;24(2):359‐373. https://doi.org/10.1002/nme.1620240207 · Zbl 0602.73091 · doi:10.1002/nme.1620240207 |
| [39] | ChengGD, GuoX. ε‐relaxed approach in structural topology optimization. Struct Optim. 1997:258‐266. https://doi.org/10.1007/BF01197454 · doi:10.1007/BF01197454 |
| [40] | BirginE, MartínezJ. Practical Augmented Lagrangian Methods for Constrained Optimization. Philadelphia, PA: Society for Industrial and Applied Mathematics; 2014. https://doi.org/10.1137/1.9781611973365 · Zbl 1339.90312 · doi:10.1137/1.9781611973365 |
| [41] | KleiberM, HienTD. The Stochastic Finite Element Method: Basic Perturbation Technique and Computer Implementation. Chichester: John Wiley & sons; 1992. · Zbl 0902.73004 |
| [42] | GuestJK, PrvostJH, BelytschkoT. Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng. 2004;61(2):238‐254. https://doi.org/10.1002/nme.1064 · Zbl 1079.74599 · doi:10.1002/nme.1064 |
| [43] | BatheKJ. Finite Element Procedures. Upper Sadle River, New Jersey: Prentice Hall; 1996. |
| [44] | BendsøeMP, SigmundO. Material interpolation schemes in topology optimization. Appl Mech. 1999;69:635‐654. · Zbl 0957.74037 |
| [45] | RossS. A First Course in Probability. Eight edition.Upper Saddle River, New Jersey: Prentice Hall; 2010. · Zbl 1307.60002 |
| [46] | SudretB, Der KiureghianA. Stochastic finite element methods and reliability: a state‐of‐the‐art report. Technical Report, Berkeley, University of California; 2000. |
| [47] | SudretB. Uncertainty propagation and sensitivity analysis in mechanical models: Contributions to structural reliability and stochastic spectral methods. Technical Report, Clermont‐Ferrand, France, Université BLAISE PASCAL ‐ Clermont II; 2007. |
| [48] | StefanouG. The stochastic finite element method: past, present and future. Comput Methods Appl Mech Engrg. 2009;198:1031‐1051. https://doi.org/10.1016/j.cma.2008.11.007 · Zbl 1229.74140 · doi:10.1016/j.cma.2008.11.007 |
| [49] | GhanemRG, SpanosPD. Stochastic Finite Elements: A Spectral Approach. New York: Dover Publications; 2003. Revised edition. |
| [50] | ElishakoffI. Probabilistic Theory of Structures. Second. London: Dover; 1999. |
| [51] | BarlowJ. Optimal stress locations in finite element models. Int J Numer Methods Eng. 1976;10(2):243‐251. https://doi.org/10.1002/nme.1620100202 · Zbl 0322.73049 · doi:10.1002/nme.1620100202 |
| [52] | SigmundO. Morphology‐based black and white filters for topology optimization. Struct Multidiscipl Optim. 2007;33(4-5):401‐424. https://doi.org/10.1007/s00158-006-0087-x · doi:10.1007/s00158-006-0087-x |
| [53] | GuestJK, AsadpoureA, HaSH. Eliminating beta‐continuation from heaviside projection and density filter algorithms. Struct Multidiscipl Optim. 2011;44(4):443‐453. https://doi.org/10.1007/s00158-011-0676-1 · Zbl 1274.74337 · doi:10.1007/s00158-011-0676-1 |
| [54] | BeckAT, GomesWJS. A comparison of deterministic, reliability‐based and risk‐based structural optimization under uncertainty. Probab Eng Mech. 2012;28:18‐29. https://doi.org/10.1016/j.probengmech.2011.08.007 · doi:10.1016/j.probengmech.2011.08.007 |
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