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Structural and Multidisciplinary Optimization
Erschienen in: Structural and Multidisciplinary Optimization 4/2020

20.08.2020 | Research Paper

Topology optimization with local stress constraints: a stress aggregation-free approach

verfasst von: Fernando V. Senhora, Oliver Giraldo-Londoño, Ivan F. M. Menezes, Glaucio H. Paulino

Erschienen in: Structural and Multidisciplinary Optimization | Ausgabe 4/2020

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Abstract

This paper presents a consistent topology optimization formulation for mass minimization with local stress constraints by means of the augmented Lagrangian method. To solve problems with a large number of constraints in an effective way, we modify both the penalty and objective function terms of the augmented Lagrangian function. The modification of the penalty term leads to consistent solutions under mesh refinement and that of the objective function term drives the mass minimization towards black and white solutions. In addition, we introduce a piecewise vanishing constraint, which leads to results that outperform those obtained using relaxed stress constraints. Although maintaining the local nature of stress requires a large number of stress constraints, the formulation presented here requires only one adjoint vector, which results in an efficient sensitivity evaluation. Several 2D and 3D topology optimization problems, each with a large number of local stress constraints, are provided.
Vorheriger Artikel A review on feature-mapping methods for structural optimization
Nächster Artikel Adaptive level set topology optimization using hierarchical B-splines

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Fußnoten
1
A variation of the vanishing constraints is used in the present study to solve stress-constrained topology optimization problems.
 
2
According to Cheng and Guo (1997), the restriction on the variable lower bound to be ε2 is not necessary. They demonstrated that, in order to guarantee convergence, the lower bound on the design variables has to be a higher order term smaller than ε as \(\varepsilon \rightarrow 0.\)
 
3
However, the stress-constrained problem is not well-behaved because, given the degenerate nature of the constraints, the Lagrange multiplier set associated with a stationary point is unbounded. For optimization problems of this type, Izmailov et al. (2012) and Andreani et al. (2012) showed that the AL method exhibits global convergence properties, which suggests that this method is a viable alternative to solve stress-constrained topology optimization problems.
 
4
Notice the modular structure developed for the stress-constrained topology optimization problem. Due to this feature, different constitutive behaviors can be incorporated in the present computational mechanics framework.
 
5
As an example, suppose that for a given AL step we have g = − 0.5, λ = 1, and μ = 1. For this combination of constraint values and AL parameters, we have that, \(h=\max \limits (g,-~\lambda /\mu )=\max \limits (-~0.5, -~1)=-~0.5\), which yields \(P=\lambda h+\frac {1}{2}\mu h^{2}=-~3/8<0\), where P is the penalization term of the AL function.
 
6
The piecewise constraint given by (17) is C1(z) because, for \(\sigma _{j}^{\text {v}}/\sigma _{\lim } >1\), \(g_{j}(\textbf {z})={\rho ^{p}_{j}}(\sigma _{j}^{\text {v}}/\sigma _{\lim }-1)^{2}\), which is the finite composition of \(C^{\infty }\) functions in this domain, and for \(\sigma _{j}^{\text {v}}/\sigma _{\lim } <1\), gj(z) = 0, which is also \(C^{\infty }\). Moreover, when \(\sigma _{j}^{\text {v}}/\sigma _{\lim }=1\), both the value of \({\rho ^{p}_{j}}(\sigma _{j}^{\text {v}}/\sigma _{\lim }-1)^{2}\) and its first derivative with respect to z are equal to zero, which is the same value of gj(z) and its derivative with respect to z, when \(\sigma _{j}^{\text {v}}/\sigma _{\lim }<1\).
 
7
Because the interphase penalization F(z) in (19) is only used when we apply continuation on the filter radius, we have decided not to include it in the current derivation.
 
8
Stagnation is reached when the average change in the design variables between two consecutive iterations is smaller than a given tolerance, i.e., when Change < tol (cf. Algorithm 2) and the constraints are yet not satisfied.
 
9
When two consecutive filter matrices have similar topology, it indicates that the material distribution between two consecutive iterations has not changed significantly. Alternatively, we could use a criterion based on \(\left \|\textbf {P}_{i+1}-\textbf {P}_{i} \right \|\) to stop the filter reduction, but this means storing both Pi and Pi+ 1, which requires a substantial amount of RAM memory.
 
10
The function defined in Eq. (36) is not unique, and other functions such as ρ(1 − ρ) can be used for the same purpose. We choose the function in Eq. (36) because its slight asymmetry with respect to ρ = 0.5 tends to favor designs with lower weight.
 
11
In addition to all parameters shown in Table 1, the MMA parameters (Svanberg 1987) used in all examples for the minimization of the AL sub-problems are asyinit = 0.2, asyinc = 1.2, asydec = 0.7, move = 0.1.
 
12
The stress shown in this example, as well as those shown in subsequent examples, is the stress measure \(\widetilde {\sigma }_{e}^{\text {v}}\) (18) normalized with respect to the stress limit, \(\sigma _{\lim }\).
 
13
The isosurfaces as well as the STL files are obtained using the MATLAB-based graphical tool TOPslicer (Zegard and Paulino 2016).
 
14
The computational costs reported in this section are based on the topology optimization results obtained in a computer with an i7-4930k CPU at 3.40 GHz and 64 GB of RAM and a NVIDIA GEFORCE GTX 1080 Ti GPU running on a 64-bit operating system.
 
15
On the other hand, an inconsistent formulation refers to the case in which stresses are not treated locally, i.e., they are either aggregated or clustered.
 
16
The interested reader is referred to Bertsekas (1996, p. 161) for complete proof of the differentiability of the AL function used with (42).
 
Literatur
Achtziger W, Kanzow C (2008) Mathematical programs with vanishing constraints: optimality conditions and constraint qualifications. Math Program 114(1):69–99MathSciNetMATH
Achtziger W, Hoheisel T, Kanzow C (2013) A smoothing-regularization approach to mathematical programs with vanishing constraints. Comput Optim Appl 55(3):733–767MathSciNetMATH
Amstutz S, Novotny AA (2010) Topological optimization of structures subject to von Mises stress constraints. Struct Multidiscip Optim 41(3):407–420MathSciNetMATH
Andreani R, Haeser G, Schuverdt M, Silva P (2012) A relaxed constant positive linear dependence constraint qualification and applications. Math Program 135(1-2):255–273MathSciNetMATH
Bell E. T. (1986) Men of Mathematics. Simon and Schuster
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224MathSciNetMATH
Bendsøe MP (1995) Optimization of structural topology, shape, and material. Springer, Berlin
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9-10):635–654MATH
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, Berlin
Bertsekas DP (1996) Constrained optimization and Lagrange multiplier methods (ptimization and Neural Computation Series), 1 ed. Athena Scientific
Bertsekas DP (1999) Nonlinear programming, 2nd ed. Athena Scientific
Borrvall T, Petersson J (2001) Topology optimization using regularized intermediate density control. Comput Methods Appl Mech Eng 190(37-38):4911–4928MathSciNetMATH
Bruggi M (2008) On an alternative approach to stress constraints relaxation in topology optimization. Struct Multidiscip Optim 36(2):125–141MathSciNetMATH
Bruggi M, Duysinx P (2012) Topology optimization for minimum weight with compliance and stress constraints. Struct Multidiscip Optim 46(3):369–384MathSciNetMATH
Cauchy AL (1827) De la pression ou tension dans un corps solide. Exercices Math 2:42–56
Cheng G, Jiang Z (1992) Study on topology optimization with stress constraints. Eng Optim 20(2):129–148
Cheng G (1995) Some aspects on truss topology optimization. Struct Optim 10(3-4):173–179
Cheng G, Guo X (1997) ε-relaxed approach in structural topology optimization. Struct Optim 13(4):258–266
Christensen P, Klarbring A (2008) An introduction to structural optimization. Solid Mechanics and Its Applications. Springer, Netherlands
da Silva GA, Beck AT, Sigmund O (2019) Stress-constrained topology optimization considering uniform manufacturing uncertainties. Comput Methods Appl Mech Eng 344:512–537MathSciNetMATH
Duysinx P, Bendsøe MP (1998a) Topology optimization of continuum structures with local stress constraints. Int J Numer Methods Eng 43(8):1453–1478MathSciNetMATH
Duysinx P, Sigmund O (1998b) New developments in handling stress constraints in optimal material distribution. In: Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, vol 1, pp 1501–1509
Emmendoerfer Jr, H, Fancello EA (2014) A level set approach for topology optimization with local stress constraints. Int J Numer Methods Eng 99(2):129–156MathSciNetMATH
Emmendoerfer Jr, H, Fancello EA (2016) Topology optimization with local stress constraint based on level set evolution via reaction-diffusion. Comput Methods Appl Mech Eng 305:62–88MathSciNetMATH
Ermoliev YM, Kryazhimskii AV, Ruszczyński A (1997) Constraint aggregation principle in convex optimization. Math Program 76(3):353–372MathSciNetMATH
Gibson I, Rosen D, Stucker B (2015) Additive manufacturing technologies: 3D printing, rapid prototyping, and direct digital manufacturing, vol 2. Springer, Berlin
Guest JK, Prévost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254MathSciNetMATH
Guo X, Zhang WS, Wang MY, Wei P (2011) Stress-related topology optimization via level set approach. Comput Methods Appl Mech Eng 200(47-48):3439–3452MathSciNetMATH
Gurtin ME (1981) An introduction to continuum mechanics, vol 158. Academic Press
Hoheisel T, Kanzow C (2008) Stationary conditions for mathematical programs with vanishing constraints using weak constraint qualifications. J Math Anal Appl 337(1):292–310MathSciNetMATH
Holmberg E, Torstenfelt B, Klarbring A (2013) Stress constrained topology optimization. Struct Multidiscip Optim 48(1):33–47MathSciNetMATH
Izmailov AF, Solodov MV, Uskov EI (2012) Global convergence of augmented lagrangian methods applied to optimization problems with degenerate constraints, including problems with complementarity constraints. SIAM J Optim 22(4):1579–1606MathSciNetMATH
James K, Lee E, Martins J (2012) Stress-based topology optimization using an isoparametric level set method. Finite Elem Anal Des 58:20–30
Kirsch U (1989) Optimal topologies in truss structures. Comput Methods Appl Mech Eng 72 (1):15–28MATH
Kirsch U, Taye S (1986) On optimal topology of grillage structures. Eng Comput 1:229–243
Kirsch U (1990) On singular topologies in optimum structural design. Struct Optim 2(3):133–142
Kiyono C, Vatanabe S, Silva E, Reddy J (2016) A new multi-p-norm formulation approach for stress-based topology optimization design. Compos Struct 156:10 –19. 70th Anniversary of Professor J. N. Reddy
Kreisselmeier G, Steinhauser R (1979) Systematic control design by optimizing a vector performance index. IFAC Proc Vol 12(7):113–117. IFAC Symposium on computer Aided Design of Control Systems, Zurich, SwitzerlandMATH
Le C, Norato J, Bruns T, Ha C, Tortorelli D (2010) Stress-based topology optimization for continua. Struct Multidiscip Optim 41(4):605–620
Lee E, James KA, Martins JRRA (2012) Stress-constrained topology optimization with design-dependent loading. Struct Multidiscip Optim 46(5):647–661MathSciNetMATH
Lian H, Christiansen AN, Tortorelly DA, Sigmund O (2017) Combined shape and topology optimization for minimization of maximal von Mises stress. Struct Multidiscip Optim 55(5):1541–1557MathSciNet
Love AEH (1892) A treatise on the mathematical theory of elasticity, Vol 1. Cambridge University Press, Cambridge
Malvern LE (1969) Introduction to the mechanics of a continuous Mmedium. Prentice-Hall
Navarrina F, Muiños I, Colominas I, Casteleiro M (2005) Topology optimization of structures: a minimum weight approach with stress constraints. Adv Eng Softw 36(9):599–606MATH
Nocedal J, Wright SJ (2006) Numerical optimization, 2nd ed. Springer, Berlin
Paris J, Navarrina F, Colominas I, Casteleiro M (2009) Topology optimization of continuum structures with local and global stress constraints. Struct Multidiscip Optim 39(4):419–437MathSciNetMATH
Paris J, Navarrina F, Colominas I, Casteleiro M (2010) Block aggregation of stress constraints in topology optimization of structures. Adv Eng Softw 41(3):433–441MATH
Park YK (1995) Extensions of optimal layout design using the homogenization method. Ph.D thesis, University of Michigan, Ann Arbor
Pereira JT, Fancello EA, Barcellos CS (2004) Topology optimization of continuum structures with material failure constraints. Struct Multidiscip Optim 26(1-2):50–66MathSciNetMATH
Petersson J (2001) On continuity of the design-to-state mappings for trusses with variable topology. Int J Eng Sci 39(10):1119– 1141MathSciNetMATH
Rozvany GIN, Birker T (1994) On singular topologies in exact layout optimization. Struct Optim 8(4):228–235
Rozvany GIN (2001) On design-dependent constraints and singular topologies. Struct Multidiscip Optim 21(2):164–172MathSciNet
Sharma A, Maute K (2018) Stress-based topology optimization using spatial gradient stabilized XFEM. Struct Multidiscip Optim 57(1):17–38MathSciNet
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):227–239MATH
Stolpe M, Svanberg K (2001) On the trajectories of the epsilon-relaxation approach for stress-constrained truss topology optimization. Struct Multidiscip Optim 21(2):140–151
Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373MathSciNetMATH
Sved G, Ginos Z (1968) Structural optimization under multiple loading. Int J Mech Sci 10(10):803–805
Talischi C, Paulino GH, Pereira A, Menezes IFM (2012a) PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidiscip Optim 45(3):309–328MathSciNetMATH
Talischi C, Paulino GH, Pereira A, Menezes IFM (2012b) PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Struct Multidiscip Optim 45(3):329–357MathSciNetMATH
Talischi C, Paulino GH (2013) An operator splitting algorithm for Tikhonov-regularized topology optimization. Comput Methods Appl Mech Eng 253:599–608MathSciNetMATH
Timoshenko S, Goodier JN (1951) Theory of elasticity, 2 ed. McGraw-Hill
Verbart A, Langelaar M, van Keulen F (2016) Damage approach: a new method for topology optimization with local stress constraints. Struct Multidiscip Optim 53(5):1081–1098MathSciNet
Verbart A, Langelaar M, Keulen F. v. (2017) A unified aggregation and relaxation approach for stress-constrained topology optimization. Struct Multidiscip Optim 55(2):663–679MathSciNet
Wang MY, Wang S (2005) Bilateral filtering for structural topology optimization. Int J Numer Methods Eng 63(13):1911–1938MathSciNetMATH
Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784MATH
Wong KV, Hernandez A (2012) A review of additive manufacturing. ISRN Mechanical Engineering 2012, pp 1–10
Xia Q, Shi T, Liu S, Wang MY (2012) A level set solution to the stress-based structural shape and topology optimization. Comput Struct 90:55–64
Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear density filter based on Heaviside functions. Struct Multidiscip Optim 41(4):495–505MathSciNetMATH
Yang RJ, Chen CJ (1996) Stress-based topology optimization. Struct Optim 12(2):98–105
Zegard T, Paulino GH (2016) Bridging topology optimization and additive manufacturing. Struct Multidiscip Optim 53(1):175–192
Metadaten
Titel
Topology optimization with local stress constraints: a stress aggregation-free approach
verfasst von
Fernando V. Senhora
Oliver Giraldo-Londoño
Ivan F. M. Menezes
Glaucio H. Paulino
Publikationsdatum
20.08.2020
Verlag
Springer Berlin Heidelberg
Erschienen in
Structural and Multidisciplinary Optimization / Ausgabe 4/2020
Print ISSN: 1615-147X
Elektronische ISSN: 1615-1488
DOI
https://doi.org/10.1007/s00158-020-02573-9

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