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On projection methods, convergence and robust formulations in topology optimization

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Abstract

Mesh convergence and manufacturability of topology optimized designs have previously mainly been assured using density or sensitivity based filtering techniques. The drawback of these techniques has been gray transition regions between solid and void parts, but this problem has recently been alleviated using various projection methods. In this paper we show that simple projection methods do not ensure local mesh-convergence and propose a modified robust topology optimization formulation based on erosion, intermediate and dilation projections that ensures both global and local mesh-convergence.

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Notes

  1. The difference between the present robust formulation and the one introduced in Sigmund (2009) is the representation of the intermediate design and the volume bound which is applied on the dilated structure instead of the intermediate. In Sigmund (2009) the intermediate design is simply the design variable field ρ. However, in the present approach the intermediate design has gone through the same density filtering as the dilated and eroded designs and it is obtained as the projection for η = 0.5. The modifications eliminate numerical artifacts associated with the old approach and result in much more stable convergence of the optimization problem.

  2. It is interesting to note that there are many similarities between the threshold projection schemes and the level-set approach to topology optimization (Allaire et al. 2004; Wang et al. 2003; Kawamoto et al. 2010). In the presented scheme the filtered density field \(\widetilde{\rho}\) correspond to the level-set function φ and the projections correspond to zero and ±Δη level curves. Hence it is obvious that the proposed robust optimization scheme can be implemented using a level-set approach as well.

References

  • Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194(1):363–393

    Article  MathSciNet  MATH  Google Scholar 

  • Ambrosio L, Buttazzo G (1993) An optimal design problem with perimeter penalization. Calc Var Partial Differ Equ 1:55–69

    Article  MathSciNet  MATH  Google Scholar 

  • Amir O, Bendsoe MP, Sigmund O (2009) Approximate reanalysis in topology optimization. Int J Numer Methods Eng 78(12):1474–1491

    Article  MathSciNet  MATH  Google Scholar 

  • Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224

    Article  Google Scholar 

  • Bendsøe MP, Sigmund O (2004) Topology optimization—theory, methods and applications. Springer, Berlin

    MATH  Google Scholar 

  • Borel PI, Frandsen LH, Harpøth A, Kristensen M, Jensen JS, Sigmund O (2005) Topology optimised broadband photonic crystal Y-splitter. Electron Lett 41(2):69–71

    Article  Google Scholar 

  • Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158

    Article  MathSciNet  MATH  Google Scholar 

  • Bruns T, Tortorelli D (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459

    Article  MATH  Google Scholar 

  • Díaz A, Sigmund O (1995) Checkerboard patterns in layout optimization. Struct Multidisc Optim 10:40–45

    Google Scholar 

  • Guest J (2009) Topology optimization with multiple phase projection. Comput Methods Appl Mech Eng 199(1–4):123–135. doi:10.1016/j.cma.2009.09.023

    Article  MathSciNet  Google Scholar 

  • Guest J, Prevost J, 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

    Article  MathSciNet  MATH  Google Scholar 

  • Haber RB, Jog CS, Bendsøe MP (1996) A new approach to variable-topology shape design using a constraint on the perimeter. Struct Optim 11(1):1–11

    Article  Google Scholar 

  • Jensen JS, Sigmund O (2005) Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide. J Opt Soc Am B Opt Phys 22(6):1191–1198

    Article  Google Scholar 

  • Jog CS, Haber RB (1996) Stability of finite element models for distributed-parameter optimization and topology design. Comput Methods Appl Mech Eng 130(3–4):203–226

    Article  MathSciNet  MATH  Google Scholar 

  • Kawamoto A, Matsumori T, Yamasaki S, Nomura T, Kondoh T, Nishiwaki S (2010) Heaviside projection based topology optimization by a pde-filtered scalar function. Struct Multidisc Optim. doi:10.1007/s00158-010-0562-2

    MATH  Google Scholar 

  • Kirsch U (2008) Reanalysis of structures. Springer, Berlin

    MATH  Google Scholar 

  • Lazarov B, Sigmund O (2009) Sensitivity filters in topology optimisation as a solution to Helmholtz type differential equation. In: Proc. of the 8th world congress on structural and multidisciplinary optimization. Lisbon, Portugal

    Google Scholar 

  • Lazarov B, Sigmund O (2010) Filters in topology optimizat ion based on Helmholtz type differential equations. Int J Numer Methods Eng. doi:10.1002/nme.3072

  • Luo J, Luo Z, Chen S, Tong L, Wang MY (2008) A new level set method for systematic design of hinge-free compliant mechanisms. Comput Methods Appl Mech Eng 198(2):318–331

    MATH  Google Scholar 

  • Poulsen TA (2003) A new scheme for imposing a minimum length scale in topology optimization. Int J Numer Methods Eng 57(6):741–760

    Article  MathSciNet  MATH  Google Scholar 

  • Sardan Ö, Petersen DH, Mølhave K, Sigmund O, Bøggild P (2008) Topology optimized electrothermal polysilicon microgrippers. Microelectron Eng 85:1096–1099. doi:10.1016/j.mee.2008.01.049

    Article  Google Scholar 

  • Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25(4):493–524

    Article  Google Scholar 

  • Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33(4–5):401–424

    Article  Google Scholar 

  • Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mech Sin 25(2):227–239. doi:10.1007/s10409-009-0240-z

    Article  Google Scholar 

  • Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16(1):68–75

    Article  Google Scholar 

  • Stainko R, Sigmund O (2007) Tailoring dispersion properties of photonic crystal waveguides by topology optimization. Waves Random Complex Media 17:477–489

    Article  MathSciNet  MATH  Google Scholar 

  • Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373

    Article  MathSciNet  MATH  Google Scholar 

  • Wang F, Jensen J, Sigmund O (2010) Robust topology optimization of photonic crystal waveguides with tailored dispersion properties (submitted)

  • Wang MY, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1–2):227–246

    Article  MATH  Google Scholar 

  • Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear density filter based on heaviside funtions. Struct Multidisc Optim 41:495–505

    Article  MathSciNet  Google Scholar 

  • Yoon GH, Sigmund O (2008) A monolithic approach for topology optimization of electrostatically actuated devices. Comput Methods Appl Mech Eng 197:4062–4075. doi:10.1016/j.cma.2008.04.004

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was financially supported by Villum Fonden (via the NATEC Centre of Excellence), a Eurohorcs/ESF European Young Investigator Award (EURYI), a Center of Advanced User Support (CAUS) grant from the Danish Center of Scientific Computing (DCSC), and by the Elite Research Prize from the Danish Minister of Research.

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Correspondence to Boyan Stefanov Lazarov.

Appendix

Appendix

The thresholds for the dilated and the eroded designs can be chosen independently, if different length scales need to be imposed on the void and the solid phases. The procedure for determining the minimum length scale is similar to the one presented in Section 6. The estimation is based on the assumption that the topology does not change between the three designs—the eroded, the intermediate and the dilated. Two cases for the filtered designs are shown in Fig. 19. In the first case, Fig. 19a, the maximum of the filtered design \({\bar{\tilde{\rho}}}\) is equal to the threshold for the eroded design and 0/1 projection with threshold η e results in point with zero length scale. In the second case, Fig. 19b, the maximum of the filtered design is below η e , and the projection with threshold η e results in void phase. Therefore, even if the projections with thresholds η i and η d result in intervals with finite lengths, the topology for \({\bar{\tilde{\rho}}}^e, {\bar{\tilde{\rho}}}^i\) and \({\bar{\tilde{\rho}}}^d\) is different and the presented minimum length scale estimate is not valid for this case.

Fig. 19
figure 19

Filtered density field and with 0/1 projection based on different thresholds

If the topologies are the same for all three thresholds η e, η i and η d, \(\tilde{\rho}_{\max}\) is always \(\tilde{\rho}_{\max}\geq \eta_e\). Then the minimum length scale for the intermediate and the dilated designs can be estimated numerically by finding the lengths b and b d of the intervals obtained by crossing the threshold lines with the filtered density. The length scale for the void phase can be estimated in a similar way. The difference between η e and η i determines the length scale in the solid phase and the difference between η d and η i determines the length scale in the void phase. Graphs for the normalized length scales of the solid and void phases as functions of the thresholds are shown in Figs. 20 and 21.

Fig. 20
figure 20

Length scale of the solid phase as functions of the threshold for the eroded designs for η i  = 0.4. Dashed line shows the minimum length scale for the solid phase in the dilated design for η d  = 0.2

Fig. 21
figure 21

Length scale of void phase as functions of the threshold for the dilated designs for η i  = 0.4. Dotted line shown the minimum length scale for the void phase in the eroded design for η e  = 0.7

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Wang, F., Lazarov, B.S. & Sigmund, O. On projection methods, convergence and robust formulations in topology optimization. Struct Multidisc Optim 43, 767–784 (2011). https://doi.org/10.1007/s00158-010-0602-y

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