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A projection approach for topology optimization of porous structures through implicit local volume control

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Abstract

Porous structures are of valuable importance in additive manufacturing. They can also be exploited to improve damage tolerance and fail-safe behavior. This paper presents a projection approach to design optimized porous structures in the framework of density-based topology optimization. In contrast to conventional constraint approach, the maximum local volume limitation is integrated into the material interpolation model through a filtering and projection process. This paper also presents two extensions of the basic approach, including a robust formulation for improving weak structural features and mesh/design refinement for enhancing computational stability and efficiency. The applicability of the proposed methodology is demonstrated by a set of numerical minimum compliance problems. This approach can be used in a wider range of applications concerning porous structures.

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Notes

  1. These six scalar fields (ψ, \( \tilde{\psi} \), ϕ, \( \overline{\phi} \), φ, ρ) are denoted as X, X1, X2, X3, X4 and X5 in the code filtersub.m shown in the appendix.

References

  • Aage N, Andreassen E, Lazarov BS (2015) Topology optimization using PETSc: an easy-to-use, fully parallel, open source topology optimization framework. Struct Multidiscip Optim 51(3):565–572

    MathSciNet  Google Scholar 

  • Aage N, Andreassen E, Lazarov BS, Sigmund O (2017) Giga-voxel computational morphogenesis for structural design. Nature 550(7674):84–86

    Google Scholar 

  • Alexandersen J, Lazarov BS (2015) Topology optimisation of manufacturable microstructural details without length scale separation using a spectral coarse basis preconditioner. Comput Methods Appl Mech Eng 290:156–182

    MathSciNet  MATH  Google Scholar 

  • Andreassen E, Clausen A, Schevenels M, Lazarov BS, Sigmund O (2011) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optim 43(1):1–16

    MATH  Google Scholar 

  • Bai W, Li QH, Chen WJ, Liu ST (2017) A novel projection based method for imposing maximum length scale in topology optimization. Eng Mech 34(9):18–26 (in Chinese)

    Google Scholar 

  • Bendsøe MP (1989) Optimal shape design as a material distribution problem. Structural Optimization 1(4):193–202

    Google Scholar 

  • Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9–10):635–654

    MATH  Google Scholar 

  • Cheng L, Zhang P, Biyikli E, Bai J, Robbins J, To A (2017) Efficient design optimization of variable-density cellular structures for additive manufacturing: theory and experimental validation. Rapid Prototyp J 23(4):660–677

    Google Scholar 

  • Clausen A, Aage N, Sigmund O (2015) Topology optimization of coated structures and material interface problems. Comput Methods Appl Mech Eng 290:524–541

    MathSciNet  MATH  Google Scholar 

  • Clausen A, Aage N, Sigmund O (2016) Exploiting additive manufacturing infill in topology optimization for improved buckling load. Engineering 2(2):250–257

    Google Scholar 

  • Gaynor AT, Guest JK (2016) Topology optimization considering overhang constraints: eliminating sacrificial support material in additive manufacturing through design. Struct Multidiscip Optim 54(5):1157–1172

    MathSciNet  Google Scholar 

  • Groen JP, Sigmund O (2018) Homogenization-based topology optimization for high-resolution manufacturable microstructures. Int J Numer Methods Eng 113(8):1148–1163

    MathSciNet  Google Scholar 

  • Groen J, Wu J, Sigmund O (2019) Homogenization-based stiffness optimization and projection of 2D coated structures with orthotropic infill. Comput Methods Appl Mech Eng 349:722–742

    MathSciNet  MATH  Google Scholar 

  • Guest JK (2009) Imposing maximum length scale in topology optimization. Struct Multidiscip Optim 37(5):463–473

    MathSciNet  MATH  Google Scholar 

  • Guest J, Prévost J (2006) A penalty function for enforcing maximum length scale criterion in topology optimization. In: 11th AIAA/ISSMO multidisciplinary analysis and optimization conference, Portsmouth, Virginia, pp AIAA 2006–6938

  • 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):238–254

    MathSciNet  MATH  Google Scholar 

  • Guest JK, Asadpoure A, Ha SH (2011) Eliminating beta-continuation from Heaviside projection and density filter algorithms. Struct Multidiscip Optim 44(4):443–453

    MathSciNet  MATH  Google Scholar 

  • Guo X, Zhang W, Zhong W (2014) Doing topology optimization explicitly and geometrically–a new moving morphable components based framework. J Appl Mech 81(8):081009

    Google Scholar 

  • Jansen M, Lombaert G, Schevenels M, Sigmund O (2014) Topology optimization of fail-safe structures using a simplified local damage model. Struct Multidiscip Optim 49(4):657–666

    MathSciNet  Google Scholar 

  • Keshavarzzadeh V, Kirby RM, Narayan A (2019) Parametric topology optimization with multiresolution finite element models. Int J Numer Methods Eng 119(7):567–589

    MathSciNet  Google Scholar 

  • Kim TS, Kim JE, Jeong JH, Kim YY (2004) Filtering technique to control member size in topology design optimization. KSME International Journal 18(2):253–261

    Google Scholar 

  • Langelaar M (2017) An additive manufacturing filter for topology optimization of print-ready designs. Struct Multidiscip Optim 55(3):871–883

    MathSciNet  Google Scholar 

  • Lazarov BS, Sigmund O (2011) Filters in topology optimization based on Helmholtz-type differential equations. Int J Numer Methods Eng 86(6):765–781

    MathSciNet  MATH  Google Scholar 

  • Lazarov BS, Wang F (2017) Maximum length scale in density based topology optimization. Comput Methods Appl Mech Eng 318:826–844

    MathSciNet  MATH  Google Scholar 

  • Liao Z, Zhang Y, Wang Y, Li W (2019) A triple acceleration method for topology optimization. Struct Multidiscip Optim 60(2):727–744

    MathSciNet  Google Scholar 

  • Liu C, Du Z, Zhang W, Zhu Y, Guo X (2017) Additive manufacturing-oriented design of graded lattice structures through explicit topology optimization. J Appl Mech 84(8):081008

    Google Scholar 

  • Liu J, Gaynor AT, Chen S, Kang Z, Suresh K, Takezawa A, Li L, Kato J, Tang J, Wang CC, Cheng L, Liang X, To AC (2018a) Current and future trends in topology optimization for additive manufacturing. Struct Multidiscip Optim 57(6):2457–2483

    Google Scholar 

  • Liu J, Yu H, To AC (2018b) Porous structure design through Blinn transformation-based level set method. Struct Multidiscip Optim 57(2):849–864

    Google Scholar 

  • Qian X (2017) Undercut and overhang angle control in topology optimization: a density gradient based integral approach. Int J Numer Methods Eng 111(3):247–272

    MathSciNet  Google Scholar 

  • Querin OM, Steven GP, Xie YM (1998) Evolutionary structural optimisation (ESO) using a bidirectional algorithm. Eng Comput 15(8):1031–1048

    MATH  Google Scholar 

  • Schury F, Stingl M, Wein F (2012) Efficient two-scale optimization of manufacturable graded structures. SIAM J Sci Comput 34(6):B711–B733

    MathSciNet  MATH  Google Scholar 

  • Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidiscip Optim 21(2):120–127

    Google Scholar 

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

    Google Scholar 

  • Sigmund O, Maute K (2013) Topology optimization approaches: a comparative review. Struct Multidiscip Optim 48(6):1031–1055

    MathSciNet  Google Scholar 

  • Sivapuram R, Dunning PD, Kim HA (2016) Simultaneous material and structural optimization by multiscale topology optimization. Struct Multidiscip Optim 54(5):1267–1281

    MathSciNet  Google Scholar 

  • Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22(2):116–124

    Google Scholar 

  • Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373

    MathSciNet  MATH  Google Scholar 

  • Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784

    MATH  Google Scholar 

  • Wang W, Wang TY, Yang Z, Liu L, Tong X, Tong W, Deng J, Chen F, Liu X (2013) Cost-effective printing of 3D objects with skin-frame structures. ACM Trans Graph 32(6):1–10

    Google Scholar 

  • Wang X, Xu S, Zhou S, Xu W, Leary M, Choong P, Qian M, Brandt M, Xie YM (2016) Topological design and additive manufacturing of porous metals for bone scaffolds and orthopaedic implants: a review. Biomaterials 83:127–141

    Google Scholar 

  • Wang B, Zhou Y, Zhou Y, Xu S, Niu B (2018) Diverse competitive design for topology optimization. Struct Multidiscip Optim 57(2):891–902

    MathSciNet  Google Scholar 

  • Wu J, Dick C, Westermann R (2016) A system for high-resolution topology optimization. IEEE Trans Vis Comput Graph 22(3):1195–1208

    Google Scholar 

  • Wu J, Clausen A, Sigmund O (2017) Minimum compliance topology optimization of shell-infill composites for additive manufacturing. Comput Methods Appl Mech Eng 326:358–375

    MathSciNet  MATH  Google Scholar 

  • Wu J, Aage N, Westermann R, Sigmund O (2018) Infill optimization for additive manufacturing—approaching bone-like porous structures. IEEE Trans Vis Comput Graph 24(2):1127–1140

    Google Scholar 

  • Wu J, Wang W, Gao X (2019) Design and optimization of conforming lattice structures. IEEE Trans Vis Comput Graph. https://doi.org/10.1109/TVCG.2019.2938946

  • Yang XY, Xie YM, Steven GP, Querin OM (1999) Bidirectional evolutionary method for stiffness optimization. AIAA J 37(11):1483–1488

    Google Scholar 

  • Yang K, Zhao ZL, He Y, Zhou S, Zhou Q, Huang W, Xie YM (2019) Simple and effective strategies for achieving diverse and competitive structural designs. Extreme Mech Lett 30:100481

    Google Scholar 

  • Zhang W, Yuan J, Zhang J, Guo X (2016) A new topology optimization approach based on moving morphable components (MMC) and the ersatz material model. Struct Multidiscip Optim 53(6):1243–1260

    MathSciNet  Google Scholar 

  • Zhang W, Chen J, Zhu X, Zhou J, Xue D, Lei X, Guo X (2017) Explicit three dimensional topology optimization via moving morphable void (MMV) approach. Comput Methods Appl Mech Eng 322:590–614

    MathSciNet  MATH  Google Scholar 

  • Zhang W, Li D, Zhou J, Du Z, Li B, Guo X (2018) A moving morphable void (MMV)-based explicit approach for topology optimization considering stress constraints. Comput Methods Appl Mech Eng 334:381–413

    MathSciNet  MATH  Google Scholar 

  • Zhao ZL, Zhou S, Feng XQ, Xie YM (2018) On the internal architecture of emergent plants. Journal of the Mechanics and Physics of Solids 119:224–239

    MathSciNet  Google Scholar 

  • Zhou M, Fleury R (2016) Fail-safe topology optimization. Struct Multidiscip Optim 54(5):1225–1243

    Google Scholar 

  • Zhou M, Rozvany G (1991) The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1):309–336

    Google Scholar 

  • Zhou M, Lazarov BS, Wang F, Sigmund O (2015) Minimum length scale in topology optimization by geometric constraints. Comput Methods Appl Mech Eng 293:266–282

    MathSciNet  MATH  Google Scholar 

  • Zhu Y, Li S, Du Z, Liu C, Guo X, Zhang W (2019) A novel asymptotic-analysis-based homogenisation approach towards fast design of infill graded microstructures. J Mech Phys Solids 124:612–633

    MathSciNet  Google Scholar 

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Acknowledgments

The author is grateful to Boyan Lazarov at University of Manchester (currently at Lawrence Livermore National Laboratory), Jun Wu at Delft University of Technology, and Fengwen Wang and Mathias Stolpe at Technical University of Denmark for their valuable feedback. The author is also grateful to the anonymous reviewers for their valuable suggestions and comments.

Funding

This work was supported by Independent Research Fund Denmark [grant no. 7017-00084A]: Fail-Safe Structural Optimization (SELMA).

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Correspondence to Suguang Dou.

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Replication of results

The approach and the algorithm are described in the paper. Details are provided concerning the procedures to generate the designs together with the specific values for the parameters. The Matlab code to compute the physical densities and the sensitivities is included in the Appendix. The codes to generate the optimized designs in this paper will be available on gitlab.windenergy.dtu.dk.

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Responsible Editor: Xu Guo

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Appendices

Appendix 1

The derivatives in (16) and (17) are given as

$$ {\displaystyle \begin{array}{l}\frac{\partial c}{\partial {\rho}_i}=-p\;{\rho}_i^{p-1}\left({E}_{\mathrm{max}}-{E}_{\mathrm{min}}\right){\mathbf{u}}_i^T{\mathbf{k}}_0{\mathbf{u}}_i,\\ {}\frac{\partial {\rho}_i}{\partial {\phi}_i}={\varphi}_i,\kern0.6em \frac{\partial {\rho}_i}{\partial {\varphi}_i}={\phi}_i,\kern0.72em \frac{\partial g}{\partial {\rho}_i}=\frac{1}{N_e},\\ {}\frac{\mathrm{d}{\varphi}_i}{\mathrm{d}{\overline{\phi}}_i}=-\frac{\beta_2\left(1-{\tanh}^2\left({\beta}_2\left({\overline{\phi}}_i-{\eta}_2\right)\right)\right)}{\tanh \left({\beta}_2{\eta}_2\right)+\tanh \left({\beta}_2\left(1-{\eta}_2\right)\right)},\kern0.84em \frac{\partial {\overline{\phi}}_j}{\partial {\phi}_i}=\frac{1}{\sum_{k\in {\mathbbm{I}}_{j,2}}1}\\ {}\frac{\mathrm{d}{\phi}_i}{\mathrm{d}{\tilde{\psi}}_i}=\frac{\beta_1\left(1-{\tanh}^2\left({\beta}_1\left(\tilde{\psi}-{\eta}_1\right)\right)\right)}{\tanh \left({\beta}_1{\eta}_1\right)+\tanh \left({\beta}_1\left(1-{\eta}_1\right)\right)},\kern0.7em \frac{\partial {\tilde{\psi}}_i}{\partial {\psi}_e}=\frac{w_{ei}}{\sum_{k\in {\mathbbm{I}}_{i,1}}{w}_{ki}}\end{array}} $$
(20)

Appendix 2

figure b

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Dou, S. A projection approach for topology optimization of porous structures through implicit local volume control. Struct Multidisc Optim 62, 835–850 (2020). https://doi.org/10.1007/s00158-020-02539-x

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