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Structural topology optimization with smoothly varying fiber orientations

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Abstract

In recent years, the field of additive manufacturing (AM), often referred to as 3D printing, has seen tremendous growth and radically changed the means by which we describe valid 3D models for production. In particular, it is now conceivable to produce composite structures consisting of smoothly varying oriented anisotropic constitutive materials. In the present work, we propose a sensitivity driven method for the generation of transverse isotropic fiber reinforced structures having smooth spatially varying orientations. Our approach builds upon finite element analysis (FEA) and density-based topology optimization (TO). The local material orientations are formulated as design variables in a stiffness maximization problem, and solved with a non-convex gradient-based optimization scheme. Length-scale control is achieved through the use of filters for regularization. We demonstrate the ability of the proposed approach to handle large-scale 3D problems with synchronous optimization of material densities and orientations yielding millions of design variables on multiple load case scenarios. The method is shown to be compatible with compliant mechanism optimization as well as local volume constraints. Finally, the approach is extended with an additional design variable dictating the ratio of anisotropy for each element, thereby delegating the choice of material type to the optimization scheme.

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References

  • Allaire G, Geoffroy-Donders P, Pantz O (2019) Topology optimization of modulated and oriented periodic microstructures by the homogenization method. Comput Math Appl 78(7):2197–2229

    Article  MathSciNet  Google Scholar 

  • Bendsøe MP (1983) On obtaining a solution to optimization problems for solid, elastic plates by restriction of the design space. Mech Based Des Struct Mach 11:501–521

    Article  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  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    MATH  Google Scholar 

  • Bendsøe MP, Sigmund O (2003) Topology optimization - theory, methods and applications. Springer, Germany

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Díaz A, Sigmund O (1995) Checkerboard patterns in layout optimization. Struct Optim 10 (1):40–45

    Article  Google Scholar 

  • Fernandez F, Compel WS, Lewicki JP, Tortorelli DA (2019) Optimal design of fiber reinforced composite structures and their direct ink write fabrication. Comput Methods Appl Mech Eng 353:277–307

    Article  MathSciNet  Google Scholar 

  • Gao J (2018) Optimal motion planning in redundant robotic systems for automated composite lay-up process. Theses, École centrale de Nantes

  • GrabCAD (2013) GrabCAD GE jet engine bracket challenge. https://grabcad.com/challenges/ge-jet-engine-bracket-challenge. Accessed: 2019-12-28

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

    Article  Google Scholar 

  • Huang Y, Zhang J, Hu X, Song G, Liu Z, Yu L, Liu L (2016) Framefab: Robotic fabrication of frame shapes. ACM Trans Graph 35(6)

  • Jiang D (2017) Three dimensional topology optimization with orthotropic material orientation design for additive manufacturing structures. Master’s Thesis, Baylor University

  • Jiang D, Hoglund R, Smith DE (2019) Continuous fiber angle topology optimization for polymer composite deposition additive manufacturing applications. Fibers 7(2)

  • Kabir SMF, Mathur K, Seyam A-FM (2020) A critical review on 3d printed continuous fiber-reinforced composites: history, mechanism, materials and properties. Compos Struct 232:111476

    Article  Google Scholar 

  • Kutta M (1901) Beitrag zur näherungweisen integration totaler differentialgleichungen. Z Math Phys 46:435–453

    MATH  Google Scholar 

  • Lee J, Kim D, Nomura T, Dede EM, Yoo J (2018) Topology optimization for continuous and discrete orientation design of functionally graded fiber-reinforced composite structures. Compos Struct 201:217–233

    Article  Google Scholar 

  • Liu J, Yu H (2017) Concurrent deposition path planning and structural topology optimization for additive manufacturing. Rapid Prototyp J 23(5):930–942

    Article  Google Scholar 

  • Michell A (1904) The limits of economy of material in frame-structures. Phil Mag 8:589–597

    Article  Google Scholar 

  • Nomura T, Dede E, Matsumori T, Kawamoto A (2015) Simultaneous optimization of topology and orientation of anisotropic material using isoparametric projection method. World Congress on Structural and Multidisciplinary Optimization

  • Pedersen P (1989) On optimal orientation of orthotropic materials. Struct Optim 1(2):101–106

    Article  Google Scholar 

  • Runge CDT (1895) ÜBer die reduktion der positiven quadratischen formen mit drei unbestimmten ganzen zahlen. Math Ann 46:167–178

    Article  MathSciNet  Google Scholar 

  • Safonov AA (2019) 3d topology optimization of continuous fiber-reinforced structures via natural evolution method. Compos Struct 215:289–297

    Article  Google Scholar 

  • Schmidt MP, Pedersen C, Gout C (2019) On structural topology optimization using graded porosity control. Struct Multidiscip Optim 60:1437–1453

    Article  MathSciNet  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 

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

    Article  Google Scholar 

  • Sigmund O, Maute K (2012) Sensitivity filtering from a continuum mechanics perspective. Struct Multidiscip Optim 46(4):471–475

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  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

    Article  MathSciNet  Google Scholar 

  • Tan D, Chen Z (2012) On a general formula of fourth order Runge-Kutta method. Struct Multidiscip Optim 46(7):1–10

    MathSciNet  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

    Article  Google Scholar 

  • Wu J, Wang W, Gao X (2019) Design and optimization of conforming lattice structures. CoRR, arXiv:1905.02902

Download references

Acknowledgments

The authors are thankful to the reviewers for their insightful comments helping improve the paper. L. Couret and C. Gout thank M2SiNum project (co-financed by the European Union and by the Normandie Regional Council) and CIEMME OpenMod platform (INSA Rouen) for their support.

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Correspondence to Martin-Pierre Schmidt.

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Replication of results

Unless explicitly stated, all optimized designs of the present paper used the following parameters: γ = 3, Ymin = 10− 6, ρmin = 10− 6, p = 6, n = 6, Gdyn,E = 0.6 ∀EΩ, a sensitivity filter of radius 1.3 with elements of size 1 arranged in a regular 2D or 3D grid of squares or cubes, respectively.

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Schmidt, MP., Couret, L., Gout, C. et al. Structural topology optimization with smoothly varying fiber orientations. Struct Multidisc Optim 62, 3105–3126 (2020). https://doi.org/10.1007/s00158-020-02657-6

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  • DOI: https://doi.org/10.1007/s00158-020-02657-6

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