Advertisement

On structural topology optimization using graded porosity control

  • Martin-Pierre SchmidtEmail author
  • Claus B. W. Pedersen
  • Christian Gout
Research Paper
  • 11 Downloads

Abstract

In recent years, the field of additive manufacturing (AM), often referred to as 3D printing, has seen tremendous growth and radically changed the way we describe valid 3D models for fabrication. While not free of constraints, AM offers an unprecedented level of freedom in geometrical complexity for manufacturable feasible designs. One example of such design freedom is the creation of intricate, robust, and lightweight internal structures. Our approach builds upon and extends the recent works on topology optimization for the so-called infill structures. In order to have more design control over these infill structures, we present a new formulation allowing the generation of mixed design patterns containing bulk and porous regions using a guiding constraint parameterized by non-uniform constraining fields. Secondly, we demonstrate multiple methods of generating such non-uniform fields to leverage the present formulation and analyze their effect on the geometrical and physical properties of the obtained designs.

Keywords

Topology optimization Finite element analysis Mathematical programming Local volume constraints Guided porosity control Additive manufacturing 

Notes

Acknowledgements

The authors are thankful to the reviewers for their insightful comments helping improve the paper and to the members of the Department of Mechanical Engineering and Department of Electrical Engineering of the Technical University of Denmark for the organization of the 2017 PhD course “Topology Optimization - Theory, Methods and Applications.” C. Gout thanks M2SiNum project (co-financed by the European Union and by the Normandie Regional Council) and CIEMME OpenMod platform (INSA Rouen) for their support.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

Replication of results

Unless explicitly stated, all optimized designs of the present paper used the following parameters: γ = 3, Y 0 = 1Ymin = 10− 6, \(G_{l}^{\star }= 0.6\), Poisson ratio of 0.3, isotropic local neighborhood of radius rE = 4, 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.

References

  1. Amir O, Aage N, Lazarov BS (2014) On multigrid-CG for efficient topology optimization. Struct Multidiscip Optim 49(5):815–829MathSciNetCrossRefGoogle Scholar
  2. 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 11:501–521CrossRefGoogle Scholar
  3. Bendsøe MP (1989) Optimal shape design as a material distribution problem. Structural Optimization 1 (4):193–202CrossRefGoogle Scholar
  4. Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9):635–654zbMATHGoogle Scholar
  6. Bendsøe MP, Sigmund O (2003) Topology optimization - Theory, methods and applications. Springer, GermanyzbMATHGoogle Scholar
  7. Clausen A, Aage N, Sigmund O (2016) Exploiting additive manufacturing infill in topology optimization for improved buckling load. Engineering 2(2):250–257CrossRefGoogle Scholar
  8. Clausen A, Andreassen E, Sigmund O (2017) Topology optimization of 3D shell structures with porous infill. Acta Mech Sinica, pp 1–14Google Scholar
  9. Díaz A., Sigmund O (1995) Checkerboard patterns in layout optimization. Structural Optimization 10 (1):40–45CrossRefGoogle Scholar
  10. Dick C, Georgii J, Westermann R (2011) A real-time multigrid finite hexahedra method for elasticity simulation using CUDA. Simul Model Pract Theory 19(2):801–816CrossRefGoogle Scholar
  11. Gao W, Zhang Y, Ramanujan D, Ramani K, Chen Y, Williams C, Wang C, Shin Y, Zhang S, Zavattieri P (2015) The status, challenges, and future of additive manufacturing in engineering. Comput Aided Des 69:65–89CrossRefGoogle Scholar
  12. Liu L, Shamir A, Wang C, Whitening E (2014) 3D printing oriented design: geometry and optimization. In: SIGGRAPH Asia 2014 Courses, SA ’14, ACM, New YorkGoogle Scholar
  13. Schmidt S, Schulz V (2011) A 2589 line topology optimization code written for the graphics card. Comput Vis Sci 14(6):249–256MathSciNetCrossRefzbMATHGoogle Scholar
  14. Sigmund O, Maute K (2012) Sensitivity filtering from a continuum mechanics perspective. Struct Multidiscip Optim 46(4):471–475MathSciNetCrossRefzbMATHGoogle Scholar
  15. Sigmund O, Maute K (2013) Topology optimization approaches a comparative review. Struct Multidiscip Optim 48:1031–1055MathSciNetCrossRefGoogle Scholar
  16. Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural optimization 16(1):68–75CrossRefGoogle Scholar
  17. Sigmund O, Aage N, Andreassen E (2016) On the (non-)optimality of Michell structures. Struct Multidiscip Optim 54(2):361–373MathSciNetCrossRefGoogle Scholar
  18. Svanberg K (1987) The method of moving asymptotes - a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373MathSciNetCrossRefzbMATHGoogle Scholar
  19. von Neumann J, Goldstine H (1947) Numerical inverting of matrices of high order. Bull Am Math Soc 53 (11):1021–1099MathSciNetCrossRefzbMATHGoogle Scholar
  20. Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784CrossRefzbMATHGoogle Scholar
  21. Wu J, Aage N, Westermann R, Sigmund O (2016a) Infill optimization for additive manufacturing - approaching bone-like porous structures. IEEE Trans Vis Comput Graph 24:1127–1140Google Scholar
  22. Wu J, Dick C, Westermann R (2016b) A system for high-resolution topology optimization. IEEE Trans Vis Comput Graph 22(3):1195–1208Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Martin-Pierre Schmidt
    • 1
    Email author
  • Claus B. W. Pedersen
    • 2
  • Christian Gout
    • 1
  1. 1.LMIINSA Rouen NormandieSaint-Étienne-du-RouvrayFrance
  2. 2.HamburgGermany

Personalised recommendations