Abstract
Spatial gradient calculations are regularly applied in topology optimization and typically used for detecting material distribution boundaries, interfaces, and overhanging features. While adopted for a variety of applications, the current approaches in literature are restricted to perfectly uniform mapped-meshing strategies or require element shape function vectors to complete computations, both of which are potential limitations when solving practical problems with commercial finite element solvers. To address these drawbacks, this brief note presents a technique for calculating spatial gradients using a neighbouring element search strategy, using the gradient norm, thinning, and thresholding calculations to determine a nearly discrete interface indicator. The proposed technique is generalized for a 3D unstructured mesh and applicable when using black-box finite element solvers. The technique is validated on three sample problems with increasing geometric and mesh discretization complexity to demonstrate and suggest its effectiveness for topology optimization in 2D and 3D.
References
Altair Engineeering Inc. (2019) Altair OptiStruct 2019 User Guide. https://connect.altair.com/CP/downloads.html. Accessed 17 July 2020
Beckers M (1999) Topology optimization using a dual method with discrete variables. Struct Optim 17(1):14–24. https://doi.org/10.1007/BF01197709
Clausen A, Aage N, Sigmund O (2015) Topology optimization of coated structures and material interface problems. Comput Methods Appl Mech Eng 290:524–541. https://doi.org/10.1016/j.cma.2015.02.011
Costa JCA Jr, Alves MK (2003) Layout optimization with h-adaptivity of structures. Int J Numer Methods Eng 58(1):83–102. https://doi.org/10.1002/nme.759
Florea V, Pamwar M, Sangha B, Kim IY (2020) Simultaneous single-loop multimaterial and multijoint topology optimization. Int J Numer Methods Eng 121(7):1558–1594. https://doi.org/10.1002/nme.6279
Nana A, Cuilliere JC, Francois V (2016) Towards adaptive topology optimization. Adv Eng Softw 100:290–307. https://doi.org/10.1016/j.advengsoft.2016.08.005
Qian XP (2017) Undercut and overhang angle control in topology optimization: a density gradient based integral approach. Int J Numer Methods Eng 111(3):247–272. https://doi.org/10.1002/nme.5461
Ryan L, Kim IY (2019) A multiobjective topology optimization approach for cost and time minimization in additive manufacturing. Int J Numer Methods Eng 118(7):371–394. https://doi.org/10.1002/nme.6017
Sabiston G, Kim IY (2019) 3D topology optimization for cost and time minimization in additive manufacturing. Struct Multidiscip Optim 61:731–748. https://doi.org/10.1007/s00158-019-02392-7
Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4–5):401–424. https://doi.org/10.1007/s00158-006-0087-x
Stainko R (2006) An adaptive multilevel approach to the minimal compliance problem in topology optimization. Commun Numer Methods Eng 22(2):109–118. https://doi.org/10.1002/cnm.800
Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22(2):116–124. https://doi.org/10.1007/s001580100129
Woischwill C, Kim IY (2018) Multimaterial multijoint topology optimization. Int J Numer Methods Eng 115(13):1552–1579. https://doi.org/10.1002/nme.5908
Yang KK, Fernandez E, Niu C, Duysinx P, Zhu JH, Zhang WH (2019) Note on spatial gradient operators and gradient-based minimum length constraints in SIMP topology optimization. Struct Multidiscip Optim 60(1):393–400. https://doi.org/10.1007/s00158-019-02269-9
Zhou M, Shyy YK, Thomas HL (2001) Checkerboard and minimum member size control in topology optimization. Struct Multidiscip Optim 21(2):152–158. https://doi.org/10.1007/s001580050179
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Replication of results
A pseudo code is included in Appendix to assist in replicating the results presented in this note. The code is implemented in 3D and necessary input parameters are specified in the associated numerical examples.
Additional information
Responsible Editor: Ming Zhou
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Pseudo code
Pseudo code
Rights and permissions
About this article
Cite this article
Crispo, L., Bohrer, R., Roper, S.W.K. et al. Spatial gradient interface detection in topology optimization for an unstructured mesh. Struct Multidisc Optim 63, 515–522 (2021). https://doi.org/10.1007/s00158-020-02688-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-020-02688-z