Abstract
One of the straightforward definitions of structural topology optimization is to design the optimal distribution of the holes and the detailed shape of each hole implicitly in a fixed discretized design domain. However, typical numerical instability phenomena of topology optimization, such as the checkerboard pattern and mesh dependence, all take the form of an unexpected number of holes in the optimal result in standard density-type design methods, such as SIMP and ESO. Typically, the number of holes is indirectly controlled by tuning the value of the radius of the filter operator during the optimization procedure, in which the choice of the value of the filter radius is one of the most opaque and confusing issues for a beginner unfamiliar with the structural topology optimization algorithm. Based on the soft-kill bi-directional evolutionary structural optimization (BESO) method, an optimization model is proposed in this paper in which the allowed maximal number of holes in the designed structure is explicitly specified as an additional design constraint. The digital Gauss-Bonnet formula is used to count the number of holes in the whole structure in each optimization iteration. A hole-filling method (HFM) is also proposed in this paper to control the existence of holes in the optimal structure. Several 2D numerical examples illustrate that the proposed method cannot only limit the maximum number of holes in the optimal structure throughout the whole optimization procedure but also mitigate the phenomena of the checkerboard pattern and mesh dependence. The proposed method is expected to provide designers with a new way to tangibly manage the optimization procedure and achieve better control of the topological characteristics of the optimal results.
Similar content being viewed by others
Abbreviations
- A1 :
-
The set consisting of all elements of a connected structure
- Ai :
-
The i-th set of solid elements that belong to a connected structure
- b1 :
-
A solid element
- B:
-
The set of solid elements
- c:
-
The optimization objective
- C:
-
The set consisting of solid elements connected by an edge to b1 or an element of D
- D:
-
The set consisting of solid elements included in both C and B
- er:
-
The evolutionary volume ratio
- E1 :
-
The Young’s modulus of the solid material
- F :
-
The force vectors
- g:
-
The number of holes in the connected structure
- G:
-
The cell consisting of holes in order of area size from largest to smallest, consisting of finite elements
- Gj :
-
The hole at the j-th position in G
- h:
-
The number of holes in the topological structure in each iteration
- h0:
-
The peak value of the number of holes during the topology optimization process
- H:
-
The allowed maximum number of holes in the optimal structure as defined by the user
- k 0 :
-
The element stiffness matrix for an element with unit Young’s modulus
- K :
-
The global stiffness matrix
- Mi :
-
(i = 4, 2) The set of digital points with i neighboring edges
- n:
-
The number of connected structures in the current iteration of topology optimization
- N:
-
The number of elements used to discretize the design domain
- p:
-
The penalization power
- r:
-
The filter radius, relative to the unit length
- Si :
-
The area of the i-th hole
- S∗ :
-
A prescribed minimum area of the hole in final structure. In this paper, we set S∗ = 1
- u e :
-
The element displacement vector
- U :
-
The global displacement
- VA1 :
-
The set consisting of void elements connected to each other by an edge
- VA:
-
The cell consisting of void elements, where each element of VA is a hole
- vb1 :
-
A void element
- VB:
-
The set consisting of all void elements
- VC:
-
The set consisting of void elements connected by an edge to a void element of VD or vb1
- VD:
-
The set consisting of void elements included in both VC and VB
- Ve :
-
The volume of an individual element
- V*:
-
The prescribed volume of the final structure
- xh:
-
The numbers of elements in the horizontal direction of design domain
- yv:
-
The numbers of elements in the vertical direction of design domain
- ρ :
-
The design variables
- ρ e :
-
The design variable of the e-th element
- ρ min :
-
A fixed value equal to 0.001
References
Andreassen E, Clausen A, Schevenels M, Lazarov B, Sigmund O (2011) Efficient topology optimization in MATLAB using 88 lines of code. Struct Multidiscip Optim 43:1–16. https://doi.org/10.1007/s00158-010-0594-7
BendsØe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69:635–654. https://doi.org/10.1007/s004190050248
BendsØe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1:193–202. https://doi.org/10.1007/BF01650949
Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50:2143–2158. https://doi.org/10.1002/nme.116
Borrvall T, Petersson J (2003) Topology optimization of fluids in stokes flow. Int J Numer Methods Fluids 41:77–107. https://doi.org/10.1002/fld.426
Buhl T, Pedersen CBW, Sigmund O (2000) Stiffness design of geometrically nonlinear structures using topology optimization. Struct Multidiscip Optim 19:93–104. https://doi.org/10.1007/s001580050089
Chen SK, Chen W, Lee SH (2010) Level set based robust shape and topology optimization under random field uncertainties. Struct Multidiscip Optim 41:507–524. https://doi.org/10.1007/s00158-009-0449-2
Chen L, Rong Y (2010) Digital topological method for computing genus and the Betti numbers. Topol Appl 157(12):1931–1936. https://doi.org/10.1016/j.topol.2010.04.006
Chen L (2004) Discrete surfaces and manifolds. Sp Computing, Rockville
Deng Y, Korvink JG (2018) Self-consistent adjoint analysis for topology optimization of electromagnetic waves. J Comput Phys 361:353–376. https://doi.org/10.1016/j.jcp.2018.01.045
Deng Y, Liu Z, Zhang P, Liu Y, Wu Y (2011) Topology optimization of unsteady incompressible Navier_Stokes flows. J Comput Phys 230:6688–6708. https://doi.org/10.1016/j.jcp.2011.05.004
Evgrafov A, Pingen G, Maute K (2008) Topology optimization of fluid domains: kinetic theory approach. ZAMM 88:129–141. https://doi.org/10.1002/zamm.200700122
Gu DX, Yau ST (2008) Computational conformal geometry. HIGHER EDUCATION PRESS. International Press, Somerville, Massachusetts, U.S.A.; Higher Education Press, Beijing, China
Gersborg-Hansen A, Sigmund O, Haber RB (2005) Topology optimization of channel flow problems. Struct Multidiscip Optim 20:181–192. https://doi.org/10.1007/s00158-004-0508-7
Guest JK, Prevost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61:238–254. https://doi.org/10.1002/nme.1064
Huang X, Xie YM (2009) Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Comput Mech 43:393–401. https://doi.org/10.1007/s00466-008-0312-0
Lazarov BS, Sigmund O (2011) Filters in topology optimization based on Helmholtz-type differential equations. Int J Numer Methods Eng 86:765–781. https://doi.org/10.1002/nme.3072
Lazarov BS, Wang FW, Sigmund O (2016) Length scale and manufacturability in density-based topology optimization. Arch Appl Mech 86:189–218. https://doi.org/10.1007/s00419-015-1106-4
Labbe T, Dehez B (2011) Topology optimization method based on the Maxwell stress tensor for the design of ferromagnetic parts in electromagnetic actuators. IEEE Trans Magn 47:2188–2193. https://doi.org/10.1109/TMAG.2011.2138151
Mlejnek HP (1992) Some aspects of the genesis of structures. Struct Optim 5:64–69. https://doi.org/10.1007/BF01744697
Okamoto Y, Wakao S, Sato S (2016) Topology optimization based on regularized level-set function for solving 3-D nonlinear magnetic field system with spatial symmetric condition. IEEE Trans Magn 52:3. https://doi.org/10.1109/tmag.2015.2492978
Osher S, Sethian JA (1988) Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 79:12–49. https://doi.org/10.1016/0021-9991(88)90002-2
Rozvany GIN (2001) Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics. Struct Multidiscip Optim 21:92–108. https://doi.org/10.1007/s001580050174
Rozvany GIN (2009) A critical review of established methods of structural topology optimization. Struct Multidiscip Optim 37:217–237. https://doi.org/10.1007/s00158-007-0217-0
Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mech Sinica 25:227–239. https://doi.org/10.1007/s10409-009-0240-z
Sigmund O (2001a) Design of multiphysics actuators using topology optimization - part I: one-material structures. Comput Methods Appl Mech Eng 190:6577–6604. https://doi.org/10.1016/S0045-7825(01)00251-1
Sigmund O (2001b) Design of multiphysics actuators using topology optimization - part II: two-material structures. Comput Methods Appl Mech Eng 190:6605–6627. https://doi.org/10.1016/S0045-7825(01)00252-3
Sigmund O (2001c) A 99 line topology optimization code written in Matlab. Struct Multidiscip Optim 21:120–127. https://doi.org/10.1007/s001580050176
Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16:68–75. https://doi.org/10.1007/BF01214002
Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33:401–424. https://doi.org/10.1007/s00158-006-0087-x
Talischi C, Paulino GH, Pereira A, Menezes IFM (2012) PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidiscip Optim 45:309–328. https://doi.org/10.1007/s00158-011-0706-z
Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43:767–784. https://doi.org/10.1007/s00158-010-0602-y
Xie YM, Steven GP (1992) A simple evolutionary procedure for structural optimization. Comput Struct 49(5):885–896. https://doi.org/10.1016/0045-7949(93)90035-C
Xie YM, Steven GP (1997) Evolutionary structural optimization. Springer, London
Yamada T, Izui K, Nishiwaki S (2010) A topology optimization method based on the level set method incorporating a fictitious interface energy. Comput Methods Appl Mech Eng 199:2876–2891. https://doi.org/10.1016/j.cma.2010.05.013
Yang KK, Fernandez E, Cao N, Duysinx P, Zhu JH, Zhang W (2019) Note on spatial gradient operators and gradient-based minimum length constraints in SIMP topology optimization. Struct Multidiscip Optim 60:393–400. https://doi.org/10.1007/s00158-019-02269-9
Zuo WJ, Saitou K (2017) Multi-material topology optimization using ordered SIMP interpolation. Struct Multidiscip Optim 55:477–491. https://doi.org/10.1007/s00158-016-1513-3
Zhou M, Rozvany GIN (1991) The COC algorithm, part II: topological, geometry and generalized shape optimization. Comp Meth Appl Mech Engrng 89:197–224. https://doi.org/10.1016/0045-7825(91)90046-9
Zhou MD, Lazarov BS, Wang FW, Sigmund O (2015) Minimum length scale in topology optimization by geometric constraints. Comput Methods Appl Mech Eng S0045-7825(15):00169–00163. https://doi.org/10.1016/j.cma.2015.05.003
Acknowledgments
The authors are grateful to Yimin Xie and Xiaodong Huang for providing the MATLAB codes for BESO. The authors thank Prof. Xianfeng David Gu (Stony Brook University) for valuable discussions about algebra topology.
Funding
This research was funded by the National Science Foundation of China under grant no. 51675506 and under National Science and Technology Major Project 2017ZX10304403.
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
The original Soft-BESO MATLAB code by Yimin Xie and Xiaodong Huang can be downloaded at http://www.isg.rmit.edu.au.
The results presented in Section 3 were obtained via the HFM using a MATLAB function defined as follows:
Soft_BESO_HFM (xh, yv, V, r, er, H, flag).
The complete MATLAB code is given as an Appendix file.
Additional information
Responsible Editor: Ji-Hong Zhu
Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Han, H., Guo, Y., Chen, S. et al. Topological constraints in 2D structural topology optimization. Struct Multidisc Optim 63, 39–58 (2021). https://doi.org/10.1007/s00158-020-02771-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-020-02771-5