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.
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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
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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.
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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.
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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
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DOI: https://doi.org/10.1007/s00158-020-02771-5