Struct Multidisc Optim DOI 10.1007/s00158-015-1342-9
BRIEF NOTE
Topology optimization of periodic structures using BESO based on unstructured design points Guanqiang He 1 & Xiaodong Huang 2 & Hu Wang 1 & Guangyao Li 1
Received: 22 April 2014 / Revised: 27 May 2015 / Accepted: 2 June 2015 # Springer-Verlag Berlin Heidelberg 2015
Abstract A topology optimization method is proposed for periodic structures when unit cells have different geometries and irregular FE meshes. The relationships between the elements and unstructured design points are established according to the Shepard interpolation functions. Then, the BESO method is applied by switching the density of the design points between solid and void iteratively until an optimized solution is achieved. Due to the separation of finite element analysis and design variables, the optimized topology of periodic structures can be clearly described by unstructured design points. Finally, numerical examples are presented to demonstrate the validity and effectiveness of the proposed heuristic method. Keywords Topology optimization . Periodic structures . Unstructured design points . BESO
1 Introduction An important branch of structural topology optimization problems is periodic structure, and they are investigated
* Hu Wang
[email protected] * Guangyao Li
[email protected] Guanqiang He
[email protected] 1
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China
2
Center for Innovative Structures and Materials, School of Civil, Environmental and Chemical Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Australia
extensively at the microscopic level (Sigmund 1994; Guest and Prévost 2007), as well as at the macroscopic level (Moses et al. 2002; Zhang and Sun 2006). In the research above, the unit cells within a periodic structure is assumed to have identical geometry and FE mesh. Conversely, the lack of the subject of periodic macrostructure optimization with unit cells containing different geometries as shown in Fig. 1 has motivated the author’s effort in investigating topology optimization based on unstructured design points, which emloying the idea of separating the design variables and analysis space in topology otpimization (Ha and Cho 2008; Stanford and Ifju 2009; Guest and Smith Genut 2010). In Fig. 1, it can be seen that both unit cells have different geometries and it is difficult to obtain identical FE mesh for both unit cells, so the elemental mapping reationship between these two cells is hard to trace by using traditional topology optimzation method based on elemental variables. Another motivation of using unstructured design points is to take the advantage of nodal design variables. And the research of topology optimization based on nodal design variables has ranged from finite element method (Guest and Smith Genut 2010; Kang and Wang 2011) to meshless method (Luo et al. 2013), in order to avoid some numerical difficulties which are highly related to the use of elemental design variables in topology optimization, even when higher-order finite elements (Díaz and Sigmund 1995; Sigmund and Petersson 1998) and non-conforming finite elements (Jang et al. 2003) are employed. In this paper, we will propose a novel topology optimization method based on unstructured design points for periodic structures with different geometries and irregular FE meshes, we choose BESO method since it can be easily integrated with commercial FEM packages like ANSYS used as FEM solver in this work.
Struct Multidisc Optim
of elemental density design variables. xi,j is the binary design variable where xi,j = 1 represents a solid element and xi,j =xmin represents a void element. xmin is normally set to be a small value for avoiding any numerical difficulties. It can be seen that, for the optimization of periodic structures, the elemental density values at the same position of all unit cells must set to be the same so as to ensure the periodicity of the design. Since the sensitivity number is defined as (Yang et al. 1999; Querin et al. 1998). 8 > < 1 ui K0 ui ð x i ¼ 1Þ i ð2Þ αi ¼ 2x min > : ui K0i ui ðxi ¼ xmin Þ 2
Fig. 1 Complex geometry
2 Optimal design of periodic structure using conventional BESO For a typical 2D periodic structure shown in Fig. 2, the design domain is divided into m=nx ×ny unit cells where nx =3 and ny =2 denote the cell numbers along direction x and direction y, respectively. The design of the periodic structure requires an identical topology for all unit cells. Thus, the optimization problem for the maximum stiffness of the periodic structure using BESO method can be formulated with the following equation in terms of the binary design variable xi,j, where i and j represent the number of unit cell and the element number in its corresponding cell, respectively (Huang and Xie 2007). find xi; j ði ¼ 1; 2; ⋯; m; j ¼ 1; 2; ⋯; nÞ 1 min c ¼ f T u 2 V ≤V f s:t: V0 Ku ¼ f x1; j ¼ x2; j ¼ ⋯ ¼ xm; j xi; j ¼ f1; xmin g
m X 1 αj ¼ xi; j uTi; j K0i; j ui; j 2 i¼1
ð3Þ
It can be seen that the traditional BESO method assuming all unit cells to be identical.
3 Optimal design of periodic structure based on unstructured design points ð1Þ
Here c is known as the mean compliance, K is the global stiffness matrix, u is the displacement vector, f is the external load vector, V0 denotes the volume of the design domain, V represents the actual material volume, Vf is the specified volume fraction ratio and x is the vector Fig. 2 A typical 2D periodic structure with six unit cells
where xmin is set to be a small value such as 10−6 in this work. Then the sensitivity number of the jth element in representative unit cell (RUC) is
In the presented method, the design variables are defined at a given set of unstructured points (Kang and Wang 2012) as shown in Fig. 3. To establish the relationship between finite elements and design variable points, a dual-level Shepard interpolation is employed (Luo et al. 2013; Kang and Wang 2012). Considering the different elemental size, the sensitivity number of the jth design point is defined by summing jth sensitivity number of all unit cells and can be expressed by
Topology optimization of periodic structures using BESO
αpoint j
m X X αi; j ¼ w xi; j V i; j i¼1 j∈Ω
ð4Þ
1
where αi,j is the sensitivity number of the jth element in ith unit cell and Vi,j denotes the volume of the jth element in ith unit cell. w(xi,j) denotes the Shepard function. Dividing by the volume of the element makes all elements related to the RUC have equal contribution to the sensitivity, regardingless of total strain energy in the elements. The resulting sensitivity numbers of points will be used to update the topology of the structure for RUC. With the updated design variables of all unstructured points, the relative density of jth element in ith unit cell is determined by the density values of design points within a circular influence domain Ω2. X point point xi; j ¼ w xi; j xi; j ð5Þ
Fig. 3 Interpolation for sensitivity number and density
j∈Ω2
(a) Model of 2D trapezoidal bridge
(b) FE mesh of the design domain
(c) Distribution of desin points
(d) Optimal design of 2D bridge
4000
1
3650
0.9 0.8
3300 Volfrac
0.7
2600
0.6
2250
0.5
1900
0.4
1550
0.3
1200
0
10
20
30
40
50
60
70
volfrac
C (N-mm))
C
2950
0.2 80
iteration
(e) Evolution histories of mean compliance and volume fraction Fig. 4 The topology optimization problem for 2D trapezoidal bridge
Struct Multidisc Optim
With the convensional ESO/BESO methods, large oscillations are often observed in the evolutionary history of the objective functions. The reason for such chaotic behaviour is that the sensitivity numbers of solid (1) and void (0) elements are based on discrete design varibales of element presence (1) and abpresense (0). This makes the objective function and the topology difficult to converge. And it has found that averaging the sensitivity number with its historical information is an effective way to solve this problem.
(a) Model of 3D cuneiform beam
The sensitivity numbers of the jth point in k iteration is averaged to avoid chaos. point
point
^j α
V kþ1 ¼ max V k ð1−ERÞ; V *
(d) Optimal design of 3D beam
5 x 10
1
3.2
0.9
2.9
0.8
0.6
2
0.5
1.7
0.4
1.4
0.3
1.1
0.2
0.8
0.1 0
10
20
30
40 iteration
50
60
70
volfrac
0.7 C Volfrac
2.3
0.5
ð7Þ
(b) FE mesh of the design domain
2.6 C (N-mm))
ð6Þ
2
The volume for the next iteration is determined by
(c) Distribution of desin points 3.5
¼
^ j;k−1 αpoint j;k þ α
0 80
(e) Evolution histories of mean compliance and volume fraction Fig. 5 The optimization problem for 3D cuneiform beam
Topology optimization of periodic structures using BESO
where ER is evolution rate and V* denotes the objective volume. The threshold of the sensitivity number, αth, is determined by using the bi-section method, and the design variable of the jth point is then updated by ( ^ j ≥ αth 1 when α point xj ¼ ð8Þ ^ j < αth xmin when α The following convergence criterion is applied. τmax is an allowable convergence error and it is set to be 0.1 % in this paper. k denotes the current iteration number and N is 5 to obtain a stable compliance in ten successive iterations. N X ðC k−iþ1 −C k−N −iþ1 Þ i¼1 τ¼ ≤ τ max ð9Þ N X C k−iþ1 i¼1
4 Numerical examples For both numerical examples, the volume fraction is 30 % of the design domain and the evolution ratio is 2 %. 4.1 Topology optimization of a periodic 2D trapezoidal bridge The proposed method is applied to find the optimal design of a periodic 2D trapezoidal bridge as shown in Fig. 4. 4.2 Topology optimization of a 3D cuneiform beam As shown in Fig. 5, a topology optimization problem for a 3D cuneiform beam is conducted.
5 Conclusions A novel topology optimization method for periodic structures with different geometies and irregular meshes is proposed in this paper, and its effectiveness is demonstrated by the numerical examples.
Acknowledgments This work is supported by the Key Project of NSFC (61232014) and the Science Fund of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body No. 31115010.
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