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作者: 許銘修(2002-10-17);推薦:徐業良(2002-10-18)
附註:本文發表於第二十六屆全國力學會議。

Integrating 2-D topology and shape optimization using the density contour approach

Abstract

In this research, a process for integrating 2-D structural topology optimization and shape optimization using the density contour approach is presented. The critical issue is how to interpret topology optimization result into a representation that can be used in shape optimization. In the interpretation process presented in this research, a continuity analysis is first used to filter out the so-called “porous topology” formed by the elements with intermediate densities in the topology optimization result. Next, trivial solids and trivial voids are also filtered out. Finally the black-and-white finite element topology optimization result is transferred into a smooth density contour represented by B-spline curves, which will be used as the initial design of shape optimization. 14 two-dimensional structural topology optimization examples commonly seen in literature are used to demonstrate the integration process.

Keywords: Topology optimization, shape optimization, density contour approach

The topology optimization process

In structural shape optimization, as in most design optimization problems, an initial design model has to be defined first. During the optimization process, the size and shape of the structure may be altered, but the topology of the structure is not changed. If the topology of the initial design model is not optimal, the result after shape optimization is only the optimal design under this non-optimal topology. Therefore, using the topology optimization result as the initial design of the shape optimization is a natural concept.

After the important advance by Bendsfe and Kikuchi [1] in the field of topology optimization, many researchers paid attention to integrating structural topology optimization and shape optimization since early 1990. Some researchers treat structural design optimization as a three-phase design process, and aim at developing an automated design process [2, 3, 4]. Phase I is the topology generation process. The optimal topology of the structure is generated in this phase. Phase II is the topology interpretation process. In this process various approaches are used to interpret the topology optimization result. Phase III is the detail design phase. The shape and size optimization are implemented in phase III.

The approaches used in the Phase II interpretation process can be roughly divided into three categories: image interpretation approach [2, 3, 4, 5, 6, 7], density contour approach [8, 9, 10], and geometric reconstruction approach [11]. The image interpretation approach uses graphic facilities or computer vision technologies to represent the boundary of the black-and-white finite element topology optimization result. The density contour approach generates the boundaries of the structure by redistributing densities from the topology optimization result. In the geometric reconstruction approach, the boundaries are represented by mathematical geometric reconstruction technique.

In this research, the concept of three-phase design process is adopted. Density contour approach is used. Figure 1 shows the phase II interpretation process that connects topology optimization and shape optimization. The normalized material densities ri,final are obtained from topology optimization. The continuity analysis is implemented in order to filter out the porous topologies. In the analysis, a density value rcontinuous is to be decided so that a continuous topology can be obtained if only elements whose normalized material densities are equal or greater than rcontinuous are considered. The normalized material densities of these elements are set to be 1. On the other hand, the normalized material densities which are less than rcontinuous are set to be 0.

Figure 1 The interpretation process

 

After the continuity analysis, two “filters” are implemented to filter out the trivial solids and trivial voids in the topology result. A density redistribution algorithm then evaluates the nodal density values from the element normalized material densities. Spline curves are used to generate the boundary curves using the boundary nodes as control points. Thus, the black-and-white finite element topology result is represented by smooth B-spline curves, which will be used as the initial design of shape optimization.

The continuity of the topology optimization result

Topology optimization with penalty function attempts to generate the result with elements whose normalized material densities are either 0 or 1. But practically the “porous topology” [12], which is formed by elements with intermediate normalized material densities between 0 and 1, often occurs. From engineering point of view, the porous topology is meaningless because materials with intermediate densities do not exist.

Practically the most convenient way is to force the intermediate normalized material densities which are equal to or grater than a given threshold value to 1 directly. On the other hand, the densities that are less than the given threshold value are forced to be 0. The continuity of the structure should be the most important consideration in deciding this threshold value. As shown in Figure 2, if the threshold value is high and a portion of porous topology A is filtered out, a discontinuous structure will be generated, which may greatly affect the compliance of the structure. On the other hand, the continuity of the structure is not affected if porous topology B is filtered out.

Figure 2 The illustration of the porous topology

A structure with continuous topology is stiffer than discontinuous one, in other words, the compliance of the structure with continuous topology is lower. Thus, a proper threshold density value rcontinuous is to be decided by comparing the compliances of the structure using different threshold density values.

Figure 3 shows the compliances of the cantilever beam example using different threshold density values. The x-axis is the threshold density value starting from 0.1 to 1.0, with increment 0.1. Using the compliance when the threshold density is 0.1 as the base, the increasing percentages of the compliances are plotted. The respective topology results are shown in Table 1.

As shown in Figure 3, there are 6 “jumps” in compliance when the threshold density value increases. Compare with the figures in Table 1, there is a more than 60% increase (jump 2) in compliance when the threshold density value increase from 0.5 to 0.6, because a discontinuity of the topology occurs. Other discontinuities occur at jump 4, 5 and 6. Although there is no discontinuity occurs in jump 1 and jump 3, but the material in critical connection of the topology decreases. In this research, if the increase in compliance is more than 10% at certain threshold density value, this threshold density value is set to be rcontinuous. In the cantilever beam example, rcontinuous is decided to be 0.2.

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Figure 3 The compliances of the cantilever beam example using different threshold densities

Figure 4 shows the compliances of the suspension arm example using different threshold density values. The respective topology results are shown in Table 2. As shown in Figure 4, there is a 24% increase in compliance when the threshold density value increase from 0.1 to 0.2, because a discontinuity of the topology occurs. Therefore, rcontinuous is decided to be 0.1.

Table 1 The topology of different threshold density

Topology

Topology

0.1

0.6

0.2

0.7

0.3

0.8

0.4

0.9

0.5

1.0

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Figure 4 The compliances of the suspension arm example using different threshold densities

Table 2 The topology of different threshold density of suspension arm example

Topology

Topology

0.1

0.6

0.2

0.7

0.3

0.8

0.4

0.9

0.5

1.0

 

Table 2 shows the rcontinuous values and the topology results after continuity analysis for the 14 examples discussed in the previous chapter. Note that in example 5, 6, 11, 12, 13, some structural connections in topology optimization results are deleted after the continuity analysis because they have very small influence on the compliance of the structure.

Table 3 The rcontinuous values and topology results after continuity analysis

 

The topology optimization result

Compliance analysis

rcontinuous

The topology results after continuity analysis

1

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1.0

2

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0.9

3

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1.0

4

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1.0

5

MATLAB Handle Graphics

1.0

6

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1.0

7

MATLAB Handle Graphics

0.1

8

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1.0

9

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1.0

10

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1.0

11

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1.0

12

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0.9

13

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1.0

14

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0.1

 

The trivial solid and void filter

After the continuity analysis, it is observed that trivial solids and voids appear in the topology result. Figure 5 shows the typical trivial solids, such as islands or small salient, in the cantilever beam example. These trivial solids are useless in mechanics and will complicate the implementation of the shape optimization. Thus, the trivial solids should be filter out before implementing shape optimization.

Trivial voids are the voids containing few elements whose normalized material densities are 0. Figure 5 and 6 show the trivial voids in the cantilever beam and suspension arm examples. Similar to trivial solids, trivial voids are useless in mechanics, will complicate the implementation of the shape optimization, and should also be filtered out.

Figure 5 The trivial solids and void in cantilever beam example

Figure 6 The trivial voids in the suspension arm example

In this research, a trivial solid and void filter is developed to filter those trivial topologies. The trivial solid is defined as the solid containing a group of elements that does not connect with other groups, or connects with other groups in only one node. Figure 7 is the result of the cantilever beam example after the trivial solid filter is applied.

Figure 7 The result of the cantilever beam example after the trivial solid filter

There can be many voids in the topology optimization result. In this research, if the number of element contained in a void is less than 1% of the total number of elements multiplied by r,0, the ratio of the amount of usable material, it is defined as a trivial void. Figure 4.8 shows the topology result of the suspension arm example after the trivial void filter. The checkerboard like trivial voids shown in Figure 6 are filtered out.

Figure 8 The result of suspension arm example after the trivial void filter

The density contour approach

Finally, we hope to transfer the topology optimization result into a smooth CAD model that can be used in shape optimization. As discussed earlier, Kumar and Gossard [8], Youn and Park [9] and Hsu et al. [10] use the density contour approach to interpret the topology optimization result to obtain smooth boundary. In their works, the element densities of topology optimization result are redistributed into nodal densities, and the nodal density contour is generated at a specified density value. This contour is then transferred into a smooth CAD representation (such as B-splines) of the structure. This research also adopts the density contour approach, as discussed below.

Youn and Park [9] evaluate the nodal density (x) by averaging the normalized material densities of the neighboring elements as follows:

                                                                                               (1)

where xk is the density of the k-th node, W is the number of neighboring elements at this node, and rk,e  is the normalized material density of the e-th neighboring element of the k-th node. In two dimensional topology optimization, almost all nodes have four neighboring elements, and thus W=4.

Figure 9 shows the three possibilities of nodal density values using Equation (1), x=0, x=1 and 0<x<1. If the node is inside a void, the normalized material densities of all neighboring elements are 0, and x=0. On the other hand, if the node is inside a solid, then x=1. For the nodes on the boundary between solid and void, the nodal density value is between 0 and 1. Figure 9 also shows three possible cases, x=0.25, x=0.5 and x=0.75.

Figure 9 The three types of nodal density value

Figure 10 shows the nodal densities and nodal density contours of the Case 1 in Figure 9. It is clear that the contour for x=0.5 properly describes the boundary of the solid. The nodal density contours are linearly interpolated from nodal densities. Spline curves, which are commonly used in CAD models, are generated to approximate the nodal density contour for x=0.5 to represent the structure. The nodes whose nodal density is 0.5 are control points of the spline curves directly. As shown in Figure 10, for the nodes whose nodal densities are 0.25 or 0.75, the positions of the corresponding control points are moved inward or outward from the node to the nodal density contour for x=0.5, using simple linear interpolation. For the cantilever beam example, figure 11 compares the nodal density contour for x=0.5 and the approximate nodal density contour using spline curves, which can be used in shape optimization.

MATLAB Handle Graphics

Figure 10 The nodal densities and nodal density contour of Case 1

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(a) The nodal density contour

(b) The approximate nodal density contour

Figure 11 Representing the topology result of the cantilever beam example using Spline curves

Figure 12 shows the resulting CAD model of the suspension arm example using the approximate nodal density contour. Table 4 shows the resulting CAD models of the 14 examples.

Figure 12 Representing the topology result of the suspension arm example using Spline curves

Table 4 Representing the topology result of the 14 examples using Spline curve

 

The topology result after continuity analysis

The topology result after filtering trivial voids out

Representing the topology result using Spline curve

1

2

3

4

5

6

7

8

9

10

11

12

13

14

 

Conclusion

A process for integrating 2-D structural topology optimization and shape optimization using the density contour approach is presented in this research. There are four steps to interpret the topology optimization result in the interpretation process. The continuity analysis is first used to filter the porous topology out. Second, trivial solids and trivial voids are also filtered out. Next, the nodal density value is evaluated from the normalized material density of elements by density redistribution algorithm. In the last step of the process, the black-and-white finite element topology optimization result is transferred into a smooth density contour represented by B-spline curves, which will be used as the initial design of shape optimization. In this research, 14 two-dimensional structural topology optimization examples are demonstrated to show the generality of this integration process. This integration process can be easily expanded to deal with 3-D structures.

References

[1]    Bendsfe, M. P., Kikuchi, N., 1988, “Generating Optimal Topologies in Structural Design Using Homogenization Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 71, pp. 197-244.

[2]    Bremicker, M., Chirehdast, M., Kikuchi, N., Papalambros, P. Y., 1991, “Integrated Topology and Shape Optimization in Structural Design,” Mechanics of Structures and Machines, Vol. 19, pp.551-587.

[3]    Chirehdast, M., Gea, H-C., Kikuchi, K., Papalambros, P. Y., 1994, “Structural Configuration Examples of an Integrated Optimal Design Process,” Journal of Mechanical Design, Vol. 116, pp. 997-1004.

[4]    Lin, C.-Y., Chao, L.-S., 2000, “Automated Image Interpretation for Integrated Topology and Shape Optimization,” Structural and Multidisciplinary Optimization, Vol. 20, pp.125-137.

[5]    Papalambros, P., Chirehdast, M., 1990, “An Integrated Environment for Structural Configuration Design,” Journal of Engineering Design, Vol. 1, pp. 73-96.

[6]    Bendsfe, M. P., Rodrigues, H. C., 1991, “Integrated Topology and Boundary Shape Optimization of 2-D Solid,” Computer Methods in Applied Mechanics and Engineering, Vol. 87, pp.15-34.

[7]    Olhoff, N., Bendsfe, M. P., Rasmussen, J., 1991, “On CAD-Integrated Structural Topology and Design Optimization,” Computer Methods in Applied Mechanics and Engineering, Vol. 89, pp. 259-279.

[8]    Kumar, A. V., Gossard, D. C., 1996, “Synthesis of Optimal Shape and Topology of Structures,” Journal of Mechanical Design, Vol. 118, pp. 68-74.

[9]    Youn, S. K., Park, S-H., 1997, “A Study on the Shape Extraction Process in the Structural Topology Optimization Using Homogenized Material,” Computers & Structures, Vol. 62, No. 3, pp. 527-538.

[10]Hsu, Y. L., Hsu, M. S., Chen, C. T., 2001, “Interpreting Results from Topology Optimization Using Density Contours,” Computers & Structures, Vol. 79, pp. 1049-1058.

[11]Tang, P.-S., Chang, K.-H., 2001, “Integration of Topology and Shape Optimization for Design of Structural Components,” Structural and Multidisciplinary Optimization, Vol. 22, pp.65-82.

[12]      Rozvany, G.I.N., Zhou, M., Birker, T., 1992, “Generalized Shape Optimization without Homogenization,” Structural Optimization, Vol. 4, pp. 250-252.

利用密度等高線法整合二維結構形式幾何最佳化與型態最佳化

摘要

本研究發展了一套利用密度等高線法整合二維結構形式幾何最佳化與型態最佳化的程序,研究的重點在於如何闡述形式幾何最佳化的結果,使其能為型態最佳化所使用。於闡述形式幾何最佳化的過程中,首先對其進行了連續性分析,用以找到一適合的門檻值,並強制所有元素密度值為01。接著,於形式幾何最佳化結果中,對整體結構剛性沒有貢獻之結構以及孔洞將予以去除;最後,將元素密度值完全為01的形式幾何最佳化結果,以B-spline曲線描繪出其密度等高線,此B-spline曲線所表示的結構即可作為型態最佳化的初始設計。本研究對14個文獻中常出現的二維結構最佳化範例進行了整合設計,用以展示此整合程序的可行性。

關鍵詞: 形式幾何最佳化, 型態最佳化, 密度等高線法