Hole Appearance Constraint Method in 2D Structural Topology Optimization
Abstract
:1. Introduction
2. Implementation of Topology Optimization
2.1. The Topological Optimization Model
2.2. Workflow
3. Hole Extraction and Feature Shape Measurement Algorithm
3.1. Extraction of Hole Shape Information
3.2. Calculating the Area and Centroid of Holes
3.3. Calculating the Elongation of Holes
3.4. Best Equivalent Shape Matching
3.4.1. Measurement Method for Shape Similarity of Equivalent Shapes
- Rectangularity Measurement
- Isosceles Triangle Measurement
- Ellipticity Measurement
3.4.2. Method for Calculating Equivalent Shapes
- The centroids of the two shapes should coincide;
- The areas of the two shapes should be identical;
- The two shapes should possess equivalent aspect ratios;
- The two shapes should have the maximum possible overlapping area.
Algorithm 1: Algorithm to obtain the MES of a hole. |
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4. Hole Number Control
5. Appearance Target Image and Minimum Scale between Holes Control
5.1. Generate Appearance Target Image
- Obtain the grayscale image, as depicted in Figure 10a, using classical topology optimization methods.
- Process the obtained image through binarization to achieve Figure 10b.
- Fill the redundant holes. At this point, we defined .
- After determining the MES for all holes, obtain the results depicted in Figure 10c.
- Map the MES onto the binary image to obtain the appearance target image, as shown in Figure 10d, where interference between holes becomes apparent.
- Before performing hole shrinking on interfering holes, expand the holes in the appearance target image by k pixels using morphological dilation operations to generate an interference assessment map, as illustrated in Figure 10e. This step aims to prevent the holes being too close to or touching each other.
- Identify all interfering holes based on the interference assessment map and replace the equivalent shapes of interfering holes in the appearance target image with their original hole shapes. Use morphological erosion operations to reduce the size of the holes by one pixel. This process results in an updated appearance target image shown in Figure 10f.
5.2. Appearance Constraints
6. Results and Discussion
6.1. Example 1
6.2. Example 2
6.3. Example 3
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Zhu, L.; Zuo, T.; Wang, C.; Wang, Q.; Yu, Z.; Liu, Z. Hole Appearance Constraint Method in 2D Structural Topology Optimization. Mathematics 2024, 12, 2645. https://doi.org/10.3390/math12172645
Zhu L, Zuo T, Wang C, Wang Q, Yu Z, Liu Z. Hole Appearance Constraint Method in 2D Structural Topology Optimization. Mathematics. 2024; 12(17):2645. https://doi.org/10.3390/math12172645
Chicago/Turabian StyleZhu, Lei, Tongxing Zuo, Chong Wang, Qianglong Wang, Zhengdong Yu, and Zhenyu Liu. 2024. "Hole Appearance Constraint Method in 2D Structural Topology Optimization" Mathematics 12, no. 17: 2645. https://doi.org/10.3390/math12172645
APA StyleZhu, L., Zuo, T., Wang, C., Wang, Q., Yu, Z., & Liu, Z. (2024). Hole Appearance Constraint Method in 2D Structural Topology Optimization. Mathematics, 12(17), 2645. https://doi.org/10.3390/math12172645