NURBS-based Optimization For Bolt Hole Geometry Design
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摘要:
基于非均匀有理B样条（NURBS）曲线，提出了一种螺栓孔设计优化方法，采用极点式双段NURBS曲线构建螺栓孔三维模型，以降低螺栓孔最大当量应力为优化目标，以极点坐标为设计优化变量，约束挡板最大径向位移、螺栓孔最大周向应力和螺栓压紧面积，建立了螺栓孔优化数学模型，采用强化学习差分进化（QLDE）算法开展优化，获得了螺栓孔的一种优化结构，计算表明：优化后的螺栓孔最大当量应力降低了19.0%，最大周向应力降低了12.7%，挡板最大径向位移减小了0.3%，螺栓压紧面积增大了0.5%，挡板低循环疲劳寿命提高了122%。试验验证了设计优化方法的有效性，低循环疲劳试验寿命显著提高，解决了挡板螺栓孔因低循环疲劳试验寿命低导致过早出现裂纹的失效问题。
Abstract:A bolt hole design optimization method based on Non-Uniform Rational B-Splines (NURBS) curves was proposed. The three-dimensional model of the bolt hole was constructed using a two-segment NURBS curve with pole-based control, with the optimization objective of minimizing the maximum equivalent stress. The pole coordinates served as design variables, while constraints included the maximum radial displacement of the baffle, maximum circumferential stress of the bolt hole, and bolt clamping area. A mathematical optimization model for the bolt hole was established, and the enhanced learning differential evolution (QLDE) algorithm was employed for optimization. The results showed that the optimized bolt hole achieved a 19.0% reduction in maximum equivalent stress, a 12.7% decrease in maximum circumferential stress, a 0.3% reduction in maximum radial displacement of the baffle, and a 0.5% increase in bolt clamping area, while the low-cycle fatigue life of the baffle improved by 122%. Experimental validation confirmed the effectiveness of the proposed method, with significantly enhanced low-cycle fatigue life. This addressed the failure issue of premature crack initiation in baffle bolt holes due to inadequate low-cycle fatigue life.
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Key words:
- Bolt holes /
- NURBS curves /
- design optimization /
- enhanced learning /
- differential evolution algorithm
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图 5 螺栓孔优化前后结构对比
Figure 5. Structural comparison before and after bolt hole optimization
图 6 螺栓孔优化前后上孔边当量应力分布（单位：兆帕）
Figure 6. Equivalent stress distribution at the upper hole edge before and after bolt hole optimization （Unit：MPa）
图 7 螺栓孔优化前后上孔边周向应力分布（单位：兆帕）
Figure 7. Circumferential stress distribution at the upper hole edge before and after bolt hole optimization （Unit：MPa）
图 8 螺栓孔优化前后下孔边当量应力分布（单位：兆帕）
Figure 8. Equivalent stress distribution at the lower hole edge before and after bolt hole optimization （Unit：MPa）
图 9 螺栓孔优化前后下孔边周向应力分布（单位：兆帕）
Figure 9. Circumferential stress distribution at the lower hole edge before and after bolt hole optimization （Unit：MPa）
图 10 螺栓孔优化前后主应力分布（单： 兆帕）
Figure 10. Principal stress distribution before and after bolt hole optimization （Unit：MPa）
图 11 后挡板优化前后径向变形分布（单位：毫米）
Figure 11. Radial deformation distribution of the backplate before and after optimization （Unit：mm）
图 14 故障件荧光显示及裂纹形貌
Figure 14. Fluorescent indication and crack morphology of the failed component
表 1 测试函数集
Table 1. Test functions
函数 范围 维度 $ {F}_{2} （\boldsymbol{x}） ={10}^{6}{x}_{1}^{2}+\displaystyle\sum\limits_{i=2}^{d}{x}_{i}^{2} $ $ {F}_{2} （\boldsymbol{x}） ={10}^{6}{x}_{1}^{2}+\displaystyle\sum\limits_{i=2}^{d}{x}_{i}^{2} $ 30 $ {F}_{2} （\boldsymbol{x}） ={10}^{6}{x}_{1}^{2}+\displaystyle\sum\limits_{i=2}^{d}{x}_{i}^{2} $ $ {F}_{3} （\boldsymbol{x}） =\displaystyle\sum\limits_{i=1}^{d}\left| {x}_{i}\right| +\prod\limits_{i=1}^{d}\left| {x}_{i}\right| $ 30 $ {F}_{3} （\boldsymbol{x}） =\displaystyle\sum\limits_{i=1}^{d}\left| {x}_{i}\right| +\prod\limits_{i=1}^{d}\left| {x}_{i}\right| $ $ {F}_{4} （\boldsymbol{x}） =\displaystyle\sum\limits_{i=1}^{d}{ （{{x}_{i}}+0.5） }^{2} $ 30 $ {F}_{4} （\boldsymbol{x}） =\displaystyle\sum\limits_{i=1}^{d}{ （{{x}_{i}}+0.5） }^{2} $ $ {F}_{5} （\boldsymbol{x}） ={\left| \displaystyle\sum\limits_{i=1}^{d}{x}_{i}^{2}-d\right| }^{1/4}+\left(0.5\displaystyle\sum\limits_{i=1}^{d}{x}_{i}^{2}+\displaystyle\sum\limits_{i=1}^{d}{x}_{i}\right)/d+0.5 $ 30 $ {F}_{5} （\boldsymbol{x}） ={\left| \displaystyle\sum\limits_{i=1}^{d}{x}_{i}^{2}-d\right| }^{1/4}+\left(0.5\displaystyle\sum\limits_{i=1}^{d}{x}_{i}^{2}+\displaystyle\sum\limits_{i=1}^{d}{x}_{i}\right)/d+0.5 $ $ {F}_{6} （\boldsymbol{x}） =\displaystyle\sum\limits_{i=1}^{d}\dfrac{x_{i}^{2}}{4\;000}-\prod\limits_{i=1}^{d}\cos \left(\dfrac{{x}_{i}}{\sqrt{i}}\right)+1 $ 30 $ {F}_{6} （\boldsymbol{x}） =\displaystyle\sum\limits_{i=1}^{d}\dfrac{x_{i}^{2}}{4\;000}-\prod\limits_{i=1}^{d}\cos \left(\dfrac{{x}_{i}}{\sqrt{i}}\right)+1 $ $ {F}_{7} （\boldsymbol{x}） =\displaystyle\sum\limits_{i=1}^{d}i{x}_{i}^{4}+{\mathrm{random}} （0,1） $ 30 $ {F}_{7} （\boldsymbol{x}） =\displaystyle\sum\limits_{i=1}^{d}i{x}_{i}^{4}+{\mathrm{random}} （0,1） $ $ {F}_{8} （\boldsymbol{x}） =4{x}_{1}^{2}-2.1{x}_{1}^{4}+\dfrac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4} $ 30 $ {F}_{8} （\boldsymbol{x}） =4{x}_{1}^{2}-2.1{x}_{1}^{4}+\dfrac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4} $ $ {F}_{9} （\boldsymbol{x}） =\displaystyle\sum\limits_{i=1}^{11}{\left[{{a}_{i}}-\dfrac{{x}_{1} （{b}_{i}^{2}+{b}_{i}{x}_{2}） }{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}\right]}^{2} $ 2 $ {F}_{9} （\boldsymbol{x}） =\displaystyle\sum\limits_{i=1}^{11}{\left[{{a}_{i}}-\dfrac{{x}_{1} （{b}_{i}^{2}+{b}_{i}{x}_{2}） }{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}\right]}^{2} $ $ X={[{{x}_{1}},{{x}_{2}},\cdots ,{{x}_{16}}]}^{{\mathrm{T}}},X\subset {R}^{16} $ 4 
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表 2 测试结果
Table 2. Test results
FUN QLDE DE PSO SA F1 平均值 4.69×10−5 9.38×10−1 1.66×10−1 6.43×10−4 平均误差 4.69×10−5 9.38×10−1 1.66×10−1 6.43×10−4 中位数 1.47×10−5 9.16×10−1 7.76×10−2 1.49×10−4 最好值 1.88×10−7 4.65×10−1 1.20×10−2 5.80×10−6 标准差 1.06×10−4 2.67×10−1 1.99×10−1 1.40×10−3 F2 平均值 2.53×10−4 2.17×100 4.00×102 1.44×10−3 平均误差 2.53×10−4 2.17×100 4.00×102 1.44×10−3 中位数 2.70×10−5 2.02×100 2.39×10−1 2.89×10−4 最好值 1.10×10−6 1.18×100 2.89×10−2 6.45×10−6 标准差 5.69×10−4 5.87×10−1 1.98×103 3.25×10−3 F3 平均值 1.15×10−3 2.04×10−1 1.28×100 2.70×10−3 平均误差 1.15×10−3 2.04×10−1 1.28×100 2.70×10−3 中位数 6.56×10−4 2.00×10−1 6.32×10−2 2.23×10−3 最好值 6.89×10−5 1.32×10−1 2.86×10−2 5.16×10−4 标准差 1.39×10−3 2.93×10−2 3.27×100 1.73×10−3 F4 平均值 1.14×10−4 8.96×10−1 1.48×10−1 6.98×10−4 平均误差 1.14×10−4 8.96×10−1 1.48×10−1 6.98×10−4 中位数 1.10×10−5 9.22×10−1 8.64×10−2 2.28×10−4 最好值 5.41×10−7 4.12×10−1 1.29×10−2 3.21×10−5 标准差 3.81×10−4 2.07×10−1 1.93×10−1 1.07×10−3 F5 平均值 5.55×10−1 6.72×10−1 6.86×10−1 5.26×10−1 平均误差 5.55×10−1 6.72×10−1 6.86×10−1 5.26×10−1 中位数 5.29×10−1 6.65×10−1 6.63×10−1 5.12×10−1 最好值 3.28×10−1 5.10×10−1 4.03×10−1 2.88×10−1 标准差 1.40×10−1 6.67×10−2 1.55×10−1 1.44×10−1 F6 平均值 1.27×10−2 8.48×10−1 2.22×10−1 6.98×10−2 平均误差 1.27×10−2 8.48×10−1 2.22×10−1 6.98×10−2 中位数 7.46×10−3 8.39×10−1 1.62×10−1 5.89×10−2 最好值 1.74×10−6 7.03×10−1 3.81×10−2 4.15×10−5 标准差 1.57×10−2 6.31×10−2 1.44×10−1 5.77×10−2 F7 平均值 2.79×10−2 9.84×10−2 5.36×10−2 2.28×10−1 平均误差 2.79×10−2 9.84×10−2 5.36×10−2 2.28×10−1 中位数 2.60×10−2 9.81×10−2 5.18×10−2 2.14×10−1 最好值 7.36×10−3 5.69×10−2 2.56×10−2 8.01×10−2 标准差 1.11×10−2 1.84×10−2 1.79×10−2 9.24×10−2 F8 平均值 −1.03×100 −1.03×100 −1.03×100 −1.03×100 平均误差 2.25×10−15 2.23×10−15 2.30×10−15 7.56×10−11 中位数 −1.03×100 −1.03×100 −1.03×100 −1.03×100 最好值 −1.03×100 −1.03×100 −1.03×100 −1.03×100 标准差 2.67×10−16 2.44×10−16 3.19×10−16 8.46×10−11 F9 平均值 3.62×10−4 7.82×10−4 1.44×10−3 1.39×10−2 平均误差 5.49×10−5 4.74×10−4 1.14×10−3 1.36×10−2 中位数 3.07×10−4 7.69×10−4 6.37×10−4 1.76×10−3 最好值 3.07×10−4 6.14×10−4 3.09×10−4 5.17×10−4 标准差 2.20×10−4 1.28×10−4 3.91×10−3 1.80×10−2
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表 3 弗里德曼检验统计结果
Table 3. The Friedman test results
Friedman QLDE DE PSO SA F1 1 4 3 2 F2 1 3 4 2 F3 1 3 4 2 F4 1 4 3 2 F5 2 3 4 1 F6 1 4 3 2 F7 1 3 2 4 F8 2.5 2.5 2.5 2.5 F9 1 2 3 4 Average 1.3 3.2 3.2 2.4 Ranks 1 3.5 3.5 2
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表 4 优化设计变量
Table 4. Optimization design variables
序号 参数名 初始值 下限 上限 1 x1/mm 55.5 53.0 57.0 2 x2/mm 56.3 54.0 59.0 3 x3/mm 57.7 56.0 58.0 4 x4/mm 59.0 57.0 61.0 5 x5/mm 60.0 58.0 63.0 6 x6/mm 32.5 32.0 33.8 7 x7/mm 53.2 51.0 56.0 8 x8/mm 52.0 50.0 54.0 9 x9/mm 51.0 49.0 53.0 10 y1/mm 5.6 5.0 6.0 11 y2/mm 5.0 4.7 5.5 12 y3/mm 4.7 4.5 5.2 13 y4/mm 4.2 4.0 4.5 14 y5/mm 5.9 5.2 6.3 15 y6/mm 5.8 5.5 6.4 16 y7/mm 4.5 4.0 5.0
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表 5 优化结果计算值
Table 5. Calculated values of the optimization result
参数 优化前 优化后 变化率/% 螺栓压紧面积/mm2 68.001 68.312 +0.5 螺栓孔最大当量应力/MPa 1754.2 1420.1 −19.0 螺栓孔最大主应力/MPa 1800.4 1427.4 −20.7 螺栓孔最大周向应力/MPa 1615.7 1410.9 −12.7 挡板最大径向位移/mm 0.6968 0.6950 −0.3 低循环疲劳寿命/次 6204 13789 +122
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表 6 低循环疲劳试验阶段
Table 6. Low-cycle fatigue life test phase
试验阶段 试验次数 第1阶段 0～ 4000 第2阶段 4000 ～8000 第3阶段 8000 ～12000
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表 7 优化前后低循环疲劳寿命次数
Table 7. Low-cycle fatigue life cycles before and after optimization
参数 优化前 优化后 变化率 低循环疲劳寿命/次 < 8000 > 12000 >+50%
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[1] 韩佳欣. 基于连续/离散变量的异型孔多目标均衡优化研究[D]. 南京: 南京航空航天大学, 2018. HAN Jiaxin. Balanced multi-objective optimization of non-circular hole based on continuous & discrete variables[D]. Nanjing: Nanjing University of Aeronautics and Astronautics, 2018. (in ChineseHAN Jiaxin. Balanced multi-objective optimization of non-circular hole based on continuous & discrete variables[D]. Nanjing: Nanjing University of Aeronautics and Astronautics, 2018. (in Chinese) [2] 王银虎. 发动机典型螺栓连接结构综合优化设计研究[D]. 沈阳: 沈阳航空航天大学, 2023. WANG Yinhu. Comprehensive optimization design research on typical bolted connections in engines[D]. Shenyang: Shenyang Aerospace University, 2023. (in ChineseWANG Yinhu. Comprehensive optimization design research on typical bolted connections in engines[D]. Shenyang: Shenyang Aerospace University, 2023. (in Chinese) [3] 陈景阳, 李百洋. 冲击载荷下法兰连接螺栓结构强度评估方法[J]. 航空动力学报, 2025, 40(5): 20230560. CHEN Jingyang, LI Baiyang. Strength evaluation method of flange connection bolt structure under impact load[J]. Journal of Aerospace Power, 2025, 40(5): 20230560. (in Chinese doi: 10.13224/j.cnki.jasp.20230560CHEN Jingyang, LI Baiyang. Strength evaluation method of flange connection bolt structure under impact load[J]. Journal of Aerospace Power, 2025, 40(5): 20230560. (in Chinese) doi: 10.13224/j.cnki.jasp.20230560 [4] 刘丁. 基于薄层单元法的航空发动机螺栓连接结构仿真方法研究[D]. 天津: 中国民航大学, 2019. LIU Ding. Research on simulation method of aero-engine bolted connection structure based on thin-layer element method[D]. Tianjin: Civil Aviation University of China, 2019. (in ChineseLIU Ding. Research on simulation method of aero-engine bolted connection structure based on thin-layer element method[D]. Tianjin: Civil Aviation University of China, 2019. (in Chinese) [5] 焦俊杰, 莫蓉, 徐广庆, 等. 螺栓孔的位置度误差对短精密螺栓连接结构装配力学特性的影响[J]. 航空动力学报, 2021, 36(5): 935-947. JIAO Junjie, MO Rong, XU Guangqing, et al. Influence of position error of bolt hole on assembly mechanical characteristics of short precision bolted connection structure[J]. Journal of Aerospace Power, 2021, 36(5): 935-947. (in ChineseJIAO Junjie, MO Rong, XU Guangqing, et al. Influence of position error of bolt hole on assembly mechanical characteristics of short precision bolted connection structure[J]. Journal of Aerospace Power, 2021, 36(5): 935-947. (in Chinese) [6] 罗丰, 邓旺群, 吕彪, 等. 航空发动机螺栓止口连接转子动力学建模及特性[J]. 哈尔滨工业大学学报, 2025, 57(12): 219-228. LUO Feng, DENG Wangqun, LÜ Biao, et al. Modeling and characteristics of aero-engine rotor with bolted-rabbet joints[J]. Journal of Harbin Institute of Technology, 2025, 57(12): 219-228. (in ChineseLUO Feng, DENG Wangqun, LÜ Biao, et al. Modeling and characteristics of aero-engine rotor with bolted-rabbet joints[J]. Journal of Harbin Institute of Technology, 2025, 57(12): 219-228. (in Chinese) [7] 龚梦贤, 黄庆东, 肖育祥, 等. 某Ⅰ级涡轮盘低循环疲劳寿命试验研究[J]. 航空动力学报, 1999, 14(4): 361-365, 450-451. GONG Mengxian, HUANG Qingdong, XIAO Yuxiang, et al. An experimental study on low cycle fatigue life of 1st turbine disk in aeroengine[J]. Journal of Aerospace Power, 1999, 14(4): 361-365, 450-451. (in ChineseGONG Mengxian, HUANG Qingdong, XIAO Yuxiang, et al. An experimental study on low cycle fatigue life of 1st turbine disk in aeroengine[J]. Journal of Aerospace Power, 1999, 14(4): 361-365, 450-451. (in Chinese) [8] 杨俊, 李承彬, 谢寿生. 涡轮盘多轴低循环疲劳寿命预测及试验验证[J]. 航空动力学报, 2011, 26(10): 2220-2226. YANG Jun, LI Chengbin, XIE Shousheng. Multiaxial low cycle fatigue life prediction and test verification for turbine disk[J]. Journal of Aerospace Power, 2011, 26(10): 2220-2226. (in ChineseYANG Jun, LI Chengbin, XIE Shousheng. Multiaxial low cycle fatigue life prediction and test verification for turbine disk[J]. Journal of Aerospace Power, 2011, 26(10): 2220-2226. (in Chinese) [9] 李伟, 董立伟, 蔡向晖, 等. 某型发动机涡轮盘销钉孔结构分析与寿命评估[J]. 航空动力学报, 2009, 24(8): 1699-1706. LI Wei, DONG Liwei, CAI Xianghui, et al. Structure analysis and life evaluation of the pin holes in a turbine disc of a type of aero-engine[J]. Journal of Aerospace Power, 2009, 24(8): 1699-1706. (in ChineseLI Wei, DONG Liwei, CAI Xianghui, et al. Structure analysis and life evaluation of the pin holes in a turbine disc of a type of aero-engine[J]. Journal of Aerospace Power, 2009, 24(8): 1699-1706. (in Chinese) [10] 陆山, 黄其青. 涡轮盘销钉孔损伤容限分析新方法及其应用[J]. 航空动力学报, 2002, 17(1): 87-92. LU Shan, HUANG Qiqing. New method for damage tolerance analysis of turbine disk and its application[J]. Journal of Aerospace Power, 2002, 17(1): 87-92. (in ChineseLU Shan, HUANG Qiqing. New method for damage tolerance analysis of turbine disk and its application[J]. Journal of Aerospace Power, 2002, 17(1): 87-92. (in Chinese) [11] 高阳, 白广忱, 张瑛莉. 涡轮盘多轴低循环疲劳寿命可靠性分析[J]. 航空学报, 2009, 30(9): 1678-1682. GAO Yang, BAI Guangchen, ZHANG Yingli. Reliability analysis of multiaxial low cycle fatigue life for turbine disk[J]. Acta Aeronautica et Astronautica Sinica, 2009, 30(9): 1678-1682. (in ChineseGAO Yang, BAI Guangchen, ZHANG Yingli. Reliability analysis of multiaxial low cycle fatigue life for turbine disk[J]. Acta Aeronautica et Astronautica Sinica, 2009, 30(9): 1678-1682. (in Chinese) [12] 陈光. 航空发动机结构设计分析[M]. 北京: 北京航空航天大学出版社, 2006. CHEN Guang. Structural design analysis of aero-engine[M]. Beijing: Beijing University of Aeronautics & Astronautics Press, 2006. (in ChineseCHEN Guang. Structural design analysis of aero-engine[M]. Beijing: Beijing University of Aeronautics & Astronautics Press, 2006. (in Chinese) [13] 陈秋任, 郭海丁, 刘小刚. 涡轮盘双轴对称异形孔结构建模与优化[J]. 航空动力学报, 2013, 28(6): 1250-1256. CHEN Qiuren, GUO Haiding, LIU Xiaogang. Modeling and optimization for the structure of biaxial symmetry non-circular hole of turbine disk[J]. Journal of Aerospace Power, 2013, 28(6): 1250-1256. (in ChineseCHEN Qiuren, GUO Haiding, LIU Xiaogang. Modeling and optimization for the structure of biaxial symmetry non-circular hole of turbine disk[J]. Journal of Aerospace Power, 2013, 28(6): 1250-1256. (in Chinese) [14] 韩佳欣, 郭海丁. 轮盘超椭圆异型螺栓孔均衡优化设计[J]. 航空发动机, 2018, 44(2): 57-63. HAN Jiaxin, GUO Haiding. Equilibrium optimize design of hyper-elliptic non-circular bolt hole of disk[J]. Aeroengine, 2018, 44(2): 57-63. (in ChineseHAN Jiaxin, GUO Haiding. Equilibrium optimize design of hyper-elliptic non-circular bolt hole of disk[J]. Aeroengine, 2018, 44(2): 57-63. (in Chinese) [15] RIESENFELD R F. Applications of B-Spline approximation to geometric problems of CAD, Ph. D Thesis, New York: Syracuse University, March 1973 [16] RIESENFELD R F. Non-uniform B-spline[C]// Proceeding of 2nd USA-JAPAN Computer Conference, AFIPS, 1975: 551-555 [17] VERSPRILLE K J. Computer-aided design applications of rational B-Spline approximation form, Ph, D Disernation, New York: Syracuse University, Februray, 1974 [18] GORDON W J, RIESENFELD R F. B-spline curves and surfaces[C]//Computer Aided Geometry Design. New York: Academtric Press, 1974. [19] FUHR R D. Rational B-spline representation of curves and surfaces[J]. IEEE Computer Graphics and Applications, 1981, 3(6): 61-69. [20] PIEGL L A, TILLER W. The NURBS Book[M]. Berlin: Springer, 1995. [21] STORN R. Differential evolution design of an IIR-filter[C]//Proceedings of IEEE International Conference on Evolutionary Computation. Piscataway, US: IEEE, 2002: 268-273. [22] STORN R, PRICE K. Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces[J]. Journal of Global Optimization, 1997, 11(4): 341-359. [23] WATKINS C J C H, DAYAN P. Q-learning[J]. Machine Learning, 1992, 8(3): 279-292. [24] 朱心雄. 自由曲线曲面造型技术[M]. 北京: 科学出版社, 2000. ZHU Xinxiong. Free-form curve and surface modeling technology[M]. Beijing: Science Press, 2000. (in ChineseZHU Xinxiong. Free-form curve and surface modeling technology[M]. Beijing: Science Press, 2000. (in Chinese) [25] ZAMLI K Z, DIN F, AHMED B S, et al. A hybrid Q-learning sine-cosine-based strategy for addressing the combinatorial test suite minimization problem[J]. PLoS One, 2018, 13(5): e0195675. [26] HAMAD Q S, SAMMA H, SUANDI S A, et al. Q-learning embedded sine cosine algorithm (QLESCA)[J]. Expert Systems with Applications, 2022, 193: 116417. [27] DERRAC J, GARCÍA S, MOLINA D, et al. A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms[J]. Swarm and Evolutionary Computation, 2011, 1(1): 3-18. [28] COFFIN L F Jr. A study of the effects of cyclic thermal stresses on a ductile metal[J]. Journal of Fluids Engineering, 1954, 76(6): 931-949. [29] SMITH K N, WATSON P, TOPPER T H. A stress-strain function for the fatigue of materials[J]. Journal of Materials, 1970, 5: 767-778. [30] 陈欢欢, 胡殿印, 王贵灿, 等. 基于三参数临界距离模型的轮盘偏心孔低周疲劳寿命预测[J/OL]. 航空动力学报. (2026-01-06)[2026-01-12]. https://doi.org/10.1322/j.cnki,jasp.20250366. CHEN Huanhuan, HU Dianyin, WANG Guican, et al. Prediction of low-cycle fatigue life of eccentric holes in disk based on a three-parameter critical distance model. Journal of Aerospace Power. (2026-01-06)[2026-01-12]. https://doi.org/10.1322/j.cnki,jasp.20250366. (in ChineseCHEN Huanhuan, HU Dianyin, WANG Guican, et al. Prediction of low-cycle fatigue life of eccentric holes in disk based on a three-parameter critical distance model. Journal of Aerospace Power. (2026-01-06)[2026-01-12]. https://doi.org/10.1322/j.cnki,jasp.20250366. (in Chinese) -
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