1 Introduction
2 Crack Growth Models
2.1 Forman’s Model
2.2 Elber’s Model
2.3 Kujawski’s Model
2.4 Huang’s Model
2.5 Zhan’s Model
2.6 NASGRO’s Model
3 Symbolic Regression
3.1 High-Performance Symbolic Regression in Python
3.2 Domain Knowledge-Guided
Model name | Equation |
---|---|
Paris’ model | \(\ln C + m \cdot \ln \Delta K\) |
Forman’s model | \(\ln C + m \cdot \ln \Delta K - \ln \left[ {(1 - R) \cdot K_{{\text{C}}} - {\Delta }K} \right]\) |
Elber’s model | \(\ln C + m \cdot \ln \left( {\frac{\Delta K}{{1 - R}} - K_{{{\text{op}}}} } \right)\) |
Kujawski’s model | \(\ln C + m \cdot \alpha_{{\text{K}}} \cdot \ln \frac{\Delta K}{{1 - R}} + (1 - \alpha_{{\text{K}}} )m \cdot {\text{ln}}\Delta K^{ + }\) |
Huang’s model | \(\ln C + m \cdot \ln \Delta K + m \cdot \ln M\) |
Zhan’s model | \(\ln C + m \cdot \ln {\Delta }K + m \cdot \alpha_{{\text{Z}}} \cdot R\) |
NASGRO model | \(\ln C + {\text{m}} \cdot \ln \Delta K + {\text{m}} \cdot \ln \frac{1 - f}{{1 - R}} + p \cdot \ln (1 - \frac{{\Delta K_{{{\text{th}}}} }}{\Delta K}) - q \cdot \ln (1 - \frac{{K_{\max } }}{{K_{{\text{C}}} }})\) |
Simplified NASGRO model | \(\ln C + {\text{m}} \cdot \ln \Delta K + {\text{m}} \cdot \ln \frac{1 - f}{{1 - R}} + p \cdot \ln (1 - \frac{{\Delta K_{{{\text{th}}}} }}{\Delta K})\) |
Parameters | Value |
---|---|
Material | Al-7055-T7511 |
Niterations | 2000 |
Complexity | 20 |
Operators | ‘+’, ‘-’, ‘×’, ‘/’, ‘ln_abs()’ |
Constraints | ‘ln_abs’:(1) |
Loss | \({\text{MSE}} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left( {\hat{y}_{i} - \overline{y}} \right)^{2} }\) |
Input characters | \(\ln \Delta K,\ln \left( {1 - R} \right){\text{ or (1 - }}R{\text{) or }}R,\ln (1 - \frac{{\Delta K_{{{\text{th}}}} }}{\Delta K})\) |
Output character | \(\ln \frac{{{\text{d}}a}}{{{\text{d}}N}}\) |
4 Results and Extensions
4.1 Symbolic Regression Results and Analysis
Complexity | Loss | SCORE | Equation |
---|---|---|---|
1 | 11.95728320 | 0 | \(- 16.64\) |
3 | 6.85792887 | 0.506034 | \(x_{0} - 18.65\) |
4 | 5.12093165 | 1.015250 | \(- \ln (\left| {x_{2} } \right|) - 17.52\) |
5 | 1.62683224 | 5.138852 | \({{ - 0.79} \mathord{\left/ {\vphantom {{ - 0.79} {x_{2} }}} \right. \kern-0pt} {x_{2} }} - 20.06\) |
6 | 0.38504615 | 7.903765 | \(- 2.78{ \times }\ln (\left| {x_{2} } \right|) - 19.08\) |
8 | 0.26074687 | 1.355014 | \(- 3.66{ \times }\ln (\left| {x_{2} } \right|) - x_{2} - 20.76\) |
10 | 0.18593335 | 1.515445 | \(- 3.33{ \times }\ln (\left| {x_{2} } \right|) - 0.62{ \times }x_{2} - 20.12\) |
11 | 0.16588967 | 1.196780 | \(\ln (\left| {x_{2} } \right|){ \times ((} - 0.31{ \times }\ln (\left| {x_{2} } \right|)) + 2.38) - 19.44\) |
12 | 0.15085690 | 1.091710 | \({{{ - 4}{.43}} \mathord{\left/ {\vphantom {{{ - 4}{.43}} {(x_{0} - }}} \right. \kern-0pt} {(x_{0} - }}x_{2} ) - 2.79{ \times }\ln (\left| {x_{2} } \right|) - 17.41\) |
14 | 0.12528518 | 1.204919 | \({{{( - 6}{.19 - }x_{1} )} \mathord{\left/ {\vphantom {{{( - 6}{.19 - }x_{1} )} {(x_{0} - }}} \right. \kern-0pt} {(x_{0} - }}x_{2} ) - 2.80{ \times }\ln (\left| {x_{2} } \right|) - 16.90\) |
16 | 0.12092982 | 0.264973 | \({{{6}{.19}} \mathord{\left/ {\vphantom {{{6}{.19}} {(x_{0} - }}} \right. \kern-0pt} {(x_{0} - }}x_{2} ) - 0.44{ \times }x_{1} - 2.80{ \times }\ln (\left| {x_{2} } \right|) - 16.90\) |
17 | 0.11267048 | 1.166901 | \({{ - 0.02} \mathord{\left/ {\vphantom {{ - 0.02} (}} \right. \kern-0pt} (}x_{1} + 0.58){ \times }(0.31{ \times }\ln (\left| {x_{2} } \right|) - 2.38) + \ln (\left| {x_{2} } \right|) - 19.39\) |
19 | 0.11246359 | 0.016524 | \({{ - 0.02} \mathord{\left/ {\vphantom {{ - 0.02} (}} \right. \kern-0pt} (}x_{1} + 0.59){ \times }(0.31{ \times }\ln (\left| {x_{2} } \right|) - 2.38) - 0.02 + \ln (\left| {x_{2} } \right|) - 19.39\) |
4.1.1 Symbolic Regression Results by ln(1-R)
4.1.2 Symbolic Regression Results by R
Complexity | Loss | SCORE | Equation |
---|---|---|---|
1 | 11.95728320 | 0 | \(- 16.64\) |
3 | 6.85792887 | 0.506034 | \(x_{0} - 18.65\) |
4 | 5.12093165 | 1.015250 | \(- \ln (\left| {x_{2} } \right|) - 17.52\) |
5 | 1.62683224 | 5.138852 | \({{ - 0.79} \mathord{\left/ {\vphantom {{ - 0.79} {x_{2} }}} \right. \kern-0pt} {x_{2} }} - 20.06\) |
6 | 0.38504615 | 7.903765 | \(- 2.78{ \times }\ln (\left| {x_{2} } \right|) - 19.08\) |
8 | 0.26074687 | 1.355014 | \(- 3.66{ \times }\ln (\left| {x_{2} } \right|) - x_{2} - 20.76\) |
9 | 0.20797211 | 1.920022 | \(3.97{ \times }x_{0} + 2.99{ \times }x_{1} - 24.82\) |
10 | 0.18593337 | 1.063165 | \(- 3.33{ \times }\ln (\left| {x_{2} } \right|) - 0.62{ \times }x_{2} - 20.12\) |
11 | 0.16588967 | 1.196781 | \(\ln (\left| {x_{2} } \right|){ \times ((} - 0.31{ \times }\ln (\left| {x_{2} } \right|)) + 2.38) - 19.44\) |
12 | 0.14459242 | 1.579150 | \(1.65{ \times }\left( {x_{0} - \ln (\left| {x_{2} } \right|)} \right) + 1.12{ \times }x_{1} - 18.17\) |
14 | 0.11595452 | 1.431851 | \(2.59{ \times }x_{0} + 1.87{ \times }x_{1} - 1.03{ \times }\ln (\left| {x_{2} } \right|) - 22.85\) |
16 | 0.11349541 | 0.160530 | \(2.61{ \times }x_{0} + x_{1} + x_{1} - {{0.02} \mathord{\left/ {\vphantom {{0.02} {x_{1} }}} \right. \kern-0pt} {x_{1} }} - \ln (\left| {x_{2} } \right|) - 22.85\) |
17 | 0.11347807 | 0.002519 | \(2.62{ \times }x_{0} + x_{1} + x_{1} + 0.10{ \times }\ln (\left| {x_{1} } \right|) - \ln (\left| {x_{2} } \right|) - 22.83\) |
18 | 0.10986783 | 0.565648 | \(2.67{ \times }x_{0} + {{0.16} \mathord{\left/ {\vphantom {{0.16} {x_{0} }}} \right. \kern-0pt} {x_{0} }} + 1.88{ \times }x_{1} - \ln (\left| {x_{2} } \right|) + 0.44 - 22.69\) |
20 | 0.10930385 | 0.048846 | \(2.69{ \times }x_{0} + {{0.13} \mathord{\left/ {\vphantom {{0.13} {x_{0} }}} \right. \kern-0pt} {x_{0} }} + x_{1} + x_{1} - {{0.02} \mathord{\left/ {\vphantom {{0.02} {x_{1} }}} \right. \kern-0pt} {x_{1} }} - \ln (\left| {x_{2} } \right|) - 23.11\) |
4.1.3 Symbolic Regression Results by (1-R)
Complexity | Loss | SCORE | Equation |
---|---|---|---|
1 | 11.95728320 | 0 | \(- 16.64\) |
3 | 6.85792887 | 0.506034 | \(x_{0} - 18.65\) |
4 | 5.12093165 | 1.015250 | \(- \ln (\left| {x_{2} } \right|) - 17.52\) |
5 | 1.62683224 | 5.138852 | \({{ - 0.79} \mathord{\left/ {\vphantom {{ - 0.79} {x_{2} }}} \right. \kern-0pt} {x_{2} }} - 20.06\) |
6 | 0.38504615 | 7.903765 | \(- 2.78{ \times }\ln (\left| {x_{2} } \right|) - 19.08\) |
8 | 0.26074687 | 1.355014 | \(- 3.66{ \times }\ln (\left| {x_{2} } \right|) - x_{2} - 20.76\) |
9 | 0.20797211 | 1.920022 | \(3.97{ \times }x_{0} - 2.99{ \times }x_{1} - 21.83\) |
10 | 0.18593335 | 1.063166 | \(- 3.33{ \times }\ln (\left| {x_{2} } \right|) - 0.62{ \times }x_{2} - 20.12\) |
11 | 0.16588967 | 1.196780 | \(\ln (\left| {x_{2} } \right|){ \times ((} - 0.31{ \times }\ln (\left| {x_{2} } \right|)) + 2.38) - 19.44\) |
12 | 0.14459242 | 1.579150 | \(1.65{ \times }\left( {x_{0} - \ln (\left| {x_{2} } \right|)} \right) + 1.12{ \times }x_{1} - 20.32\) |
13 | 0.13260567 | 1.081161 | \(3.65{ \times }x_{0} - 2.72{ \times }x_{1} + 0.30{ \times }x_{2} - 21.17\) |
14 | 0.11579778 | 1.828878 | \(2.55{ \times }x_{0} - {{(1.79{ \times }x_{1} + \ln (\left| {x_{2} } \right|))} \mathord{\left/ {\vphantom {{(1.79{ \times }x_{1} + \ln (\left| {x_{2} } \right|))} {0.97}}} \right. \kern-0pt} {0.97}} - 20.95\) |
16 | 0.11292975 | 0.187817 | \(2.55{ \times }x_{0} - (\ln (\left| {x_{2} } \right|) + x_{1} { \times }(1.79 - 0.05{ \times }x_{2} )) - 20.95\) |
18 | 0.11002615 | 0.221151 | \(2.64{ \times }x_{0} - {{{{0.04} \mathord{\left/ {\vphantom {{0.04} {x_{0} }}} \right. \kern-0pt} {x_{0} }}} \mathord{\left/ {\vphantom {{{{0.04} \mathord{\left/ {\vphantom {{0.04} {x_{0} }}} \right. \kern-0pt} {x_{0} }}} {x_{1} }}} \right. \kern-0pt} {x_{1} }} + 1.84{ \times }x_{1} - \ln (\left| {x_{2} } \right|) + 0.44 - 21.17\) |
20 | 0.11002614 | 0.000001 | \(2.64{ \times (}x_{0} - 0.07) - {{{{0.04} \mathord{\left/ {\vphantom {{0.04} {x_{0} }}} \right. \kern-0pt} {x_{0} }}} \mathord{\left/ {\vphantom {{{{0.04} \mathord{\left/ {\vphantom {{0.04} {x_{0} }}} \right. \kern-0pt} {x_{0} }}} {x_{1} }}} \right. \kern-0pt} {x_{1} }} + 1.84{ \times }x_{1} - \ln (\left| {x_{2} } \right|) + 0.44 - 20.99\) |
4.2 Equation Selection and Extension
4.3 Performance Evaluation and Model Comparison
Materials | lnC | m | α | q |
---|---|---|---|---|
Ti-10V-2Fe-3Al | − 23.08 | 3.29 | − 2.40 | 0.47 |
Ti-6Al-4V | − 24.90 | 3.61 | − 2.07 | 0.38 |
Cr-Mo-V steel | − 24.55 | 3.16 | − 2.76 | 0.26 |
LC9cs | − 20.28 | 3.23 | − 2.29 | 0.49 |
Al-2324-T3 | − 22.85 | 4.78 | − 4.85 | 0.55 |
Al-6013-T651 | − 22.93 | 3.76 | − 2.63 | 0.36 |
SR model | Kujawski’s model | Huang’s model | Zhan’s model | |
---|---|---|---|---|
Ti-6Al-4V | 0.08999753 | 0.34794424 | 1.13046593 | 1.03957789 |
Ti-10V-2Fe-3Al | 0.24147716 | 0.99731395 | 0.38633674 | 0.61096132 |
Cr-Mo-V | 0.12460164 | 0.39392591 | 0.33399462 | 0.28708033 |
LC9cs | 0.17511087 | 0.32597147 | 0.35360496 | 0.35360907 |
Al-2324-T3 | 0.32543334 | 1.26745959 | 0.94963109 | 0.82653289 |
Al-6013-T651 | 0.12429032 | 0.29828217 | 0.36357324 | 0.42400604 |
Al-7055-T7511 | 0.13260567 | 0.21224501 | 0.26963431 | 0.27983963 |
Average | 0.17155109 | 0.51196363 | 0.46937770 | 0.50304010 |