A novel inverse strain range-based adaptive Kriging method for analyzing the combined fatigue life reliability

Abstract

In this paper, a novel combined fatigue life reliability analysis model is constructed from the perspective of inverse analysis of Manson-Coffin equation. By the derived equivalent threshold of low cycle fatigue life, the failure event that the combined fatigue life is less than or equal to the presupposed threshold is equivalently transformed into the event that the actual strain range in the low cycle fatigue mode is larger than or equal to the inverse strain range threshold. The inverse strain range threshold corresponds to the equivalent threshold of low cycle fatigue life derived by the presupposed threshold of combined fatigue life. Then, the inverse strain range-based limit state function is constructed to analyze the fatigue life reliability, where solution of the exponential Manson-Coffin equation which is used to determine the low cycle fatigue life is avoided. A combination of the inverse strain range-based limit state function and adaptive Kriging (AK) model is constructed first to estimate the combined fatigue life reliability where the AK model directly surrogates the inverse strain range-based limit state function, and this algorithm is defined as a full-surrogate algorithm. The inverse strain range-based limit state function consists of two nested parts. The first part is the structural analysis which is usually an implicit function and the second part is the life analysis which is usually an explicit function. In this regard, another combination of the inverse strain range-based limit state function and AK model is constructed to estimate the combined fatigue life reliability, where the AK model only surrogates a part of inverse strain range-based limit state function, i.e., the implicit structural analysis part, and this algorithm is regarded as a semi-surrogate algorithm. Two aero-engine structures are analyzed to validate the effectiveness of the proposed method.

Appendix

1.1 The basic theory of the Kriging model

The Kriging model includes two parts. The first part is the parametric linear regression part and the second part is the nonparametric stochastic process part. The Kriging model of an unknown function g(X) is described as follows (Sacks et al. 1989; Kersaudy et al. 2015).

$$ {g}_K\left(\boldsymbol{X}\right)=\sum \limits_{i=1}^p{B}_i\left(\boldsymbol{X}\right){\beta}_i+S\left(\boldsymbol{X}\right)={\boldsymbol{B}}^T\left(\boldsymbol{X}\right)\boldsymbol{\beta} +\boldsymbol{S}\left(\boldsymbol{X}\right) $$

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where B(X) = [B₁(X), B₂(X), …, B_p(X)]^T are the base functions of vector X, β = [β₁, β₂, …, β_p]^T is the regression coefficient vector, and p is the number of base function. S(X) is a stationary Gaussian process with zero mean and covariance which is defined as follows:

$$ \operatorname{cov}\left[S\left({\boldsymbol{x}}_i\right),S\left({\boldsymbol{x}}_j\right)\right]={\sigma}^2R\left({\boldsymbol{x}}_i,{\boldsymbol{x}}_j\right)i,j=1,2,\dots, {N}_0 $$

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where N_T denotes the number of training points.

Define $\boldsymbol{R}=\left[\begin{array}{ccc}R\left({\boldsymbol{x}}_1,{\boldsymbol{x}}_1\right)& \cdots & R\left({\boldsymbol{x}}_1,{\boldsymbol{x}}_{N_T}\right)\\ {}\vdots & \ddots & \vdots \\ {}R\left({\boldsymbol{x}}_{N_T},{\boldsymbol{x}}_1\right)& \cdots & R\left({\boldsymbol{x}}_{N_T},{\boldsymbol{x}}_{N_T}\right)\end{array}\right]$, $\boldsymbol{B}=\left[B{\left({\boldsymbol{x}}_1\right)}^T,B{\left({\boldsymbol{x}}_2\right)}^T,\dots, B{\left({\boldsymbol{x}}_{N_T}\right)}^T\right]$ and g is the corresponding vector of the function g(X) calculated at each experiment points x_i(i = 1, 2, …, N_T), the unknown β and σ² can be estimated by the following equations, i.e.,

$$ \hat{\boldsymbol{\beta}}={\left({\boldsymbol{B}}^T{\boldsymbol{R}}^{-1}\boldsymbol{B}\right)}^{-1}{\boldsymbol{B}}^T{\boldsymbol{R}}^{-1}\boldsymbol{g} $$

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$$ \hat{\sigma^2}=\frac{1}{N_f}{\left(\boldsymbol{g}-\boldsymbol{B}\hat{\boldsymbol{\beta}}\right)}^T{\boldsymbol{R}}^{-1}\left(\boldsymbol{g}-\boldsymbol{B}\hat{\boldsymbol{\beta}}\right) $$

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Therefore, the Kriging mean and Kriging variance are given as follows:

$$ {\mu}_{g_K}\left(\boldsymbol{X}\right)={\boldsymbol{B}}^T\left(\boldsymbol{X}\right)\hat{\boldsymbol{\beta}}+{\boldsymbol{r}}^T\left(\boldsymbol{X}\right){\boldsymbol{R}}^{-1}\left(\boldsymbol{g}-\boldsymbol{B}\hat{\boldsymbol{\beta}}\right) $$

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$$ {\sigma}_{g_K}^2\left(\boldsymbol{X}\right)={\sigma}^2\left(1-{\boldsymbol{r}}^T\left(\boldsymbol{X}\right){\boldsymbol{R}}^{-1}\boldsymbol{r}\left(\boldsymbol{X}\right)+{\left[{\boldsymbol{B}}^T{\boldsymbol{R}}^{-1}\boldsymbol{r}\left(\boldsymbol{X}\right)-\boldsymbol{B}\left(\boldsymbol{X}\right)\right]}^T{\left({\boldsymbol{B}}^T{\boldsymbol{R}}^{-1}\boldsymbol{B}\right)}^{-1}{\left[{\boldsymbol{B}}^T{\boldsymbol{R}}^{-1}\boldsymbol{r}\left(\boldsymbol{X}\right)-\boldsymbol{B}\left(\boldsymbol{X}\right)\right]}^T\right) $$

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where ${\boldsymbol{r}}^T\left(\boldsymbol{X}\right)={\left[R\left(\boldsymbol{X},{\boldsymbol{x}}_1\right),\dots, R\left(\boldsymbol{X},{\boldsymbol{x}}_{N_T}\right)\right]}^T$.