Reliability sensitivity analysis of thermal protection system

Abstract

This paper carries on a reliability sensitivity analysis on the non-ablative thermal protection system (TPS) of spacecraft during the conceptual design. In the previous work on probabilistic estimation of TPS, the temperature dependency of material properties has not yet been investigated. In this paper, however, the temperature dependency of material properties is characterized and considered during the thermal analysis and reliability sensitivity analysis. Compared to general black-box problems, three special challenges of uncertainty analysis for TPS in real practice are a generally high dimension and multiple outputs on massive meshing nodes, a high level of reliability design target, and a fast evaluation process due to the requirement of the conceptual design. In order to cope with these challenges, a unified reliability sensitivity analysis methodology including multi-input and multi-output support vector machines (MIMO-SVMs), a space-partition (SP) method, and a generalized subset simulation (GSS) is proposed for the conceptual design of TPS with temperature-dependent materials. MIMO-SVMs are used to approximate the thermal responses to save calculation costs. The variance-based global sensitivity indices are calculated by SP to make full use of the information within samples. Based on the sensitivity indices, a dimension reduction process is introduced. In the reduced space, GSS is used to simultaneously evaluate all the failure probabilities by fully exploring the correlation among all the LSFs. Two application examples including a lifting body vehicle model and a spacecraft model are used to demonstrate the performance of the proposed methodology.

Appendix 1 MIMO-SVM technique

For a system with md outputs, the training data set S is defined as

$$ {\displaystyle \begin{array}{l}S=\left\{{\left(x{x}_i,{y}_i\right)}_{i=1}^l\right\},x{x}_i\in {R}^{nd},{y}_i\in {R}^{md},\\ {}\left(i=1,2,\cdots, l\right),\end{array}} $$

(20)

where nd is the dimension of inputs, l is the training sample size, xx_i is the input parameters, and y_i is the scalar output, respectively. The control parameter w_i in SVM regression expression is divided as w_i = w₀ + v_i (Arora et al. 1998).

The regression parameters w₀, v_i, and b_i are solved by minimizing the following objective function with constraints

$$ {\displaystyle \begin{array}{l}\min \kern0.24em \frac{1}{2}{w}_0^T{w}_0+\frac{1}{2}\frac{\lambda }{md}\sum \limits_{i=1}^{md}{v}_i^T{v}_i+c\sum \limits_{i=1}^{md}{\xi}_i^T{\xi}_i\\ {}s.t.{y}_i={Z}_i^T\left({w}_0+{v}_i\right)+{b}_i\;{1}_l+{\xi}_i\kern0.24em \left(i=1,2,\cdots, md\right),\end{array}} $$

(21)

where λ and c are two positive real regularized parameters; 1_l = (1, 1, …, 1)^T∈R^l; Z = (ϕ(x_i,1), ϕ(x_i,2), …, ϕ(x_i,l)); ξ_i = (ξ_i,1, ξ_i,2, …, ξ_i,1)^T; and${\sum}_{i=1}^{md}{\xi}_i^T{\xi}_i$ is a quadratic loss function.

The optimization problem in (21) is formulated by the following Lagrange function

$$ {\displaystyle \begin{array}{l}L\left({w}_0,{v}_i,b,{\xi}_i,{\alpha}_i\right)=\frac{1}{2}{w}_0^T{w}_0+\frac{1}{2}\frac{\lambda }{md}\sum \limits_{i=1}^{md}{v}_i^T{v}_i+c\sum \limits_{i=1}^{md}{\xi}_i^T{\xi}_i-\\ {}\kern0.6em \sum \limits_{i=1}^{md}{\alpha}_i^T\left({Z}_i^T\left({w}_0+{v}_i\right)+{b}_i\;{1}_l+{\xi}_i-{y}_i\right)\;\left(i=1,2,\cdots, md\right),\end{array}} $$

(22)

where α_i = (α_i,1, α_i,2,…,α_i,l)^T are the Lagrange multipliers. The corresponding Karush-Kuhn-Tucker conditions are given by

$$ {\displaystyle \begin{array}{l}\frac{\partial L}{\partial {w}_0}=0,\frac{\partial L}{\partial {v}_i}=0,\frac{\partial L}{\partial {b}_i}=0,\frac{\partial L}{\partial {\xi}_i}=0,\\ {}\frac{\partial L}{\partial {\alpha}_i}=0\kern0.24em \left(i=1,2,\cdots, md\right).\end{array}} $$

(23)

The following linear equations can be deduced as

$$ \Big\{{\displaystyle \begin{array}{l}{w}_0= Z\alpha, \kern0.75em {v}_i=\frac{md}{\lambda }{Z}_i{\alpha}_i,\\ {}\sum \limits_{i=1}^m{\alpha}_i=0,\kern0.5em {\alpha}_i=2c{\xi}_i,\\ {}{y}_i={Z}_i^T\left({w}_0+{v}_i\right)+{b}_i\;{1}_l+{\xi}_i,\end{array}}\kern0.36em \left(i=1,2,\cdots, md\right), $$

(24)

where Z = (Z₁, Z₂,…, Z_md) and α = (α₁^T, α₂^T,…, α_md^T)^T. The following matrix equation is further formulated as

$$ \left[\begin{array}{cc}{\mathbf{0}}_{md\times md}& {\mathbf{O}}^T\\ {}\mathbf{O}& \mathbf{H}\end{array}\right]\cdot \left[\begin{array}{l}\mathbf{b}\\ {}\boldsymbol{\upalpha} \end{array}\right]=\left[\begin{array}{l}{\mathbf{0}}_{md}\\ {}\mathbf{y}\end{array}\right], $$

(25)

where O = (1_l1, 1_l2,…, 1_lmd) is a block diagonal matrix. The positive definite matrix H = Z^TZ+(1/2c) I_l + (md/λ)B. I_l is a unitary matrix. B = (K₁, K₂,…, K_md) is a block diagonal matrix, in which the i-th element satisfies K_i = Z_i^TZ_i. Supposed that the solution to (25) is α^* = (α₁^*T, α₂^*T,…, α_md^*T)^T and b^* = (b₁^*,b₂^*,…,b_md^*)^T, where α_i^* = (α_i,1^*, α_i,2^*,…, α_i,l^*)^T. The regression functions can be expressed as

$$ {\displaystyle \begin{array}{c}{f}_i(xx)=\phi {(xx)}^T\left({w}_0^{\ast }+{v}_i^{\ast}\right)+{b}_i^{\ast}\\ {}=\phi {(xx)}^T\left(Z{\alpha}^{\ast }+\frac{md}{\lambda }{Z}_i{\alpha}_i^{\ast}\right)+{b}_i^{\ast}\\ {}=\sum \limits_{i^{\hbox{'}}=1}^{md}\sum \limits_{j=1}^l{\alpha}_{i^{\hbox{'}},j}^{\ast }K\left(x{x}_{i^{\hbox{'}},j}, xx\right)+\frac{md}{\lambda}\sum \limits_{j=1}^l{\alpha}_{i^{\hbox{'}},j}^{\ast }K\left(x{x}_{i,j}, xx\right)+{b}_i^{\ast}\\ {}\kern1.5em \left(i=1,2,\cdots, md\right),\end{array}} $$

(26)

where ϕ(⋅) is the specified kernel function. More details about the MIMO-SVM technique can be referred to (Xu et al. 2014; Xu et al. 2013).