Adaptive-sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems

Abstract

This paper presents an adaptive-sparse polynomial chaos expansion (adaptive-sparse PCE) method for performing engineering reliability analysis and design. The proposed method combines three ideas: (i) an adaptive-sparse scheme to build sparse PCE with the minimum number of bivariate basis functions, (ii) a new projection method using dimension reduction techniques to effectively compute the expansion coefficients of system responses, and (iii) an integration of copula to handle nonlinear correlation of input random variables. The proposed method thus has three positive features for reliability analysis and design: (a) there is no need for response sensitivity analysis, (b) it is highly efficient and accurate for reliability analysis and its sensitivity analysis, and (c) it is capable of handling a nonlinear correlation. In addition to the features, an error decomposition scheme for the proposed method is presented to help analyze error sources in probability analysis. Several engineering problems are used to demonstrate the three positive features of the adaptive-sparse PCE method.

Appendix

The derivation of the error decomposition

• Error source I: Truncation

  $$\begin{array}{rll} \varepsilon _P^2 &=&\int_\Gamma \big[{g-\emph{w}_{UB}^p} \big]^2f({\rm {\bf x}})d{\rm {\bf x}} \\[12pt] &=&\int_\Gamma {\left[ {\begin{array}{@{}l} \displaystyle \sum\limits_k {\sum\limits_{j>p} {s_j^k \cdot \Phi _j ({\zeta _k})} } \\[18pt] + \displaystyle\sum\limits_{[x_k ,x_l ]\notin {\rm {\bf B}}} \sum\limits_{j_1 +j_2 \le p} s_{j_1 ,j_2 }^{k,l} \cdot \Phi _{j_1 ,j_2}({\zeta _k ,\zeta _l}) \\[18pt] + \displaystyle\sum\limits_{k<l} \sum\limits_{j_1 +j_2 >p} s_{j_1 ,j_2}^{k,l} \cdot \Phi _{j_1 ,j_2 } ({\zeta _k ,\zeta _l}) \\[18pt] \displaystyle +\sum\limits_{k<l<m} \sum\limits_{j_1 ,j_2 ,j_3 \ge 1} s_{j_1 ,j_2 ,j_3 }^{k,l,m} \\[18pt] \cdot \Phi _{j_1 ,j_2 ,j_3 } \left( {\zeta _k ,\zeta _l ,\zeta _m } \right) +\cdots \\ \end{array}}\!\!\! \right]^2 \!\! f({\rm {\bf x}})d{\rm {\bf x}}} \end{array}$$

• Error source II: Bivariate decomposition

  $$\begin{array}{rll} \varepsilon _B^2 &=&\int_\Gamma {\big[ {\emph{w}_{UB}^p -\emph{w}_U^p} \big]^2f({\rm {\bf x}})d{\rm {\bf x}}} \\[-1.5pt] &=&\int_\Gamma {\left[ {\sum\limits_{[{x_k ,x_l}]\in {\rm {\bf B}}} {\sum\limits_{j_1 +j_2 \le p} {\big( {s_{j_1 ,j_2 }^{k,l} -\hat {s}_{j_1 ,j_2 }^{k,l}} \big)\Phi _{j_1 ,j_2 } ({\zeta _k ,\zeta _l})} } } \right]^2f({\rm {\bf x}})d{\rm {\bf x}}} \\[-1.5pt] &=&\int_\Gamma {\left[ {\sum\limits_{[{x_k ,x_l}]\in {\rm {\bf B}}} {\sum\limits_{j_1 +j_2 \le p} {\frac{E\left[ {\big( {g({\rm {\bf x}})-g\big( {x_k ,x_l ,\boldsymbol{\upmu}^{k,l}} \big)} \big)\cdot \Phi _{j_1 ,j_2} ({\zeta _k ,\zeta _l})} \right]}{E\big[ {\Phi _{j_1 ,j_2 }^2 ({\zeta _k ,\zeta _l})} \big]}} } \cdot \Phi _{j_1 ,j_2 } ({\zeta _k ,\zeta _l})} \right]^2f ({\rm {\bf x}})d{\rm {\bf x}}} \end{array}$$

  where

  $$\begin{array}{rll} && E\left[ {\left( {g({\rm {\bf x}})-g\left( {x_k ,x_l ,\boldsymbol{\upmu}^{k,l}} \right)} \right)\Phi _{j_1 ,j_2} ({\zeta _k ,\zeta _l})} \right] \\[-1.5pt] &&{\kern10pt} =\frac{1}{2!1!1!}\sum\limits_{i\ne k,l} {\frac{\partial ^4g}{\partial x_i^2 \partial x_k \partial x_l }(\boldsymbol{\upmu})} \cdot E\big[{({x_i -\mu _i})^2({x_k -\mu _k})({x_l -\mu _l})\Phi _{j_1 ,j_2 } ({\zeta _k ,\zeta _l })} \big]+\cdots \end{array}$$

• Error source III: Univariate decomposition

  $$\begin{array}{rll} \varepsilon _U^2 &=& \int_\Gamma {\big[ {\emph{w}_U^p -\emph{w}_I^p } \big]^2f( {\rm {\bf x}} )d{\rm {\bf x}}} = \int_\Gamma {\left[ {\sum\limits_{k=1}^N {\sum\limits_{j=1}^p {\left( {s_j^k -\hat {s}_j^k } \right)\Phi _j \left( {\zeta _k } \right)} } } \right]^2f\left( {\rm {\bf x}} \right)d{\rm {\bf x}}} \\ &=&\int_\Gamma {\left[ {\sum\limits_{k=1}^N {\sum\limits_{j=1}^p {\frac{E\left[ {\left( {g\left( {\rm {\bf x}} \right)-g\left( {x_k ,\boldsymbol{\upmu}^k} \right)} \right)\Phi _j \left( {\zeta _k } \right)} \right]}{E\big[ {\Phi _j^2 ({\zeta _k})}\big]}} } \cdot \Phi _j ({\zeta _k})} \right]^2f({\rm {\bf x}})d{\rm {\bf x}}} \end{array}$$

  where

  $$ E\left[ {({g({\rm {\bf x}})-g ({x_k ,\boldsymbol{\upmu}^k})})\Phi _j ({\zeta _k})} \right] = \frac{1}{2!1!}\sum\limits_{i\ne k} {\frac{\partial ^3g}{\partial x_i^2 \partial x_k }(\boldsymbol{\upmu})} E\big[ {( {x_i -\mu _i } )^2({x_k -\mu _k})\Phi _j ({\zeta _k})} \big]+\cdots $$

• Error source IV: Aliasing error

  $$\begin{array}{rll} \varepsilon _I^2 &=&\int_\Gamma {\big[ {\emph{w}_I^p -\emph{w}^p} \big]^2} f({\rm {\bf x}})d{\rm {\bf x}} \\[-1.5pt] &=&\int_\Gamma {\left[ {\sum\limits_{k=1}^N {\sum\limits_{j=1}^p {\left( {\hat {s}_j^k -\hat {\hat {s}}_j^k } \right)\Phi _j ({\zeta _k}) +\sum\limits_{[{x_k ,x_l}]\in {\rm {\bf B}}} {\sum\limits_{j_1 +j_2 \le p} {\left( {\hat {s}_{j_1 ,j_2 }^{k,l} -\hat {\hat {s}}_{j_1 ,j_2 }^{k,l}}\right)\Phi _{j_1 ,j_2} ({\zeta _k ,\zeta _l})}}}}} \right]^2f ({\rm {\bf x}})d{\rm {\bf x}}} \\[-1.5pt] &=&\int_\Gamma {\left[ {\begin{array}{l} \displaystyle \sum\limits_{k=1}^N {\sum\limits_{j=1}^p {\frac{\big({E-\hat {E}} \big)\left[ {g\big({x_k ,\boldsymbol{\upmu}^k}\big)\Phi _j ({\zeta _k})} \right]}{E\big[ {\Phi _j^2 \left( {\zeta _k } \right)} \big]}} } \cdot \Phi _j ({\zeta _k}) \\ \displaystyle{\kern12pt} +\sum\limits_{[{x_k ,x_l}]\in {\rm {\bf B}}} {\sum\limits_{j_1 +j_2 \le p} {\frac{\big({E-\hat {E}}\big)\left[ {g( {x_k ,x_l ,\boldsymbol{\upmu}^{k,l}})\cdot \Phi _{j_1 ,j_2 } ({\zeta _k ,\zeta _l})} \right]}{E\big[ {\Phi _{j_1 ,j_2 }^2 ({\zeta _k ,\zeta _l})} \big]}} } \cdot \Phi _{j_1 ,j_2 } ({\zeta _k ,\zeta _l }) \\ \end{array}} \right]^2f ({\rm {\bf x}})d{\rm {\bf x}}} \end{array}$$

  where $\hat {E}(\cdot)$ denotes the approximate expectation by using the SMLS and Gaussian quadrature integration.