Structural reliability analysis based on analytical maximum entropy method using polynomial chaos expansion

Jianbin Guo; Jianyu Zhao; Shengkui Zeng

doi:10.1007/s00158-018-1961-z

Structural and Multidisciplinary Optimization

pp 1–17 | Cite as

Structural reliability analysis based on analytical maximum entropy method using polynomial chaos expansion

Authors
Authors and affiliations

Jianbin Guo
Jianyu Zhao
Shengkui ZengEmail author

RESEARCH PAPER

First Online: 17 April 2018

Received: 09 October 2017

Revised: 16 January 2018

Accepted: 20 February 2018

3 Downloads

Abstract

The maximum entropy (ME) method is a promising tool for structural reliability analysis by estimating the unknown probability density function (PDF) of given model response from its moment constraints. However, the classic ME algorithm has to resort to an iterative procedure due to non-linear constraints, and the required high order moment estimations may have large statistical error. In this paper, we (i) propose an analytical ME method based on integration by parts algorithm to transform the non-linear constraints to a system of linear equations and (ii) derive the polynomial chaos expansion (PCE) multiplication for improving higher order moment calculation required in the previous step efficiently. Thus, an analytical formula of response PDF is obtained directly without intensively iterative procedure and associated convergence error, and it is followed by probability failure estimation using numerical integration computation. Two structural engineering cases are implemented to illustrate the accuracy and efficiency of the proposed method.

Keywords

Structural reliability analysis Maximum entropy method Polynomial chaos expansion Multiplication of orthogonal polynomials

This is a preview of subscription content, to check access

Notes

Acknowledgements

The authors would like to extend their sincere thanks to anonymous reviewers for their valuable comments.

Appendix 1

Chebyshev polynomials

The definition of Chebyshev polynomials is

$$ {\displaystyle \begin{array}{l}{T}_0(x)=1,\\ {}{T}_n(x)=\frac{n}{2}\sum \limits_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }{\left(-1\right)}^k\frac{\left(n-k-1\right)!}{k!\left(n-2k\right)!}{(2x)}^{n-2k},\kern0.6em n\ge 1\end{array}} $$

(38)

and the most well-known form of Chebyshev polynomials is given by the recurrence relation

$$ {\displaystyle \begin{array}{l}{T}_0(x)=1,\\ {}{T}_1(x)=x,\\ {}{T}_{n+1}(x)=2{xT}_n(x)-{T}_{n-1}(x),\kern0.8000001em n\ge 2\end{array}} $$

(39)

where x∈[− 1, 1]. This definition is also referred as Chebyshev polynomials of the first kind, and there is another definition of Chebyshev polynomials of the second kind:

$$ {\displaystyle \begin{array}{l}{U}_0(x)=1,\\ {}{U}_1(x)=2x,\\ {}{U}_{n+1}(x)=2{xU}_n(x)-{U}_{n-1}(x),\kern0.8000001em n\ge 2\end{array}} $$

(40)

The relationship between these two kinds of Chebyshev polynomials are given as (41)-(44):

$$ \frac{dT_n(x)}{dx}={nU}_{n-1}(x) $$

(41)

$$ {T}_j(x){U}_k(x)=\left\{\begin{array}{cc}\frac{1}{2}\left({U}_{j+k}(x)+{U}_{k-j}(x)\right),& k\ge j-1,\\ {}\frac{1}{2}\left({U}_{j+k}(x)-{U}_{j-k-2}(x)\right),& k\le j-2.\end{array}\right. $$

(42)

with the convention U₋₁ ≡ 0.

$$ {T}_n(x)=\frac{1}{2\left(n+1\right)}\frac{dT_{n+1}(x)}{dx}-\frac{1}{2\left(n-1\right)}\frac{dT_{n-1}(x)}{dx},\kern0.9000001em \forall i\ge 1. $$

(43)

$$ \left(1-{x}^2\right){U}_{n-1}(x)={xT}_n(x)-{T}_{n+1}(x)=\frac{T_{n-1}(x)-{T}_{n+1}(x)}{2} $$

(44)

Hermite polynomials

The Hermite polynomials {H_n(x)}_n ≥ 0 are defined as

$$ {H}_n(x)={\left(-1\right)}^n{e}^{\frac{x^2}{2}}\frac{d^n}{dx^n}\left({e}^{-\frac{x^2}{2}}\right) $$

(45)

and the well-known recursive relation of the Hermite polynomials is

$$ \left\{\begin{array}{l}{H}_0(x)\equiv 1,\\ {}{H}_1(x)=2x,\\ {}{H}_{n+1}(x)=2{xH}_n(x)-2{nH}_{n-1}(x)\kern0.2em \end{array}\right. $$

(46)

Appendix 2

In this section, the derivation details of (12) is provided.

Consider (10), and take $ {F}_1^{(i)}(y)={\int}_{-1}^1\left(1-{y}^2\right)\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right)\frac{1}{2\left(i+1\right)}{dT}_{i+1}(y) $ and $ {F}_2^{(i)}(y)={\int}_{-1}^1\left(1-{y}^2\right)\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right)\frac{1}{2\left(i-1\right)}{dT}_{i-1}(y) $ for simplicity. Then, we would solve the two parts separately.

First of all, F₁⁽ⁱ⁾(x) could be transformed as

$$ {\displaystyle \begin{array}{l}{F}_1^{(i)}(y)={\left[\frac{1}{2\left(i+1\right)}\left(1-{y}^2\right)\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right)\right]}_{-1}^1\\ {}\kern3.099999em -{\int}_{-1}^1\frac{1}{2\left(i+1\right)}{T}_{i+1}(y)\left[-2x+\left(1-{y}^2\right)\frac{d}{dy}\left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right)\right]\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right) dy\\ {}\kern2.4em =\frac{1}{i+1}{\int}_{-1}^1{yT}_{i+1}(y)\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right) dy\\ {}\kern3.099999em +\frac{1}{2\left(i+1\right)}{\int}_{-1}^1{T}_{i+1}(y)\left(1-{y}^2\right)\frac{d}{dy}\left(\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right)\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right) dy\end{array}} $$

(47)

Take (39) into consideration, the first integral term above is derived as

$$ {\displaystyle \begin{array}{l}\frac{1}{i+1}{\int}_{-1}^1{yT}_{i+1}(y)\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right) dy\\ {}=\frac{1}{i+1}{\int}_{-1}^1\frac{T_{i+1}(y)+{T}_i(y)}{2}\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right) dy\\ {}=\frac{1}{2\left(i+1\right)}\left({M}_{i+2}+{M}_i\right)\end{array}} $$

(48)

Then, take (41) and (42) into consideration, the second integral term in (47) is derived as

$$ {\displaystyle \begin{array}{l}\frac{1}{2\left(i+1\right)}{\int}_{-1}^1{T}_{i+1}(y)\left(1-{y}^2\right)\frac{d}{dy}\left(\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right)\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right) dy\\ {}=\frac{1}{2\left(i+1\right)}{\int}_{-1}^1{T}_{i+1}(y)\left(1-{y}^2\right)\left(\sum \limits_{j=1}^mj{\lambda}_j{U}_{j-1}(y)\right)\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(x)\right) dy\\ {}=\frac{1}{2\left(i+1\right)}{\int}_{-1}^1\left(1-{y}^2\right)\left(\sum \limits_{j=1}^mj{\lambda}_j{T}_{i+1}(y){U}_{j-1}(y)\right)\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right) dy\\ {}=\frac{1}{2\left(i+1\right)}{\int}_{-1}^1\left(1-{y}^2\right)\left[\begin{array}{l}\left(\sum \limits_{j=1}^ij{\lambda}_j\frac{U_{i+j}(y)-{U}_{i-j}(y)}{2}\right)+\left(i+1\right){\lambda}_{i+1}\frac{U_{2i+1}}{2}\\ {}+\left(\sum \limits_{j=i+2}^mj{\lambda}_j\frac{U_{i+j}(y)+{U}_{j-i-2}(y)}{2}\right)\end{array}\right]\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right) dy\end{array}} $$

(49)

where the Chebyshev polynomials of the second kind come as intermediate polynomials.

By using (44), the polynomial multiplication of Chebyshev polynomials in (49) is given as the following:

$$ {\displaystyle \begin{array}{l}\left(1-{y}^2\right)\frac{U_{i+j}(y)-{U}_{i-j}(y)}{2}=\frac{T_{i+j}(y)-{T}_{i+j+2}(y)}{2}-\frac{T_{i-j}(y)-{T}_{i-j+2}(y)}{2}\\ {}\left(1-{y}^2\right)\frac{U_{2i+1}}{2}=\frac{T_{2i+1}(y)-{T}_{2i+3}(y)}{2}\\ {}\left(1-{y}^2\right)\frac{U_{i+j}(y)+{U}_{j-i-2}(y)}{2}=\frac{T_{i+j}(y)-{T}_{i+j+2}(y)}{2}+\frac{T_{j-i-2}(y)-{T}_{j-i}(y)}{2}\end{array}} $$

(50)

Substitute (50) into (49), the second integral term in (47) is derived as follows:

$$ {\displaystyle \begin{array}{l}\frac{1}{2\left(i+1\right)}{\int}_{-1}^1{T}_{i+1}(y)\left(1-{y}^2\right)\frac{d}{dy}\left(\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right)\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right) dy\\ {}=\frac{1}{2\left(i+1\right)}{\int}_{-1}^1\left[\sum \limits_{j=1}^i\frac{j}{2}{\lambda}_j\left(\frac{T_{i+j}(y)-{T}_{i+j+2}(y)}{2}-\frac{T_{i-j}(y)-{T}_{i-j+2}(y)}{2}\right)\right]\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right) dy\\ {}\kern0.7em +\frac{1}{2\left(i+1\right)}{\int}_{-1}^1\frac{i+1}{2}{\lambda}_{i+1}\frac{T_{2i+1}(y)-{T}_{2i+3}(y)}{2}\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right) dy\\ {}\kern0.7em +\frac{1}{2\left(i+1\right)}{\int}_{-1}^1\sum \limits_{j=i+2}^m\frac{j}{2}{\lambda}_j\left(\frac{T_{i+j}(y)-{T}_{i+j+2}(y)}{2}+\frac{T_{j-i-2}(y)-{T}_{j-i}(y)}{2}\right)\exp \left(-1-\sum \limits_{j=0}^m{\lambda}_j{T}_j(y)\right) dy\\ {}=\frac{1}{8\left(i+1\right)}\sum \limits_{j=1}^ij\left({M}_{i+j}-{M}_{i+j+2}-{M}_{i-j}+{M}_{i-j+2}\right){\lambda}_j+\frac{1}{8}\left({M}_{2i+1}-{M}_{2i+3}\right){\lambda}_{i+1}\\ {}\kern0.9000001em +\frac{1}{8\left(i+1\right)}\sum \limits_{j=i+2}^mj\left({M}_{i+j}-{M}_{i+j+2}+{M}_{j-i-2}-{M}_{j-i}\right){\lambda}_j\end{array}} $$

(51)

By Combining (48) and (51), we have the following relationship:

$$ {\displaystyle \begin{array}{l}{F}_1^{(i)}(y)=\frac{1}{2\left(i+1\right)}\left({M}_{i+2}+{M}_i\right)+\frac{1}{8\left(i+1\right)}\sum \limits_{j=1}^ij\left({M}_{i+j}-{M}_{i+j+2}-{M}_{i-j}+{M}_{i-j+2}\right){\lambda}_j\\ {}\kern3.099999em +\frac{1}{8}\left({M}_{2i+1}-{M}_{2i+3}\right){\lambda}_{i+1}+\frac{1}{8\left(i+1\right)}\sum \limits_{j=i+2}^mj\left({M}_{i+j}-{M}_{i+j+2}+{M}_{j-i-2}-{M}_{j-i}\right){\lambda}_j\end{array}} $$

(52)

Secondly, when it comes to F₂⁽ⁱ⁾(x), it is a little more complicated. When i = 1, since T₀(x) is a constant according to the definition of Chebyshev polynomials. Otherwise, when i ≥ 2, we could derive F₂⁽ⁱ⁾(x) with a similar procedure as (47)-(51) with the following result:

$$ {\displaystyle \begin{array}{l}{F}_2^{(i)}(y)=\frac{1}{2\left(i-1\right)}\left({M}_i+{M}_{i-2}\right)+\frac{1}{8\left(i-1\right)}\sum \limits_{j=1}^{i-2}j\left({M}_{i+j-2}-{M}_{i+j}-{M}_{i-j-2}+{M}_{i-j}\right){\lambda}_j\\ {}\kern3.099999em +\frac{1}{8}\left({M}_{2i-3}-{M}_{2i-1}\right){\lambda}_{i+1}+\frac{1}{8\left(i-1\right)}\sum \limits_{j=i+2}^mj\left({M}_{i+j-2}-{M}_{i+j}+{M}_{j-i}-{M}_{j-i+2}\right){\lambda}_j\end{array}} $$

(53)

What is more, consider (8) and (9), we would establish the following equations. When i = 1, we have

$$ {\displaystyle \begin{array}{l}{\int}_{-1}^1\left(1-{y}^2\right){T}_1(y)\exp \left(-1-\sum \limits_{j=1}^m{\lambda}_j{T}_j(y)\right) dy={F}_1^{(1)}(y)\\ {}{\int}_{-1}^1\left(1-{y}^2\right){T}_i(y)\exp \left(-1-\sum \limits_{j=1}^m{\lambda}_j{T}_j(y)\right) dy={F}_1^{(i)}(y)-{F}_2^{(i)}(y)\end{array}} $$

(54)

and it could be rewritten as

$$ \frac{1}{16}\left(2{M}_2-{M}_4-{M}_0\right){\lambda}_1+\frac{1}{8}\left({M}_3-{M}_5\right){\lambda}_2+\frac{1}{16}\sum \limits_{j=3}^mj\left({M}_{j+1}-{M}_{j+3}+{M}_{j-3}-{M}_{j-1}\right){\lambda}_j=\frac{1}{4}\left({M}_1-{M}_3\right) $$

(55)

and for i ≥ 2

$$ {\displaystyle \begin{array}{l}\frac{1}{8\left(i+1\right)}\sum \limits_{j=1}^ij\left({M}_{i+j}-{M}_{i+j+2}-{M}_{i-j}+{M}_{i-j+2}\right){\lambda}_j+\frac{1}{8}\left({M}_{2i+1}-{M}_{2i+3}\right){\lambda}_{i+1}\kern0.1em \\ {}+\frac{1}{8\left(i+1\right)}\sum \limits_{j=i+2}^mj\left({M}_{i+j}-{M}_{i+j+2}+{M}_{j-i-2}-{M}_{j-i}\right){\lambda}_j-\frac{1}{8\left(i-1\right)}\sum \limits_{j=1}^{i-2}j\left({M}_{i+j-2}-{M}_{i+j}-{M}_{i-j-2}+{M}_{i-j}\right){\lambda}_j\\ {}-\frac{1}{8}\left({M}_{2i-3}-{M}_{2i-1}\right){\lambda}_{i+1}-\frac{1}{8\left(i-1\right)}\sum \limits_{j=i+2}^mj\left({M}_{i+j-2}-{M}_{i+j}+{M}_{j-i}-{M}_{j-i+2}\right){\lambda}_j\\ {}=\left(\frac{1}{2}-\frac{1}{2\left(i+1\right)}+\frac{1}{2\left(i-1\right)}\right){M}_i-\left(\frac{1}{4}+\frac{1}{2\left(i+1\right)}\right){M}_{i+2}-\left(\frac{1}{4}-\frac{1}{2\left(i-1\right)}\right){M}_{i-2}\end{array}} $$

(56)

Appendix 3

In this section, we would prove the formula of two PCEs multiplication.

First of all, for any non-negative integer α and β, the product of two Hermite polynomials can be expressed as (Savin and Faverjon 2017)

$$ {H}_{\alpha }(x){H}_{\beta }(x)=\sum \limits_{r\le \min \left(\alpha, \beta \right)}{\left[\left(\begin{array}{l}\alpha \\ {}r\end{array}\right)\left(\begin{array}{l}\beta \\ {}r\end{array}\right)\left(\begin{array}{c}\alpha +\beta -2r\\ {}\alpha -r\end{array}\right)\right]}^{\frac{1}{2}}{H}_{\alpha +\beta -2r}(x) $$

(57)

Then, this formula could be generalized to multi-dimensional form as (58).

$$ {\psi}_{\boldsymbol{\alpha}}\left(\boldsymbol{\xi} \right){\psi}_{\boldsymbol{\beta}}\left(\boldsymbol{\xi} \right)=\sum \limits_{r\le \min \left(\boldsymbol{\alpha}, \boldsymbol{\beta} \right)}B\left(\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{r}\right){\psi}_{\boldsymbol{\alpha} +\boldsymbol{\beta} -2\boldsymbol{r}}\left(\boldsymbol{\xi} \right) $$

(58)

where $ B\left(\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{r}\right)={\left[\left(\begin{array}{l}\boldsymbol{\alpha} \\ {}\boldsymbol{r}\end{array}\right)\left(\begin{array}{l}\boldsymbol{\beta} \\ {}\boldsymbol{r}\end{array}\right)\left(\begin{array}{c}\boldsymbol{\alpha} +\boldsymbol{\beta} -2\boldsymbol{r}\\ {}\boldsymbol{\alpha} -\boldsymbol{r}\end{array}\right)\right]}^{\frac{1}{2}} $, and the meaning of the notations above is shown in Section 3.2.

With the preparation, we could derive the multiplication of two PCEs as follows.

Suppose u and v have PCE formula with the same n-dimensional standardized random variables ξ = (ξ₁, ⋯, ξ_n)^T but different order p_α and p_β respectively. Namely, we have $ u=\sum \limits_{\left|\boldsymbol{\alpha} \right|\le {p}_{\alpha }}{u}_{\boldsymbol{\alpha}}{\psi}_{\boldsymbol{\alpha}}\left(\boldsymbol{\xi} \right) $, $ v=\sum \limits_{\left|\boldsymbol{\beta} \right|\le {p}_{\beta }}{v}_{\boldsymbol{\beta}}{\psi}_{\boldsymbol{\beta}}\left(\boldsymbol{\xi} \right) $. So the multiplication of u and v is

$$ {\displaystyle \begin{array}{l} uv=\sum \limits_{\left|\boldsymbol{\alpha} \right|\le {p}_{\alpha }}\sum \limits_{\left|\boldsymbol{\beta} \right|\le {p}_{\beta }}{u}_{\boldsymbol{\alpha}}{v}_{\boldsymbol{\beta}}{\psi}_{\boldsymbol{\alpha}}\left(\boldsymbol{\xi} \right){\psi}_{\boldsymbol{\beta}}\left(\boldsymbol{\xi} \right)\\ {}=\sum \limits_{\left|\boldsymbol{\alpha} \right|\le {p}_{\alpha }}\sum \limits_{\left|\boldsymbol{\beta} \right|\le {p}_{\beta }}{u}_{\boldsymbol{\alpha}}{v}_{\boldsymbol{\beta}}\sum \limits_{r\le \min \left(\boldsymbol{\alpha}, \boldsymbol{\beta} \right)}B\left(\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{r}\right){\psi}_{\boldsymbol{\alpha} +\boldsymbol{\beta} -2\boldsymbol{r}}\left(\boldsymbol{\xi} \right)\\ {}=\sum \limits_{\left|\boldsymbol{\alpha} \right|\le {p}_{\alpha }}\sum \limits_{\left|\boldsymbol{\beta} \right|\le {p}_{\beta }}{u}_{\boldsymbol{\alpha}}{v}_{\boldsymbol{\beta}}\sum \limits_{\boldsymbol{r}\le \min \left(\boldsymbol{\alpha}, \boldsymbol{\beta} \right)}\left(\begin{array}{c}\boldsymbol{\alpha} \\ {}\boldsymbol{r}\end{array}\right)\left(\begin{array}{c}\boldsymbol{\beta} \\ {}\boldsymbol{r}\end{array}\right)r!\frac{\sqrt{\left(\boldsymbol{\alpha} +\boldsymbol{\beta} -2\boldsymbol{r}\right)!}}{\sqrt{\boldsymbol{\alpha} !\boldsymbol{\beta} !}}{\psi}_{\boldsymbol{\alpha} +\boldsymbol{\beta} -2\boldsymbol{r}}\left(\boldsymbol{\xi} \right)\end{array}} $$

(59)

Let $ \tilde{\boldsymbol{\alpha}}=\boldsymbol{\alpha} -\boldsymbol{r} $, $ \tilde{\boldsymbol{\beta}}=\boldsymbol{\beta} -\boldsymbol{r} $, then r = min(α, β) is equivalent to $ \tilde{\boldsymbol{\alpha}},\tilde{\boldsymbol{\beta}}\ge 0 $. Alternatively, $ \boldsymbol{\alpha} =\tilde{\boldsymbol{\alpha}}+\boldsymbol{r} $, $ \boldsymbol{\beta} =\tilde{\boldsymbol{\beta}}+\boldsymbol{r} $ and the above summation can be rewritten as

$$ uv=\sum \limits_{\left|\tilde{\boldsymbol{\alpha}}\right|\le {p}_{\alpha }}\sum \limits_{\left|\tilde{\boldsymbol{\beta}}\right|\le {p}_{\beta }}\sum \limits_{\begin{array}{l}\left|\boldsymbol{r}+\tilde{\boldsymbol{\alpha}}\right|\le {p}_{\alpha },\\ {}\left|\boldsymbol{r}+\tilde{\boldsymbol{\beta}}\right|\le {p}_{\beta}\end{array}}{u}_{\tilde{\boldsymbol{\alpha}}+\boldsymbol{r}}{v}_{\tilde{\boldsymbol{\beta}}+\boldsymbol{r}}\left(\begin{array}{c}\tilde{\boldsymbol{\alpha}}+\boldsymbol{r}\\ {}\boldsymbol{r}\end{array}\right)\left(\begin{array}{c}\tilde{\boldsymbol{\beta}}+\boldsymbol{r}\\ {}\boldsymbol{r}\end{array}\right)r!\frac{\sqrt{\left(\tilde{\boldsymbol{\alpha}}+\tilde{\boldsymbol{\beta}}\right)!}}{\sqrt{\left(\tilde{\boldsymbol{\alpha}}+\boldsymbol{r}\right)!\left(\tilde{\boldsymbol{\beta}}+\boldsymbol{r}\right)!}}{\psi}_{\tilde{\boldsymbol{\alpha}}+\tilde{\boldsymbol{\beta}}}\left(\boldsymbol{\xi} \right) $$

(60)

For simplicity, we still denote $ \boldsymbol{\alpha} =\tilde{\boldsymbol{\alpha}} $ and $ \boldsymbol{\beta} =\tilde{\boldsymbol{\beta}} $. Then, let θ = α + β, thus α = θ − β ≥ 0 and 0 ≤ β ≤ θ. The above summation is equivalent to

$$ {\displaystyle \begin{array}{l} uv=\sum \limits_{\left|\boldsymbol{\theta} \right|\le {p}_{\alpha }+{p}_{\beta }}\sum \limits_{\boldsymbol{\alpha} +\boldsymbol{\beta} =\boldsymbol{\theta}}\sum \limits_{\begin{array}{l}\left|\boldsymbol{\alpha} +\boldsymbol{r}\right|\le {p}_{\alpha },\\ {}\left|\boldsymbol{\beta} +\boldsymbol{r}\right|\le {p}_{\beta}\end{array}}{u}_{\boldsymbol{\alpha} +\boldsymbol{r}}{v}_{\boldsymbol{\beta} +\boldsymbol{r}}\left(\begin{array}{c}\boldsymbol{\alpha} +\boldsymbol{r}\\ {}\boldsymbol{r}\end{array}\right)\left(\begin{array}{c}\boldsymbol{\beta} +\boldsymbol{r}\\ {}\boldsymbol{r}\end{array}\right)r!\frac{\sqrt{\boldsymbol{\theta} !}}{\sqrt{\left(\boldsymbol{\alpha} +\boldsymbol{r}\right)!\left(\boldsymbol{\beta} +\boldsymbol{r}\right)!}}{\psi}_{\boldsymbol{\theta}}\left(\boldsymbol{\xi} \right)\\ {}=\sum \limits_{\left|\boldsymbol{\theta} \right|\le {p}_{\alpha }+{p}_{\beta }}\sum \limits_{0\le \boldsymbol{\beta} \le \boldsymbol{\theta}}\sum \limits_{\begin{array}{l}\left|\boldsymbol{\theta} -\boldsymbol{\beta} +\boldsymbol{r}\right|\le {p}_{\alpha },\\ {}\left|\boldsymbol{\beta} +\boldsymbol{r}\right|\le {p}_{\beta}\end{array}}{u}_{\boldsymbol{\theta} -\boldsymbol{\beta} +\boldsymbol{r}}{v}_{\boldsymbol{\beta} +\boldsymbol{r}}\left(\begin{array}{c}\boldsymbol{\theta} -\boldsymbol{\beta} +\boldsymbol{r}\\ {}\boldsymbol{r}\end{array}\right)\left(\begin{array}{c}\boldsymbol{\beta} +\boldsymbol{r}\\ {}\boldsymbol{r}\end{array}\right)r!\frac{\sqrt{\boldsymbol{\theta} !}}{\sqrt{\left(\boldsymbol{\theta} -\boldsymbol{\beta} +\boldsymbol{r}\right)!\left(\boldsymbol{\beta} +\boldsymbol{r}\right)!}}{\psi}_{\boldsymbol{\theta}}\left(\boldsymbol{\xi} \right)\end{array}} $$

(61)

For simplicity, denote

$$ C\left(\boldsymbol{\theta}, \boldsymbol{\beta}, \boldsymbol{r}\right)=\left(\begin{array}{c}\boldsymbol{\theta} -\boldsymbol{\beta} +\boldsymbol{r}\\ {}\boldsymbol{r}\end{array}\right)\left(\begin{array}{c}\boldsymbol{\beta} +\boldsymbol{r}\\ {}\boldsymbol{r}\end{array}\right)r!\frac{\sqrt{\boldsymbol{\theta} !}}{\sqrt{\left(\boldsymbol{\theta} -\boldsymbol{\beta} +\boldsymbol{r}\right)!\left(\boldsymbol{\beta} +\boldsymbol{r}\right)!}} $$

(62)

Thus, we have

$$ uv=\sum \limits_{\left|\boldsymbol{\theta} \right|\le {p}_{\alpha }+{p}_{\beta }}\sum \limits_{0\le \boldsymbol{\beta} \le \boldsymbol{\theta}}\sum \limits_{\begin{array}{l}\left|\boldsymbol{\theta} -\boldsymbol{\beta} +\boldsymbol{r}\right|\le {p}_{\alpha },\\ {}\left|\boldsymbol{\beta} +\boldsymbol{r}\right|\le {p}_{\beta}\end{array}}C\left(\boldsymbol{\theta}, \boldsymbol{\beta}, \boldsymbol{r}\right){u}_{\boldsymbol{\theta} -\boldsymbol{\beta} +\boldsymbol{r}}{v}_{\boldsymbol{\beta} +\boldsymbol{r}}{\psi}_{\boldsymbol{\theta}}\left(\boldsymbol{\xi} \right) $$

(63)

which completes the proof.

Appendix 4

In this section, we would prove the statement in Section 3.2 that the multiplication of two PCEs converges in the L₂ sense. Let us consider a system as y = g(x) with independent distribution input x. And assume that the PCE approximation of this system is expressed as y_p = g_PCE(ξ), where p is the order of PCE, and ξ is the standard normal variable associated with input x. Then, since PCE is L₂ convergence, we have

$$ \underset{p\to \infty }{\lim }E{\left({y}_p-y\right)}^2=\underset{p\to \infty }{\lim }{\int}_{\Omega}{\left({y}_p-y\right)}^2f\left(\boldsymbol{x}\right)d\boldsymbol{x}=0 $$

(64)

where Ω is the domain of input x, and f(x) is the joint PDF that subjects to ∫_Ωf(x)dx = 1.

Therefore, ∀ 0 < ε < 1, ∃ n > N where N is a positive integer,

$$ \left|{\int}_{\Omega}{\left({y}_p-y\right)}^2f\left(\boldsymbol{x}\right)d\boldsymbol{x}\right|<\varepsilon <1 $$

(65)

Now considering the fact that (y_p − y)² ≥ 0, if (y_p − y)² ≥ 1, we would derive that

$$ \left|{\int}_{\Omega}{\left({y}_p-y\right)}^2f\left(\boldsymbol{x}\right)d\boldsymbol{x}\right|\ge \left|{\int}_{\Omega}f\left(\boldsymbol{x}\right)d\boldsymbol{x}\right|=1 $$

(66)

which is in contradiction with (65). So, we would have 0 ≤ (y_p − y)² < 1. This means that |y_p − y| is bounded.

Noting that the absolute value of response of given system is less than 1 in our proposed method, namely, |y| ≤ 1. So, we could obtain that

$$ \left|{y}_p\right|\le 2 $$

(67)

Now, we would prove the convergence of PCE multiplication. First of all, we would prove a basis inequality as

$$ {\displaystyle \begin{array}{l}{\left({y}_p^k-{y}^k\right)}^2={\left({y}_p-y\right)}^2{\left({y}_p^{k-1}+{y}_p^{k-2}y+\cdots {y}_p{y}^{k-2}+{y}^{k-1}\right)}^2\\ {}\kern4.599998em \le {\left({y}_p-y\right)}^2{\left(\left|{y}_p^{k-1}\right|+\left|{y}_p^{k-2}\right|\left|y\right|+\cdots \left|{y}_p\right|\left|{y}^{k-2}\right|+\left|{y}^{k-1}\right|\right)}^2\\ {}\kern4.499998em \le {\left({y}_p-y\right)}^2{\left({2}^{k-1}+{2}^{k-2}+\cdots 2+1\right)}^2\\ {}\kern4.499998em ={\left({y}_p-y\right)}^2{\left({2}^k-1\right)}^2\end{array}} $$

(68)

where k means the order of required moment.

Then, with this preparation, we could derive that for a limited number k,

$$ {\displaystyle \begin{array}{l}\underset{p\to \infty }{\lim }E{\left({y}_p^k-{y}^k\right)}^2=\underset{p\to \infty }{\lim }{\int}_{\Omega}{\left({y}_p^k-{y}^k\right)}^2f\left(\boldsymbol{x}\right)d\boldsymbol{x}\\ {}\le \underset{p\to \infty }{\lim }{\int}_{\Omega}{\left({y}_p-y\right)}^2f\left(\boldsymbol{x}\right)d\boldsymbol{x}\\ {}\le {\left({2}^k-1\right)}^2\underset{p\to \infty }{\lim }{\int}_{\Omega}{\left({y}_p-y\right)}^2f\left(\boldsymbol{x}\right)d\boldsymbol{x}\\ {}=0\end{array}} $$

(69)

This means that if y_p is L₂ convergence with given system, the k-power of y_p also converges to y^k in L₂ sense. Thus, the approach of PCE multiplication could provide an accurate estimation of high order moment information. So, the proof completes.

References

Abramov RV (2007) An improved algorithm for the multidimensional moment-constrained maximum entropy problem. J Comput Phys 226(1):621–644MathSciNetCrossRefMATHGoogle Scholar
Abramov RV (2009) The multidimensional moment-constrained maximum entropy problem: a BFGS algorithm with constraint scaling. J Comput Phys 228(1):96–108MathSciNetCrossRefMATHGoogle Scholar
Bandyopadhyay K, Bhattacharya AK et al (2005) Maximum entropy and the problem of moments: a stable algorithm. Phys Rev E Stat Nonlinear Soft Matter Phys 71(2):057701CrossRefGoogle Scholar
Bucher C, Most T (2008) A comparison of approximate response functions in structural reliability analysis. Probab Eng Mech 23(2):154–163CrossRefGoogle Scholar
Chakraborty S, Chowdhury R (2015) A semi-analytical framework for structural reliability analysis. Comput Methods Appl Mech Eng 289:475–497MathSciNetCrossRefGoogle Scholar
Dai H, Zhang H et al (2016) A new maximum entropy-based importance sampling for reliability analysis. Struct Saf 63:71–80CrossRefGoogle Scholar
Fan J, Liao H et al (2018) Local maximum-entropy based surrogate model and its application to structural reliability analysis. Struct Multidiscip Optim 57:373–392MathSciNetCrossRefGoogle Scholar
Gherlone M, Mattone MC et al (2013) A comparative study of uncertainty propagation methods in structural problems. Comput Methods Appl Sci 26:87–111CrossRefMATHGoogle Scholar
Gotovac H, Gotovac B (2009) Maximum entropy algorithm with inexact upper entropy bound based on Fup basis functions with compact support. J Comput Phys 228(24):9079–9091MathSciNetCrossRefMATHGoogle Scholar
Huang W, Mao J et al (2014) The maximum entropy estimation of structural reliability based on Monte Carlo simulation. ASME 2014 33rd International Conference on Ocean, Offshore and Arctic EngineeringGoogle Scholar
Isukapalli SS (1999) Uncertainty analysis of transport-transformation models. The State University of New Jersey, RutgersGoogle Scholar
Jamshidi AA, Kirby MJ (2010) Skew-radial basis function expansions for empirical modeling. SIAM J Sci Comput 31(6):4715–4743MathSciNetCrossRefMATHGoogle Scholar
Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106(4):620–630MathSciNetCrossRefMATHGoogle Scholar
Kaymaz I (2005) Application of kriging method to structural reliability problems. Struct Saf 27(2):133–151CrossRefGoogle Scholar
Kelley CT (1999) Iterative methods for optimization. Society for Industrial and Applied MathematicsGoogle Scholar
Lasota R, Stocki R et al (2015) Polynomial chaos expansion method in estimating probability distribution of rotor-shaft dynamic responses. Bull Pol Acad Sci Tech Sci 63(2):413–422Google Scholar
Li G, Zhang K (2011) A combined reliability analysis approach with dimension reduction method and maximum entropy method. Struct Multidiscip Optim 43(1):121–134MathSciNetCrossRefMATHGoogle Scholar
Luo W (2006) Wiener chaos expansion and numerical solutions of stochastic partial differential equations. California Institute of TechnologyGoogle Scholar
Mason JC, Handscomb DC (2002) Chebyshev polynomials. CRC PressGoogle Scholar
Meruane V, Ortiz-Bernardin A (2015) Structural damage assessment using linear approximation with maximum entropy and transmissibility data. Mech Syst Signal Process 54-55:210–223CrossRefGoogle Scholar
Petrov V, Soares CG et al (2013) Prediction of extreme significant wave heights using maximum entropy. Coast Eng 74(5):1–10CrossRefGoogle Scholar
Rabitz HAAC (1999) General foundations of high-dimensional model representations. J Math Chem 25(2):197–233MathSciNetCrossRefMATHGoogle Scholar
Rahman S, Xu H (2004) A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probab Eng Mech 19(4):393–408CrossRefGoogle Scholar
Savin E, Faverjon B (2017) Higher-order moments of generalized polynomial chaos expansions for intrusive and non-intrusive uncertainty quantification. 19th AIAA Non-Deterministic Approaches ConferenceGoogle Scholar
Shi X, Teixeira AP et al (2014) Structural reliability analysis based on probabilistic response modelling using the maximum entropy method. Eng Struct 70(9):106–116CrossRefGoogle Scholar
Sudret B (2008) Global sensitivity analysis using polynomial chaos expansions. Reliab Eng Syst Saf 93(7):964–979CrossRefGoogle Scholar
Sudret B, Der Kiureghian A (2002) Comparison of finite element reliability methods. Probab Eng Mech 17(4):337–348CrossRefGoogle Scholar
Xiu D, Em Karniadakis G (2002) Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput Methods Appl Mech Eng 191(43):4927–4948MathSciNetCrossRefMATHGoogle Scholar
Zhang X, Pandey MD (2013) Structural reliability analysis based on the concepts of entropy, fractional moment and dimensional reduction method. Struct Saf 43(9):28–40CrossRefGoogle Scholar
Zhao YG, Ono T (1999) A general procedure for first/second-order reliabilitymethod (FORM/SORM). Struct Saf 21(2):95–112CrossRefGoogle Scholar
Zio E (2013) The Monte Carlo simulation method for system reliability and risk analysis. Springer, LondonCrossRefGoogle Scholar

Copyright information

Authors and Affiliations

Jianbin Guo
- 1
- 2
Jianyu Zhao
- 1
Shengkui Zeng
- 1
- 2
Email author

1.School of Reliability and Systems EngineeringBeihang UniversityBeijingChina
2.Science and Technology on Reliability and Environmental Engineering LaboratoryBeihang UniversityBeijingChina

Structural reliability analysis based on analytical maximum entropy method using polynomial chaos expansion

Abstract

Keywords

Notes

Acknowledgements

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article

Cookies