Non-probabilistic uncertain inverse problem method considering correlations for structural parameter identification

Abstract

This paper presents an effective sequence interval and correlation inverse strategy for the uncertain inverse problem, aiming to identify the uncertainties and non-probabilistic correlations of the structural parameters simultaneously. First, an ellipsoidal convex model is adopted to quantify the uncertainty boundary of the measured responses with limited samples. Then, the uncertain inverse problem based on the ellipsoidal convex model is decoupled into an interval inverse problem and a correlation inverse problem. For the interval inverse problem, a subinterval decomposition analysis method constrained by the ellipsoidal convex model is developed to evaluate the intervals of the structural responses with a low computational cost. For the correlation inverse problem, the correlation propagation equations are derived to calculate the non-probabilistic correlation coefficient matrix of the structural responses. After that, by using optimization algorithms to circularly reduce the errors of the intervals and the correlation coefficients between the measured responses and calculated structural responses, the intervals and the non-probabilistic correlation coefficient matrix of the structural parameters are identified effectively, and an ellipsoidal convex model of the structural parameters can be established eventually. Two numerical examples and one experimental example are investigated to verify the effectiveness and accuracy of the proposed sequence interval and correlation inverse strategy.

Appendix

Assuming there are n-dimensional structural parameters X_i i = 1, 2, ⋯, n with a few numbers of the measured samples X_i(s) s = 1, 2, ⋯, n_s, an EC model can be constituted as (2). Set

$$ {U}_i={X}_i-{X}_i^C\kern0.5em i=1,2,\cdots, n $$

(22)

where ${U}_i=\left[-{X}_i^W,{X}_i^W\right]$, and ${U}_i^C=0$. Considering (3) and (Yang et al. 2019), the uncertainty domain described in (2) can be transformed as

$$ {\Omega}_U=\left\{\mathbf{U}|{\mathbf{U}}^{\mathrm{T}}\mathbf{Co}{\mathbf{v}}_X^{-1}\mathbf{U}\le 1\right\} $$

(23)

Multiplying matrix U from the left side and U^T from the right side of (23), there is

$$ {\Omega}_U=\left\{\mathbf{U}|{\mathbf{U}\mathbf{U}}^{\mathrm{T}}\mathbf{Co}{\mathbf{v}}_X^{-1}\mathbf{U}{\mathbf{U}}^{\mathrm{T}}\le \mathbf{U}{\mathbf{U}}^{\mathrm{T}}\right\} $$

(24)

Moving the right side to the left side, it can be achieved as

$$ {\Omega}_U=\left\{\mathbf{U}|{\mathbf{U}\mathbf{U}}^{\mathrm{T}}\left(\mathbf{Co}{\mathbf{v}}_X^{-1}\mathbf{U}{\mathbf{U}}^{\mathrm{T}}-\mathbf{E}\right)\le \mathbf{0}\right\} $$

(25)

where E is the unit matrix. Due to ${U}_i=\left[-{X}_i^W,{X}_i^W\right]$, the two sides of (25) are equal only when $\mathbf{Co}{\mathbf{v}}_X^{-1}\mathbf{U}{\mathbf{U}}^{\mathrm{T}}-\mathbf{E}=\mathbf{0}$. Thus, there is

$$ \mathbf{Co}{\mathbf{v}}_X^{-1}\mathbf{U}{\mathbf{U}}^{\mathrm{T}}=\mathbf{E} $$

(26)

Multiplying Cov_X from the left side on both sides of (26) and considering (Yang et al. 2019), it can obtain as

$$ \mathbf{U}{\mathbf{U}}^{\mathrm{T}}={\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X $$

(27)

where the diagonal matrix ${\mathbf{W}}_X=\operatorname{diag}\left[{X}_1^W,{X}_2^W,\cdots, {X}_n^W\right]$. Meanwhile, assume there are n nonlinear models as

$$ {Y}_i={g}_i\left({X}_1,{X}_2,\cdots, {X}_n\right)\kern0.5em i=1,2,\cdots, n $$

(28)

Using the Taylor expansion to extend (28) at middle points, it leads to

$$ {Y}_i={g}_i\left(\mathbf{X}\right)={g}_i\left({\mathbf{X}}_0^C\right)+{\boldsymbol{\upalpha}}_i^{\mathrm{T}}\left(\mathbf{X}-{\mathbf{X}}^C\right)+{\left(\mathbf{X}-{\mathbf{X}}^C\right)}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_i}{2!}\left(\mathbf{X}-{\mathbf{X}}^C\right)+{\vartheta}_i $$

(29)

where α_i is the first-order derivative vector of the structural model g_i(X). β_i is the second-order Hessian matrix of g_i(X). ϑ_i is the higher-order term. Then

$$ {\displaystyle \begin{array}{l}\left({Y}_i-{\hat{Y}}_i^C\right)\left({Y}_j-{\hat{Y}}_j^C\right)={\mathrm{g}}_i\left(\mathbf{X}\right){\mathrm{g}}_j\left(\mathbf{X}\right)\\ {}=\left({\boldsymbol{\upalpha}}_i^{\mathrm{T}}\left(\mathbf{X}-{\mathbf{X}}^C\right)+{\left(\mathbf{X}-{\mathbf{X}}^C\right)}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_i}{2!}\left(\mathbf{X}-{\mathbf{X}}^C\right)\right){\left({\boldsymbol{\upalpha}}_j^{\mathrm{T}}\left(\mathbf{X}-{\mathbf{X}}^C\right)+{\left(\mathbf{X}-{\mathbf{X}}^C\right)}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_j}{2!}\left(\mathbf{X}-{\mathbf{X}}^C\right)\right)}^{\mathrm{T}}\end{array}} $$

(30)

where ${\hat{Y}}_i^C=g\left({\mathbf{X}}^C\right)+{\vartheta}_i$ and ${\hat{Y}}_j^C=g\left({\mathbf{X}}^C\right)+{\vartheta}_j\kern0.5em j=1,2,\cdots, n$. Similarly, we employ an EC model as (Wang 2019) to quantify the uncertainty boundary of the structural responses, and take (22), (27), and (30) into account. There is

$$ \mathbf{Co}{\mathbf{v}}_Y=\left(\mathbf{Y}-{\hat{\mathbf{Y}}}^C\right){\left(\mathbf{Y}-{\hat{\mathbf{Y}}}^C\right)}^{\mathrm{T}}=\left[\begin{array}{c}{\boldsymbol{\upalpha}}_1^{\mathrm{T}}\mathbf{U}+{\mathbf{U}}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_1}{2!}\mathbf{U}\\ {}{\boldsymbol{\upalpha}}_2^{\mathrm{T}}\mathbf{U}+{\mathbf{U}}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_2}{2!}\mathbf{U}\\ {}\vdots \\ {}{\boldsymbol{\upalpha}}_n^{\mathrm{T}}\mathbf{U}+{\mathbf{U}}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_n}{2!}\mathbf{U}\end{array}\right]{\left[\begin{array}{c}{\boldsymbol{\upalpha}}_1^{\mathrm{T}}\mathbf{U}+{\mathbf{U}}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_1}{2!}\mathbf{U}\\ {}{\boldsymbol{\upalpha}}_2^{\mathrm{T}}\mathbf{U}+{\mathbf{U}}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_2}{2!}\mathbf{U}\\ {}\vdots \\ {}{\boldsymbol{\upalpha}}_n^{\mathrm{T}}\mathbf{U}+{\mathbf{U}}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_n}{2!}\mathbf{U}\end{array}\right]}^{\mathrm{T}} $$

(31)

where ${\hat{\mathbf{Y}}}^C=\mathbf{g}\left({\mathbf{X}}^C\right)+\vartheta$. Equation (31) can be rewritten as

$$ {\displaystyle \begin{array}{c}{Cov}_{Y_i{Y}_j}=\left({\boldsymbol{\upalpha}}_i^{\mathrm{T}}\mathbf{U}+{\mathbf{U}}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_i}{2!}\mathbf{U}\right){\left({\boldsymbol{\upalpha}}_j^{\mathrm{T}}\mathbf{U}+{\mathbf{U}}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_j}{2!}\mathbf{U}\right)}^{\mathrm{T}}\\ {}={\boldsymbol{\upalpha}}_i^{\mathrm{T}}\mathbf{U}{\mathbf{U}}^{\mathrm{T}}{\boldsymbol{\upalpha}}_j+{\boldsymbol{\upalpha}}_i^{\mathrm{T}}\mathbf{U}{\mathbf{U}}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_j^{\mathrm{T}}}{2!}\mathbf{U}+{\mathbf{U}}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_i}{2!}\mathbf{U}{\mathbf{U}}^{\mathrm{T}}{\boldsymbol{\upalpha}}_j+{\mathbf{U}}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_i}{2!}\mathbf{U}{\mathbf{U}}^{\mathrm{T}}\frac{{\boldsymbol{\upbeta}}_j^{\mathrm{T}}}{2!}\mathbf{U}\end{array}} $$

(32)

where j = 1, 2, …, n. Substituting (27) into (32) and applying trace operation on both sides of (32) simultaneously, it can be obtained that

$$ {\displaystyle \begin{array}{c}{\mathrm{Cov}}_{Y_i{Y}_j}={\boldsymbol{\upalpha}}_i^{\mathrm{T}}{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X{\boldsymbol{\upalpha}}_j+\frac{1}{2}\mathrm{Trace}\left({\boldsymbol{\upalpha}}_i^{\mathrm{T}}{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X{\boldsymbol{\upbeta}}_j^{\mathrm{T}}\mathbf{U}\right)+\\ {}\frac{1}{2}\mathrm{Trace}\left({\mathbf{U}}^{\mathrm{T}}{\boldsymbol{\upbeta}}_i{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X{\boldsymbol{\upalpha}}_j\right)+\frac{1}{4}\mathrm{Trace}\left({\boldsymbol{\upbeta}}_i{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X{\boldsymbol{\upbeta}}_j^{\mathrm{T}}\mathbf{Co}{\mathbf{v}}_X\right)\end{array}} $$

(33)

Integrating U into the uncertainty domain Ω_U, there is

(34)

Due to the integral domain Ω_U is symmetrical to the original point, the following equations can be obtained.

(35)

and

(36)

Substituting (35) and (36) into (34), there is

$$ {Cov}_{Y_i{Y}_j}={\boldsymbol{\upalpha}}_i^{\mathrm{T}}{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X{\boldsymbol{\upalpha}}_j+\frac{1}{4}\mathrm{Trace}\left({\boldsymbol{\upbeta}}_i{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X{\boldsymbol{\upbeta}}_j^{\mathrm{T}}{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X\right) $$

(37)

Combining (Yang et al. 2019) with (37), the correlation coefficient ${\rho}_{Y_i{Y}_j}$ can be achieved as

$$ {\displaystyle \begin{array}{c}{\rho}_{Y_{\mathrm{i}}{Y}_{\mathrm{j}}}=\frac{Cov_{Y_i{Y}_j}}{\sqrt{Cov_{Y_i{Y}_i}}\sqrt{Cov_{Y_j{Y}_j}}}\\ {}=\frac{{\boldsymbol{\upalpha}}_i^{\mathrm{T}}{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X{\boldsymbol{\upalpha}}_j+\frac{1}{4}\mathrm{Trace}\left({\boldsymbol{\upbeta}}_i{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X{\boldsymbol{\upbeta}}_j^{\mathrm{T}}{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X\right)}{\sqrt{{\boldsymbol{\upalpha}}_i^{\mathrm{T}}{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X{\boldsymbol{\upalpha}}_i+\frac{1}{4}\mathrm{Trace}\left({\boldsymbol{\upbeta}}_i{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X{\boldsymbol{\upbeta}}_i^{\mathrm{T}}{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X\right)}\cdot \sqrt{{\boldsymbol{\upalpha}}_j^{\mathrm{T}}{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X{\boldsymbol{\upalpha}}_j+\frac{1}{4}\mathrm{Trace}\left({\boldsymbol{\upbeta}}_j{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X{\boldsymbol{\upbeta}}_j^{\mathrm{T}}{\mathbf{W}}_X{\boldsymbol{\uprho}}_X{\mathbf{W}}_X\right)}}\end{array}} $$

(38)

Similarly, we can also derive the correlation propagation equations when the nonlinear system models are expanded to n-order Taylor expansion.