A modified stochastic perturbation algorithm for closely-spaced eigenvalues problems based on surrogate model

Abstract

Aiming at uncertainty propagation and dynamic reanalysis of closely-spaced eigenvalues, with consideration of uncertainties in design variables, a modified stochastic perturbation method is proposed. Concerning quasi-symmetric or partial-symmetric structures that frequently appear, one of their primary features is closely-distributed natural frequencies. For structure with closely-spaced eigenvalues, due to its instability and sensitivity to the changes of design variables and its excessively concentrated adjacent eigenvalues, conventional uncertainty analysis or dynamic reanalysis methods for distinct eigenvalue are no longer available. Initially, the spectral decompositions of stiffness and mass matrices are provided; by transfer technique, the eigen-problem of closely-spaced eigenvalues is converted to that of repeated eigenvalues with two perturbation parts appended; then the perturbed closely-spaced eigenvalue is rewritten as the sum of original closely-spaced eigenvalues’ mean value and surrogate model which approximates the first-order perturbation term by polynomial chaos expansions. According to this method, statistical quantities of perturbed closely-spaced eigenvalues are calculated directly and accurately, which contributes to its uncertainty analysis and dynamic reanalysis. Furthermore, the capability of proposed method in dealing with relatively large uncertainties and complex engineering structure is demonstrated. The accuracy and efficiency of proposed method have been verified sufficiently by numerical examples.

Appendices

Appendix

1.1 Discrimination of closely-spaced eigenvalues

The perturbation method for closely-spaced eigenvalues differs from the method for distinct eigenvalue and repeated eigenvalues. Hence, before the perturbation analysis is implemented on an eigenvalue and its eigenvector, we have to distinguish the type of these eigenvalues.

There are some existing methods about the discrimination of closely-spaced eigenvalues, concerning two adjacent eigenvalues λ _i and λ _i+1, the index of intensive degree δ is defined as

$$ \delta =\left({\lambda}_{i+1}-{\lambda}_i\right)/\left({\lambda}_{i+1}+{\lambda}_i\right). $$

(1)

The smaller δ is, the closer λ _i and λ _i+1 are. According to (1), δ measures the gap between adjacent eigenvalues.

Nevertheless, there are some limitations on the adoption of δ. For instance, if two vibration systems are investigated, in which we have δ ₁ = 0.03 and δ ₂ = 0.09, respectively, one cannot draw a conclusion that the system corresponding to δ ₁ = 0.03 is associated with closely-spaced eigenvalues while the system corresponding to δ ₂ = 0.09 is associated with distinct eigenvalue. Therefore, the aforementioned index δ doesn’t reflect the essence of closely-spaced eigenvalues, which leads to the difficulty in application.

In this work, the angel θ _i between the original eigenvector u ⁱ₀ and the eigenvector u _i after perturbation is utilized for measuring the intensive degree of adjacent eigenvalues, which is defined as

$$ {\theta}_i=1- c o{s}^2\left\langle {\boldsymbol{u}}_0^i,{\boldsymbol{u}}_i\right\rangle =1-\left|{{\boldsymbol{u}}_0^i}^{\mathrm{T}}{\boldsymbol{u}}_i\right|/{\left(\left({{\boldsymbol{u}}_0^i}^{\mathrm{T}}{\boldsymbol{u}}_0^i\right)\left({{\boldsymbol{u}}_i}^{\mathrm{T}}{\boldsymbol{u}}_i\right)\right)}^{\frac{1}{2}}, $$

(2)

with regard to u ⁱ₀ is a complex eigenvector, θ _i is defined as

$$ {\theta}_i=1- c o{s}^2\left\langle {\boldsymbol{u}}_0^i,{\boldsymbol{u}}_i\right\rangle =1-\left|{{\boldsymbol{u}}_0^i}^{\mathrm{H}}{\boldsymbol{u}}_i\right|/{\left(\left({{\boldsymbol{u}}_0^i}^{\mathrm{H}}{\boldsymbol{u}}_0^i\right)\left({{\boldsymbol{u}}_i}^{\mathrm{H}}{\boldsymbol{u}}_i\right)\right)}^{\frac{1}{2}}. $$

(3)

If θ _i approaches 0, then the perturbations of parametric matrices don’t have much influence on u ⁱ₀ ; If θ _i approaches 1, then the perturbations of parametric matrices generate considerable influence on u ⁱ₀ . As it is well known, the closely-spaced eigenvectors are very sensitive to the parametric perturbations. So if θ _i is approximately equal to 1, we can preliminarily conclude that λ ⁱ₀ has a relatively small gap from the adjacent eigenvalues. Furthermore, if every element in the set (θ _j , θ _j + 1, ⋯, θ _k ) obtained from (3) approaches 1, it’s for sure that (λ _j , λ _j + 1, ⋯, λ _k ) is a set of clustered eigenvalues.

In conclusion, the discrimination of closely-spaced eigenvalues is summarized as:

• 1) Calculating each and every θ _i in the system;

  $$ {\theta}_i=1- c o{s}^2\left\langle {\boldsymbol{u}}_0^i,{\boldsymbol{u}}_i\right\rangle =1-\left|{{\boldsymbol{u}}_0^i}^{\mathrm{T}}{\boldsymbol{u}}_i\right|/{\left(\left({{\boldsymbol{u}}_0^i}^{\mathrm{T}}{\boldsymbol{u}}_0^i\right)\left({{\boldsymbol{u}}_i}^{\mathrm{T}}{\boldsymbol{u}}_i\right)\right)}^{\frac{1}{2}}, $$

• 2) Identifying a cluster (θ _j , θ _j + 1, ⋯, θ _k ) in which every element is large in value from θ = {θ ₁, θ ₂, ⋯, θ _n };

• 3) The cluster of eigenvalues (λ _j , λ _j + 1, ⋯, λ _k ) which corresponds to (θ _j , θ _j + 1, ⋯, θ _k ) is a set of closely-spaced eigenvalues.

Difficulties in applying the distinct eigenvalue’s perturbation method on closely-spaced eigenvalues

For clarity, we start the investigation from the first-order and second-order perturbation formulae in terms of the distinct eigenvalue as follows:

$$ {\boldsymbol{K}}_0{\boldsymbol{u}}_0^i={\lambda}_0^i{\boldsymbol{M}}_0{\boldsymbol{u}}_0^i, $$

(1)

$$ {{\boldsymbol{u}}_0^i}^{\mathrm{T}}{\boldsymbol{M}}_0{\boldsymbol{u}}_0^i=1, $$

(2)

where K ₀ and M ₀ are n×n symmetric stiffness and mass matrices, respectively; λ ⁱ₀ and u ⁱ₀ represent the distinct eigenvalue and its eigenvector, respectively.

After the perturbations, K ₀ and M ₀ will become K ₀ + ε K ₁ and M ₀ + ε M ₁, respectively; the eigenpairs λ _i  , u _i corresponding to λ ⁱ₀ , u ⁱ₀ are expressed as

$$ {\lambda}_i={\lambda}_0^i+\varepsilon {\lambda}_1^i+{\varepsilon}^2{\lambda}_2^i+ O\left({\varepsilon}^3\right), $$

(3)

$$ {\boldsymbol{u}}_i={\boldsymbol{u}}_0^i+\varepsilon {\boldsymbol{u}}_1^i+{\varepsilon}^2{\boldsymbol{u}}_2^i+ O\left({\varepsilon}^3\right), $$

(4)

in which ε denotes a minimum parameter.

In the light of matrix perturbation method, simultaneously for brevity in expression, the first-order and second-order perturbation of eigenvalue λ _i are obtained as

$$ \left\{\begin{array}{l}{\lambda}_1^i={{\boldsymbol{u}}_0^i}^{\mathrm{T}}\left({\boldsymbol{K}}_1-{\lambda}_0^i{\boldsymbol{M}}_1\right){\boldsymbol{u}}_0^i\\ {}{\lambda}_2^i={{\boldsymbol{u}}_0^i}^{\mathrm{T}}\left({\boldsymbol{K}}_1-{\lambda}_0^i{\boldsymbol{M}}_1\right){\boldsymbol{u}}_1^i-{\lambda}_1^i{{\boldsymbol{u}}_0^i}^{\mathrm{T}}{\boldsymbol{M}}_0{\boldsymbol{u}}_1^i-{\lambda}_1^i{{\boldsymbol{u}}_0^i}^{\mathrm{T}}{\boldsymbol{M}}_1{\boldsymbol{u}}_0^i\end{array}\right., $$

(5)

on the basis of modal superposition method, we have

$$ \begin{array}{ll}{\boldsymbol{u}}_1^i={\displaystyle \sum_{j=1}^n{C}_{i j}{\boldsymbol{u}}_0^j}\kern0.5em ,\hfill & {\boldsymbol{u}}_2^i={\displaystyle \sum_{j=1}^n{D}_{i j}{\boldsymbol{u}}_0^j}\hfill \end{array}, $$

(6)

in conjunction with

$$ \begin{array}{l}{C}_{i j}=\left[\begin{array}{ll}-\frac{1}{2}{{\boldsymbol{u}}_0^i}^{\mathrm{T}}{\boldsymbol{M}}_1{\boldsymbol{u}}_0^i\hfill & \left( i= j\right)\hfill \\ {}\frac{1}{\lambda_0^i-{\lambda}_0^j}{{\boldsymbol{u}}_0^j}^{\mathrm{T}}\left({\boldsymbol{K}}_1-{\lambda}_0^i{\boldsymbol{M}}_1\right){\boldsymbol{u}}_0^i\hfill & \left( i\ne j\right)\hfill \end{array}\right.,\hfill \\ {}{D}_{i j}=\left[\begin{array}{ll}-\frac{1}{2}{{\boldsymbol{u}}_1^i}^{\mathrm{T}}\left({\boldsymbol{M}}_0{\boldsymbol{u}}_1^i+{\boldsymbol{M}}_0{\boldsymbol{u}}_0^i+{\boldsymbol{M}}_1{\boldsymbol{u}}_0^i\right)\hfill & \kern0.2em \left( i= j\right)\hfill \\ {}\frac{1}{\lambda_0^i-{\lambda}_0^j}\left[{{\boldsymbol{u}}_0^j}^{\mathrm{T}}\left({\boldsymbol{K}}_1-{\lambda}_1^i{\boldsymbol{M}}_1\right){\boldsymbol{u}}_0^i-\frac{\lambda_1^j}{\lambda_0^i-{\lambda}_0^j}{{\boldsymbol{u}}_0^j}^{\mathrm{T}}\left({\boldsymbol{M}}_0{\boldsymbol{u}}_1^i-{\boldsymbol{M}}_1{\boldsymbol{u}}_0^i\right)\right]\hfill & \left( i\ne j\right)\hfill \end{array}\right..\hfill \end{array} $$

• (1) Truncation error in the perturbation expansions

  Note that in perturbation (5)-(6), when the clustered eigenvalues occur, (λ ⁱ₀  − λ ^j₀ ) will become extremely small, and then C _ij is no longer a small coefficient. Thus u ⁱ₁ is not small compared to u ⁱ₀ , due to the fact that u ⁱ₁ is included in the expression of λ ⁱ₂ , which means λ ⁱ₂ is no longer a minimum as well.

  Additionally, 1/(λ ⁱ₀  − λ ^j₀ ) and u ⁱ₁ exist in D _ij , which means u ⁱ₂ is also not small compared to u ⁱ₁ . Videlicet, when there comes the clustered natural frequencies, the existing perturbation method for distinct eigenvalue, no matter first-order or second-order perturbation, is probably invalid. On the premise that perturbations of design variables are very small, although the convergence of perturbation expansions could be guaranteed, the calculative result is lack of reliability because of truncation error.

• (2) Difficulties in determining the individual eigenvector correspond to clustered eigenvalues

  Each and every eigenvector which belongs to a set of clustered eigenvalues is hard to determine precisely, but the eigen-subspace spanned by those entire eigenvectors can be determined precisely. According to the theory of algebraic eigenvalue, the distinct eigenvalue of real symmetric matrix is well-conditioned. However, its eigenvector is probably ill-conditioned, and the ill-conditioned degree depends on how intensive the natural frequencies are.

Perturbation analysis for repeated eigenvalues

Assuming that λ ⁱ₀ are repeated eigenvalues with m multiplicities, which belongs to a degenerate system. Accordingly, there should be m eigenvectors w ⁱ(i = 1, 2, ⋯, m), which are pairwise orthogonal, subject to

$$ {\boldsymbol{K}}_0{\boldsymbol{w}}^i={\lambda}_0^i{\boldsymbol{M}}_0{\boldsymbol{w}}^i, $$

(1)

in which K ₀, M ₀ denote original stiffness matrix and original mass matrix of the system with repeated eigenvalues, respectively. Moreover, the linear combination of w ⁱ is also the eigenvector of repeated eigenvalue λ ⁱ₀ , which can be expressed as

$$ \begin{array}{l}{\boldsymbol{u}}_0^i={\alpha}_1{\boldsymbol{w}}^1+{\alpha}_2{\boldsymbol{w}}^2+\cdots +{\alpha}_m{\boldsymbol{w}}^m\hfill \\ {}\kern1.62em =\left[{\boldsymbol{w}}^1,{\boldsymbol{w}}^2,\cdots, {\boldsymbol{w}}^m\right]\cdot {\left[{\alpha}_1,{\alpha}_2,\cdots, {\alpha}_m\right]}^{\mathrm{T}}\hfill \\ {}\kern1.62em =\boldsymbol{W}\cdot \boldsymbol{\alpha} \hfill \end{array}, $$

(2)

where α is an undetermined vector.

The structural parameters of degenerate system will be changed under the external perturbation, such as variations of serving environment and differences in raw materials, which finally results in the variations of mass matrix and stiffness matrix as follows:

$$ \left\{\begin{array}{l}\boldsymbol{K}\kern0.6em =\kern0.5em {\boldsymbol{K}}_0\kern0.3em +\varepsilon {\boldsymbol{K}}_1\hfill \\ {}\boldsymbol{M}={\boldsymbol{M}}_0+\varepsilon {\boldsymbol{M}}_1\hfill \end{array}\right., $$

(3)

in which ε is a minimum parameter, and the original system, known as the system corresponding to ε = 0. ε M ₁, ε K ₁ indicate the first-order perturbation term of M ₀, K ₀, respectively.

Repeated eigenvalues no longer exist in the system after perturbation, which means there should be m different eigenvalues λ ¹, λ ², ⋯, λ ^m corresponding to m different eigenvectors u ⁱ(i = 1, 2, ⋯, m). It is similar to the circumstance of distinct eigenvalue. We expand the eigenvalue λ ⁱ and eigenvector u ⁱ as ε -series form

$$ {\lambda}^i={\lambda}_0^i+\varepsilon {\lambda}_1^i+{\varepsilon}^2{\lambda}_2^i+\cdots, $$

(4)

$$ {\boldsymbol{u}}^i={\boldsymbol{u}}_0^i+\varepsilon {\boldsymbol{u}}_1^i+{\varepsilon}^2{\boldsymbol{u}}_2^i+\cdots . $$

(5)

By virtue of the perturbation method, (3), (4), (5) are substituted into Ku = λ Mu, then we expand it with O(ε ³) terms omitted. After that, comparing the coefficients of the same power terms of ε on each side of the equation, we have

$$ {\boldsymbol{K}}_0{\boldsymbol{u}}_1^i+{\boldsymbol{K}}_1{\boldsymbol{u}}_0^i={\lambda}_0^i{\boldsymbol{M}}_0{\boldsymbol{u}}_1^i+{\lambda}_0^i{\boldsymbol{M}}_1{\boldsymbol{u}}_0^i+{\lambda}_1^i{\boldsymbol{M}}_0{\boldsymbol{u}}_0^i, $$

(6)

substituting (2) into (6), yielding

$$ {\boldsymbol{K}}_0{\boldsymbol{u}}_1^i+{\boldsymbol{K}}_1{\displaystyle \sum_{j=1}^m{\alpha}_j{\boldsymbol{w}}^j={\lambda}_0^i{\boldsymbol{M}}_0{\boldsymbol{u}}_1^i+{\lambda}_0^i{\boldsymbol{M}}_1{\displaystyle \sum_{j=1}^m{\alpha}_j{\boldsymbol{w}}^j+{\lambda}_1^i{\boldsymbol{M}}_0{\displaystyle \sum_{j=1}^m{\alpha}_j}{\boldsymbol{w}}^j}}, $$

(7)

premultiplying (7) with [w ^k]^T gives

$$ {\left[{\boldsymbol{w}}^k\right]}^{\mathrm{T}}{\boldsymbol{K}}_0{\boldsymbol{u}}_1^i+{\displaystyle \sum_{j=1}^m{\left[{\boldsymbol{w}}^k\right]}^{\mathrm{T}}{\boldsymbol{K}}_1{\boldsymbol{w}}^j{\alpha}_j={\lambda}_0^i{\left[{\boldsymbol{w}}^k\right]}^{\mathrm{T}}{\boldsymbol{M}}_0{\boldsymbol{u}}_1^i+{\lambda}_0^i{\displaystyle \sum_{j=1}^m{\left[{\boldsymbol{w}}^k\right]}^{\mathrm{T}}{\boldsymbol{M}}_1{\boldsymbol{w}}^j{\alpha}_j+{\lambda}_1^i{\displaystyle \sum_{j=1}^m{\left[{\boldsymbol{w}}^k\right]}^{\mathrm{T}}{\boldsymbol{M}}_0{\boldsymbol{w}}^j{\alpha}_j}}}, $$

(8)

taking the following conditions into consideration

$$ {\left[{\boldsymbol{w}}^k\right]}^{\mathrm{T}}{\boldsymbol{K}}_0{\boldsymbol{u}}_1^i={\left[{\boldsymbol{u}}_1^i\right]}^{\mathrm{T}}{\boldsymbol{K}}_0{\boldsymbol{w}}^k={\lambda}_0^i{\left[{\boldsymbol{u}}_1^i\right]}^{\mathrm{T}}{\boldsymbol{M}}_0{\boldsymbol{w}}^k, $$

(9)

$$ {\lambda}_0^i{\left[{\boldsymbol{w}}^k\right]}^{\mathrm{T}}{\boldsymbol{M}}_0{\boldsymbol{u}}_1^i={\lambda}_0^i{\left[{\boldsymbol{u}}_1^i\right]}^{\mathrm{T}}{\boldsymbol{M}}_0{\boldsymbol{w}}^k, $$

(10)

and the orthogonality

$$ {\left[{\boldsymbol{w}}^k\right]}^{\mathrm{T}}{\boldsymbol{M}}_0{\boldsymbol{w}}^j={\delta}_{k j}, $$

(11)

Equation (8) is rewritten as

$$ \begin{array}{ll}{\displaystyle \sum_{j=1}^m\left({\left[{\boldsymbol{w}}^k\right]}^{\mathrm{T}}{\boldsymbol{K}}_1{\boldsymbol{w}}^j-{\lambda}_0^i{\left[{\boldsymbol{w}}^k\right]}^{\mathrm{T}}{\boldsymbol{M}}_1{\boldsymbol{w}}^j\right){\alpha}_j={\lambda}_1^i{\alpha}_j}\hfill & \left( k=1,2,\cdots, m\right)\hfill \end{array}. $$

(12)

Assuming that

$$ \left\{\begin{array}{l}{a}_{k j}={\left[{\boldsymbol{w}}^k\right]}^{\mathrm{T}}{\boldsymbol{M}}_1{\boldsymbol{w}}^j\\ {}{b}_{k j}={\left[{\boldsymbol{w}}^k\right]}^{\mathrm{T}}{\boldsymbol{K}}_1{\boldsymbol{w}}^j\end{array}\right., $$

(13)

then (12) is equivalent to

$$ \begin{array}{ll}{\displaystyle \sum_{j=1}^m\left({b}_{j k}-{\lambda}_0^i{a}_{k j}\right){\alpha}_j={\lambda}_1^i{\alpha}_k,}\hfill & k=1,2,\cdots, m\hfill \end{array}, $$

(14)

the matrix form of (14) can be represented as

$$ \left(\boldsymbol{D}-{\lambda}_1^i{\boldsymbol{I}}^{m\times m}\right)\boldsymbol{\alpha} =0, $$

(15)

where the element of matrix D is subject to

$$ \begin{array}{ll}{d}_{kj}={b}_{kj}-{\lambda}_0^i{a}_{kj},\hfill & \left( k, j=1,2,\cdots, m\right)\hfill \end{array}. $$

(16)

Equation (15) is a standard eigenvalue equation with m order, by solving this eigenvalue problem we will acquire the first-order perturbation term λ ^j₁ (j = 1, 2, ⋯, m), which corresponds to the original repeated eigenvalue λ ⁱ₀ , and the undetermined coefficient α _j .

λ ^j₁ (j = 1, 2, ⋯, m) are arranged in ascending order, if (15) has no repeated eigenvalues, then both λ ^j₁ and α _j are unique. For simplicity, we assume that (15) has no repeated eigenvalues.

Solving (15) equals to solve the following equation

$$ \det \left[\boldsymbol{D}-{\lambda}_1^j{\boldsymbol{I}}^{m\times m}\right]=0, $$

(17)

then λ ^j₁  (j = 1, 2, ⋯, m) is obtained, by substituting it back into (15), coefficients α _j  (j = 1, 2, ⋯, m) are acquired.

After that, according to (2), we will get the eigenvector which correspond to the repeated eigenvalues λ ⁱ₀ as follows

$$ \begin{array}{ll}{u}_0^j=\boldsymbol{W}{\alpha}_j\hfill & \left( j= i, i+1,\cdots, i+ m-1\right)\hfill \end{array} $$

(18)

Construction of surrogate model based on PCE method

Essentially speaking, it is the uncertainties of structural parameters, such as statistical variability of material property and geometry condition, errors caused by equipment in manufacture, variations in usage condition, that lead to the perturbation of mass matrix M ₁ and perturbation of stiffness matrix K ₁, i.e.,

$$ \begin{array}{ll}{\boldsymbol{M}}_1={\boldsymbol{M}}_1\left(\boldsymbol{\xi} \right)={\boldsymbol{M}}_1\left({\xi}_1,{\xi}_2,\cdots, {\xi}_n\right),\hfill & {\boldsymbol{K}}_1={\boldsymbol{K}}_1\left(\boldsymbol{\xi} \right)={\boldsymbol{K}}_1\left({\xi}_1,{\xi}_2,\cdots, {\xi}_n\right)\hfill \end{array}, $$

(1)

where ξ = ξ ₁, ξ ₂, ⋯, ξ _n denote the continuous random variables in system. It is worth mentioning that, to avoid ambiguity in derivation, we will use superscript q to stand for the serial number of closely-spaced eigenvalues’ first-order perturbation term. Assuming the original system as

$$ {\lambda}_1^q= f\left({\boldsymbol{M}}_1,{\boldsymbol{K}}_1\right)= f\left({\xi}_1,{\xi}_2,\cdots, {\xi}_n\right). $$

(2)

Let the probability space of original system be smooth adequately, in addition f(ξ ₁, ξ ₂, ⋯, ξ _n ) cannot be expressed explicitly. But it’s available to obtain the corresponding responses of original system by means of numerical approximation.

The surrogate model of original system is established by PCE method hereinafter. Homogeneous Hermite polynomial expansion is employed in terms of Gaussian random process (Dongbin and Em 2003; Field and Grigoriu 2004; Wang and Wang 2015), and the random response Y(ξ) is expressed as

$$ Y\left(\boldsymbol{\xi} \right)={c}_0+{\displaystyle \sum_{i_1=1}^n{c}_{i_1}{H}_1\left({\xi}_{i_1}\right)+{\displaystyle \sum_{i_1=1}^n{\displaystyle \sum_{i_2=1}^{i_1}{c}_{i_1{i}_2}{H}_2\left({\xi}_{i_1},{\xi}_{i_2}\right)}+{\displaystyle \sum_{i_1=1}^n{\displaystyle \sum_{i_2=1}^{i_1}{\displaystyle \sum_{i_3=1}^{i_2}{c}_{i_1{i}_2{i}_3}{H}_3\left({\xi}_{i_1},{\xi}_{i_2},{\xi}_{i_3}\right)+\cdots }}}}}, $$

(3)

in which Y(ξ) is a second-order random variable with finite variance. According to the Cameron-Martin theorem (Cameron and Martin 1947), it can approximate any functionals in L ₂(C) and converges in the L ₂(C) sense, where C is the space of real functions which are continuous on the interval [0, 1] and vanish at 0. In which $\boldsymbol{c}=\left({c}_0,{c}_{i_1},\cdots, {c}_{i_1{i}_2},\cdots, {c}_{i_1{i}_2{i}_3},\cdots \right)$ is a vector of undetermined coefficients, ξ = (ξ ₁, ξ ₂, ⋯, ξ _n ) denotes the Gaussian random variables, ${H}_n\left({\xi}_{i_1},{\xi}_{i_2},\cdots, {\xi}_{i_n}\right)$ stands for the n th order multidimensional Hermite polynomial.

We truncate (3) by finite terms, assuming the remaining number of terms is s, then (3) is simplified as

$$ Y\left(\boldsymbol{\xi} \right)\approx {\displaystyle \sum_{j=0}^{s-1}{\overset{\frown }{\boldsymbol{c}}}_j{\varGamma}_j\left(\boldsymbol{\xi} \right)}, $$

(4)

where ${\overset{\frown }{\boldsymbol{c}}}_j$ stands for the undetermined coefficients, ${\varGamma}_j\left(\boldsymbol{\xi} \right)={\varGamma}_j\left({\xi}_{i_1},{\xi}_{i_2},\cdots, {\xi}_{i_n}\right)$ denotes the jth order generalized Wiener-Askey polynomial chaos. Simultaneously, there is a one-to-one correspondence between the functions ${H}_n\left({\xi}_{i_1},{\xi}_{i_2},\cdots, {\xi}_{i_n}\right)$ and Γ _j (ξ), and their coefficients ${c}_0,{c}_{i_1},\cdots, {c}_{i_1{i}_2},\cdots, {c}_{i_1{i}_2{i}_3},\cdots$ and ${\overset{\frown }{\boldsymbol{c}}}_j$.

For clarity, the n-dimensional Hermite polynomial is shown as

$$ {H}_n\left({\xi}_{i_1},\cdots, {\xi}_{i_n}\right)={e}^{1/2{\boldsymbol{\xi}}^{\mathrm{T}}\boldsymbol{\xi}}{\left(-1\right)}^n\frac{\partial^n}{\partial {\xi}_{i_1}\cdots \partial {\xi}_{i_n}}{e}^{-1/2{\boldsymbol{\xi}}^{\mathrm{T}}\boldsymbol{\xi}}, $$

and one-dimensional Hermite polynomials are demonstrated as

$$ \begin{array}{l}{H}_0=1,\kern0.4em {H}_1\left(\xi \right)=\xi, \kern0.4em {H}_2\left(\xi \right)={\xi}^2-1,\kern0.4em {H}_3\left(\xi \right)={\xi}^3-3\xi, \hfill \\ {}{H}_{i+1}\left(\xi \right)=\xi {H}_i\left(\xi \right)- i{H}_{i-1}\left(\xi \right),\cdots \hfill \end{array}. $$

The polynomial chaos Γ _j (ξ) forms a complete orthogonal basis in the L ₂ space of the Gaussian random variables, i.e.,

$$ \left\langle {\varGamma}_i,{\varGamma}_j\right\rangle =\left\langle {\varGamma}_i^2\right\rangle {\delta}_{i j}, $$

where δ _ij is the Kronecker Delta operator and 〈⋅, ⋅ 〉 denotes the ensemble average which is also the inner product in the Hilbert space of Gaussian random variables ξ,

$$ \left\langle f\left(\boldsymbol{\xi} \right) g\left(\boldsymbol{\xi} \right)\right\rangle ={\displaystyle \int f\left(\boldsymbol{\xi} \right) g\left(\boldsymbol{\xi} \right) W\left(\boldsymbol{\xi} \right) d\boldsymbol{\xi}}, $$

where the weighting function is

$$ W\left(\boldsymbol{\xi} \right)=\frac{1}{\sqrt{{\left(2\pi \right)}^n}}{e}^{-1/2{\boldsymbol{\xi}}^{\mathrm{T}}\boldsymbol{\xi}}, $$

in this way, a complete set of orthogonal basis which is square-integrable is established with respect to n-dimensional Hermite polynomial. Moreover, the polynomials in (3) are convergence in mean square.

If the surrogate model is formed by second order Hermite polynomial chaos expansion, then the random response Y(ξ) can be expressed as

$$ {Y}_2\left(\boldsymbol{\xi} \right)={c}_{0,2}+{\displaystyle \sum_{i=1}^n{c}_{i,2}{\xi}_i+{\displaystyle \sum_{i=1}^n{c}_{i i,2}\left({\xi}_i^2-1\right)+{\displaystyle \sum_{i=1}^{n-1}{\displaystyle \sum_{j> i}^n{c}_{i j,2}{\xi}_i{\xi}_j}}}}, $$

(5)

where n denotes the dimensionality of random variables, c _0,2 , c _i,2 , c _ii,2 , c _ij,2 stand for the undetermined coefficients of Hermite polynomial expansion. By virtue of (5), one can induce the number of undetermined coefficients in second order Hermite polynomial expansion for approximate random response Y ₂(ξ) as s = (n + p) !/(n ! p !), in which p stands for the highest order in polynomial expansion.

Note that in (5), the key of constructing the surrogate model, which is formed by PCE method, is the calculation of undetermined coefficients c _0,2 , c _i,2 , c _ii,2 , c _ij,2, ⋯. Concerning the PCE surrogate model which consists of n-dimensional continuous random variables, the corresponding response functions will be generated upon the specified collocation points groups, and each collocation points group corresponds to a set of sample $\tilde{\boldsymbol{\xi}}=\left({\tilde{\xi}}_1,{\tilde{\xi}}_2,\cdots, {\tilde{\xi}}_n\right)$, which belongs to continuous random variable ξ = (ξ ₁, ξ ₂, ⋯, ξ _n ).

As mentioned before, all the continuous random variables in this paper are subject to Gaussian distribution. Besides that, Hermite polynomials are chosen as the orthogonal basis. It is a routine that if the highest order of Hermite polynomial is p, then roots of the (p + 1) th order Hermite polynomial are adopted as collocation points (Isukapalli S.S. Uncertainty analysis of transport-transformation models. PhD thesis and New Jersey 1999; Villadsen and Michelsen 1978; Sudret 2008; Berveiller et al. 2006; Menner A. Tatang. Direct Incorporation of Uncertainty in Chemical and Environmental Engineering Systems. PhD thesis and Technology 1995). After preparing all the collocation points, one substitutes the previous collocation points into the original system one by one. The corresponding response function will be generated sequently, and the following formula is established

$$ \left[\begin{array}{cccc}\hfill \begin{array}{c}\hfill {\varGamma}_0\left({\boldsymbol{\xi}}_0\right)\hfill \\ {}\hfill {\varGamma}_0\left({\boldsymbol{\xi}}_1\right)\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {\varGamma}_0\left({\boldsymbol{\xi}}_N\right)\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill {\varGamma}_1\left({\boldsymbol{\xi}}_0\right)\hfill \\ {}\hfill {\varGamma}_1\left({\boldsymbol{\xi}}_1\right)\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {\varGamma}_1\left({\boldsymbol{\xi}}_N\right)\hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill \cdots \hfill \\ {}\hfill \cdots \hfill \\ {}\hfill \cdots \hfill \\ {}\hfill \cdots \hfill \end{array}\hfill & \hfill \begin{array}{c}\hfill {\varGamma}_{s-1}\left({\boldsymbol{\xi}}_0\right)\hfill \\ {}\hfill {\varGamma}_{s-1}\left({\boldsymbol{\xi}}_1\right)\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {\varGamma}_{s-1}\left({\boldsymbol{\xi}}_N\right)\hfill \end{array}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {c}_0\hfill \\ {}\hfill {c}_1\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill {c}_{s-1}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill f\left({\boldsymbol{\xi}}_0\right)\hfill \\ {}\hfill f\left({\boldsymbol{\xi}}_1\right)\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill f\left({\boldsymbol{\xi}}_{\mathrm{N}}\right)\hfill \end{array}\right], $$

(6)

where ξ ₀, ξ ₁, ⋯, ξ _N denote the sampling point groups, N stands for the number of sampling point groups, s is the number of undetermined coefficients. By means of regression analysis based on the least squares method, the undetermined coefficients of polynomial chaos expansion could be calculated from (6).

Construction of M ₁, K ₁ in section 3

According to the stochastic perturbation method, which is truncated to first-order perturbation term in this paper, the system response g(x) = g(x ₁, x ₂, ⋯, x _n ) can be expressed as

$$ g\left(\boldsymbol{x}\right)= g\left(\overline{\boldsymbol{x}}\right)+{g}_1\left(\boldsymbol{x}\right), $$

(1)

viz. g(x ₁, x ₂, ⋯, x _n ) = g(μ ₁, μ ₂, ⋯, μ _n ) + g ₁(x ₁, x ₂, ⋯, x _n ),where x = [x ₁, x ₂, ⋯, x _n ] denotes the continuous random variables in structural system, $\overline{\boldsymbol{x}}=\left[{\mu}_1,{\mu}_2,\cdots, {\mu}_n\right]$ stands for the mean values of continuous random variables, g ₁(x) is the first-order perturbation term of system response.

On the other aspect, in the light of high dimensional model representation (HDMR), the first-order approximation of system response also can be denoted as

$$ g\left(\boldsymbol{x}\right)\equiv g\left({x}_1,{x}_2,\cdots, {x}_n\right)={\displaystyle \sum_{i=1}^n g\left({\mu}_1,\cdots, {\mu}_{i-1},{x}_i,{\mu}_{i+1},\cdots, {\mu}_n\right)-\left( n-1\right) g\left(\overline{\boldsymbol{x}}\right)}, $$

(2)

Taking (1) and (2) into consideration, the first-order perturbation term g ₁(x) is obtained as

$$ \begin{array}{l}{g}_1\left(\boldsymbol{x}\right)={\displaystyle \sum_{i=1}^n g\left({\mu}_1,\cdots, {\mu}_{i-1},{x}_i,{\mu}_{i+1},\cdots, {\mu}_n\right)- ng\left(\overline{\boldsymbol{x}}\right)}\hfill \\ {}={\displaystyle \sum_{i=1}^n\left[ g\left({\mu}_1,\cdots, {\mu}_{i-1},{x}_i,{\mu}_{i+1},\cdots, {\mu}_n\right)- g\left(\overline{\boldsymbol{x}}\right)\right]}\hfill \end{array}, $$

(3)

on the basis of the preceding derivation, the first-order perturbation term of mass matrix and stiffness matrix can be expressed as follows:

$$ \left\{\begin{array}{l}\kern0.1em {\boldsymbol{M}}_1={\boldsymbol{M}}_{\beta 1}={\displaystyle \sum_{i=1}^n\left[\boldsymbol{M}\left({\mu}_1,{\mu}_2,\cdots, {\mu}_{i-1},{\xi}_{\beta i},{\mu}_{i+1},\cdots {\mu}_n\right)-\boldsymbol{M}\left({\mu}_1,{\mu}_2,\cdots, {\mu}_{i-1},{\mu}_i,{\mu}_{i+1},\cdots, {\mu}_n\right)\right]\kern0.1em }\\ {}\kern0.1em {\boldsymbol{K}}_1={\boldsymbol{K}}_{\beta 1}={\displaystyle \sum_{i=1}^n\left[\boldsymbol{K}\left({\mu}_1,{\mu}_2,\cdots, {\mu}_{i-1},{\xi}_{\beta i},{\mu}_{i+1},\cdots {\mu}_n\right)-\boldsymbol{K}\left({\mu}_1,{\mu}_2,\cdots, {\mu}_{i-1},{\mu}_i,{\mu}_{i+1},\cdots, {\mu}_n\right)\right]\kern0.1em }\end{array}\right.. $$