A hybrid method for stochastic response analysis of a vibrating structure

Abstract

Response analysis of a linear structure with uncertainties in both structural parameters and external excitation is considered here. When such an analysis is carried out using the spectral stochastic finite element method (SSFEM), often the computational cost tends to be prohibitive due to the rapid growth of the number of spectral bases with the number of random variables and the order of expansion. For instance, if the excitation contains a random frequency, or if it is a general random process, then a good approximation of these excitations using polynomial chaos expansion (PCE) involves a large number of terms, which leads to very high cost. To address this issue of high computational cost, a hybrid method is proposed in this work. In this method, first the random eigenvalue problem is solved using the weak formulation of SSFEM, which involves solving a system of deterministic nonlinear algebraic equations to estimate the PCE coefficients of the random eigenvalues and eigenvectors. Then the response is estimated using a Monte Carlo (MC) simulation, where the modal bases are sampled from the PCE of the random eigenvectors estimated in the previous step, followed by a numerical time integration. It is observed through numerical studies that this proposed method successfully reduces the computational burden compared with either a pure SSFEM of a pure MC simulation and more accurate than a perturbation method. The computational gain improves as the problem size in terms of degrees of freedom grows. It also improves as the timespan of interest reduces.

Appendix

1.1 The Jacobian of the NR iterations

In this section, the analytical form of the Jacobian matrix is derived from Eqs. (13, 14). Note that these expressions are the modified forms of the expressions derived in [30], to include the mass matrix into consideration. For a given m between 0 and $P-1$ let $F_{m}$, denote the system of P equations in Eq. (13). Then its derivatives with respect to the components of the unknown chaos coefficients can be expressed as

$$\begin{aligned} \frac{\partial {F_{m}}}{\partial {\lambda ^{(i)}}}= & {} -\sum _{k=0}^{L_2-1}\sum _{j=0}^{P-1}{\mathbb {E}}\{\psi _i\psi _j\psi _k\psi _m\} \mathbf{M}^{(k)}{\varvec{\phi }}^{(j)} , \quad m = 0,\ldots , P-1 , \quad i = 0,\ldots , P-1, \end{aligned}$$

(30)

$$\begin{aligned} \frac{\partial {F_{m}}}{\partial {{\varvec{\phi }}^{(j)}}}= & {} \sum _{i=0}^{L_1-1}{\mathbb {E}}\{\psi _i\psi _j\psi _m\} \mathbf{K}^{(i)} - \sum _{i=0}^{P-1}\sum _{k=0}^{L_2-1}{\mathbb {E}}\{\psi _i\psi _j\psi _k\psi _m\} \lambda ^{(i)} \mathbf{M}^{(k)}, \nonumber \\ m= & {} 0,\ldots , P-1, \quad j = 0,\ldots , P-1. \end{aligned}$$

(31)

Equation (31) results in a matrix whose $(i_1,i_2)$th term is the derivative of the $i_1$-th function in $F_{m}$ with respect to the $i_2$-th element of ${\varvec{\phi }}^{(j)}$.

Similarly, from Eq. (14),

$$\begin{aligned}&\frac{\partial {f_m}}{\partial {\lambda ^{(j)}}}=0,\quad m = 0,\ldots , P-1 , j = 0,\ldots , P-1 ,\end{aligned}$$

(32)

$$\begin{aligned}&\frac{\partial {f_m}}{\partial {{\varvec{\phi }}^{(j)}}}=2\sum _{i=0}^{P-1}\sum _{k=0}^{L_2-1}\mathbf{M}^{(k)}{\varvec{\phi }}^{(i)}{\mathbb {E}}\{\psi _i\psi _j\psi _k\psi _m\},\quad m = 0,\ldots , P-1 , j = 0,\ldots , P-1 . \end{aligned}$$

(33)