The proposed ABC-SubSim algorithm for structural model updating using modal data

Input

M, K0: the mass and nominal stiffness matrix

y = {${\widehat{f}}_r^i$, ${\widehat{\varphi }}_r^i$}: the measured frequencies and mode shapes, i = 1:Ns, r = 1:Nm

θ0, p(θ): the nominal value and prior PDF of model parameters

Output

θopt, Θpost: the optimal value and posterior samples of model parameters

/*Initialization*/

$R^{tol}$, $S^{tol}$: tolerances for stopping criterion of subset simulation

P0, N: the conditional probability and the number of samples in each level of subset simulation

w0: calculate the initial values for the weighting factors based on nominal value θ0

εw: tolerances for stopping criterion of weighting factors iteration

k = 1

while (k==1 or $\left|w_k-w_{k-1}\right|/w_{k-1}>{\epsilon }_w$) do /*External loop for weighting factors iteration*/

Sample $\left[({\theta }_0^{\left(1\right)},x_0^{\left(1\right)}),\dots ,({\theta }_0^{\left(n\right)},x_0^{\left(n\right)}),\dots ,({\theta }_0^{\left(N\right)},x_0^{\left(N\right)})\right]$, where $\left(\theta ,x\right)~p\left(\theta \right)p(x|\theta )$

j = 1

while (j==1 or Rj >$R^{tol}$ and Sj > $S^{tol}$) do /*Internal loop for subset simulation*/

for n: 1, …, N do

Evaluate ${\rho }_j^{\left(n\right)}=\rho (\eta (x_{j-1}^{\left(n\right)}),\eta \left(y\right))$

end for

Sort and renumber the samples $\left[({\theta }_{j-1}^{\left(n\right)},x_{j-1}^{\left(n\right)}),n:1,\dots ,N\right]$ so that ${\rho }_j^{\left(1\right)}\le {\rho }_j^{\left(2\right)}\le \cdots {\rho }_j^{\left(N\right)}$

$${\mathsf{ϵ}}_j=\frac{1}{2}\left({\rho }_j^{\left(N{P}_0\right)}+{\rho }_j^{\left(N{P}_0+1\right)}\right)$$

for l = 1, …, $N{P}_0$ do

Select as a seed $({\theta }_{j-1}^{\left(l\right)},x_{j-1}^{\left(l\right)})\in D_j$

Run a self-regulating modified Metropolis algorithm [29] to generate 1/$P_0$ states of a Markov chain lying in $D_j$: $\left[({\theta }_j^{\left(l\right),1},x_j^{\left(l\right),1}),\dots ,({\theta }_j^{\left(l\right),1/P_0},x_j^{\left(l\right),1/P_0})\right]$

end for

Renumber $\left[({\theta }_j^{\left(l\right),m},x_j^{\left(l\right),m}),l=1,\dots ,N{P}_0;m=1,\dots ,1/P_0\right]$ as $\left[({\theta }_j^{\left(1\right)},x_j^{\left(1\right)}),\dots ,({\theta }_j^{\left(N\right)},x_j^{\left(N\right)})\right]$

j = j + 1

Update Rj and Sj

end while

k = k + 1

Take the optimal value in the last level of subset simulation samples as the initial guess for a local optimization search operator, and conduct the local optimization search to obtain a fine optimum.

Update the weighting factors $w_k$ based on the newly obtained optimal model parameters

end while