![th10 hypercube th10 hypercube](https://cocbases.com/wp-content/uploads/th10-dark-farming-base-2-air-sweepers-and-bomb-tower.jpg)
The updated regions (see Figure 6) generated by the interval method are generally greater than those calculated from the covariance method. In the interval model, the mean value of the updating parameters is not necessarily located in the mid-point between the upper and lower bounds, whereas in the covariance method, this is a constraint on parameters represented by normal distributions. It can be seen from Figure 6 that the updated regions of output data, produced by both covariance and interval methods, enclose the measured samples. Nevertheless the output clouds (shown in Figure 6) are larger from the interval method than the covariance method which can be explained by the unbounded correlation for the interval method. In most cases the 3 s intervals for the covariance method are larger than the intervals determined by the interval method. The bar plot shows that the updated values are very similar for both approaches. To compare both methods only 3 s is shown for each parameter of the updated covariance matrix. The updated parameter bounds with respect to the initial parameter mean value from the finite element model are presented in Figure 5. The initial hypercube of updating parameters was constructed within the range of 0.1–2.0 of the mean values. Convergence was achieved within three iterations together with further reduction of the objective function. A final adjustment was then carried out by interval updating using inverse propagation of data samples corresponding to the 14 frequencies (but not the mode-shapes) using the Kriging model by inverse propagation of measured samples. The starting values were therefore obtained using both the 14 frequency and 14 mode- shape residuals, as explained previously. The parameter starting values used in the interval model updating were the converged means from deterministic model updating described in Section 4.1. The size and orientation of the 105 frequency clouds show a very good agreement between test (red) and updated analysis clouds (green) for the active range. This type of figure can also be seen as a graphical representation of off-diagonal values of a covariance matrix. The mean values before and after updating are shown in Table 2 The results presented in Figure 6 include all combinations of the active frequency samples for the test and for both updating methods. The parameter means were updated using eigenvalue and mode-shape residuals, but for the covariances only the eigenvalue residuals were used. The selected modes are also called active mode set while the other ones (passive mode set) are taken to test the prediction quality of the updated model. Stochastic model updating was carried out for the 18 parameters listed in Table 1 using residuals comprising modes 1-8, 10-12, 14, 19 and 20. Further details on the mathematical development of the method are provided by Haddad Khodaparast et al (2011). The need for a highly computationally efficient meta-model is illustrated by the fact that a large number of inversions are required for interval model updating. 6 may then be solved by taking all the measured data (represented by circles in Figure 4 and by mapping them in the reverse direction (using the Kriging predictor) to find the vertices of the updated parameter hypercube. This can be achieved by increasing the number of training samples until the Mean Squared Error (MSE) at an unsampled point falls below a threshold, when the Kriging model is deemed to be accurate enough. Care should be taken to make sure that the mapping is good enough to represent the relationship between inputs and outputs with sufficient accuracy. In the next step, an initial hypercube around this point is constructed and the meta-model (the Kriging predictor in the present case) is then used to map the space of the initial hypercube of updating parameters to the space of outputs. If the solution is unique, the vector of updated parameter mean values can be represented by a point in the parameter space. Firstly, the mean values of updating parameters are found by deterministic model updating using the mean values of measured data. The interval model updating procedure is illustrated in Figure 4. The accuracy of the technique depends on the type of meta-model, the sampling used and the behaviour of the outputs within the range of parameter variation. (2011) based on the use of a meta-model which acts as a highly efficient surrogate for the deterministic finite element model. (6) that avoids the use of interval arithmetic (known to be unduly conservative) was developed by Haddad Khodaparast et al. ɶ represents an interval vector or matrix.