29 Non-unique Estimates in Material Parameter Identification of Nonlinear. . . 261 Fig. 29.3 FE model hierarchy: (a) Structure level, (b) Element level, and (c) Material level stress tensor, and I denotes the fourth order identity tensor). In this paper, the eleven time-invariant parameters of the cap plasticity model define the unknown parameter vector θ as θ =[G, K, X0, D, W, R, λ, θ, β, α, T] T ∈R11×1. 29.4.1 Simulation The numerically simulated seismic response data with added Gaussian white noise (to simulate measurement noise) is assumed to represent the data measured from a real-world dam and is used in developing and validating the Bayesian filtering framework considered herein. In the simulation (data generation) phase, the 2D FE model of the dam characterized by a realistic set of material parameter values (obtained through calibrating the cap plasticity model to the Colorado concrete test data [7] at the material level) is subjected to the first 20 s of the 360◦ horizontal component of the 1994 Northridge earthquake (M6.7) recorded at Sylmar Hospital station scaled by factor 2. The set of material parameter values used in the simulation phase are referred to as θtrue from here on and are reported in Eq. (29.2). θtrue : G=1700 ksi, K=2100 ksi, D=0.0032 ksi−1 , W=0.42, X0 =16 ksi, R=4.43, λ =1.16 ksi, β =0.44 ksi−1 , θ =0.11, α =3.86 ksi, T =−0.3 ksi (29.2) Then, the absolute acceleration and relative displacement (with respect to the base of the dam) response time histories at locations A–G (see Fig. 29.3a) are obtained from the dynamic seismic response analysis. These response data are referred to as true measured responses and are now polluted with Gaussian white noise of root mean square (RMS) 1.0% g and 0.075 in, for the acceleration and displacement responses, respectively, to simulate the measurement noise. Therefore, the output response vector is defined as y =[aA, aB, . . . aF, dA, dB, . . . dF] T ∈R14×N, where N denotes the number of time steps (or sample points) in the simulated response histories, ai and di are a vector of the absolute acceleration and the relative displacement response, respectively, at location i ∈[A, B, . . . , F] (see Fig. 29.3a). The input (ground motion record) is also polluted with 1.0%g RMS Gaussian white noise to simulate measurement noise. Note that this input white noise transforms to a colored noise at the system output. Therefore, the total output noise (transformed input noise together with the added output measurement noise) is non-white which violates the white noise assumption for vk in Eq. (29.1) and may result in biased estimation of the model parameters [2]. Therefore, heterogeneous sensors (absolute acceleration and relative displacement response histories) are considered in this study to enhance the estimation accuracy.
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