29 Non-unique Estimates in Material Parameter Identification of Nonlinear. . . 259 29.3.1 Unscented Kalman Filtering The process of recursive Bayesian filtering involves sequentially computing the probability distribution of k given the current and previous measurements y1:k ={y1, y2, . . . . , yk}: k|k ∼ p(θk| y1:k), k =1, 2, 3, . . . . This involves initializing the filter with an initial probability distribution of the unknown parameter vector 0|0 ∼ p(θ0) to sequentially compute p(θk| y1:k) for every k, i.e., recursively update the probability distribution of unknown parameter vector considering new measurements. At each time k, the process of recursive updating involves computing p(θk| y1:k −1), referred to as predicted parameter distribution at time step tk, using p(θk −1| y1:k −1) and then updating the predicted distribution to p(θk| y1:k), referred to as the posterior probability distribution of k given measurements y1:k or updated parameter distribution at time step tk, after observing measurement vector yk. Note that the FE model hk is a nonlinear function of θk. Therefore, the nonlinear Kalman filter, a nonlinear Bayesian filtering technique, is used in this study to compute the probability distribution of the random vectors k|k, k =1, 2, 3, . . . . The nonlinear Kalman filter is a special type of nonlinear Bayes filter for which: 1. The initial distribution of the unknown parameter vector is modeled as Gaussian. Therefore, initial parameter vector 0|0 is modeled as a Gaussian random vector with estimated mean vector θ0|0 and estimated covariance matrix Pθθ 0|0 , i.e., 0|0 ∼N θ0|0, Pθθ 0|0 . 2. Both the process noise and measurement noise are modeled as zero-mean Gaussian white noise processes, i.e., wk ∼ N(0, Qk) and vk ∼N(0, Rk) for all k, where Qk and Rk are the process and measurement noise covariance matrices, respectively, at time step tk. 3. The process noise, wk, and measurement noise, vk, across all time steps tk (k =0, 1, 2, . . . ), along with the initial parameter vector, 0|0, are assumed to be mutually statistically independent. 4. The posterior distribution of the unknown parameter vector is assumed to be Gaussian. Therefore, k|k, at any time tk, is a Gaussian random vector with estimated mean vector θk|k and estimated covariance matrix Pθθ k|k . The unscented Kalman filter (UKF) is a special type of nonlinear Kalman filter which uses a minimal set of deterministically chosen sample points V(i), also known as sigma-points (SPs), to represent a random vector E ∈ RnE (Gaussian or non-Gaussian). The SPs are selected such that they accurately capture the true mean vector and the true covariance matrix of the random vector E. These SPs when propagated through any nonlinear function, F=g(E), capture the true mean vector and covariance matrix of the transformed random vector Faccurately up to second order (third order if Efollows a Gaussian distribution) [9]. A deterministic sampling technique known as scaled Unscented transformation (SUT) is used in this paper. Figure 29.1 summarizes the algorithm to recursively estimate the mean vector and the covariance matrix of the unknown parameter vector of a nonlinear FE model using the UKF. 29.4 Application Example Parameter estimation or system identification of concrete gravity dams is a subject of interest to many researchers [10]. Such studies were conducted by Chopra and co-workers using forced vibration test data to estimate the linear elastic material parameters such as Young’s modulus of concrete [11] but without accounting for uncertainty. In the present work, the multiaxial material model parameters which govern both the elastic and plastic behavior of the concrete of a dam are estimated using numerically simulated seismic response data contaminated with added Gaussian white noise (to mimic real-world data). The uncertainty due to the input and output measurement noise and the unknown model parameters is accounted for, but the effects of modeling uncertainty are not considered here. To achieve this, an idealized 2D model of Pine Flat dam is developed in OpenSees and the FE simulated noisy seismic response data are used to recursively estimate the time-invariant parameters of the cap plasticity material model (used to model the plain concrete of the dam) using the UKF. Figure 29.2 shows a picture of Pine Flat dam and illustrates the 2D plane strain nonlinear FE model developed. In this FE model, the dam is assumed to be sitting on a rigid foundation, i.e., the boundary conditions at the bottom of the dam are assumed fixed. The concrete is assumed to be isotropic and homogeneous (i.e., characterized by the same material model and same set of material parameter values) over the entire cross-section of the dam. Each finite element consists of a bilinear quadrilateral element with material behavior at each integration point governed by the cap plasticity model (see Fig. 29.3). The dynamic interaction between the water reservoir and the dam (i.e., fluid-structure interaction) is not accounted for in this study; the hydrostatic water pressure distribution along the upstream face of the dam, see Fig. 29.2, is applied statically and kept constant during the dynamic seismic response analysis.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==