Lambros S. Katafygiotis

A probabilistic approach to structural model updating

The problem of updating a structural model and its associated uncertainties by utilizing measured dynamic response data is addressed. A Bayesian probabilistic formulation is followed to obtain the posterior probability density function (PDF) of the uncertain model parameters for given measured data. The present paper discusses the issue of identifiability of the model parameters and reviews existing asymptotic approximations for identifiable cases. The focus of the paper is on the treatment of the general unidentifiable case where the earlier approximations are not applicable. In this case the posterior PDF of the parameters is found to be concentrated in the neighborhood of an extended and extremely complex manifold in the parameter space. The computational difficulties associated with calculating the posterior PDF in such cases are discussed and an algorithm for an efficient approximate representation of the above manifold and the posterior PDF is presented. Numerical examples involving noisy data are presented to demonstrate the concepts and the proposed method.

Substructure Identification and Health Monitoring Using Noisy Response Measurements Only

A probabilistic substructure identification and health monitoring methodology for linear systems is presented using measured response time histories only. A very large number of uncertain parameters have to be identified if one considers the updating of the entire structure. For identifiability, one then would require a very large number of sensors. Furthermore, even when such a large number of sensors are available, process- ing of vast amount of the corresponding data raises com- putational difficulties. In this article a substructuring ap- proach is proposed, which allows for the identification and monitoring of some critical substructures only. The proposed method does not require any interface measure- ments and/or excitation measurements. No information regarding the stochastic model of the input is required. Specifically, the method does not require the response to be stationary and does not assume any knowledge of the parametric form of the spectral density of the input. There- fore, the method has very wide applicability. The proposed approach allows one to obtain not only the most probable values of the updated model parameters but also their as- sociated uncertainties using only one set of response data. The probability of damage can be computed directly using data from the undamaged and possibly damaged struc- ture. A hundred-story building model is used to illustrate the proposed method.

Bayesian Fast Fourier Transform Approach for Modal Updating Using Ambient Data

The problem of identification of the modal parameters of a structural model using measured ambient response time histories is addressed. A Bayesian Fast Fourier Transform approach (BFFTA) for modal updating is presented which uses the statistical properties of the Fast Fourier transform (FFT) to obtain not only the optimal values of the updated modal parameters but also their associated uncertainties, calculated from their joint probability distribution. Calculation of the uncertainties of the identified modal parameters is very important when one plans to proceed with the updating of a theoretical finite element model based on modal estimates. The proposed approach requires only one set of response data in contrast to many of the existing frequency-based approaches which require averaging. It is found that the updated PDF can be well approximated by a Gaussian distribution centred at the optimal parameters at which the posterior PDF is maximized. Examples using simulated data are presented to illustrate the proposed method.

Bayesian time–domain approach for modal updating using ambient data

Abstract The problem of identification of the modal parameters of a structural model using measured ambient response time histories is addressed. A Bayesian time–domain approach for modal updating is presented which is based on an approximation of a conditional probability expansion of the response. It allows one to obtain not only the optimal values of the updated modal parameters but also their associated uncertainties, calculated from their joint probability distribution. Calculation of the uncertainties of the identified modal parameters is very important if one plans to proceed in a subsequent step with the updating of a theoretical finite-element model based on modal estimates. The proposed approach requires only one set of response data. It is found that the updated PDF can be well approximated by a Gaussian distribution centered at the optimal parameters at which the updated PDF is maximized. Examples using simulated data are presented to illustrate the proposed method.

Optimal Sensor Placement Methodology for Identification with Unmeasured Excitation

A methodology is presented for designing cost-effective optimal sensor configurations for structural model updating and health monitoring purposes. The optimal sensor configuration is selected such that the resulting measured data are most informative about the condition of the structure. This selection is based on an information entropy measure of the uncertainty in the model parameter estimates obtained using a statistical system identification method. The methodology is developed for the uncertain excitation case encountered in practical applications for which data are to be taken either from ambient vibration tests or from other uncertain excitations such as earthquake and wind. Important issues related to robustness of the optimal sensor configuration to uncertainties in the structural model are addressed. The theoretical developments are illustrated by designing the optimal configuration for a simple 8-DOF chain-like model of a structure subjected to an unmeasured base excitation and a 40-DOF truss model subjected to wind/earthquake excitation. @DOI: 10.1115/1.1410929#

Unified Probabilistic Approach for Model Updating and Damage Detection

A probabilistic approach for model updating and damage detection of structural systems is presented using noisy incomplete input and incomplete response measurements. The situation of incomplete input measurements may be encountered, for example, during low-level ambient vibrations when a structure is instrumented with accelerometers that measure the input ground motion and the structural response at a few instrumented locations but where other excitations, e.g., due to wind, are not measured. The method is an extension of a Bayesian system identification approach developed by the authors. A substructuring approach is used for the parameterization of the mass, damping and stiffness distributions. Damage in a substructure is defined as stiffness reduction established through the observation of a reduction in the values of the various substructure stiffness parameters compared with their initial values corresponding to the undamaged structure. By using the proposed probabilistic methodology, the probability of various damage levels in each substructure can be calculated based on the available dynamic data. Examples using a single-degree-of-freedom oscillator and a 15-story building are considered to demonstrate the proposed approach.

Spectral density estimation of stochastic vector processes

A spectral density matrix estimator for stationary stochastic vector processes is studied. As the duration of the analyzed data tends to infinity, the probability distribution for this estimator at each frequency approaches a complex Wishart distribution with mean equal to an aliased version of the power spectral density at that frequency. It is shown that the spectral density matrix estimators corresponding to different frequencies are asymptotically statistically independent. These properties hold for general stationary vector processes, not only Gaussian processes, and they allow efficient calculation of updated probabilities when formulating a Bayesian model updating problem in the frequency domain using response data. A three-degree-of-freedom Duffing oscillator is used to verify the results.

Approximate analysis of response variability of uncertain linear systems

A probabilistic methodology is presented for obtaining the variability and statistics of the dynamic response of multi-degree-of-freedom linear structures with uncertain properties. Complex mode analysis is employed and the variability of each contributing mode is analyzed separately. Low-order polynomial approximations are first used to express modal frequencies, damping ratios and participation factors with respect to the uncertain structural parameters. Each modal response is then expanded in a series of orthogonal polynomials in these parameters. Using the weighted residual method, a system of linear ordinary differential equations for the coefficients of each series expansion is derived. A procedure is then presented to calculate the variability and statistics of the uncertain response. The technique is extended to the stochastic excitation case for obtaining the variability of the response moments due to the variability of the system parameters. The methodology can treat a variety of probability distributions assumed for the structural parameters. Compared to existing analytical techniques, the proposed method drastically reduces the computational effort and computer storage required to solve for the response variability and statistics. The performance and accuracy of the method are illustrated by examples.

Treatment of unidentifiability in structural model updating

The present study addresses the issues of non-uniqueness and unidentifiability arising in structural model updating. A Bayesian probabilistic framework is used for model updating which properly handles the uncertainties due to model error and measurement noise associated with model updating. Uncertainties in the model parameters are quantified by probability density functions (PDF) specifying the relative plausibilities of the possible values of the parameters. The Bayesian formulation is well-suited for updating the PDF of the uncertain model parameters taking into account engineering experience and measured dynamic data. Methods are presented for approximating this updated PDF for the general unidentifiable case for which the region of significant probabilities is concentrated in the neighborhood of a manifold of lower dimension than the original parameter space. This PDF is useful for both model updating and structural damage predictions. Asymptotic approximations are also developed for computing the uncertainties in the model response predictions. It is demonstrated that unidentifiable cases are not treatable by existing results valid only for identifiable cases for which the dimension of the manifold is exactly zero. Two examples involving simulated model error and measurement noise are presented to demonstrate the advantages of the new proposed method in effectively addressing unidentifiability issues.

DYNAMIC RESPONSE VARIABILITY OF STRUCTURES WITH UNCERTAIN PROPERTIES

A modal-based analysis of the dynamic response variability of multiple degree-of-freedom linear structures with uncertain parameters subjected to either deterministic or stochastic excitations is considered. A probabilistic methodology is presented in which random variables with specified probability distributions are used to quantify the parameter uncertainties. The uncertainty in the response due to uncertainties in the structural modelling and loading is quantified by various probabilistic measures such as mean, variance and coefficient of excess. The computation of these probabilistic measures is addressed. A series expansion involving orthogonal polynomials in terms of the system parameters is first used to model the response variability of each contributing mode. Linear equations for the coefficients of each series expansion are derived using the weighted residual method. Mode superposition is then used to derive analytical expressions for the variability and statistics of the uncertain response in terms of the coefficients of the series expansions for all contributing modes. A primary-secondary system and a ten-story building subjected to deterministic and stochastic loads are used to demonstrate the methodology, as well as evaluate its performance by comparing it to existing methods, including the computationally cost-efficient perturbation method.