Finite element method based Monte Carlo filters for structural system identification

Abstract

The paper proposes a strategy for combining two powerful computational procedures, namely, the finite element method (FEM) for structural analysis and particle filtering for dynamic state estimation, to tackle the problem of structural system parameter identification based on a set of noisy measurements on static and (or) dynamic structural responses. The proposed identification method automatically inherits the wide ranging capabilities of both FEM and particle filtering procedures thereby enabling the treatment of many complicated features of system identification problems, namely, imperfections in mathematical models for structural behavior, noisy and spatially incomplete measurements, nonlinear models for process and measurement equations, possible non-Gaussian nature of system and measurement noises, and dynamic and (or) static behaviors of the structure. Additionally, the authors propose a method that permits assimilation of measurements from multiple static and (or) dynamic tests and from multiple sensors in the system identification procedure in a unified manner. The paper describes how finite element models residing in commercially available softwares can be made to communicate with a database of measurements via a particle filtering algorithm developed on the Matlab platform. This leads to a probabilistic description of system parameters to be identified. Illustrative examples consider measurements from computational models, laboratory and field tests. These illustrations include studies on a rubber sheet with a hole undergoing large deformations, laboratory investigations on a single span beam and field investigations on an existing multi-span masonry arch bridge subjected to diagnostic moving loads.

Introduction

The problem of structural system identification constitutes an important and difficult class of inverse problems in structural mechanics. Various methods for identification of parameters of dynamical systems exist and the works of Juang [1], Ljung [2], Maia and De Silva [3], Heylen et al. [4], Bendat [5], Ewins [6], Pintelon and Schoukens [7], Peeters and Roeck [8], Lieven and Ewins [9], Worden and Tomlinson [10], Nelles [11] and Kerschen et al. [12] provide extensive overviews of the state of the art. Several issues related to non-uniqueness of solutions, spatio-temporal incompleteness of response measurements, presence of measurement noise, modeling uncertainties, possible presence of structural nonlinearities, and difficulties arising out of complete/partial lack of measurement of inputs have been addressed in these studies. The related problems of finite element model updating and vibration based structural damage detection have also received wide attention [13], [14], [15]. The focus of the present study is on exploring the application of identification methods that are based on dynamic state estimation (DSE) techniques. Specifically, we aim to develop identification procedures that account for following problem features: (a) structural modeling using the finite element (FE) method (especially when these models reside in commercially available FE analysis packages), (b) availability of a set of (noisy, spatially incomplete, static/dynamic) measurement data on strains, displacements and (or) accelerations, and (c) possibility of the structure behaving nonlinearly.

The problem of DSE is typically stated in the discrete state space form and consists of determination of conditional probability density function (pdf) of system states given a set of measurements. The errors in formulating the mathematical models governing the system states and measurements and also the sensor noises in acquiring measurements are treated as a sequence of identical independent random vectors. Consequently the vector of system states is modeled as a Markov vector and the application of Bayes’ theorem leads to a set of recursive relations for evolution of the posterior pdf conditioned on available measurements [16]. These relations are formulated in terms of a set of multi-dimensional integrals which, most often, are not amenable for closed-form solutions nor can they be practically be evaluated using rules of numerical quadrature. For linear state space models with additive Gaussian noises, an exact solution to this problem was developed by Kalman [17] and this formulation forms the cornerstone of many practical engineering applications [18], [19], [20], [21]. For nonlinear processes and (or) measurement equations with additive Gaussian noises, one could linearize the evolving states along a reference trajectory and apply the Kalman filter on the resulting approximate equations and such an algorithm is known as the extended Kalman filter (EKF) [19]. For a more general class of problems involving nonlinear state space models and (or) non-Gaussian additive/multiplicative noises, Monte Carlo simulation strategy could be used to recursively evaluate the multi-dimensional integrals and these methods are styled variously as particle filters, Monte Carlo filters, Bayesian filters or population Monte Carlo algorithms [22], [23], [24], [25], [26], [27], [28], [29], [30], [16], [31], [32].

The problem of structural system identification can be brought into the folds of DSE problems and different options are available in the existing literature to achieve this. One option is to declare the vector of system parameters to be identified as additional state variables and augment the original state vector by these artificial state variables. The solution to the resulting state estimation problem also contains the estimates of the posterior pdf-s of the system parameters. It may be noted that, since system parameters often multiply the system states, the problem of DSE here becomes nonlinear in nature even when the original process and measurement equations are linear. Consequently, the DSE problem here could be solved using extended Kalman filtering or particle filtering strategies. The studies by Yun and Shinozuka [33], Hoshiya and Saito [34], Imai et al. [35] are perhaps some of the earliest investigations on the application of the EKF algorithm for identification of parameters of linear and nonlinear structural systems. More recently, the question of identification of parameters of deteriorating structures using EKFs combined with finite element discretization, has been discussed by Corigliano and Mariani [36]. The work of Ghosh et al. [37] describes the development of novel forms of EKFs which are based on derivative-free transversal linearization schemes for handling the nonlinear process equations. The idea of using particle filters to estimate the combined state vector of system states and system parameters is being explored in the area of structural system identification over the last five years [38], [39], [40], [41], [42].

The next option to estimate system parameters, within the ambit of DES methods, is to combine maximum likelihood estimation along with state estimation. Here an assumption on joint pdf of the system parameters to be estimated needs to be made and often recourse is made to Gaussian models [43], [44]. The study by Namdeo and Manohar [45] offers another alternative for system parameter identification using DSE techniques. Here the parameters of the system, which are to be identified, are treated as a set of random variables with finite number of discrete states. The authors consider nonlinear systems and develop a procedure that combines a bank of self-learning particle filters with a global iteration strategy to estimate the probability distribution of the system parameters to be identified and no ad hoc assumption of pdf of parameters to be identified is made.

In the broader context of signal processing and applied statistics, the problem of identification of time invariant parameters of dynamical systems using DSE tools has been widely researched upon (see, for example, the works of Liu and West [46], Storvik [47], Chopin [48], Doucet and Tadic, [49], and Ionides et al. [50]). To the best of present authors’ knowledge the ramifications of this body of knowledge in the context of structural system identification has not been explored in the existing literature.

The recent work by the present authors [51], [52] consider system identification problem in which measurement data emanate from multiple sensor types and multiple test scenarios. The authors consider situations in which measurements on strains and displacements at a set of points are available from static tests (load–displacement tests and quasi-static moving load tests) and dynamic tests (measurement of frequency response functions (FRF-s) for linear systems). A dummy independent variable which takes values on a unit interval is introduced and the entire set of measurement data from multiple tests and multiple sensors are assimilated into the mathematical model using a pseudo-sequencing approach. Both the Kalman filter (after using a one-term Neumann’s expansion on structural stiffness matrix) and particle filtering strategies have been proposed for the purpose of data assimilation. The studies conducted so far have shown promise for further extension to cover dynamical behavior of nonlinear systems and also for extension to study large scale problems. These extensions would be greatly facilitated if the identification procedure readily employs existing FE analysis softwares and avoids the duplication of FE code development while performing system identification. Accordingly, the present study is taken up with the following objectives:

• embedding finite element procedures into the system identification algorithm for time domain analysis of linear/nonlinear dynamical systems,

• combining the particle filtering algorithm with FEM by developing interfaces between FE models residing in readily available commercial codes and DSE algorithms developed on the Matlab platform,

• uncoupling the problems of state estimation and parameter estimation which offers particular advantage in fusing measurement data from multiple tests (and hence assimilation into multiple mathematical models) and in avoiding treatment of a large number of state variables in the DSE problem,

• consider a modification to the filtering algorithm that takes into account the fact that the system parameters being estimated are essentially time invariant in nature and it is desirable to lessen the number of calls to the FE code especially when assimilating large amount of vibration measurement data acquired at high sampling rates, and

• illustration of proposed strategy for problems of SSI when measurement data emanate from computational codes, laboratory experiments and field investigations.

The range of examples covered include a rubber sheet with hole undergoing large amplitude static/dynamic displacements; laboratory studies on a single span beam; and, a multi-span arch railway bridge structure which has been tested for its static behavior. The last example is drawn from ongoing studies being conducted by the present authors and their colleagues on conditional assessment of existing railway bridges in India. This investigation itself is motivated by the long term objective of Indian Railways to increase the axle loads of the freight formations, increase movement speeds and lengths of formations and the concomitant need to ascertain the capabilities of the existing bridge structures to cope up with these enhanced demands.

As has been already noted, the literature in the area of mathematical statistics contains several alternate versions of DSE tools to treat the problem of estimation of static parameters of dynamical models. In the present study we are adopting one of the earlier versions of the particle filtering method with focus of the study being on exploring the parameters of FE structural models based on multiple static and (or) dynamic test data. The overall framework of combining DSE tools and FE models could be retained even if more sophisticated particle filtering tools are used.