Hierarchical Strategies of Optimization for Structural System Identification Based on the Condensation Method

Summary Identifiability of parameters in structural system identification (SSI) is of primary importance in any SSI method. It depends on the number and the location of the measurements, which is linked with sensor configuration. In this paper, under the framework of SSI by observability method (OM), the number of necessary measurements to identify all parameters of structural system was clarified first. Then, an example was solved step by step to show the lacking constraints among unknowns in SSI by OM. In a frame example, it was found that no measurement set having as many measurements as the number of unknowns was able to identify all parameters. To further understand this phenomenon, the observability of a simply supported beam was analyzed in an exhaustive way using 252 possible measurement sets. Three quarters of these sets were not able to identify all the parameters. In order to solve this issue, for the very first time, SSI by constrained observability method (COM), which appends the nonlinear constraints to SSI by OM, was proposed. With SSI by COM applied, the observability of the structural parameters with respect to the 252 sets was greatly improved. Finally, the efficacy of SSI by COM was verified by a 13-story frame building.

Calculating the displacement vector for a harmonically vibrating structure becomes computationally expensive for larger Degree of Freedom (DOF) systems. If a portion of the system is modified, then the stiffness and mass of the system are perturbed and a new displacement vector must be calculated. In cases where an optimal solution must be determined or in cases where a parameter is uncertain many perturbations may need to be calculated, which becomes computationally expensive. Different methods exist to more quickly estimate the perturbed displacement using information from the nominal system. Here the Bickford method is selected to perturb the eigenvalues and eigenvectors for the system, then these values are used in a modal analysis to calculate the displacement vector. In addition, the Condensation method is selected to calculate the perturbed displacement more quickly using nominal eigenvectors. This research proposes a method to further accelerate the Bickford and Condensation methods, which consists of breaking the structure into segments and scaling a segment of the structure to perturb the system. This segmenting approach allows for the computations needed for a perturbation by the Condensation and Bickford methods to be simplified.

A direct and effective approach is presented for the inverse problem of dynamic structural systems, which is related to structural optimization, system identification, and damage detection. The structural modifications are sought for the characteristic changes assigned from the design goals or modal measurements. A finite element method is used for the system analysis and inverse problem. Mathematical programming techniques are applied for the minimization of the deviation of the finite element model from the desired inverse system, along with an objective function of least structural change. The modal method is based on the perturbation equations of a set of selected degrees of freedom and the energy equation associated with the frequency change. The mode shape change is expressed as the sum of the baseline mode shape and complementary vector, which plays a very important role in the search for the inverse solution. The linear perturbation equation is employed to get an initial approximation, which can be improved through iterations with the nonlinear perturbation equation. The proposed method does not involve the system reduction. Therefore, it is believed to be superior to conventional methods, which suffer from the residual error due to transformation of the eigenproblem.

Sparse structural system identification method for nonlinear dynamic systems with hysteresis/inelastic behavior

Recently, blind source separation (BSS) methods have gained significant attention in the area of signal processing. Independent component analysis (ICA) and second-order blind identification (SOBI) are two popular BSS methods that have been applied to modal identification of mechanical and structural systems. Published results by several researchers have shown that ICA performs satisfactorily for systems with very low levels of structural damping, for example, for damping ratios of the order of 1% critical. For practical structural applications with higher levels of damping, methods based on SOBI have shown significant improvement over ICA methods. However, traditional SOBI methods suffer when nonstationary sources are present, such as those that occur during earthquakes and other transient excitations. In this paper, a new technique based on SOBI, called the modified cross-correlation method, is proposed to address these shortcomings. The conditions in which the problem of structural system identification can be posed as a BSS problem is also discussed. The results of simulation described in terms of identified natural frequencies, mode shapes, and damping ratios are presented for the cases of synthetic wind and recorded earthquake excitations. The results of identification show that the proposed method achieves better performance over traditional ICA and SOBI methods. Both experimental and large-scale structural simulation results are included to demonstrate the applicability of the newly proposed method to structural identification problems.

The problem of system identification is an inverse problem of difficult solution. Currently, difficulties lie in the development of algorithms that use measured data from the system to characterize it without significant a priori knowledge of the system. In this paper, a parameter estimation technique based on an evolution strategy (an optimization algorithm inspired by natural evolution) is presented to overcome some of the difficulties encountered in the field. Using this method, a set of direct problems is solved instead of directly tackling the inverse problem. If the uniqueness of the identification solution is guaranteed for the assumed model and the available data, this heuristic method is able to find a solution without incurring restrictions of other classical optimization methods, like the need for reliable initial estimates and convergence to local optima. Some results obtained with this algorithm are presented for the identification of 3 degrees of freedom (DOF) and a 10 DOF structural system under conditions including limited input/output data, noise polluted signals, and no prior knowledge of mass, damping, or stiffness of the system.

Structural system identification is a critical task in resilience assessments, especially following a natural hazard. In this paper, we propose PIDynNet, a novel physicsinformed approach that produces an ordinary differential equation-constrained neural network model for structural system identification. PIDynNet improves the estimation of structural parameters of nonlinear structural systems by embedding an auxiliary physicsbased loss term into the overall loss function as well as a supervised data-driven loss term. The proposed framework has the generalization capability to predict nonlinear structural response given unseen ground excitations. Two nonlinear numerical experiments are conducted to demonstrate the advantage of PIDynNet over other identification methods in problems with or without latent variables.

Motivated by a smart structures philosophy of self-health monitoring in structural systems, a damage detection system for beam structures using subspace rotation techniques is presented. In doing so, the subspace rotation is extended from displacement-based data to strain-based data. Methods for accommodating the additional degrees of freedom of beam elements are presented. Using simulated data, the ability to locate and quantify single and multiple damage events is investigated. It has been found that the higher-order vibration modes are required to locate damage events in beams because the frequency response of beams is much less sensitive to a damage event than that of truss structures. It has also been found that condensation methods cannot be used to remove rotational degrees of freedom in the beam because of the coupled nature of the rotational degrees of freedom to the translational degrees of freedom.

Estimating the governing equation parameter values is essential for integrating experimental data with scientific theory to understand, validate, and predict the dynamics of complex systems. In this work, we propose a new method for structural system identification (SI), uncertainty quantification, and validation directly from data. Inspired by generative modeling frameworks, a neural network maps random noise to physically meaningful parameters. These parameters are then used in the known equation of motion to obtain fake accelerations, which are compared to real training data via a mean square error loss. To simultaneously validate the learned parameters, we use independent validation datasets. The generated accelerations from these datasets are evaluated by a discriminator network, which determines whether the output is real or fake, and guides the parameter-generator network. The uncertainty in each parameter is quantified by fitting a Gaussian distribution to each parameter, then computing the 95% confidence bounds. Analytical and real experiments show the parameter estimation accuracy and model validation for different nonlinear structural systems.

Structural system identification using Least Mean Square (LMS) adaptive technique

The common assumption in the so-called linear inverse vibration problem, which provides the mass/stiffness/damping matrices of second order dynamic models, is the availability of a full set of sensors and actuators. In “reduced-order” problems (with limited number of instrumentation), only the components of the eigenvector matrix regarding the measured degrees of freedom can be successfully identified while nothing can be said about the components connected to the unmeasured degrees of freedom. This paper presents a recently developed “reduced-order” model and expands such a model to a “full-order” one that is quite useful in damage detection. The five representative categories of “reduced-order” problems, defined by considering different types of geometrical conditions, are analyzed and a discussion on their solution space has been performed. The effectiveness and robustness of this approach is shown by means of a numerical example.

Concurrent multi-scale design optimization of fiber-reinforced composite material based on an adaptive normal distribution fiber optimization scheme for minimum structural compliance and additive manufacturing

With this work, we present the application of a damage distribution function to finite element (FE) model updating with the goal to detect, locate and quantify structural damage using a prestressed concrete tower as an example. The choice of design variables greatly influences the quality of the model updating procedure. Thereby, the parameterization connected to the design variables has to be able to model the structural damage with a high resolution. At the same time, the number of design variables should be as low as possible, since a high amount of design variables can result in an objective value space with many local minima, making numerical optimization unfeasible. A common approach consists of directly fitting the sectional stiffness of FEs to measured behavior. Thereby, the number of design variables increases linearly to the element count of the model. In order to reduce the number of design variables, they are commonly assigned to groups of FEs supposedly having similar mechanical properties. An additional method to even further reduce the amount of design variables is the determination of susceptible regions based on experience, prior knowledge or inspection findings. However, the position of damage is unknown in many applications. To alleviate this problem, we introduce a damage distribution function, which is described by few parameters. As these parameters represent the design variables of the optimization process, the proposed model updating procedure is independent of the FE mesh resolution as well as prior assumptions about the defect location. We demonstrate the application of the damage distribution function using a prestressed concrete tower as an example. First, we apply a three-parameter damage distribution function in order to detect, locate and quantify an artificially induced damage in a simple beam model of the considered tower. In this application, the defect location is determined along the height. Second, we extend the damage distribution function to a three-dimensional shell model of the concrete tower. This time, the location of the fictitious damage is also determined along the perimeter. The proposed parameterization keeps the amount of design variables low and enables a high numerical performance as well as stability, while maintaining the ability to detect, locate and quantify damage.

*† ‡ § An efficient and high-fidelity design approach is proposed by combining global and local optimization methods for wing planform and surface design. For enhanced design results, aerodynamic shape optimization process is carried out via 2-stage with different optimization strategy. In the first stage, global optimization techniques are applied to planform design with a few geometric design variables. In the second stage, local optimization techniques are used for wing surface design with a lot of design variables to maintain a sufficient design space with high DOF (Degree of Freedom) geometric change. For global optimization, meta-modeling techniques such as RS (Response Surface) and Kriging methods are used in conjunction with Genetic Algorithm (GA). For local optimization, a discrete adjoint variable method is used. By the successive combination of global and local optimization techniques, drag minimization is performed for a multi-body aircraft configuration while maintaining the baseline lift and the wing weight at the same time. Through the design process, performances of the test models are remarkably improved in comparison with the single stage design approach. The capability of proposed design framework including wing planform design variables can be evaluated by the drag decomposition method which can provide improvement of induced drag and wave drag, respectively.

Bayesian inference provides a powerful approach to system identification and damage assessment for structures. The application of Bayesian method is motivated by the fact that inverse problems in structural engineering, including structural health monitoring, are typically ill-conditioned and ill-posed when using noisy incomplete data because of various sources of modeling uncertainties. One should not just search for a single “optimal” value for the vector of model parameters but rather attempt to describe the whole family of plausible model parameters based on measured data using a Bayesian probabilistic framework. In this article, the fundamental principles of Bayesian analysis and computation are summarized; then a review is given of recent state-of-the-art practices of Bayesian inference in system identification and damage assessment for civil infrastructure. Discussions of the benefits and deficiencies of these approaches, as well as potentially useful avenues for future studies, are also provided. Our focus is on meeting challenges that arise from system identification and damage assessment for the civil infrastructure but our presented theories also have a considerably broader applicability for inverse problems in science and technology.