An adaptive inverse iteration algorithm using interpolating multiwavelets for structural eigenvalue problems

The rapid computation of random eigenvalue problems of uncertain structures is the key point in structural dynamics, and it is prerequisite to the efficient dynamic analysis and optimal design of structures. In this paper, by combining finite element-transfer matrix method (FE-TMM) with perturbation method, a new method named as perturbation FE-TMM is presented for random eigenvalue problems of uncertain structures. By using the proposed method, the rapid computation of random eigenvalue problems of uncertain structures with complicated shapes and boundaries can be achieved, and the repeated eignvalues and characteristic vectors can be solved conveniently. Compared with stochastic finite element method, this method has the low memory requirement, high computational efficiency and high computational stability. It has more advantages for dynamic design of uncertain structures. Formulations as well as some numerical examples are given to validate the method.

The smooth orthogonal decomposition (SOD) method is one of the output only modal analysis methods, which is originally suggested for the undamped/lightly damped systems. In this paper, the error sources of the SOD method for damped vibrating systems have been investigated. Then, in order to improve the accuracy of SOD for modal analysis of damped systems, some modifications have been made on the traditional SOD algorithm, and instead of the forward two-point differential operator D, the central eight-point differential operator D m is used. In order to compare the accuracy of improved SOD and traditional SOD, the two methods are applied for vibration analysis of a discrete system subjected to the random white noise with the same excitation arrangement. The estimated results are compared with the exact solution obtained by the structural eigenvalue problem. The comparison reveals that the improved SOD is more accurate than the traditional SOD method. In addition, the effects of the excitation arrangement and the measurement noise on the estimated modal parameters are investigated.

Within the context of optimization of the structural dynamics properties of finite element models, methodology is developed for the tracking of eigenpairs through changes in the structural eigenvalue problem. The goal is to eliminate difficulties caused by switching (i.e., frequency crossing). Out of several candidate methods, two methods for mode tracking are successful. The first method, the higher order eigenpair perturbation algorithm, is based on a perturbation expansion of the eigenproblem. It iteratively computes changes in the eigenpairs due to parameter perturbations with the important feature of maintaining the correspondence between the baseline and perturbed eigenpairs. The second method is a cross-orthogonality check method, which uses mass orthogonality to reestablish correspondence after a standard reanalysis. Modified eigenpair extraction routines (Lanczos, subspace iteration, inverse power) were unsuccessful in tracking modes. Applications of mode tracking technology that are presented are frequency-constrained optimization and optimization with mode shape constraints. Each application procedure is outlined and examples are given. Recommendations are made based on method efficiency and robustness in the example problems.

A method to derive approximate explicit solutions for structural mechanics problems

Natural gas pipeline simulation technology is the core technology for pipeline management departments to acquire and monitor the operational status of pipelines. It enables managers to observe changes in the pipeline network, thereby improving the safety and reliability of pipeline operations. When solving large-scale and complex natural gas pipeline models using the traditional Newton-Raphson algorithm, issues such as slow convergence speed and low computational efficiency may arise. In this paper, an improved Newton-Raphson adaptive iteration algorithm based on Logistic Regression is proposed. Based on the variation law of iteration error when solving the model using the Newton-Raphson algorithm, an adaptive iteration step size control and convergence judgment mechanism are introduced based on the Logistic Regression function to improve the convergence speed and stability of the algorithm. The improved algorithm effectively improves the solution efficiency while maintaining high accuracy. Computational results show that when using the improved algorithm to solve steady-state simulation models of pipelines of different scales and complexities, it can maintain the same level of accuracy and stability as commercial software SPS. Compared with the traditional Newton-Raphson algorithm, the improved algorithm can achieve an acceleration effect of over 12% in terms of solution time.

Badly-scaled matrices or matrix pencils can reduce the reliability and accuracy of computed results for various numerical problems, including the computation of spectra and bases of invariant or deflating subspaces, which are used in many basic procedures for optimal and robust control, model reduction, spectral factorization, and other domains. Standard balancing techniques can often improve the results, but sometimes the solution of the scaled problem is much worse than that for the original problem. This paper presents an improved balancing technique for general or structured matrices and matrix pencils, and illustrates its good performance in solving eigenvalue problems and algebraic Riccati equations for large sets of examples from well-known benchmark collections.

Solution of the generalized eigenvalue problem using overlapping finite elements

Impact of time synchronization error on the mode-shape identification and damage detection/localization in WSNs for structural health monitoring

AN ITERATIVE METHOD FOR DYNAMIC CONDENSATION OF STRUCTURAL MATRICES

In this paper, we present a finite element lumped mass scheme for eigenvalue problems of circular arch structures, and give error estimates for the approximation. They assert that approximate eigenvalues and eigenfuctions converge to the exact ones. Some numerical examples are also given to illustrate our results.

The speedy computation of eigenvalue problems is the key point in structure dynamics. In this paper, by combining transfer matrix method and finite element method, the modified finite element-transfer matrix method and its algorithm for eigenvalue problems are presented. By using this method, the speedy computation of eigenvalue problem of flexible structures can be realized, and the repeated eignvalue problem can be solved simply and conveniently. This method has the low order of system matrix, high computational efficiency, and stability. Formulations of this method, as well as some numerical examples, are given to validate the method.

Damping matrix modeling and identification has important applications in many engineering fields such as vibration analysis and control, modal analysis, condition monitoring and structural dynamic modifications. A damping model should represent both the mechanism and spatial distribution of the energy loss in the system. In contrast to the mass and stiffness matrices, formulation of the damping matrix still stands as a big challenge in modeling a linear dynamic system. Several methods have been proposed in the literature to identify the damping and the parameters of a damping matrix from measurements on a vibrating system. It is felt that a review of the various approaches developed would help to compare their main features and their relative advantages or limitations to allow for choosing the most suitable method for a particular application. In view of this, this paper presents a review of the methods of damping identification in general, but with more emphasis on the methods developed in the framework of finite element model updating.

In a solution to large eigenvalue problems, the number of required eigenvalues and corresponding vectors is usually much smaller than the order of matrices. For small bandwidth problems, the determinant search method is very efficient and the oldest. Its from a family of nonlinear Eigen solution techniques also referred to as frequency scanning methods. By using the shift technique, it can also calculate the largest eigenvalues. This calculation process involves three steps: polynomial iteration, inverse iteration, and polynomial deflation. The polynomial iteration determines the eigenvalues of interest by looking at a plot of characteristic polynomial versus the eigenvalue λ, which involves evaluating the determinant |K–λM| at fine intervals of λ selected from accelerated secant iteration. The method was developed in the 1970s and was extensively used for only small-size problems due to its limitations. During that time, the memory available was very low for the in-core solver. Also, the numbers generated in polynomial iteration overflows the float point variables in a computer program. Of course, scaling was the obvious choice. However, the method still suffered the numerical scaling difficulties as a determinant of the matrix |K–λM| in structural problems is generally a very fast varying function. Despite different scaling, the number gets overflows after dozens of eigenvalues which made the technique unattractive. The variables type float, double and long double in a computer program can store values up to 1049, 10308, and 104932, respectively, which is not comparatively much smaller than the determinant of small size structural models. In the present paper, a power number representation is proposed in the polynomial iteration. The determinant |K–λM| can be stored in the power form ρD¯n, where prefactor ρ stores the determinant sign, n is the size of the matrix, and D¯ can be calculated and stored accordingly. The order analysis shows the storage of D¯ requires a variable of range 108 to 1010, which can be stored by float or double type variable. The determinant search method gets to be free from limitations and may easily estimate each large eigenpair at higher frequencies independently from all those previously calculated. The determinant search method can also estimate the clustered eigenvalues accurately. The models’ practical significance is in the FEA analysis of beams, frames, buckling analysis, and stress analysis of pipes in the industry. In the present work, the proposed approach has been applied to the piping model. The results show the determinant search algorithm can extract modes in a higher frequency range without any limitations. The modal analysis is the basis of dynamic analyses of structures such as Response spectrum analysis. The inclusion of higher modes ensures that more than 95% of modal mass has been included in the calculation.

Square steel tube, which is widely used in civil engineering, can suffer from a wide variety of damages and aging defects, thus its nondestructive testing (NDT) has attracted wide attention. This work proposes an ultrasonic guided wave (UGW)-based damage detection method for square steel tubes using structure symmetry. Firstly, the dispersion characteristics of square steel tubes are obtained using the semi-analytical finite element (SAFE) method, after which the optimal guided wave modes for damage located on the long and short edges of the steel tube are selected by modal analysis. Then, using the symmetry of the square steel tube’s section, the symmetric layout scheme of the transmitters and the receivers is designed; on this basis, a signal processing strategy for damage detection is proposed by subtracting the receiving signals obtained from symmetric positions. Finally, the effectiveness of the proposed damage detection method is verified by numerical simulations and laboratory experiments. The results show that the proposed method has good inspection accuracy for crack and hole damages on both the long and short edges of square steel tube because the dispersion effect and clutters can be reduced utilizing structure symmetry. It is worth mentioning that the inspection effect for corner holes can be enhanced by modifying the position of the receivers.

The results of numerical simulations to test a method to locate and estimate the severity of damage in a structure in the blind mode, by using the simulated data from a four-story steel frame structure analyzed at the Univ. of British Columbia, are reported here. The blind simulated data include the response in the time domain of the undamaged structure and the responses of the structure with two sets of simulated damage conditions. Elements of the theory utilized here for damage detection and localization (the so-called damage index method) are first summarized. Modal analyses are performed to extract the modal characteristics (i.e., resonant frequencies and mode shapes) for the undamaged structure (designated, Case 1) and the damaged structures (here designated, Damage Cases 2 and 3). The results of the modal analyses are reported in detail. Using the proposed nondestructive damage evaluation technique, the predicted damage locations, and estimation of the damage severity are provided for the two damage cases. The actual locations of damage in the test structure for each of the damage cases were revealed subsequent to the damage predictions proposed by the writers during the 16th International Modal Analysis Conference. The predicted damage locations and severity estimates are compared with the damage inflicted in the original test cases. The current method correctly predicts the inflicted damage locations and provides realistic estimates of the associated damage severities.