Dynamic sensitivity analysis and optimum design of aerospace structures

A numerical method of structural dynamic shape optimization is presented by using reproducing kernel particle method (RKPM), by which the mesh distortion that exists in shape optimal method based on finite element can be eliminated completely and the optimal model for structural dynamic optimization design is built. The discreteness-based design sensitivity analysis in both natural frequency and dynamic response is proposed by using direct differentiation method and discrete derivatives on the basis of structural dynamic analysis, in which the penalty method is employed into imposing the essential boundary conditions, and the derivatives of shape functions with respect to design variables are derived. The algorithm of dynamic sensitivity analysis is testified by numerical example, and the numerical results obtained are in good agreement with those obtained using semi-analytical method and global finite differences method. Finally, by integrating the algorithm mentioned based on RKPM with parameterized descriptive method of boundary shape, two examples for structural dynamic shape optimization are performed.

In the present study, the free and forced vibration responses of the composite sandwich plate with carbon nanotube reinforced honeycomb as the core material and laminated composite plates as the top and bottom face sheets are investigated. The governing equations of motion of hybrid composite honeycomb sandwich plates are derived using higher order shear deformation theory and solved numerically using a four-noded rectangular finite element with nine degrees of freedom at each node. Further, various elastic properties of honeycomb core materials with and without reinforcement of carbon nanotube and face materials are evaluated experimentally using the alternative dynamic approach. The effectiveness of the finite element formulation is demonstrated by performing the results evaluated experimentally on a prototype composite sandwich plate with and without carbon nanotube reinforcement in core material. Various parametric studies are performed numerically to study the effects of carbon nanotube wt% in core material, core thickness, ply orientations, and various boundary conditions on the dynamic properties of composite honeycomb sandwich plate. Further, the transverse vibration responses of hybrid composite sandwich plates under harmonic force excitation are analyzed at various wt% of carbon nanotubes and the results are compared with those obtained without addition of carbon nanotubes to demonstrate the effectiveness of carbon nanotube reinforcement in enhancing the stiffness and damping characteristics of the structures. The study provides the guidelines for the designer on enhancing both the stiffness and damping properties of sandwich structures through carbon nanotube reinforcement in core materials.

Design sensitivity analysis of structures deals with the calculation of the response derivatives with respect to the design variables. These derivatives, called the sensitivity coefficients, are used in the solution of various problems. In design optimization, the sensitivity coefficients are often required to select a search direction. These coefficients are used also in generating approximations for the response of a modified system. In addition, the sensitivities are required for assessing the effects of uncertainties in the structural properties on the system response. Calculation of the sensitivities involves much computational effort, particularly in large structural systems with many design variables. As a result, there has been much interest in efficient procedures for calculating the sensitivity coefficients. Developments in methods for sensitivity analysis are discussed in many studies (e.g. Haug et. al. 1986; Haftka and Adelman, 1989; Haftka and Gurdal, 1993; van Keulen et. al. in press). Methods of sensitivity analysis for discretized systems can be divided into the following classes: a. Finite-difference methods, which are easy to implement but might involve numerous repeated analyses and high computational cost, particularly in problems with many design or response variables. In addition, finite-difference approximations might have accuracy problems. The efficiency can be improved by using fast reanalysis techniques. b. Analytical methods, which provide exact solutions but might not be easy to implement in some problems such as shape optimization. c. "Semi-analytical" methods, which are based on a compromise between finite-difference methods and analytical methods. These methods use finite-difference evaluation of the right-hand-side vector. They are easy to implement but might provide inaccurate results. In general, the following factors are considered in choosing a suitable sensitivity analysis method for a specific application: the accuracy of the calculations, the computational effort involved and the ease-of-implementation. The implementation effort is weighted against the performance of the algorithms as reflected in their computational efficiency and accuracy. The quality of the results and efficiency of the calculations are usually two conflicting factors. That is, higher accuracy is often achieved at the expense of more computational effort. Dynamic sensitivity analysis has been demonstrated by several authors. Using the mode superposition approach and assuming harmonic loading, the response sensitivities were evaluated by direct differentiation of the equations of motion in the generalized coordinates (Kramer and Grierson, 1989). In cases of earthquake loading the ground acceleration is usually given in discrete time steps, thus the loading is not given analytically. In several studies (Kim and Choi, 2000; van Keulen et. al, in press) the unconditionally stable implicit numerical equation was directly derived. It was found that the analysis equations and the sensitivity equations have the same left-hand side expression. Thus, it was possible to use the available factorized coefficient matrix. A numerical procedure was applied for calculation of the sensitivity of the response.

This paper presented the dynamic characteristics and sensitivity analysis of the shaft of test machine based on finite element method and orthogonal design method.It is concluded that the top eight order natural frequencies (Hz) are 1673.3,3666.6,4421.8,5713.8,6612.9, 6656.8,9550.9, 9617.5 based on modal analysis.The harmonic analysis results indicate that the third natural frequency is more likely to intensify initiation and propagation of the bending crack on the interference joint between specimen and shaft.It is concluded that the radius dimension parameters and the length parameters affecting more greatly the dynamic characteristics of shaft are R3, R1, R2 and L3, L4, L5, L1 through dynamic sensitivity analysis.The results above provide reference for dynamic optimization and design of the shaft and other shaft parts in the future.

Longitudinal data is almost always burdened with missing data. However, in educational and psychological research, there is a large discrepancy between methodological suggestions and research practice. The former suggests applying sensitivity analysis in order to the robustness of the results in terms of varying assumptions regarding the mechanism generating the missing data. However, in research practice, participants with missing data are usually discarded by relying on listwise deletion. To help bridge the gap between methodological recommendations and applied research in the educational and psychological domain, this study provides a tutorial example of sensitivity analysis for latent growth analysis. The example data concern students’ changes in learning strategies during higher education. One cohort of students in a Belgian university college was asked to complete the Inventory of Learning Styles–Short Version, in three measurement waves. A substantial number of students did not participate on each occasion. Change over time in student learning strategies was assessed using eight missing data techniques, which assume different mechanisms for missingness. The results indicated that, for some learning strategy subscales, growth estimates differed between the models. Guidelines in terms of reporting the results from sensitivity analysis are synthesised and applied to the results from the tutorial example.

To investigate the dynamic behavior of offshore structures to irregular waves, we conducted model tests using frame and column models, and took field measurements of the vibrations of an offshore structure installed in 80 m of water. We also conducted theoretical analyses of the dynamic response of offshore structures to irregular waves, and compared calculated values with the results of model tests and actual field measurements. From these studies to evaluate the dynamic response characteristics of offshore structures to irregular waves, we obtained data necessary for the dynamic design of deep-water offshore structures. INTRODUCTION The design of large offshore structures to be installed in deep water must be based on dynamic analysis, but strict analysis of their dynamic behavior requires frequent use of large computers. To avoid this costly procedure, it is necessary to employ either hypotheses or the approximation method in analyzing phenomena ranging from sea-surface fluctuations to vibration responses. To be analyzed by the approximation method, vibration phenomena must be clearly defined, so we analyzed the dynamic behavior of offshore structures under the influence of irregular waves by conducting model tests. We also conducted field measurements of the vibrations of an offshore platform. Further, we determined the responses by linear analysis using relatively simple models and evaluated the dynamic behavior characteristics by comparing calculated and measured results. The model tests comprised free vibration tests in which column and frame models were tested both in the submerged state and in air and dynamic response tests in which the models were tested using two dimensional irregular waves. In the former tests, the natural periods, damping constants and vibration modes of the models were determined in the latter, the fluid force coefficients of irregular waves and the response displacements and stress range frequency distribution characteristics of offshore structures were obtained. In the field vibration measurements, the natural period and damping constant of the platform were determined from its response power spectra in the tremor state. Our field measurements also revealed wave response characteristics of offshore structures during the passage of a storm center and their vibration characteristics during drilling. In the theoretical analyses, a lumped-mass model was substituted for the structure using the natural periods and damping constants obtained in the model and field tests. Borgman's wave force spectrum1 was used for the wave force spectrum in determining the response values. Then these theoretical response values were compared with the measured values. OUTLINE OF THE MODEL TESTS The column and frame models shown in Fig. 1 were used in the model tests. They were made of Acrylite. In the free vibration tests, plates each weighing about 5 kg were used as a load on the deck of each model.

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.

A dynamic industrial design optimization requires high-quality optimization algorithms as well as adaptive representations to find the global solution for a given problem. For adapting the representation to changing environments or to new input we utilize the concept of evolvability, which in our interpretation consists of three criteria: variability, regularity, and improvement potential, where regularity and improvement potential characterize conflicting goals between exploration and exploitation. Our goal is the efficient adaptation of the representation according to a given preference weight between regularity and improvement potential. We propose a combination of two heuristics, Lloyd sampling and orthogonal least squares sampling, to initialize the adaptation process for a given preference weight. We show that this initialization improves the convergence speed of the adaptation process as well as the resulting fitness. We then realize a stepwise design optimization procedure by alternating the adaptation of the representation with optimization of the design. During the design optimization process we extract information which we exploit in the next adaptation phase. We show that an intermediate preference weight, balancing between regularity and improvement potential, allows to exploit this information and is robust to erroneous initial information. Thereby, we increase the performance of the whole design optimization process.

Structural design optimization under dynamic reliability constraints based on the probability density evolution method and highly-efficient sensitivity analysis

The classical step-by-step integration method is commonly used for solving structural dynamic responses, but its performance is highly sensitive to the choice of time step. A large time step can significantly reduce accuracy and stability, while a very small time step greatly diminishes computational efficiency. Data-driven modeling and analysis methods are particularly favored for their potential to replace traditional structural analysis methods. However, traditional data-driven machine learning approaches require substantial amounts of training data to predict structural dynamic responses. Moreover, these methods often function as black boxes, lacking physical interpretability, and acquiring the necessary data can be challenging in practice. To address these limitations, this paper employs a physics-informed neural network (PINN) method for structural dynamic response analysis. The PINN approach incorporates the governing equations of structural dynamics into the neural network’s loss function as physical constraints, endowing the model with physical interpretability. The feasibility and effectiveness of this method are demonstrated through case studies on structural dynamic responses under different excitation scenarios. Numerical results show that the PINN method can accurately analyze structural dynamic responses without relying on system response data, a key advantage over traditional data-driven models. Moreover, unlike the classical step-by-step integration method, the PINN approach operates as a spatial domain optimization process and is independent of time step. Given the critical role of hyperparameters in the performance of the PINN model, this paper systematically investigates the influence of various hyperparameter settings on accuracy and efficiency, uncovering patterns of their impact on structural response analysis. Replication code for this paper will be available on GitHub:https://github.com/duke-iem/AIStructDynSolve.

Nonlinear dynamic analysis of harmonically excited debonded sandwich plates using finite element modelling

This paper presents the implementation of the boundary element method to shape sensitivity analysis of elastic structures with stress concentrators. An elastic body which contains a number of voids (internal boundaries), playing the role of stress concentrators, is considered. We are interested in calculating the first order sensitivity of shape–dependent functionals with respect to the shape variation of the body domain. This task is accomplished using the adjoint variable method. As it has been shown by Dems and Mróz [10], for a basic transformation (i.e. a translation, a rotation or a scale change) of the body, the sensitivity of the considered functional takes the form of a path-independent integral (PII) whose integrand depends on the primary and adjoint state fields, along an arbitrary path (curve in 2D, surface in 3D), enclosing the transformed stress concentrator (void). It is very important for numerical computations, because we can compute this integral along the path placed far from the stress concentrator to eliminate the negative influence of stress concentrations on the accuracy of calculations. The boundary element method (BEM) is used to solve both primary and adjoint problems. Some important cases of adjoint problems related to functionals are analyzed in the paper. A thorough numerical verification of the proposed method is performed in this work. The presented method of sensitivity analysis is utilised in gradient-based optimization and identification problems. Numerical examples of optimization and identification are shown in this paper.

The present work illustrates a dynamic nonlinear analysis of structures characterized by plan irregularity and base isolated by friction devices. Base isolation systems with friction pendulums are analysed to properly regularize the dynamic nonlinear behaviour of structures. Different isolation solutions with friction pendulums are investigated to optimize and regularize the dynamic nonlinear behaviour of structures with irregularities in plan. The nonlinear behaviour of the base-isolated structure is investigated when subject to representative dynamic events. Criteria for regularizing the dynamic behaviour of plan irregular structures are presented. The beneficial effects are illustrated for the adopted optimized isolation solution of friction pendulum system. The nonlinear behaviour of the isolation devices is determined as well as the nonlinear dynamic behaviour of the irregular in plan structure. A comparative dynamic nonlinear analysis is presented for the structure base isolated by the adopted solution of friction isolation system and the corresponding fixed base structure. In the comparative analysis, recorded accelerograms associated with dynamic events are considered and the nonlinear hysteretic cycles of isolation devices are presented, along with the time history of the base acceleration, the base shear, the base displacement and the inter-storey drifts of the structure.

Point-wise impulse (blast) response of a composite sandwich plate including core compressibility effects

Free vibration of composite sandwich plates and cylindrical shells