To improve the efficiency of midfrequency analysis of built-up structure systems with interval parameters, the second-order interval and subinterval perturbation methods are introduced into the hybrid finite element/statistical energy analysis (FE/SEA) framework in this paper. Based on the FE/SEA for built-up structure systems and the second-order interval perturbation method, the response variables are expanded with the second-order Taylor series and nondiagonal elements of the Hessian matrices are neglected. Extreme values of the expanded variables are searched by using efficient search algorithm. For large parameter intervals, the subinterval perturbation method is introduced. Numerical results verify the effectiveness of the proposed methods.
In the last two decades, predicting the response of a system with uncertainties has got more and more attention in the engineering design. There are several ways to describe the parametric uncertainties of a system, such as random variables and intervals [
Before performing the uncertain analysis, we must select an approach to model the system. The finite element method (FEM) [
As stated above, the low- and high-frequency problems can be efficiently solved by using FEM and SEA, respectively. But for a system consisting of both the low- and high-frequency subsystems, which is the so-called “midfrequency” system, neither method is suitable: for pure FEM, the system must be modeled by lots of degrees of freedom and the computational efficiency is too low; for pure SEA, the tenets that the system must be “highly random” may not be met. In recent years, a variety of methods have been proposed for the analysis of midfrequency system. The variational theory of complex rays (VTCR) [
As previously mentioned, the SEA subsystems of the FE-SEA model are assumed to be highly random, and the randomness of them is modeled as nonparametric uncertainty, while the FE components are assumed to be fully deterministic; in other words, the uncertainties of the FE components are ignored. However, uncertainties in properties caused by manufacturing imperfections or aggressive environmental factors are unavoidable, and it is important to take the uncertainties of the FE components into consideration in engineering design. Recently, Cicirello and Langley have introduced parametric uncertainty into the FE components by considering the parameters of them as probabilistic or interval rather than deterministic [
In this paper, to improve the efficiency of the midfrequency analysis of built-up structure systems with interval parametric uncertainty, the second-order interval perturbation method and subinterval analysis technique are introduced into the hybrid FE-SEA framework. Firstly, the second-order interval perturbation method combined with FE-SEA (SIPFEM/SEA) is proposed for the response prediction of the built-up structure systems with nonparametric and small interval parametric uncertainties; secondly, the subinterval perturbation method based on the SIPFEM/SEA is introduced to predict the response of the built-up structure systems with nonparametric and large interval parametric uncertainties. The procedure of the SIPFEM/SEA method is as follows: at first, the ensemble averaged energy of the SEA components and the cross-spectrum of the response of the FE components are expanded with the second-order Taylor series; for the sake of simplicity and efficiency, the nondiagonal elements of the Hessian matrices are neglected; then, by searching the target positions of interval parameters that maximize or minimize the objective functions, the bounds of the expanded responses can be obtained. For large parameter intervals, the subinterval perturbation method based on the SIPFEM/SEA is introduced. Effectiveness of the proposed methods is verified by the numerical results of two built-up structure models. Benchmark comparisons are made with the Monte Carlo simulations of the hybrid FE/SEA models.
This section is intended to summarize the hybrid FE/SEA equations for built-up structure systems with fixed FE properties as presented by Langley et al. The main procedure for the hybrid FE/SEA method for built-up structure systems can be summarized as follows. At first, a built-up structure system is partitioned into the long-wavelength subsystems and the short-wavelength subsystems, which are modeled by the FEM and the SEA, respectively. Secondly, the response of each SEA subsystem is described as the superposition of a series of ingoing waves and reflection waves, which are called the “direct field” and “reverberant field,” respectively. Finally, a diffuse field reciprocity relation between the reverberant force loading and the energy responses of the SEA subsystems are established for the coupling of the FE components and SEA subsystems. The detailed equations of the hybrid FE/SEA method will be presented in the following sections.
The master system consists of the FE components which can be described by a set of degrees of freedom
By combining (
The SEA subsystem is described by the ensemble averaged energy
According to [
As previously mentioned, the coupling between the FE and SEA components is achieved by using the diffuse field reciprocity relation, which can be expressed as
By combining (
It can be seen from (
In this section, the interval parametric uncertainty is introduced into the FE components within the hybrid FE–SEA model for built-up structure systems, and the interval formulations for the responses of the built-up structure systems are discussed as follows.
Assume that the parameter vector
In this section, the second-order interval perturbation method is introduced into the hybrid FE-SEA framework, and the SIPFEM/SEA method is proposed for the midfrequency analysis of built-up structure systems with small interval parametric uncertainty.
For the sake of convenience, (
Based on the second-order Taylor expansion, the interval response variables in (
Equation (
Similarly, the
If
For the purpose of calculating the extreme values of the expanded response variables, (
It can be seen from (
The detailed searching algorithm for the calculation of the extreme values of
If
If
Thus, the upper and lower bounds of
It can be seen from (
It can be seen from (
Thus, by submitting the mean value vector
Similarly, by applying the first- and second-order partial differential operators to (
The main steps of SIPFEM/SEA for the midfrequency analysis of built-up structure systems with interval parameters are summarized as follows.
Partition the built-up structure system into a combination of the master system and the subsystem and establish the equations of FE-SEA (equations (
Introduce interval parametric uncertainty into the master system within the hybrid FE–SEA framework and establish the interval equations (equations (
Expand the interval response variables (
Calculate the first- and second-order partial derivatives of the interval response variables (equations (
Search the target positions of interval parameters that maximize or minimize the objective functions, and calculate the bounds of the expanded responses by submitting the target values into the objective functions (equations (
Because of the neglect of the higher order terms of Taylor series, SIPFEM/SEA is limited to the midfrequency analysis of the built-up structure systems with small interval parametric uncertainty. In order to predict the response of the built-up structure systems with large interval parametric uncertainty, the subinterval perturbation method based on SIPFEM/SEA is introduced in this section.
By dividing the large interval parameters
By applying the SIPFEM/SEA method to each subinterval combination, the corresponding subintervals of the response variables can be obtained and expressed as
By assembling the subintervals of the response variables with the interval union operation, the global intervals of the response variables can be obtained and expressed as
From the convergence condition of perturbation theory, we can see that if the number of the subintervals for each parameter is sufficiently large, the bounds of the response variables of the built-up structure systems with large interval parametric uncertainty will be predicted accurately.
Figure
The oscillator-plate system.
To describe the system via the FE-SEA method, the oscillator is modeled as the FE component, and the plate is modeled as the SEA subsystem. Considering the parametric uncertainty of the FE component, the stiffness of the spring
To investigate the accuracy of the proposed SIPFEM/SEA method for predicting the response of the built-up structure system with interval parameters, we calculate the lower and upper bounds of the plate energy and the autospectra of the mass displacements. Benchmark results are made by the Monte Carlo simulations of the hybrid FE/SEA model with 10,000 samples, and all of the results for this oscillator-plate system are obtained by using MATLAB R2014a on a 3.60 GHz Intel(R) Core (TM) CPU i7-4790. Figures
The bounds of the plate energy: (a)
The bounds of the autospectra of the mass displacements: (a)
The bounds of the plate energy: (a) one subinterval, (b) two subintervals, (c) four subintervals, and (d) eight subintervals.
To investigate the accuracy of the proposed SIPFEM/SEA method for predicting the response of the built-up structure system with interval parameters more clearly, the relative errors of the bounds of the plate energy and the autospectra of the mass displacements obtained by SIPFEM/SEA for
The relative errors of the bounds of the plate energy and the autospectrum of the mass displacements for different uncertainty levels (
Uncertainty level |
Lower bounds | Upper bounds | |||||
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MCS | SIPFEM/SEA | Relative errors (%) | MCS | SIPFEM/SEA | Relative errors (%) | ||
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The relative errors of the bounds of the plate energy and the autospectrum of the mass displacements for different uncertainty levels (
Uncertainty level |
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MCS | SIPFEM/SEA | Relative errors (%) | MCS | SIPFEM/SEA | Relative errors (%) | ||
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To extend SIPFEM/SEA to the midfrequency analysis of the built-up structure system with large interval parameters, the subinterval perturbation method is introduced based on SIPFEM/SEA(S-SIPFEM/SEA). To illustrate the accuracy of S-SIPFEM/SEA, the uncertainty level
The relative errors of the bounds of the plate energy and the autospectra of the mass displacements yielded by S-SIPFEM/SEA with different subinterval numbers (
The number of subintervals | Lower bounds | Upper bounds | |||||
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MCS | S-SIPFEM/SEA | Relative errors (%) | MCS | S-SIPFEM/SEA | Relative errors (%) | ||
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The relative errors of the bounds of the plate energy and the autospectra of the mass displacements yielded by S-SIPFEM/SEA with different subinterval numbers (
The number of subintervals | Lower bounds | Upper bounds | |||||
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MCS | S-SIPFEM/SEA | Relative errors (%) | MCS | S-SIPFEM/SEA | Relative errors (%) | ||
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50.90 |
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1.46 |
The bounds of the autospectra of the mass displacements: (a) one subinterval, (b) two subintervals, (c) four subintervals, and (d) eight subintervals.
Execution times of the S-SIPFEM/SEA with eight subintervals and the Monte Carlo simulations of the hybrid FE/SEA model for calculating the plate energy are shown in Table
Execution times of S-SIPFEM/SEA and the Monte Carlo simulation of the hybrid FE/SEA model.
MCS | S-SIPFEM/SEA | |
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Execution time(s) | 9.22 | 0.19 |
Beam-plate systems are typical built-up structures in practical engineering, and the beams are usually much stiffer than the plates [
The beam-plate system.
In this numerical example, Young’s modulus
All simulations of the beam-plate system are carried out by using MATLAB R2014a on a 3.60 GHz Intel(R) Core (TM) CPU i7-4790. To illustrate the proposed SIPFEM/SEA method, the bounds of the plate energy and the autospectra of the driving point displacements are calculated, and benchmark results are obtained by the Monte Carlo simulations of the hybrid FE/SEA model with 10,000 samples. The bounds of the plate energy and the autospectra of the driving point displacements in the frequency band
The relative errors of the bounds of the plate energy and the autospectrum of the driving point displacements for different uncertainty levels (
Uncertainty level |
Lower bounds | Upper bounds | |||||
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MCS | SIPFEM/SEA | Relative errors (%) | MCS | SIPFEM/SEA | Relative errors (%) | ||
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24.86 |
The relative errors of the bounds of the plate energy and the autospectrum of the driving point displacements for different uncertainty levels (
Uncertainty level |
Lower bounds | Upper bounds | |||||
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MCS | SIPFEM/SEA | Relative errors (%) | MCS | SIPFEM/SEA | Relative errors (%) | ||
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The bounds of the plate energy: (a)
The bounds of the autospectra of the driving point displacements: (a)
To illustrate the effectiveness of S-SIPFEM/SEA for the analysis with large uncertainty level, we take the uncertainty level
The relative errors of the bounds of the plate energy and the autospectrum of the driving point displacements with three subintervals.
Frequency (Hz) | Lower bounds | Upper bounds | |||||
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MCS | S-SIPFEM/SEA | Relative errors (%) | MCS | S-SIPFEM/SEA | Relative errors (%) | ||
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The bounds of the plate energy with three subintervals.
The bounds of the autospectra of the driving point displacements with three subintervals.
Execution times of the S-SIPFEM/SEA with three subintervals and the Monte Carlo simulation of the hybrid FE/SEA model for calculating the plate energy at the same frequency are listed in Table
Execution times of S-SIPFEM/SEA and the Monte Carlo simulation of the hybrid FE/SEA model.
MCS | S-SIPFEM/SEA | |
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Execution time(s) | 5,954 | 123 |
In order to improve the efficiency of midfrequency analysis of built-up structure systems with interval parameters, the second-order interval perturbation method and the subinterval technique are introduced into the FE/SEA framework in this study. Based on the FE/SEA equations for built-up structure systems and the second-order interval perturbation method, the frequency response variables (the SEA subsystem energy and the cross-spectrum of the response of the FE components, over the interval parametric and nonparametric uncertainties) are expanded with the second-order Taylor series at the mean values of interval parameters. The nondiagonal elements of the Hessian matrices are neglected for the sake of simplicity and efficiency and thus the expanded response variables can be considered as the sum of a series of quadratic functions with respect to the interval parameters. By searching the target positions of the interval parameters that maximize or minimize the quadratic functions, the bounds of the ensemble averaged responses can be obtained. Due to the neglect of the higher order terms of Taylor series, SIPFEM/SEA is limited to the interval analysis of built-up systems with narrow parameter intervals. For the interval analysis of built-up systems with larger parameter intervals, the subinterval perturbation method based on the SIPFEM/SEA is introduced.
The proposed methods are illustrated by application to two built-up structure models, and reference results are obtained by the Monte Carlo simulations of the hybrid FE/SEA models. From the numerical results on the two built-up structure models, we conclude that
The authors declare that there is no conflict of interests regarding the publication of this paper.
The paper is supported by National Natural Science Foundation of China (no. 11402083) and Independent Research Projects of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body in Hunan University (Grants no. 734215002 and no. 51375002).