Shock and Vibration

Volume 2015 (2015), Article ID 712428, 18 pages

http://dx.doi.org/10.1155/2015/712428

## Efficient Midfrequency Analysis of Built-Up Structure Systems with Interval Parameters

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, Hunan 410082, China

Received 22 May 2015; Accepted 18 August 2015

Academic Editor: Ranjan Banerjee

Copyright © 2015 Hui Yin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1–3]. If the objective information about the uncertain parameters is adequate to establish the probability density functions of them, the random variable model can be the prior way to describe the uncertainties. Many approaches have been proposed to deal with the probabilistic uncertainty recently, such as the Monte Carlo method, the spectral stochastic method, and the perturbation stochastic method [4, 5]. Unfortunately, in the early stage of design, there may be no sufficient statistical information to establish the probability density functions of the uncertain parameters. Under this circumstance, the nonprobabilistic model, such as the interval model, may be an advisable model to represent the uncertain parameters. In this paper, the interval model is selected to describe the parametric uncertainty.

Before performing the uncertain analysis, we must select an approach to model the system. The finite element method (FEM) [6] is the most commonly used technique to model a system in engineering practice, owing to its simplicity and accuracy. However, because the computational efficiency of FEM typically decreases exponentially with the increase of frequency, it is improper to analyze mid- to high-frequency system by using FEM. Thus, the application of FEM is limited to the so-called low-frequency range [7]. Statistical energy analysis (SEA) [8] is a statistical technique that was developed specifically to solve high-frequency problems. This approach is established on the assumption that the system is highly random. In contrast with FEM, the computational efficiency of SEA model is much better due to the much fewer degrees of freedom of it.

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) [9, 10] and the wave based method (WBM) [11] are deterministic methods based on the Trefftz approach [12] for midfrequency analysis, and they are both aiming to improve the computational efficiency by modeling the system with fewer degrees of freedom than that of FEM. Another method for midfrequency analysis is the hybrid approach by dividing the system into deterministic subsystems and highly random subsystems, such as the fuzzy structure theory [13]. Based on this deterministic/random partitioning idea, Langley and coworkers have recently developed a hybrid finite element/statistical energy method (FE-SEA) [14]. In this proposed method, the deterministic subsystem is called the “master system” and modeled by FEM; the random subsystem is called the “subsystem” and modeled by SEA. The randomness of the “subsystem” is modeled as nonparametric uncertainty. The coupling of the two systems is achieved by using a diffuse field reciprocity relation [15]. A lot of research works about FE-SEA have been done by Langley and coworkers [16–19].

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 [20]. Thus, a hybrid uncertain model with parametric and nonparametric uncertainties is yielded. The distribution of the response of this hybrid model can be obtained by dealing with the parametric uncertainty with Monte Carlo simulations and the nonparametric uncertainty analytically. This method will be efficient when the FE components have few uncertain parameters and degrees of freedom. However, for large scale engineering systems with many uncertain parameters, it is computationally intensive to employ the Monte Carlo simulations to deal with the parametric uncertainty. Developing efficient MCS techniques [21] or alternative algorithms [22, 23] is a direction to improve the efficiency of the analysis method for the hybrid model with parametric and nonparametric uncertainties. Recently, Cicirello and Langley [24] have proposed two different asymptotic statistical techniques to target this problem, namely, the hybrid FE/SEA method combined with the first-order reliability method and the hybrid FE/SEA method combined with Laplace’s method, which allow the evaluation of the failure probability of a complex built-up system with probabilistic input parameters of the FE components. The two methods are much more efficient than the FE Monte Carlo simulations and the accuracy of them was good. Another powerful tool for solving the stochastic problems is the stochastic finite element method (SFEM) [2], which mainly includes the perturbation stochastic finite element method (PSFEM) [25–27] and the spectral stochastic finite element method (SSFEM) [28]. These numerical analysis methods are all probabilistic techniques for propagating the probabilistic parametric uncertainty, while for the interval analysis, many other approaches have been proposed, such as the Gaussian elimination scheme [29], the vertex method [30], and the interval perturbation method (IPM) [31]. IPM is an efficient technique for interval analysis proposed by Qiu et al. In this method, the interval matrices and the interval vectors were expanded to a first-order Taylor series. To improve the accuracy of the IPM, an interval perturbation method based on the second-order Taylor expansion (SIPM) [32, 33] was recently developed. Because of the neglect of the higher order terms of Taylor series, IPM is limited to the interval analysis with narrow parameter intervals. To release this restriction, the subinterval analysis technique was introduced into the interval perturbation method [34]. The interval and subinterval perturbation methods have been widely applied to the interval analysis of vibroacoustic response due to their simplicity and efficiency [35–37].

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.

#### 2. Basic Principle of the Hybrid FE/SEA Theory for Built-Up Structure Systems with Fixed FE Properties

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.

##### 2.1. The Dynamic Equilibrium Equation of the FE Components

The master system consists of the FE components which can be described by a set of degrees of freedom . For a specific frequency , the equations of motion for the FE components can be written aswhere is the dynamic stiffness matrix of the FE components which can be obtained by FEM, is the external force vector applied directly to the FE components, and is the force vector exerted on the FE components by the subsystem . is considered to be the sum of two parts and can be written aswith the “reverberant field force vector” arising from the reflected waves and being the “direct field dynamic stiffness matrix” for subsystem .

By combining (1) and (2), one can get where can be expressed as

##### 2.2. The Power Balance Equation of the SEA Subsystems

The SEA subsystem is described by the ensemble averaged energy , which can be calculated via the power balance equation expressed aswhere , , and are the modal density, the ensemble averaged energy, and the loss factor of the subsystem , respectively, is the coupling loss factor between the subsystem and the master system, is the coupling loss factor between the subsystem and the subsystem , and is the number of the SEA subsystems. and are the power input to the subsystem arising from the forces applied to the master system and directly to the subsystem , respectively. can be calculated by the conventional SEA method, and other terms in (5) can be calculated bywhere the superscript stands for the Hermitian transpose, is the cross-spectral matrix of the external loadings , and with being the ensemble average. is a factor in consideration of the local concentrations in the wavefield, and the details about it are discussed in [38]. If the subsystem is excited predominantly through the master system, the value of is close to 2; in other cases, equals 1.

According to [15], there is a relationship between and , which can be expressed asThus, (5) can be rewritten as the following matrix form:where , , and are the vectors made up of , , and , respectively, and is the coefficient matrix, the th element of which can be written as

##### 2.3. The Coupling between the FE and SEA Components

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 (3) and (12), the cross-spectrum of the response of the FE components can be calculated by

It can be seen from (13) that the response of the FE components is controlled by both the forces applied directly to the FE components and the reverberant forces arising from the SEA subsystems.

#### 3. Introducing Interval Parametric Uncertainty into the FE Components within the Hybrid FE–SEA Model for Built-Up Structure Systems

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 stands for the set of the interval parameters of the FE components, and it can be written as where the subscripts and stand for the lower and upper bounds of the interval parameters, respectively. is the number of the interval parameters. Because of the interval description of the input parameters, the responses of the built-up structure systems in (5) and (13) become interval variables, which can be expressed as whereThe bounds in (15) can be obtained by using the minimization/maximization analysis shown in (16), which can be implemented by the Monte Carlo simulations of the hybrid FE-SEA model. Also, the Monte Carlo simulations will be used to verify the effectiveness of the proposed methods discussed in the following sections.

#### 4. SIPFEM/SEA for the Midfrequency Analysis of Built-Up Structure Systems with Interval Parameters

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.

##### 4.1. Basic Formulation for SIPFEM/SEA

For the sake of convenience, (14) can be rewritten aswhere

Based on the second-order Taylor expansion, the interval response variables in (15) can be expanded about the mean value vector and expressed aswith and being the values of and at the mean value vector .

Equation (19) can be rewritten aswhere and are the gradient vector and the Hessian matrix at the mean value vector , respectively, and they can be expressed as

Similarly, the th element of can be written aswhere

If is large, the computation of the Hessian matrices in (19) and (21) will be intensive. Thus, for the sake of computational efficiency, the nondiagonal elements of the Hessian matrices are neglected, and (21) and (23) are simplified aswhere

##### 4.2. Algorithm for Calculating the Extreme Values of the Expanded Response Variables

For the purpose of calculating the extreme values of the expanded response variables, (25) are rewritten aswhere

It can be seen from (29) that the expanded equations can be treated as the sum of a series of quadratic functions with respect to . Therefore, we can calculate the max/min values of and at the points , , or , the values of which can be calculated by for (27) andfor (28).

The detailed searching algorithm for the calculation of the extreme values of and is discussed as follows.

If , the max/min values of and can be calculated by

If , the values of and can be calculated by

Thus, the upper and lower bounds of and can be calculated by

##### 4.3. Computation of the First- and Second-Order Partial Derivatives of the Interval Response Variables

It can be seen from (27) and (28) that the key to establish the expanded equations is the first- and the second-order partial derivatives of the response variables. To compute the partial derivatives of with respect to , we apply the first- and second-order partial differential operators to (10):Given that the term is independent of , that is to say, both and equal zero, thus, the first- and second-order partial derivatives of the energy response vector with respect to can be calculated byBy combining (6)~(7) and (10), the th element of and can be expressed as where From (8), we can see that the th element of and can be expressed as

It can be seen from (38)~(39) that the key to calculating the first- and the second-order partial derivatives of the energy response vector is the first- and second-order partial derivatives of , which can be calculated by

Thus, by submitting the mean value vector into (36), the partial derivatives of with respect to can be written as

Similarly, by applying the first- and second-order partial differential operators to (13), we getIt can be seen from (43) that the key to calculating the first- and the second-order partial derivatives of the response cross-spectrum matrix is the first- and second-order partial derivatives of and , and they can be calculated by combining (40)~(42). Thus, the th element of and can be expressed as

##### 4.4. The Procedure of the SIPFEM/SEA Method for the Midfrequency Analysis of Built-Up Structure Systems with Interval Parameters

The main steps of SIPFEM/SEA for the midfrequency analysis of built-up structure systems with interval parameters are summarized as follows.

*Step 1. *Partition the built-up structure system into a combination of the master system and the subsystem and establish the equations of FE-SEA (equations (5) and (13)).

*Step 2. *Introduce interval parametric uncertainty into the master system within the hybrid FE–SEA framework and establish the interval equations (equations (15) and (16)).

*Step 3. *Expand the interval response variables ( and ) of the built-up structure system with the second-order Taylor series at the mean values of interval parameters, and for the sake of simplicity and efficiency, the nondiagonal elements of the Hessian matrices are neglected (equations (25)~(26)).

*Step 4. *Calculate the first- and second-order partial derivatives of the interval response variables (equations (41)~(43)).

*Step 5. *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 (32)~(34)).

#### 5. The Formulation of the Subinterval Perturbation Method Based on SIPFEM/SEA for the Midfrequency Analysis of Built-Up Structure Systems with Interval Parameters

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 into small subintervals, one can getwhere is the th subinterval of the th interval parameter . According to permutation and combination theory, we can see that the number of the subinterval combinations is , and each subinterval combination can be expressed as

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.

#### 6. Numerical Examples

##### 6.1. An Oscillator-Plate System

Figure 1 shows an oscillator-plate system in which the oscillator is attached to the simply supported plate. The dimensions of the plate are , Young’s modulus is , the density is , Poisson’s ratio is , and the modal density is . The oscillator consists of a spring and a mass, and it is attached at the point (0.882, 0.772). The spring is fixed at the other end and a vertical unit force is applied to the mass. The stiffness of the spring is expressed as , and the mass value is expressed as . The damping loss factors of the plate and the oscillator are both .