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

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.


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][2][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 highfrequency 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][17][18][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 firstorder 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][26][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][36][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.

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.

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 q.For a specific frequency , the equations of motion for the FE components can be written as where D  is the dynamic stiffness matrix of the FE components which can be obtained by FEM, f is the external force vector applied directly to the FE components, and f  is the force vector exerted on the FE components by the subsystem .f  is considered to be the sum of two parts and can be written as with f () rev the "reverberant field force vector" arising from the reflected waves and D ()  dir being the "direct field dynamic stiffness matrix" for subsystem .
By combining (1) and ( 2), one can get where D tot can be expressed as where   ,   , 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. ext in, and  in, are the power input to the subsystem  arising from the forces applied to the master system and directly to the subsystem , respectively. in, can be calculated by the conventional SEA method, and other terms in ( 5) can be calculated by where the superscript  stands for the Hermitian transpose, S  is the cross-spectral matrix of the external loadings f, and 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 as Thus, ( 5) can be rewritten as the following matrix form: where Ε, P ext in , and P in are the vectors made up of   ,  ext in, , and  in, ( = 1, 2, . . .,   ), respectively, and C is the coefficient matrix, the th element of which can be written as

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.

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 a  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 where The 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.

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 as where Based on the second-order Taylor expansion, the interval response variables in (15) can be expanded about the mean value vector a  and expressed as with   (a  ) and S  (a  ) being the values of    and S   at the mean value vector a  .Equation ( 19) can be rewritten as where E   (a  ) and E   (a  ) are the gradient vector and the Hessian matrix at the mean value vector a  , respectively, and they can be expressed as Similarly, the th element of S   can be written as where 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 as where

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 as where It can be seen from ( 29) that the expanded equations can be treated as the sum of a series of quadratic functions with Shock and Vibration respect to Δã  .Therefore, we can calculate the max/min values of    and   , at the points −Δ  , Δ  , or    , the values of which can be calculated by for (27) and Thus, the upper and lower bounds of   (a  ) and  , (a  ) can be calculated by (34)

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 P in is independent of   , that is to say, both P in /  and  2 P in / 2  equal zero, thus, the first-and second-order partial derivatives of the energy response vector with respect to   can be calculated by By combining (6)∼( 7) and ( 10), the th element of C/  and  2 C/ 2  can be expressed as Shock and Vibration 7 where From (8), we can see that the th element of P ext in /  and  2 P ext in / 2  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 D −1  tot , which can be calculated by Thus, by submitting the mean value vector a  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 get It 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 D 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 S   ) 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 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)).

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    ( = 1, 2, . . ., ) into   small subintervals, one can get where     , 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 ∏  =1   , 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 (  = 1, 2, . . .,   ,  = 1, 2, . . ., ) .
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.

Numerical Examples
6.1.An Oscillator-Plate System.Figure 1 shows an oscillatorplate system in which the oscillator is attached to the simply supported plate.The dimensions of the plate are 2.1 m × 1.9 m × 1.25 mm, Young's modulus is   = 7.2 × 10 4 MPa, the density is   = 2800 kg/m 3 , Poisson's ratio is ] = 0.3, and the modal density is   = 1.05 modes/Hz.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  = 0.01.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  and the mass value  are set as the interval parameters.By introducing the uncertainty level , the intervals of the spring  and the mass value  can be expressed as where the mean values of  and  are   = 3.2 × 10 6 N/m,   = 2 kg.
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 2 and 3 show the bounds of the plate energy and the autospectra of the mass displacements for frequencies 100∼300 Hz, respectively.The considered uncertainty levels are  = 0.01, 0.02, 0.03, and 0.04.From Figures 3 and 4, we can see that there is an excellent agreement between the results yielded by the SIPFEM/SEA method and by the Monte Carlo simulation of the hybrid FE/SEA model when the uncertainty level  is not more than 0.02, and there is a significant deviation between the results yielded by the two methods when the uncertainty level  is increased to 0.04.Due to the effect of the inherent flaw of the perturbation method, the bounds calculated by SIPFEM/SEA at the neighborhood of the peaks corresponding to the resonance frequencies of the oscillator are unreliable and valueless.
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  = 0.01, 0.02, 0.03, and 0.04 are calculated and listed in Tables 1 and 2; the considered frequencies are  = 170 Hz and  = 230 Hz, respectively.It can be seen from Tables 1 and  2 that the relative errors of SIPFEM/SEA increase gradually with the increase of the uncertainty level .From Figures 1 and 2 and Tables 1 and 2, we can conclude that the proposed SIPFEM/SEA method is very accurate for predicting the response of the built-up structure system with small interval parametric uncertainty, and when the uncertainty level gets larger, the accuracy of it decreases and unreliable results will be obtained.This is mainly because of the neglect of the higher order terms of Taylor series, which may bring unpredictable and uncontrollable effect on the results.
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  is set as 0.08; other parameters are kept the same.Figures 4 and 5 show the bounds of the plate energy and the autospectra of the mass displacements for  = 0.08, respectively.The considered numbers of subintervals for each parameter are one, two, four, and eight.It can be seen from Figures 4 and 5 that the bounds of the plate energy and the autospectra yielded by S-SIPFEM/SEA match the reference bounds more closely with the increase of the number of the subintervals, and the bounds yielded by S-SIPFEM/SEA with eight subintervals match the reference bounds perfectly.The relative errors of the bounds yielded by S-SIPFEM/SEA with different subinterval numbers are shown in Tables 3 and 4, and the considered frequencies are  = 170 Hz and  = 230 Hz, respectively.From Tables 3 and 4, we can see that the relative errors of S-SIPFEM/SEA get smaller with the increase of the number of subintervals.Therefore, we can conclude that S-SIPFEM/SEA will be accurate for the midfrequency analysis of the built-up structure system with large interval parameters if the number of the subintervals is sufficiently large.
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 5; we can see that the S-SIPFEM/SEA is much more efficient than the Monte Carlo simulation of the hybrid FE/SEA model.

A Beam-Plate
System.Beam-plate systems are typical built-up structures in practical engineering, and the beams are usually much stiffer than the plates [39].Figure 6 depicts a beam-plate system in which a plate and a beam are coupled via three point connections.The plate is rectangular with dimensions of 2 m × 0.8 m × 5 mm, and Young's modulus of it is   = 2.1 × 10 5 MPa, the density is   = 7850 kg/m 3 , Poisson's ratio is ]  = 0.3, and the modal density is   = 0.102 modes/Hz.The beam with a length of   = 2m is simply supported at both ends, and the cross section of it is rectangular with dimensions of 0.03 m × 0.04 m; Poisson's ratio of the beam is ]  = 0.3; Young's modulus and the density of it are expressed as   and   , respectively.The damping loss factors of the beam and the plate are both  = 0.01.The coupling points are located along the beam length at 0.27  , 0.47  , and 0.69  , and a force  is vertically exerted at 0.37  of the beam.In order to employ FE-SEA to model this system, the beam is set as the master system and modeled by using FEM with 200 beam elements; the plate is set as the subsystem and modeled by SEA.
−1tot and   , and they can be calculated by combining (40)∼(42).Thus, the th element of S  (a  )/  and  2 S  (a  )/ 2  can be expressed as  , (a  )

Table 1 :
The relative errors of the bounds of the plate energy and the autospectrum of the mass displacements for different uncertainty levels ( = 170 Hz).

Table 2 :
The relative errors of the bounds of the plate energy and the autospectrum of the mass displacements for different uncertainty levels ( = 230 Hz).

Table 3 :
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 ( = 170 Hz).

Table 4 :
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 ( = 230 Hz).

Table 6 :
The relative errors of the bounds of the plate energy and the autospectrum of the driving point displacements for different uncertainty levels ( = 330 Hz).bounds of the plate energy and the autospectra of the driving point displacements in the frequency band f = 200∼ 600 Hz calculated by S-SIPFEM/SEA; the considered number of subintervals for each interval parameter is three.The corresponding relative errors of the bounds for frequencies  = 330 Hz and  = 440 Hz are shown in Table8.From Figures7(d), 8(d), 9, and 10 and Tables 6∼8, we can conclude that if the uncertainty level of the interval parameters increases to a certain value, SIPFEM/SEA will yield unreliable results.When SIPFEM/SEA yields unreliable results, S-SIPFEM/SEA can be used to improve the accuracy by gradually increasing the number of subintervals.If the number of subintervals is sufficiently large, S-SIPFEM/SEA will yield perfect results. the

Table 7 :
The relative errors of the bounds of the plate energy and the autospectrum of the driving point displacements for different uncertainty levels ( = 440 Hz).

Table 8 :
The relative errors of the bounds of the plate energy and the autospectrum of the driving point displacements with three subintervals.