Perturbation Finite Element Method Research Articles

Antioptimization of mass-in-mass acoustic metamaterials based on interval analysis

In this study, an integrated analytical and experimental study on a type of one-dimensional periodic acoustic metamaterial with mass-in-mass unit cells is presented. Then, this paper proposes an antioptimization strategy to study the effect of the design parameters of the local resonators of the mass-in-mass acoustic metamaterial model on the band gap behavior. In this antioptimization strategy, the interval model is used to deal with the inner mass, spring stiffness and damping coefficient of the local resonators, and an interval mass-in-mass acoustic metamaterial model is established. Both the rational series expansion-interval perturbation finite element method (RSE-IPFEM) and interval affine finite element method (IAFEM) are used to evaluate the lower and upper bounds of the dynamic response interval vector, based on which an interval vibration transmission analysis can be implemented. Meanwhile, Monte Carlo method and experiments are also presented to demonstrate the efficiency and accuracy of the RSE-IPFEM and IAFEM. Results show that the design parameters of the local resonators significantly affect the beginning frequency of the band gap. Compared with the traditional optimization algorithm, the calculation accuracy of the antioptimization algorithm based on RSE-IPFEM and IAFEM can meet the engineering requirements, and the calculation efficiency is also significantly improved. Since IAFEM is more effective than RSE-IPFEM in suppressing conservatism, IAFEM is a better antioptimization strategy for dealing with the interval mass-in-mass acoustic metamaterial model. Therefore, the proposed antioptimization scheme is expected to inspire future researchers to analyze regulation rules of the band gap following the same procedures.

Band Gap Properties in Metamaterial Beam with Spatially Varying Interval Uncertainties

First, this study proposed a metamaterial beam model with spatially varying interval density. The interval dynamic equation of this model could be established by incorporating the decomposition results of the interval field based on Karhunen–Loeve expansion into the finite element method. An interval perturbation finite element method was developed to evaluate the bounds of the dynamic response interval vector. Then, an interval vibration transmission analysis could be performed, and the frequency range of the safe band gap could be determined. Meanwhile, Monte Carlo simulations and the vertex method are also presented to provide reference solutions. By comparison, it was found that the calculation accuracy of the interval perturbation finite element method was acceptable. The numerical results also showed that the safe band gap range was significantly smaller than that of the deterministic band gap.

HYPAD-UQ: A Derivative-Based Uncertainty Quantification Method Using a Hypercomplex Finite Element Method

Abstract A derivative-based uncertainty quantification (UQ) method called HYPAD-UQ that utilizes sensitivities from a computational model was developed to approximate the statistical moments and Sobol' indices of the model output. Hypercomplex automatic differentiation (HYPAD) was used as a means to obtain accurate high-order partial derivatives from computational models such as finite element analyses. These sensitivities are used to construct a surrogate model of the output using a Taylor series expansion and subsequently used to estimate statistical moments (mean, variance, skewness, and kurtosis) and Sobol' indices using algebraic expansions. The uncertainty in a transient linear heat transfer analysis was quantified with HYPAD-UQ using first-order through seventh-order partial derivatives with respect to seven random variables encompassing material properties, geometry, and boundary conditions. Random sampling of the analytical solution and the regression-based stochastic perturbation finite element method were also conducted to compare accuracy and computational cost. The results indicate that HYPAD-UQ has superior accuracy for the same computational effort compared to the regression-based stochastic perturbation finite element method. Sensitivities calculated with HYPAD can allow higher-order Taylor series expansions to be an effective and practical UQ method.

A novel univariate dimension‐reduction based interval finite element method for static response prediction of uncertain structures

AbstractTo eliminate the errors caused by the conventional interval perturbation finite element method due to classic interval arithmetic and neglect of higher‐order terms, we propose a novel univariate dimension‐reduction based interval finite element method to predict the static response bounds of structures with uncertain but bounded parameters. First, a univariate dimension‐reduction algorithm is derived using the generalized Taylor expansion. The global stiffness matrix is expressed as the sum of the median and the univariate disturbance radius. Compared with Taylor expansion approximation, the univariate dimension‐reduction approximation has higher accuracy and does not increase the amount of calculation. Then the inverse of the interval global stiffness matrix is approximated as an improved Neumann series. Higher‐ order terms are included by summing up the geometric terms in the Neumann series. Finally, the improved interval algorithm is used to solve the upper and lower bounds of the structural displacement response and the element stress response. The dependence between the interval parameters is accounted in comparison with the classic interval algorithm. The accuracy and effectiveness of the new method are validated by numerical cases on 2D truss, 3D frame and truck frame with multiple interval parameters.

Interval and subinterval perturbation finite element-boundary element method for low-frequency uncertain analysis of structural-acoustic systems

Uncertainties exist widely in the structural-acoustic systems, due to the physical imperfections, model inaccuracies and system complexities. To deal with the uncertainties, the interval perturbation and subinterval perturbation techniques, for the first time, are extended and integrated into the hybrid finite element-boundary element method (IPFEM/BEM) in this work. Firstly, the structural part of structural-acoustic systems is modeled using finite element method (FEM) and acoustic cavity is modeled by boundary element method (BEM). Then the interval perturbation technique is introduced to establish the interval perturbation equations of the system. For large uncertainty levels, the interval of parameters is divided into smaller sub-intervals to improve the accuracy of proposed IPFEM/BEM (sub-interval perturbation finite element-boundary element method, SIPFEM/BEM). The proposed IPFEM/BEM and SIPFEM/BEM are verified by two numerical examples. The results obtained by IPFEM/BEM, SIPFEM/BEM and Monte-Carlo method are compared. It is found that IPFEM/BEM and SIPFEM/BEM is reliable for the frequency response analysis of structural-acoustic systems with small or large uncertain levels.

Finite Element Analysis of the Uncertainty of Physical Response of Acoustic Metamaterials with Interval Parameters

The physical response of acoustic metamaterials may change due to the variation of the material properties in the manufacture process. Thus, an interval perturbation finite element method is formulated to study the mechanical response of acoustic metamaterials with the interval parameters, which includes the uncertainty effects on the band structure, resonance mode and frequency response of acoustic metamaterials. By virtue of the first-order Taylor series expansion and sensitivity analysis of dynamic properties of acoustic metamaterials with respect to the interval parameters, the interval perturbation finite element method established in this work can predict the upper and lower bounds of the dynamic properties of the acoustic metamaterials. Three numerical examples are studied to validate the effectiveness of the interval perturbation finite element method to analyze the physical response of acoustic metamaterials with the interval parameters, and the results calculated by Monte Carlo method are regarded as the reference results to validate the interval perturbation finite element method. The uncertainty model constructed by interval perturbation finite element method provides a great help in the design of acoustic metamaterials.

A stochastic perturbation finite element-least square point interpolation method for the analysis of uncertain structural-acoustics problems with random variables

In this paper, a novel stochastic perturbation finite element-least square point interpolation method (SP-FE-LSPIM) is introduced to improve the calculation accuracy for analyzing structural-acoustics problems. Inherited the element-compatibility of finite element method and the quadratic polynomial completeness of LSPIM, the present method obtains the global shape function for partition of unity (PU) and the least square point interpolation for local approximation. Besides, a first-order perturbation technique is also introduced into this theory for probabilistic analyzing. Thus, the response of the coupled systems can be expressed simply as a linear function of all pre-defined input variables by using the change-of-variable techniques. Due to the linear relationships between variables and the response, the probability density function and the cumulative probability density function of response can be obtained based on a simple mathematical transformation of probability theory. So the proposed approach not only improves the numerical accuracy of deterministic output quantities with respect to a given random variable, but also handles the randomness well in the systems. One numerical example for frequency response analysis of random structural-acoustics is presented and verified by Monte Carlo (MC) simulation and stochastic perturbation finite element method (SP-FEM) to demonstrate the effectiveness of the present method.

Perturbation Finite Element Method for Efficient Copper Losses Calculation in Switched Reluctance Machines

Copper losses dissipated in the windings of electric machines are the sum of classical ohmic dc losses and additional ac eddy current losses. In fact, the level of eddy current losses is strongly correlated with the manner of disposition of coil conductors in machine slots. Then, to improve the efficiency in electric machines, the selection of an optimal winding configuration becomes substantial. Since eddy current losses derive from the strong electromagnetic coupling between current density and time-dependent magnetic field, which cannot be solved easily, numerical analyses, such as particularly the one using the finite element (FE) method, are often used. As for the FE modeling, it can employ moving band technique to perform the rotor motion and Newton-Raphson iterations to deal with the nonlinear behavior of magnetic circuits. It leads then to a substantial computational time that hinders any process of conception or optimization of winding geometries. To overcome this issue, a 2-D FE model reduction based on the perturbation method is proposed. It starts from one approximate FE solution of a simplified complete machine modeling to find fast but accurate solutions in slots subdomains when any variation of geometrical or physical data occurs. It allows adapting nonconforming meshes and provides clear advantages in repetitive analyses when we search the optimized winding configuration for a given number of turns.

Finding Better Solutions to Reduce Computational Effort of Large-Scale Engineering Eddy Current Fields

In the finite element analysis of the engineering eddy current fields in electrical machines and transformers there are the problems such as the huge scale of computation, too long computing time and poor precision which could not meet the demand of engineering accuracy. The current research situation and difficulties of these problems are analyzed in this paper mainly from the aspect of computation methodology. The methods to deal with these problems, e.g., homogenization models of the laminated iron core, the sub-domain perturbation finite element method, domain decomposition method, and EBE (Element by Element) parallel finite element method are described. Their advantages and limitations are discussed, and the authors’ suggestions for the further research strategies are also included.

Hybrid uncertainty propagation of coupled structural–acoustic system with large fuzzy and interval parameters

Based on the finite element framework and uncertainty analysis theory, this paper proposes a first-order subinterval perturbation finite element method (FSPFEM) and a modified subinterval perturbation finite element method (MSPFEM) to solve the uncertain structural–acoustic problem with large fuzzy and interval parameters. Fuzzy variables are used to represent the subjective uncertainties associated with the expert opinions; whereas, interval variables are adopted to quantify the objective uncertainties with limited information. By using the level-cut strategy and subinterval methodology, the original large fuzzy and interval parameters are decomposed into several subintervals with small uncertainty level. In both FSPFEM and MSPFEM, the subinterval matrix and vector are expanded by the Taylor series. The inversion of subinterval matrix in FSPFEM is approximated by the first-order Neumann series, while the modified Neumann series with higher order terms is adopted in MSPFEM. The eventual fuzzy interval frequency responses are reconstructed by the interval union operation and fuzzy decomposition theorem. A numerical example evidences the remarkable accuracy and effectiveness of the proposed methods to solve engineering structural–acoustic problems with hybrid uncertain parameters.

Uncertain temperature field prediction of heat conduction problem with fuzzy parameters

This paper proposes a first-order fuzzy perturbation finite element method (FFPFEM) and a modified fuzzy perturbation finite element method (MFPFEM) for the uncertain temperature field prediction with fuzzy parameters in material properties, external loads and boundary conditions. Fuzzy variables are used to represent the subjective uncertainties associated with the expert opinions. Based on the α-cut method, the finite element equilibrium equation is equivalently transformed into groups of interval equations. The interval matrices and vectors in both methods are expanded by the first-order Taylor series. First-order Neumann series is employed in FFPFEM for the inversion of interval matrix, while some high-order terms of the Neumann series are retained in MFPFEM. Based on the fuzzy decomposition theorem, the fuzzy solutions to the original uncertain heat conduction problem are eventually derived. By comparing the results with traditional Monte Carlo simulation and vertex method, two numerical examples evidence the feasibility and effectiveness of the proposed methods at predicting the fuzzy temperature field in engineering.

Hybrid finite element/statistical energy method for mid-frequency analysis of structure−acoustic systems with interval parameters

The hybrid Finite Element/Statistical Energy Analysis (FE-SEA) model for mid-frequency analysis consists of two parts: the FE components and the SEA components, which are called as the master system and the subsystem respectively. The hybrid Finite Element/Statistical energy analysis approach is established on the assumption that the FE components have fully deterministic properties, while the SEA subsystems have a high degree of randomness. This method has been recently extended to allow the properties of the FE components of built-up structures to be uncertain like interval or probabilistic, and the hybrid model with parametric and non-parametric uncertainties is obtained. By dealing with the non-parametric uncertainty analytically and the parametric uncertainty with Monte Carlo Simulations, the distribution of the responses of the hybrid model can be obtained. In this paper, the interval parametric uncertainty is introduced into the hybrid FE–SEA framework for structure−acoustic systems, and the interval dynamic equilibrium equations are established, thus a hybrid model with non-parametric and interval parametric uncertainties for structure−acoustic systems is obtained; to improve the computational efficiency, the second-order interval perturbation finite element method is introduced into the hybrid FE-SEA framework and the second-order interval perturbation finite element/statistical energy analysis (SIPFEM/SEA) method is proposed. For the structure−acoustic system with interval parameters modeled by FE-SEA, the parameters of FE components are considered as interval parameters instead of deterministic ones, thus the expectations of the second order quantities such as the vibrational energy and the response cross-spectrum in the FE-SEA framework become interval variables, too. In the SIPFEM/SEA, the expectations of the second order response quantities of a structure−acoustic system are expanded with the second-order Taylor series at the nominal values of interval parameters, and for the sake of simplicity and efficiency, the non-diagonal 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 vibrational energy and the response cross-spectrum can be obtained. Because of the neglect of the higher order terms of Taylor series, SIPFEM/SEA is limited to the interval analysis with narrow parameter intervals. For larger parameter intervals, the sub-interval perturbation method based on the SIPFEM/SEA is introduced. The proposed methods are illustrated by applications to two example structure−acoustic systems in which the acoustic cavities are modeled by using the interval FE method and the structures are modeled by using the SEA. Reference comparisons are made with the Monte Carlo simulations of the hybrid FE/SEA model. The numerical results verify the accuracy and efficiency of the proposed methods.

Decomposed Interval Matrix Perturbation Finite Element Method for the Response Analysis of the Acoustic Field with Uncertain Parameters

To further improve the computational accuracy and efficiency of the modified interval perturbation finite element method(MIPFEM),a decomposed interval matrix perturbation finite element method(DIMPFEM) is proposed. In the proposed method,the dynamic stiffness matrix of an acoustic system is decomposed into the sum of several sub-matrices whose perturbation matrix can be expressed as the products of perturbation factors and determine matrices,thus the errors arising from the first-order Taylor expansion can be avoided. To achieve a higher computational efficiency,the inverse perturbation matrix,approximated by the modified Neumann series expansion is calculated by the epsilon-algorithm. Numerical examples on a 2D acoustic tube and a 2D acoustic cavity of a multi-purpose vehicle(MPV) with interval parameters verify that the computational accuracy and efficiency of DIMPFEM are higher than those of MIPFEM.

A Modified Computational Scheme for the Stochastic Perturbation Finite Element Method

A MODIFIED COMPUTATIONAL SCHEME OF THE STOCHASTIC PERTURBATION FINITE ELEMENT METHOD (SPFEM) IS DEVELOPED FOR STRUCTURES WITH LOW-LEVEL UNCERTAINTIES. THE PROPOSED SCHEME CAN PROVIDE SECOND-ORDER ESTIMATES OF THE MEAN AND VARIANCE WITHOUT DIFFERENTIATING THE SYSTEM MATRICES WITH RESPECT TO THE RANDOM VARIABLES. WHEN THE PROPOSED SCHEME IS USED, IT INVOLVES LIMITED ANALYSES OF DETERMINISTIC SYSTEMS. IN THE CASE OF ONE RANDOM VARIABLE WITH A SYMMETRIC PROBABILITY DENSITY FUNCTION, THE PROPOSED COMPUTATIONAL SCHEME CAN EVEN PROVIDE A RESULT WITH FIFTH-ORDER ACCURACY. COMPARED WITH THE TRADITIONAL COMPUTATIONAL SCHEME OF SPFEM, THE PROPOSED SCHEME IS MORE CONVENIENT FOR IMPLEMENTATION. THREE NUMERICAL EXAMPLES DEMONSTRATE THAT THE PROPOSED SCHEME CAN BE USED IN LINEAR OR NONLINEAR STRUCTURES WITH CORRELATED OR UNCORRELATED RANDOM VARIABLES

Interval finite element analysis and reliability-based optimization of coupled structural-acoustic system with uncertain parameters

A modified interval parameter perturbation finite element method (MIPPM) and a reliability-based optimization model are proposed for the coupled structural-acoustic field prediction and structural design with uncertainties in both the physical parameters and boundary conditions. Interval variables are used to quantitatively describe all the uncertain parameters with limited information. The interval matrix and vector are expanded by the modified Taylor series. Compared with the traditional perturbation method, the proposed MIPPM can yield more accurate ranges of the uncertain structural-acoustic field, in which the higher order terms of Neumann series are employed to approximate the interval matrix inverse. The reliability idea is introduced to establish an interval optimization model relying on the satisfaction degree of interval. The uncertain constraints can be transformed into deterministic ones if given the confidence level. The proposed MIPPM is used to predict the intervals of the constraints, and whereby eliminate the optimization nesting. Numerical results about a 3D car are given to demonstrate the feasibility and efficiency of the proposed model and algorithm.

Random Response Analysis of Vibration Transfer Path Systems with Translational and Rotational Motions <sup></sup>

This paper based on the generalized probabilistic perturbation finite element method solves the random response analysis problem of vibration transfer path systems with translational and rotational motions. The effective random response analysis approaches are achieved using Kronecker algebra, matrix calculus, generalized second moment technique of vector-valued functions and matrix-valued functions. For the vibration transfer path system with multi-dimensional paths, the random response is described correctly and expressly in time domain as uncertain factors, which include mass, damping, stiffness and position, are considered. The mathematical expressions of the first order and second order moments for the random vibration response of vibration transfer path are obtained. According to the corresponding numerical example, the results of calculation are consistent with the results of Monte-Carlo simulation, which shows the method is feasible theoretically.

Modified Interval Perturbation Finite Element Method for a Structural-Acoustic System With Interval Parameters

For the frequency response analysis of the structural-acoustic system with interval parameters, a modified interval perturbation finite element method (MIPFEM) is proposed. In the proposed method, the interval dynamic equilibrium equation of the uncertain structural-acoustic system is established. The interval structural-acoustic dynamic stiffness matrix and the interval force vector are expanded by using the first-order Taylor series; the inversion of the invertible interval structural-acoustic dynamic stiffness matrix is approximated by employing a modified approximate interval-value Sherman–Morrison–Woodbury formula. The proposed method is implemented at an element-by-element level in the finite element framework. Numerical results on a shell structural-acoustic system with interval parameters verify the accuracy and efficiency of the proposed method.

Interval analysis of acoustic field with uncertain-but-bounded parameters

This paper presents an interval perturbation finite element method (IPFEM) and a modified interval perturbation finite element method (MIPFEM) for the uncertain acoustic field prediction with uncertain-but-bounded parameters. The interval matrix and vector in both methods are expanded by the first-order Taylor series. The interval matrix inverse in IPFEM is approximated by the first-order Neumann series; however, the interval matrix inverse in MIPFEM is approximated by a modified Neumann expansion in which the higher order terms of Neumann series are considered. Numerical results demonstrate the accuracy of MIPFEM to evaluate the intervals of sound pressure response in acoustic field.

Modified sub-interval perturbation finite element method for 2D acoustic field prediction with large uncertain-but-bounded parameters

Based on the sub-interval perturbation analysis, a modified sub-interval perturbation finite element method is proposed to determine the bounds of sound pressure in the 2D acoustic field with large uncertain-but-bounded parameters. In the proposed method, the inversion of the invertible sub-interval dynamic stiffness matrix is approximated by a modified approximate interval-value Sherman–Morrison–Woodbury formula to overcome the drawbacks arising from the dependency phenomenon of parameters and the unpredictable effect of neglecting the higher order terms in Neumann series. The modified sub-interval perturbation procedures are implemented in a numerical finite element framework. Numerical results on a 2D acoustic tube and a 2D acoustic cavity of a car with large uncertain-but-bounded parameters evidence the remarkable accuracy and effectiveness of the proposed method.

Robust Optimization of Dynamic Response of Structures with Uncertain Parameters

On the base of traditional robust optimal design in statics, in this paper, by considering the system under the incentive force of the arbitrary time function, the dynamic response robust optimal design problem, when system dynamic response can only be calculated using numerical integration. Considering the environmental interference factors in a system, derived the expression of response with a small parameter perturbation finite element method. Through the tiny fluctuation frame system in a post and beam’s elastic modulus, realize the robust optimal design of the structural dynamic response, and using New-mark method for dynamic analysis, robust optimal design is applied in dynamic response optimization. Compared to the traditional optimal design’s result in a frame system, robust optimal design of the frame displayed a significant improvement in performance.