First-order Perturbation Technique Research Articles

Free vibration analysis of the geometrically nonlinear functionally graded porous curved panels in deterministic and stochastic domains considering various boundary conditions

This study focuses on the analysis of free vibrations in functionally graded (FG) panels with characteristics such as porosity, curvature, and geometric nonlinearity. The analysis encompasses both deterministic and stochastic domains while considering a range of boundary conditions. The nonlinear finite element formulation adopted in this study is based on a higher-order nonlinear shell theory that incorporates the von Karman geometric nonlinearity. The material properties of the FG curved panels are graded using a power law distribution. The investigation includes four different boundary conditions and various types of curved panels, namely spherical, hyperbolic paraboloidal, elliptical, and cylindrical. Stochastic analysis employs the first-order perturbation technique (FOPT), while Monte Carlo simulation (MCS) is employed solely for validating the FOPT model. The study explores the influence of boundary conditions, panel types, porosity distribution, volume fraction index, and other factors in both deterministic and stochastic domains, taking into account randomness in material properties. The stochasticity in the geometric properties like dimensions, and boundary conditions can be the potential future scope of this study.

Detecting Outliers Using Modified Recursive PCA Algorithm For Dynamic Streaming Data

Outlier analysis has been widely studied and has produced many methods. However, there is still rare a method to detect outliers for dynamically streaming batch data (online learning). In the present research, a novel online algorithm to detect outliers in such dataset is proposed. Data points are proceeded by applying a modified recursive PCA to predict sequentially parameters of the model; eigenvalues and eigenvectors of the statistical detection model are recursively updated using approximate values by perturbation methods. More specifically, the recursive eigenstructure is obtained from the derivation of the covariance matrix using the first-order perturbation technique. The Mahalanobis distance is then used as an outlier score. Our algorithm performances are evaluated using some metrics, namely accuration, precision, recall, F1-score, AUC-PR, and the execution time. Results show that the proposed online outlier detection is computationally efficient in time and the algorithm's performance effectiveness is comparable to that of the offline outlier detection algorithm via classical PCA.

A study of the natural frequencies of the functionally graded curved beams including porosities: Deterministic and stochastic modeling

In this research article, an investigation into the natural frequencies of functionally graded porous curved beams is presented, covering both deterministic and stochastic domains. The stochastic finite element analysis is based on a three-node, C0 continuous element, and higher-order shear deformation theory, utilizing the first-order perturbation technique (FOPT). The study includes a comparative analysis of three porosity distribution models, including a recently proposed sinusoidal model. Additionally, the study examines the influence of various factors, such as porosity distributions, boundary conditions, volume fraction indices, and geometric configurations, on the natural frequencies. The analysis encompasses uncertain natural frequencies in functionally graded curved beams, accounting for material stochasticity through FOPT. Monte Carlo simulation (MCS) is used to validate the accuracy of the FOPT model, increasing the approach’s reliability. Potential future research in this area could investigate the effects of stochastic variations in geometric parameters, including dimensions and boundary conditions.

A simplified model for the impact of dielectric polarization of a charged droplet on its diffusiophoresis

This study aims to quantify the impact of the dielectric permittivity of a droplet on its diffusiophoresis in different types of electrolytes. The dielectric droplet polarizes by the diffusion field along with the local electric field created by the interactions of the double layer with the imposed ionic concentration gradient, which generates an induced surface charge density anti-symmetrically distributed on the droplet surface. This induced surface charge influences both electrophoresis and chemiphoresis parts. Based on a low imposed concentration gradient, a simplified model is derived through a first-order perturbation technique. Dielectric polarization of the droplet attenuates the spinning force at the interface. This creates the mobility of a droplet of higher dielectric permittivity in the presence of a stronger diffusion field significantly higher than that of a perfectly dielectric droplet, and its value depends on the polarity of the droplet surface charge. In the absence of the diffusion field, the mobility of a conducting droplet remains a positive immaterial of the polarity of its surface charge density. We find that the impact of the dielectric polarization becomes significant as the surface charge density increases and attenuates with the increase in droplet viscosity. For a dielectric droplet at a thinner Debye length, a step-jump in mobility occurs at a higher value of the surface charge density. Such a type of step-jump in mobility does not appear for the conducting droplet due to the absence of the Maxwell stress at the interface.

Stochastic Higher-order Finite Element Model for the Free Vibration of a Continuous Beam resting on Elastic Support with Uncertain Elastic Modulus

This paper deals with a continuous beam resting on elastic support with elastic modulus derived from a random process. Governing equations of the stochastic higher-order finite element method of the free vibration of the continuous beam were derived from Hamilton's principle. The random process of elastic modulus was discretized by averaging random variables in each element. A solution for the stochastic eigenvalue problem for the free vibration of the continuous beam was obtained by using the perturbation technique, in conjunction with the finite element method. Spectral representation was used to generate a random process and employ the Monte Carlo simulation. A good agreement was obtained between the results of the first-order perturbation technique and the Monte Carlo simulation.

On the stochastic vibration analysis of the geometrically nonlinear graded cellular curved panels with material stochasticity

In this paper, an effort was made to study the free flexural vibration analysis of the geometrically nonlinear graded cellular curved panels in a stochastic domain. The nonlinear finite element formulations were developed using von Karman nonlinearity hypotheses based on higher-order shear deformation theory, including Lame's parameter. The first-order perturbation technique (FOPT) was employed to assess the stochastic vibration behavior of the graded cellular panels considering randomness in the material properties. The graded cellular curved panels are considered to have three distinguished porosity distributions. The properties of the graded cellular curved panels are continuously varying in the thickness direction as a function of porosity coefficient and mass density. The influences of material stochasticity, porosity coefficient, porosity distribution type, and geometric parameters have been included in the formulations. The detailed convergence and comparison studies were conducted to demonstrate the accuracy and efficiency of the present formulation and highlight the importance of the present study in stochastic environments. The various numerical examples for the second-order statistics (i.e., mean and standard deviation) of the linear frequency parameters (λ) and nonlinear frequencies of the graded cellular structures have been presented which may be considered as the guidelines for future studies in this area of research.

On the stochastic natural frequency of graphene reinforced functionally graded porous panels with unconventional boundary conditions

The present study explores the stochastic natural frequency of graphene reinforced functionally graded porous panels with unconventional boundary conditions. The material uncertainty is considered in the parameters associated with constituents that is, metal and graphene nanoplatelets (GPLs) individually and simultaneously. The metal matrix is reinforced with ultra-lightweight and high stiffness carbonaceous nanofiller, that is, GPLs. Halpin-Tsai micromechanics model is adopted to estimate effective Young’s modulus of the metal-GPLs composite whereas effective mass density and Poisson’s ratio are calculated using the Voigt model. The micro-structural gradation by creating pores is accomplished along the thickness direction of the panel employing certain continuous functions. These functions describe symmetric and asymmetric porosity distributions and associated patterns of GPLs. The equation of motion is developed using Euler-Lagrange’s equation which is based on C 0-continuous structural kinematics with 7 degrees of freedom. Finite element modeling in conjunction with the first-order perturbation technique has been applied to quantify the stochastic natural frequency in terms of the mean and standard deviation of the natural frequency. Various results have been examined to highlight the influence of parameters especially related to porosity, and graphene nanoplatelets on the stochastic natural frequency of FG-GPLs porous panel constrained with conventional and unconventional boundary conditions.

Influence of material uncertainty on higher-order FG-GPLs reinforced porous spherical panels under blast loading

The present study investigates the influence of material uncertainty on the dynamic response of functionally graded graphene nanoplatelets (FG-GPLs) reinforced porous spherical panels under air blast loading. The uncertainty is included in various parameters such as Young’s moduli, mass density, Poisson’s ratios, porosity coefficient, and GPLs’ weight fraction. The microstructural gradation is accomplished by creating pores followed by symmetric, asymmetric, and uniform porosity distributions which lead to variation in the properties along the thickness direction. The metal matrix i.e., Aluminum is reinforced with graphene nano-platelets, and the effective Young’s modulus is estimated using the Halpin–Tsai model whereas the rule of the mixture is employed to determine effective mass density and Poisson’s ratio. A higher-order C0 — continuous structural kinematics is employed to find the dynamic response of the spherical panels. The Governing relations are developed using Euler–Lagrange’s equation and the finite element method in conjunction with the first-order perturbation technique and Newmark’s time integration scheme is applied to find the numerical results in terms of mean and standard deviation of center displacement. The influence of various parameter such as porosity coefficient, GPLs weight fraction, TNT-weight, stand-off stand, etc. are also presented in detail.

Stochastic Buckling Analysis of Non-Uniform Columns Using Stochastic Finite Elements with Discretization Random Field by the Point Method

This study examined the discretization random field of the elastic modulus by a point method to develop a stochastic finite element method for the stochastic buckling of a non-uniform column. The formulation of stochastic analysis of a non-uniform column was constructed using the perturbation method in conjunction with the finite element method. The spectral representation was used to generate a random field to employ the Monte Carlo simulation for validation with a stochastic finite element approach. The results of the stochastic buckling problem of non-uniform columns with the random field of elastic modulus by comparing the first-order perturbation technique were in good agreement with those obtained from the Monte Carlo simulation. The numerical results showed that the response of the coefficient of variation of critical loads increased when the ratio of the correlation distance of the random field increased.

Influence of Material Uncertainty on Vibration Characteristics of Higher-Order Cracked Functionally Gradient Plates Using XFEM

In this study, the influence of material uncertainty on the vibration characteristics of the cracked functionally graded materials (FGM) plates is investigated. Extended stochastic finite element formulation is implemented to model the cracked FGM plate with material uncertainty using higher-order shear deformation theory (HSDT). The level set function is employed to track the crack in the FGM domain. The concept of partition of unity technique is implemented to enrich the primary variable with additional functions. The gradation of the material properties along the thickness direction is done using the power-law distribution. The first-order perturbation technique (FOPT) is incorporated in the methodology for stochastic vibration analysis. The convergence and validation study has been performed to verify the efficacy and accuracy of the formulation. Numerical results are obtained to show the effects of various influential parameters like crack length, gradient index, thickness ratio, and boundary condition on the covariance of the square of natural frequencies. The presented computational approach is accurate, efficient, and robust enough to investigate the vibration response of cracked FGM plates with material randomness.

Stochastic Extended Finite Element Implementation for Natural Frequency of Cracked Functionally Gradient and Bi-Material Structures

In this work, stochastic perturbation-based vibration characteristics of cracked bi-material and functionally graded material (FGM) domain with uncertain material properties are investigated using the extended finite element method. The level set function is implemented to track the geometrical discontinuities. The partition of unity-based extrinsic enrichment technique is employed to model the crack and material interface. The exponential law is used to model the graded material properties of FGM. The First-order perturbation technique (FOPT) is implemented to predict the standard deviation of natural frequency for the given uncertainties in the material properties. The numerical results are presented to show the effect of geometrical discontinuities and material randomness on vibration characteristics.

Influence of Material Stochasticity on Buckling Characteristics of Initially Imperfect Higher-Order Shear Deformable Gradient Plates

This paper demonstrates the influence of material stochasticity on buckling characteristics of higher-order shear deformable gradient plates with initial geometric imperfections. The gradient plates are assessed by smooth variation in the volume fraction of the constituents (i.e. ceramic and metal) as power-law distribution function in the thickness direction. The effective material properties are achieved by means of the Voigt model. Plate kinematic based on Reddy’s higher-order shear deformation theory (HSDT) associated with initial geometric imperfection in the transverse direction is employed. The governing differential equation is produced using a variational approach. The mean and standard deviation of the critical buckling load are evaluated using finite element method and a mean-centered first-order perturbation technique in order to highlight the variation in buckling response. Numerical results are compared both in deterministic and probabilistic frameworks along with convergence in support of efficacy and performance of the proposed model. Based on the results, it can be concluded that the combined influence of geometric imperfection and uncertain material properties prominently affect the buckling response of the gradient plates.

Cable Angle and Minimum Resultant Force Response Analysis of Lower Limb Traction Device for Rehabilitation Robot With Interval Parameters

Abstract This paper proposes a hybrid uncertainty analysis method (HUAM) based on the first-order interval perturbation method (FIPM) and Monte Carlo method (MCM) for minimum resultant force response analysis of the lower limb traction device (LLTD) of a hybrid-driven parallel waist rehabilitation robot (HDPWRR) with interval parameters. Based on the analysis of cable angles by using the interval algorithm, the problem of non-uniqueness of the force solution in redundant constraint mechanisms is solved. The force response domain prediction with interval parameters on rehabilitation patients is estimated by using the HUAM which combining the first-order interval perturbation technique with direct Monte Carlo method in different stages, and it reduces the calculation amount. First, the kinematic and static models of the LLTD with deterministic information are established according to its work principle. Then, the interval matrices with interval parameters are calculated by using the FIPM and the response of cable angles is combined with the static model. Third, by numerical examples, the accuracy and efficiency of the HUAM for solving the force response domain problem with interval parameters are verified. The bounds of cable angle response domain of the interval LLTD model are determined. Finally, the minimum resultant force response domain prediction with interval parameters on rehabilitation patients is estimated by combining the FIPM and MCM.

Stochastic Perturbation-Based Finite Element for Free Vibration of Functionally Graded Beams with an Uncertain Elastic Modulus

The stochastic eigenvalue problem for free vibrations of functionally graded beams with a random elastic modulus is investigated. The effective material properties and beam cross section are assumed to vary continuously in different directions according to the exponential law. The governing equations for the natural frequency of the functionally graded beams are derived from Hamilton’s principle. In the stochastic finite-element method, the random process was discretized by averaging random variables in each element to increase the accuracy of the natural frequency found. A solution of the stochastic eigenvalue problem formulated was obtained using the perturbation technique in conjunction with the finite-element method. The spectral representation was used to generate a random process to employ the Monte Carlo simulation. A good agreement was obtained between the results of the first-order perturbation technique and the Monte Carlo simulation.

Electro-elastic analysis of functionally graded piezoelectric variable thickness cylindrical shells using a first-order electric potential theory and perturbation technique

In this research, an analytical solution is presented for the functionally graded piezoelectric cylindrical variable wall thickness that is subjected to mechanical and electrical loading. The non-homogeneous distribution of materials is considered as a power function. The first-order electric potential theory, first-order shear deformation theory, and the energy method are used for extracting the system of governing equations. The solution is accomplished using the matched asymptotic expansion method of the perturbation technique. The effects of non-homogeneous properties on the electromechanical are discussed. Since the intensity of variations in the distribution of properties in functionally graded piezoelectric cylinders can be changed using non-homogeneity constant, the electromechanical behavior of the cylinder can be changed by non-homogeneity constant. By reducing the electric or displacement field in functionally graded piezoelectric cylinders, de-polarization or loss of piezoelectric properties may be averted. Results indicate that non-homogeneity constant has a significant effect on the electromechanical behavior. However, in some cases, the effects of non-homogeneity constant may be neglected. Comparing these results with those predicted by the plane elasticity theory and finite element method shows good agreement. In fact, the present solution can be considered as an objective function to optimize the properties and behavior.

An analytical solution for heat conduction of FGM cylinders with varying thickness subjected to non-uniform heat flux using a first-order temperature theory and perturbation technique

In this research, thermal analysis in cylinders with varying thickness constructed from functionally graded materials (FGMs) which are under non-uniform heat fluxes in their inner and outer layers is presented. The heterogeneous distribution of properties is a modelled power function along the thickness and the length of the cylinder. Using first-order temperature theory (FTT) and the energy method, the governing equations are extracted. The system of governing differential equations is a system of ordinary differential equations with variable coefficients that is performed by using the analytical method of matched asymptotic expansion of the perturbations technique for solving it. Results relating to temperature distribution, heat flux, and thermal strain for different heat fluxes and heterogeneous constants are discussed. The results show that the heterogeneity has a magnificent effect on the temperature field and thermal strain inside FG cylinders. Moreover, it is observed that the heterogeneity has no effect on the direction of heat flux vector inside the body but any changes in the heterogeneity would change the magnitude of heat flux. The results achieved from the analytical method are compared with those predicted by finite element method (FEM). A good agreement was found between the predicted results of both methods.

Higher-Order Stabilized Perturbation for Recursive Eigen-Decomposition Estimation

Abstract Eigen-decomposition remains one of the most invaluable tools for signal processing algorithms. Although traditional algorithms based on QR decomposition, Jacobi rotations and block Lanczos tridiagonalization have been proposed to decompose a matrix into its eigenspace, associated computational expense typically hinders their implementation in a real-time framework. In this paper, we study recursive eigen perturbation (EP) of the symmetric eigenvalue problem of higher order (greater than one). Through a higher order perturbation approach, we improve the recently established first-order eigen perturbation (FOP) technique by creating a stabilization process for adapting to ill-conditioned matrices with close eigenvalues. Six algorithms were investigated in this regard: first-order, second-order, third-order, and their stabilized versions. The developed methods were validated and assessed on multiple structural health monitoring (SHM) problems. These were first tested on a five degrees-of-freedom (DOF) linear building model for accurate estimation of mode shapes in an automated framework. The separation of closely spaced modes was then demonstrated on a 3DOF + tuned mass damper (TMD) problem. Practical utility of the methods was probed on the Phase-I ASCE-SHM benchmark problem. The results obtained for real-time mode identification establishes the robustness of the proposed methods for a range of engineering applications.

Thermomechanically induced post-buckling analysis of functionally graded material plates with circular cut-outs resting on elastic foundations

Post-buckling analysis of functionally graded material (FGM) plates resting on Winkler and Pasternak elastic foundations subjected to thermomechanical loadings with circular cut-outs at centre and random material properties is presented. The material properties of each constituent’s materials, volume fraction index, thermal expansion coefficients, foundation stiffness parameters and thermal conductivities are taken as independent basic random input variables. The basic formulation is based on applying Reddy’s higher order shear deformation theory, which requires C1 continuous element approximation. A modified form C0 continuity is applied in the present investigation. A serum-free expansion medium with mean-centred first-order regular perturbation technique for composite plates is extended for FGM plates to solve the random eigenvalue problem. Typical numerical results are presented to examine the second-order statistics for effect of the volume fractions index, plate length-to-thickness ratios, plate aspect ratios, types of loadings, amplitude ratios, support conditions and various shape and size of holes with random thermomechanical properties. The results obtained by the present solution approach are validated with published papers and the robust method of simulation. It is found that the laminates with round cuts (FGM plates resting on Winkler and Pasternak elastic foundations) have a significant influence on the post-buckling response under Thermomechanical loading conditions. Present investigations are useful for the applications and further research.

First-Order Eigen-Perturbation Techniques for Real-Time Damage Detection of Vibrating Systems: Theory and Applications

AbstractThis manuscript provides a detailed synopsis of the contemporary advancements in the nascent area of real-time structural damage detection for vibrating systems. The paper mainly focuses on the theoretical development and engineering applications of algorithms that are based on first-order perturbation (FOP) techniques applied to vibration responses. The importance of this work stems from the fact that recent developments in the field of online structural health monitoring (SHM) have given rise to algorithms that are computationally complex and, consequently, are not amenable to real-time implementation. In this paper, we discuss and demonstrate the FOP-based algorithms in the light of all the contemporary nonadaptive/nonrecursive techniques to establish their relevance. We review 216 papers in this regard. The efficacy, efficiency, robustness, and the applicability of the FOP family of algorithms are highlighted in light of several experimental, theoretical, and field studies.

Stochastic isogeometric analysis method for plate structures with random uncertainty

This paper proposes the stochastic isogeometric analysis (IGA) method in conjunction with the perturbation technique for random response analysis of plate structures. Specifically, the benchmark solutions of stochastic deflection responses for rectangular plate are achieved. Firstly, the random field is represented by Karhunen-Loève expansion, and the corresponding analytical formulas of the Fredholm integral equation for rectangular plate are illustrated to obtain the benchmark stochastic responses. Subsequently, the stochastic IGA framework is suggested by combining isogeometric analysis with the first-order perturbation technique, especially for the bending problems of circular Mindlin plates and Kirchhoff plates. Moreover, the first two moments of stochastic responses of plate structures are formulated. Finally, the efficiency and applicability of the stochastic IGA method are demonstrated by three numerical examples. To compare with the efficiency and accuracy of proposed method, the mean values, standard deviations and coefficients of variation of stochastic responses are calculated by Monte Carlo simulation. For the rectangular and circular Kirchhoff thin plates, the effects of different correlation lengths, boundary conditions and loading cases on stochastic responses and uncertainty propagation are scrutinized. It is also indicated that the proposed stochastic IGA method presents high efficiency and acceptable accuracy of random structural analysis of plates.