Isoparametric Elements Research Articles

Nonlinear vibrations of a dielectric elastomer cantilever combined-stepped-plate actuator

A dielectric elastomer (DE) cantilever combined-stepped-plate actuator is proposed and its nonlinear vibration enhancement behaviors are investigated. The irregular geometry of the actuator is transformed into a rectangular computational domain with the ideal of the iso-parametric element transformation; then the governing equations describing the nonlinear vibration of the actuator are formulated by the variational method. The nonlinear periodic responses are solved through the harmonic balance method and the arc-length continuation method. The results indicate that the single side (upper or lower) voltage excitation is more efficient in the actuating performance. The asymmetrical combined plate presents a better motion performance comparing with the symmetrical one. Not only can more drive modes be achieved (For instance, both symmetrical flapping mode and asymmetrical torsional mode can be excited by changing the voltage frequency), but also higher responses can be obtained. Additionally, isolated responses are found for some particular dimension parameters, which provides a potential design strategy to regulate the frequency domain of significant responses. This research concludes that, the asymmetrical combined plate subjected to an asymmetrical voltage loading would show a more efficient locomotion performance, which can provide an insight into the structural design and the excitation strategy for DE actuators.

Dynamic analysis of laminated shell panels with cutout and cracked corners carrying concentrated and distributed mass

This article employs Mindlin’s first-order shear deformation theory to analyze the dynamic behavior of laminated composite shells with various geometries, featuring central cutouts and cracked corners that carry concentrated and distributed mass. The formulation considers an isoparametric element with nine nodes and five degrees of freedom per node. Both in-plane and transverse effects of mass are addressed by incorporating lumped mass with rotational inertia in the formulation. The obtained results are compared to previously published data, showing an excellent agreement with a variation of less than ±2.5% for all cases. Following validation, the study examines shell vibration for various geometries carrying concentrated mass, distributed mass, and distributed patch mass. Additionally, the dynamic analysis of shell panels with central cutouts and cracked corners with different cracked dimensions is conducted. Unique boundary conditions are introduced by applying them to both the outer and cutout edges (i.e., SSSS-CCCC, FFFF-CCCC).

Deflection of FGM sandwich conoidal shell with porous core

The maximum deflection of functionally graded sandwich conoidal (FGSC) shell with porous core is studied in this paper. Even and uneven porosity distribution is considered to investigate the effect of porosity in the core of the FGSC shell. The inclusion of porosity (even or uneven) in the core makes the sandwich shell lighter. The modified power law is used to account the porosity distribution. The governing equations are developed based on improved mathematical model. An isoparametric finite element (FE) having 9 nodes and 63 degrees of freedom have been used. The improved mathematical model is executed in the present FE code. The numerical results in terms of maximum deflections are compared with available results in the literature. A broad parametric study is carried out to check the effect of porosity, volume fraction index and material properties on the maximum deflection of FGSC shell with porous core.

NONLINEAR LATERAL RESPONSE OF PILE GROUP IN CLAY USING THE MODIFIED CAM CLAY SOIL MODEL

Lateral pile response of the 3 × 3 square pile group has been investigated in terms of lateral displacement and maximum bending moment using three-dimensional finite element analysis. Soil is represented using 8-node isoparametric elements. Piles and pile cap are modelled using 20-node isoparametric elements. The 16-node isoparametric interface elements are used to establish the continuity between the pile and surrounding soil. Soil is represented by the modified Cam clay criterion. The entire code has been developed in FORTRAN 90. The parametric study has been performed to investigate the effect of yield criteria, soil modulus, pile spacing, pile diameter and pile length-to-diameter ratio on the response of the 3 × 3 pile group embedded in clay. A considerable effect of these parameters is observed. It is found that the maximum bending moment in the middle row of the pile group is higher than the front and rear rows for all cases considered in the study. The pile displacement and bending moments in the pile group reduce with an increase in soil modulus, pile spacing and pile diameter. As pile slenderness is increased, it causes an increase in displacement of the pile and a decrease in the maximum bending moment. The modified Cam clay model predicts greater displacements as compared to the Mohr-Coulomb model highlighting the impact of p0 on the yield surface. From the results obtained, the ultimate loads are predicted at a displacement of 5 mm, 10 % of diameter and 20% of diameter.

Nonlinear deflection characteristics of damaged composite structure theoretical prediction and experimental verification

This research derived a nonlinear computational model for the prediction of deflection responses of the damaged layered composite structures. The model validity was verified using in-house experimentation. The composite panel model has been derived mathematically using the higher-order polynomial kinematics without non-zero shear stress conditions and Green’s strain for the geometrical nonlinearity. The nonlinear deflection responses of combined damage curved shell structures are evaluated numerically utilizing the nonlinear isoparametric finite element technique in conjunction with the direct iterative method. By setting the necessary convergence criteria, the model accuracy has been explored by comparing the deflections of the published literature (the maximum deviation observed is 6.97%). Additionally, the experimentation is done on damaged laminated shell structures, and results are compared to show the proposed model’s efficacy (maximum deviation observed is 2.17%). Finally, a number of numerical examples are solved to assess the effects of the material, geometry and damage-related parameters on the nonlinear deflections of the shell structure.

Dynamic Analysis of Mindlin Plate Subjected to a Moving Mass-Spring-Damper System Using the Moving Element Method

The moving element method (MEM) is applied in this paper to study the dynamic behavior of the Mindlin plate subjected to a moving mass-spring-damper system. Reissner- Mindlin plate theory is utilized to study the plate structure. The plate is modeled by the Isoparametric quadrilateral nine-node element. The governing equation of motion is built into a coordinate system which moves with the moving mass-spring-damper system and is based on the principle of virtual work. The MEM is proposed to solve the governing equation of motion of the plate resting on a Pasternak foundation under the moving mass-spring-damper system including roughness of plate surface which is a dynamic source. Numerical results are investigated and verified by comparing with the published results. The effects of different parameters, such as spring stiffness, damping coefficient and velocity of the moving load system, plate thickness, foundation parameters and roughness of the plate surface on the dynamic response of the Mindlin plate are investigated.

Application of the finite element moment scheme to the investigation of thin elastic shells of inhomogeneous structure

The work is devoted to the problem of developing a universal method for studying the deformation, buckling, postbuckling behavior and vibrations of thin and medium-thickness shells of complex shape and structure under the action of mechanical and thermal loads. A wide class of shells is considered: of constant and smooth-variable thickness, with ribs and cover plates, channels and cavities, holes, sharp bends of the mid-surface, and with a multilayer structure of the material. The method is based on the positions of the 3D geometrically nonlinear theory of thermoelasticity without the use of simplifying hypotheses of the theory of shells. The development of the computational model is based on the use of a universal isoparametric spatial finite element with multilinear shape functions, which is the same for all sections of the shell with stepwise variable thickness. The governing equations are constructed in accordance with the requirements of the finite element moment scheme. The mid-surface of the shell’s casing is taken as a single reference surface. All matrices of governing equations for a universal spatial multilayer finite element are obtained in explicit form by analytical integration. This speeds up the execution of calculations in the algorithmic implementation of the method. Such an approach, based on a unified methodology, makes it possible to study the behavior of multilayer shells with different geometric features in terms of thickness under complex thermo-mechanical loading.

Vibration response of functionally graded material sandwich plates with elliptical cutouts and geometric imperfections under the mixed boundary conditions

Abstract The present article investigates the effect of elliptical cutouts and geometric imperfections on the vibrational response of functionally graded material (FGM) sandwich plates. Generalised governing equations for the sandwich FGM (SFGM) plate are derived based on non-polynomial higher-order shear deformation theory. Geometric discontinuities have been incorporated as elliptical cutouts in the plates, and the various geometric imperfections are modelled using the generic function. The mathematical modelling has been carried out using the C0 continuity isoparametric finite element formulation by considering four-noded elements with seven degrees of freedoms per node. Convergence and validation studies have been performed to demonstrate the efficiency and accuracy of the present methodology. The influence of volume fraction index, geometric imperfections, and elliptical cutouts on the vibrational frequency of SFGM plates have been analysed under the mixed boundary conditions.

A new finite element for the analysis of functionally graded shells

This paper presents the development of a finite element model for functionally graded shells. The formulation solves the equations of the Stress Approach Model for Functionally Graded shells (SAM-FG) developed and validated by Rubio Rascón et al. (2020). SAM-FG involves only 5 generalized displacements and its equations differ from those of classical first order shear deformation theories mainly in the expressions of the generalized stiffnesses and the approximation of stresses. A 10-node isoparametric triangular element using cubic shape functions and 5 degrees of freedom per node is proposed. The equations of the numerical method were implemented in a program called FESAM-FG. The finite element code enables to calculate and visualize generalized fields on the middle surface of the shell and stress components at a given position across the thickness of the structure. To test the accuracy and convergence of FESAM-FG results, the numerical tool is applied to the structural analysis of four families of functionally graded shells with various distributions of material properties and thickness-to-radius ratios. Results are compared to those obtained by solid finite elements, a higher order shear deformation theory developed by Viola et al. (2014) and two shell models used for laminated composite structures: an equivalent single layer model and a layerwise model, both using 30 fictitious layers.

Error estimates for a linear folding model

Abstract An interior penalty discontinuous Galerkin method is devised to approximate minimizers of a linear folding model by discontinuous isoparametric finite element functions that account for an approximation of a folding arc. The numerical analysis of the discrete model includes an a priori error estimate in case of an accurate representation of the folding curve by the isoparametric mesh. Additional estimates show that geometric consistency errors may be controlled separately if the folding arc is approximated by piecewise polynomial curves. Various numerical experiments are carried out to validate the a priori error estimate for the folding model.

Modal analysis of power law functionally graded material plates with rectangular cutouts

This article aims to analyze rectangular functionally graded material plates with rectangular cutouts of diverse sizes, numbers, and positions for free vibration using Mindlin’s first-order shear deformation theory (FSDT). Isoparametric plate elements of nine nodes and five degrees of freedom at each node were used for the present finite element formulation. The Poisson ratio is assumed to be constant throughout the plate. At first, results piled up from the present finite element formulation were compared with existing results collected from various previous journals for different isotropic plates and functionally graded material plates. After verification of finite element formulation, some new results (nondimensional fundamental frequency) were obtained considering various shapes, sizes, numbers, positions of cutouts, thickness ratio, aspect ratio, FGM power law index, and different edge conditions. Dynamic analysis of FGM plate with interior support along rectangular cutout edges is an integral part of this study.

Spatial wind speed interpolation using Isoparametric shape functions for structural loading

This study developed a spatial interpolation technique using Isoparametric finite element shape functions to derive wind speed records for all unsampled Kansas counties from actual data recorded at 17 city locations within and around Kansas. A computational method using the Kaimal spectrum is presented for generating artificial time histories of wind speeds. This is done to extract wind-cycle distribution using the Rainflow counting technique, which can be used as input for fatigue analysis procedures of highway sign structures. Comprehensive validation, verification and benchmarking has been performed in this paper at all spatial and time span levels to guarantee the accuracy and efficiency of the algorithm developed. The comparison of the validation measures strongly suggests a very accurate interpolation scheme that can be confidently used in further fatigue computations and analysis. A user-friendly software was designed using C# to extract wind-speed cycles for all Kansas counties related to different time spans. This software is expected to facilitate fatigue-life prediction because it generates a full range of wind-loading.

On the time-dependent mechanics of membranes via the nonlinear finite element method

In this work, the problem of finite generalized and viscoelastic deformation of thin membranes with different geometries, made of incompressible hyperelastic materials, is formulated. The multiplicative decomposition of the deformation gradient tensor into elastic and viscous parts, and making use of dissipation inequality, nonlinear evolution equations for the internal variables of the models are obtained. The mechanical behavior of the dampers is assumed to be linearly viscous. Therefore, the Cauchy-like stress in the dampers is similar to that in Newtonian fluids and includes terms for the hydrostatic pressure and viscosity. The implicit and second-order accurate trapezoidal method is employed for the time integration of the evolution equations. Due to the highly nonlinear governing differential equations including the effects of geometric nonlinearity and viscoelasticity, a nonlinear finite element formulation based on isoparametric elements is developed. The accuracy and performance of the developed formulation and time-dependent solutions are verified by studying several numerical examples. The obtained results are compared with theoretical and experimental data available in the literature. The proposed formulation can appropriately predict the experimental results of viscoelastic membranes for both in-plane and out-of-plane deformations.

The viscoplastic constitutive model of TC4 alloy under high temperature

TC4 alloy is widely used in the high temperature environment. This paper proposed an ununified viscoplastic constitutive model according to the uniaxial creep experiment and high temperature tensile experiment of TC4 alloy. The inelastic strain rate was decomposed into two parts: creep strain rate and plastic strain rate. The creep strain was calculated by multiaxial ductility exhaustion creep damage law, and the plastic strain was get from the yield function containing the damage and the kinematic hardening effect. The implicit stress integration algorithm and the consistent tangent modulus of this model were derived. The material constants of the proposed model were determined by using genetic algorithm. The proposed constitutive model was compiled in a UMAT subroutine of finite element software Abaqus and 3D eight nodes isoparametric brick element was employed to simulate the creep and the tensile behavior of TC4 alloy. Form the result, the proposed constitutive model can predict the three stages of creep curves and the softening stage of tensile curves accurately.

Development of analytical and FEM solutions for static and dynamic analysis of smart piezoelectric laminated composite plates on elastic foundation

This paper proposes new analytical and finite element solutions for studying the effects of elastic foundations on the uncontrolled and controlled static and vibration responses of smart multi-layered laminated composite plates with integrated piezoelectric layers, acting as actuators and sensors. A non-polynomial higher-order plate theory with zigzag kinematics involving a trigonometric function and a local segmented zigzag function is adopted for the first time for modeling the deformation of a smart piezoelectric laminated composite plate supported on an elastic foundation. This model has only five independent primary variables like that of the first-order shear deformation theory, yet it considers the realistic parabolic behavior of the transverse shear stresses across the thickness of the laminated composites plates, and also maintains the continuity conditions of transverse shear stresses at the interfaces of the laminated plates. A two-parameter foundation model, namely Pasternak’s foundation, is used to model the deformation and shear interactions of the elastic foundation. The governing set of equations is derived by implementing Hamilton’s principle and variational calculus. Two different solution methods, namely, a generalized closed-form analytical solution of Navier-type, and a C0 isoparametric finite element (FE) formulation, are developed for solving the governing set of equations. The solutions in the time domain are obtained with Newmark’s average acceleration method. Comprehensive parametric studies are presented to investigate the influence of elastic foundation parameters, piezoelectric layers, loading, and boundary conditions on the static and dynamic responses of the smart composite plates with piezoelectric layers. The effects of the elastic foundations on the vibration control of the smart composite plates are also presented by coupling the piezoelectric actuator and sensor with a feedback controller. Several benchmark results are presented to show the influence of the various material and geometrical parameters on the controlled and uncontrolled responses of the smart plates, and also the significant effect of the elastic foundations on the static and dynamic responses of the smart structures. The results obtained are in very good agreement with the available literature, and it can be concluded that the proposed analytical solution and FE formulation can be efficiently used to model the static and dynamic electro-elastic behavior of smart laminated plates supported on elastic foundations.

Comparative Study of Dispersion Curves for LAMB Waves Using Analytical Solutions and Semi-Analytical Methods

Lamb wave dispersion curves are useful for optimizing the inspection scanning distance that can be covered with good sensitivity in many current applications. However, one of the main problems concerning this calculation lies in selecting a numerical method that is computationally accurate and efficient. In this paper, Lamb waves dispersion curves are generated by the Scaled Boundary Finite Element Method, and by the Rayleigh–Lamb equation. For the semi-analytical case, waveguide cross-section discretization was performed using isoparametric elements and high-order spectral elements. The semi-analytical formulations lead to an eigenvalue problem that can be solved efficiently by calculating the couples of wavenumbers and frequencies that guarantee the wave mode propagation, the basis for generating the dispersion curves. These are compared with those obtained from the analytical solution for the symmetric and antisymmetric modes; in both cases, homogeneous plates of constant thickness are considered. The numerical results show good agreement when using a low number of isoparametric elements, or a single spectral element with shape functions of the order of six for computing the dispersion curves and wave structure. The calculation is given with low computational effort, and the relative variation with respect to the analytical reference values is less than 2%.

A computationally efficient isoparametric tangled finite element method for handling inverted quadrilateral and hexahedral elements

The finite element method (FEM) requires elements to be tangle-free i.e. the Jacobian must be positive throughout the element. In particular, quadrilateral and hexahedral elements are required to be convex. However, generating high quality tangle-free meshes, especially 3D hexahedral meshes, remains an open challenge. Recently, the tangled finite element method (TFEM) was proposed to handle concave (tangled) quadrilateral elements. However, even in 2D, it was found to be computationally expensive and programmatically complex.Here, we present a computationally efficient isoparametric-TFEM (i-TFEM) framework for inverted 2D quadrilateral and 3D hexahedral elements. i-TFEM employs the properties of isoparametric elements to make the formulation computationally much more efficient. In i-TFEM, the constraint on full invertibility (convexity) is replaced by partial invertibility by modifying the elemental stiffness matrices of the concave elements, and by incorporating certain piecewise-compatibility conditions. The proposed i-TFEM is simple, efficient, and provides accurate solutions with optimal convergence rate even in the presence of severely inverted elements. Moreover, i-TFEM requires minimal changes to the standard FEM framework and reduces to standard FEM for meshes without tangled elements. The accuracy and efficiency of i-TFEM are demonstrated by solving elastostatics problems on several 2D and 3D tangled meshes.

An analysis of the isoparametric bilinear finite volume element method by applying the Simpson rule to quadrilateral meshes

<abstract><p>In this work, we construct and study a special isoparametric bilinear finite volume element scheme for solving anisotropic diffusion problems on general convex quadrilateral meshes. The new scheme is obtained by employing the Simpson rule to approximate the line integrals in the classical isoparametric bilinear finite volume element method. By using the cell analysis approach, we suggest a sufficient condition to ensure the coercivity of the new scheme. The sufficient condition has an analytic expression, which only involves the anisotropic diffusion tensor and the geometry of quadrilateral mesh. This yields that for any diffusion tensor and quadrilateral mesh, we can directly judge whether this sufficient condition is satisfied. Specifically, this condition covers the traditional $ h^{1+\gamma} $-parallelogram and some trapezoidal meshes with any full anisotropic diffusion tensor. An optimal $ H^1 $ error estimate of the proposed scheme is also obtained for a quasi-parallelogram mesh. The theoretical results are verified by some numerical experiments.</p></abstract>

Finite element analysis of thermoelastic free vibration behaviour of hardcore higher-order doubly curved sandwich shell panel

Free vibration analysis is carried through for functionally graded material (FGM) sandwich structure under uniform thermal loading and the material property variation according to the power-law distribution. A self-prepared computer code in MATLAB numeric computing environment based on finite element scheme using higher-order kinematics and replicating quadratic function is equipped for the computation of responses for symmetric as well as unsymmetric doubly curved sandwich structure under diverse support conditions. For numerical approximation while deriving the system of equations, Hamilton's principle is utilised to evaluate the thermo-elastic natural frequencies and the critical buckling temperature. An isoparametric Lagrangian element with zero order Hermitian interpolation function is utilised for model discretisation. After establishing the convergence and validity, the present higher-order model is further hold out for solving diverse numerical illustrations and functional inferences those will assist in designing the imminent graded structures serving under intense thermal loading in high-performance engineering applications.

Acoustic radiation efficiencies of laminated composite structure under mechanical excitation and hygrothermal loading: Numerical analysis

In this research, the sound radiation efficiencies of multi-layer composite flat as well as curved structure under harmonic mechanical excitation in hygrothermal environment is numerically investigated using a coupled micromechanical finite element and boundary element scheme. The higher-order mid-plane kinematics including the stretching effect is equipped for the structural modeling and the Hamilton’s principle is utilized to evaluate the hygro-thermo-elastic natural frequencies alongside corresponding modes of the structures lying on an infinite rigid-baffle. An isoparametric Lagrangian element with zero order Hermitian interpolation function (ten degrees of freedom per node) is utilized for transforming the continuous structural equations into discrete counterparts. The effective composite material properties due to combined hygrothermal loading are considered via a micromechanical model. Then, an indirect boundary element method is employed to attain the structure-surrounding coupling and the sound radiation efficiencies are computed by solving the Helmholtz wave equation. The effectiveness of the model is established by converging it with the similar results available in the open literature for vibrational as well as sound radiation efficiencies and the results reveal the confirmation of the model accuracy. Upon solving diverse numerical illustrations, the sound radiation efficiency found to decrease with intensity of hygrothermal loads and the cross-ply spherical panels with less volume fraction of fiber depicted the least RE values over the entire frequency range considered.