Nine-node Element Research Articles

Analysis of nonlinear problems using the Dual Mesh Control Domain Method with arbitrary meshes

The Dual Mesh Control Domain Method (DMCDM), proposed and developed by the second author, is a recent advancement in the field of numerical methods. This method represents a blend of the Finite Element Method (FEM) and the Finite Volume Method (FVM), each having its primary domains of popularity. The DMCDM incorporates the concepts of interpolation, duality, and numerical integration from the former and the idea of satisfying the integral (global) form of the differential (or balance) equations from the latter. In this paper, we present a fresh and systematic approach to solving two-dimensional nonlinear problems, including advection–diffusion problems, with arbitrary primal meshes. We develop an approach where element-level equations are derived and assembled in a manner similar to that of the FEM, as opposed to using nodal equations, as has been the convention in all of the previous works involving the DMCDM. Additionally, we extend the previous works to include higher-order interpolation, specifically quadratic interpolation. In short, we provide formulations for four-node bilinear and nine-node biquadratic elements of quadrilateral shape, enabling the application of the DMCDM to arbitrary primal and dual meshes. Finally, we explore the Newton iteration method for linearizing the nonlinear system of equations that arises from the DMCDM discretization. We present numerical results and comparisons against available exact and numerical solutions to verify the accuracy and robustness of the DMCDM.

Grey wolf optimizer based-natural frequency optimization of laminated composite shells carrying distributed mass

In this study, a grey wolf optimizer is proposed to find the optimum stacking sequence to maximize the fundamental frequency of shells. The vibration analysis of the shells is done using first-order shear deformation theory and a nine-node isoparametric element with five degrees of freedom per node. A mass lumping scheme with rotary inertia is considered in the finite element formulation to include the rotary inertia effect. The design variable considered is the fiber angle orientation of the shells. The present approaches are compared with previously published results using layerwise optimization theory, the Ritz method, and classical lamination theory. A comparison study is also performed, comparing the GWO-based finite element model against six different metaheuristic approaches, and it demonstrates that the Grey Wolf Optimizer consistently outperforms these competing algorithms. The optimum fiber angle orientation of shells carrying distributed mass for various boundary conditions and shell types is analyzed for the first time, to the best of the author’s knowledge. This novel discovery will benefit future studies.

Comparative study of thermal effect on thermoelastic double porous solid by finite element method and finite-infinite element method

In the present study, we have made a comparative study between the well-established finite element method (FEM) and finite-infinite element method (FIEM). Nine-noded infinite elements have been developed to model the half-space. Further, we assume a thermal shock which is applied on the double porosity medium. This medium is assumed to be unbounded in terms of x and y directions. For sake of simplicity, we have converted our model equations in non-dimensional form. Then after, suitable boundary conditions have been imposed in the present study. Numerical validation of the present mathematical model has been made for copper-like material. Displacement components of solid particles, volume fractions fields of double porosity solid, temperature field and stress components have been computed for the domain . Due to limitation and a better understanding of the comparative study between these two methods, we are giving 2D-graphical representation of field variables with respect to x-space for fixed value of y=1 and 3D-graphical representation for fixed value of x = 1, respectively. From the results, we find advantage of use of infinite element method over finite element method.

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.

Higher-order models for the passive damping analysis of variable-angle-tow composite plates

In this paper, the damped free-vibration and frequency response analysis of variable-angle-tow composite plates embedding viscoelastic layers with frequency-dependent properties are performed. The governing equations are derived from the Principle of Virtual Displacements and higher-order Layer-Wise models are used for the unknown variables description in the thickness direction, furthermore a nine-node finite plate element is employed to solve them. The plate formulation presented in this paper is more efficient with respect to three-dimensional approaches, typically used for this kind of structures, and it is more accurate than existing reduced order methods thanks to the use of the mixed interpolation of tensorial components technique, employed to solve the problem of the shear and membrane locking phenomena. Different multilayered structures have been considered, laminated with unidirectional cross-ply layers or with curvilinear fibers path ones. The use of viscoelastic soft sheets permits to have a passive damping of the structural vibration. The frequency dependent properties of the viscoelastic materials are described through the use of a Kelvin-Voigt model. Numerical solutions are presented for the analysis of variable-angle-tow composites with different boundary-conditions and various lamination schemes.

Thermoelectric nonlinear vibration and buckling analysis of the smart porous core sandwich plate (SPCSP) resting on the elastic foundation

This paper analyses the vibration and buckling responses analysis of the smart porous core sandwich plate (SPCSP) resting on the elastic foundation under thermoelectric and thermomechanical loading. The plate is a constraint with conventional and unconventional boundary conditions. The thermo-mechanical properties of the plate are graded through the thickness direction with the sigmoid and exponential laws. The sandwich plate considered geometrical nonlinearity and assumed quadratic electric potential function across the thickness direction. The Hamilton principle derives the formulation with the first-order shear deformation theory (FSDT) displacement field. A nine-node quadratic element-based higher-order finite element method with seven degrees of freedom (DOFs) is used to convert the governing equation into a set of algebraic forms and solve it computationally (using MATLAB) with a modified Newton-Raphson scheme. Buckling and vibration responses are analysed under varying plate parameters like material exponent N, thickness ratio (a/h), porous exponent with different porosity dispersion, conventional and unconventional conditions, applied electric voltage V and temperature change [Formula: see text]. It is found that the presence of the Pasternak parameter gives additional stability to structures and overcomes the instability due to porosity. Also, the proposed work is not only precise but also simple in predicting the vibration and buckling response of smart sandwich porous plates under thermo-electric mechanical loading.

Application of UPV-Instrument in Health Monitoring of Indian Rail Section Using AE Technique

In present work, buckling analysis of laminated composite plates is carried out using recently proposed higher-order zigzag theory (HOZT). Third order variation of in-plane displacement field is taken across the thickness of the plate. For predicting the behavior efficiently for thick plates, quadratic variation of transverse displacement field is assumed for core layer and constant for face layers. The present theory satisfies inter-laminar transverse shear stress continuity condition at interface along with zero value at top and bottom surface of the plate. In present study, nine-noded finite element having eleven degrees of freedom per node is used. Present model is free from requirement of any kind of penalty function or C-1 condition.

Buckling Analysis of Laminated Composite Plates under Thermal Conditions

In present work, buckling analysis of laminated composite plates is carried out using recently proposed higher-order zigzag theory (HOZT). Third order variation of in-plane displacement field is taken across the thickness of the plate. For predicting the behavior efficiently for thick plates, quadratic variation of transverse displacement field is assumed for core layer and constant for face layers. The present theory satisfies inter-laminar transverse shear stress continuity condition at interface along with zero value at top and bottom surface of the plate. In present study, nine-noded finite element having eleven degrees of freedom per node is used. Present model is free from requirement of any kind of penalty function or C-1 condition.

A Triangular Shell Element Based on Higher-order Strains for the Analysis of Static and Free Vibration

This research paper proposes a new triangular cylindrical finite element for static and free vibration analysis of cylindrical structures. The formulation of the proposed element is based on deep shell theory and uses assumed strain functions instead of displacement functions. The assumed strain functions satisfy the compatibility equations. This finite element possesses only the five necessary degrees of freedom for each of the three corner nodes. The element's displacement field, which contains higher-order terms, satisfies the requirement of rigid-body displacement. The element's performance is evaluated using various numerical static and free vibration tests for cylindrical shell problems, including an analysis of the effect of shell openings on natural frequencies. The results of the developed element are evaluated in comparison with published analytical and numerical solutions. The new cylindrical element's formulation is straightforward. Compared to the degenerate nine-node shell element and other elements, the results of the present element have shown excellent accuracy and efficiency in predicting static and free vibration of curved structures. This element only requires the use of very coarse meshes to converge. In addition, the triangular shape of this element is more advantageous than the quadrilateral shape when the geometric domain of the structure is deformed or complicated. Doi: 10.28991/CEJ-2022-08-10-06 Full Text: PDF

Static Analysis of Skew Functionally Graded Plate Using Novel Shear Deformation Theory.

In this article, the static response of a functionally graded material (FGM) plate is studied via hybrid higher-order shear deformation theory which uses hyperbolic and polynomial shape functions and includes the effect of thickness stretching. The composition of the plate comprises metallic and ceramic phases. The ceramic volume fraction varies gradually along with the thickness following the power law. The mechanical properties of the FGM plate are determined by the rule of mixtures and the Mori–Tanaka homogenization scheme. The displacement fields are defined to satisfy the requirement of traction-free boundary conditions at the bottom and top surfaces of the plate surface removing the need for determination of shear correction factor. A C0 continuity FE model is developed for the present mathematical model. Nine-node isoparametric elements with eight nodal unknowns at each node are developed. The present model comparison with existing literature is completed and found to be coherent. Inhouse MATLAB code is developed for the present work. Sinusoidal and uniformly distributed loading is analyzed in the present work. The parametric study is undertaken to explore the effect of the side-to-thickness ratio, aspect ratio, thickness, and volume fraction index on stresses and transverse displacements.

Flutter characteristics of multi-layered composite shell panels under supersonic flow with curvature effect

ABSTRACT This study presents the finite element (FE) solutions to the aeroelastic flutter behaviour of multi-layered composite shell panels with flat and cylindrical geometry under supersonic flow based on third-order shear deformation mid-plane kinematics. The shell panel is discretized by a nine-noded isoparametric shell element with nine nodal degrees of freedom. The effect of the aerodynamic load is incorporated through first-order piston theory. The numerical model's accuracy and validity are justified by comparing the present solutions (free vibration frequency, coalescence frequency, and the corresponding critical aerodynamic pressure) with the existing numerical results. Also, the current natural frequencies are compared with the values from in-house experimental tests. The influence of parameters such as the flow angle, fibre orientation, material anisotropy, thickness ratio, and curvature ratio on the supersonic flutter boundaries of the shell panels subjected to simply supported and clamped conditions is investigated. The results are deliberated in detail, and it is observed that the fibre orientation and material anisotropy affect the flutter boundary differently under the simply supported and clamped conditions. Moreover, the first coalescence is greatly influenced by the flow angle instead of the curvature of the curved shell panels.

Dynamic Analysis of FGM Plate Under Moving Load Considering Effect of Temperature

The paper used the moving element method (MEM) to analyze the dynamics of the functionally graded material (FGM) plate under the moving load considering the effect of temperature. To calculate the displacement of the plate, the study applied the Mindlin plate theory. A nine-node isoparametric element, each with five degrees of freedom, is used to model the plate element. According to the MEM, the equation of motion of the FGM plate is established based on the principle of virtual work and on a coordinate system that moves along with the moving load. The temperature field is assumed to be constant in the plane and varies across the plate thickness. By solving the governing equation of temperature transfer, it is possible to obtain a temperature distribution function. Both mechanical strain and temperature-induced strain are considered to determine plate strain. Numerical results were surveyed with different parameters and compared with published results to verify the reliability of the model. It is found that temperature significantly affects the dynamic response of the FGM plate. This study shows that the displacement of the plate increases when the temperature increase.

Free vibration response of cnt-reinforced multiscale functionally graded plates using the modified shear deformation theory

ABSTRACT This article represents the free vibration analysis of multiscale functionally graded plates, reinforced with a carbon nanotube. The modified third-order shear deformation theory with a variation in transverse displacement is employed in the present work. MATLAB code is developed for C0 finite element (FE) formulation. A rectangular nine-noded element containing 117 nodal unknowns at each element was considered. The top and bottom transverse shear stress of the plate is considered zero. No shear correction factor is required in the present model. In this study, the FGM plates are assumed to be reinforced with MWCNT and SWCNT and also graded with fibre material throughout the thickness. The effective elastic properties of the multiscale composite plate are estimated by the combination of the Halpin–Tsai equation and the homogenisation scheme. The effects of various parameters such as the volume fraction of fibre material, a/h ratio, aspect ratio and weight percentage of both types of CNTs on frequency are studied.

Nonlinear dynamic analysis of sandwich shell panels with auxetic honeycomb core and curvilinear fibre reinforced facesheets

The nonlinear dynamic behaviour of sandwich shell panels having auxetic honeycomb core and facesheets composed of composite material with curvilinear fibres is investigated for the first time. A finite element study is performed considering a third-order shear deformation theory incorporating Murakami zig-zag effects for the analysis of the variable stiffness composite laminate (VSCL) auxetic honeycomb sandwich panels. A nine-noded C0 isoparametric quadratic shell element with eleven degrees of freedom per node is considered and von Kármán nonlinear strain-displacement equations are used to include the geometrical nonlinearity. The linear free vibration and nonlinear transient response of the panels are obtained using the eigenvalue solution and Newmark's time integration scheme, respectively. Most of the auxetic sandwich structures considered in the literature are made up of aluminium metal and therefore, the authors explored the viability of the VSCL based auxetic sandwich panels. A thorough numerical study including the effects of the geometrical parameters of the honeycomb unit cell, curvilinear fibre path angles and boundary conditions on the nonlinear dynamic response of auxetic sandwich panel with negative Poisson's ratio is presented.

Influence of initial geometrical imperfections on thermal stability of laminated composite plates using layerwise finite element

In this paper, the influence of initial geometrical imperfections on thermal stability of laminated composite plates using Layer wise plate model is presented. The mathematical model assumes layer wise variation of in-plane displacements, non-linear strain-displacement relations (in von Karman sense) and linear thermo mechanical material properties. The Koiter’s model for initial geometrical imperfection is adopted. Principle of virtual displacements (PVD) is used to derive Euler-Lagrange differential equations of linearized buckling problem. The weak form is discretized using nine-node Lagrangian isoparametric finite element. The original MATLAB program is coded for finite element solution. The effects of imperfection mode and amplitude, temperature distribution, side to thickness ratio b/h, aspect ratio b/a, boundary conditions and lamination scheme on critical buckling temperature are analysed. The accuracy of the numerical model is verified by comparison with the available results from the literature and new results are presented.

Geometrically nonlinear dynamic analysis of variable stiffness composite laminated and sandwich shell panels

The nonlinear free and forced vibration analyses of variable stiffness composite laminated (VSCL) and sandwich shell panels having curvilinear fibres, not yet reported in literature, are conducted based on a higher-order theory. The present structural model considers a third-order shear deformation theory incorporating Murakami zig-zag effects. The analyses are performed using C0 nine-noded isoparametric element with thirteen degrees of freedom and geometrical nonlinearity is included through von Kármán nonlinear strain–displacement relations. The free and forced vibrational responses of the VSCL are established using the eigenvalue solutions and Newmark’s time integration technique, respectively. The results obtained using the present finite element formulation are compared with the available three-dimensional exact and two-dimensional analytical and numerical solutions for establishing the validity of the model. The fundamental frequencies and nonlinear transient deflections of VSCL and sandwich spherical and cylindrical shell panels are evaluated considering the effects of curvilinear fibre path, curvature ratio, number of layers and lamination configuration.

The displacement view of a multilayered HSDT plate

This paper presents the state of displacement of a multilayered composite laminate subjected to transverse static load with varying balance, symmetric and anti-symmetric angle-ply and cross-ply staking sequences. Higher-order shear deformation theory (HSDT) is considered in the finite element formulation of nine-noded isoparametric element with seven degrees of freedom at each node. The finite element formulation is transformed into computer codes. A convergence study is carried out first to obtain the optimal mesh size for minimizing the computational time. The maximum deflection at the center of plate for both fixed and simply supported edges is verified with reported literature and a good conformity is found. An attempt has been made to observe the minimum value of maximum deflection in the laminate for attaining the maximum strength of laminate with a suitable combination of stacking sequences with a constant volume of material.

Influence of local stiffeners and cutout shapes on the vibration and stability characteristics of quasi-isotropic laminates under hygro-thermo-mechanical loadings

Abstract s The perforated stiffened panel is generally found as a sub-component of sophisticated structures. The fundamental purpose of this panel is to withstand against buckling under complicated loading and environmental conditions. Hence, an accurate knowledge of critical buckling behaviour of stiffened panels is very much essential for a reliable and lightweight structural design. In this paper, the focus is on quasi-laminated panels with different cutout shapes of various sizes and their responses to hygrothermal environments under nonlinearly varying edge loads and is compared with the locally stiffened panels. Towards this, the modelling of the panel and stiffener is done by adopting nine-noded heterosis plate elements and three noded beam elements respectively. The stiffener formulation is suitably modified in order to take the torsional effect also into consideration along with the effect of shear deformation. Initially, the plate and the stiffener elements are treated separately, and then the displacement compatibility is maintained between them by using the transformation matrix. For a given loading and geometric discontinuity, the stress distribution within the perforated panel is highly non-uniform in nature and hence a dynamic approach has been used to calculate buckling loads by adopting two sets of boundary conditions, one set for pre-buckling stress analysis and the second set for buckling analysis. Four different quasi-isotropic stacking sequences are deliberated in this work by varying different ply-orientation in each scheme. The study also addresses the effect of various parameters such as nonlinear loads, hygro-thermal loads, cutout size and shapes, position of cutout, stiffener parameters, stacking sequences, thickness of plate and boundary conditions.

A size-dependent nonlinear finite element free vibration analysis of multilayer FG-GPLRC toroidal micropanels in thermal environment

A nonlinear finite element model based on the modified strain gradient theory (MSGT) in combination with the first-order shear deformation theory (FSDT) of shells for the free vibration of multilayer functionally graded graphene platelets reinforced composite (FG-GPLRC) toroidal micropanels is developed. The nine-noded shell elements with five degrees of freedom per node are used to accurately simulate thin as well as moderately thick toroidal micropanels without shear locking phenomena. In addition to initial thermal stresses, the influences of nonlinear elastic foundation and the elastic rotational springs at the micropanel edges are considered. The formulation and corresponding computer codes are validated by performing convergence study and doing comparison studies with some existing results in the literature. Then, the impacts of amplitude ratio, geometric parameters, ratio of thickness-to-material length scale parameter (MLSP), temperature rise, coefficients of elastic foundation and rotational springs, and different GPLs distribution patterns on the nonlinear vibrational behaviors of the multilayer FG-GPLRC toroidal micropanels are investigated. The results show that the size dependence of material properties increases the impact of geometric nonlinearity on the vibrational behavior of the micropanels. In addition, it affects the nonlinear vibrational behavior more than the linear one in a widespread range of thickness-to-MLSP ratio.

A phase field model for fracture based on the strain gradient elasticity theory with hybrid formulation

In this paper, a novel phase field (PF) model for fracture is developed in the framework of strain gradient elasticity. The strain energy decomposition methods initially proposed for linear elastic fracture problems are extended to the gradient elasticity situation. The PF model is numerically implemented through Abaqus subroutine UEL (user defined element) with nine-node quatrilateral C1-continous element. The finite element implementation is validated by studying the stress field near the static crack tip. When the length-scale parameter of the gradient elasticity is zero, the stress field is consistent with the analytical solution predicted by linear elastic fracture mechanics (LEFM). For non-zero length-scale parameters, the Cauchy stress at the crack tip is found to be non-singular and the numerical results exhibit less mesh sensitivity in the crack tip region compared with the linear elastic case. After that, four different strain energy decomposition methods (Method I-IV) are compared and discussed by studying Mode I fracture behavior under tensile load. It is found that Method I can properly characterize the toughening effect of gradient elasticity. Based on Method I, the influences of the fracture length-scale parameter lc and the strain gradient length-scale parameter l on the characteristics of the smeared crack model and the load–displacement curves are systematically discussed. The specimen size effect of Mode I crack propagation is also investigated. It is found that the length-scale parameter l has a larger influence on the load–displacement curve as the specimen size decreases. Finally, crack propagation behavior in pure shear test, three point bending test and tensile test of a notched plate with hole is investigated. It is demonstrated that the proposed PF model can well predict the curvilinear crack propagation path and properly characterize the effect of gradient elasticity. Thus, the PF model developed in this paper can be applied to study complex fracture behaviors in the framework of gradient elasticity, where the effect of internal structure on the mechanical responses become significant.