Isoparametric Finite Element Research Articles

Time-dependent interaction between superstructure, raft and layered cross-anisotropic viscoelastic saturated soils

By introducing the elastic-viscoelastic correspondence principle and the integral transform technique, the extended precise integral solution to Biot's consolidation equation is derived for cross-anisotropic viscoelastic saturated soils based on the Merchant model. A displacement-time solution for a slowly changing load is obtained based on the above solution in the Laplace transformed domain, which is taken as the kernel function for the boundary element method (BEM). The Mindlin plate model is established by 8-node isoparametric finite element method (FEM), and the substructure condensation technique is employed to couple the stiffness matrices of the superstructure and the plate. With the BEM-FEM coupling method, a semi-analytical and semi-numerical solution is proposed for the interaction between layered cross-anisotropic viscoelastic saturated soils and raft foundation considering the stiffness contributions of the superstructure. Numerical examples are performed to study the influences of viscoelastic parameters, cross-anisotropic parameters, raft thickness and superstructure stiffness on the time-dependent behavior of the settlement and bending moment for the raft.

Large deflection response-based geometrical nonlinearity of nanocomposite structures reinforced with carbon nanotubes

This paper deals with the nonlinear large deflection analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates and panels using a finite element method. Based on the first-order shear deformation theory (FSDT), the proposed model takes into account the transverse shear deformations and incorporates the geometrical nonlinearity type. A C0 isoparametric finite shell element is developed for the present nonlinear model with the description of large displacements and finite rotations. By adopting the extended rule of mixture, the effective material properties of FG-CNTRCs are approximated with the introduction of some efficiency parameters. Four carbon nanotube (CNT) distributions, labeled uniformly distributed (UD)-CNT, FG-V-CNT, FG-O-CNT, and FG-X-CNT, are considered. The solution procedure is carried out via the Newton-Raphson incremental technique. Various numerical applications in both isotropic and CNTRC composite cases are performed to trace the potential of the present model. The effects of the CNT distributions, their volume fractions, and the geometrical characteristics on the nonlinear deflection responses of FG-CNTRC structures are highlighted via a detailed parametric study.

Study of deformation due to thermal shock in porous thermoelastic material with reference temperature dependent properties

The present study is focused on the numerical investigation in time domain of the deformation in thermoelastic material with one relaxation time (Lord and Shulman theory) with voids under dependence of modulus of elasticity and thermal conductivity on reference temperature. For dependency on reference temperature, we have taken two cases: (i) linear case and (ii) quadratic case. For numerical investigation, the finite element method with eight-/nine-noded isoparametric finite elements has been used. The results demonstrate that finite element method can reliably predict the deformation of the medium. Finite element method has an advantage over Laplace/Fourier transformation as the equations are solved directly in the time domain, but in Laplace/Fourier transformation, numerical inversion is quite complicated with possible errors. The comparison of these two cases has been discussed in result and discussion section for the temperature distribution, the displacement component, volume fraction field and the stresses. A comparison of Newmark method and Galerkin’s method for time domain has been given though three-dimensional graphs. An appreciable effect of thermal relaxation times t 0 is observed from the various results.

Experimental Validation of Role of Cut-Out Parameters on Modal Responses of Laminated Composite — A Coupled FE Approach

The free vibration frequency responses of the laminated composite structure with a cut-out of variable shapes (square/circular/elliptical), position (center/eccentric) and orientation (parallel/inclined) are investigated for the first time in this research including geometrical shapes. The eigenvalues are obtained computationally for the cut-out borne structure via a linear isoparametric finite element model of the composite structure in association with cubic-order of displacement kinematics. Also, a coupled code is prepared in MATLAB environment by joining the higher-order formulation and the simulation model (ABAQUS) to achieve the generic form to investigate the influential cut-out parameter (shape, size and position) on their eigenvalues. Further, a series of experimentations are carried out using the cut-out borne composite panel and compared with the computational frequency, including the experimental properties. Finally, the key behavior is surveyed through different kinds of numerical examples for various design constraint parameters including the cut-out relevant factors (shape, position and orientation) to show the subsequent inclusiveness of the proposed model.

Analysis of non-skew and skew laminated composite and sandwich plates under hygro-thermo-mechanical conditions including transverse stress variations

A new fourth-order in-plane displacement field based refined higher-order zigzag theory is proposed for analysis of laminated sandwich plate (for both skew and non-skew shaped) subjected to hygro-thermal conditions. In order to predict behavior of thick laminated sandwich plates, third-order transverse displacement field is taken. Zig-zag effects are introduced using linear unit Heaviside step function. The theory satisfies zeros transverse normal and shear stress condition at the bottom and top surface of the plate along with continuity condition at interface. The proposed model is free from any kind of C-1 or penalty function requirements. Nine-noded isoparametric finite element having twelve degrees of freedom per node is used during analysis. Since, the present theory incorporates transverse displacement field along with continuity conditions, is able to predict the behavior of thick sandwich plates more efficiently. In literature no results are present for skew laminated composite and sandwich plates therefore, present results for skew plates are entirely new and will serve as benchmark for future studies.

Modal Analysis of the Axial Compressor Blade: Advanced Time-Dependent Electronic Interferometry and Finite Element Method

Abstract Natural frequencies and vibration modes of axial compressor blades are investigated. A refined mathematical model based on the usage of an eight-nodal curvilinear isoparametric finite element was applied. The verification of the model is carried out by finding the frequencies and vibration modes of a smooth cylindrical shell and comparing them with experimental data. A high-precision experimental setup based on an advanced method of time-dependent electronic interferometry was developed for this aim. Thus, the objective of the study is to verify the adequacy of the refined mathematical model by means of the advanced time-dependent electronic interferometry experimental method. The divergence of the results of frequency measurements between numerical calculations and experimental data does not exceed 5 % that indicates the adequacy and high reliability of the developed mathematical model. The developed mathematical model and experimental setup can be used later in the study of blades with more complex geometric and strength characteristics or in cases when the real boundary conditions or mechanical characteristics of material are uncertain.

Numerical eigenfrequency and experimental verification of variable cutout (square/rectangular) borne layered glass/epoxy flat/curved panel structure

The variation of modal responses due to the change in shapes (square/rectangular) and sizes of the cutout impregnated composite panels (Glass/Epoxy) are predicted numerically using higher-order finite element formulation. The degree of accuracy of the proposed model is verified by comparing the modal values with the experimentally recorded data for the validation purpose, including the experimental composite properties. The isoparametric finite element formulation is adopted to model the layered panel configuration in association with the higher-order polynomial kinematic model. The modeling of the cutout is achieved through the equations of one-quarter of the plate and extended further to join for all four quarters. The model generosity is checked through analyzing the frequencies for variable geometrical shapes (plate, cylindrical, spherical, hyperboloid and elliptical), material properties and boundary conditions. Also, the results are verified for the variable cutout shapes (square and rectangular) and the sizes by comparing the results with published numerical modal values. This is the maiden experimental verification to show the inevitability of higher-order numerical model for the computation of eigenvalues of the laminated composite components with cutout. Lastly, the conclusions are deliberating the effect either individual or the combined design-dependent structural parameters (aspect ratios, thickness ratios, curvature ratios and different end boundary constraints including the sizes of cutout) by solving different examples.

Numerical investigation and experimental verification of thermal frequency of carbon nanotube-reinforced sandwich structure

The modal responses of multi-walled carbon nanotube-reinforced composite sandwich structural plate are computed under the elevated temperature environment using a higher-order polynomial kinematic model and the isoparametric finite element steps. The proposed model accuracy has been verified with experimental modal values under the influence of elevated temperature and ambient conditions. To perform the modal experiment the nanotube-reinforced composite sandwich panel filled with the epoxy core is fabricated. Further, the experimental elastic properties of epoxy, nanotube/epoxy composite and the sandwich are obtained individually for the current computational purpose. A tailor-made finite element computer code (MATLAB environment) is prepared using the multiscale mathematical formulation for the evaluation of thermal frequencies of the nanotube sandwich panel. The impact type vibration analyser has been utilized for the current testing purpose with the help of three components i.e. hardware (compact data acquisition system, cDAQ-9178), software (LAB-VIEW) and a microcontroller controlled thermal chamber (to maintain temperature profile). Finally, wide varieties of numerical examples are solved using the proposed computational model for the different design-related parameters. The intrinsic behaviour of each parameter on the epoxy-filled nanotube sandwich construction including the elevated temperature loading is discussed in details.

Features of the software implementation of the numerical solution of stationary heat equation taking into account the effects of nonlocal finite element method

The features of the numerical solution of the two-dimensional equation of stationary heat conduction are considered taking into account nonlocal effects. The finite element method using isoparametric finite elements was chosen as the numerical solution scheme. A method for approximating the zone of nonlocal influence was proposed, and an estimate of the time for calculating the coefficients of the thermal conductivity matrix was also constructed. The influence of the main parameters of the model on the final results is studied, and a polynomial family of functions of non-local influence is also studied. The results are compared with the results obtained with the classical approach.

Arbitrary Lagrangian–Eulerian finite element method for curved and deforming surfaces: I. General theory and application to fluid interfaces

An arbitrary Lagrangian–Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity need not depend on the in-plane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf–sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to non-axisymmetric perturbations, and unstable with respect to long-wavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors.

Finite element discretization of semilinear acoustic wave equations with kinetic boundary conditions

We consider isoparametric finite element discretizations of semilinear acoustic wave equations with kinetic boundary conditions and derive a corresponding error bound as our main result. The difficulty is that such problems are stated on domains with curved boundaries and this renders the discretizations nonconforming. Our approach is to provide a unified error analysis for nonconforming space discretizations for semilinear wave equations. In particular, we introduce a general, abstract framework for nonconforming space discretizations in which we derive a-priori error bounds in terms of interpolation, data, and conformity errors. The theory applies to a large class of problems and discretizations that fit into the abstract framework. The error bound for wave equations with kinetic boundary conditions is obtained from the general theory by inserting known interpolation and geometric error bounds into the abstract error result of the unified error analysis.

КОНЕЧНО-ЭЛЕМЕНТНАЯ МЕТОДИКА ЧИСЛЕННОГО МОДЕЛИРОВАНИЯ УПРУГОПЛАСТИЧЕСКОГО ДЕФОРМИРОВАНИЯ ДРЕВЕСИНЫ ПРИ УДАРНОМ НАГРУЖЕНИИ

The 3D problem of wood deformation under shock loading is considered. The governing system of equations is formulated in Lagrange variables. A defining system of equations in a three-dimensional formulation is presented. The equation of motion is derived from the balance of the virtual powers of work. Wood is modeled as a unidirectionally reinforced material with a description of the descending branch of the deformation diagram. Deformations and stresses are determined in a local basis, the position of which in space is related to the direction of the wood grain. Wood material is represented as a combination of reinforcing fibers and a matrix, the elastoplastic deformation of which is described by the relations of the theory of flow with combined kinematic and isotropic strengthening. The deformation characteristics of the matrix and fibers are determined on the basis of a computational and experimental study of the mechanical properties of wood along and across the fibers. In numerical simulation, the moment scheme of the finite element method and an explicit time integration scheme of the “cross” type are used. Discretization of the computational domain is based on an eight-node isoparametric finite element adapted to the specifics of the problem under consideration. Software realization of the developed mathematical model and numerical methodology is implemented within the computing complex “Dynamics-3”. Computer simulation of compression of an experimental specimen of spruce along and across the fibers has been performed. The reliability of the calculation results is confirmed by good agreement with the experimental data.

Effect of nano glass cenosphere filler on hybrid composite eigenfrequency responses - An FEM approach and experimental verification

The effect of an increasing percentage of nanofiller (glass cenosphere) with Glass/Epoxy hybrid composite curved panels modeled mathematically using the multiscale concept and subsequent numerical eigenvalues of different geometrical configurations (cylindrical, spherical, elliptical, hyperboloid and flat) predicted in this research article. The numerical model of Glass/Epoxy/Cenosphere is derived using the higher-order polynomial type of kinematic theory in association with isoparametric finite element technique. The multiscale mathematical model utilized for the customized computer code for the evaluation of the frequency data. The numerical model validation and consistency verified with experimental frequency data and convergence test including the experimental elastic properties. The experimental frequencies of the multiscale nano filler-reinforced composite are recorded through the impact hammer frequency test rig including CDAQ-9178 (National Instruments) and LABVIEW virtual programming. Finally, the nano cenosphere filler percentage and different design associated geometrical parameters on the natural frequency data of hybrid composite structural configurations are illustrated through a series of numerical examples.

Optimal deflection and stacking sequence prediction of curved composite structure using hybrid (FEM and soft computing) technique

The bending deflections and the corresponding optimal fiber angle sequences of the subsequent layers have been predicted in this article using a hybrid technique. The structural static responses are computed numerically via the isoparametric finite element steps in association with Reddy’s higher order mid-plane theory. The final stacking sequences of individual layers are further predicted through two types of soft computing techniques (particle swarm optimization, PSO; teaching–learning-based optimization, TLBO). The responses (deflection and optimal angle) are obtained via a customized computer code (MATLAB) using the current mathematical model in association with two different optimization algorithms. The accuracy of the currently derived higher order hybrid model is established by conducting a few numerical experimentations. The study indicates the superiority of TLBO technique over PSO for any particular problem when compared to the minimum deflection constraint whereas not many deviations for stacking sequences. Finally, the influences of the different structural parameter are explored by solving a variety of numerical examples and the corresponding inferences provided in detail.

Development of a 2D Isoparametric Finite-Element Model Based on Reddy’s Third-Order Theory for the Bending Behavior Analysis of Composite Laminated Plates

The objective of this work is to propose a simple efficient finite element, based on Reddy’s third-order shear deformation theory, for analyzing the bending behavior of isotropic and composite laminated plates. It is a 2D isoparametric quadrilateral C0 four-node element with seven degrees of freedom at each node, three translation components, two rotations, and two higher-order rotational degrees. A selective numerical integration technique is introduced for the numerical formulation in order to improve results and to alleviate the locking problem. The formulation is able to take into account the parabolic distribution of transverse shear across the plate thickness in the bending behavior of isotropic and laminated composite plates, both thin and thick, without a need for correction factors. The performance and reliability of the formulation proposed are confirmed by comparing the author’s results with those obtained using the 3D elasticity theory, analytical solutions, and other advanced finite-element models.

АНАЛИЗ ВЛИЯНИЯ КОНСЕРВАТИВНОГО СГЛАЖИВАНИЯ НА ТОЧНОСТЬ ЧИСЛЕННОГО МОДЕЛИРОВАНИЯ ДИНАМИЧЕСКОГО УПРУГОПЛАСТИЧЕСКОГО ВЫПУЧИВАНИЯ СФЕРИЧЕСКИХ ОБОЛОЧЕК

Large changes of a lead spherical shell enclosed in an aluminum spacesuit under the action of an overload pulse are considered. The defining system of equations is formulated in Lagrange variables in a two-dimensional (axisymmetric) formulation. Strain and stress rates are determined in the local coordinate system. Kinematic relations are recorded in the metric of the current state. The relations of the flow theory with isotropic hardening are used as state equations. The contact interaction of the shell and the spacesuit is modeled by non-penetration conditions taking into account friction. The numerical solution of the problem under given boundary and initial conditions is based on the finite element method moment scheme and the explicit time integration “cross” type scheme. 4-node isoparametric finite elements with bilinear form functions are used to discretize the defining system of equations for spatial variables. To suppress the numerical solution high-frequency oscillations, the procedure of nodal displacements rates conservative smoothing is used. As shown by the results of numerical research spherical shell in the process of intensive dynamic loading undergoes large deformation and rotation angles as a rigid whole. The calculation results reliability is confirmed by a good correspondence to the experimental data. The influence of conservative smoothing procedure and moment components of deformations and stresses on the solution accuracy is analyzed. It is shown that without conservative smoothing procedure using, the shape of the spherical shell buckling obtained in the calculation does not correspond to the experimental data. Neglect of the moment components of strains and stresses leads to the development of instability of the “hourglass” type.

Нестационарные процессы в подкрепленном тонкостенной оболочкой вязкоупругом цилиндре при импульсном нагружении

Solid propellant rocket motor is considered as hollow viscoelastic cylinder inserted in multilayered elastic shell-like case. The material of propellant is considered to be compressible. An estimation of maximum unsteady stresses on cylinder-shell boundary and shell under growing pressure on interior or external cylindrical surface were calculated by FEM. Four corner isoparametric finite element is utilized. Numark method to integrate by time the dynamic equations is used. The problem of linear viscoelasticity have been employing of the Schapery method. `In the case of internal pressure, the possibility of tensile radial stresses on the contact surface of the propellant-shell during the transition process has been established. The dependence of the maximum contact stresses as well as circumferential stresses in the shell on the shell thickness is established. In the case of external pressure pulse, the presence of significant tensile radial stresses on the propellant-shell interface is shown. Insignificant tensile circumferential stresses in the transient wave process are possible in the shell.

Analysis of functionally graded carbon nanotube-reinforced laminates

Abstract In present work, analysis of functionally graded carbon nanotube-reinforced composite laminates is investigated. Different types of functionally graded aligned reinforcement distributions of carbon nanotubes along the thickness of laminates are considered. A micro mechanical model using some effective parameters estimates material properties. The governing equations are developed based on third order shear deformation theory. A nine-node isoparametric finite element with seven unknowns at each node is used. The obtained results in terms of deflection and stresses are compared with available results in literature. The effects of carbon nanotube volume fraction. Length-to-thickness ratio, boundary conditions and other geometrical parameters are also examined.

МЕТОД ЧИСЕЛЬНОГО МОДЕЛЮВАННЯ ГІДРОДИНАМІЧНИХ ПРОЦЕСІВ

This article presents a numerical simulation method that is used in the study of the dynamics of continuous viscous weakly compressed fluids based on the system of equations of continuity and Navier-Stokes. The proposed method uses a complex approach using the numerical solution of the continuity equation by the finite-volume method, and for solving the Navier-Stokes equation the splitting method by physical factors. The article shows that the finite volume method, which was used to describe the flow of both compressed and uncompressed liquids, has such important advantages as the presence of good conservative properties and assumptions of discretization of complex computational domains into simpler ones than isoparametric finite element formulation of the problem or the introduction of generalized coordinates. A component is introduced into the method of splitting according to physical factors, which takes into account the artificial compressibility of the test liquid, which allows you to first calculate the intermediate velocity field, which is then corrected taking into account the pressure gradients. The difference scheme of this method allows one to calculate the flow field without using the values ​​of the vortex and pressure on a solid surface. In the framework of the proposed approach, it is not necessary to calculate the value of the vortex on a solid surface. The latter can be found from the calculated velocity field using any of the difference representations of the expression for the vortex at the boundary points. To confirm the effectiveness of the proposed method, in the FlowVision CFD program, solutions were obtained for a number of problems in external hydrodynamics, using the example of a cylindrical surface optics, which confirmed the stability of the results obtained. This method allows one to compute the flow of flat, axisymmetric, and three-dimensional bodies of a complex configuration with a flow of a viscous weakly compressed fluid, as well as internal flows in a wide range of Reynolds numbers using a single algorithm.

АНАЛИЗ ВЛИЯНИЯ КОНСЕРВАТИВНОГО СГЛАЖИВАНИЯ НА ТОЧНОСТЬ ЧИСЛЕННОГО МОДЕЛИРОВАНИЯ ДИНАМИЧЕСКОГО УПРУГОПЛАСТИЧЕСКОГО ВЫПУЧИВАНИЯ СФЕРИЧЕСКИХ ОБОЛОЧЕК

Large changes of a lead spherical shell enclosed in an aluminum spacesuit under the action of an overload pulse are considered. The defining system of equations is formulated in Lagrange variables in a two-dimensional (axisymmetric) formulation. Strain and stress rates are determined in the local coordinate system. Kinematic relations are recorded in the metric of the current state. The relations of the flow theory with isotropic hardening are used as state equations. The contact interaction of the shell and the spacesuit is modeled by non-penetration conditions taking into account friction. The numerical solution of the problem under given boundary and initial conditions is based on the finite element method moment scheme and the explicit time integration “cross” type scheme. 4-node isoparametric finite elements with bilinear form functions are used to discretize the defining system of equations for spatial variables. To suppress the numerical solution high-frequency oscillations, the procedure of nodal displacements rates conservative smoothing is used. As shown by the results of numerical research spherical shell in the process of intensive dynamic loading undergoes large deformation and rotation angles as a rigid whole. The calculation results reliability is confirmed by a good correspondence to the experimental data. The influence of conservative smoothing procedure and moment components of deformations and stresses on the solution accuracy is analyzed. It is shown that without conservative smoothing procedure using, the shape of the spherical shell buckling obtained in the calculation does not correspond to the experimental data. Neglect of the moment components of strains and stresses leads to the development of instability of the “hourglass” type.