P-version Of Finite Element Method Research Articles

Thermal buckling analysis of bi-directional FGM doubly curved shell panels using a TSDT p-version finite element method

Thermal buckling analysis of bi-directional FGM doubly curved shell panels using a TSDT p-version finite element method

Micromechanical analysis of fiber-reinforced ceramic matrix composites by a geometrically nonlinear hierarchical quadrature element model

The stress fields of fiber-reinforced ceramic matrix composites (CMCs) under longitudinal tension are predicted using a non-linear hierarchical quadrature element method (HQEM) in this work. The HQEM combines the differential quadrature method (DQM) with the hierarchical finite element method (HFEM), which is a typical p-version finite element method (FEM). High-order interpolation functions of the HQEM allow accurate evaluation of stress distribution. Fiber, matrix, and interfaces of the CMCs are modeled as linear elastic materials that may have large displacements. The nonlinear HQEM estimation of CMCs stress distributions are validated by comparing with analytical results obtained using classical BHE shear-lag model. Stress distribution in three characteristic stages in the damage evolution process, namely, interface perfectly bonded, interface debonding, and fiber failure are investigated. It is shown that when fiber failure happens, geometric non-linearity must be considered to avoid excessive stress concentration. Furthermore, the stress distribution within three-dimensional cylindrical CMCs cell is studied, which sheds light on the future exploration of three-dimensional CMCs analyses.

Free vibration and bending analysis of porous bi-directional FGM sandwich shell using a TSDT p-version finite element method

Free vibration and bending analysis of porous bi-directional FGM sandwich shell using a TSDT p-version finite element method

Vibrational analysis of a coupled damaged Timoshenko beam-arch mechanical system with von Kármán nonlinearities and layer discontinuity

The present study investigates vibrations in the nonlinear regime of a partially elastically connected beam with damage and either a curved or straight beam, focusing on the time domain. The mechanical system under consideration consists of two beams with varying discontinuities: one in an elastic layer and the other in a narrow cross-section area of the beam. To analyze the vibrations in this coupled system, a p-version finite element method is employed. This method allows for the examination of shear deformable beam-beam or beam-arch systems with small or large discontinuities in the elastic layer and cross-section area of the beam. The main contribution of this work lies in the dynamic analysis of a novel coupled geometrically nonlinear mechanical system. It has been discovered that in specific cases of combinations of discontinuities in the upper beam and elastic layer, a different energy transmission occurs, caused by geometrically nonlinear couplings of the nonlinearity of the elastic layer, to the lower arch, resulting in multiple cycles of oscillations executed by the system in the steady-state regime of vibrations. The analysis considers various scenarios, including the effects of the discontinuity in the viscoelastic connection layer, beam damage, and the varying curvature of the lower arch. General mode shapes are determined, and forced damped vibrations in the time domain are analyzed using the Newmark method. The findings from this dynamic analysis have broad applications in engineering and technical practice.

Structures with varying discontinuities and curvatures: A dynamic analysis approach by the p-version finite element method

The present paper investigates the geometrically nonlinear time domain dynamic analysis of a coupled Timoshenko beam-beam or beam-arch mechanical system. The coupled structure is modelled with a variable discontinuity in an elastic layer, which represents a real case from technical practice where there is no continuous distribution of the elastic layer or the stiffness of the layer is changed by other influences at different locations. A modified p-version finite element method is developed for the vibrations of a shear deformable coupled beam system with a discontinuity in an elastic layer. The main contribution of this work is the discovery of coupled effects and phenomena in the simultaneous vibration analysis of varying discontinuity and varying curvature of the newly modelled coupled mechanical system. New general mode shapes are presented, and the forced vibrations in the time domain are analyzed using the Newmark method. The study compares the results for various locations and types (size) of discontinuity in an elastic coupling layer and various curvatures of the lower supported beam. It explains the cases in which the influence of curvature takes precedence over discontinuity and vice versa in achieving a steady-state regime of vibrations. The analysis results are valuable and have broader applications in the field of solids and structures.

Maximisation of Bending and Membrane Frequencies of Vibration of Variable Stiffness Composite Laminated Plates by a Genetic Algorithm

PurposeIn comparison to traditional, constant stiffness laminates, variable stiffness composite laminates (VSCL) with curvilinear fibres represent an extra analysis effort. It is the purpose of this work to present and test a relatively simple optimisation procedure, in order to find the maximum fundamental frequency of vibration in bending and in in-plane vibrations. It is also intended to explain why certain fibre paths lead to higher frequencies.MethodsThe optimisation is performed using a genetic algorithm (GA), which is described in detail. The bending and the in-plane plate models are based on the p-version Finite Element Method. Each model requires a small number of degrees of freedom, an important feature because applying the GA involves the solution of a large number of eigenvalue problems. In order to support the physical interpretation of the optimal designs, mode shapes and stress fields corresponding to some optimal solutions are illustrated.ResultsSingle- and multi-layer plates with different boundary conditions and fibre path types are studied. Fibre paths that lead to maximum fundamental frequencies are found and justified. The consequences that maximising the first frequency has on the higher-order modes of vibration are studied.ConclusionThe proposed optimisation and modelling methods are effective. Curvilinear fibres with the characteristics considered led to the maximum first natural frequency of vibration in a few cases, but not all. Particularly in in-plane vibrations, curvilinear fibres can provide major gains in comparison to straight fibres. The increase in the vibration frequency is accompanied by, overall, larger stresses.

Variable-kinematic finite elements for the aero-thermo-elastic analysis of variable stiffness composite laminates

This paper explores the aero-thermo-elastic properties of variable stiffness composite laminates (VSCLs) with fibers steered along curved trajectories using a refined 1 D model. High-order solutions are generated by the p-version finite element method in conjunction with the Carrera Unified Formulation (CUF), which is applied to structural modeling by employing the improved hierarchical Legendre expansion (IHLE) in particular. The merit of using such a kind of expansion lies in that both the Equivalent Single Layer (ESL) and Layer-Wise (LW) theories can be formulated in a compact and straightforward manner. The linear piston theory is utilized to model the aerodynamic force, whereas the steady-state temperature field is applied during the flutter analysis. The weak-form governing equations are derived by taking a perturbation of the equilibrium configuration, and divergence and flutter instabilities are determined by solving the related eigenvalue problem. The accuracy of the proposed model has been verified through several test cases involving flutter, thermal buckling, and vibration of symmetric and antisymmetric VSCLs. The obtained results were compared to those obtained using the ABAQUS software and the available literature. Finally, the effect of the kinematic order, shear coefficient, and temperature variation on the critical dynamic pressure of VSCLs is thoroughly investigated.

Micromechanical analysis of fiber-reinforced ceramic matrix composites by a hierarchical quadrature element method

This work performed stress predictions of unidirectional fiber-reinforced ceramic matrix composites (CMCs) subjected to longitudinal tension using a hierarchical quadrature element method (HQEM). The HQEM is a p-version finite element method that can present highly accurate results using only a few sampling points. Parametric analysis was conducted to investigate the sensitivity of the HQEM model to interface properties and the detailed process of determining the optimal interface parameters for micromechanical analysis of damaged CMCs has been elaborated. By comparison with the analytical results based on the classical shear-lag model which is commonly adopted to analyze the stress distributions of the damaged fiber-reinforced CMCs, the HQEM estimates of fiber and matrix stress distributions were validated. For uniformly loaded SiCf/SiC composites with a single matrix crack, the micromechanical behaviors of three typical cases during failure process, namely, interface perfectly bonded, interface debonding and fiber failure were analyzed, illustrating the characteristics of CMC failure and providing insight into the mechanisms relating the microstructural behavior to global failure. The present work offers the foundations of the extension of a promising approach for highly accurate and efficient fracture analysis of CMCs.

Unlocking the secrets of locking: Finite element analysis in planar linear elasticity

Finite element methods have been the subject of active research for the last six decades. However, gaps remain in the understanding of even the simplest applications, including linear elasticity — the area where finite elements were first developed as a serious technique for the approximation of partial differential equations by engineers interested in the stress analysis of structures. For nearly incompressible materials, such as rubber, the standard, conforming finite element method sometimes exhibits suboptimal convergence rates for the energy and/or stresses. This type of behavior, termed “locking” or “non-robustness”, is still not completely understood despite receiving a great deal of investigation. This paper reviews the concept of locking in conforming finite element approximations to planar linear elasticity and seeks to quantify the effects of locking for the h- and p-version finite element method with a particular emphasis on quantifying the effect of mesh topology and geometry. As a by-product, we show that the inf–sup constant, which is responsible for locking, is independent of the mesh size h and polynomial degree p when p≥4. Numerical examples are provided throughout to illustrate the theoretical results.

Direct Evaluation of the Stress Intensity Factors for the Single and Multiple Crack Problems Using the P-Version Finite Element Method and Contour Integral Method

Stress intensity factor (SIF) is one of three important parameters in classical linear elastic fracture mechanics (LEFM). The evaluation of SIFs is of great significance in the field of engineering structural and material damage assessment, such as aerospace engineering and automobile industry, etc. In this paper, the SIFs of a central straight crack plate, a slanted single-edge cracked plate under end shearing, the offset double-edge cracks rectangular plate, a branched crack in an infinite plate and a crucifix crack in a square plate under bi-axial tension are extracted by using the p-version finite element method (P-FEM) and contour integral method (CIM). The above single- and multiple-crack problems were investigated, numerical results were compared and analyzed with results using other numerical methods in the literature such as the numerical manifold method (NMM), improved approach using the finite element method, particular weight function method and exponential matrix method (EMM). The effectiveness and accuracy of the present method are verified.

Layer-wise solutions for variable stiffness composite laminated sandwich plate using curvilinear fibers

This paper deals with the vibration analysis of variable stiffness composite laminated (VSCL) sandwich plates. The VSCL sandwich plate is composed by soft core and two laminated face sheets, in which the face sheets are considered with curvilinear fiber. The layer-wise plate theory and p-version of finite element method are used to solve the problem formulation. In layer-wise theory, each layer modeled separately as a one p-element and the continuity of displacement fields is based on a linear zig-zag variation of displacements through the thickness of the layers. The obtained results of present model are compared and validated with available solutions, such as equivalent single layer theories, layer-wise theory and three-dimensional theory for sandwich plate with constant stiffness. The frequencies and mode shapes are examined for different physical and mechanical parameters such as plate thickness, core to face sheets thickness ratio, orientation angle of curvilinear fibers, stacking sequence and boundary conditions. New layer-wise solutions for thin and thick VSCL sandwich plate are offered, in which can be used to establish benchmarks for future comparisons.

Direct Computation of 3-D Stress Intensity Factors of Straight and Curved Planar Cracks with the P-Version Finite Element Method and Contour Integral Method.

This paper presents direct computations of 3-D fracture parameters including stress intensity factors (SIFs) and T-stress for straight and curved planar cracks with the p-version finite element method (P-FEM) and contour integral method (CIM). No excessive singular element or enrichment function is required for the computation. To demonstrate the accuracy and efficiency of the proposed approaches, several benchmark numerical models of 3-D planar straight and curved cracks with single and mixed-mode fractures are considered and analyzed: a through thickness edge straight crack in a homogeneous material, a through thickness inclined straight crack, a penny-shaped crack embedded in a cube and a central ellipse shaped crack in a homogeneous cube. Numerical results are analyzed and compared with analytical solutions and those reported by the extended finite element method (XFEM) and the scaled boundary finite element method (SBFEM) in the available literature. Numerical experiments show the accuracy, robustness and effectiveness of the present method.

Implicit large-eddy simulations of turbulent flow in a channel via spectral/hp element methods

The spectral/hp element method [which is the hp-version finite element method, where h denotes the h-version finite element method and p denotes the p-version finite element method (or the spectral element method) with elementwise expansion based on (modified) orthogonal polynomials up to pth-order in each element] together with the regularized spectral vanishing viscosity (SVV) is employed to perform implicit large eddy simulation (iLES) of the turbulent separated flows in a channel with streamwise periodic hill-shaped constriction. The simulations are conducted at a Reynolds number of 10 595 based on the hill height and the bulk velocity magnitude above the crest, where the standard benchmark was presented with abundant experimental and numerical data. The flow statistical properties are discussed in detail, including mean velocities, Reynolds stresses, anisotropy measures, and spectra, which are in good agreement with the available numerical and experimental data in the literature. It is demonstrated that the SVV-iLES model performs at least as well as the established explicit models and therefore, the high-order spectral/hp element method via the calibrated model-free iLES is well-prepared for highly resolved wall-bounded turbulent simulations with large-scale separations and certainly for industrial complex flows.

Hybrid integral transform and p-version FEM for thermo-mechanical analysis of a functionally graded piezoelectric hollow cylinder subjected to asymmetric loads

As the first endeavor, a combination of fast Fourier transform and p-version of finite element method is proposed for electro-thermo-elastic analysis of a thick hollow cylinder under asymmetric thermal loadings. Especially in shells of revolution, the proposed FFT-pFE method is accompanied by a significant decrease in the computational costs. Due to the problem periodicity in such structures, the fast Fourier transform technique is used to discretize the governing equations into a set of harmonics in circumferential direction. Each harmonic is then partitioned using higher order finite elements. Hierarchical finite elements based on Legendre polynomial interpolation functions are utilized to discretize 2D governing equations of a functionally graded piezoelectric (FGP) cylinder. 3D governing equations of a FGP hollow cylinder are then discretized by using the higher-order Lagrangian finite elements. The effects of FFT grid-size as well as the order of the interpolation functions are investigated on convergence behavior of the proposed mixed FFT-pFE method. The piezoelectric material properties, with the exception of the Poisson’s ratio, are considered to vary along the radius of the cylinder and pursue the power function. The governing equations are derived using the principle of virtual displacements. For a 3D FGP hollow cylinder, the influence of ...

Finding a weak solution of the heat diffusion differential equation for turbulent flow by Galerkin's variation method using p-version finite elements

The stochastic turbulence model developed by Professor Czibere provides a means of clarifying the flow conditions in pipes and of describing the heat evolution caused by shear stresses in the fluid. An important part of the theory is a consideration of the heat transfer-diffusion caused by heat generation. Most of the heat is generated around the pipe wall. One part of the heat enters its environment through the wall of the tube (heat transfer), the other part spreads in the form of diffusion in the liquid, increasing its temperature. The heat conduction differential equation related to the model contains the characteristics describing the turbulent flow, which decisively influence the resulting temperature field, appear. A weak solution of the boundary value problem is provided by Bubnov-Galerkin’s variational principle. The axially symmetric domain analyzed is discretized by a geometrically graded mesh of a high degree of p-version finite elements, this method is capable of describing substantial changes in the temperature gradient in the boundary layer. The novelty of this paper is the application of the p-version finite element method to the heat diffusion problem using Czibere’s turbulence model. Since the material properties depend on temperature, the problem is nonlinear, therefore its solution can be obtained by iteration. The temperature states of the pipes are analyzed for a variety of technical parameters, and useful suggestions are proposed for engineering designs.

Curved p-version C1 finite elements for the finite deformation analysis of isotropic and composite laminated thin shells in irregular shape

In this work, we generalize our recently developed p-version curved C1 plate elements to the finite deformation analysis of general isotropic and laminated thin shells with arbitrary curvature and shape. The kinematic formulations are based on the geometrically-exact nonlinear thin shell model where the Kirchhoff-Love hypothesis is strictly implemented during the deformation. Since high accurate curved coarse meshes are required in the implementation of p-version finite element method, the quasi regional mapping technique is employed and modified to obtain the high-order meshes with fitted boundaries. Through this method, shells with irregular shapes can be easily modeled. To avoid dealing the difficult problem of constructing fully C1 conforming displacement fields on shell surface, the C1 conformity of present elements is implemented on the selected Gauss-Lobatto nodes, which had been found to be able to produce very fast convergence. Nodal variables including second-order directional derivatives on surfaces are derived and used for element assembly. In this manner, direct construction of displacement fields on shell mid-surface defined by the quasi regional mapping is allowed. Several benchmark problems of large deformation analysis of isotropic and laminated thin shells are presented to illustrate the accuracy as well as robustness of the present method.

Extraction of Stress Intensity Factors by Using the P-Version Finite Element Method and Contour Integral Method

The stress intensity factors (SIFs) for two-dimensional cracks are extracted using the p-version finite element method (P-FEM) and the contour integral method. Several numerical experiments, e.g., crack initiating from the edge of a circular hole under an unidirectional uniform tension and two equal-length, unequal-length hole–edge cracks, respectively, at a rectangular plate, an inclined centered crack under uniaxial tension at a square plate and a pipeline crack model, are used to demonstrate the accuracy and effectiveness of the approaches. SIFs are presented for the effects of various crack lengths and length–width ratio. Numerical results are analyzed and compared with reference solutions and results obtained by the Voronoi cell finite element method, boundary element method, high-order extended finite element method (high-order XFEM) and commercial finite element software ABAQUS in the available literature. Numerical results are in good agreement with the benchmark problems and show faster convergence rate, higher accuracy and better numerical stability.

Temperature Variation at Solid-Fluid Interface of Thin Film Lubricated Contact Problems

Many calculating methods have been already developed for solving contact problems of parts such as gears, cams, and followers under fluid film lubrication conditions considering the temperature and pressure dependence. Similarly, the determination of the elasto-hydrodynamic pressure distribution the processes taking place in the lubricant and the contacting bodies, as well as in their environment, have to be dealt with simultaneously for the determination of the temperature field. A system of equation for the modelling of thermo-elastohydrodynamic lubrication between two contacting bodies containing hydrodynamic, thermodynamic, and strength problems is a highly non-linear system which becomes even more so if the temperature and pressure dependence of the material properties are considered. To solve this system, scientists started to use the finite element formulation in the 1960s and it was found to be a promising and reliable method. Earlier, the lubrication analysts used only the h-version finite element method (h-FEM) till 1991, when the first usage of the p-version finite element method (p-FEM) was published in the literature. In order to reduce the problem, in case of point or line contact, the contact bodies can be handled as semi-infinite ones. Following this simplification that had been successfully applied for the gap size determination, a substructure model was defined using analytical solution of the moving heat source. Instead of an iterative way between the solid and fluid problem, in this paper we present an efficient solution when thermal model for lubricant and surfaces were coupled and solved by a direct numerical method in one step.

Vibration analysis of variable stiffness laminated composite sandwich plates

This paper presents the free vibration analysis of a variable stiffness laminated composite sandwich plates. The fiber orientation angle of the face sheets (Skin) is assumed to vary linearly with the x-axis. The problem formulation is based on the higher-order shear deformation plate theory HDST C0 coupled with p-version of finite element method. The elements of the stiffness and mass matrices are calculated analytically. The sandwich plate is presented with a uniform mesh of four p-elements and the convergence properties are achieved by increasing the degree p of the hierarchical shape functions. A calculation program is developed to determine the fundamental frequencies for different physical and mechanical parameters such as plate thickness, core to face sheets thickness ratio, orientation angle of curvilinear fibers and boundary conditions. The results obtained show a good agreement with the available solutions in the literature. New comparison study of vibration response of laminated sandwich plate between the straight and curvilinear fibers is presented.

Experimental and numerical investigation on the dynamic characteristics of thick laminated plant fiber-reinforced polymer composite plates

Although considerable attention has been devoted to the experimental research works of plant fiber-reinforced composites, theoretical determination of their structural characteristics is also much needed one in their evolution. But it is still considered as a difficult process due to the several factors such as short and discontinuous fiber form, plant fiber’s intrinsic characteristics, fiber anisotropy and porosity. In this study, vibration characteristics of woven jute/epoxy and woven aloe/epoxy laminated composite plate structures are investigated theoretically and experimentally. The elastic constants of the woven jute/epoxy and woven aloe/epoxy laminated composites are obtained through experimental testing. The governing differential equations of motion for the uniform woven jute/epoxy and woven aloe/epoxy laminated thick composite plates are carried out using the p-version finite element method based on higher-order shear deformation theory. The effectiveness of the developed finite element formulations is demonstrated by comparing the natural frequencies obtained using the present finite element method with those obtained from the experimental measurements. Also a comparative study is carried out between the results of h-version FEM and p-version FEM to check their accuracies and efficiencies. The effects of aspect ratio and angle of fiber orientation under various end conditions on the free vibration responses of the woven jute/epoxy laminated composite plate are also investigated. The forced vibration response of the woven jute/epoxy laminated composite plate under the harmonic force excitation is carried out under various end conditions.