P-version Finite Element 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.

Nonlinear flutter of composite laminates with curvilinear fibres using a full linearized aerodynamic theory

This paper studies nonlinear flutter of rectangular variable stiffness composite laminates (VSCLs) subjected to a low supersonic airflow. The VSCLs are formed by different individual laminae where a polymeric matrix is reinforced by curvilinear carbon fibres. The plate’s displacement field follows a Third-order Shear Deformation Theory (TSDT); a p-version finite element is employed to discretize the displacement components. A full linearized (inviscid, potential flow) aerodynamic theory is used to approximate the aerodynamic pressure. The self-exciting vibrational model is formed using the principle of virtual work. The equations of motion representing the model are solved using Newmark method. Limit cycle oscillations (LCOs) of these plates are studied using time histories, phase-plane plots and FFT spectra. A damage onset criterion is utilized to control the structural health of the laminates during nonlinear flutter. Effects of curvilinear fibre path angles on linear and nonlinear flutter behaviour, as well as the damage onset in VSCLs are investigated.

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.

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.

Linear aeroelastic analysis of cantilever hybrid composite laminated plates with curvilinear fibres and carbon nanotubes

In this paper, cantilever laminated composite plates, subjected to supersonic airflow, are studied in the linear elastic regime. The considered material is a multiscale three-phase composite, constituted by epoxy resin reinforced with multi-walled carbon nanotubes and curvilinear carbon fibres. The orientation of the fibres is defined using three different functions. A model based on a third-order shear deformation theory and a p-version finite element is applied. Furthermore, a hierarchical approach is applied to characterise the mechanical properties of the composite material. A combination of adequate micromechanics based models, including a modified version of the Halpin-Tsai model, the Rule of Mixtures, a unit cell-based model, and the Chamis model, is implemented. The aeroelastic analysis is performed considering the linear piston theory to evaluate flutter (dynamic instability) and divergence (static instability) in such structures. The improvements achieved through the combination of carbon nanotubes and curvilinear fibres on the instabilities are explored.

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.

Stability and bifurcations in oscillations of composite laminates with curvilinear fibres under a supersonic airflow

Nonlinear flutter of variable stiffness composite plates—using a reduced-order model benefiting from a large-enough number of modes or degrees of freedom—is the subject of this research. Dynamic of such plates is enriched by different scenarios including period-n (un)stable limit cycle oscillations, quasi-periodic or chaotic solutions, along with Hopf, symmetry breaking and period doubling bifurcations. The fibre path orientation, in any individual ply of these rectangular plates, changes linearly from the left edge to the right one. The plates are subjected to a supersonic airflow. The plate’s displacement field is modelled using a third-order shear deformation theory; a p-version finite element is used to discretize the displacement components. Aerodynamic pressure due to the airflow is approximated using linear piston theory. The self-exciting vibrational full-order model (FOM) is formed by employing the principle of virtual work, and then, the ROM is extracted from the FOM using two reduction techniques, namely static condensation and modal summation method. The inclusion of a large-enough number of modes of vibration in vacuum in the latter technique is critical for the observation of, often overlooked, complex dynamic behaviour of these plates. The equations of motion representing the ROM are solved using the shooting method (to obtain stable and unstable LCOs) and using Newmark method (for stable or non-oscillatory solutions). The rich dynamics of these plates are explored with the help of vibration and flutter mode shapes, phase-plane plots, bifurcation diagrams, fast Fourier transform diagrams and Poincare sections.

Three-dimensional buckling analysis of variable angle tow composite laminated plates

This paper presents three-dimensional solutions for the buckling of variable angle tow (VAT) composite laminated plates. The p-version finite element with hierarchical basis functions is adopted to determine the buckling of VAT composite laminates with different parameters. Inter-element compatibility is achieved by matching the generalized displacements at the vertices, edges, and faces shared by neighboring elements. The buckling loads of symmetrical and anti-symmetrical laminates with various boundary conditions are obtained. A convergence study is performed and the accuracy of the method is confirmed by comparing the numerical results with those in the literatures. The verified results indicate that the present 3D p-FEM method is accurate for the buckling analysis of VAT composite laminated plates, and that these three-dimensional results may serve as a benchmark for future studies.

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.