Higher-order Finite Element Formulation Research Articles

An enhanced higher-order finite element formulation for wrinkling analysis of thick sandwich plates

Sandwich structures are susceptible to wrinkling when subject to in-plane compressive loads, which threatens structural safety. To accurately investigate the failure behavior, the following work is to be required. Firstly, the wrinkling behavior of sandwich plates is a typical three-dimensional (3D) issue. To consider a 3D effect, an enhanced higher-order finite element formulation including transverse normal strain has been proposed for the first time. Utilizing the proposed finite element formulation, a refined quadrilateral element model has been constructed. Secondly, the buckling and wrinkling behaviors of square sandwich plates have been investigated. The performance of the proposed model has been evaluated by comparing it with the quasi-3D solutions, the results obtained from the 3D finite element method (3D-FEM), and other models. The differences between the buckling and wrinkling failure behaviors have been carefully explained. Thirdly, the effects of boundary conditions on buckling and wrinkling deformation have been explored in detail. By utilizing the proposed model, the translation mechanism between the buckling and wrinkling deformations of sandwich plates under different boundary conditions has been revealed, which can help other investigators to well understand the buckling and wrinkling behaviors of sandwich plates. It can be shown that the wrinkling behaviors of thick sandwich plates occur earlier than the buckling behaviors when the thickness ratio of core to face panel is more than 35. Besides, it also can be shown that the buckling and wrinkling loads decrease gradually with the relaxation of the boundary restriction.

An extensive study of shear thinning flow around a spherical particle for power-law and Carreau fluids

We develop and implement a high-order finite element formulation to solve incompressible shear thinning flows using power-law and Carreau rheology models. We verify the implementation using the Method of Manufactured Solutions (MMS) and demonstrate that the implementation preserves the order of accuracy of the FEM scheme. We run high-order flow-past-a-sphere simulations for Re ∈ [ 0 . 1 , 100 ] for Newtonian and shear thinning flows. Power-law simulations cover n ∈ [ 0 . 3 , 0 . 9 ] , Carreau simulations cover the same range of n and dimensionless relaxation times Λ ∈ [ 0 . 1 , 100 ] . We use 3D Cartesian grids, adopting a high-order sharp-interface immersed boundary method (IBM) to impose the no-slip boundary condition on the surface of the sphere. We use dynamic mesh adaptation with a Kelly error estimator to adaptively refine the grid. Newtonian drag coefficients are compared to Clift et al.’s model. Using the power-law model for shear thinning flow, results show that when decreasing n : the drag coefficient increases for Re ⪅ 5, it decreases for Re ⪆ 5, and that for higher Reynolds the wake region is longer and the separation point is moved upstream. These drag coefficient results are in adequation with previously established simulation results. We provide a new drag coefficient correlation for power-law fluids that is valid for a wide range of n and for Re ∈ [ 0 . 1 , 100 ] , with R 2 = 0 . 998 . The power-law model is singular in the zero-shear limit, and this limit is bound to occur in a flow-past-a-sphere geometry. For that matter, the Carreau model is well-posed. As the Carreau-modeled results show, the drag force on the sphere is considerably affected by the initial Newtonian plateau of the fluid. Drag coefficient results in the creeping flow regime are compared to experimental results, allowing validation of the numerical model. This work provides a better understanding of shear thinning flow past a sphere, for which the relationship between drag and flow regimes are highly nonlinear. • The effect of shear thinning flow past a sphere on the drag force is studied. • The open-source Computational Fluid Dynamics (CFD) software Lethe using a Finite Element Method (FEM) is employed. • A correlation for the drag coefficient for power-law fluid flow past a sphere is given, with a coefficient of determination R 2 = 0.998. • Power-law and Carreau drag coefficients are compared.

On pressure robustness and independent determination of displacement and pressure in incompressible linear elasticity

We investigate the possibility to determine the divergence-free displacement uindependently from the pressure reaction p for a class of boundary-value problems in incompressible linear elasticity. If not possible, we investigate if it is possible to determine it pressure robustly, i.e. pollution free from the pressure reaction.For convex domains there is but one variational boundary value problem among the investigated that allows the independent determination. It is the one with essential no-penetration conditions combined with homogeneous tangential traction conditions.Further, for the investigated cases, the weakly divergence-free displacement can be computed pressure robustly provided the total body force is decomposed into its direct sum of divergence- and rotation-free components using the Helmholtz decomposition. The elasticity problem is solved using these components as separate right-hand sides. The total solution is obtained using the superposition principle.We employ a (u,p) higher-order finite element formulation with discontinuous pressure elements. It is inf–sup stable for polynomial degree p≥2 but not pressure robust by itself. We propose a three step procedure to solve the elasticity problem which includes by the Helmholtz decomposition of the total body force. The extra cost for the three-step procedure is essentially the cost for the Helmholtz decomposition of the assembled total body force, and the small cost of solving the elasticity problem with one extra right-hand side. The results are corroborated by theoretical derivations as well as numerical results.

Optimization of damping characteristics of tapered MWCNT/glass fiber reinforced composite rectangular sandwich plates: A higher-order finite element formulation

The research work presented in this paper is focused on the investigation of damping characteristics and structural optimization of the tapered composite hybrid smart rectangular sandwich plates. The composite rectangular hybrid sandwich plate contains the constraining and base layers made of multi-walled carbon nanotubes reinforced composite (CNTRC) face sheets and the core layer made of multi-walled carbon nanotubes reinforced magnetorheological elastomer. Higher-order shear deformation theory (HSDT) and finite element (FE) formulations are employed to derive the governing equations of the CNTRC rectangular hybrid smart sandwich plate. The performance of the derived HSDT model is validated by comparing the results in terms of natural frequencies evaluated using experimental tests on the composite hybrid smart sandwich structures under different support conditions. The influence of CNT reinforcement on the damping performance of the tapered rectangular hybrid smart sandwich plates are explored. Also, the optimal ply orientations of the tapered CNTRC rectangular sandwich plates with hybrid core is identified using the developed HSDT model coupled with Genetic Algorithm (GA) to enhance the natural frequencies and loss factors. Further, it can be observed that the reinforcement of CNTs in composites significantly influences the stiffness and damping characteristics of the tapered composite rectangular smart sandwich plates.

Stability of Layered Structures with Hybridized Configuration by Means of a Reddy-Type Higher-Order Finite Element Formulation

To make use of the merit of designability, each lamina in layered structures may possess diverse materials and geometry to realize specific application. For the hybridized structures, geometry and material properties relative to the middle surface are generally unsymmetrical, which have a significant impact on stability. Some models might lose capability to deal with such issues, so that these issues are less reported. Within the developed models, Reddy’s model possesses merit of simplicity and efficiency, so a Reddy-type higher-order zig-zag model is constructed by utilizing the proposed zig-zag function (ZZF). Instead of the standard finite element formulation using the principle of minimum potential energy, the three-field Hu–Washizu (HW) mixed variational principle is employed to acquire the finite element formulation which can meet the harmonious conditions of transverse shear stress at the interface of adjacent layers. By investigating buckling behaviors of hybridized structures, performance of the proposed finite element formulation is appraised by comparing with the results obtained from the three-dimensional (3D) model as well as other models. Effect of boundary conditions (BCs), material properties, and span-to-thickness ratio on the buckling loads is also studied in detail. Numerical results show that buckling loads of hybridized structures are significantly impacted by the chosen parameters. The results acquired from proposed model are in very good agreement with those obtained from the layerwise (LW) model and the 3D finite element results.

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.

Reduced modal state-space approach for low-velocity impact analysis of sandwich beams

Abstract The convergence problems of nonlinear dynamic impact analysis may be in some extent overcome by using the linear forms of contact laws. However, the full-scale finite element analysis of structures with linearized contact laws need high computational efforts. In this study, an efficient reduced modal state-space approach with low computational cost was introduced to predict the dynamic structural response of sandwich beams. A one degree of freedom spring-mass system was used to model the dynamics of projectile. Furthermore, the fully dynamic equations of the sandwich beam were obtained using a new high-order finite element formulation . The dimensions of the dynamic system were reduced based on the method of truncated combination of modes. Finally, the reduced dynamic equations were solved in the framework of state-space approach. The results obtained from the present formulation were compared with the available analytical and numerical results and good agreements found between them.

Modal analysis of FG sandwich doubly curved shell structure

The modal frequency responses of functionally graded (FG) sandwich doubly curved shell panels are investigated using a higher-order finite element formulation. The system of equations of the panel structure derived using Hamilton\'s principle for the evaluation of natural frequencies. The present shell panel model is discretised using the isoparametric Lagrangian element (nine nodes and nine degrees of freedom per node). An in-house MATLAB code is prepared using higher-order kinematics in association with the finite element scheme for the calculation of modal values. The stability of the opted numerical vibration frequency solutions for the various shell geometries i.e., single and doubly curved FG sandwich structure are proven via the convergence test. Further, close conformance of the finite element frequency solutions for the FG sandwich structures is found when compared with the published theoretical predictions (numerical, analytical and 3D elasticity solutions). Subsequently, appropriate numerical examples are solved pertaining to various design factors (curvature ratio, core-face thickness ratio, aspect ratio, support conditions, power-law index and sandwich symmetry type) those have the significant influence on the free vibration modal data of the FG sandwich curved structure.

Theoretical and experimental investigation of vibration characteristic of carbon nanotube reinforced polymer composite structure

The vibration frequencies of multi-walled carbon nanotube-reinforced polymer composite structure are examined numerically via a generic higher-order shear deformation kinematics for different panel geometries. The extensive behaviour of the current higher-order model is demonstrated by comparing the results with the published data including the own in-house experimental values. In this analysis, the required elastic properties of the randomly distributed nanotube-reinforced polymer composite panel are evaluated numerically using Mori–Tanaka scheme. Firstly, the equation of motion of the vibrated nanotube composite panel derived via the classical Hamilton's principle and the isoparametric finite element steps are implemented for the numerical purpose. Further, the modal responses are obtained computationally using an original computer code (MATLAB) with the help of the higher-order finite element formulation. The necessary convergence and subsequent comparison have been made for the presently developed numerical model with those available published results including the values obtained via commercial package (ANSYS). Additionally, the model validation has been established by comparing the present numerical frequency values with the lab-scale free vibration experimental data. The specific conclusions are drawn by examining different numerical examples for various structural parameters using the experimental properties.

Delamination detection in buckling laminated composite plates

This paper introduces a new damage detection procedure for laminated composite plates with multiple through-the-width delaminations under compressive loading. The higher-order shear deformation theory is used for the higher-order finite-element formulation. A 13-node triangular element is introduced for finite-element modelling, which includes a parabolic variation of the transverse shear strains through the thickness. As a result, there is no need to use shear correction coefficients in computing the shear stresses. A virtual iso-parametric element is employed to avoid overlapping between delaminated areas. Moreover, w,n is used as an independent degree of freedom to obtain accurate responses and accelerate the rate of convergence in the finite-element analysis. A two-step genetic algorithm approach for delamination identification is also proposed, which can identify the location and magnitude of delamination in laminated composite plates under compressive loading. By minimising the difference between the analytical responses, resulting from the suggested formulation, and the measured responses, a genetic algorithm identifies the delamination. In this study, some examples are provided to illustrate the accuracy of the formulation.

Geometrically nonlinear transient analysis of functionally graded shell panels using a higher-order finite element formulation

Nonlinear transient analysis of functionally graded curved panels is carried out employing a higher-order C0 finite element formulation. The element consists of nine degrees-of-freedom per node with higher-order terms in the Taylor’s series expansion which represents the higher-order transverse cross sectional deformation modes. The formulation includes Sanders’ approximation for doubly curved shells considering the effects of rotary inertia, transverse shear and moderately large rotations in the von Karman sense. A realistic parabolic distribution of transverse shear strains through the shell thickness is assumed and the use of shear correction factor is avoided. The accuracy of the formulation is validated by comparing the results with those available in the literature. The transient dynamic responses of the functionally graded shell panels are investigated by varying the volume fraction index using a simple power law distribution. Material properties are assumed to be temperature-independent and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Heat conduction between ceramic and metal constituents is neglected. Effects of different panel geometry parameters, boundary conditions and loadings are studied. Key words: Functionally graded materials, higher-order formulation, geometric nonlinearity, transient analysis.

Buckling Analysis of Tapered Composite Beams using a Higher Order Finite Element Formulation

Tapered composite beams are increasingly being used in various engineering applications such as helicopter yoke, robot arms, and turbine blades. In the present work, the buckling analysis of laminated tapered composite beams is conducted using a higher order finite element formulation. In tapered laminates, the material and geometric discontinuities at ply drop-off locations lead to significant discontinuities in stress distributions. Higher order formulation ensures the continuity of the stress distribution through the thickness of a laminate as well as across the element interfaces, which is very important for the analysis of tapered laminates. In addition, higher order finite element formulation requires lesser number of elements in order to achieve reasonably accurate results in buckling analysis. The stiffness coefficients of the tapered laminated beam are determined in the present work based on the stress and strain transformations and the classical laminate theory. A detailed parametric study on various types of tapered composite beams is conducted.

Free vibration analysis of functionally graded curved panels using a higher-order finite element formulation

Free vibration analysis of functionally graded curved panels is carried out using a higher-order formulation. A C 0 finite element formulation is used to carry out the analysis. The element consists of nine degrees of freedom per node with higher-order terms in the Taylor's-series expansion, which represents the higher-order transverse cross-sectional deformation modes. The formulation includes Sanders’ approximation for doubly curved shells considering the effects of rotary inertia and transverse shear. A realistic parabolic distribution of transverse shear strains through the shell thickness is assumed and the use of shear correction factor is avoided. Material properties are assumed to be temperature independent and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. Heat conduction between ceramic and metal constituents is neglected. The accuracy of the formulation is validated by comparing the results with those available in the literature. Effects of panel geometry parameters and boundary conditions are studied.

A New Locking Free Higher Order Finite Element Formulation for Composite Beams

A refined $2$-node, $7$ DOF/node beam element formulation is presented in this paper. This formulation is based on higher order shear deformation theory with lateral contraction for axial-flexural-shear coupled deformation in asymmetrically stacked laminated composite beams. In addition to axial, transverse and rotational degrees of freedom, the formulation also incorporates the lateral contraction and its higher order counterparts as degrees of freedom. The element shape functions are derived by solving the static part of the governing equations. The element considers general ply stacking and the numerical results shows that the element exhibits super convergent property. The efficiency of the element in capturing both the static and dynamic inter-laminar stresses is demonstrated. The accuracy of the element to capture free vibration and wave propogation responses with small problem sizes is also demonstrated.

A stabilized finite element method based on SGS models for compressible flows

We present an appropriate extension of the stabilized finite element formulation, introduced in [A. Corsini, F. Rispoli, A. Santoriello, A variational multiscale high-order finite element formulation for turbomachinery flow computations, Comput. Methods Appl. Mech. Engrg. 194 (2005) 4797–4823] for the prediction of incompressible flows, aimed at compressible flows. The stabilized formulation is the so-called variable subgrid scale method (V-SGS) based on an approximation of the class of subgrid scale models (SGS) derived from the Hughes variational multiscale method (Hughes-VMS) introduced in [T.J.R. Hughes, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg. 127 (1995) 387–401]. It is characterized by a variable stabilization parameter within the domain of element. We also propose an innovative procedure for computing the stabilization parameter by using the one-dimensional element Green’s functions corresponding to an advective–diffusive differential operator. The stabilization parameter is defined as the sum of two components, the first providing its mean value and the second its element-wise space dependency. On this basis the present work proposes the stabilization matrix for the multidimensional advective–diffusive system of equations governing compressible flows. The procedure developed for computing the stabilization parameter can be applied to other differential operators as it has been shown in [A. Corsini, F. Rispoli, A. Santoriello, A variational multiscale high-order finite element formulation for turbomachinery flow computations, Comput. Methods Appl. Mech. Engrg. 194 (2005) 4797–4823]. The proposed stabilization device is validated by simulation of inviscid and viscous supersonic flow configurations.

An rp-adaptive finite element method for the deformation theory of plasticity

In this paper, we present an rp-discretization strategy for physically non-linear problems based on a high order finite element formulation. In order to achieve convergence, the p-version leaves the mesh unchanged and increases the polynomial degree of the shape functions locally or globally, whereas the r-method moves nodes and edges of an existing FE-mesh. The basic idea of our rp-version approach is to adjust the finite element mesh to the shape of the elastic–plastic interface in order to take into account the loss of regularity which arises along the curve of the plastic front. Numerical examples will demonstrate that this approach leads to an exponential rate of convergence and highly accurate results.

Vibration analysis of tapered composite beams using a higher-order finite element. Part II: parametric study

Tapered composite beams are increasingly being used in various engineering applications. Different types of ply drop-off can be achieved in tapered beams in practice depending on the application. The objective of the present work is to conduct an investigation of the undamped free vibration response of such tapered composite beams. A higher-order finite element formulation has been developed in Part I of the present work for vibration analysis of tapered composite beams based on classical laminated plate theory. In the present part of the work, the developed formulation is used for the analysis of various types of tapered composite beams. A parametric study on tapered composite beams is conducted, considering the effects of taper angle, types of taper, boundary conditions, laminate configuration and geometric parameters on the undamped natural frequencies.

Vibration analysis of tapered composite beams using a higher-order finite element. Part I: Formulation

Tapered composite beams are increasingly being used in various engineering applications such as helicopter yoke, robot arms and turbine blades. The objective of the present work is to conduct an investigation of the free undamped vibration response of such tapered composite beams, using the finite element method. Conventional cubic Hermitian finite element formulation requires a large number of elements to obtain reasonably accurate results in the analysis of tapered laminated beams. Since the continuity of curvature at element interfaces cannot be guaranteed with the use of conventional formulation, the stress distribution across the thickness is not continuous at element interfaces. The material and geometric discontinuities at ply drop-off locations leads to additional discontinuities in stress distributions. As a result, efficient and accurate calculation of natural frequencies becomes very difficult. In order to overcome these limitations, a higher-order finite element formulation is developed in Part I of the present work. The stiffness coefficients of the tapered laminated beam are determined based on the stress and strain transformations and classical laminate theory. In Part II of the present work, the developed formulation is used for the free undamped vibration analysis of various types of tapered composite beams and a parametric study is conducted.

High-Order Finite Elements for Inhomogeneous Acoustic Guiding Structures

Silvester's high-order finite-element formulation for potential problems is extended to enable the analysis of acoustic wave propagation in lossless isotropic, uniform, and inhomogeneous guiding structures. The formulation allows a large class of problems to be solved using elements of any desired order, with only minimal computer coding. Three examples are cited---one involving a simple homogeneous region having an analytic solution, and two inhomogeneous problems. Good agreement with other methods and with limiting cases is obtained in each case.