Isoparametric Quadrilateral Element Research Articles

Free vibration behavior in functionally graded material plates with dual efficient quadrilateral finite elements

Functionally graded materials represent advanced composites with typically two materials that exhibit gradual variations in their mechanical properties throughout the structure’s thickness in general, as determined by a presumed mathematical model. The current investigation focuses on analyzing free vibrations of functionally graded material plate structures with two efficient isoparametric quadrilateral finite elements (Lagrange and Serendipity) together for the first time and different material combinations based on first-order shear deformation theory. The analysis was carried out on relatively thin as well as medium thick functionally graded plates with varying power law indices and at varied boundary conditions. The framework equation was formed with Hamilton’s principle, and solutions were done with finite element method, considering a shear correction factor for the behavioral study of varying material properties. The findings align well with the information presented in prior literature. After validating the present model, parametric analysis was conveyed to show the impact of length-to-thickness ratio, aspect ratio, volume fractions, and skew angles for two novel materials on the fundamental frequencies and mode shapes. The results thus obtained with the two quadrilateral elements have a significant effect on the changes in these parameters. Also, the natural frequencies of different functionally graded plates, including two novel materials, were compared and discussed.

Verification of a High-Order FEM-based CFD Code using the Method of Manufactured Solutions

A high-order computational fluid dynamics (CFD) code capable of solving compressible turbulent flow problems was developed. The CFD code employs the Flowfield Dependent Variation (FDV) scheme implemented in a Finite Element Method (FEM) framework. The FDV scheme is basically derived from the Lax-Wendroff Scheme (LWS) involving the replacement of LWS’s explicit time derivatives with a weighted combination of explicit and implicit time derivatives. The code utilizes linear, quadratic and cubic isoparametric quadrilateral and hexahedral Lagrange finite elements with corresponding piecewise shape functions that have formal spatial accuracy of second-order, third-order and fourth-order, respectively. In this paper, the results of observed order-of-accuracy of the implemented FDV FEM-based CFD code involving grid and polynomial order refinements on uniform Cartesian grids are reported. The Method of Manufactured Solutions (MMS) is applied to governing 2-D Euler and Navier-Stokes equations for flow cases spanning both subsonic and supersonic flow regimes. Global discretization error analyses using discrete 𝐿2 norm show that the spatial order-of-accuracy of the FDV FEM-based CFD code converges to the shape function polynomial order plus one, in excellent agreement with theory. Uniquely, this procedure establishes the wider applicability of MMS in verifying the spatial accuracy of a high-order CFD code.

Behavior of Partially Saturated Cohesive Soil under Strip Footing

In this paper, a shallow foundation (strip footing), 1 m in width is assumed to be constructed on fully saturated and partially saturated Iraqi soils, and analyzed by finite element method. A procedure is proposed to define the H – modulus function from the soil water characteristic curve which is measured by the filter paper method. Fitting methods are applied through the program (SoilVision). Then, the soil water characteristic curve is converted to relation correlating the void ratio and matric suction. The slope of the latter relation can be used to define the H – modulus function. The finite element programs SIGMA/W and SEEP/W are then used in the analysis. Eight nodded isoparametric quadrilateral elements are used for modeling both the soil skeleton and pore water pressure. A parametric study was carried out and different parameters were changed to study their effects on the behavior of partially saturated soil. These parameters include the degree of saturation of the soil (S) and depth of water table. The study reveals that when the soil becomes partially saturated by dropping water table at different depths with different degrees of saturation, the bearing capacity of shallow foundation increases about (4 – 7) times higher than the bearing capacity of the same soil under saturated conditions. This result is attributed to matric suction value (i.e negative pore water pressure). The behavior of soil in partially saturated condition is likethat of fully saturated condition but with smaller values of displacement. It is found that the settlement is reduced when the water table drops to a depth of 2 m (i.e. twice the foundation width) by about (92 %).

Dynamic Analysis of Mindlin Plate Subjected to a Moving Mass-Spring-Damper System Using the Moving Element Method

The moving element method (MEM) is applied in this paper to study the dynamic behavior of the Mindlin plate subjected to a moving mass-spring-damper system. Reissner- Mindlin plate theory is utilized to study the plate structure. The plate is modeled by the Isoparametric quadrilateral nine-node element. The governing equation of motion is built into a coordinate system which moves with the moving mass-spring-damper system and is based on the principle of virtual work. The MEM is proposed to solve the governing equation of motion of the plate resting on a Pasternak foundation under the moving mass-spring-damper system including roughness of plate surface which is a dynamic source. Numerical results are investigated and verified by comparing with the published results. The effects of different parameters, such as spring stiffness, damping coefficient and velocity of the moving load system, plate thickness, foundation parameters and roughness of the plate surface on the dynamic response of the Mindlin plate are investigated.

Study on the Vector Quadrilateral Isoparametric Shell Element with Shear Deformation

Based on the linear superposition of the quadrilateral isoparametric membrane element and quadrilateral Mindlin plate element, basic vector formulas for the quadrilateral isoparametric shell element with the shear deformation is derived. Two different coordinate modes for updating the particle displacement and the element node internal force are put forward. The uniform numerical solution for an irregular quadrilateral element is realized by the isoparametric integral scheme along the element plane. Then, the bilinear elastoplastic constitutive of material is introduced into the vector finite shell element, and the nonlinear effect of the material is realized based on the integral method along the thickness. On this basis, the nonlinear calculation and analysis program of the quadrilateral isoparametric shell element is developed. Numerical example results show that the static and dynamic analysis, large deformation and large rotation analysis, and buckling analysis can all be well performed for the shell structures by the developed program, which verifies the validity of theoretical derivation and computer program.

Thermal-Mechanical Coupling Model Based on the Hybrid Finite Element Method for Solving Bipolar the Plate Deformation of Hydrogen Fuel Cells

New energy is the focus of attention all over the world, and research into new energy can inject new vitality into the industrial system. Hydrogen fuel cells are not only environmentally friendly, but also rich in reserves that can be used as a strategic resource for the entire country. The difficulty lies in the safe design of application equipment and the batch generation and storage of hydrogen. In addition, fuel cells have the disadvantage of a slow start-up. Based on the above problems, this paper proposes a hybrid-element method to solve the thermal-mechanical coupling model of fuel cell plate, which can effectively solve the thermal stress change, temperature field distribution and displacement change of the battery plate when working. Firstly, the hybrid-element algorithm is given for 2D plate deformation. Then, the deformation application of a 3D fuel cell plate is given. The 2D numerical results show that the hybrid finite element method (FEM) is more flexible for realizing the flexible combination of sub-mesh and finite element basis functions, and has a better mesh quality compared to the traditional constant strain triangular element (CST) adaptive FEM and quadrilateral isoparametric element (Q4) adaptive FEM. This method achieves a balance between numerical accuracy and solving efficiency for the multi-porous elastic plate. In addition, a deformation control formula is given which can display the displacement deformation and stress merge to same graph, since it is convenient to quickly compare the regions where the displacement and stress extremum appear. In short, the hybrid finite element method proposed in this paper has good mesh evaluation results, and when the number of discrete elements is equivalent, the hybrid element converges faster and the solution efficiency is higher. This paper also provides a good numerical theory and simulation reference for industrial mechanics and new energy applications.

An Arbitrarily High-order Spectral Difference Method with Divergence Cleaning (SDDC) for Compressible Magnetohydrodynamic Simulations on Unstructured Grids

This paper reports a recent development of the high-order spectral difference method with divergence cleaning (SDDC) for accurate simulations of both ideal and resistive magnetohydrodynamics (MHD) on curved unstructured grids consisting of high-order isoparametric quadrilateral elements. The divergence cleaning approach is based on the improved generalized Lagrange multiplier, which is thermodynamically consistent. The SDDC method can achieve an arbitrarily high order of accuracy in spatial discretization, as demonstrated in the test problems with smooth solutions. The high-order SDDC method combined with the artificial dissipation method can sharply capture shock interfaces with the oscillation-free property and resolve small-scale vortex structures and density fluctuations on relatively sparse grids. The robustness of the codes is demonstrated through long time simulations of ideal MHD problems with progressively interacting shock structures, resistive MHD problems with high Lundquist numbers, and viscous resistive MHD problems on complex curved domains.

Computational analysis of a sandwich beam with FGM face sheets under flexural loading

Functionally graded materials (FGM) are nonhomogeneous materials at the microscopic level, which have a gradation in their structure and composition with the properties of the material varying along one or more directions. The applications of these materials are in the fields: Aerospace, Automotive, Biomedical-Medical, Optoelectronics-electronics, Machinery, Energy, Marine, etc. In the present work the problem of a sandwich beam with metal core and FGM face sheets and a sandwich beam with ceramic core and FGM face sheets under flexural loading is studied numerically via the Finite Element Method. The load is a distributed load on the upper face sheet of the beam. The elements that will be used are 8-node two-dimensional plane quadrilateral isoparametric elements. The elements used for the FGM layers, are graded finite elements where as for the core are homogeneous elements. The advantage of this study is the utilization of 8-node two-dimensional plane quadrilateral graded isoparametric elements in the FGM face sheets, for the study of a sandwich beam. The effect of the volume fraction index on the stress fields is examined.

Size-dependent vibration response of porous graded nanostructure with FEM and nonlocal continuum model

In the present paper, a refined trigonometric higher-order shear deformation theory has been presented with the conjunction of nonlocal theory for the vibrational response of functionally graded (FG) porous nanoplate. The displacement field is chosen based on assumptions that the out of the plane displacement consists of bending and shear components whereas the transverse shear-strain has nonlinear variation along the thickness direction. The number of unknown variables is four, as against five in other renowned shear deformation theories. The governing equations have been derived using Hamilton's principle. A generalized porosity model has also been developed to accommodate both even and uneven type of distribution of porosity in the FG nanoplates. The closed-form solution of simply-supported FG porous nanoplates is obtained and the results are compared with the available reported results. In finite element solution, a C0 continuous isoparametric quadrilateral element has been used with various conventional and unconventional boundary conditions. The effects of various parameters like small-scale effect, aspect ratio, volume fraction index, porosity volume fraction, and thickness ratio have been investigated. The significant influence of small-scale effects and porosity inclusions have been observed in the reported results. It has been reported that both closed-form and finite element solutions with the present theory can make accurate predictions of the free vibration response.

Isoparametric Element Analysis of Two-Dimensional Unsaturated Transient Seepage

A FEM for unsaturated transient seepage is established by using a quadrilateral isoparametric element, considering the fact that the main permeability does not coincide with the axis situation. It creates a function by using the element’s node hydraulic head and shape function instead of the real head in the Richard seepage control equation. With the help of the Galerkin weighted residual method, a FEM equation is given for analyzing 2-dimensional transient seepage problem. Further, based on the Jacobi matrix and Gauss numerical integral, it determines the elements of stiffness and capacitance matrices. This FEM equation considers not only the anisotropic of soil but also the uncoincidence between permeability and the axis. It is a common form of transient seepage. In the end, two examples illustrate the node accuracy of the quadrilateral element and the correctness of this FEM equation.

Thermal buckling strength of smart nanotube-reinforced doubly curved hybrid composite panels

The eigenvalue buckling responses of smart carbon nanotube-reinforced hybrid composite shell structure are analyzed under the influence of uniform thermal loading using a multiscale material model. The hybrid nanotube shell structural model is formulated mathematically via a cubic-order shear deformation theory introducing the material nonlinearity due to the shape memory alloy fiber. Additionally, the nanotube-reinforced composite properties are evaluated via two material modeling techniques (Mori–Tanaka technique and rule of mixture) considering the variable scale effect due to hybridization. The final form of the eigenvalue buckling equation is obtained via Hamilton’s principle (a dynamic version of the variational technique) including the temperature-dependent properties and thermal loading. The structural model is derived considering the distortion due to the in-plane thermal loading via the generic type of strain kinematics, i.e. Green–Lagrange nonlinearity. The thermal load values are predicted further by solving the derived eigenvalue equation using a nine-node isoparametric quadrilateral element from the finite element technique. The novelty of this research is that first time the shape memory alloy type functional material has been introduced with nanotube-reinforced composite shell structure to highlight the shape memory effect on the improvement of thermal buckling temperature. The derived numerical model is engaged further to solve varieties of examples for comprehensive testing (accuracy and reliability). Finally, a series of parametric analyzes has been performed for different design considerations associated with geometry as well as the material to show the model applicability.

Free vibration analysis of laminated composite hybrid and GFRP shells based on higher order zigzag theory with experimental validation

Free vibration response behavior of laminated composite hybrid and Glass Fiber Reinforced Plastic (GFRP) shells is analyzed using a modified higher order zigzag theory (HOZT). The laminated shell model is developed by considering the thickness co-ordinate to radius of curvature ratio (z/R) as well as zero shear conditions at free surfaces of the shell. The C0 finite element formulations with nine-noded quadrilateral isoparametric shell element having seven degrees of freedom per node are adopted to compute the natural frequency of the shells. The hybrid laminate is modeled by placing carbon fibers in the outermost layers and glass fibers in the rest to improve the flexural behavior of the shell. Experimental studies are performed to find out the mechanical properties of hybrid and GFRP laminates such as Young modulus, shear modulus and poison's ratio. The experimental procedure of free vibration is carried out by Bruel & Kjaer (B&K) modal analysis instruments along with RT Pro modal analysis software and the results are compared with the theoretical findings. Moreover the present theoretical results have been examined by comparing with 2D solutions as well as 3D elasticity solutions available in the published literature to verify the accuracy of the present HOZT. An extensive parametric study on laminated composite hybrid and GFRP shells with various laminations is undertaken to investigate the effect of reinforcement scheme on the free vibration behavior of such laminated shells.

Thermal frequency analysis of FG sandwich structure under variable temperature loading

The thermal eigenvalue responses of the graded sandwich shell structure are evaluated numerically under the variable thermal loadings considering the temperature-dependent properties. The polynomial type rule-based sandwich panel model is derived using higher-order type kinematics considering the shear deformation in the framework of the equivalent single-layer theory. The frequency values are computed through an own home-made computer code (MATLAB environment) prepared using the finite element type higher-order formulation. The sandwich face-sheets and the metal core are discretized via isoparametric quadrilateral Lagrangian element. The model convergence is checked by solving the similar type published numerical examples in the open domain and extended for the comparison of natural frequencies to have the final confirmation of the model accuracy. Also, the influence of each variable structural parameter, i.e. the curvature ratios, core-face thickness ratios, end-support conditions, the power-law indices and sandwich types (symmetrical and unsymmetrical) on the thermal frequencies of FG sandwich curved shell panel model. The solutions are helping to bring out the necessary influence of one or more parameters on the frequencies. The effects of individual and the combined parameters as well as the temperature profiles (uniform, linear and nonlinear) are examined through several numerical examples, which affect the structural strength/stiffness values. The present study may help in designing the future graded structures which are under the influence of the variable temperature loading.

Dynamic plastic impact behavior of CNTs/fiber/polymer multiscale laminated composite doubly curved shells

In the present paper, the dynamic elastoplastic response of carbon nanotube (CNT) fiber/polymer multiscale laminated composite (CNTFPC) doubly curved shell subjected to low-velocity impact of the projectile is investigated. For this purpose, the effective material properties of the multiphase composite are calculated by combining the Halpin-Tsai model and a micromechanical approach that strengths are estimated by an empirical equation and modified through a mixture rule. The elastoplastic constitutive equations are obtained incrementally using Hoffman’s criterion and isotropic hardening model. Also, the Green-Lagrange type of nonlinear kinematics is extracted in terms of displacement terms based on Reddy’s higher-order shear deformation theory for the CNTFPC shell. All incremental relations are regulated in the framework of the updated Lagrangian formulations to solve by an iterative finite element procedure through conforming isoparametric quadrilateral elements. Moreover, an elastoplastic Brake’s contact model is developed to determine truly the contact force between the CNTFPC plate and impactor. Temporal discretization is done by the Newmark’s average acceleration scheme. The results are validated with the data available in the literature to demonstrate the accuracy of the model. A parametric study was carried out to analysis stress and investigate the yield conditions and the effects of plastic deformation, weight percentage of CNTs, fiber volume fraction and curvature radius on the contact force history, indentation, permanent indentation depth, central deflection. Numerical results show that in the low velocity impact analysis, the stresses occurring in the shell for the areas further away from the contact area are also yielded.

On the Computational Precision of Finite Element Algorithms in Slope Stability Problems

Although the finite element method (FEM) has been used extensively to analyse the slope stability problems, the computational precision and definition of failure are still two main key concepts of finite element algorithms that attract the attention of researchers. In this paper, the modified Euler algorithm and the explicit modified Euler algorithm with stress corrections are used to analyse two dimensional (2D) slope stability problems with the associated flow rule, based on the shear strength reduction method. The rounded hyperbolic Mohr-Coulomb (M-C) yield surface is applied. Effects of the element type and various definitions of failure on the computational precision of 2D slope stability problems are evaluated. Conclusions can be drawn that the modified Euler scheme is applicable when the factor of safety (FOS) is small; however, the explicit modified Euler algorithm with stress corrections is more precise if the factor of safety is relatively large. The fully integrated quadrilateral isoparametric element is better than the triangular element in terms of the precision. With respect to the definition of failure, the displacement mutation of the characteristic point combining with the continuums of the plastic zone can be regarded as a reliable definition of failure and can be widely used to perform and analyse numerical simulations of slope stability problems.

Study on transient heat flux intensity factor with interaction integral

Aim at determining of the heat flux intensity factor (HFIF) near a crack tip in structures under transient heat flux load, the transient JT integral is defined firstly and proved to be path-independent. Based on the path-independence of the transient JT integral, a transient interaction integral utilized to extract the transient HFIF is established by introducing a known auxiliary field. A singular isoparametric quadrilateral element with eight nodes, which is used to discretize the structure around the crack tip, by moving the midside nodes to one-fourth of the sides is presented. With this singular element, the 1/r singularity of heat flux near the crack tip can be exactly depicted. Numerical examples are carried out to verify the path-independence of the transient JT integral and the interaction integral method to extract the transient HFIF. Moreover, the transient HFIF in a structure with multiple cracks are numerically investigated by means of the presented method. The method proposed in this paper can be extended to investigate the thermo-mechanical problems.

Anisotropic interpolation error estimate for arbitrary quadrilateral isoparametric elements

The aim of this paper is to show that, for any $$p \in [1,\infty )$$ , the $$W^{1,p}$$ -anisotropic interpolation error estimate holds on quadrilateral isoparametric elements verifying the maximum angle condition (MAC) and the property of comparable lengths for opposite sides (clos), i.e., on all those quadrilaterals with interior angles uniformly bounded away from $$\pi $$ and with both pairs of opposite sides having comparable lengths. For rectangular elements our interpolation error estimate agrees with the usual one whereas for perturbations of rectangles (the most general quadrilateral elements previously considered as far as we know) our result has some advantages with respect to the pre-existing ones: the interpolation error estimate that we proved is written by using two neighboring sides of the element instead of the sides of the unknown perturbed rectangle and, on the other hand, conditions MAC and clos are much simpler requirements to verify and with a clearer geometrical sense than those involved in the definition of perturbations of a rectangle.

A novel element differential method for solid mechanical problems using isoparametric triangular and tetrahedral elements

A novel strong form numerical method, Element Differential Method (EDM), is developed to solve geometrically complex mechanics problems based on triangular or tetrahedral meshes. The discretization of the structure under investigation has been based on Lagrange isoparametric quadrilateral or hexahedral elements while applying EDM. In this paper, a new family of isoparametric triangular and tetrahedral elements with a central node is proposed for EDM. A set of shape functions with analytical expressions for their first and second order partial derivatives is constructed for these triangular and tetrahedral elements, respectively. Moreover, a new element collocation scheme is proposed to establish a system of equations directly from the governing differential equations for internal nodes and traction-equilibrium equations for nodes on edges of an element. In this collocation scheme, no variational principles or virtual energy principles are required to set up the solution scheme, while no integration is needed when forming the coefficients of the system of equations. Numerical examples including standard patch tests and more practical problems are given to demonstrate the correctness of the constructed elements and the efficiency of the proposed element collocation method.

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.

Finite Element Analysis of Rectangular Beams

This paper is concerned with the analysis of simply supported beam using MATLAB programming language and structural analysis program SAP2000. The beam was discretized into rectangular elements using finite element method. Three patterns of different dimensions and numbers of rectangular elements were used to verify the results of vertical displacements and stresses obtained by MATLAB and SAP 2000.The development of four noded isoparametric quadrilateral membrane elements in MATLAB programming language is presented. The membrane elements developed are plane strain condition. The considered patterns were analyzed as shell elements using SAP2000. A finite element program is also developed using MATLAB to check the accuracy of the developed elements.