Isoparametric Quadrilateral Element Research Articles

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.

Deformation characteristics of sinusoidally-corrugated laminated composite panel – A higher-order finite element approach

In this article, the deformation characteristics of sinusoidally-corrugated laminated composite panels are examined under uniform and sinusoidally distributed transverse loads. The sinusoidal corrugation in laminate structure is achieved using the general curvature method. The displacement field is based on the third-order shear deformation mid-plane kinematics theory with nine degrees-of-freedom. The minimum total potential energy method is employed to obtain the governing equation. The solutions are obtained using two-dimensional finite element approximation via nine-noded isoparametric quadrilateral Lagrangian element. The convergence and comparison studies reveal the consistency and accuracy of present model. The new numerical illustrations reflect the significance of corrugation ratios, edge constraints, side-to-thickness ratios and aspect ratios on the deformation behavior of the corrugated and non-corrugated laminated composite panels.

A Three-Node Triangular Element with Continuous Nodal Stress (Trig3-CNS) for Geometry Nonlinear Solid Mechanics Problems

This paper investigates the performance of the three-node triangular element with continuous nodal stress (Trig3-CNS) for geometry nonlinear solid mechanic problems. This Trig3-CNS element was recently proposed to improve accuracy of the finite element method (FEM). By synergizing the individual strengths of meshfree method and FEM, the Trig3-CNS element achieves higher accuracy and convergence rate. Furthermore, Trig3-CNS presents high tolerance to mesh distortion. Therefore, it is potentially useful for geometry nonlinear solid mechanics problems in which mesh distortion takes place. Compared with the traditional hybrid “FE-Meshfree” elements, Trig3-CNS naturally processes CNS without requiring any extra operation in post-processing. Numerical tests in the present work show that for geometry nonlinear analysis, the results of the Trig3-CNS element are better than the 3-node triangular element (Trig3) and 4-node isoparametric quadrilateral element (Quad4). In addition, the performance of Trig3-CNS is comparable to that of traditional hybrid “FE-Meshfree” elements.

Comparison between the stiffness method and the hybrid method applied to a circular ring

This study deals with the comparison of stresses and displacements of a circular ring obtained by two numerical formulations, namely the stiffness method and the hybrid method, applied to isoparametric quadrilateral elements. After explaining the formulation of the hybrid method in finite elements, an orthogonalization method proposed previously for hybrid elements is applied. Then, the computational cost per element of the stiffness method and of the hybrid method with and without orthogonalization has been evaluated for the first time. A circular ring loaded by two opposing forces is analyzed in order to compare the solution obtained by the hybrid method and the stiffness method with experimental and analytical results. The agreement between analytical and experimental results with those numerical is better in the case of the hybrid method than in the case of the stiffness method. It is observed that the element ratio needed to obtain a given relative error is one magnitude order greater in the stiffness method than in the case of the hybrid method.

FE sloshing modelling in bidimensional cavity using wave equation

Tuned liquid column damper (TLCD) is a kind of passive damper that use water sloshing motion to control structural vibration in tall buildings, wind turbines and ships. This paper presents finite element analysis of sloshing in 2D water tank using a pressure based Eulerian approach. The fluid domain is discretised by an isoparametric quadrilateral finite element. The cavity was modelled with rigid contours and free surface in order to study the decoupled liquid reservoir. Next, the cavity was coupled to a flexible structure with free vibration and forced vibration. The numerical results obtained by the created program were validated using experimental results [1] presenting acceptable errors with less than 5k elements.

Numerical Analysis of Transient Responses of Delaminated Layered Structure Using Different Mid-plane Theories and Experimental Validation

The present article mainly focuses on the fundamental frequency and the time-dependent deflection responses of the layered composite plate structure considering the effect of debonding between the adjacent layers introduced via sub-laminate approach. The responses are mainly obtained numerically using a newly developed finite element model of the delaminated composite plate using two types of higher-order mid-plane shear deformable kinematics. The debonded plate model is discretized using a nine-noded quadrilateral isoparametric Lagrangian element to obtain the elemental equation and assembled subsequently for the evaluation of the final form of equation of motion. The structural responses (natural frequency and time-dependent deflection) are obtained computationally using a specialized computer code via MATLAB with the help of currently proposed model. Further, the convergence and subsequent comparison study have been performed using the similar examples from the earlier published literature. In addition, the present results are also compared with those of the experimental responses of the fabricated glass-/epoxy-laminated composite plate including the different size of delamination. The elastic material properties and the frequency responses of the fabricated laminate are evaluated using tensile test and vibration measuring experimental set-up. Finally, the dynamic responses of debonded plate have been measured for different sets of structural parameters to show the influence of each parameter as well as the size of debonding on the final responses.

Four-Node Quadrilateral Element with Continuous Nodal Stress for Geometrical Nonlinear Analysis

In this paper, the performance of a hybrid ‘FE-Meshfree’ quadrilateral element with continuous nodal stress (Quad4-CNS) is investigated for geometrical nonlinear solid mechanic problems. By combining finite element method (FEM) and meshfree method, this Quad4-CNS synergizes the individual strengths of these two methods, which leads to higher accuracy, better convergence rate, as well as high tolerance to mesh distortion. Therefore, Quad4-CNS is attractive for geometrical nonlinear solid mechanic problems where excessive distorted meshes occur. For geometrical nonlinear analysis, numerical results show that the results of Quad4-CNS element are much better than those of four-node isoparametric quadrilateral element (Quad4), and are comparable to quadratic quadrilateral element (Quad8) and other hybrid ‘FE- Meshfree’ elements.

A partition-of-unity based three-node triangular element with continuous nodal stress using radial-polynomial basis functions

A partition-of-unity (PU) based “FE-Meshfree” three-node triangular element (Trig3-RPIM) was recently developed for linear elastic problems. This Trig3-RPIM element employs hybrid shape functions that combine the shape functions of three-node triangular element (Trig3) and radial-polynomial basis functions for the purpose of synergizing the merits of both finite element method and meshfree method. Although Trig3-RPIM element is capable of obtaining higher accuracy and convergence rate than the Trig3 element and four-node iso-parametric quadrilateral element without adding extra nodes or degrees of freedom (DOFs), the nodal stress field through Trig3-RPIM element is not continuous and extra stress smooth operations are still needed in the post processing stage. To further improve the property of Trig3-RPIM element, a new PU-based triangular element with continuous nodal stress, called Trig3-RPIMcns, is developed. Numerical examples including several linear, free vibration and forced vibration test problems, have confirmed the correctness and feasibility of the proposed Trig3-RPIMcns element.

Free transverse vibration of shear deformable super-elliptical plates

Free transverse vibration of shear deformable super-elliptical plates with uniform thickness was studied based on Mindlin plate theory using finite element method. Quadrilateral isoparametric elements were used in the paper. Sensitivity analysis was made to determine the influence of the thickness, the aspect ratio, and the shape of the plate on the natural frequency. Accuracy of the results computed in the current study was validated by comparing them with the solutions available in the literature. The results reveal that the frequencies of clamped super-elliptical plates lie in the range bounded by elliptical and rectangular plates irrespective of the aspect ratio, and furthermore, the frequency decreases if the super-elliptical power increases. A similar trend was observed for simply supported plates with high aspect ratio. The free vibration response for the first and the second symmetric-antisymmetric (SA) modes were found to be different for high aspect ratio. The results reveal that using insufficient number of degrees of freedom results in finding a totally different relation between the super-elliptical power and the frequency.

Application of the ‘FE-Meshfree’ QUAD4 with continuous nodal stress using radial-polynomial basis functions for vibration and geometric nonlinear analyses

A hybrid ‘FE-Meshfree’ four-node quadrilateral element with continuous nodal stress using radial-polynomial basis functions (Quad4-RPIMcns), was recently proposed for static analysis. The Quad4-RPIMcns element can be considered as a development of the previous partition-of-unity (PU) based ‘FE-Meshfree’ QUAD4 element (Quad4-RPIM) which uses FE shape functions to construct the PU and radial-polynomial basis functions to construct the local approximation (LA), so as to synergize the individual strengths of finite element and meshfree methods. As a result, high order global approximations in Quad4-RPIMcns element could be easily constructed without adding extra nodes and DOFs, thereby achieving high accuracy and convergence rate. In this paper, the element is further applied to conduct free vibration, forced vibration and geometric nonlinear analyses of two-dimensional solids. Several numerical test problems are solved and the performance of the element is compared with that of the three-node triangular element (Trig3) and four-node isoparametric quadrilateral element (Quad4). Numerical results show that Quad4-RPIMcns element has higher tolerance to mesh distortion and gives more accurate solution as compared to Trig3 and Quad4 elements.

Piecewise shear deformation theory and finite element formulation for vibration analysis of laminated composite and sandwich plates in thermal environments

In this paper, a piecewise shear deformation theory for laminated composite and sandwich plates is presented by integrating the advantages of the layerwise theory and equivalent single layer theory. A C0 continuous four-noded quadrilateral isoparametric plate element based on this theory is developed for free and forced vibration analysis of laminated composite and sandwich plates in thermal environments. The accuracy and effectiveness of piecewise shear deformation theory and finite element formulation are validated by comparing the numerical results obtained by the present finite element formulation with the analytical and exact results available in literatures as well as the numerical results computed by MSC.Nastran software. It is demonstrated that the piecewise shear deformation theory and finite element formulation are suitable for the vibration analysis of thin and thick laminated composite and sandwich plates. As compared with the layerwise theory, high-order zigzag theory and high-order equivalent single layer theory, the piecewise shear deformation theory is able to produce a sufficiently accurate result at a very low computational cost. The work reported in this paper provides an efficient modeling approach for thermal vibration analysis of laminated composite and sandwich plates in practical engineering.

A four-node quadrilateral element fitted to numerical manifold method with continuous nodal stress for crack analysis

Formulations of a four-node quadrilateral element fitted to numerical manifold method (NMM) with continuous nodal stress called Quad4-CNS (NMM) element for two-dimensional (2D) crack analysis are presented. This Quad4-CNS (NMM) element can be considered as a development of the recently published four-node quadrilateral element with continuous nodal stress (Quad4-CNS). In contrast to the four-node iso-parametric quadrilateral element (Quad4), the Quad4-CNS element has higher order of global approximations, much better accuracy and continuous nodal stress. Moreover, it is free from the “linear dependence” which otherwise cripples many of the partition of unity (PU) based methods with high order global approximations. Due to the adoption of two cover systems, namely, the mathematical cover and physical cover, the NMM is capable of solving continuous and discontinuous problems in a unified way. The purpose of this paper is to synergize the advantages of both the Quad4-CNS element and the NMM to precisely model linear elastic fracture problems. A number of numerical examples indicate the accuracy and robustness of the present Quad4-CNS (NMM) element.

Nonlinear transient analysis of smart laminated composite plate integrated with PVDF sensor and AFC actuator

In this article, the geometrically nonlinear transient response of the smart laminated composite plate is investigated under the coupled electromechanical load. The plate has been modeled using the higher-order kinematic model in conjunction with Green-Lagrange nonlinear strains. The electric potential across the sensor/actuator layer is modeled as the quadratic function of the thickness coordinate. The domain is discretized with the help of a nine noded isoparametric quadrilateral element with thirteen degrees of freedom per node. The desired transient responses are obtained using the direct iterative method in association with Newmark’s time integration scheme. In order to achieve distributed actuator and the sensor behaviour the active fibre composite and Polyvinylidene fluoride fibre reinforced composite are utilized along the poled fibre direction and through the thickness, respectively in this present investigation. In this analysis, two types of configurations of the sensor and actuator (collocated and non-collocated) are considered to compute the active stiffening responses. The convergence and the validity of the present numerical model have been checked. Finally, the effects of various geometrical parameters along with the sensor and the actuator arrangements on the nonlinear transient responses of the laminated composite plate are examined by solving various numerical examples.

IMPLEMENTATIONS OF BOUNDARY–DOMAIN INTEGRO-DIFFERENTIAL EQUATION FOR DIRICHLET BVP WITH VARIABLE COEFFICIENT

In this paper, we present the numerical results of the Boundary-Domain Integro-Differential Equation (BDIDE) associated to Dirichlet problem for an elliptic type Partial Differential Equation (PDE) with a variable coefficient. The numerical constructions are based on discretizing the boundary of the problem region by utilizing continuous linear iso-parametric elements while the domain of the problem region is meshed by using iso-parametric quadrilateral bilinear domain elements. We also use a semi-analytic method to handle the integration that exhibits logarithmic singularity instead of using Gauss-Laguare quadrature formula. The numerical results that employed the semi-analytic method give better accuracy as compared to those when we use Gauss-Laguerre quadrature formula. The system of equations that obtained by the discretized BDIDE is solved by an iterative method (Neumann series expansion) as well as a direct method (LU decomposition method). From our numerical experiments on all test domains, the relative errors of the solutions when applying semi-analytic method are smaller than when we use Gauss-Laguerre quadrature formula for the integration with logarithmic singularity. Unlike Dirichlet Boundary Integral Equation (BIE), the spectral properties of the Dirichlet BDIDE is not known. The Neumann iterations will converge to the solution if and only if the spectral radius of matrix operator is less than 1. In our numerical experiment on all the test domains, the Neumann series does converge. It gives some conclusions for the spectral properties of the Dirichlet BDIDE even though more experiments on the general Dirichlet problems need to be carried out.

An Enhanced Tire Model for Dynamic Simulation based on Geometrically Exact Shells

Abstract In the present work, a tire model is derived based on geometrically exact shells. The discretization is done with the help of isoparametric quadrilateral finite elements. The interpolation is performed with bilinear Lagrangian polynomials for the mid-surface as well as for the director field. As time stepping method for the resulting differential algebraic equation a backward differentiation formula is chosen. A multilayer material model for geometrically exact shells is introduced, to describe the anisotropic behavior of the tire material. To handle the interaction with a rigid road surface, a unilateral frictional contact formulation is introduced. Therein a special surface to surface contact element is developed, which rebuilds the shape of the tire.

Free and forced vibration analyses using the four-node quadrilateral element with continuous nodal stress

The recently published four-node quadrilateral element with continuous nodal stress (Quad4-CNS) is extended to free and forced vibration analyses of two-dimensional solids. The Quad4-CNS element can be regarded as a partition-of-unity (PU) based ‘FE-Meshfree’ element which inherits better accuracy, higher convergence rate, and high tolerance to mesh distortion from the meshfree methods, while preserving the Kronecker-delta property of the finite element method (FEM). Moreover, the Quad4-CNS element is free from the linear dependence problem which otherwise cripples many of the PU based finite elements. Several free and forced vibration problems are solved and the performance of the element is compared with that of the four-node isoparametric quadrilateral element (Quad4) and eight-node isoparametric quadrilateral element (Quad8). The results show that, for regular meshes, the performance of the element is superior to that of Quad4 element, and comparable to that of Quad8 element. For distorted meshes, the present element has better mesh-distortion tolerance than Quad4 and Quad8 elements.

Development of a 2D isoparametric finite element model based on the layerwise approach for the bending analysis of sandwich plates

The aim of this work is the development of a 2D quadrilateral isoparametric finite element model, based on a layerwise approach, for the bending analysis of sandwich plates. The face sheets and the core are modeled individually using, respectively, the first order shear deformation theory and the third-order plate theory. The displacement continuity condition at the interfaces 'face sheets-core' is satisfied. The assumed natural strains method is introduced to avoid an eventual shear locking phenomenon. The developed element is a four-nodded isoparametric element with fifty two degrees-of-freedom (52 DOF). Each face sheet has only two rotational DOF per node and the core has nine DOF per node: six rotational degrees and three translation components which are common for the all sandwich layers. The performance of the proposed element model is assessed by six examples, considering symmetric/unsymmetric composite sandwich plates with different aspect ratios, loadings and boundary conditions. The numerical results obtained are compared with the analytical solutions and the numerical results obtained by other authors. The results indicate that the proposed element model is promising in terms of the accuracy and the convergence speed for both thin and thick plates.

A partition-of-unity based ‘FE-Meshfree’ triangular element with radial-polynomial basis functions for static and free vibration analysis

A new ‘FE-Meshfree’ three-node triangular element (Trig3-RPIM) is developed based on the partition of unity (PU) concept. The Trig3-RPIM element employs the shape function of classical three-node triangular element (Trig3) to construct the PU and the radial-polynomial basis function which is free from the possible singularity of the moment matrix to construct the nodal approximation. The Trig3-RPIM element synergizes the individual strengths of finite element method and meshfree method. Moreover, it is free from the linear dependence problem which otherwise cripples many of the PU based finite elements. Several linear, nonlinear and free vibration test problems are solved and the performance of the element is compared with those of the well-known three-node triangular element (Trig3) and four-node iso-parametric quadrilateral element (Quad4). The results show that, for regular meshes, the performance of the element is superior to those of Trig3 and Quad4 elements. For distorted meshes, the present element has better mesh-distortion tolerance than Trig3 and Quad4 elements.