Quadratic Quadrilateral Elements Research Articles

Static Response of Functionally Graded Porous Circular Plates via Finite Element Method

AbstractThe main purpose of this paper is to investigate the axisymmetric bending response of functionally graded porous (FGP) circular plates. The material properties are changed continuously in the thickness direction of the plate. Three distinct porosity distributions uniform, symmetric and monolithic are employed. The effect of porosity on the axisymmetric bending analysis of circular plates is examined parametrically. In this study, clamped and roller supports which commonly serve to achieve ideal boundary conditions in numerous engineering applications are used. The finite element method is employed for numerical analysis. The principal of the potential energy is used to obtain the governing equations. To generate the model of the FGP circular plates, an eight-node quadratic quadrilateral element with two degrees of freedom on each node is utilized. The results of this study are confirmed by the existing published literature. A good agreement between the results of the presented model and the previous literature has been observed. The results of the present study show that plate deflection increases with the increase of the porosity coefficient and the ratio of radius to thickness of circular plates. By increasing the porosity coefficient, the displacement values of the plates made of uniform porosity distribution is effected more than those of other porosity distributions.

Buckling optimization of variable-stiffness composite plates via variable stiffness optimization algorithm

A variable stiffness optimization (VSO) algorithm is presented for optimizing variable-stiffness composite (VSC) plates with linear fiber path functions. A new definition of lamination parameters characterizing the distances between stacking configurations of VSC plates is presented. Using the inherent sensitivity of bending stiffness of composite laminates, the concept of “through thickness design” is introduced. The first step is to determine a good initial point for the VSC plates using three-dimensional sampling optimization (3DSO). The fiber angle design variables are then categorized into three groups. By optimizing the three groups' design variables sequentially and iteratively, the stiffness of the VSC plates is stiffened part by part until an optimum is reached. Lastly, the obtained optimum is redesigned accounting for the curvature constraint. The finite element method (FEM) is developed for the buckling analysis of VSC plates, and both Q4 and Q8 elements are employed to verify the accuracy and convergence of the FEM. Under a variety of boundary conditions and loading cases, FEM and VSO algorithm are used to maximize the buckling load of square and rectangular symmetrical VSC plates. Optimal results are compared with those available in the literature, demonstrating the effectiveness, robustness, and efficiency of the VSO algorithm.

Numerical analysis of the magnetic shape memory effect based on the absolute nodal coordinate formulation

In this paper, the absolute nodal coordinate formulation (ANCF) is applied to simulate the magnetic shape memory effect. Using the absolute nodal coordinate formulation makes it possible to describe complicated or large deformation cases. The nonlinear bidirectional coupling terms between the mechanical and magnetic field are taken into account in the analysis of the single-crystalline Ni-Mn-Ga sample. A two-loop iteration procedure with variable steps is implemented to predict the magnetic-field-induced strain (MFIS) in the specimen under a changing external magnetic field and a constant auxiliary compression. In addition, the proposed approach is used to track the superelastic behavior of the magnetic shape memory alloy when subjected to a constant magnetic field. The approach effectively describes the hysteresis and superelastic phenomenon of the shape memory effect. The solution is compared here with solutions obtained using classical linear and quadratic quadrilateral elements. The deviation observed in the solution is discussed, and its cause is further clarified from a two-domain pure magnetostatic analysis of a permanent magnet. It is found that the accurate solution of such problems is associated with discontinuity of the normal component of the magnetic potential gradient across the domain interface. Special measures must be taken to make the absolute nodal coordinate formulation element compatible with the discontinuity. A mixed FEs strategy, which adopts ANCF FE in the displacement field solver and classical FE in the magnetic field solver, is proposed as an alternative option to rectify the problem, which is verified by predicting the MFIS and the dynamic mechanical response of a sample under cyclic compression.

Stiffness Optimization Through a Modified Greedy Algorithm

Due to the 3D printing techniques, it is possible to create and design components with soft and rigid combinations of materials with different elastic properties which may satisfy multiple material requirements. A soft (blue) and a rigid (purple) material with moduli of elasticity in a ratio 1:10 can be innkjet printed together following an initial pattern. The objective function was formulated as to maximize the effective stiffness on both in-plane directions. Finite element analyses were performed by using the PyAnsys software under an MIT License in a domain with 16 Q8 elements which represents by symmetry a representative volume element (RVE). The objective function to be maximized can also reduce the anisotropy of the printed topology. We proved that the formulation of three objective functions affects the results obtained with a Greedy algorithm.

A simulated annealing algorithm for stiffness optimization

A domain of 1x1 mm was chosen and divided in 4x4 squares on each side. A soft (purple) and a rigid (blue) material distribution with an elastic modulus ratio of 1:10 can be inkjet printed together in equal parts, following an initial pattern. The simulated annealing algorithm used in this work is similar to the Greedy algorithm but accepts less favorable solutions at a current iteration to make possible further calculations. The objective function is formulated as maximizing the effective stiffness on both in-plane directions. Finite element analyses are performed by using the PyAnsys software under an MIT License and a domain with 16 Q8 elements which represents by symmetry a quarter of RVE. The simulated annealing algorithm provided better results than the Greedy algorithm in all analyzed cases, but it does not always converge to a global optimum.

Accurate and efficient thermal stress analyses of functionally graded solids using incompatible graded finite elements

Functionally graded materials have found a wide usage in high temperature applications. The smooth transition from one material to another, in graded materials, may reduce thermal stresses, residual stresses and stress concentration factors as well as utilize properties of both materials. To perform accurate and efficient finite element analysis for heat transfer and transient thermal stress analyses in two-dimensional functionally graded materials, incompatible graded finite elements are developed and verified. User-defined subroutines in ABAQUS are developed to address the gradation of material properties within an element. An emphasis is made on an incompatible six-node graded finite element (QM6) which is accurate and efficient compared to linear four-node (Q4) and quadratic eight-node (Q8) elements. With the help of posteriori error estimation, a critical comparison is made among three types of solid elements. Modified 6-node (QM6) incompatible graded elements provide better accuracy than Q4 elements and take less computational time than Q8 elements, thereby showing QM6 as an optimal element for engineering analysis.

Nodal integral method for convection-diffusion transport using linear and higher order quadrilateral elements

Generic nodal integral method (NIM) based scheme, utilizing nine noded 2D quadratic elements along with four noded 2D linear elements, is developed to solve the fluid flow and heat transfer equations in complex geometries. Non-linear (quadratic) quadrilateral elements are used for discretization of boundary region and linear quadrilateral elements are used for interior region. Lagrange interpolation functions are used to map both type of elements to corresponding square computational elements. The scheme for Neumann and mixed type of boundary conditions are also developed for quadratic elements. C1 type continuity condition is imposed at the interfaces of adjacent elements. Numerical results are compared with analytical solutions for diffusion and advection-diffusion equations. Results for Navier–Stokes equations in curved domain are compared with previously reported experimental as well as numerical results. The comparative study has also been done between presently developed scheme using quadratic and linear elements (referred as scheme-1) and scheme based on complete discretization with linear elements (referred as scheme-2). The comparison shows that both the schemes are of nearly second order accurate while the scheme-1 is more accurate than scheme-2. The results show that the efficient mapping of curved surface with quadratic elements improves the accuracy of NIM schemes.

Numerical simulation of two-dimensional laminar unsteady flow past a right trapezoidal cylinder at low Reynolds number: study of sharpening angle, time step, grid independence and domain size

Numerical simulation of two-dimensional laminar unsteady flow past a right trapezoidal cylinder at low Reynolds number (Re = 100), zero of the flow approaching angle and sharpening angle of the right trapezoidal of 22.5° with a side ratio B/A = 1 are carried out to provide moreapplicable data for engineering design of barred tee in aspect of structural integrity. A finite volume method, non-uniform meshing with second-order implicit time discretization into eight-node quadratic quadrilateral finite elements is employed. An incompressible flow SIMPLEC code with constant fluid properties is used. The convective terms using a third-order QUICK scheme. The numerical simulation result is compared against the published results of flow past a square cylinder. The effect of sharpening angle on the response of the right trapezoidal cylinder is investigated. A special study of the effects of flow on significant factor for time step, grid independence, blockage ratio, domain size, upstream and downstream extents, size domain next to cylinder and size domain extent are performed systematically. The Strouhal number and RMS lift coefficients of fully saturated flow are calculated. The result shown that increasing of sharpening angle, the Strouhal number is negligible changed whilst the RMS lift coefficients significantly increased.

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.

An efficient multi-stage optimization approach for damage detection in plate structures

The paper presents an efficient multi-stage optimization approach for damage detection in plate-like structures. In this approach, the damage identification process is achieved by minimizing an objective function established via flexibility changes of the structure. The vector of design variables represents correspondingly the damage extent of elements discretized by the finite element model. For analyzing the response of plate structures, the finite element model using 9-node quadratic quadrilateral elements is applied. For solving the optimization problem, a modified differential evolution (MDE) algorithm, which can help enhance the balance of global and local searches in each generation, is used for many stages of damage detection, in which the low damage variables in each stage are gradually eliminated after several generations to reduce the dimension of searching space and to increase the convergence rate of the problem. The efficiency of the proposed method is investigated through two numerical examples for isotropic and laminated composite plates. The obtained results indicate that the proposed method not only successfully detects the location and severity of multi-damage cases in the plate structures, but also show the better efficiency in term of computational cost.

Least-squares finite element solution of Euler equations with H-type mesh refinement and coarsening on triangular elements

Purpose – The purpose of this paper is to demonstrate successful use of least-squares finite element method (LSFEM) with h-type mesh refinement and coarsening for the solution of two-dimensional, inviscid, compressible flows. Design/methodology/approach – Unsteady Euler equations are discretized on meshes of linear and quadratic triangular and quadrilateral elements using LSFEM. Backward Euler scheme is used for time discretization. For the refinement of linear triangular elements, a modified version of the simple bisection algorithm is used. Mesh coarsening is performed with the edge collapsing technique. Pressure gradient-based error estimation is used for refinement and coarsening decision. The developed solver is tested with flow over a circular bump, flow over a ramp and flow through a scramjet inlet problems. Findings – Pressure difference based error estimator, modified simple bisection method for mesh refinement and edge collapsing method for mesh coarsening are shown to work properly with the LSFEM formulation. With the proper use of mesh adaptation, time and effort necessary to prepare a good initial mesh reduces and mesh independency control of the final solution is automatically taken care of. Originality/value – LSFEM is used for the first time for the solution of inviscid compressible flows with h-type mesh refinement and coarsening on triangular elements. It is shown that, when coupled with mesh adaptation, inherent viscous dissipation of LSFEM technique is no longer an issue for accurate shock capturing without unphysical oscillations.

Super-convergent second derivative recovery for lower-order strain gradient plasticity

An element patch based super-convergent second derivative recovery technique is developed in this paper for evaluating the first and second strain derivatives for a lower-order strain gradient plasticity model. The element based patches are defined and the patch least square fitting process is introduced. A classical shear band simulation is then conducted to test the new technique with various unstructured meshes with mixed quadratic quadrilateral and triangular C0 elements. The result shows that a direct recovery from the patch least square fitting process is more reliable than that from the nodal averaging process usually necessary for the super-convergent second derivative recovery methods. The proposed technique is proven to be effective for recovering both first and second order strain derivatives and has the potential for even higher order derivative recovery.

Bar와 Beam 구조물의 기본적인 유한요소 모델의 수치해석

The finite element analysis (FEA) is a numerical technique to find solutions of field problems. A field problem is approximated by differential equations or integral expressions. In a finite element, the field quantity is allowed to have a simple spatial variation in terms of linear or polynomial functions. This paper represents a review and an accuracy-study of the finite element method comparing the FEA results with the exact solution. The exact solutions were calculated by solid mechanics and FEA using matrix stiffness method. For this study, simple bar and cantilever models were considered to evaluate four types of basic elements - constant strain triangle (CST), linear strain triangle (LST), bi-linear-rectangle(Q4),and quadratic-rectangle(Q8). The bar model was subjected to uniaxial loading whereas in case of the cantilever model moment loading was used. In the uniaxial loading case, all basic element results of the displacement and stress in x-direction agreed well with the exact solutions. In the moment loading case, the displacement in y-direction using LST and Q8 elements were acceptable compared to the exact solution, but CST and Q4 elements had to be improved by the mesh refinement.

High-Order Spectral Stochastic Finite Element Analysis of Stochastic Elliptical Partial Differential Equations

This study presents an experiment of improving the performance of spectral stochastic finite element method using high-order elements. This experiment is implemented through a two-dimensional spectral stochastic finite element formulation of an elliptic partial differential equation having stochastic coefficients. Deriving this spectral stochastic finite element formulation couples a two-dimensional deterministic finite element formulation of an elliptic partial differential equation with generalized polynomial chaos expansions of stochastic coefficients. Further inspection of the performance of resulting spectral stochastic finite element formulation with adopting linear and quadratic (9-node or 8-node) quadrilateral elements finds that more accurate standard deviations of unknowns are surprisingly predicted using quadratic quadrilateral elements, especially under high autocorrelation function values of stochastic coefficients. In addition, creating spectral stochastic finite element results using quadratic quadrilateral elements is not unacceptably time-consuming. Therefore, this study concludes that adopting high-order elements can be a lower-cost method to improve the performance of spectral stochastic finite element method.

A ‘FE-Meshfree’ TRIA3 element based on partition of unity for linear and geometry nonlinear analyses

A new 3-node triangular element is developed on the basis of partition of unity (PU) concept. The formulation employs the parametric shape functions of classical triangular element (TRIA3) to construct the PU and the least square point interpolation method to construct the local displacement approximation. The proposed element synergizes the individual merits of finite element method and meshfree method. Moreover, the usual linear dependence problem associated with PU finite elements is eliminated in the present element. Application of the element to several linear and geometric nonlinear problems shows that the proposed element gives a performance better than that of classical linear triangular as well as linear quadrilateral elements, and comparable to that of quadratic quadrilateral element. The proposed element does not necessitate a new mesh or additional nodes in the mesh. It uses the same mesh as the classical TRIA3 element and is able to give more accurate solution than the TRIA3 element.

A nine-node displacement-based finite element for Reissner–Mindlin plates based on an improved formulation of the NIPE approach

The nodally integrated plate element formulation is an assumed strain finite element technique for shear deformable plates that enforce weakly the balance and the kinematic equations. The strain–displacement operators are derived via nodal integration satisfying a priori the kinematic weighted residual statement. The present work analyzes the NIPE technique, including the new element, by testing thoughtfully the sensitivity of the elements to severe geometry distortions. We propose an improvement that confers robustness to all element shapes developed by the NIPE formulation.We present also a new nine-node NIP element configuration. The new nine-node element uses bi-quadratic interpolations of the transverse displacement and rotations and is computed by means of a nine-node quadrature rule.A brief review of the improved triangular and quadrangular NIPEs is reported for elastic plate analyses. A few challenging benchmarks carried out on extreme distorted meshes illustrate the performance of the introduced NIP-Q9 element. We detail that the new NIPE formulation confers insensitivity to extreme distortions for the quadratic quadrilateral element and allows to solve for severe distortion with the NIPE family.

2D and 3D finite element analysis of central incisor generated by computerized tomography

The purpose of this study was to compare the results of different hierarchical models in engineering analysis applied to dentistry with 2D and 3D models of a tooth and its supporting structures under 100 N occlusal loading at 45° and examine the reliability of simplified 2D models in dental research. Five models were built from computed-tomography scans: four 2D models with Plane Strain and Plane Stress State with linear triangular and quadratic quadrilateral elements and one 3D model. The finite element results indicated that the stress distribution was similar qualitatively in all models but the stress magnitude was quite different. It was concluded that 2D models are acceptable when investigating the biomechanical behavior of upper central incisor qualitatively. However, quantitative stress analysis is less reliable in 2D-finite element analysis, because 2D models overestimate the results and do not represent the complex anatomical configuration of dental structures. Therefore 3D finite element analyses of dental biomechanics cannot be simplified.

A partition of unity‐based ‘FE–Meshfree’ QUAD4 element for geometric non‐linear analysis

AbstractThe recently published ‘FE–Meshfree’ QUAD4 element is extended to geometrical non‐linear analysis. The shape functions for this element are obtained by combining meshfree and finite element shape functions. The concept of partition of unity (PU) is employed for the purpose. The new shape functions inherit their higher order completeness properties from the meshfree shape functions and the mesh‐distortion tolerant compatibility properties from the finite element (FE) shape functions. Updated Lagrangian formulation is adopted for the non‐linear solution. Several numerical example problems are solved and the performance of the element is compared with that of the well‐known Q4, QM6 and Q8 elements. The results show that, for regular meshes, the performance of the element is comparable to that of QM6 and Q8 elements, and superior to that of Q4 element. For distorted meshes, the present element has better mesh‐distortion tolerance than Q4, QM6 and Q8 elements. Copyright © 2009 John Wiley & Sons, Ltd.

BEM Solutions of Frequency Domain Gradient Elastodynamic 3-D Porblems

A boundary element methodology is presented for the frequency domain elastodynamic analysis of three-dimensional solids characterized by a linear elastic material behavior coupled with microstructural effects taken into account with the aid of the simple gradient elastic theory of Aifantis. A variational statement is established to determine all possible classical and non-classical (due to gradient terms) boundary conditions of the general boundary value problem. The gradient frequency domain elastodynamic fundamental solution is explicitly derived and used to construct the boundary integral representation of the solution with the aid of a reciprocal integral identity. In addition to a boundary integral representation for the displacement, a boundary integral representation for its normal derivative is also necessary for the complete formulation of a well posed problem. All the kernels in the integral equations are explicitly provided. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. The solution procedure is described in detail. A numerical example serves to illustrate the method and demonstrate its accuracy. The present version of the method does not provide explicit expressions for the computation of interior stresses.

Integral equation approach for 3D multiple-crack problems

In this paper, a traction integral equation containing no hypersingular integrals is presented to study the interaction of multiple cracks in an infinite elastic medium. 8-node quadratic quadrilateral elements are used to discretize general crack surfaces, and special crack tip elements are employed along surface boundaries to model the r variation of displacements near the crack fronts. Thus, the method possesses the merits of the traction integral equation without hypersingular integrals and those of the special crack tip elements for modeling r variation of displacements near the crack tips. The stress intensity factors at the crack front are evaluated using one point formulation and the results are compared with available solutions.