Isoparametric Quadrilateral Element Research Articles

Further discussion on four-node isoparametric quadrilateral elements in plane bending

The aim of the study is: first—to show that the finite element method for plane bending is not sensitive to the distorted geometry of isoparametric quadrilateral elements, second—to close out further investigations on irregular elements based on Wilson incompatible functions employing ad hoc methods to force elements to pass the patch test, regardless of the type of the variational formulation used for the problem. A systematic review of papers dealing with finite four-node plane elements with two degrees of freedom at each node is presented. Selected results of the benchmark example of the mesh distortion test used by all cited authors are compared suggesting that no improvement has been made in more than two decades of intensive development of new elements and methods in the field. An attempt has been made to localize the pathological behaviour of irregular elements under consideration by calculating the areas of the elements after deformation for different values of the distortion parameter. It made possible to observe the mass density of the whole deformed system. The tip deflections at the lower and upper edge at the free end of the cantilever beam were depicted as well, illustrating unexpected physical phenomena of the longitudinal fibers deviating from being parallel. The horizontal displacement components at two common nodes of the adjacent elements have been compared with the analytical solution of the problem for the varying distortion parameter. The resulting horizontal displacement distribution along the common side of the elements has been shown to be the source of unreal behavior of all classes of irregular elements based on Wilson incompatible functions. Increasing the order of the interpolation functions for the displacement field, i.e. application of subparametric elements, makes the finite element method insensitive to element distortion. Copyright © 2000 John Wiley & Sons, Ltd.

Further discussion on four‐node isoparametric quadrilateral elements in plane bending

The aim of the study is: first—to show that the finite element method for plane bending is not sensitive to the distorted geometry of isoparametric quadrilateral elements, second—to close out further investigations on irregular elements based on Wilson incompatible functions employing ad hoc methods to force elements to pass the patch test, regardless of the type of the variational formulation used for the problem. A systematic review of papers dealing with finite four-node plane elements with two degrees of freedom at each node is presented. Selected results of the benchmark example of the mesh distortion test used by all cited authors are compared suggesting that no improvement has been made in more than two decades of intensive development of new elements and methods in the field. An attempt has been made to localize the pathological behaviour of irregular elements under consideration by calculating the areas of the elements after deformation for different values of the distortion parameter. It made possible to observe the mass density of the whole deformed system. The tip deflections at the lower and upper edge at the free end of the cantilever beam were depicted as well, illustrating unexpected physical phenomena of the longitudinal fibers deviating from being parallel. The horizontal displacement components at two common nodes of the adjacent elements have been compared with the analytical solution of the problem for the varying distortion parameter. The resulting horizontal displacement distribution along the common side of the elements has been shown to be the source of unreal behavior of all classes of irregular elements based on Wilson incompatible functions. Increasing the order of the interpolation functions for the displacement field, i.e. application of subparametric elements, makes the finite element method insensitive to element distortion. Copyright © 2000 John Wiley & Sons, Ltd.

Eigenvalues for waveguides containing re-entrant corners by a finite-element method with superelements

Superelements have been developed to enable the finite-element method to be used for computing accurate eigenvalues of the Laplacian over domains containing re-entrant corners of arbitrary angle. A truncated asymptotic expansion of the solution is employed in the region of the corner and linear blending is used over the remainder of the superelement to provide a smooth transition to piecewise quadratics over the superelement boundary. The superelement thus conforms with the usual triangular or quadrilateral isoparametric elements used over the remainder of the domain, and can be easily incorporated into a general finite-element program. The scheme has been tested on various waveguides containing one or more angles of size 3/spl pi//2 or 2/spl pi/, and also on domains containing various other angles, and the results agree well with those obtained by other methods, mostly of less general applicability.

Dynamic response and damage of composite shell under impact

The dynamic response and damage of laminated composite cylindrical shell subjected to impact load is numerically investigated using the finite element method. A nine-node isoparametric quadrilateral element based on Sander's shell theory is developed, in which the transverse shear deformation is considered. A semi-empirical contact law that accounts for the permanent indentation is incorporated into the finite element program to evaluate the contact force. The Newmark time ingegration algorithm is used for solving the time dependent equations of the shell and the impactor. The Tsai-Wu failure criterion is used to estimate the failure of the laminated shell. Numerical results, including the contact force history, interlaminate damage zone, and failure indices in the shell are presented. Effects of curvature, impact velocity and mass of impactor on the composite shell behaviors are discussed.

Rotational dependence of the superconvergent patch recovery and its remedy for 4-node isoparametric quadrilateral elements

The superconvergent patch recovery (SPR) with bilinear interpolation functions usually gives good values of recovered stresses in an element patch. However, when 4-node quadrilateral elements meeting at a node are rigidly rotated with the essential and natural boundary conditions unchanged, the recovered stresses obtained by the SPR change and depend upon the local rotation of the patch. This can be remedied either by including higher-order terms in the polynomials for the assumed stress distribution in an element patch, or by using linear interpolation functions, which gives inferior accuracy of the recovered stresses near the boundaries of the domain. Additional sampling points are suggested to compute the higher-order terms.

FREE VIBRATION OF LAMINATED COMPOSITE PLATES WITH CUTOUT

The present investigation is concerned with free vibration analysis of composite plates in the presence of cutouts undergoing large amplitude oscillations. The Ritz finite element model using a nine-nodedC0continuity, isoparametric quadrilateral element along with a higher order displacement theory which accounts for parabolic variation of transverse shear stresses is used to predict the dynamic behavior. Results have been obtained for laminated plates with various cutout geometries such as square, rectangle, circle and ellipse in the large amplitude range. Backbone curves are drawn for various boundary conditions and aspect ratios of the cutout.

Analysis of Thermal Stresses at Dissimilar Metal Pipe Welds

Due to the differencees in the coefficients of thermal expansion between stainless steel and carbon steel, under the same change in temperature, stainless steel will expand 25–32% more than carbon steel. When these dissimilar metals are welded together and used in high temperature environments, both the interface between the pipes and the pipes themselves experience thermal stresses. A piping system analysis based on beam-type elements does not calculate all these additional thermal stresses. In this paper, an elastic, linear, finite-element analysis, using isoparametric quadrilateral elements, attempts to quantify these thermal stresses at butt-welded joint of dissimilar metals by considering an example of a stainless steel pipe welded to a carbon steel pipe.

Two-dimensional Fourier representation used in the bioelectric forward problem.

The objective of this paper is the application of two-dimensional discrete Fourier transformation for solving the integral equation of the bioelectric forward problem. Therefore, the potential, the source term, and the integral equation kernel are assumed to be sampled at evenly spaced intervals. Thus the continuous functions of the problem domain can be expressed by their two-dimensional discrete Fourier transform in the spatial frequency domain. The method is applied to compute the surface potential generated by an eccentric dipole in a homogeneous spherical conducting medium. The integral equation for the potential is solved in the spatial frequency domain and the value of the potential at the sampling points is obtained from inverse Fourier transformation. The solution of the presented method is compared to both, an analytic solution and a solution gained from applying the boundary element method. Isoparametric quadrilateral boundary elements are used for modeling the spherical volume conductor in the boundary element solution, while in the two-dimensional Fourier transformation method the volume conductor is represented by a parametric boundary surface approximation.

Optimum design of laminated composite plates with cutouts using a genetic algorithm

Optimum design of the laminated composite plate is attempted in the presence of elliptical cutouts. Orientation of the ellipse with respect to the reference axis, aspect ratio of the cutout, orientation of plies, thickness of plies and material of the plies are used as design parameters with a constraint on natural frequencies. Thus an investigation of the free vibrational response of composite plates with an elliptical cutout is studied, and some results presented. First-order shear deformation theory (FSDT) is used to account for the transverse shear stresses. A nine noded quadrilateral isoparametric element is used in the finite element formulation. A genetic algorithm (GA) is employed to locate the optimum. Numerical results are presented for square plates with simply supported and clamped edge conditions.

A study of the accuracy of some FEM stress recovery schemes for 2D stress concentration problems

The FEM stress recovery for plane quadrilateral isoparametric element is investigated when used in 2D geometric stress concentration problems. When there is no local mesh refinement the element at the fillet has a typical tapered shape which can be parametrically defined. An explicit expression for the determinant of the Jacobian matrix which, when approaching zero is responsible for FEM stress singularities, is presented. For a specific problem with a known theoretical solution it is shown that the 4-node and 8-node isoparametric elements can recover exactly the maximum stress at the node of stress concentration, provided the theoretical nodal displacements are used. For two typical shapes of the elements the radial and circumferential distribution of stress are compared with the theoretical ones. The performance of three stress recovery schemes is tested by a convergence study of the maximum stress at the stress concentrator when coarse mesh refinement and the 8-node isoparametric element are used. The nodal stresses in an element are calculated in three ways: 1. directly at the nodes; 2. by an extrapolation from the stresses at the 2×2 Gauss points; 3. by an empirical formula, using stresses computed by (1) and (2). By means of numerical examples it is shown that scheme (2), which is used in most commercial FEM programs, is much inferior compared to the other two schemes when used in stress concentration problems.

Analysis of shear wall structures on elastic foundations

A method for analyzing shear wall structures on elastic foundations is presented. The shear walls are modeled using a nine-noded isoparametric quadrilateral plane stress element and the soil is modeled using a three-noded quadratic element that includes the vertical subgrade reaction and soil shear stiffness. It is observed that analyzing shear wall structures as fixed cantilevers, i.e. ignoring soil-structure interaction, significantly underestimates the wall drift. It is shown that including soil shear stiffness in the analysis reduces the maximum drift and increases the normal stresses in the wall, especially for poor soil. It is also shown that ignoring the soil deformation underestimates the bending moment in the lintel beams of coupled shear walls, and the transverse bending moment in core structures.

The patch tests and convergence for nonconforming Mindlin plate bending elements

In this paper, the classical Irons\' patch tests which have been generally accepted for the convergence proof of a finite element are performed for Mindlin plate bending elements with a special emphasis on the nonconforming elements. The elements considered are 4-node and 8-node quadrilateral isoparametric elements which have been dominantly used for the analyses of plate bending problems. It was recognized from the patch tests that some nonconforming Mindlin plate elements pass all the cases of patch tests even though nonconforming elements do not preserve conformity. Then, the clues for the Mindlin plate element to pass the Irons\' patch tests are investigated. Also, the convergent characteristics of some nonconforming Mindlin plate elements that do not pass the Irons\' patch tests are examined by weak patch tests. The convergence tests are performed on the benchmark numerical problems for both nonconforming elements which pass the patch tests and which do not. Some conclusions on the relationship between the patch test and convergence of nonconforming Mindlin plate elements are drawn.

Elastic-plastic finite element analysis of metal matrix plates with edge notches

Abstract A two-dimensional finite element program has been developed for elastic-plastic analysis. Isoparametric quadrilateral element with four nodes is used and Lagrange polynomial has been chosen as interpolation function. In the finite element solution, an initial stress method (modified Newton-Raphson method) is used. Residual stresses have been shown in figures and the spread of plastic zones has been illustrated for steel-aluminum composite plates with edge notches.

Non-linear static and dynamic analysis of stiffened laminated plates

The static and dynamic analysis of eccentrically stiffened laminated plates is presented. For large deformations, the von Karman kinematic relations of the plate and stiffener are considered in the formulation. Also, the formulation accounts for transverse shear deformation effects of the plate and stiffener, and the model is applicable for the analysis of both thin and thick stiffened laminated plates. The finite element model consists of representing the plate by nine-noded isoparametric quadrilateral elements and the stiffener by three-noded isoparametric beam elements. Numerical examples are presented for the non-linear responses of stiffened laminated plates. The influences of stiffeners such as depth, number, and location are examined.

Finite element model for active control of intelligent structures

A generalized finite element formulation for active vibration control of a laminated plate integrated with piezoelectric polymer layers acting as distributed sensors and actuators is presented. An eight-noded two-dimensional quadratic quadrilateral isoparametric element is derived for modeling the global coupled electroelastic behavior of the overall structure using higher-order shear deformable displacement theory. The procedure is illustrated with a simply supported plate in which the substrate is a symmetric (0/90/0 deg) graphite-epoxy laminate, and the sensor and actuator layers are made of polyvinylidene fluoride. Results show that significant reduction in vibration amplitude occurs because of increased damping through feedback.

Mesh generation and flow calculations in highly contorted geometries

We present a method to define the solid-liquid boundaries in complex dendritic geometries and construct finite element meshes using quadrilateral isoparametric elements. Calculations of flow at low Reynolds numbers through the dendritic network are also performed. The results of the calculations are used to estimate the average pressure drop as a function of the fraction of liquid in the region, from which permeabilities can be obtained to model the full domain as a porous medium.

Generalized modified crack closure integral (GMCCI) and its application to interface crack problems

It is well known that at an interface crack tip, mode mixity exists even under single mode loading situations. The modified crack closure integral (MCCI) technique together with finite element analysis has been extensively used in the past to estimate the mixed mode strain energy release rate (SERR) components at these crack tips. One of the aspects which attracted considerable attention in these problems is the convergence of SERR components as the virtual crack extension size is progressively decreased to zero. Such studies have proved to be computationally expensive as the problem has to be solved each time for various sizes of virtual crack extension. Also it was not possible to estimate the effect of preventing the crack face interpenetration on SERR estimates for large sizes of virtual crack extension greater than the contact zone size. To overcome these in this paper, a generalized modified crack closure integral (GMCCI) algorithm is proposed for four and eight-noded isoparametric quadrilateral elements, which can estimate the SERR components for several sizes of virtual crack extension through a single finite element analysis. This is achieved through a numerical integration of the crack closure integral (CCI) and for various sizes of virtual crack extension covering a large number of equal or unequal size elements ahead and behind the crack tip. The procedure developed is demonstrated with application to typical interface crack problems under remote tension and shear.

FINITE ELEMENT FREE VIBRATION ANALYSIS OF DOUBLY CURVED LAMINATED COMPOSITE SHELLS

A finite element analysis for the free vibration behaviour of doubly curved shells is presented in which eight-noded curved quadrilateral isoparametric finite elements are used. The first order shear deformation theory for thin and shallow shells is used in the formulation. Results are obtained for comparison with those in the existing literature and to investigate the effects of various composite parameters relevant to doubly curved shells, such as fibre orientations and lamination schemes and several geometrical parameters like aspect ratio, smaller height to greater height ratio (for conoids), thickness to radius ratio (for hyperbolic and elliptic paraboloids), and radii of curvature ratio (for elliptic paraboloids).

A Posteriori Error Estimation and Adaptive Mesh Generation in Two-dimensional Elasto-plastic Finite Element Analyses for An Integrated System

Based on a series of works finished by our research group this paper presents a systematic method of element-wise a posteriori error estimation and adaptive mesh generation in finite element analysis in two-dimensional elasto-plastic problems for an integrated system. In this method, the error is estimated element by element for the arbitrary shaped elements, such as triangle and quadrilateral isoparametric elements. A technique to partition the error force of traction jump between adjacent elements is discussed. The mesh is originally generated according to the geometric condition of the configure model and regenerated automatically according to the error norm and its distribution with the objective that the total error energy norm in the whole domain becomes minimum. The high convergent rates of error energy norm are obtained in two numerical examples. Since the calculation is done with the arbitrary shaped elements and the mesh is regenerated automatically, the method developed in this paper has high abilities and effectiveness for two-dimensional elasto-plastic problems in the real engineering field.

Coupling of two- and three-dimensional composite shell elements in linear and non-linear applications

A numerical model for the non-linear analysis of laminates is presented. The developed element is based on piecewise polynomial interpolation in thickness direction. Although the total number of degrees of freedom is comparable to a discretization with 3D brick elements this approach provides several advantages, e.g. a simple mesh generation due to a 2D-type data structure and a better bending behavior. The developed isoparametric quadrilateral finite element allows prediction of the complete stress state. Furthermore, a transition element is presented which is used to couple the developed element with 5-parameter shell elements. Several linear and non-linear examples are presented to illustrate the performance of the developed numerical model.