Nine-node Element Research Articles

A field-consistent and variationally correct representation of transverse shear strains in the nine-noded plate element

We present a biquadratic Lagrangian plate bending element with consistent fields for the constrained transverse shear strain functions. A technique involving expansion of the strain interpolations in terms of Legendre polynomials is used to redistribute the kinematically derived shear strain fields so that the field-consistent forms (i.e. avoiding locking) are also variationally correct (i.e. do not violate the variational norms). Also, a rational method of isoparametric Jacobian transformation is incorporated so that the constrained covariant shear strain fields are always consistent in whatever general quadrilateral form the element may take. Finally the element is compared with another formulation which was recently published. The element is subjected to several robust bench mark tests and is found to pass all the tests efficiently.

Some considerations on a four-node finite element for composites with the drilling degrees of freedom

A plate-type finite element with six degrees of freedom (DOF) for each of the four nodes is introduced. To obtain this result a Mindlin-type bending-plate element is superimposed on a three DOF per node plane-stress element. Composites are simulated using the YNS theory. A few considerations on some plate-shell elements already tested are reported and the introduction of the drilling DOF is justified. A comparison with the results obtained by using a nine-node reduced-integration isoparametric element is also supplied.

Buckling of Laminated Cylindrical Panels Under Uniaxial Compression

Buckling characteristics of anisotropic laminated cylindrical panels are presented using an isoparametric nine-noded rectangular element. Kinematic relations, extended to include the first-order shear deformation theory, are utilized in the formulation. Numerical results are presented for laminated cylindrical panels under uniaxial compression loading to illustrate the influence of panel sizes, orientation of layers, material properties, and boundary conditions on the buckling loads. Results for plates are also presented as special cases and are compared with the available solutions.

Non-linear dynamics of laminated plates with a higher-order theory and C 0 finite elements

Impact responses of composite laminates are investigated with C 0 finite elements. A nine-node isoparametric quadrilateral element based on a higher-order theory and the von Kármán large deflection assumptions is developed. An experimentally established contact law which accounts for the permanent indentation is incorporated into the finite element program to evaluate the impact force. In the time integration, the explicit central difference technique is used in conjunction with the special mass matrix diagonalization scheme. Numerical results, including the contact force histories, deflections and strains in the plate, are presented.

A Numerical Solution for Calculating Elastic Deformation in Elliptical-Contact EHL of Rough Surface

The analytical expressions are developed for elastic deformation provided the pressure function on a nine-node element of any nonequidistant rectangle is approximated by a biquadratic polynomial. Using the present scheme, the coefficients Aij showing in pressure solution are simply calculated once enough in the full iterations. The computational works are greatly reduced.

Chebyshev collocation solutions of flow problems

Chebyshev collocation methods are reviewed. For general second-order elliptic equations, the algebraic system obtained through the collocation process is ill-conditioned. Recent work on finite element preconditioning shows that bilinear elements give full satisfaction. For Stokes problems, as interpolants of different degree are used for the velocity and the pressure, the classical nine-node Lagrangian element with biquadratic velocities and bilinear pressures constitutes the best choice. In order to treat the non-linear terms, a Newton's method is designed. Domain decomposition is set up with the jumps of the stress vector across interfaces between adjacent subdomains. Two-dimensional curvy geometries are handled by a coordinate mapping. Problems with singularities are treated by a mixed finite element and spectral approximation. Finally, current developments for Non-Newtonian fluids are evoked.

Accurate and efficient nonlinear formulation of a nine-node shell element with spurious mode control

A nine-node Lagrangian shell finite element is presented for full nonlinear analysis of shell-type structures. The formulation is based on the assumption of large displacements, finite rotations, and small strains. The major focus is on the robustness, accuracy, and efficiency of the finite element. Uniform reduced (2 x 2) quadrature is used for evaluation of the element stiffness and recovery of element nodal forces, and an improved ‘ projection operator ’ procedure is employed for stabilization of the resulting spurious zero-energy modes. Efficient techniques are described for calculation of shape function derivatives, performance of local-to-global transformations, and determination of the ‘ stabilization coefficients ’ of the artificial mode control procedure. Numerical integration is employed through the thickness of the element for accurate evaluation of the plastic response in moderately thick shells. The projection operator employed for the mode stabilization is orthogonal to constant, linear, and quadratic displacement fields in the local Cartesian ( x′, y′) coordinates at each of the 2 × 2 Gauss quadrature stations. As a result, the flat-plate pure-bending patch tests are not affected for arbitrary element sizes. The resulting element is essentially free of membrane and shear locking problems, and it has not shown any signs of spurious zero energy mode effects for a large number of linear and nonlinear tests.

Transient/pseudo-transient finite element small/large deformation analysis of two-dimensional problems

A unified approach is presented for the static and transient linear and geometrically non-linear analyses of two-dimensional problems (plane stress/strain and axisymmetric). A finite element idealization with four-, eight- and nine-noded isoparametric quadrilateral elements is used for space discretization. An explicit central difference time-marching scheme is employed for time integration of the resulting discrete ordinary differential equations. Results of several numerical examples are presented and compared with the available data. A comparative study of the performance of various elements with different damping factors is also presented.

Testing of shell finite element accuracy and robustness

This paper presents the authors experiences with a suite of tests employed to determine the accuracy and robustness of shell finite elements for linear elastic and geometric nonlinear problems. Particular emphasis is placed on testing element sensitivity to distortions, validation of spurious zero-energy mode stabilization procedures, and use of linear elastic tests for selection of “good” nonlinear elements. Specific linear elastic test results are presented for several variations of the Semiloof quadrilateral and triangle shell element as well as a nine-node Lagrangian ( Lag9) degenerate-based shell element with uniformly-reduced integration and spurious mode control. The Lag9 element is shown to exhibit superior overall performance compared to that of the Semiloof elements in these linear elastic tests. The performance of the LAG9 element is also validated for a number of sensitive tests for detection of zero- or low-energy mode problems. Lastly, the performance of the nine-node element is investigated for several geometric nonlinear test cases.

Buckling of Laminated Plates with Holes

This paper deals with buckling analysis of rectangular composite laminates with circular holes under inplane static loadings. The first-order shear deformation theory and the variational energy method are employed in mathematical formulation, and the nine-node Lagrangian finite element method is used for finding critical loads. The effects on critical load by hole size, plate thickness ratio, material modulus ratio, ply lamination geometry, loading types, and boundary conditions are investigated. Numerical solutions are shown in graphical form where some comparisons are made with those of given litera tures.

Simulation of the newtonian flow through an abruptly contracted tube using a mixed finite element method

The solutions of the Navier-Stokes equation for a Newtonian flow through a 4/1 contraction tube were obtained numerically using the Galerkin finite element method with the nine-node Lagrangian element which was believed to be one of the most accurate tools for mixed-type interpolating formulations. It was proved from this study that the vortex occurrence in the entrance corner region were confirmed but its size was gradually decreased with the increase of Reynolds numbers, and that the velocity profiles and pressure distributions along the applied mesh layers were in agreement with the experimental and the previously reported numerical results.

A higher-order theory for free vibration of unsymmetrically laminated composite and sandwich plates—finite element evaluations

This paper presents a refined higher-order theory for free vibration analysis of unsymmetrically laminated multilayered plates. The theory accounts for parabolic distribution of the transverse shear strains through the thickness of the plate and rotary inertia effects. A simple C 0 finite element formulation is presented and the nine-noded Lagrangian element is chosen with seven degrees of freedom per node. Numerical results are presented showing the parametric effects of aspect ratio, length/thickness ratio, number of layers, and lamination angle. The present theory predicts the frequencies more accurately when compared with first-order and classical plate theories.

Hygrothermal effects on the stability of a cylindrical composite shell panel

The finite element method is applied to study the problem of moisture and temperature effects on the stability of a general orthotropic cylindrical composite shell panel subjected to axial or in-plane shear loading. The element employed is a nine-node isoparametric shell element. As the hygrothermal effects on the elastic properties of the matrices and the fibers are very different, the degradation of elastic moduli, the transverse shear effect and the induced initial stress are all considered. Numerical investigation shows that if the temperature increases from 300 to 422 K and the moisture concentration is saturated, the buckling load with both the degradation of elastic moduli and transverse shear deformation considered is ca. 12% lower than Snead and Papazotto's result where only the degradation of elastic moduli was considered. In addition, it is shown that the influence of the initial stress, induced by the same environmental variations, on the buckling load is far less significant.

Flexural analysis of laminated composites using refined higher-order [formula omitted] plate bending elements

A finite element formulation for flexure of a symmetrically laminated plate based on a higher-order displacement model and a three-dimensional state of stress and strain is presented here. The present higher-order theory incorporates linear variation of transverse normal strains and parabolic variation of transverse shear strains through the plate thickness, and as a result it does not require shear correction coefficients. A nine-noded Lagrangian parabolic isoparametric plate bending element is described. The applications of the element to bending of laminated plates with various loading, boundary conditions, and lamination types are discussed. The numerical evaluations also include the convergence study of the element used. The present solutions for deflections and stresses are compared with those obtained using the three-dimensional elasticity theory, closed-form solutions with another high-order shear deformation theory, and the Mindlin's theory. In addition, numerical results for a number of new problems, not available in the literature, are presented for future reference.

A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates

A higher-order theory which satisfies zero transverse shear stress conditions on the bounding planes of a generally laminated fibre-reinforced composite plate subjected to transverse loads is developed. The displacement model accounts for non-linear distribution of inplane displacement components through the plate thickness and the theory requires no shear correction coefficients. A C∘ continuous displacement finite element formulation is presented and the coupled membrane-flexure behaviour of laminated plates is investigated. The nodal unknowns are the three displacements, two rotations and two higher-order functions as the generalized degrees of freedom. The simple isoparametric formulation developed here is capable of evaluating transverse shears and transverse normal stress accurately by using the equilibrium equations. The accuracy of the nine-noded Lagrangian quadrilateral element is then established by comparing the present results with the closed-form, three-dimensional elasticity and other finite element available solutions.

On singular, nine-noded, distorted, isoparametric elements in linear elastic fracture mechanics

Distorted, nine-noded, Lagrangian, isoparametric elements are studied to determine if stress intensity factors can be calculated more accurately with these elements than with eight-noded quarter-point elements. To achieve square-root singular stress behavior in the nine-noded element, the mid-side nodes adjacent to the crack tip are moved to the quarter-point and the nodal point within the element is positioned along the element diagonal, at least at the quarter-point. Several optimal positions for this node are determined. Through two numerical examples, it is seen that the eight-noded element performs nearly as well, and sometimes better than the nine-noded element.

A nine-node assumed-strain finite element for composite plates and shells

A new finite element for modeling fiber-reinforced composite plates and shells is developed and its performance for static linear problems is evaluated. The element is a nine-node degenerate solid shell element based on a modified Hellinger-Reissner principle with independent inplane and transverse shear strains. Several numerical examples are solved and the solutions are compared with other available finite solutions and with classical lamination theory. The results show that the present element yields accurate solutions for the test problems presented. Convergence characteristics are good, and the solution is relatively insensitive in element distortion. The element is also shown to be free of locking even for thin laminates.

Covariant shear strains interpolation in a nine-node degenerated plate element

A rectangular, thin or moderately thick nine-node Lagrangian plate element following the Mindlin Theory is presented. The covariant components of shear deformation are approximated by lower-order polynomials than those resulting directly from the cinematic equations. As a consequence, locking and spurious modes are avoided.

Casing Headloss in Centrifugal Slurry Pumps

The headlosses in slurry pump casings are caused by the energy dissipation in the flowing liquid-solid mixture, and the particle-casing wall interaction. This paper presents a quasi-three-dimensional flow analysis, in two steps: (1) the casing width-averaged parameters are obtained from the two-dimensional flow simulation in a normal plane to the pump axis, (2) the results are then used for flow computation in the casing radial cross-sections. The governing equations are averaged over the typical time interval for the periodic flow in the casing caused by the finite number of impeller blades. Finite element techniques with linear three-node and quadratic nine-node elements are used in the numerical simulation. The effect of the casing shape on the headlosses is illustrated by comparing the performances of three centrifugal slurry pumps with quasi-spiral, semi-annular and annular casings at various flowrates. The flow recirculated between the casing tongue and impeller shrouds is shown to play a major role in the overall headloss. The flow results may also be used for determining the wear in the casing and thrust on the impeller, as well as for optimizing the casing dimensions.

Numerical Solutions for Flow in a Partially Filled, Rotating Cylinder

An efficient least-squares method with finite elements is used to simulate the viscous flow with surface tension in a partially liquid-filled, horizontal, rotating cylinder. Mixed interpolations are used for the method: the free boundary is interpolated by periodic cubic splines, the velocity by nine-node isoparametric biquadratic elements, and the pressure by four-node isoparametric bilinear elements. The free boundary is located by minimizing the sum of squares of the boundary residuals. Advantages of the method over the often used collocation method are shown. Numerical solutions are validated to some extent by comparing with results from perturbation methods and physical experiments.