Thin Plate Element Research Articles

An inverse approach in obtaining shape functions for a superconvergent thin plate element

This article presents an inverse approach to provide shape functions associate with a superconvergent thin plate element formulation. In the proposed approach, candidates for shape functions of an element are selected using a series of functions such as, trigonometric series, simple and hierarchical polynomials or a combination of them. Next, one imposes all the physical, geometrical, compatibility and completeness constraints associated with the concerned element on these function. In the final stage, the unknown parameters of the shape functions are determined by minimizing the discrimination errors in the element formulation. The proposed method is employed to determine the shape functions associate with the superconvergent plate element formulation. The accuracy of the obtained formulation is examined against previously developed plate models using several numerical examples. These comparisons indicate the developed model provides more accurate results both in local and global coordinates system.

Use of ANCF Surface Geometry in the Rigid Body Contact Problems: Application to Railroad Vehicle Dynamics

In the analysis of multibody system (MBS) dynamics, contact between two arbitrary rigid bodies is a fundamental feature in a variety of models. Many procedures have been proposed to solve the rigid body contact problem, most of which belong to one of the two categories: offline and online contact search methods. This investigation will focus on the development of a contact surface model for the rigid body contact problem in the case where an online three-dimensional nonconformal contact evaluation procedure, such as the elastic contact formulation—algebraic equations (ECF-A), is used. It is shown that the contact surface must have continuity in the second-order spatial derivatives when used in conjunction with ECF-A. Many of the existing surface models rely on direct linear interpolation of profile curves, which leads to first-order spatial derivative discontinuities. This, in turn, leads to erroneous spikes in the prediction of contact forces. To this end, an absolute nodal coordinate formulation (ANCF) thin plate surface model is developed in order to ensure second-order spatial derivative continuity to satisfy the requirements of the contact formulation used. A simple example of a railroad vehicle negotiating a turnout, which includes a variable cross-section rail, is tested for the cases of the new ANCF thin plate element surface, an existing ANCF thin plate element surface with first-order spatial derivative continuity, and the direct linear profile interpolation method. A comparison of the numerical results reveals the benefits of using the new ANCF surface geometry developed in this investigation.

Free vibration analysis of circular thin plates with stepped thickness by the DSC element method

Novel formulation is presented by using the discrete singular convolution (DSC) for free vibration analysis of circular thin plates with uniform and stepped thickness. Different from the commonly used ones in literature, regularity conditions are not needed at the circular plate center point to avoid singularity. DSC circular and annular thin plate elements are established. For the DSC circular plate element with radius of R1, the stiffness equation is first formulated in region [−R1, R1] with even number of nodes and then reduced to region [0, R1] by using either symmetric or anti-symmetric conditions. The proposed DSC circular and annular plate elements are used for obtaining frequencies of uniform/stepped circular thin plates or annular thin plates with different boundary conditions. Comparison of the present DSC results to existing analytic and numerical solutions verifies the proposed formulations. The present research extends the DSC method to free vibration of circular thin plates with stepped thicknesses.

Methods of modeling slope discontinuities in large size wind turbine blades using absolute nodal coordinate formulation

This paper describes and evaluates the use of the Absolute Nodal Coordinate Formulation (ANCF) in modeling large size wind turbine blades. Modern blade model can be divided into two regions classified by aerodynamic and structural function. The aerodynamic region, blade-span, is utilizing the thinnest possible airfoil section. On the other hand, the transition between the circular mount and the first airfoil profile is referred as blade-root region, which carries highest loads along the blade. In this investigation, an efficient procedure is developed for mapping NACA airfoil wind-turbine blades into ANCF thin plate models. The procedure concerns a complete wind turbine blade structure, blade-root as well as the blade-span regions with non-uniform and twisted nature. As a result, the slope discontinuity problem arises in both chord-wise and span-wise directions, and consequently presents numerical errors in dynamic simulation. The paper investigates the methods of modeling slope discontinuity resulting from the variations of the cross-sectional layouts across the blade. The developed method is applied for the gradient-deficient thin plate element in order to account for structural discontinuity. In addition, the aerodynamic loads are precisely expressed and the aerodynamic characteristics of such blades are examined with the ANCF and with the classical finite element method. The static and dynamic solutions of different operating conditions are obtained and results are compared with those obtained using ANSYS code. Both the limitations and advantages of using the ANCF in modeling large size wind turbine blades are concluded and discussed. A Dynamics for Design (DFD) procedure is presented with numerical example concerning large-rotation, large deformation wind turbine blades.

Reduction of system matrices of plate in absolute nodal coordinate formulation by component mode synthesis method

In this paper, we present a reduction method of the system matrix of a flexible thin plate element of 48 degrees of freedom, which is formulated on the basis of the absolute nodal coordinate method (ANCF) by use of the component mode synthesis method. We focus that the bending stiffness matrix expressed by the ANCF plate element is constant when the continuum mechanics approach is adopted and the in-plane strain is not so large. This feature allows applying the Craig-Bampton method to the equation of motion of the plate expressed by ANCF, and then it enable to reduce the dimension size of the system matrix of the plate. Three numerical examples that compare the presented method and the conventional ANCF plate are demonstrated to examine the performance and the accuracy of the presented method. From these examples, it is verified that the presented method can describe the large deformation and rotation effects as well as the conventional ANCF plate does. The use of the presented method is also verified to be able to reduce the computing time against the conventional ANCF plate. This feature is effective for carrying out numerical simulations for the structure composed of multiple plate elements when the number of elements increases.

Sliding and Nonsliding Joint Constraints of B-Spline Plate Elements for Integration With Flexible Multibody Dynamics Simulation

In this investigation, a numerical procedure for modeling sliding and nonsliding joint constraints for the B-spline thin plate element is developed for the large deformation analysis of multibody systems. A concept of intermediate reference coordinates proposed for the absolute nodal coordinate formulation is generalized for B-spline elements such that a wide variety of joint constraints can be modeled using existing joint constraint libraries already implemented in multibody dynamics codes. This procedure allows for modeling sliding joints for B-spline elements that requires a solution to moving boundary problems by introducing time-variant surface parameters in the B-spline parametric domain. Since surface parameters treated as knot variables in the basis function are defined in the entire parametric domain rather than the element domain, the location of the constraint definition point can be determined without knowing in which elements the sliding point is located. Furthermore, using the B-spline recurrence formula, control points used for describing the constraint equations can be systematically extracted. It is shown that many types of nonsliding joints fixed on the flexible body can also be modeled as a special case of the sliding joint formulation developed in this investigation, leading to a unified joint constraint formulation for B-spline elements. Several numerical examples are presented in order to demonstrate the use of the numerical procedure developed in this investigation.

A Thick/Thin Plate Element Based on the Area Coordinate Analytical Trial Functions

In this paper, a generalized conforming thick/thin plate element based on the quadrilateral area coordinate system called AATF-PQ4 is developed. Based on the governing equations of Mindlin-Reissner plate theory, the fundamental analytical solutions are first derived, then using this trial functions to formulate element AATF-PQ4. In the case of thin plate, this thick/thin plate element is degraded into corresponding thin plate element automatically and is free from shear locking. Numerical examples show that the proposed element, AATF-PQ4, has a good precision for thick and thin plate.

Three-Dimensional Fully Parameterized Triangular Plate Element Based on the Absolute Nodal Coordinate Formulation

In this work, a new three-dimensional fully parameterized triangular plate element based on the absolute nodal coordinate formulation (ANCF) is introduced. This plate element has 12 coordinates per node; therefore, it can be used in thick plate applications. The proposed 12 shape functions are obtained by adding three shape functions to the nine shape functions that were previously used with the ANCF thin triangular plate element. Unlike the existing ANCF thin triangular plate element, which allows only the use of classical Kirchoff's plate theory, the fully parameterized ANCF triangular plate element proposed in this work allows for the use of a general continuum mechanics approach and also allows for a straight forward implementation of general nonlinear constitutive equations. Moreover, all deformation modes including thickness deformation can be captured using the fully parameterized ANCF triangular plate element proposed in this paper. The numerical results obtained in this investigation show that in case of negligible deformation, the fully parameterized ANCF triangular plate element behaves like a rigid body. Moreover, it is found that there is a good agreement between the solutions obtained using the proposed fully parameterized ANCF triangular plate element and the theoretical model in the case of small deformations. Furthermore, it is shown that the results of the proposed element agree well with the results obtained using the existing fully parameterized ANCF rectangular plate element when large deformation conditions are applied. The twist behavior of the proposed element is verified by comparison with the results obtained using a conventional nonlinear rectangular plate element.

Finite Element Modeling of the Contact Geometry and Deformation in Biomechanics Applications1

The main contribution of this paper is to demonstrate the feasibility of using one computational environment for developing accurate geometry as well as performing the analysis of detailed biomechanics models. To this end, the finite element (FE) absolute nodal coordinate formulation (ANCF) and multibody system (MBS) algorithms are used in modeling both the contact geometry and ligaments deformations in biomechanics applications. Two ANCF approaches can be used to model the rigid contact surface geometry. In the first approach, fully parameterized ANCF volume elements are converted to surface geometry using parametric relationship that reduces the number of independent coordinate lines. This parametric relationship can be defined analytically or using a spline function representation. In the second approach, an ANCF surface that defines a gradient deficient thin plate element is used. This second approach does not require the use of parametric relations or spline function representations. These two geometric approaches shed light on the generality of and the flexibility offered by the ANCF geometry as compared to computational geometry (CG) methods such as B-splines and NURBS (Non-Uniform Rational B-Splines). Furthermore, because B-spline and NURBS representations employ a rigid recurrence structure, they are not suited as general analysis tools that capture different types of joint discontinuities. ANCF finite elements, on the other hand, lend themselves easily to geometric description and can additionally be used effectively in the analysis of ligaments, muscles, and soft tissues (LMST), as demonstrated in this paper using the knee joint as an example. In this study, ANCF finite elements are used to define the femur/tibia rigid body contact surface geometry. The same ANCF finite elements are also used to model the MCL and LCL ligament deformations. Two different contact formulations are used in this investigation to predict the femur/tibia contact forces; the elastic contact formulation which allows for penetrations and separations at the contact points, and the constraint contact formulation in which the nonconformal contact conditions are imposed as constraint equations, and as a consequence, no separations or penetrations at the contact points are allowed. For both formulations, the contact surfaces are described in a parametric form using surface parameters that enter into the ANCF finite element geometric description. A set of nonlinear algebraic equations that depend on the surface parameters is developed and used to determine the location of the contact points. These two contact formulations are implemented in a general MBS algorithm that allows for modeling rigid and flexible body dynamics.

A thin plate element based on the combined arbitrary Lagrange–Euler and absolute nodal coordinate formulations

The goal of this study is to expand the combination of the arbitrary Lagrange–Euler (ALE) and absolute nodal coordinate formulation (ANCF) to plate elements. In the ALE–ANCF method, nodal coordinates are not associated with any specific material points. This means that nodal positions in a finite element mesh can be varied during simulations forward in time. This article contains a description of the kinematics and equation of motion of a thin ALE–ANCF plate element. The element kinematics is described using C1 continuous shape functions and elastic forces are evaluated using a combined membrane and curvature approach. The presented plate element formulation is validated by comparison with a conventional ANCF plate element.

Use of B-spline surface to model large-deformation continuum plates: procedure and applications

The absolute nodal coordinate formulation (ANCF) has been used in the analysis of large deformation of flexible multibody systems that encompass belt drive, rotor blade, and cable applications. As demonstrated in the literature, the ANCF finite elements are ideal for isogeometric analysis. The purpose of this investigation is to establish a relationship between the B-splines, which are widely used in the geometric modeling, and the ANCF finite elements in order to construct continuum models of large-deformation geometries. This paper proposes a simplified approach to map the B-spline surfaces into ANCF thin plate elements. Matrix representation of the mapping process is established and examined through numerical examples successfully. The matrix representation of the mapping process is used because of its suitability of computer coding and to minimize the calculation time. The error estimation is carried out by analyzing the gap between the points of each ANCF element and the corresponding points of the portion of the B-spline surface. The Hausdorff distance is used to study the effect of the number of control points, the degree of interpolation, and the knot multiplicity on the mapped geometry. It is found that cubic interpolation is recommended for optimizing the accuracy of mapping the B-spline surface to ANCF thin plate elements. It is found that thin plate element in ANCF missing a number of basis functions which considered a source of error between the two surfaces, as well as it does not allow to converting the ANCF thin plate elements model to B-spline surface. In this investigation, an application example of modeling large-size wind turbine blade with uniform structure is illustrated. The use of the continuum plate elements in modeling flexible blades is more efficient because of the relative scale between the plate thickness and its length and width and the high flexibility of its structure. The numerical results are compared with the results of ANSYS code with a good agreement. The dynamic simulation for mapped surface model shows a numerical convergence, which ensures the ability of using the proposed approach for applications of dynamics for design and computer-aided design.

A Continuum Based Three-Dimensional Modeling of Wind Turbine Blades

Accurate modeling of large wind turbine blades is an extremely challenging problem. This is due to their tremendous geometric complexity and the turbulent and unpredictable conditions in which they operate. In this paper, a continuum based three dimensional finite element model of an elastic wind turbine blade is derived using the absolute nodal coordinates formulation (ANCF). This formulation is very suitable for modeling of large-deformation, large-rotation structures like wind turbine blades. An efficient model of six thin plate elements is proposed for such blades with non-uniform, and twisted nature. Furthermore, a mapping procedure to construct the ANCF model of NACA (National Advisory Committee for Aeronautics) wind turbine blades airfoils is established to mesh the geometry of a real turbine blade. The complex shape of such blades is approximated using an absolute nodal coordinate thin plate element, to take the blades tapering and twist into account. Three numerical examples are presented to show the transient response of the wind turbine blades due to gravitational/aerodynamics forces. The simulation results are compared with those obtained using ANSYS code with a good agreement.

Structural Intensity Modeling and Simulations for Damage Detection

Structural health monitoring (SHM) techniques have previously been proposed based on structural intensity (SI) due to its sensitivity to changes in boundary and loading conditions, and impedance, as well as to various damage mechanisms. In this paper, computational techniques for SI-based SHM are presented. Finite element solvers combined with SI equations can yield intensity maps over structures to determine characteristic changes in power flow due to damage. Numerical techniques for structural surface intensity (SSI) are also introduced using two alternative methods: A time domain approach that directly uses SSI equations that are valid at the surface of any elastic solid, and a frequency domain technique, which computes SI for very thin plate elements located at the surface of the structure. Advanced contact features such as nonlinearity can also be included in the model to increase the damage detection sensitivity. A plate model is used to illustrate these capabilities using SSI maps at nonlinear harmonics (NSSI). The results show both improved damage sensitivity and more global detection capabilities in a NSSI-based SHM system. A complex structure is also included to show global and local changes in SSI due simulated damage scenario. The techniques developed can be applied to general SI/SSI assessments and the design of SI-based SHM systems.

Analysis of thin plates by the weak form quadrature element method

The recently proposed weak form quadrature element method (QEM) is applied to flexural and vibrational analysis of thin plates. The integrals involved in the variational description of a thin plate are evaluated by an efficient numerical scheme and the partial derivatives at the integration sampling points are then approximated using differential quadrature analogs. Neither the grid pattern nor the number of nodes is fixed, being adjustable according to convergence need. The C1 continuity conditions characterizing the thin plate theory are discussed and the robustness of the weak form quadrature element for thin plates against shape distortion is examined. Examples are presented and comparisons with analytical solutions and the results of the finite element method are made to demonstrate the convergence and computational efficiency of the weak form quadrature element method. It is shown that the present formulation is applicable to thin plates with varying thickness as well as uniform plates.

The addition of transverse shear flexibility to triangular thin plate elements

A method is described for incorporating shear flexibility into existing three-node triangular thin plate elements with particular relevance to finite element templates. When the parent thin plate elements satisfy some necessary conditions the resulting shear flexible elements have correct rank and are free of shear locking. Numerical results indicate that the convergence characteristics of the shear flexible elements are similar to those of the parent thin plate elements.

PRO/MECHANICA-based structural and random vibration analysis of picosatellite structure

This paper focuses on the structural and vibration analysis of a low-cost picosatellite structure to the launch load environment. This picosatellite complies with the CubeSat program design specifications, where the satellite is less than one kilogram in mass with a dimension of 10 cm3. The satellite structure is treated as a combination of beams and thin plate elements. A detailed finite element model is developed to observe its responses to the quasi-static accelerations induced by the launch vehicle. The natural frequencies of the prototype were computed numerically through finite element based methods and compared with the natural frequencies induced by the launch vehicle. It was also found that the maximum stress experienced by the structure during launch was well within the stress limits of the material.

Absolute nodal coordinate formulation of large-deformation piezoelectric laminated plates

The Absolute Nodal Coordinate Formulation (ANCF) has been initiated in 1996 by Shabana (Computational Continuum Mechanics, 3rd edn., Cambridge: Cambridge University Press, 2008). It introduces large displacements of planar and spatial finite elements relative to the global reference frame without using any local frame. A sub-family of beam, plate and cable finite elements with large deformations are proposed and employed the 3D theory of continuum mechanics. In the ANCF, the nodal coordinates consist of absolute position coordinates and gradients that can be used to define a unique rotation and deformation fields within the element. In contrast to other large deformation formulations, the equations of motion contain constant mass matrices as well as zero centrifugal and Coriolis inertia forces. The only nonlinear term is a vector of elastic forces. This investigation concerns a way to generate new finite element in the ANCF for laminated composite plates. This formulation utilizes the assumption that the bonds between the laminae are thin and shear is non-deformable. Consequently, the Equivalent Single Layer, ESL model, is implemented. In the ESL models, the laminate is assumed to deform as a single layer, assuming a smooth variation of the displacement field across the thickness. In this paper, the coupled electromechanical effect of Piezoelectric Laminated Plate is imposed within the ANCF thin plate element, in such a way as to achieve the continuity of the gradients at the nodal points, and obtain a formulation that automatically satisfies the principle of work and energy. Convergence and accuracy of the finite-element ANCF Piezoelectric Laminated Plate is demonstrated in geometrically nonlinear static and dynamic test problems, as well as in linear analysis of natural frequencies. The computer implementation and several numerical examples are presented in order to demonstrate the use of the formulation developed in this paper. A comparison with the commercial finite element package COMSOL MULTIPHYSICS (http://www.comsol.com/) is carried out with an excellent agreement.

Modal Analysis of a Rotating Thin Plate via Absolute Nodal Coordinate Formulation

Modal analysis of a rotating thin plate is made in this paper through the use of the thin plate elements described by the absolute nodal coordinate formulation (ANCF). The analytical expressions of elastic forces and their Jacobian matrices of the thin plate elements are derived and expressed in a computationally efficient way. The static analysis of a cantilever thin plate and the modal analysis of a square thin plate with completely free boundaries are made to validate the derived formulations. The modal analysis of a rotating cantilever thin plate based on the ANCF is studied. The effect of rotating angular velocity on the natural frequencies is investigated. The eigenvalue loci veering and crossing phenomena along with the corresponding modeshape variations are observed and carefully discussed. Finally, the effects of dimensional parameters on the dimensionless natural frequencies of the thin plate are studied.

A Universal Finite Element of Thick and Thin Plate without Shear Locking

In order to prevent shear locking, a method using theory of deep beam is proposed. A universal finite element for thick and thin plates is constructed. When the plate thickness approaches to the limit of thin plate, the universal element degenerates to the thin plate element automatically. As a results, the shear locking phenomenon will not appear. The computational results indicate that the current element has high-accuracy and good usefulness.

A refined nonconforming quadrilateral element for couple stress/strain gradient elasticity

AbstractC0−1 patch test (Int. J. Numer. Meth. Engng 2004; 61:433–454) proposed by Soh and Chen is a reliable method to ensure convergence of nonconforming finite element for the couple stress/strain gradient elasticity. The C0−1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations. To pass the C0−1 patch test, the element displacement functions used to calculate strains must satisfy C0 continuity (or weak C0 continuity) and quadratic completeness. In this paper, a 24‐DOF (degrees of freedom) quadrilateral element (CQ12+RDKQ) for the couple stress/strain gradient elasticity is developed by combining the refined thin plate element RDKQ and the nonconforming element CQ12. The element RDKQ, which satisfies weak C1 continuity, is used to calculate strain gradients, whereas strains are computed by the element CQ12, which is established based on an extended variational functional and satisfies weak C0 continuity and quadratic completeness. Numerical examples show that the element (CQ12+RDKQ) passes the C0−1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects. Copyright © 2010 John Wiley & Sons, Ltd.