Rational ANCF Thin Plate Finite Element

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Carmine M. Pappalardo ◽  
Zuqing Yu ◽  
Xiaoshun Zhang ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Thin Plate ◽  
Geometric Modeling ◽  
Absolute Nodal Coordinate Formulation ◽  
Position Vector ◽  
Plate Element ◽  
Conic Sections ◽  
Absolute Nodal Coordinate ◽  
Linear Algebraic Equations

In this paper, a rational absolute nodal coordinate formulation (RANCF) thin plate element is developed and its use in the analysis of curved geometry is demonstrated. RANCF finite elements are the rational counterpart of the nonrational absolute nodal coordinate formulation (ANCF) finite elements which employ rational polynomials as basis or blending functions. RANCF finite elements can be used in the accurate geometric modeling and analysis of flexible continuum bodies with complex geometrical shapes that cannot be correctly described using nonrational finite elements. In this investigation, the weights, which enter into the formulation of the RANCF finite element and form an additional set of geometric parameters, are assumed to be nonzero constants in order to accurately represent the initial geometry and at the same time preserve the desirable ANCF features, including a constant mass matrix and zero centrifugal and Coriolis generalized inertia forces. A procedure for defining the control points and weights of a Bezier surface defined in a parametric form is used in order to be able to efficiently create RANCF/ANCF FE meshes in a straightforward manner. This procedure leads to a set of linear algebraic equations whose solution defines the RANCF coordinates and weights without the need for an iterative procedure. In order to be able to correctly describe the ANCF and RANCF gradient deficient FE geometry, a square matrix of position vector gradients is formulated and used to calculate the FE elastic forces. As discussed in this paper, the proposed finite element allows for describing exactly circular and conic sections and can be effectively used in the geometry and analysis modeling of multibody system (MBS) components including tires. The proposed RANCF finite element is compared with other nonrational ANCF plate elements. Several numerical examples are presented in order to demonstrate the use of the proposed RANCF thin plate element. In particular, the FE models of a set of rational surfaces, which include conic sections and tires, are developed.

Analysis of Thin Plate Structures Using the Absolute Nodal Coordinate Formulation

2005 ◽  
Vol 219 (4) ◽  
pp. 345-355 ◽  
Author(s):  
K Dufva ◽  
A A Shabana
Keyword(s):  
Finite Element ◽  
Thin Plate ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Plate Element ◽  
Shell Elements ◽  
Absolute Nodal Coordinate ◽  
Plate And Shell ◽  
Spatial Parameters ◽  
The Absolute

The absolute nodal coordinate formulation can be used in multibody system applications where the rotation and deformation within the finite element are large and where there is a need to account for geometrical non-linearities. In this formulation, the gradients of the global positions are used as nodal coordinates and no rotations are interpolated over the finite element. For thin plate and shell elements, the plane stress conditions can be applied and only gradients obtained by differentiation with respect to the element mid-surface spatial parameters need to be defined. This automatically reduces the number of element degrees of freedoms, eliminates the high frequencies due to the oscillations of some gradient components along the element thickness, and as a result makes the plate element computationally more efficient. In this paper, the performance of a thin plate element based on the absolute nodal coordinate formulation is investigated. The lower dimension plate element used in this investigation allows for an arbitrary rigid body displacement and large deformation within the element. The element leads to a constant mass matrix and zero Coriolis and centrifugal forces. The performance of the element is compared with other plate elements previously developed using the absolute nodal coordinate formulation. It is shown that the finite element used in this investigation is much more efficient when compared with previously proposed elements in the case of thin structures. Numerical examples are presented in order to demonstrate the use of the formulation developed in this paper and the computational advantages gained from using the thin plate element. The thin plate element examined in this study can be efficiently used in many applications including modelling of paper materials, belt drives, rotor dynamics, and tyres.

Three-Dimensional Solid Brick Element Using Slopes in the Absolute Nodal Coordinate Formulation

2013 ◽  
Vol 9 (2) ◽  
Author(s):  
Alexander Olshevskiy ◽  
Oleg Dmitrochenko ◽  
Chang-Wan Kim
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Test Problems ◽  
Absolute Nodal Coordinate ◽  
Highly Nonlinear ◽  
Brick Element ◽  
The Absolute

The present paper contributes to the field of flexible multibody systems dynamics. Two new solid finite elements employing the absolute nodal coordinate formulation are presented. In this formulation, the equations of motion contain a constant mass matrix and a vector of generalized gravity forces, but the vector of elastic forces is highly nonlinear. The proposed solid eight node brick element with 96 degrees of freedom uses translations of nodes and finite slopes as sets of nodal coordinates. The displacement field is interpolated using incomplete cubic polynomials providing the absence of shear locking effect. The use of finite slopes describes the deformed shape of the finite element more exactly and, therefore, minimizes the number of finite elements required for accurate simulations. Accuracy and convergence of the finite element is demonstrated in nonlinear test problems of statics and dynamics.

Application of Plasticity Theory and Absolute Nodal Coordinate Formulation to Flexible Multibody System Dynamics

2003 ◽  
Vol 126 (3) ◽  
pp. 478-487 ◽  
Author(s):  
Hiroyuki Sugiyama ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Plasticity Theory ◽  
Position Vector ◽  
Element Formulation ◽  
Absolute Nodal Coordinate ◽  
Flexible Multibody ◽  
The Absolute ◽  
Return Algorithm ◽  
Finite Element Formulations

The objective of this investigation is to develop a nonlinear finite element formulation for the elastic-plastic analysis of flexible multibody systems. The Lagrangian plasticity theory based on J2 flow theory is used to account for the effect of plasticity in flexible multibody dynamics. It is demonstrated that the principle of objectivity that is an issue when existing finite element formulations using rate-type constitutive equations are used is automatically satisfied when the stress and strain rate are directly calculated in the Lagrangian descriptions using the absolute nodal coordinate formulation employed in this investigation. This is attributed to the fact that, in the finite element absolute nodal coordinate formulation, the position vector gradients can completely define the state of rotation and deformation within the element. As a consequence, the numerical algorithm used to determine the plastic deformations such as the radial return algorithm becomes much simpler when the absolute nodal coordinate formulation is used as compared to existing finite element formulations that employ incrementally objective algorithms. Several numerical examples are presented in order to demonstrate the use of the formulations presented in the paper.

A novel absolute nodal coordinate formulation thin plate tire model with fractional derivative viscosity and surface integral-based contact algorithm

2018 ◽  
Vol 233 (3) ◽  
pp. 583-597
Author(s):  
Peng Lan ◽  
Yaqi Cui ◽  
Zuqing Yu
Keyword(s):  
Fractional Derivative ◽  
Contact Pressure ◽  
Thin Plate ◽  
Absolute Nodal Coordinate Formulation ◽  
Tire Model ◽  
Plate Element ◽  
Absolute Nodal Coordinate ◽  
Contact Patch ◽  
Contact Algorithm ◽  
The Absolute

A new absolute nodal coordinate formulation thin plate tire model, which includes the damping property of the rubber and an efficient tire–road contact algorithm is given. The fractional derivative viscosity constitutive model is proposed in this paper based on the complete form of the absolute nodal coordinate formulation thin plate element, which is created to describe the stress-free initially curved configuration of the tire. A new contact algorithm based on the integration of the contact pressure within the contact patch is developed. By solving the simultaneous equations of the tire geometry and road profile, the dimensionless coordinates for the boundary points of contact patch could be obtained directly. A self-adaptable Gauss integration strategy is introduced to perform the integration of the contact pressure within the varying region, so the integration could reach high precision by few integration points. The calculation of contact force is determined based on penalty method and smoothed Coulomb friction model. The application of fractional derivative viscosity on the absolute nodal coordinate formulation thin plate element is demonstrated by numerical results. A pressurized Golf tire model is given to show the feasibility of the proposed tire–ground contact algorithm.

Improved Description of Elastic Forces for the Absolute Nodal Coordinate Based Plate Element

2005 ◽  
Author(s):  
Marko K. Matikainen ◽  
Aki M. Mikkola
Keyword(s):  
Finite Element ◽  
Continuum Mechanics ◽  
Absolute Nodal Coordinate Formulation ◽  
Three Dimensional ◽  
Plate Element ◽  
Absolute Nodal Coordinate ◽  
Finite Element Software ◽  
Elastic Forces ◽  
The Absolute ◽  
The Continuum

In this study, the improved description of elastic forces for the absolute nodal coordinate based plate element is introduced. The absolute nodal coordinate formulation, which utilizes global displacements and slope coordinates as nodal variables, can be used in large rotation and deformation dynamic analysis of beam and plate structures. The formulation avoids difficulties that arise when a rotation is interpolated in three-dimensional applications. In the absolute nodal coordinate formulation, a continuum mechanics approach has become the dominating procedure when elastic forces are defined. It has recently been perceived, however, that the continuum mechanics based absolute nodal coordinate elements suffer from serious shortcomings, including Poisson’s locking and poor convergence rate. These problems can be circumvented by modifying the displacement field of a finite element in the definition of elastic forces. This allows the use of the mixed type interpolation technique, leading to accurate and efficient finite element formulations. This approach has been previously applied to two- and three-dimensional absolute nodal coordinate based finite elements. In this study, the improved approach for elastic forces is extended to the absolute nodal coordinate plate element. The introduced plate element is compared in static examples to the continuum mechanics based absolute nodal coordinate plate element, as well as to commercial finite element software.

Use of Plasticity Theory in Flexible Multibody System Dynamics

2003 ◽  
Author(s):  
Hiroyuki Sugiyama ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Flexible Multibody Dynamics ◽  
Plasticity Theory ◽  
Position Vector ◽  
Absolute Nodal Coordinate ◽  
Flexible Multibody ◽  
The Absolute ◽  
Return Algorithm ◽  
Finite Element Formulations

The objective of this investigation is to develop a general nonlinear finite deformation formulation for the elastic-plastic analysis of flexible multibody systems. The Lagrangian plasticity theory based on J2 flow theory is used to account for the effect of plasticity in flexible multibody dynamics. In addition, it is demonstrated that the principle of objectivity that is an issue when existing finite element formulations using ratetype constitutive equations are used can be fully satisfied when the stress and strain rate are directly calculated in the Lagrangian descriptions using the absolute nodal coordinate formulation employed in this investigation. This is attributed to the fact that, in the finite element absolute nodal coordinate formulation, the position vector gradients can completely define the state of rotation and deformation within the element. As a consequence, the numerical algorithm used to determine the plastic deformations such as the Radial Return Algorithm becomes much simpler when the absolute nodal coordinate formulation is used as compared to existing finite element formulations that employ incrementally objective algorithms. Several numerical examples are presented in order to demonstrate the use of the formulations presented in the paper.

Finite element analysis of the geometric stiffening effect. Part 2: Non-linear elasticity

2005 ◽  
Vol 219 (2) ◽  
pp. 203-211 ◽  
Author(s):  
D García-Vallejo ◽  
H Sugiyama ◽  
A A Shabana
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Elasticity Theory ◽  
Linear Elasticity ◽  
Critical Speed ◽  
Absolute Nodal Coordinate Formulation ◽  
Linear Elasticity Theory ◽  
Rotating Beams ◽  
Absolute Nodal Coordinate ◽  
Non Linear

In the first part of this paper, the relationship between the number of finite elements used to model the dynamics of rotating beams and the critical speed at which an incorrect solution is obtained when using linear elasticity theory is discussed. The increase in the number of finite elements leads to an increase in the critical speed when linear elasticity is used and no measures are taken, as recommended in the literature, to account for the effect of the coupling between the bending and axial displacements. In this part of the paper, a non-linear finite element model based on the absolute nodal coordinate formulation is used to study the dynamics of rotating beams. It is shown that, when the non-linear elasticity theory is used, a stable solution is always obtained regardless of the number of finite elements used. Numerical results of various simulations are presented in order to compare the solution of a three-dimensional rotating beam that is obtained using the absolute nodal coordinate formulation with the results previously reported in the literature. A finite element numerical study of the dynamics of a helicopter rotor blade is also presented in this investigation. It is shown that, when the finite element absolute nodal coordinate formulation is used in the analysis of helicopter blades, the problem of ill-conditioning that characterizes many of the existing formulations is not encountered.

Development of Elastic Forces for a Large Deformation Plate Element Based on the Absolute Nodal Coordinate Formulation

2005 ◽  
Vol 1 (2) ◽  
pp. 103-108 ◽  
Author(s):  
Aki M. Mikkola ◽  
Marko K. Matikainen
Keyword(s):  
Finite Element ◽  
Dynamic Analysis ◽  
Continuum Mechanics ◽  
Absolute Nodal Coordinate Formulation ◽  
Three Dimensional ◽  
Plate Element ◽  
Absolute Nodal Coordinate ◽  
Elastic Forces ◽  
The Absolute ◽  
The Continuum

Dynamic analysis of large rotation and deformation can be carried out using the absolute nodal coordinate formulation. This formulation, which utilizes global displacements and slope coordinates as nodal variables, make it possible to avoid the difficulties that arise when a rotation is interpolated in three-dimensional applications. In the absolute nodal coordinate formulation, a continuum mechanics approach has become the dominating procedure when elastic forces are defined. It has recently been perceived, however, that the continuum mechanics based absolute nodal coordinate elements suffer from serious shortcomings, including Poisson’s locking and poor convergence rate. These problems can be circumvented by modifying the displacement field of a finite element in the definition of elastic forces. This allows the use of the mixed type interpolation technique, leading to accurate and efficient finite element formulations. This approach has been previously applied to two- and three-dimensional absolute nodal coordinate based finite elements. In this study, the improved approach for elastic forces is extended to the absolute nodal coordinate plate element. The introduced plate element is compared in static examples to the continuum mechanics based absolute nodal coordinate plate element, as well as to commercial finite element software. A simple dynamic analysis is performed using the introduced element in order to demonstrate the capability of the element to conserve energy.

Cross-Section Deformation in the Absolute Nodal Coordinate Formulation

2005 ◽  
Author(s):  
Hiroyuki Sugiyama ◽  
Johannes Gerstmayr ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Absolute Nodal Coordinate Formulation ◽  
Position Vector ◽  
High Frequencies ◽  
Absolute Nodal Coordinate ◽  
Strain Components ◽  
Nodal Coordinates ◽  
Material Element ◽  
Deformation Modes

In this investigation, the deformation modes defined in the finite element absolute nodal coordinate formulation using several strain definitions are discussed. In order to accurately define strain components that can have easy physical interpretation, a material coordinate system is introduced to define the material element rotation and deformation. The results obtained in this study clearly show cross-section deformation modes eliminated when the number of the finite element nodal coordinates is systematically and consistently reduced. Using the procedure discussed in this paper, one can obtain a reduced order dynamic model, eliminate position vector gradients that introduce high frequencies to the solution of some problems, achieve the continuity of the remaining gradients at the nodal points, and obtain a formulation that automatically satisfies the principle of work and energy.

Finite Element Analysis of the Geometric Stiffening Effect Using the Absolute Nodal Coordinate Formulation

2005 ◽  
Author(s):  
Daniel Garci´a-Vallejo ◽  
Hiroyuki Sugiyama ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Linear Elasticity ◽  
Absolute Nodal Coordinate Formulation ◽  
Coupling Effect ◽  
Mass Matrix ◽  
Element Analysis ◽  
Linear Elasticity Theory ◽  
Absolute Nodal Coordinate ◽  
The Absolute

In this paper, the limitations of the linear elasticity finite element solutions in describing the coupling between the extensional and bending displacements are discussed. The fact that incorrect unstable solutions are obtained for models with more than one finite element using the linear elasticity theory motivates the analytical study of the rotating beam presented in this paper. It is shown, as documented in the literature, that the instability of the incorrect solution is directly related to the singularity of the stiffness matrix, and such an instability occurs when the angular velocity reaches the first bending fundamental frequency of the beam. The increase in the number of finite elements only leads to an increase of the critical speed. Crucial in the analysis presented in this paper is the fact that the mass matrix and the form of the elastic forces obtained using the absolute nodal coordinate formulation remain the same under orthogonal coordinate transformation. The absolute nodal coordinate formulation, in contrast to conventional finite element formulations, does account for the effect of the coupling between bending and extension. A similar concept can be incorporated into the finite element floating frame of reference formulation in order to introduce coupling between the axial and bending displacements. Nonetheless, when the linear theory of elasticity is used and no special measures are taken to account for the coupling effect as proposed in the literature, there always exists a critical velocity regardless of the number of finite elements used.

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