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.

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.

Elastic Forces and Coordinate Systems in the Finite Element Formulations

1999 ◽  
Author(s):  
Marcello Berzeri ◽  
Marcello Campanelli ◽  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Frame Of Reference ◽  
Coordinate Systems ◽  
Absolute Nodal Coordinate ◽  
Floating Frame Of Reference ◽  
Elastic Forces ◽  
Floating Frame ◽  
The Absolute ◽  
Finite Element Formulations

Abstract The equivalence of the elastic forces of finite element formulations used in flexible multibody dynamics is the focus of this investigation. Two conceptually different finite element formulations that lead to exact modeling of the rigid body dynamics will be used. These are the floating frame of reference formulation and the absolute nodal coordinate formulation. It is demonstrated in this study that different element coordinate systems, which are used for the convenience of describing the element deformations in the absolute nodal coordinate formulation, lead to similar results as the element size is reduced. The equivalence of the elastic forces in the absolute nodal coordinate and the floating frame of reference formulations is shown. The result of this analysis clearly demonstrates that the instability observed in high speed rotor analytical models due to the neglect of the geometric centrifugal stiffening is not a problem inherent to a particular finite element formulation but only depends on the beam model that is used. Fourier analysis of the solutions obtained in this investigation also sheds new light on the fundamental problem of the choice of the deformable body coordinate system in the floating frame of reference formulation. A new method is presented and used to obtain a simple expression for the elastic forces in the absolute nodal coordinate formulation. This method, which employs a nonlinear elastic strain-displacement relationship, does not result in an unstable solution when the angular velocity is increased.

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.

Finite element analysis of the geometric stiffening effect. Part 1: A correction in the floating frame of reference formulation

2005 ◽  
Vol 219 (2) ◽  
pp. 187-202 ◽  
Author(s):  
D García-Vallejo ◽  
H Sugiyama ◽  
A A Shabana
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Frame Of Reference ◽  
Reference Formulation ◽  
Absolute Nodal Coordinate ◽  
Floating Frame Of Reference ◽  
Elastic Forces ◽  
Floating Frame ◽  
The Absolute ◽  
Finite Element Formulations

The fact that incorrect unstable solutions are obtained for linearly elastic models motivates the analytical study presented in this paper. The increase in the number of finite elements only leads to an increase in 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. Based on the analytical results obtained using the absolute nodal coordinate formulation, a new correction is proposed for the finite element floating frame of reference formulation in order to introduce coupling between the axial and bending displacements. In this two-part paper, two- and three-dimensional finite element models are used to study the problem of rotating beams. The models are developed using the absolute nodal coordinate formulation that allows for accurate representation of the axial strain, thereby avoiding the ill-conditioning problem that arises when classical displacement-based finite element formulations are used. In the first part of the paper, the case of linear elasticity is considered and assumptions used in the finite element floating frame of reference formulation are investigated. In the second part of the paper, non-linear elasticity is considered. A rotating helicopter blade is simulated, and the complexity of the motion suggests the inclusion of rotary inertia, shear deformation, and non-linear elastic forces in order to obtain an accurate solution that does not suffer from the instability problem regardless of the number of finite elements used.

Studies on the Stiffness Properties of the Absolute Nodal Coordinate Formulation for Three-Dimensional Beams

2003 ◽  
Author(s):  
Jussi T. Sopanen ◽  
Aki M. Mikkola
Keyword(s):  
Cross Section ◽  
Absolute Nodal Coordinate Formulation ◽  
Three Dimensional ◽  
Beam Element ◽  
Element Formulation ◽  
Absolute Nodal Coordinate ◽  
Numerical Examples ◽  
Elastic Forces ◽  
Nodal Coordinates ◽  
The Absolute

The objective of this study is to investigate the accuracy of elastic force models that can be used in the absolute nodal coordinate finite element formulation for the analysis of threedimensional beams. The elastic forces of the absolute nodal coordinate formulation can be derived using a continuum mechanics approach. This study investigates the accuracy and usability of such an approach for the three-dimensional absolute nodal coordinate beam element. This study also presents an improvement proposal for the use of a continuum mechanics approach in deriving the expression of the elastic forces of the beam element. The improvement proposal is verified using several numerical examples. Numerical examples show that the proposed elastic force model of the beam element agrees with analytical results as well as with solutions obtained using existing finite element formulation. The results also imply that the beam element does not suffer from the phenomenon called shear locking. In the beam element under investigation, global displacements and slopes are used as the nodal coordinates, which resulted in a large number of nodal degrees of freedom. This study provides a physical interpretation of the nodal coordinates used in the absolute nodal coordinate beam element. It is shown that the beam element based on the absolute nodal coordinate formulation relaxes the assumption of the rigid cross-section and is capable of representing a distortional deformation of the cross-section.

A Spatial Thin Beam Finite Element Based on the Absolute Nodal Coordinate Formulation Without Singularities

2011 ◽  
Author(s):  
Karin Nachbagauer ◽  
Peter G. Gruber ◽  
Yury Vetyukov ◽  
Johannes Gerstmayr
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Displacement Vector ◽  
Three Dimensional ◽  
Axial Rotation ◽  
Euler Beam ◽  
Cross Sectional ◽  
Absolute Nodal Coordinate ◽  
The Absolute ◽  
Element Free

A three-dimensional nonlinear finite element for thin beams is proposed within the absolute nodal coordinate formulation (ANCF). The deformation of the element is described by means of displacement vector, axial slope and axial rotation parameter per node. The element is based on the Bernoulli-Euler theory and can undergo coupled axial extension, bending and torsion in the large deformation case. Singularities — which are typically caused by such parameterizations — are overcome by a director per element node. Once the directors are properly defined, a cross sectional frame is defined at any point of the beam axis. Since the director is updated during computation, no singularities occur. The proposed element is a three-dimensional ANCF Bernoulli-Euler beam element free of singularities and without transverse slope vectors. Detailed convergence analysis by means of various numerical examples and comparison to analytical solutions shows the performance and accuracy of the element.

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

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
Abdel-Nasser A. Mohamed
Keyword(s):  
Rectangular Plate ◽  
Absolute Nodal Coordinate Formulation ◽  
Plate Theory ◽  
Three Dimensional ◽  
Plate Element ◽  
Shape Functions ◽  
Absolute Nodal Coordinate ◽  
Triangular Plate ◽  
The Absolute ◽  
Rectangular Plate Element

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.

Performance of the Incremental and Non-Incremental Finite Element Formulations in Flexible Multibody Problems

1999 ◽  
Vol 122 (4) ◽  
pp. 498-507 ◽  
Author(s):  
Marcello Campanelli ◽  
Marcello Berzeri ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Multibody Systems ◽  
Absolute Nodal Coordinate Formulation ◽  
Flexible Multibody Systems ◽  
Large Displacement ◽  
Body Motion ◽  
Absolute Nodal Coordinate ◽  
Flexible Multibody ◽  
The Absolute ◽  
Finite Element Formulations

Many flexible multibody applications are characterized by high inertia forces and motion discontinuities. Because of these characteristics, problems can be encountered when large displacement finite element formulations are used in the simulation of flexible multibody systems. In this investigation, the performance of two different large displacement finite element formulations in the analysis of flexible multibody systems is investigated. These are the incremental corotational procedure proposed in an earlier article (Rankin, C. C., and Brogan, F. A., 1986, ASME J. Pressure Vessel Technol., 108, pp. 165–174) and the non-incremental absolute nodal coordinate formulation recently proposed (Shabana, A. A., 1998, Dynamics of Multibody Systems, 2nd ed., Cambridge University Press, Cambridge). It is demonstrated in this investigation that the limitation resulting from the use of the infinitesmal nodal rotations in the incremental corotational procedure can lead to simulation problems even when simple flexible multibody applications are considered. The absolute nodal coordinate formulation, on the other hand, does not employ infinitesimal or finite rotation coordinates and leads to a constant mass matrix. Despite the fact that the absolute nodal coordinate formulation leads to a non-linear expression for the elastic forces, the results presented in this study, surprisingly, demonstrate that such a formulation is efficient in static problems as compared to the incremental corotational procedure. The excellent performance of the absolute nodal coordinate formulation in static and dynamic problems can be attributed to the fact that such a formulation does not employ rotations and leads to exact representation of the rigid body motion of the finite element. [S1050-0472(00)00604-8]

Comparison of Three-Dimensional Flexible Beam Elements for Dynamic Analysis: Classical Finite Element Formulation and Absolute Nodal Coordinate Formulation

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
A. L. Schwab ◽  
J. P. Meijaard
Keyword(s):  
Dynamic Analysis ◽  
Absolute Nodal Coordinate Formulation ◽  
Three Dimensional ◽  
Flexible Beam ◽  
Element Formulation ◽  
Shear Locking ◽  
Elastic Line ◽  
Absolute Nodal Coordinate ◽  
Beam Elements ◽  
Bending Mode

Three formulations for a flexible spatial beam element for dynamic analysis are compared: a Timoshenko beam with large displacements and rotations, a fully parametrized element according to the absolute nodal coordinate formulation (ANCF), and an ANCF element based on an elastic line approach. In the last formulation, the shear locking of the antisymmetric bending mode is avoided by the application of either the two-field Hellinger–Reissner or the three-field Hu–Washizu variational principle. The comparison is made by means of linear static deflection and eigenfrequency analyses on stylized problems. It is shown that the ANCF fully parametrized element yields too large torsional and flexural rigidities, and shear locking effectively suppresses the antisymmetric bending mode. The presented ANCF formulation with the elastic line approach resolves most of these problems.

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.

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