p=2 Continuous finite elements on tetrahedra with local mass matrix inversion

2012 ◽  
Vol 213-216 ◽  
pp. 289-298 ◽  
Author(s):  
Michael J. Brazell ◽  
Brian T. Helenbrook
Keyword(s):  
Finite Elements ◽  
Mass Matrix ◽  
Matrix Inversion ◽  
Local Mass

Structural and Continuum Mechanics Approaches for a 3D Shear Deformable ANCF Beam Finite Element: Application to Buckling and Nonlinear Dynamic Examples

2013 ◽  
Vol 9 (1) ◽  
Author(s):  
Karin Nachbagauer ◽  
Johannes Gerstmayr
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Continuum Mechanics ◽  
Degrees Of Freedom ◽  
Mass Matrix ◽  
Finite Element Software ◽  
High Order Convergence ◽  
Dynamics Problems ◽  
Transverse Slope ◽  
Shear Deformable

For the modeling of large deformations in multibody dynamics problems, the absolute nodal coordinate formulation (ANCF) is advantageous since in general, the ANCF leads to a constant mass matrix. The proposed ANCF beam finite elements in this approach use the transverse slope vectors for the parameterization of the orientation of the cross section and do not employ an axial nodal slope vector. The geometric description, the degrees of freedom, and a continuum-mechanics-based and a structural-mechanics-based formulation for the elastic forces of the beam finite elements, as well as their usage in several static problems, have been presented in a previous work. A comparison to results provided in the literature to analytical solution and to the solution found by commercial finite element software shows accuracy and high order convergence in statics. The main subject of the present paper is to show the usability of the beam finite elements in dynamic and buckling applications.

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.

General Method for Modeling Slope Discontinuities and T-Sections Using ANCF Gradient Deficient Finite Elements

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Ahmed A. Shabana
Keyword(s):  
Finite Elements ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Constant Mass ◽  
Centrifugal Forces ◽  
Linear Transformations ◽  
Complete Set ◽  
Coordinate Lines ◽  
General Method ◽  
Slope Discontinuities

Slope discontinuities and T-sections can be modeled in a straight forward manner using fully parameterized absolute nodal coordinate formulation (ANCF) finite elements that have a complete set of gradient vectors. Linear transformations that define the element connectivity can always be obtained and used to preserve ANCF desirable features that include constant mass matrix and zero Coriolis and centrifugal forces in the case of spinning structures. The objective of this paper is to develop a general method that allows for modeling slope discontinuities and T-sections using gradient deficient ANCF finite elements that do not have a complete set of coordinate lines and gradient vectors. Linear connectivity conditions that preserve all the ANCF desirable features including the constant mass matrix are developed at the nodes of slope discontinuities. At these nodes of discontinuity, one can always define a complete set of independent coordinate lines that lie on the structure. These coordinate lines can be used to define a complete set of independent gradient vectors at these nodes. Since the proposed method is based on linear coordinate transformations, the method can be implemented in a preprocessor computer program. The application of the proposed general method is demonstrated using ANCF gradient deficient beam element example.

Finite Element Modeling of Dynamic Behavior of Some Basic Structural Members

1992 ◽  
Vol 114 (1) ◽  
pp. 3-9 ◽  
Author(s):  
R. C. Engels
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Finite Element Modeling ◽  
Mass Matrix ◽  
Matrix Approach ◽  
Structural Members ◽  
Shape Functions ◽  
Generalized Coordinates ◽  
Stiffness Matrices ◽  
Consistent Mass Matrix

A method is described to model the dynamics of finite elements. The assumed modes method is used to show how static shape functions approximate the element mass distribution. The deterioration of the modal content of a model can be linked to the neglect of interface restrained assumed modes. Restoration of a few of these modes leads to higher accuracy with fewer generalized coordinates compared to the standard consistent mass matrix approach. Also, no need exists for subdivision of basic elements such as rods and beams. The mass and stiffness matrices for several basic elements are derived and used in demonstration problems.

Free Undamped Vibration of Geometrically Non-Linear Cable Structures

1997 ◽  
Author(s):  
Alan S. K. Kwan
Keyword(s):  
Finite Elements ◽  
Mass Matrix ◽  
Three Dimensional ◽  
Geometrically Nonlinear ◽  
Theoretical Derivation ◽  
Cable Structures ◽  
Non Linear ◽  
Membrane Models ◽  
High Degree ◽  
Axial Element

The stiffness relationship and the distributed mass matrix for a geometrically nonlinear three dimensional straight axial element is derived for use in prestressed cablenet structures. The justification for the use of a linearised stiffness relationship is provided through a theoretical derivation. Results using this simple element have shown a high degree of correlation with results to those available in the literature obtained with more complex curved finite elements, analogous membrane models and other techniques.

Closed Form Solutions to the Equation of Motion Avoiding Mass Matrix Inversion and Eigenvectors Decomposition

2001 ◽  
Author(s):  
Karine Reaud ◽  
Claude Vallee ◽  
Danielle Fortune
Keyword(s):  
Closed Form ◽  
Mass Matrix ◽  
Equation Of Motion ◽  
Matrix Inversion ◽  
Matrix Exponential ◽  
Linear Case ◽  
Closed Form Solutions ◽  
Damping Matrix ◽  
Ill Conditioned Matrices ◽  
Damping Systems

Abstract Solutions of the vibrations equations are generally obtained, in the linear case, by methods involving either matrix exponential computation or matrix eigendecomposition. However, these methods lead to a loss of symmetry because of the necessary inversion of the mass matrix. In doing so, one can introduce ill-conditioned matrices and, thus, compute eigenvectors with poor accuracy. In order to avoid these inconveniences, our method, which is an extension of Le Verrier-Souriau algorithm, provides solutions to the vibrations equations without inverting the mass matrix or computing eigenvectors. Moreover, we can solve damping systems even when the damping matrix has no specific properties such as the Basile property.

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