Oleg Dmitrochenko

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

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. [DOI: 10.1115/1.4024910]

Extended Digital Nomenclature Code for Description of Complex Finite Elements and Generation of New Elements

In recent research (Dmitrochenko and Mikkola, 2011), a digital nomenclature code in the form dncm was proposed for a systematic classification of topology of finite elements (given by the dimension d and the number of nodes n ) and their kinematics (described by the number of coordinates per node c and a so-called vectorization multiplier m ). The digital code allows the kinematics of simple finite elements to be enumerated by a few integers; allowing the elements to be reconstructed without the need of their graphical representations. More complicated elements possess a set of nodal coordinates X that formally correspond to some code dncm ; however, their kinematics require that an auxiliary element ( d η ς μ) to be created using different topology η and kinematics ς, μ with a different set of nodal coordinates . Then, a transformation T toward coordinates X leads to an element systematically denoted by code , which is proposed in the current paper and called the extended digital nomenclature code. Examples of such elements are planar triangles and rectangles with drilling degrees of freedom, quadrilaterals with extra shape functions, discrete Kirchhoff triangles and other elements, including rigid bodies. It is possible to construct a universal procedure, which is capable of generating the necessary structural matrices of a finite element by its extended code dncm ( d η ς μ){…}. By changing digits d , n , c , m , η, ς, μ it is possible to find new elements, some of those are proposed in the current paper.

Beam Elements with Trapezoidal Cross Section Deformation Modes Based on the Absolute Nodal Coordinate Formulation

In this study, higher order beam elements are developed based on the absolute nodal coordinate formulation. The absolute nodal coordinate formulation is a finite element procedure that was recently proposed for flexible multibody applications. Many different elements based on the absolute nodal coordinate formulation are introduced, but still the beam elements are not able to describe the trapezoidal cross section mode. This leads to the locking phenomena, and therefore, the beam elements based on the absolute nodal coordinate formulation with three dimensional elasticity converge to an inexact solution. In order to avoid the locking phenomena, the trapezoidal cross section deformation mode is included in the beam elements based on the absolute nodal coordinate with additional degrees of freedom. The proper description for the trapezoidal cross section deformation is important for the continuum beam elements based on three‐dimensional elasticity where the material model is often based on general continuum...

Coupled Deformation Modes in the Large Deformation Finite Element Analysis: Generalization

The effect of the absolute nodal coordinate formulation (ANCF)-coupled deformation modes on the accuracy and efficiency when higher order three-dimensional beam and plate finite elements are used is investigated in this study. It is shown that while computational efficiency can be achieved in some applications by neglecting the effect of some of the ANCF-coupled deformation modes, such modes introduce geometric stiffening/ softening effects that can be significant in the case of very flexible structures. As shown in previous publications, for stiff structures, the effect of the ANCF-coupled deformation modes can be neglected. For such stiff structures, the solution does not strongly depend on some of the ANCF-coupled deformation modes, and formulations that include these modes lead to numerical results that are in good agreement with formulations that exclude them. In the case of a very flexible structure, on the other hand, the inclusion of the ANCF-coupled deformation modes becomes necessary in order to obtain an accurate solution. In this case of very flexible structures, the use of the general continuum mechanics approach leads to an efficient solution algorithm and to more accurate numerical results. In order to examine the effect of the elastic force formulation on the efficiency and the coupling between different modes of deformation, three different models are used again to formulate the elastic forces in the absolute nodal coordinate formulation. These three methods are the general continuum mechanics approach, the elastic line (midsurface) approach, and the elastic line (midsurface) approach with the Hellinger-Reissner principle. Three-dimensional absolute nodal coordinate formulation beam and plate elements are used in this study. In the general continuum mechanics approach, the coupling between the cross section deformation and the beam centerline or plate midsurface displacement is considered, while in the approaches based on the elastic line and the Hellinger-Reissner principle, this coupling is neglected. In addition to the fully parametrized beam element used in this study, three different plate elements, two fully parametrized and one reduced order thin plate elements, are used. The numerical results obtained using different finite elements and elastic force formulations are compared in this study.

Shear Correction for Thin Plate Finite Elements Based on the Absolute Nodal Coordinate Formulation

This study is an extension of a newly introduced approach to account transverse shear deformation in absolute nodal coordinate formulation. In the formulation, shear deformation is usually defined by employing slope vectors in the element transverse direction. This leads to the description of deformation modes that, in practical problems, may be associated with high frequencies. These high frequencies, in turn, could complicate the time integration procedure, burdening numerical performance of shear deformable elements. In a recent study of this paper’s authors, the description of transverse shear deformation is accounted for in a two-dimensional beam element, based on the absolute nodal coordinate formulation without the use of transverse slope vectors. In the introduced shear deformable beam element, slope vectors are replaced by vectors that describe the rotation of the beam cross-section. This procedure represents a simple enhancement that does not decrease the accuracy or numerical performance of elements based on the absolute nodal coordinate formulation. In this study, the approach to account for shear deformation without using transverse slopes is implemented for a thin rectangular plate element. In fact, two new plate elements are introduced: one within conventional finite element and another using the absolute nodal coordinates. Numerical results are presented in order to demonstrate the accuracy of the introduced plate element. The numerical results obtained using the introduced element agree with the results obtained using previously proposed shear deformable plate elements.© 2009 ASME

Large Deformation Triangular Plate Elements for Multibody Problems

In this paper, triangular finite elements based on the absolute nodal coordinate formulation are introduced. Triangular elements employ the Kirchhoff plate theory and can, accordingly, be used in thin plate bending problems. These elements can exactly describe arbitrary rigid body motion while their mass matrices are constant. Previous plate developments in the absolute nodal coordinate formulation have focused on rectangular elements that are difficult to use when arbitrary meshes need to be described. The elements introduced in this study have overcome this problem and represent an important addition to the absolute nodal coordinate formulation. The two elements introduced are based on Specht’s and Morley’s shape functions. The numerical solutions of these elements are compared with results obtained using the previously proposed rectangular finite element and analytical results.Copyright

The Simplest 3- and 4-Noded Fully-Parameterized ANCF Plate Elements

In this research, the simplest kinematical models of triangular and rectangular plate finite elements using the absolute nodal coordinate formulation (ANCF) are presented. The ANCF is the finite-element large-displacement-and-rotation approach, which uses the inertial-frame nodal position vectors and their derivatives (slopes) only, without employing any rotation parameters or their equivalent. As a consequence, the kinematics of the elements becomes linear, simplifying the inertia part of the equations of motion, which is also linear. In contrary, due to the need for employing the Green-Lagrange strain tensor, the elastic forces normally appear in a more complicated highly-nonlinear manner than in other large-rotation formulations. In this research, to reduce the computational burden, two new plate elements are proposed that are the simplest possible triangular and rectangular elements in the fully-parameterized ANCF: they employ transverse slopes only, without using longitudinal slopes.Copyright

A Procedure for the Inclusion of Transverse Shear Deformation in a Beam Element Based on the Absolute Nodal Coordinate Formulation

In this study, a procedure to account for transverse shear deformation in the absolute nodal coordinate formulation is presented. In the absolute nodal coordinate formulation, shear deformation is usually defined by employing the slope vectors in the element transverse direction. This leads to the description of deformation modes that are, in practical problems, associated with high frequencies. These high frequencies, in turn, complicate the time integration procedure burdening numerical performance. In this study, the description of transverse shear deformation is accounted for in a two-dimensional beam element based on the absolute nodal coordinate formulation without the use of transverse slope vectors. In the introduced shear deformable beam element, slope vectors are replaced by vectors that describe the rotation of the beam cross-section. This procedure represents a simple enhancement that does not decrease the accuracy or numerical performance of elements based on the absolute nodal coordinate formulation. Numerical results are presented in order to demonstrate the accuracy of the introduced element in static and dynamic cases. The numerical results obtained using the introduced element agree with the results obtained using previously proposed shear deformable beam elements.Copyright

On Pure-Bending Non-Linear Plate Elements With Developable Surfaces

In this study, a new approach for generating beam and plate finite elements is proposed for a large-deformation and large-rotation dynamical multibody simulation of thin structures. The elements employ curvatures and/or torsions in the definition of generalized coordinates. This definition of element degrees of freedom is based on intrinsic properties of a spatial curve or a surface and it avoids longitudinal elongations of the beams and plates with associated high frequency vibrations, which helps to avoid problems during numerical simulation.Copyright

Generation of Matrices of a Finite Element by its Code dncm

A numerical procedure of generating the matrices of any finite element by its digital nomenclature code dncm [1,2] is proposed. Digit d is the dimension, i.e. the number of arguments x, y,… used in the interpolation polynomial Y(x,…) of the element; n is the number of nodes of the element; c is the number of coordinates per node, or the number of derivatives of the field variable Y per node; m is the vectorization multiplier showing how many polynomials are interpolated simultaneously. The dncm code itself only specifies the interpolation polynomials of the finite element. The physical properties of the element should additionally be given by a variation functional which employs, e.g., strain energy. In most cases, the form of the functional can be also identified by dncm code.