Position Vector Gradients Research Articles

Absolute nodal coordinate formulation for dynamic analysis of reinforced concrete structures

An efficient and accurate beam element for performing explicit dynamic analysis of reinforced concrete beam elements was built using the absolute nodal coordinate formulation (ANCF). Such a formulation is useful for performing dynamic analysis under large strains and displacement with sudden load changes, such as during impact loading or for progressive collapse analysis. Compared with traditional nodal position formulations, ANCF uses position vector gradients to describe the rotation of the body and strain state, thereby avoiding the need for interpolating non-vectoral rotation parameters. The fiber element method was applied to calculate element stiffness from section deformations by subdividing beam elements into uniaxial fibers. The combined fiber element beam-column model with ANCF was validated by traditional co-rotational analysis and the quadrature element method (QEM), and thus may be used in general nonlinear dynamic analysis.

Convergence Characteristics of Geometrically Accurate Spatial Finite Elements

Abstract The convergence characteristics of three geometrically accurate spatial finite elements (FEs) are examined in this study using an eigenvalue analysis. The spatial beam, plate, and solid elements considered in this investigation are suited for both structural and multibody system (MBS) applications. These spatial elements are based on geometry derived from the kinematic description of the absolute nodal coordinate formulation (ANCF). In order to allow for an accurate reference-configuration geometry description, the element shape functions are formulated using constant geometry coefficients defined using the position-vector gradients in the reference configuration. The change in the position-vector gradients is used to define a velocity transformation matrix that leads to constant element inertia and stiffness matrices in the case of infinitesimal rotations. In contrast to conventional structural finite elements, the elements considered in this study can be used to describe the initial geometry with the same degree of accuracy as B-spline and nonuniform rational B-spline (NURBS) representations, widely used in the computer-aided design (CAD). An eigenvalue analysis is performed to evaluate the element convergence characteristics in the case of different geometries, including straight, tapered, and curved configurations. The frequencies obtained are compared with those obtained using a commercial FE software and analytical solutions. The stiffness matrix is obtained using both the general continuum mechanics (GCM) approach and the newly proposed strain split method (SSM) in order to investigate its effectiveness as a locking alleviation technique.

Generalization of the strain-split method and evaluation of the nonlinear ANCF finite elements

This paper discusses the generalization of the strain-split method (SSM) for the locking alleviation of curved structures. The generalization is achieved by using proper definitions of the stress and strain tensors along the curved-coordinate lines using the matrix of position vector gradients in the reference configuration. This matrix, which accurately captures the element geometry at the integration points, allows using consistent gradient transformation in the calculation of the stress and strain tensors. The generalized SSM implementation is used to verify the results and evaluate the performance of the absolute nodal coordinate formulation (ANCF) finite elements (FE). The focus of this study is on the Poisson locking that characterizes fully parameterized ANCF elements that employ different orders of interpolation in different directions. ANCF beam and plate nonlinear problems are presented, and the obtained simulation results are compared with analytical solutions as well as results obtained using commercial FE computer programs. These results are also compared with the results obtained using ANCF beam and curved plate elements in the case of nonzero Poisson ratio in order to demonstrate the SSM effectiveness in alleviating the Poisson locking. It is shown that a much smaller number of ANCF plate elements is required to achieve approximately 0.9% difference from the results of commercial FE computer programs.

Development of Geometrically Accurate Continuum-Based Tire Models for Virtual Testing

AbstractIn this paper, an approach based on the integration of computer-aided design and analysis (I-CAD-A) is used to develop new continuum-based finite element (FE) tire models for the small and large deformation analyses. Based on given tire specifications, the mechanics-based geometry/analysis absolute nodal coordinate formulation (ANCF) is used to define the tire geometry with the same degree of accuracy as B-splines and nonuniform rational B-spline (NURBS), widely used in the computer-aided design (CAD) systems. In the case of large deformations, the ANCF geometry can be used directly as the analysis mesh without the need for conversion or adjustments. In order to define the material parameters that characterize the ANCF tire composite structure, a virtual test rig is developed, and the tire calibration process is performed according to the standards defined by the Society of Automotive Engineers (SAE). In order to develop small-deformation models that can be used in the prediction of the tire frequencies and mode shapes, the ANCF position vector gradients are consistently written in terms of rotation parameters, leading to geometrically accurate floating frame of reference (FFR) finite elements, referred to as ANCF/FFR elements. Using this mechanics-based geometry/analysis approach, new geometrically accurate reduced-order tire models are systematically developed and used to define vibration equations for the prediction of the tire frequencies, which are verified using a commercial FE software. The element stiffness matrix is calculated using the general continuum mechanics approach (GCM), and the effectiveness of the strain split method (SSM) for locking alleviation is tested. The results obtained in this investigation show that the I-CAD-A tire modeling approach can be used to develop geometrically accurate tire models suited for the large-deformation multibody system (MBS) problems as well as for the prediction of the tire frequencies and mode shapes.

Spatial ANCF/CRBF beam elements

In this paper, the consistent rotation-based formulation (CRBF) is used to develop new three-dimensional beam elements starting with the absolute nodal coordinate formulation (ANCF) kinematic description. While the proposed elements employ orientation parameters as nodal coordinates, independent rotation interpolation is avoided, leading to unique displacement and rotation fields. Furthermore, the proposed spatial ANCF/CRBF-based beam elements adhere to the noncommutative nature of the rotation parameters, allow for arbitrarily large three-dimensional rotation, and eliminate the need for using co-rotational or incremental solution procedures. Because the proposed elements have a general geometric description consistent with computational geometry methods, accurate definitions of the shear and bending deformations can be developed and evaluated, and curved structures and complex geometries can be systematically modeled. Three new spatial ANCF/CRBF beam elements, which use absolute positions and rotation parameters as nodal coordinates, are proposed. The time derivatives of the ANCF transverse position vector gradients at the nodes are expressed in terms of the time derivatives of rotation parameters using a nonlinear velocity transformation matrix. The velocity transformation leads to lower-dimensional elements that ensure the continuity of stresses and rotations at the element nodal points. The numerical results obtained from the proposed ANCF/CRBF elements are compared with the more general ANCF beam elements and with elements implemented in a commercial FE software.

Development and implementation of geometrically accurate reduced-order models: Convergence properties of planar beams

A geometrically accurate infinitesimal-rotation planar beam element is developed and implemented in this study. The performance of the element, which is suited for developing reduced-order models for both structural and multibody systems (MBS), is evaluated using an eigenvalue analysis. Unlike conventional infinitesimal-rotation finite elements (FE), the new element is compatible with the computer-aided design (CAD) B-spline and NURBS (Non-Uniform Rational B-Spline) representations and allows for a straightforward linear transformation of CAD solid models to FE analysis meshes. The absolute nodal coordinate formulation (ANCF) elements, which are related to B-splines and NURBS by linear mapping, are used as the basis for developing the planar beam element. The new element has a shape function matrix expressed in terms of geometric coefficients obtained using the ANCF position vector gradients in the reference configuration. The change in the position vector gradients is written in terms of infinitesimal rotation coordinates using a velocity transformation that defines constant element mass and stiffness matrices. Using this approach, initially straight and curved configurations can be modeled using the same displacement field. The eigenvalue analysis is used to evaluate the element performance and examine the effect of shear locking on the predicted frequencies. Several elastic force formulations are used to evaluate the convergence characteristics, including the direct displacement method (DDM), general continuum mechanics (GCM) approach, elastic line (EL) approach, and strain split method (SSM). The element performance is compared with the conventional Euler-Bernoulli and Timoshenko elements as well as the analytical solutions.

Geometrically accurate infinitesimal-rotation spatial finite elements

In this paper, a general procedure is used to develop new geometrically accurate infinitesimal-rotation finite elements (FE). New spatial beam, plate, and solid elements are developed in terms of constant geometric coefficients obtained using the matrix of position vector gradients. The spatial beam, plate, and solid element shape functions are developed using the displacement field of the absolute nodal coordinate formulation (ANCF). These elements are suited for developing reduced-order models for structural and multibody system (MBS) analyses, particularly when the floating frame of reference (FFR) formulation is used. The initial geometry of the new elements, referred to as the ANCF/FFR, is related to B-splines and NURBS (non-uniform rational B-splines) by a linear mapping, and therefore, their use does not lead to geometry distortion when geometry models are converted to analysis meshes. The main contribution of this technical note is the proposed general procedure for the development of the geometrically accurate spatial ANCF/FFR elements and demonstrating the use of this procedure by developing the ANCF/FFR displacement field of three different spatial elements.

Locking alleviation in the large displacement analysis of beam elements: the strain split method

This paper proposes a new locking alleviation technique for absolute nodal coordinate formulation (ANCF) beam and plate elements based on a strain split approach. The paper also surveys classical finite element (FE) and ANCF locking alleviation techniques discussed in the literature. Because ANCF beam elements, which allow for the cross-sectional stretch fully capture the Poisson effect, Poisson locking is an issue when such beam elements are considered. The two-dimensional fully parameterized ANCF beam element is primarily used in this investigation because such an element can serve as a good surrogate model for three-dimensional ANCF beams and plates as far as membrane, bending and transverse shearing behavior is concerned. In addition to proposing the strain split method (SSM) for ANCF locking alleviation, this work assesses the ANCF element performance in the cases of higher-order interpolation, enhanced assumed strain method, elastic line method, and the enhanced continuum mechanics approach, and demonstrates the design of the enhanced strain interpolation function by using the shape functions of higher-order ANCF elements. Additionally, a new higher-order ANCF two-dimensional beam element is proposed in order to compare its performance with other finite elements that require the use of other locking alleviation techniques proposed and reviewed in the paper. Finally, several numerical examples are shown to demonstrate the effectiveness of the locking alleviation methods applied to ANCF elements. The purpose of this investigation, apart from proposing a new locking alleviation technique, a new higher-order beam element, and comparing several existing locking alleviation techniques, is to show that dealing with locking in fully parameterized ANCF elements is feasible and that several methods exist to effectively improve the ANCF element performance without sacrificing important ANCF element properties and features including position vector gradient continuity. Because of the use of ANCF position vector gradients as nodal coordinates, complex stress-free initially-curved geometries can be systematically obtained. Such initially-curved geometries require special attention when attempting to solve locking problems, as will be discussed in this paper.

Use of independent volume parameters in the development of new large displacement ANCF triangular plate/shell elements

In this paper, a new unified kinematic description, obtained from Bezier geometry using linear mapping and position vector gradients associated with three independent parameters, is used to develop large displacement plate/shell finite elements (FE). Contrary to the conventional FE method, in the approach developed in this paper based on the absolute nodal coordinate formulation (ANCF), no distinction is made between plate and shell structures. The proposed ANCF triangular plate/shell elements have 12 coordinates per node: three position coordinates and nine position gradient coordinates that define vectors tangent to coordinate lines at the nodes. The fundamental differences between the conventional FE and the new ANCF parameterizations are highlighted. In this investigation, two different parameterizations, each of which employs independent coordinates, are used. In the first parameterization, called volume parameterization, coordinate lines along the sides of the triangular element in the straight (un-deformed) configuration are used in order to facilitate the development of closed-form cubic shape functions. In the second parameterization, called Cartesian parameterization, coordinate lines along the global axes of the structure (body) coordinate system are used to facilitate the element assembly. The element transformation between the volume and the Cartesian parameterizations is developed and used to define the structure inertia and elastic forces. Three new fully parameterized ANCF triangular plate/shell elements are developed in this investigation: a four-node mixed-coordinate element (FNMC) and two three-node elements (TN1 and TN2). All the elements developed in this investigation lead to a constant mass matrix and zero Coriolis and centrifugal forces. A non-incremental total Lagrangian procedure is used for the numerical solution of the nonlinear equations of motion. The performance of the proposed ANCF triangular plate/shell elements is analyzed by comparison with the ANCF rectangular plate element and conventional three-node linear (TNL) and six-node quadratic (SNQ) triangular plate elements.

Verification of a Total Lagrangian ANCF Solution Procedure for Fluid–Structure Interaction Problems

The objective of this investigation is to verify a new total Lagrangian continuum-based fluid model that can be used to solve two- and three-dimensional fluid–structure interaction problems. Large rotations and deformations experienced by the fluid can be captured effectively using the finite element (FE) absolute nodal coordinate formulation (ANCF). ANCF elements can describe arbitrarily complex fluid shapes without imposing any restriction on the amount of rotation and deformation within the finite element, ensure continuity of the time-rate of position vector gradients at the nodal points, and lead to a constant mass matrix regardless of the magnitude of the fluid displacement. Fluid inertia forces are computed, considering the change in the fluid geometry as the result of the large displacements. In order to verify the ANCF solution, the dam-break benchmark problem is solved in the two- and three-dimensional cases. The motion of the fluid free surface is recorded before and after the impact on a vertical wall placed at the end of the dam dry deck. The results are in good agreement with those obtained by other numerical methods. The results obtained in this investigation show that the number of degrees-of-freedom (DOF) required for ANCF convergence is around one order of magnitude less than what is required by other existing methods. Limitations and advantages of the verified ANCF fluid model are discussed.

A New ANCF/CRBF Fully Parameterized Plate Finite Element

In this paper, the consistent rotation-based formulation (CRBF) is used to develop a new fully parametrized plate finite element (FE) based on the kinematic description of the absolute nodal coordinate formulation (ANCF). The ANCF/CRBF plate element has a general geometric description which is consistent with the basic principles of continuum mechanics, defines a unique rotation field, ensures the continuity of the rotation and strains at the element nodes, can describe an arbitrarily large displacement, and is consistent with computational geometry methods allowing for correctly describing complex shapes as demonstrated in this paper. The proposed ANCF/CRBF finite element does not suffer from the serious and fundamental problems encountered when using other large rotation vector formulations (LRVF) including the coordinate redundancy and violation of the principle of non-commutativity of the finite rotations which cannot be treated as vectors. The proposed bi-cubic ANCF/CRBF plate element employs, as nodal coordinates, three position coordinates and three finite rotation parameters. This element is obtained from a fully parameterized ANCF plate element by writing the position vector gradients of the ANCF plate element in terms of three finite rotation parameters using a nonlinear velocity transformation that systematically reduces the number of the element coordinates. The resulting element captures stretch, bending, and twist deformation modes and it allows for systematically describing complex curved geometry. Because of the lower dimensionality of the resulting ANCF/CRBF plate element, it does not capture complex deformation modes that can be captured using the more general ANCF finite elements. Furthermore, the ANCF/CRBF element mass matrix is not constant, leading to nonlinear Coriolis and centrifugal inertia forces. The new element is validated by examining its performance using several examples that include pendulum plate, free falling plate, and tire models. The results obtained using this new element are compared with the results obtained using the bi-cubic fully parameterized ANCF plate element, the bi-linear shell element, and the conventional solid element implemented in the commercial software ANSYS.

A two-dimensional shear deformable ANCF consistent rotation-based formulation beam element

It has been demonstrated recently that the absolute nodal coordinate formulation (ANCF) can be used to develop lower-order consistent rotation-based formulations (CRBFs) that employ finite rotation parameters as nodal coordinates without the need for interpolating the rotation field. The objective of this study is to develop new planar shear deformable ANCF/CRBF beam elements and demonstrate their use. A cubic ANCF/CRBF shear deformable beam element is first developed starting with the ANCF kinematic description that employs position vector gradients as nodal coordinates. The transverse position vector gradients at the nodes are expressed in terms of finite rotation parameters, leading to a lower-dimensional beam element model that captures the shear deformation, ensures continuity of the stresses and rotations at the nodes, allows for an arbitrary large displacement, and has a kinematic description consistent with geometry methods and suitable for systematically modeling curved structures. The results, obtained using the new planar ANCF/CRBF shear deformable beam element, are compared with the original and more general ANCF shear deformable beam element. Another lower-dimension bilinear CRBF beam element which has three coordinates at each node, two translation coordinates and one rotation parameter, is also developed in this investigation. The formulations of the three finite elements, including the ANCF finite element, considered in this investigation are compared. Numerical results are presented in order to demonstrate the use of the new formulations and test their performances in the analysis of large displacements and deformations. While the ANCF/CRBF assumptions are evaluated numerically, the results obtained show, in general, a good agreement between the elements considered in this study. The results also show that the CRBF finite elements, which have nonlinear mass matrix, can be more efficient for smaller meshes. As the mesh size and the number of finite elements increase, the original higher-order ANCF finite elements, which have constant mass matrix and zero Coriolis and centrifugal forces, become more efficient. The ANCF/CRBF approach clearly demonstrates that there is no need for introducing an independent rotation field to capture the shear effect.

Rational ANCF Thin Plate Finite Element

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.

Definition of ANCF Finite Elements

Since the absolute nodal coordinate formulation (ANCF) was introduced, a large number of fully parametrized and gradient deficient finite elements were developed. Some of the finite elements (FE) proposed do not fall into the ANCF category, and for this reason, this technical brief describes the general requirements for ANCF finite elements. As discussed in this paper, some of the conventional isoparametric finite elements can describe arbitrary rigid body displacements and can be used with a nonincremental solution procedure. Nonetheless, these isoparametric elements, particularly the ones that employ position coordinates only, are of the C0 type and do not ensure the continuity of the position vector gradients. It is also shown that the position vector gradient continuity conditions can be described using homogeneous algebraic equations, and such conditions are different from those conditions that govern the displacement vector gradients. The use of the displacement vector gradients as nodal coordinates does not allow for an isoparametric representation that accounts for both the initial geometry and displacements using one kinematic description, can make the element assembly more difficult, and can complicate imposing linear algebraic constraint equations at a preprocessing stage. Understanding the ANCF geometric description will allow for the development of new mechanics-based computer-aided design (CAD)/analysis systems as briefly discussed in this paper.

Low Order Continuum-Based Liquid Sloshing Formulation for Vehicle System Dynamics

The objective of this investigation is to develop a low order continuum-based liquid sloshing model that can be successfully integrated with multibody system (MBS) algorithms. The liquid sloshing model proposed in this investigation allows for capturing the effect of the distributed inertia and viscosity of the fluid. The fluid viscous forces are defined using the Navier–Stokes equations. In order to demonstrate the use of the approach presented in this study, the assumption of an incompressible Newtonian fluid is considered with a total Lagrangian approach. Fluid properties such as the incompressibility condition are formulated using a penalty method. The low order model that captures the effect of the distributed fluid inertia on the vehicle dynamics is developed in this investigation using the floating frame reference (FFR) formulation. The use of this approach allows for developing an inertia-variant fluid model that accounts for the dynamic coupling between different modes of the fluid displacements. The matrix of position vector gradients and its derivative are formulated using the FFR kinematic description. The position and velocity gradient tensors are used to define the Navier–Stokes stress forces. The proposed liquid sloshing model is integrated with a MBS railroad vehicle model in which the rail/wheel interaction is formulated using a 3D elastic contact formulation that allows for the wheel/rail separation. Several simulation scenarios are used to examine the effect of the distributed liquid inertia on the motion of the railroad vehicle. The results, obtained using the sloshing model, are compared with the results obtained using a rigid body vehicle model. The comparative numerical study presented in this investigation shows that the effect of the sloshing tends to increase the possibility of wheel/rail separation as the forward velocity increases, thereby increasing the possibility of derailments at these relatively high speeds.

A nonlinear visco-elastic constitutive model for large rotation finite element formulations

Although all known materials have internal damping that leads to energy dissipation, most existing large deformation visco-elastic finite element formulations are based on linear constitutive models or on nonlinear constitutive models that can be used in the framework of an incremental co-rotational finite element solution procedure. In this investigation, a new nonlinear objective visco-elastic constitutive model that can be implemented in non-incremental large rotation and large deformation finite element formulations is developed. This new model is based on developing a simple linear relationship between the damping forces and the rates of deformation vector gradients. The deformation vector gradients can be defined using the decomposition of the matrix of position vector gradients. In this paper, the decomposition associated with the use of the tangent frame that is equivalent to the QR decomposition is employed to define the matrix of deformation gradients that enter into the formulation of the viso-elastic constitutive model developed in this investigation. Using the relationship between the deformation gradients and the components of the Green–Lagrange strain tensor, it is shown that the damping forces depend nonlinearly on the strains and linearly on the classical strain rates. The relationship between the damping forces and strains and their rates is used to develop a new visco-elastic model that satisfies the objectivity requirements and leads to zero strain rates under an arbitrary rigid body displacement. The linear visco-elastic Kelvin–Voigt model frequently used in the literature can be obtained as a special case of the proposed nonlinear model when only two visco-elastic coefficients are used. As demonstrated in this paper, the use of two visco-elastic coefficients only leads to viscous coupling between the deformation gradients. The model developed in this investigation can be used in the framework of large deformation and large rotation non-incremental solution procedure without the need for using existing co-rotational finite element formulations. The finite element absolute nodal coordinate formulation (ANCF) that allows for straightforward implementation of general constitutive material models is used in the validation of the proposed visco-elastic model. A comparison with the linear visco-elastic model is also made in this study. The results obtained in this investigation show that there is a good agreement between the solutions obtained using the proposed nonlinear model and the linear model in the case of small deformations.

On the integration of computer aided design and analysis using the finite element absolute nodal coordinate formulation

For almost a decade, the finite element absolute nodal coordinate formulation (ANCF) has been used for both geometry and finite element representations. Because of the ANCF isoparametric property in the cases of beams, plates and shells, ANCF finite elements lend themselves easily to the geometric description of curves and surfaces, as demonstrated in the literature. The ANCF finite elements, therefore, are ideal for what is called isogeometric analysis that aims at the integration ofcomputer aided designandanalysis (ICADA), which involves the integration of what is now split into the separate fields of computer aided design (CAD) and computer aided analysis (CAA). The purpose of this investigation is to establish the relationship between the B-spline and NURBS, which are widely used in the geometric modeling, and the ANCF finite elements. It is shown in this study that by using the ANCF finite elements, one can in a straightforward manner obtain the control point representation required for the Bezier, B-spline and NURBS geometry. To this end, a coordinate transformation is used to write the ANCF gradient vectors in terms of control points. Unifying the CAD and CAA will require the use of such coordinate transformations and their inverses in order to transform control points to position vector gradients which are required for the formulation of the element transformations in the case of discontinuities as well as the formulation of the strain measures and the stress forces based on general continuum mechanics theory. In particular, fully parameterized ANCF finite elements can be very powerful in describing curve, surface, and volume geometry, and they can be effectively used to describe discontinuities while maintaining the many ANCF desirable features that include a constant mass matrix, zero Coriolis and centrifugal forces, no restriction on the amount of rotation or deformation within the finite element, ability for straightforward implementation of general constitutive equations, and ability to capture coupled deformation modes that cannot be captured using existing finite element beam, plate and shell formulations. Because of the relationship between the ANCF finite elements and the B-splines, the development of a rational absolute nodal coordinate formulation (RANCF) is currently being explored. Furthermore, the relationship between the ANCF finite elements and B-splines and NURBS demonstrates that the use of B-splines and NURBS in the analysis will eventually lead to the same fundamental issues that have been already researched in the literature on the ANCF finite elements.

Deformation modes in the finite element absolute nodal coordinate formulation

The objective of this study is to provide interpretation of the deformation modes in the finite element absolute nodal coordinate formulation using several strain definitions. In this finite element formulation, 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 as well as at the nodal points. 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. Furthermore, the resulting dynamic model, unlike large rotation finite element formulations, leads to a unique rotation field, and as a consequence, the obtained formulation does not suffer from the problem of coordinate redundancy that characterizes existing large deformation finite element formulations. 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. Using the material coordinate system, the Timoshenko–Reissner and Euler –Bernoulli beam models can be systematically obtained as special cases of the absolute nodal coordinate formulation beam models. While a constraint approach is used in this study to eliminate the cross-section deformation modes, it is important to point out as mentioned in this paper that lower-order finite elements, some of which already presented in previous investigations, can be efficiently used in thin and stiff structure applications.

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

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.

Use of the Finite Element Absolute Nodal Coordinate Formulation in Modeling Slope Discontinuity

A large rigid body rotation of a finite element can be described by rotating the axes of the element coordinate system or by keeping the axes unchanged and change the slopes or the position vector gradients. In the first method, the definition of the local element parameters (spatial coordinates) changes with respect to a body or a global coordinate system. The use of this method will always lead to a nonlinear mass matrix and non-zero centrifugal and Coriolis forces. The second method, in which the axes of the element coordinate system do not rotate with respect to the body or the global coordinate system, leads to a constant mass matrix and zero centrifugal and Coriolis forces when the absolute nodal coordinate formulation is used. This important property remains in effect even in the case of flexible bodies with slope discontinuities. The concept employed to accomplish this goal resembles the concept of the intermediate element coordinate system previously adopted in the finite element floating frame of reference formulation. It is shown in this paper that the absolute nodal coordinate formulation that leads to exact representation of the rigid body dynamics can be effectively used in the analysis of complex structures with slope discontinuities. The analysis presented in this paper also demonstrates that objectivity is not an issue when the absolute nodal coordinate formulation is used due to the fact that this formulation automatically accounts for the proper coordinate transformations.