Absolute Nodal Coordinate Formulation Finite Elements Research Articles

Mechanically induced temperature oscillations in the coupled thermo-elasticity analysis of articulated dynamical systems

This article discusses a new approach for predicting and quantifying mechanically induced temperature oscillations in the coupled thermo-elasticity analysis of articulated mechanical systems (AMS). In this approach, the constrained equations of motion are solved simultaneously with discrete temperature equations obtained by converting heat partial-differential equation to a set of first-order ordinary differential equations. Dependence of the temperature gradients and their spatial derivatives on the position gradients, spinning motion, and curvatures is discussed. The approach captures dependence of the temperature-oscillation frequencies on the mechanical-displacement frequencies. The temperature field can be selected to ensure continuity of the temperature gradients at the nodal points. To generalize the AMS coupled thermo-elasticity formulation and capture the effect of the boundary and motion constraints (BMC) on the thermal expansion, the proposed method is based on integrating thermodynamics and Lagrange-D’Alembert principles. The absolute nodal coordinate formulation (ANCF) is used to describe continuum displacement and obtain accurate description of the reference-configuration geometry and change of this geometry due to deformations. A thermal-analysis large-displacement formulation is used to allow converting heat energy to kinetic energy, ensuring stress-free thermal expansion in case of unconstrained uniform thermal expansion. Cholesky heat coordinates are used to define an identity coefficient matrix for the efficient solution of the discretized heat equations. The approach presented is applicable to the two different forms of the heat equation used in the literature; one form is explicit function of the stresses while the other form does not depend explicitly on the stresses. Because of the need for using ANCF finite elements to achieve a higher degree of continuity in the coupled thermomechanical approach introduced in this article, the concept of the ANCF mesh topology is discussed.

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.

Evaluation of breaking wave effects in liquid sloshing problems: ANCF/SPH comparative study

This paper is focused on evaluating the effect of breaking waves in liquid sloshing problems. Two fundamentally different approaches, namely the smoothed particle hydrodynamics (SPH) and the finite element (FE) absolute nodal coordinate formulation (ANCF), are used to describe the liquid sloshing response in several sloshing scenarios. The SPH method is a mesh-free numerical technique often used to capture very large displacements in fluid and solid mechanics problems. ANCF finite elements, on the other hand, can be used to develop a non-incremental solution procedure, suited for the nonlinear analysis of flexible bodies undergoing large rotation and large deformation. The fundamental differences between the two approaches and the advantages and limitations of each are discussed. Two benchmark problems, the dam break and sloshing tank, are used to perform a detailed SPH/ANCF quantitative comparative study in different sloshing scenarios. While a good agreement is found between the ANCF and SPH converged solutions for the dam break problem, in the sloshing tank problem the SPH solution underpredicts the amplitude of oscillation of the fluid center of mass and the wave height at a selected probe point. Because one ANCF element can capture complex shapes, nearly 40 times fewer degrees of freedom than the SPH model are needed in both problems. The use of the ANCF models leads to a CPU saving of 70% and 25% in the broken dam and sloshing tank problems, respectively. In the case of the tank problem, the effect of light, moderate, and severe turbulence is examined. The position of the fluid center of mass is computed using the two approaches, and the results obtained are verified using a reference analytical solution. The power spectral density of the center of mass is evaluated using the fast Fourier transform (FFT) to study the effect of breaking waves. The results show that in the case of light and moderate turbulence, the ANCF model allows for accurately averaging the fluid inertia properties, and the obtained solutions are in good agreement with the SPH solutions. In the case of severe turbulence, on the other hand, the mechanical energy dissipation due to fluid mixing and wave breaking, which can only be captured using the SPH method, damps out the sloshing oscillations.

Integration of computer-aided design and analysis: application to multibody vehicle systems

Virtual design and durability investigations are currently performed in automotive industries using three different and incompatible systems: computer-aided design (CAD) system for geometry creation, finite element (FE) software for developing the analysis mesh, and multibody system (MBS) software for automatically generating and numerically solving the differential-algebraic equations (DAE's). This paper proposes a new computer-aided engineering (CAE) approach based on the integration of computer-aided design and analysis (I-CAD-A). The proposed mechanics-based approach achieves seamless geometry/analysis integration, and allows for solid modelling multi-component systems from the outset. The geometrically accurate mechanics-based solid models can be systematically used as the analysis meshes in the small deformation floating frame of reference (FFR) and/or in the large deformation absolute nodal coordinate formulation (ANCF) investigations. In this new approach, ANCF finite elements are used as the basis for creating the geometry for both small and large deformation analyses. In the case of small deformations, ANCF geometry is systematically converted to reduced-order consistent rotation-based formulation (CRBF) FFR mesh, which can be systematically used with standard coordinate reduction techniques to eliminate high-frequency insignificant modes of vibration. The paper discusses the fundamental differences between the proposed method and the isogeometric analysis (IGA) approach and presents illustrative pilot examples to demonstrate the new concepts and the feasibility of developing the mechanics-based design procedure.

Integration of computer-aided design and analysis: application to multibody vehicle systems

Virtual design and durability investigations are currently performed in automotive industries using three different and incompatible systems: computer-aided design (CAD) system for geometry creation, finite element (FE) software for developing the analysis mesh, and multibody system (MBS) software for automatically generating and numerically solving the differential-algebraic equations (DAE's). This paper proposes a new computer-aided engineering (CAE) approach based on the integration of computer-aided design and analysis (I-CAD-A). The proposed mechanics-based approach achieves seamless geometry/analysis integration, and allows for solid modelling multi-component systems from the outset. The geometrically accurate mechanics-based solid models can be systematically used as the analysis meshes in the small deformation floating frame of reference (FFR) and/or in the large deformation absolute nodal coordinate formulation (ANCF) investigations. In this new approach, ANCF finite elements are used as the basis for creating the geometry for both small and large deformation analyses. In the case of small deformations, ANCF geometry is systematically converted to reduced-order consistent rotation-based formulation (CRBF) FFR mesh, which can be systematically used with standard coordinate reduction techniques to eliminate high-frequency insignificant modes of vibration. The paper discusses the fundamental differences between the proposed method and the isogeometric analysis (IGA) approach and presents illustrative pilot examples to demonstrate the new concepts and the feasibility of developing the mechanics-based design procedure.

Continuum-Based Geometry/Analysis Approach for Flexible and Soft Robotic Systems

Control and stability of flexible and soft robotic systems (FSRS), which have complex geometry and experience desirable and undesirable deformations, are of major concern, particularly when lightweight soft materials are used. Nonetheless, there is no unified continuum-based geometry/analysis approach that can be used for the efficient FSRS virtual prototyping and design. The goal of this article is to propose a new FSRS geometric modeling and analysis methodology by addressing fundamental virtual prototyping challenges that include: (1) integration of the robot geometry and analysis; (2) implementation of general and unconventional material models and actuation forces; (3) use of new concepts for modeling FSRS joints; and (4) development of efficient and robust algorithms for the FSRS virtual prototyping. To address these challenges, the finite element (FE) absolute nodal coordinate formulation (ANCF) and multibody system computational algorithms are used. ANCF FEs allow for modeling arbitrarily large and coupled displacements, correctly capture complex geometries, allow for implementing general and nonconventional material models, provide accurate definitions of conventional and nonconventional actuation forces, lead to a constant inertia matrix that defines optimally sparse matrix structure of the dynamic equations, and allow for exploiting new geometry concepts to define linear and more general joint constraints instead of the less general and nonlinear joint constraints currently used for robot systems. Using the general structure of the FSRS nonlinear dynamic equations of motion, a nonmodal continuum-based approach can be developed and used to correctly capture the FSRS complex geometry and large deformations.

Modeling Method and Application of Rational Finite Element Based on Absolute Nodal Coordinate Formulation

In multibody system dynamics, the absolute nodal coordinate formulation (ANCF) uses power functions as interpolating polynomials to describe the displacement field. It can get accurate results for flexible bodies that undergo large deformation and large rotation. However, the power functions are irrational representation which cannot describe the complex shapes precisely, especially for circular and conic sections. Different from the ANCF representation, the rational absolute nodal coordinate formulation (RANCF) utilizes rational basis functions to describe geometric shapes, which allows the accurate representation of complicated displacement and deformation in dynamics modeling. In this paper, the relationships between the rational surface and volume and the RANCF finite element are provided, and the generalized transformation matrices are established correspondingly. Using these transformation matrices, a new four-node three-dimensional RANCF plate element and a new eight-node three-dimensional RANCF solid element are proposed based on the RANCF. Numerical examples are given to demonstrate the applicability of the proposed elements. It is shown that the proposed elements can depict the geometric characteristics and structure configurations precisely, and lead to better convergence in comparison with the ANCF finite elements for the dynamic analysis of flexible bodies.

Curvature Expressions for the Large Displacement Analysis of Planar Beam Motions

While several curvature expressions have been used in the literature, some of these expressions differ from basic geometry definitions and lead to kinematic coupling between bending and shear deformations. This paper uses three different elastic force formulations in order to examine the effect of the curvature definition in the large displacement analysis of beams. In the first elastic force formulation, a general continuum mechanics approach (method 1) based on the nonlinear strain–displacement relationship is used. The second approach (method 2) is based on a classical nonlinear beam theory, in which a curvature expression consistent with differential geometry and independent of the shear deformation is used. The third elastic force formulation (method 3) employs a curvature expression that depends on the shear angle. In order to examine numerically the effect of using different curvature definitions, three different planar beam elements are used. The first element (element I) is the fully parameterized absolute nodal coordinate formulation (ANCF) shear deformable beam element. The second element (element II) is an ANCF consistent rotation-based formulation (CRBF) shear deformable beam element obtained from element I by consistently replacing the position gradient vectors by rotation parameters. The third element (element III) is a low-order bilinear ANCF/CRBF finite element in which nonzero differential geometry-based curvature definition cannot be obtained because of the low order of interpolation. Numerical results are obtained using the three elastic force formulations and the three finite elements in order to shed light on the definition of bending and shear in the large displacement analysis of beams. The results obtained in this investigation show that the use of method 2, with a penalty formulation that restricts the excessive cross section deformation, can improve significantly the convergence of the ANCF finite element.

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.

Integration of localized surface geometry in fully parameterized ANCF finite elements

This paper introduces a new method for the integration of localized surface geometry with fully parameterized absolute nodal coordinate formulation (ANCF) finite elements. In this investigation, ANCF finite elements are used to create the global geometry and perform the finite element (FE)/multibody system (MBS) analysis of deformable bodies. The localized surface geometry details can be described on ANCF element surfaces without the need for mesh refinement. The localized surface is represented using a standard computational geometry method, Non-uniform rational B-spline surface (NURBS), which can describe both conic surface and freeform surface efficiently and accurately. The basic idea of the integration of localized surface geometry with ANCF elements lies in the inclusion of such detail in the element mass matrix and forces. The integration can be implemented by overlapping the localized surface geometry on the original ANCF element or by directly trimming the ANCF element domain to fit the required shape. During the integration process, a mapping between ANCF local coordinates and NURBS localized geometric parameters is used for a consistent implementation of the overlapping and trimming methods. Additionally, two numerical integration methods are compared for the rate of convergence. The results show that the proposed subdomain integration method is better, since it is optimized for dealing with complex geometry. The proposed subdomain method can be used with any fully parameterized ANCF element. In order to analyze the accuracy of the proposed method, a cantilever plate example with localized surface geometry is used, and the simulation results with the proposed method are compared with the simulation results obtained using a commercial FE code. Two other examples that include contact with ground and localized surface geometry are also provided. These examples are a simple plate structure with surface geometry and a tire with tread details. The incompressible hyperelastic Mooney–Rivlin material model is used to describe the material used in the tire tread. It is shown through the tire contact patch that the proposed method can successfully capture the effect of the tread grooves. The rate of convergence and locking of fully parameterized ANCF elements are also discussed in this paper.

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.

Contact force control in multibody pantograph/catenary systems

In this paper, a new continuum-based pantograph/catenary model based on the absolute nodal coordinate formulation (ANCF) is proposed and used to develop an effective method to control the contact force which arises from the pantograph/catenary interaction. In the proposed new model, only one ANCF gradient vector is used in the formulation of the pantograph/catenary contact conditions, thereby allowing for using the proposed approach for both fully parameterized and gradient-deficient ANCF finite elements. The proposed contact formulation can also be considered as a more general sliding joint formulation that allows for the use of the more efficient gradient-deficient ANCF finite elements in modeling very flexible cables. A three-dimensional multibody system (MBS) model of a pantograph mounted on a train is developed using a nonlinear augmented MBS formulation. In order to take into account the catenary large deformation, ANCF finite elements are used. The contact between the pantograph and the catenary system is ensured using a sliding joint constraint whereas the contact between the rail vehicle wheels and the train track is modelled using an elastic contact formulation. In addition to the use of the new MBS approach to model the pantograph/catenary interaction, the contact force between the pantograph and the catenary is computed using a simpler lumped parameter model which describes the pan-head and the plunger subsystem dynamics. In order to reduce the standard deviation of the contact force without affecting its mean value, a control actuator is used between the pan-head and the plunger. To this end, three types of control laws for the control action are designed to improve the contact quality both in the transient phase and in the steady state phase of the pantograph/catenary interaction. The first control law proposed features a feedback structure whereas the second and the third control strategies employ a feedback plus feed-forward architecture. In order to demonstrate the effectiveness of the proposed method, the results of a set of numerical simulations with and without the controllers are presented.

ANCF Continuum-Based Soil Plasticity for Wheeled Vehicle Off-Road Mobility

This technical brief describes the procedure and demonstrates the feasibility of integrating soil/tire models using the absolute nodal coordinate formulation (ANCF). The effects of both the soil plasticity and the tire elasticity are captured using ANCF finite elements (FEs). Capturing the tire/soil dynamic interaction is necessary for the construction of higher fidelity off-road vehicle models. ANCF finite elements, as will be demonstrated in this paper, can be effectively used for the modeling of tire and soil mechanics. In this investigation, the soil model is developed using ANCF hexahedral finite elements, while the tire model can be developed using different ANCF finite elements including beam, plate, or solid elements; ANCF plate elements are used in this investigation for demonstration purposes. The Drucker–Prager plastic material, which is used to model the behavior of the soil, is appropriate for the simulation of a number of types of soils and offers a good starting point for computational plasticity in terramechanics applications. Such higher fidelity simulations can be fruitfully applied toward the investigation of complex dynamic phenomena in terramechanics. The proposed ANCF/Drucker–Prager soil model is implemented in a multibody system (MBS) algorithm which allows for using the ANCF reference node (ANCF-RN) to apply linear connectivity conditions between ANCF finite elements and the rigid components of the vehicle. This new implementation is demonstrated using a tire of an off-road wheeled vehicle. The generality of the approach allows for the simulation of general vehicle maneuvers over unprepared terrain. Unlike other approaches that implement force or superelement models into an MBS simulation environment, in the approach proposed in this paper both the soil material and vehicle parameters can be altered independently. This allows for a greater degree of flexibility in the development of computational models for the evaluation of the off-road wheeled vehicle performance.

Analysis of warping deformation modes using higher order ANCF beam element

Most classical beam theories assume that the beam cross section remains a rigid surface under an arbitrary loading condition. However, in the absolute nodal coordinate formulation (ANCF) continuum-based beams, this assumption can be relaxed allowing for capturing deformation modes that couple the cross-section deformation and beam bending, torsion, and/or elongation. The deformation modes captured by ANCF finite elements depend on the interpolating polynomials used. The most widely used spatial ANCF beam element employs linear approximation in the transverse direction, thereby restricting the cross section deformation and leading to locking problems. The objective of this investigation is to examine the behavior of a higher order ANCF beam element that includes quadratic interpolation in the transverse directions. This higher order element allows capturing warping and non-uniform stretching distribution. Furthermore, this higher order element allows for increasing the degree of continuity at the element interface. It is shown in this paper that the higher order ANCF beam element can be used effectively to capture warping and eliminate Poisson locking that characterizes lower order ANCF finite elements. It is also shown that increasing the degree of continuity requires a special attention in order to have acceptable results. Because higher order elements can be more computationally expensive than the lower order elements, the use of reduced integration for evaluating the stress forces and the use of explicit and implicit numerical integrations to solve the nonlinear dynamic equations of motion are investigated in this paper. It is shown that the use of some of these integration methods can be very effective in reducing the CPU time without adversely affecting the solution accuracy.

Integration of Geometry and Analysis for Vehicle System Applications: Continuum-Based Leaf Spring and Tire Assembly

The development of new and complex vehicle models using the absolute nodal coordinate formulation (ANCF) and multibody systems (MBS) algorithms is discussed in this paper. It is shown how a continuum-based finite element (FE) leaf spring and tire assembly can be developed at a preprocessing stage and integrated with MBS algorithms, allowing for the elimination of dependent variables before the start of the dynamic simulations. Leaf springs, which are important elements in the suspension system of large vehicles, are discretized using ANCF FEs and are integrated with ANCF tire meshes to develop new models with significant details. To this end, the concept of the ANCF reference node (ANCF-RN) is used in order to systematically assemble the vehicle model using linear algebraic constraint equations that can be applied at a preprocessing stage. These algebraic constraint equations define new FE connectivity conditions that include the leaf spring shackle/chassis assembly, tire flexible tread/rigid rim assembly, tire/axle assembly, and revolute joints between different vehicle components. The approach presented in this paper allows for using both gradient deficient and fully parameterized ANCF FEs to develop the new models. In order to develop accurate leaf spring models, the prestress of the leaves and the contact forces between leaves are taken into consideration in the ANCF models developed in this investigation. Numerical results are presented in order to demonstrate the use of the computational framework described in this paper to build continuum-based leaf spring/tire assembly that can be integrated with complex vehicle models. The results of this paper also demonstrate the feasibility of developing a CAD (computer-aided design)/analysis system in which the geometry and analysis mesh of a complete vehicle can be developed in one step, thereby avoiding the incompatibility and costly process of using different codes in the flexible MBS analysis.

ANCF Analysis of Textile Systems

This paper presents a new flexible multibody system (MBS) approach for modeling textile systems including roll-drafting sets used in chemical textile machinery. The proposed approach can be used in the analysis of textile materials such as lubricated polyester filament bundles (PFBs), which have uncommon material properties best described by specialized continuum mechanics constitutive models. In this investigation, the absolute nodal coordinate formulation (ANCF) is used to model PFB as a hyperelastic transversely isotropic material. The PFB strain energy density function is decomposed into a fully isotropic component and an orthotropic, transversely isotropic component expressed in terms of five invariants of the right Cauchy–Green deformation tensor. Using this energy decomposition, the second Piola–Kirchhoff stress and the elasticity tensors can also be split into isotropic and transversely isotropic parts. The constitutive equations are used to define the generalized material forces associated with the coordinates of three-dimensional fully parameterized ANCF finite elements (FEs). The proposed approach allows for modeling the dynamic interaction between the rollers and PFB and allows for using spline functions to describe the PFB forward velocity. The paper demonstrates that the textile material constitutive equations and the MBS algorithms can be used effectively to obtain numerical solutions that define the state of strain of the textile material and the relative slip between the rollers and PFB.

A nonlinear approach for modeling rail flexibility using the absolute nodal coordinate formulation

This paper describes a new nonlinear formulation based on the absolute nodal coordinate formulation (ANCF) for modeling the dynamic interaction between rigid wheels and flexible rails. The generalized forces and spin moments at the contact points are formulated in terms of the absolute coordinates and gradients of ANCF finite elements used to model the rail. To this end, a new procedure for formulating the generalized ANCF applied moment based on a continuum mechanics approach is introduced. The generalized moment is calculated using the spin tensor defined in terms of the gradients at the contact points, and therefore, the use of this approach does not require the use of angles. In order to have an accurate definition of the creepages, the location and velocity of the contact points are updated online using the rail deformations. An elastic contact formulation is used to define the contact forces that enter into the dynamic formulation of the system equations of motion. An elastic line approach is used to define the rail stress forces, and the relative slip between the rigid wheel and the flexible rail is iteratively updated using the deformations of the ANCF finite elements. The formulation proposed in this investigation is demonstrated using a five-body railroad vehicle negotiating flexible rails. In order to validate the ANCF rail model, the obtained results are compared with previously published results obtained using the floating frame of reference formulation that employs eigenmodes. The comparative study presented in this paper shows that there is, in general, a good agreement between the results obtained using the two different formulations.

ANCF Consistent Rotation-Based Finite Element Formulation

In this technical brief, a consistent rotation-based formulation is proposed using the absolute nodal coordinate formulation (ANCF) kinematic description. The proposed formulation defines a unique rotation field, employs one interpolation, captures shear deformations, does not suffer from the redundancy problem encountered when using large rotation vector formulations, allows for systematically describing curved geometry, and leads to elastic force definitions that eliminate high-frequency modes associated with the deformation of the cross section. The drawback of this formulation, as it is the case with the large rotation vector formulations, is the nonlinearity of the inertia forces including nonzero Coriolis and centrifugal forces. Furthermore, the formulation does not capture deformation modes that can be captured using the more general ANCF finite elements. Nonetheless, the proposed method is consistent with the continuum mechanics general description, can be related to computational geometry methods, and can be used to develop beam, plate, and shell models without violation of basic mechanics principles.

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.

A new multibody system approach for tire modeling using ANCF finite elements

This paper introduces a new computational multibody system framework for developing accurate tire models using the finite element absolute nodal coordinate formulation (ANCF). Absolute nodal coordinate formulation finite elements are used to create the geometry and perform the finite element/multibody system analysis of the tires. The computational procedure used in this study allows for modeling composite tires and for using a continuum-based air pressure and contact tire force models. The absolute nodal coordinate formulation tire mesh, which allows for high spinning speed, has a constant inertia matrix and zero Coriolis and centrifugal forces. The concept of the absolute nodal coordinate formulation reference node, introduced recently, is used to develop linear connectivity conditions between the tire tread and rim, thereby allowing for imposing these linear conditions at a preprocessing stage. Using this approach, the dependent variables are eliminated at a preprocessing stage before the start of the simulation. The reference node, which is not associated with a particular finite element, is used to define the inertia of the rigid rims. The inertia coefficients associated with the rim reference nodes are first developed in terms of the absolute nodal coordinate formulation position and gradient coordinates. The rigidity of each rim is enforced during the dynamic analysis using six nonlinear algebraic constraint equations that are combined with the dynamic differential equations of motion using the technique of Lagrange multipliers. It is shown in this investigation that the concept of the absolute nodal coordinate formulation reference node can be used to develop a complete vehicle model using one absolute nodal coordinate formulation mesh in which the redundant variables are systematically eliminated at a preprocessing stage, and consequently, the number of differential and algebraic equations that need to be solved is significantly reduced. The use of the new approach proposed in this paper is demonstrated using a vehicle model described by one absolute nodal coordinate formulation mesh.