Large Rotation Vector Research Articles

A Fundamental Study for Numerical Integration Procedures for the ANCF Consistent Rotation-Based Formulation Beam Elements

This study presents a numerical integration technique for a shear deformable beam element based on the absolute nodal coordinate formulation (ANCF). In particular, the consistent rotation-based formulation (CRBF) is used to develop a plane ANCF beam element, namely, the ANCF/CRBF shear deformable element. Even though the ANCF/CRBF beam element introduces rotation parameters as nodal coordinates, it does not need for interpolating the rotation field. Therefore, the ANCF/CRBF formulation defines a unique rotation field, and does not suffer from a coordinate redundancy problem encountered when using the large rotation vector formulation (LRVF). On the other hand, ANCF/CRBF formulation leads to highly nonlinear inertia forces including non-zero Coriolis and centrifugal forces. The objective of this study is to develop a constrained formulation and a numerical integration technique for the ANCF/CRBF shear deformable beam element. In particular, this study focuses on the fact that the global position gradients can be expressed in terms of the finite rotation parameters. In the present method, such algebraic relations are treated as constraints, and the ANCF/CRBF beam element is formulated as the differential algebraic equations (DAEs). As a result, the present formulation ensures the constant mass matrix, however, DAE solvers have to be introduced for solving the resulting equations. In addtion, since the present method can be regarded as a kind of augmented formulation, the number of degrees of freedom is significant from a standpoint of numerical efficiency. In order to cope with increase of the degrees of freedom, this study develops size reduction techniques focusing on the algebraic expressions of constraints. Then, the results obtained from the proposed method are compared with the results obtained using the fully parameterized ANCF shear deformable 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.

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.

State of the Art of ANCF Elements Regarding Geometric Description, Interpolation Strategies, Definition of Elastic Forces, Validation and the Locking Phenomenon in Comparison with Proposed Beam Finite Elements

The focus of the present article lies on new enhanced beam finite element formulations in the absolute nodal coordinate formulation (ANCF) and its embedding in the available formulations in the literature. The ANCF has been developed in the past for the modeling of large deformations in multibody dynamics problems. In contrast to the classical nonlinear beam finite elements in literature, the ANCF does not use rotational degrees of freedom, but slope vectors for the parameterization of the orientation of the cross section. This leads to several advantages compared to the classical formulations, e.g. ANCF elements do not necessarily suffer from singularities emerging from the parameterization of rotations. In the classical large rotation vector formulation, the mass matrix is not constant with respect to the generalized coordinates. In the case of ANCF elements, a constant mass matrix follows, which is advantageous in dynamic analysis. In the standard geometrically exact formulation, the parameterization of rotations leads to a nonlinear term containing quadratic velocities. The so-called quadratic velocity vector vanishes for ANCF elements which is advantageous in dynamic applications. In the present article, existing beam finite element formulations are analyzed and improved to derive formulations which are able to solve industrial applications with high performance, efficiency and accuracy. The interest lies especially on finite element formulations for multibody dynamics systems that are capable of large deformations in order to derive accurate solutions of nonlinear engineering and research problems. Existing shear deformable ANCF beam finite elements show an overly stiff behavior caused by the locking phenomenon. Existing locking problems are discussed in this article and avoided in the proposed elements in order to gain accurate solutions. The state of the art of ANCF elements in literature is reviewed including the basic description of the kinematics, interpolation strategies and definition of elastic forces, but also problems and known disadvantages arising in the existing elements, as e.g. the locking phenomenon. Regarding the description of the elastic forces, the present article shows the two standard approaches in literature as well as new enhanced formulations to avoid locking: a continuum mechanics based formulation for the elastic forces with elimination of Poisson and shear locking, and an extended hybrid structural mechanics based formulation for the elastic forces including a term for penalizing shear and cross section deformation. The definition of the element kinematics and interpolation strategies in existing elements are discussed and compared to the according descriptions of the proposed elements. A comparison of the solution of the proposed finite elements to analytical solutions in the literature and to the solution retrieved from commercial finite element software have shown high accuracy and high order of convergence. The speed of convergence is evaluated regarding different interpolation strategies and different formulations for the elastic forces. The investigations show that the proposed elements have high potential for simulation of geometrically nonlinear problems arising from real-life multibody applications and therefore are highly competitive with existing elements in commercial finite element codes. It has to be mentioned here that most of the studies on nonlinear elements based on the ANCF in literature use linear constitutive laws. Regarding many applications in which geometric and material nonlinearities arise, elastic material models are not sufficient to represent the real problem accurately. For this reason, an extension of the material model is necessary in order to fulfill the requirements of current but also of future materials arising in engineering or research. An overview of existing nonlinear material models in literature can be found in the “Appendix”.

Use of independent rotation field in the large displacement analysis of beams

This paper examines the effect of using independent finite rotation field in the large displacement analysis of flexible bodies. This finite rotation description is at the core of the large rotation vector formulation (LRVF), which has been used in the dynamic analysis of bodies experiencing large rotation and deformation. The LRVF employs two independently interpolated meshes for describing the flexible body dynamics: the position mesh and the rotation mesh. The use of these two geometrically independent meshes can lead to coordinate and geometric invariant redundancy that can be the source of fundamental problems in the analysis of large deformations. It is demonstrated in this paper that the two geometry meshes can define different space curves, which can differ by arbitrary rigid-body displacements. The material points of the two meshes occupy different positions in the deformed configuration, and as a consequence, the geometries of the two meshes can differ significantly. The paper also discusses other issues including the inextensibility of the rotation mesh. Simple examples are presented in order to shed light on these fundamental issues.

Shear horizontal surface acoustic wave induced microfluidic flow

In this letter, we demonstrate that surface acoustic waves (SAWs) excited on 36°Y-X LiTaO3 have both Rayleigh and shear horizontal components, which can result in different acoustic streaming patterns in micro-droplets. The shear horizontal surface acoustic wave streaming pattern exhibits a quasi-stable counterflow in the SAW propagation direction with velocities of up to 16 mm/s, with minimal damping observed within the droplets. A pumping function was realized at a threshold power of 30 dBm with a large rotation vector on hydrophobically treated devices.

A geometrically exact beam element based on the absolute nodal coordinate formulation

In this study, Reissner’s classical nonlinear rod formulation, as implemented by Simo and Vu-Quoc by means of the large rotation vector approach, is implemented into the framework of the absolute nodal coordinate formulation. The implementation is accomplished in the planar case accounting for coupled axial, bending, and shear deformation. By employing the virtual work of elastic forces similarly to Simo and Vu-Quoc in the absolute nodal coordinate formulation, the numerical results of the formulation are identical to those of the large rotation vector formulation. It is noteworthy, however, that the material definition in the absolute nodal coordinate formulation can differ from the material definition used in Reissner’s beam formulation. Based on an analytical eigenvalue analysis, it turns out that the high frequencies of cross section deformation modes in the absolute nodal coordinate formulation are only slightly higher than frequencies of common shear modes, which are present in the classical large rotation vector formulation of Simo and Vu-Quoc, as well. Thus, previous claims that the absolute nodal coordinate formulation is inefficient or would lead to ill-conditioned finite element matrices, as compared to classical approaches, could be refuted. In the introduced beam element, locking is prevented by means of reduced integration of certain parts of the elastic forces. Several classical large deformation static and dynamic examples as well as an eigenvalue analysis document the equivalence of classical nonlinear rod theories and the absolute nodal coordinate formulation for the case of appropriate material definitions. The results also agree highly with those computed in commercial finite element codes.

Integration of Large Deformation Finite Element and Multibody System Algorithms

This paper presents an overview of research and development efforts that are currently being devoted to integrate large deformation finite element formulations with flexible multibody system algorithms. The goal is to develop computer simulation capabilities for the analysis of physics and engineering models with significant details. The successful development of such new and integrated algorithms will also allow modeling and simulation of systems that cannot be solved using existing computer algorithms and codes. One of the main difficulties encountered in this integration process is attributed to the fact that the solution procedures used in finite element codes differ significantly from those used in general-purpose flexible multibody system codes. Finite element methods employ the corotational formulations that are often used with incremental solution procedures. Flexible multibody computer codes, on the other hand, do not, in general, use incremental solution procedures. Three approaches are currently being explored by academic institutions and the software industry. In the first approach, gluing algorithms that aim at performing successful simulations by establishing an interface between existing codes are used. Using different coordinates and synchronizing the time stepping are among several challenging problems that are encountered when gluing algorithms are used. In the second approach, multibody system capabilities are implemented in existing finite element algorithms that are based on large rotation vector formulations. For the most part, corotational formulations and incremental solution procedures are used in this case. In the third approach, a new large deformation finite element formulation that can be successfully implemented in flexible multibody system computer algorithms that employ nonincremental solution procedures is introduced. The approach that is now being developed in several institutions is based on the finite element absolute nodal coordinate formulation. Such a formulation can be systematically implemented in general-purpose flexible multibody system computer algorithms. Nonlinear constraint equations that describe mechanical joints between different bodies can be formulated in terms of the absolute coordinates in a straightforward manner. The coupling between the motion of rigid, flexible, and very flexible bodies can also be accurately described. The successful integration of large deformation finite element and multibody system algorithms will lead to a new generation of computer codes that can be systematical and efficiently used in the analysis of many engineering applications.

W-1-1-4 INTRODUCTION OF DAMPING MATRIX INTO ABSOLUTE NODAL COORDINATE FORMULATION

Accurate seismic analysis of large deformable moving structures is still unsolved problems in the field of earthquake engineering. In order to analyze the problems, the dynamics of flexible beam undergoing large deformation has been studied by the authors. Generally this type problem is tackled by the finite element method. There are three basic finite element formulations which are used in the dynamics of flexible beams. There are the floating frame of reference approach [1,2,3], the finite segment method [4,5], and the large rotation vector approach [6] Recently, the absolute nodal coordinata (ANC) formulation for beam elements was proposed by A. A. Shabana [7,8]. In this procedure, there is no need to transform the element matrices into the global ones because the equations of motion are already defined in terms of the absolute nodal coordinates defined in the global reference frame. Consequently, the mass matrix becomes constant with time, whereas the stiffness matrix becomes nonlinear function of time, even in case of linear elastic deformation. One possible method to avoid such cumbersome of the absolute nodal coordinate formulation in calculating elastic forces is to assume the infinitesimal deformation theory for beams undergoing large rotation. A new time independent formulation for bending motion in calculating elastic forces of the beam, and consequently the constant stiffness matrix were derived by the authors [9]. In this formulation, the elastic force for axial deformation is still time dependent. When we calculate the dynamic problems of a flexible moving structure such as a crane during earthquakes, application of large deformation theory for the structures composing of the beam elements is unavoidable to obtain accurate results. Respose calculation, in this case, requires the damping property as well as mass and stiffness properties of the structures. This paper firstly summarizes the former researches done by the authors and then considers the introduction of damping property into the flexible system described by the absolute nodal coodinates. In the practical linear theory of structural dynamics, Rayleigh damping is often used to describe the system damping because of its simplicity and practicality. It may be attractive in the response calculation if we introduce Rayleigh type damping into nonlinear equations of motion described by ANC formulation. We have tried to develop a procedure of introduction of Rayleigh type and other type damping matrices into ANC formulation. Reasonable assumptions lead to practical damping matrices for calculating structual seismic responses. Our former formulation for ANC plays an important role for our present study.

Study on the Derivation and Application of the Mean Axis for the Deformed Beam by Means of the Absolute Nodal Coordinate Multibody Dynamics Formulation.

There are three basic finite element formulations which are used in the dynamics of flexible beams. These are the floating frame of reference approach, the finite segment method and the large rotation vector approach. Recently, the absolute nodal coordinate formulation was proposed by A. A. Shabana et al. In this formulation, elastic forces are lead by approximating the slope of the arbitrary point on the neutral axis of the beam in terms of the slope of the simple support axis of the beam. In this paper, we propose the mean axis of the planar Bernoulli-Euler beam for the absolute nodal coordinate approach. The origin and the orientation of this axis are selected so as to minimize the total deformation of the concerned beam. And the selected axis can be simply described by the nodal coordinates of the beam element. Using the mean axis instead of the simple support axis, the elastic forces of the beam element may be more precisely calculated. Finally, we show numerical examples to demonstrate effectiveness of this approach.

Study on the Elastic Force for the Deformed Beam by means of the Absolute Nodal Coordinate Multibody Dynamics Formulation. Derivation of the Elastic Force using Finite Displacement and Infinitesimal Strain.

There are three basic finite element formulations which are used in the multibody dynamics of flexible beams. These are the floating frame of reference approach, the incremental finite element approach and the large rotation vector approach. Recently, the absolute nodal coordinate formulation was proposed by A. A. Shabana et al. In this formulation, the mass matrix becomes constant, whereas the stiffness matrix becomes nonlinear function of time, even in case of linear elastic problems. In this paper, a new formulation to calculate the elastic forces is proposed. This formulation is based on the assumption that the deformations within each element remain very small. So, the expression of the resulting elastic force is simple, and this formulation is efficient for large rotation and medium deformation problems. Numerical examples are demonstrated for this formulation by using planar static buckling problems and flexible pendulum problems.

Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Implementation and Applications

This part of these two companion papers demonstrates the computer implementation of the absolute nodal coordinate formulation for three-dimensional beam elements. Two beam elements that relax the assumptions of Euler-Bernoulli and Timoshenko beam theories are developed. These two elements take into account the effect of rotary inertia, shear deformation and torsion, and yet they lead to a constant mass matrix. As a consequence, the Coriolis and centrifugal forces are identically equal to zero. Both beam elements use the same interpolating polynomials and have the same number of nodal coordinates. However, one of the elements has two nodes, while the other has four nodes. The results obtained using the two elements are compared with the results obtained using existing incremental methods. Unlike existing large rotation vector formulations, the results of this paper show that no special numerical integration methods need to be used in order to satisfy the principle of work and energy when the absolute nodal coordinate formulation is used. These results show that this formulation can be used in manufacturing applications such as high speed forming and extrusion problems in which the element cross section dimensions significantly change.

Application of the Absolute Nodal Coordinate Formulation to Large Rotation and Large Deformation Problems

There are three basic finite element formulations which are used in multibody dynamics. These are the floating frame of reference approach, the incremental method and the large rotation vector approach. In the floating frame of reference and incremental formulations, the slopes are assumed small in order to define infinitesimal rotations that can be treated and transformed as vectors. This description, however, limits the use of some important elements such as beams and plates in a wide range of large displacement applications. As demonstrated in some recent publications, if infinitesimal rotations are used as nodal coordinates, the use of the finite element incremental formulation in the large reference displacement analysis does not lead to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. In this paper, a simple non-incremental finite element procedure that employs the mathematical definition of the slope and uses it to define the element coordinates instead of the infinitesimal and finite rotations is developed for large rotation and deformation problems. By using this description and by defining the element coordinates in the global system, not only the need for performing coordinate transformation is avoided, but also a simple expression for the inertia forces is obtained. The resulting mass matrix is constant and it is the same matrix that appears in linear structural dynamics. It is demonstrated in this paper that this coordinate description leads to exact modeling of the rigid body inertia when the structures rotate as rigid bodies. Nonetheless, the stiffness matrix becomes nonlinear function even in the case of small displacements. The method presented in this paper differs from previous large rotation vector formulations in the sense that the inertia forces, the kinetic energy, and the strain energy are not expressed in terms of any orientation coordinates, and therefore, the method does not require interpolation of finite rotations. While the use of the formulation is demonstrated using a simple planar beam element, the generalization of the method to other element types and to the three dimensional case is straightforward. Using the finite element procedure presented in this paper, beams and plates can be treated as isoparametric elements.

Flexible Multibody Dynamics: Review of Past and Recent Developments

In this paper, a review of past and recent developments in the dynamics of flexible multibody systems is presented. The objective is to review some of the basic approaches used in the computer aided kinematic and dynamic analysis of flexible mechanical systems, and to identify future directions in this research area. Among the formulations reviewed in this paper are the floating frame of reference formulation, the finite element incremental methods, large rotation vector formulations, the finite segment method, and the linear theory of elastodynamics. Linearization of the flexible multibody equations that results from the use of the incremental finite element formulations is discussed. Because of space limitations, it is impossible to list all the contributions made in this important area. The reader, however, can find more references by consulting the list of articles and books cited at the end of the paper. Furthermore, the numerical procedures used for solving the differential and algebraic equations of flexible multibody systems are not discussed in this paper since these procedures are similar to the techniques used in rigid body dynamics. More details about these numerical procedures as well as the roots and perspectives of multibody system dynamics are discussed in a companion review by Schiehlen [79]. Future research areas in flexible multibody dynamics are identified as establishing the relationship between different formulations, contact and impact dynamics, control-structure interaction, use of modal identification and experimental methods in flexible multibody simulations, application of flexible multibody techniques to computer graphics, numerical issues, and large deformation problem. Establishing the relationship between different flexible multibody formulations is an important issue since there is a need to clearly define the assumptions and approximations underlying each formulation. This will allow us to establish guidelines and criteria that define the limitations of each approach used in flexible multibody dynamics. This task can now be accomplished by using the “absolute nodal coordinate formulation” which was recently introduced for the large deformation analysis of flexible multibody systems.

An Element Independent Corotational Procedure for the Treatment of Large Rotations

A new corotational procedure is developed which enables existing finite element formulations to be used in problems that contain arbitrarily large rotations. Through the use of a nonsingular large rotation vector, the contribution of the rigid body motion of the element to the total displacement field is removed before element computations are performed, with the result that almost any element can be easily upgraded to handle large rotations. This paper contains a derivation of the theory, an outline of the implementation into the STAGS code, and a demonstration of performance for problems involving large rotations and moderate strains.