Computational aspects of vector-like parametrization of three-dimensional finite rotations

Theoretical and computational aspects of vector-like parametrization of three-dimensional finite rotations, which uses only three rotation parameters, are examined in detail in this work. The relationship of the proposed parametrization with the intrinsic representation of finite rotations (via an orthogonal matrix) is clearly identified. Careful considerations of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations are presented for the chosen model problem of Reissner's non-linear beam theory. Pertaining details of numerical implementation are discussed for the simplest choice of the finite element interpolations for a 2-node three-dimensional beam element. A number of numerical simulations in three-dimensional finite rotation analysis are presented in order to illustrate the proposed approach.     

Finite rotations in the description of continuum deformation

Finite rotations in continuum mechanics are described by means of either a proper orthogonal tensor or finite rotation vectors. Some algebraic relations concerning the finite rotations are reviewed. Formulae expressing them in terms of displacements are given. Along each of the curvilinear coordinate lines the finite rotations are shown to satisfy some systems of the linear first-order differential equations. Each system of the equations is presented in four different but equivalent forms associated with an intermediate stretched basis or with an intermediate rotated basis. Integrability conditions of the system of equations provide various alternative forms of compatibility conditions in continuum mechanics. The displacement field is expressed through the stretch and rotation fields in the form of three successive line integrals. The formula describes the displacements to within a constant finite translation and a constant finite rotation. The procedure proposed here generalizes the formula derived by Cesàro (1906) within the classical linear theory of elasticity.     

Kinematics and dynamics of rigid and flexible mechanisms using finite elements and quaternion algebra

A technique for representing large finite rotations in terms of only three independent parameters, the conformal rotation vector, is described and applied to the finite element formulation of 3-D mechanisms problems. A beam finite element that takes into account large finite rotations and various types of rigid joints have been developed. Some test examples which demonstrate the applicability of the proposed technique are presented.     

Stress resultant geometrically exact form of classical shell model and vector‐like parameterization of constrained finite rotations

In this work we consider the geometrically exact shell model subjected to finite rotations, making use of rotation vector parameters for handling the corresponding constrained rotation for smooth shells. A modification of such a parameterization which is based on the incremental rotation vector and thus capable of avoiding the singularity problem is also discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

REFINED CONSISTENT FORMULATION OF A CURVED TRIANGULAR FINITE ROTATION SHELL ELEMENT

The present paper considers a finite rotation formulation for curved shell elements with rotations about the element sides as nodal degrees of freedom. Attention is mainly on the derivation of a consistent finite rotation formulation. Significant simplifications of the governing equations are presented. These simplifications lead to more efficient finite element implementations. Numerical examples demonstrate the differences between the present consistent and previous approximate formulations.     

A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization

While frame-invariant solutions for arbitrarily large rotational deformations have been reported through the orthogonal matrix parametrization, derivation of such solutions purely through a rotation vector parametrization, which uses only three parameters and provides a parsimonious storage of rotations, is novel and constitutes the subject of this paper. In particular, we employ interpolations of relative rotations and a new rotation vector update for a strain-objective finite element formulation in the material framework. We show that the update provides either the desired rotation vector or its complement. This rules out an additive interpolation of total rotation vectors at the nodes. Hence, interpolations of relative rotation vectors are used. Through numerical examples, we show that combining the proposed update with interpolations of relative rotations yields frame-invariant and path-independent numerical solutions. Advantages of the present approach vis-a-vis the updated Lagrangian formulation are also analyzed.     

An accurate time integration scheme for arbitrary rotation motion: application to metal forming simulation

Within the frame of implicit velocity based formulations with solid elements, usual time integration schemes often turn out unsatisfactory when the movement has large rotations, especially in metal forming applications such as ring rolling or cross-wedge rolling. These rotations generally require using a much higher order integration scheme with inherent difficulties in implementing such schemes. For pure rotation motions, it is possible to use a low order integration scheme by rewriting the motion equations in the cylindrical frame that is supported by the rotation axis. Accordingly, a first order scheme is sufficient to accurately integrate the movement but it is restricted to specific problems. In the more general case, it is possible to derive parts of the domain where rotations are predominant along with the governing rotation axis from the velocity field gradient. The motion equations are then rewritten in the resulting local cylindrical frame. Performances of this first order scheme are first evaluated and highlighted over simple analytical problems, before being applied to the finite element simulation of the torsion test, and then to more complex metal forming problems involving large rotations. The accuracy and efficiency of this scheme is so numerically demonstrated.     

NON-LINEAR FINITE ELEMENT ANALYSIS OF ISOTROPIC AND COMPOSITE SHELLS BY A TOTAL LAGRANGIAN DECOMPOSITION SCHEME

ABSTRACT

A total Lagrangian finite element scheme for arbitrarily large displacements and rotations is applied to a composite beam, an isotropic deep arch, and a section of an isotropic circular toroidal shell. The scheme decomposes the motion into stretches and rigid-body rotations, examining the deformed state with respect to an orthogonal, rigidly translated and rotated triad located at the deformed point of interest. The Jaumann stress and strain measures, which are resolved along the axes of this triad, are employed in the algorithm. Local and layer-wise thickness stretching and shear warping functions are used to model the three-dimensional behaviour of the shell. These functions are developed through the use of the constitutive equations, certain stress and displacement continuity requirements at ply interfaces and laminate surfaces, and the behaviour of the shell reference surface. Two finite elements are employed in the analyses: an eight-noded, 36 degree-of-freedom (DOF) element, and a four-noded, 44 DOF element. The 36 DOF element, which is not a compatible element with respect to the derivatives of in-plane deformations, proves adequate for moderate rotation problems, but fails in modelling very large rotation problems. The use of the 44 DOF element provides dramatically improved results in the large rotation problem.     

A finite deformation method for discrete modeling: particle rotation and parameter calibration

We present a finite deformation method for 3-D discrete element modeling. In this method particle rotation is explicitly represented using quaternion and a complete set of interactions is permitted between two bonded particles, i.e., normal and tangent forces, rolling and torsional torques. Relative rotation between two particles is decomposed into two sequence-independent rotations, such that an overall torsional and rolling angle can be distinguished and torques caused by relative rotations are uniquely determined. Forces and torques are calculated in a finite deformation fashion, rather than incrementally. Compared with the incremental methods our algorithm is numerically more stable while it is consistent with the non-commutativity of finite rotations. We study the macroscopic elastic properties of a regularly arranged 2-D and 3-D lattice. Using a micro-to-macro approach based on the existence of a homogeneous displacement field, we study the problem of how to choose the particle-scale parameters (normal, tangent, rolling and torsional stiffness) given the macroscopic elastic parameters and geometry of lattice arrangement. The method is validated by reproducing the wing crack propagation and the fracture patterns under uniaxial compression. This study will provide a theoretical basis for the calibration of the DEM parameters required in engineering applications.     

Rotation manifold <Emphasis Type="Italic">SO</Emphasis>(3) and its tangential vectors

In this paper, we prove that incremental material rotation vectors belong to different tangent spaces of the rotation manifold SO(3) at a different instant. Moreover, we show that the material tangent space as the tangent space at unity is not a possible definition yielding geometrically inconsistent results, although this kind of definition is widely adopted in applied mechanics community. In addition, we show that the standard Newmark integration scheme for incremental rotations neglects first order terms of rotation vector, not third order terms. Finally, we show that the rotation interpolation of extracted nodal values on the rotation manifold is not an objective interpolation under the observer transformation. This clarifies controversy about the frame-indifference of geometrically exact beam formulations in their finite element implementations.     

Effects of approximations in analyses of beams of open thin‐walled cross‐section—part I: Flexural–torsional stability

In formulating a finite element model for the flexural–torsional stability and 3‐D non‐linear analyses of thin‐walled beams, a rotation matrix is usually used to obtain the non‐linear strain–displacement relationships. Because of the coupling between displacements, twist rotations and their derivatives, the components of the rotation matrix are both lengthy and complicated. To facilitate the formulation, approximations have been used to simplify the rotation matrix. A simplified small rotation matrix is often used in the formulation of finite element models for the flexural–torsional stability analysis of thin‐walled beams of open cross‐section. However, the approximations in the small rotation matrix may lead to the loss of some significant terms in the stability stiffness matrix. Without these terms, a finite element line model may predict the incorrect flexural–torsional buckling load of a beam. This paper investigates the effects of approximations in the elastic flexural–torsional stability analysis of thin‐walled beams, while a companion paper investigates the effects of approximations in the 3‐D non‐linear analysis. It is found that a finite element line model based on a small rotation matrix may predict incorrect elastic flexural–torsional buckling loads of beams. To perform a correct flexural–torsional stability analysis of thin‐walled beams, modification of the model is needed, or a finite element model based on a second‐order rotation matrix can be used. Copyright © 2001 John Wiley & Sons, Ltd.     

A spatially curved‐beam element with warping and Wagner effects

This paper presents a new spatially curved‐beam element with warping and Wagner effects that can be used for the non‐linear large displacement analysis of members that are curved in space. The non‐linear behaviour of members curved in space shows that the Wagner effects are substantial in the large twist rotation analysis. Most existing finite beam element models, such as ABAQUS and ANSYS cannot predict the non‐linear large displacement response of members curved in space correctly because the Wagner effects, viz. the Wagner moment and the corresponding finite strain terms, have not been considered in these finite beam elements. As a consequence, these finite beam elements do not provide correct predictions for the out‐of‐plane buckling and postbuckling behaviour of arches as well. In this paper, the symmetric tangent stiffness matrix has been derived based on the finite rotations parameterized by the conventional displacements. The warping and Wagner effects: both the Wagner moment and the corresponding finite strain terms and their constitutive relationship, are included in the spatially curved‐beam element. Two components of the initial curvature, the initial twist and their interactions with the displacements are also considered in the spatially curved‐beam element. This ensures that the large twist rotation analysis for the members curved in space is accurate. Comparisons with existing experimental, analytical and numerical results show that the spatially curved‐beam element is accurate and efficient for the non‐linear elastic analysis of curved members, buckling and postbuckling analysis of arches, and in its ability to predict large deflections and twist rotations in more arbitrarily curved members. Copyright © 2005 John Wiley & Sons, Ltd.     

An Assumed Strain Triangular Solid Element for Efficient Analysis of Plates and Shells with Finite Rotation

A simple triangular solid shell element formulation is developed for efficient analysis of plates and shells undergoing finite rotations. The kinematics of the present solid shell element formulation is purely vectorial with only three translational degrees of freedom per node. Accordingly, the kinematics of deformation is free of the limitation of small angle increments, and thus the formulation allows large load increments in the analysis of finite rotation. An assumed strain field is carefully selected to alleviate the locking effect without triggering undesirable spurious kinematic modes. In addition, the curved surface of shell structures is modeled with flat facet elements to obviate the membrane locking effect. Various numerical examples demonstrate the efficiency and accuracy of the present element formulation for the analysis of plates and shells undergoing finite rotation. The present formulation is attractive in that only three points are needed for numerical integration over an element.     

Genuinely resultant shell finite elements accounting for geometric and material non-linearity

A general theoretical framework is presented for the fully non-linear analysis of shells by the finite element method. The governing equations are derived exclusively in terms of resulting quantities through a logical and straightforward descent from three-dimensional continuum mechanics without appealing to simplifying assumptions (hence the name genuinely resultant). As a result, the underlying theory is statically and geometrically exact, and it naturally includes small strain and finite strain problems of thin as well as thick shells. The underlying mathematical structure and the variational formulation of the theory are examined. This appears to be crucial for the development of computational procedures employing the Newton-Kantorovich linearization process and the Galerkin type discretization method. The treatment of finite rotations through an arbitrary parametrization of the rotation group and the interpolation procedure of SO(3)-valued functions underlying the construction of finite element basis are other issues studied in this paper. A numerical analysis is presented in order to assess the effectiveness of the proposed formulation. Small strain problems as well as finite strain deformation of rubber-like shells undergoing finite rotations are considered. Special attention is devoted to the assessment of the relevance of the drilling degree-of-freedom and highly non-uniform through-the-thickness deformation in the case of shells made of incompressible material.     

旋翼桨叶结构载荷计算方法比较研究

基于有限转角假设,建立了刚柔耦合旋翼动力学模型。该模型考虑了刚体转动与弹性变形之间的耦合效应,相较于基于小转角假设的传统有限元模型具有明显的优势。气动力以广义力形式与桨叶刚体转动及弹性变形耦合组建方程。在方程求解的单步上,分别采用力积分法、反力法以及曲率法计算桨叶剖面结构振动载荷。以BO105模型桨叶及SA349/2小铃羊直升机为仿真对象,比较研究了这三种载荷计算方法的预测精度与适用范围。对于不考虑气动力的纯结构振动载荷,三种计算方法具有相同的精度。在气弹瞬态计算中,力积分法对桨根载荷的预测精度不足。曲率法与反力法在桨叶有限元节点处得到了相近的结果。反力法预测精度取决于有限元建模精度,且只对节点处载荷有效。由于曲率法只计入弹性桨叶的弯曲曲率,该方法需要更高阶次的形函数以满足自由度二阶导数的连续性。此外,为加速收敛及减少累积误差,本文开发了基于外推法的数值积分算法。