Integration of B-spline geometry and ANCF finite element analysis

The goal of this investigation is to introduce a new computer procedure for the integration of B-spline geometry and the absolute nodal coordinate formulation (ANCF) finite element analysis. The procedure is based on developing a linear transformation that can be used to transform systematically the B-spline representation to an ANCF finite element mesh preserving the same geometry and the same degree of continuity. Such a linear transformation that relates the B-spline control points and the finite element position and gradient coordinates will facilitate the integration of computer aided design and analysis (ICADA). While ANCF finite elements automatically ensure the continuity of the position and gradient vectors at the nodal points, the B-spline representation allows for imposing a higher degree of continuity by decreasing the knot multiplicity. As shown in this investigation, a higher degree of continuity can be systematically achieved using ANCF finite elements by imposing linear algebraic constraint equations that can be used to eliminate nodal variables. The analysis presented in this study shows that continuity of the curvature vector and its derivative which corresponds in the cubic B-spline representation to zero knot multiplicity can be systematically achieved using ANCF finite elements. In this special case, as the knot multiplicity reduces to zero, the recurrence B-spline formula causes two segments to automatically blend together forming one cubic segment defined on a larger domain. Similarly in this special case, the algebraic constraint equations required for the C ³ continuity convert two ANCF cubic finite elements to one finite element, demonstrating the strong relationship between the B-spline representation and the ANCF finite element representation. For the same order of interpolation, higher degree of continuity at the finite element interface can lead to a coarser mesh and to a lower dimensional model. Using the B-spline/ANCF finite element transformation developed in this paper, the equations of motion of a finite element mesh that represents exactly the B-spline geometry can be developed. Because of the linearity of the transformation developed in this investigation, all the ANCF finite element desirable features are preserved; including the constant mass matrix that can be used to develop an optimum sparse matrix structure of the nonlinear multibody system dynamic equations.     

INTEGRATION OF NON-UNIFORM RATIONAL B-SPLINES GEOMETRY AND RATIONAL ABSOLUTE NODAL COORDINATES FORMULATION FINITE ELEMENT ANALYSIS

This investigation is intended to develop a computer procedure for the integration of NURBS geometry and the rational absolute nodal coordinate formulation （RANCF） finite element analysis. A linear transformation is given that can be used to convert the NURBS curve to RANCF cable element mesh retaining the same geometry and the same degree of continuity, including the discussion of continuity control and mesh refinement. The green strain tensor is used to establish the nonlinear dynamic equations with numerical examples to demonstrate the use of the procedure in the dynamic analysis of flexible bodies.     

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.     

Nonstructural geometric discontinuities in finite element/multibody system analysis

Existing multibody system (MBS) algorithms treat articulated system components that are not rigidly connected as separate bodies connected by joints that are governed by nonlinear algebraic equations. As a consequence, these MBS algorithms lead to a highly nonlinear system of coupled differential and algebraic equations. Existing finite element (FE) algorithms, on the other hand, do not lead to a constant mesh inertia matrix in the case of arbitrarily large relative rigid body rotations. In this paper, new FE/MBS meshes that employ linear connectivity conditions and allow for arbitrarily large rigid body displacements between the finite elements are introduced. The large displacement FE absolute nodal coordinate formulation (ANCF) is used to obtain linear element connectivity conditions in the case of large relative rotations between the finite elements of a mesh. It is shown in this paper that a linear formulation of pin (revolute) joints that allow for finite relative rotations between two elements connected by the joint can be systematically obtained using ANCF finite elements. The algebraic joint constraint equations, which can be introduced at a preprocessing stage to efficiently eliminate redundant position coordinates, allow for deformation modes at the pin joint definition point, and therefore, this new joint formulation can be considered as a generalization of the pin joint formulation used in rigid MBS analysis. The new pin joint deformation modes that are the result of C ⁰ continuity conditions, allow for the calculations of the pin joint strains which can be discontinuous as the result of the finite relative rotation between the elements. This type of discontinuity is referred to in this paper as nonstructural discontinuity in order to distinguish it from the case of structural discontinuity in which the elements are rigidly connected. Because ANCF finite elements lead to a constant mass matrix, an identity generalized mass matrix can be obtained for the FE mesh despite the fact that the finite elements of the mesh are not rigidly connected. The relationship between the nonrational ANCF finite elements and the B-spline representation is used to shed light on the potential of using ANCF as the basis for the integration of computer aided design and analysis (I-CAD-A). When cubic interpolation is used in the FE/ANCF representation, C ⁰ continuity is equivalent to a knot multiplicity of three when computational geometry methods such as B-splines are used. C ² ANCF models which ensure the continuity of the curvature and correspond to B-spline knot multiplicity of one can also be obtained. Nonetheless, B-spline and NURBS representations cannot be used to effectively model T-junctions that can be systematically modeled using ANCF finite elements which employ gradient coordinates that can be conveniently used to define element orientations in the reference configuration. Numerical results are presented in order to demonstrate the use of the new formulation in developing new chain models.     

ANCF curvature continuity: application to soft and fluid materials

The continuity of the position-vector gradients at the nodal points of a finite element mesh does not always ensure the continuity of the gradients at the element interfaces. Discontinuity of the gradients at the interface not only adversely affects the quality of the simulation results, but can also lead to computer models that do not properly represent realistic physical system behaviors, particularly in the case of soft and fluid material applications. In this study, the absolute nodal coordinate formulation (ANCF) finite elements are used to define general curvature-continuity conditions that allow for eliminating or minimizing the discontinuity of the position gradients at the element interface. For the ANCF solid element, with four-node surfaces, it is shown that continuity of the gradients tangent to an arbitrary point on a surface is ensured as the result of the continuity of the gradients at the nodal points. The general ANCF continuity conditions are applicable to both reference-configuration straight and curved geometries. These conditions are formulated without the need for using the computer-aided-design knot vector and knot multiplicity, which do not account properly for the concept of system degrees of freedom. The ANCF curvature-continuity conditions are written in terms of constant geometric coefficients obtained using the matrix of position-vector gradients that defines the reference-configuration geometry. The formulation of these conditions is demonstrated using the ANCF fully parameterized three-dimensional solid and tetrahedral elements, which employ a complete set of position gradients as nodal coordinates. Numerical results are presented in order to examine the effect of applying the curvature-continuity conditions on achieving a higher degree of smoothness at the element interfaces in the case of soft and fluid materials.

     

On the formulation of the planar ANCF triangular finite elements

In this paper, new planar isoparametric triangular finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are developed. The proposed ANCF elements have six coordinates per node: two position coordinates that define the absolute position vector of the node and four gradient coordinates that define vectors tangent to coordinate lines (parameters) at the same node. To shed light on the importance of the element geometry and to facilitate the development of some of the new elements presented in this paper, two different parametric definitions of the gradient vectors are used. The first parametrization, called area parameterization, is based on coordinate lines along the sides of the element in the reference configuration, while the second parameterization, called Cartesian parameterization, employs coordinate lines defined along the axes of the structure (body) coordinate system. The fundamental differences between the ANCF parameterizations used in this investigation and the parametrizations used for conventional finite elements are highlighted. The Cartesian parameterization serves as a unique standard for the triangular FE assembly. To this end, a transformation matrix that defines the relationship between the area and the Cartesian parameterizations is introduced for each element in order to allow for the use of standard FE assembly procedure and define the structure (body) inertia and elastic forces. Using Bezier geometry and a linear mapping, cubic displacement fields of the new ANCF triangular elements are systematically developed. Specifically, two new ANCF triangular finite elements are developed in this investigation, namely four-node mixed-coordinate and three-node ANCF triangles. The performance of the proposed new ANCF elements is evaluated by comparison with the conventional linear and quadratic triangular elements as well as previously developed ANCF rectangular and triangular elements. The results obtained in this investigation show that in the case of small and large deformations as well as finite rotations, all the elements considered can produce correct results, which are in a good agreement if appropriate mesh sizes are used.     

一种O(n)算法复杂度的递推绝对节点坐标法研究

相比于传统的浮动坐标法,绝对节点坐标法(absolute nodal coordinate formulation,ANCF)在处理柔性体非线性大变形问题上具有显著优势,但是对于ANCF的求解目前主要依据拉格朗日方程等分析力学原理建立微分代数方程(differential algebraic equation,DAE)进行,其算法复杂度为O(n2)或O(n3)(n为系统自由度),且求解过程存在位置或速度的违约问题.据此,研究了一种O(n)算法复杂度的递推绝对节点坐标法(recursive absolute nodal coordinate formulation,RANCF).该方法采用ANCF描述大变形柔性体,借鉴铰接体递推算法(articulatedbody algorithm,ABA)思路建立多柔体系统逐单元的运动学和动力学递推关系,得到微分形式的系统动力学方程(ordinary differential equation,ODE).在ODE方程中,系统广义质量阵为三对角块矩阵,通过恰当的矩阵处理,可以得到逐单元求解该方程的递推算法.在此基础上,给出了RANCF算法的详细流程,并对流程中每个步骤进行了细致的算法效率分析,证明了RANCF是算法复杂度为O(n)的高效算法.RANCF方法保留了ANCF对大转动、大变形多柔体系统精确计算的优点,同时极大地提升了算法效率,特别在处理高自由度复杂多柔体系统中具有显著优势.并且该方法采用ODE求解,无DAE的违约问题,因此具有更高的算法精度.最后,在算例部分,通过MSC.ADAMS仿真软件、能量守恒测试、算法复杂度曲线对RANCF的正确性、计算精度和计算效率进行了验证.     

Numerical investigation of the slope discontinuities in large deformation finite element formulations

In many multibody system applications, the system components are made of structural elements that can have different orientations, leading to slope discontinuities. In this paper, a numerical investigation of a new procedure that can be used to model structures with slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is presented. This procedure can be applied to model slope discontinuities in the case of commutative rotations of gradient deficient elements that are used for modeling thin beam and plate structures. An important special case to which the proposed procedure can be applied is the case of all planar gradient deficient ANCF finite elements. The use of the proposed method leads to a constant orthogonal element transformation that describes an arbitrary initial configuration. As a consequence, one obtains, in the case of large commutative rotations and large deformations, a constant mass matrix for structures which have complex geometry. The procedure used in this investigation to model slope discontinuities requires the use of the concept of the intermediate finite element coordinate system. For each finite element, a new set of gradient coordinates that define, at the discontinuity node, the element deformation with respect to the intermediate element coordinate system is introduced. These new gradient coordinates are assumed to be equal for the two finite elements at the point of intersection. That is, the change of the gradients of two elements at the intersection point from their respective intermediate initial reference configuration is assumed to be the same. This procedure leads to a set of linear algebraic equations that define the orthogonal transformation matrix for the finite element. Numerical examples are presented in order to demonstrate the use of the proposed procedure for modeling slope discontinuities.     

A new nonlinear multibody/finite element formulation for knee joint ligaments

The focus of this investigation is to study the mechanics of the human knee using a new method that integrates multibody system and large deformation finite element algorithms. The major bones in the knee joint consisting of the femur, tibia, and fibula are modeled as rigid bodies. The ligaments structures are modeled using the large displacement finite element absolute nodal coordinate formulation (ANCF) with an implementation of a Neo-Hookean constitutive model that allows for large change in the configuration as experienced in knee flexion, extension, and rotation. The Neo-Hookean strain energy function used in this study takes into consideration the near incompressibility of the ligaments. The ANCF is used in the formulation of the algebraic equations that define the ligament/bone rigid connection. A unique feature of the ANCF model developed in this investigation is that it captures the deformation of the ligament cross section using structural finite elements such as beams. At the ligament/bone insertion site, the ANCF is used to define a fully constrained joint. This model will reflect the fact that the geometry, placement and attachment of the two collateral ligaments (the LCL and MCL), are significantly different from what has been used in most knee models developed in previous investigations. The approach described in this paper will provide a more realistic model of the knee and thus more applicable to future research studies on ligaments, muscles and soft tissues (LMST). Current finite element models are limited due to simplified assumptions for the spatial and time dependent material properties inherent in the anisotropic and anatomic constraints associated with joint stability, and the static conditions inherent in the analysis. The ANCF analysis is not limited to static conditions and results in a fully dynamic model that accounts for the distributed inertia and elasticity of the ligaments. The results obtained in this investigation show that the ANCF finite elements can be an effective tool for modeling very flexible structures like ligaments subjected to large flexion and extension. In the future, the more realistic ANCF models could assist in examining the mechanics of the knee to study knee injuries and possible prevention means, as well as an improved understanding of the role of each individual ligament in the diagnosis and assessment of disease states, aging and potential therapies.     

Numerical convergence of finite element solutions of nonrational B-spline element and absolute nodal coordinate formulation

In this investigation, numerical convergence of finite element solutions obtained using the B-spline approach and the absolute nodal coordinate formulation (ANCF) is discussed. Furthermore, equivalence of the two formulations with different orders of polynomials and degrees of continuity is demonstrated by several numerical examples. The degree of continuity can be easily controlled in B-spline elements by changing knot multiplicities, while continuity conditions associated with higher order derivatives need to be imposed to achieve C ² and higher continuities in ANCF elements. In order to compare element performances of the third and quartic B-spline and ANCF elements, the three-node quartic ANCF beam element is developed. It is demonstrated in several numerical examples that use of B-spline and ANCF elements with same orders and continuities leads to identical results. Furthermore, effects of polynomial orders and continuities on the accuracy and numerical convergence are demonstrated.     

A piecewise beam element based on absolute nodal coordinate formulation

The element created in this investigation is based on the it absolute nodal coordinate formulation (ANCF) which has been successfully used in flexible multibody system dynamic and integration of computer aid design and analysis (ICADA). When modeling a B-spline curve with ANCF beam element, it is the common manner to convert this curve into a series of Bézier curves because the systematical conversion between ANCF beam element and a Bézier curve has already been built. In order to avoid the constrain equation produced in this method and to express a B-spline curve using only one element, an alternative approach is developed, leading to the piecewise ANCF (PANCF) beam element. It is demonstrated that when two ANCF beam elements are connected according to a particular continuity, they can constitute a PANCF element. Besides, a new PANCF element will be produced when an ANCF element is connected to an existing PANCF element. The continuity condition can be automatically ensured by the selection of nodal coordinates and the calculation of the piecewise continuous shape functions. The algorithm for converting a B-spline curve to a PANCF beam element is then given. There also are discussions on the features of PANCF element. When two neighboring segments of PANCF element have the same assumed length, the position vector at the interface cannot be expressed by the other coordinates so the position vector is preserved in the \(C^{2}\) continuous situation. Two examples are given to conclude the interpolation and continuity properties of the shape function and to demonstrate the feasibility of this PANCF in the ICADA.     

Implicit and explicit integration in the solution of the absolute nodal coordinate differential/algebraic equations

This investigation is concerned with the use of an implicit integration method with adjustable numerical damping properties in the simulation of flexible multibody systems. The flexible bodies in the system are modeled using the finite element absolute nodal coordinate formulation (ANCF), which can be used in the simulation of large deformations and rotations of flexible bodies. This formulation, when used with the general continuum mechanics theory, leads to displacement modes, such as Poisson modes, that couple the cross section deformations, and bending and extension of structural elements such as beams. While these modes can be significant in the case of large deformations, and they have no significant effect on the CPU time for very flexible bodies; in the case of thin and stiff structures, the ANCF coupled deformation modes can be associated with very high frequencies that can be a source of numerical problems when explicit integration methods are used. The implicit integration method used in this investigation is the Hilber–Hughes–Taylor method applied in the context of Index 3 differential-algebraic equations (HHT-I3). The results obtained using this integration method are compared with the results obtained using an explicit Adams-predictor-corrector method, which has no adjustable numerical damping. Numerical examples that include bodies with different degrees of flexibility are solved in order to examine the performance of the HHT-I3 implicit integration method when the finite element absolute nodal coordinate formulation is used. The results obtained in this study show that for very flexible structures there is no significant difference in accuracy and CPU time between the solutions obtained using the implicit and explicit integrators. As the stiffness increases, the effect of some ANCF coupled deformation modes becomes more significant, leading to a stiff system of equations. The resulting high frequencies are filtered out when the HHT-I3 integrator is used due to its numerical damping properties. The results of this study also show that the CPU time associated with the HHT-I3 integrator does not change significantly when the stiffness of the bodies increases, while in the case of the explicit Adams method the CPU time increases exponentially. The fundamental differences between the solution procedures used with the implicit and explicit integrations are also discussed in this paper.     

Use of B-spline surface to model large-deformation continuum plates: procedure and applications

The absolute nodal coordinate formulation (ANCF) has been used in the analysis of large deformation of flexible multibody systems that encompass belt drive, rotor blade, and cable applications. As demonstrated in the literature, the ANCF finite elements are ideal for isogeometric analysis. The purpose of this investigation is to establish a relationship between the B-splines, which are widely used in the geometric modeling, and the ANCF finite elements in order to construct continuum models of large-deformation geometries. This paper proposes a simplified approach to map the B-spline surfaces into ANCF thin plate elements. Matrix representation of the mapping process is established and examined through numerical examples successfully. The matrix representation of the mapping process is used because of its suitability of computer coding and to minimize the calculation time. The error estimation is carried out by analyzing the gap between the points of each ANCF element and the corresponding points of the portion of the B-spline surface. The Hausdorff distance is used to study the effect of the number of control points, the degree of interpolation, and the knot multiplicity on the mapped geometry. It is found that cubic interpolation is recommended for optimizing the accuracy of mapping the B-spline surface to ANCF thin plate elements. It is found that thin plate element in ANCF missing a number of basis functions which considered a source of error between the two surfaces, as well as it does not allow to converting the ANCF thin plate elements model to B-spline surface. In this investigation, an application example of modeling large-size wind turbine blade with uniform structure is illustrated. The use of the continuum plate elements in modeling flexible blades is more efficient because of the relative scale between the plate thickness and its length and width and the high flexibility of its structure. The numerical results are compared with the results of ANSYS code with a good agreement. The dynamic simulation for mapped surface model shows a numerical convergence, which ensures the ability of using the proposed approach for applications of dynamics for design and computer-aided design.     

A triangular plate element 2343 using second-order absolute-nodal-coordinate slopes: numerical computation of shape functions

In this paper, the process by which geometrical and structural matrices of plate finite elements employing absolute nodal coordinate formulation (ANCF) are constructed is studied. The kinematic and topological properties of an arbitrary plate finite element are described using universal digital code dncm that provides systematic enumeration of finite elements. This code is formed using the element’s dimension d, the number of nodes it possesses n, the number of scalar coordinates per node c, and a multiplier describing the process of transforming a conventional finite element to an ANCF element m. The detailed generation of a new type of triangular plate finite element 2343 using numerical computation of shape functions is also discussed in the paper. The new triangular element employs position vectors and slope vectors up to second-order mixed-derivative slope vector. A detailed derivation of the equations of motion of the element is also provided and examples of its numerical simulation and validation presented.     

Sparse matrix implicit numerical integration of the Stiff differential/algebraic equations: Implementation

The finite element absolute nodal coordinate formulation (ANCF) is often used in modeling very flexible bodies in multibody system (MBS) applications. This formulation leads to a constant mass matrix, allowing for an efficient sparse matrix implementation. Nonetheless, the use of the ANCF finite elements to model stiff structures can lead to high frequencies associated with ANCF coupled deformation modes, as discussed in the literature. Implicit numerical integration methods can be effectively used to develop efficient procedures for the solution of MBS differential/algebraic equations. Most existing implicit integration algorithms, however, require numerical differentiation of the equations of motion, and some of these integration methods do not ensure that the kinematic algebraic constraint equations are satisfied at all levels (position, velocity, and acceleration). Because of these limitations, existing implicit integration methods can be less accurate and less efficient when used to solve large scale MBS applications. In order to circumvent this problem, the two-loop implicit sparse matrix numerical integration (TLISMNI) method was proposed for the solution of MBS differential/algebraic equations. The TLISMNI method does not require numerical differentiation of the forces and allows for an efficient sparse matrix implementation. This paper discusses TLISMNI implementation issues including the step size selection, the error control, and the effect of the numerical damping. The relation between the step size selection and the structure stiffness is also discussed. The use of the computer implementation described in this paper is demonstrated by solving very stiff structure problems using the Hilber?CHughes?CTaylor (HHT) method, which includes numerical damping. An eigenvalue analysis and Fast Fourier Transform (FFT) are performed in order to identify the fundamental modes of deformation and demonstrate that the contributions of these fundamental modes can be erroneously damped out when some other implicit integration methods are used. The TLISMNI method, on the other hand, captures the contributions of these fundamental modes. The results, obtained using the TLISMNI method, are compared with the results obtained using other methods including the implicit HHT-I3 and the explicit Adams integration methods. The results obtained show that the TLISMNI method can be five times faster than the other two methods when no numerical damping is considered.     

Absolute nodal coordinate formulation of large-deformation piezoelectric laminated plates

The Absolute Nodal Coordinate Formulation (ANCF) has been initiated in 1996 by Shabana (Computational Continuum Mechanics, 3rd edn., Cambridge: Cambridge University Press, 2008). It introduces large displacements of planar and spatial finite elements relative to the global reference frame without using any local frame. A sub-family of beam, plate and cable finite elements with large deformations are proposed and employed the 3D theory of continuum mechanics. In the ANCF, 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. In contrast to other large deformation formulations, the equations of motion contain constant mass matrices as well as zero centrifugal and Coriolis inertia forces. The only nonlinear term is a vector of elastic forces. This investigation concerns a way to generate new finite element in the ANCF for laminated composite plates. This formulation utilizes the assumption that the bonds between the laminae are thin and shear is non-deformable. Consequently, the Equivalent Single Layer, ESL model, is implemented. In the ESL models, the laminate is assumed to deform as a single layer, assuming a smooth variation of the displacement field across the thickness. In this paper, the coupled electromechanical effect of Piezoelectric Laminated Plate is imposed within the ANCF thin plate element, in such a way as to achieve the continuity of the gradients at the nodal points, and obtain a formulation that automatically satisfies the principle of work and energy. Convergence and accuracy of the finite-element ANCF Piezoelectric Laminated Plate is demonstrated in geometrically nonlinear static and dynamic test problems, as well as in linear analysis of natural frequencies. The computer implementation and several numerical examples are presented in order to demonstrate the use of the formulation developed in this paper. A comparison with the commercial finite element package COMSOL MULTIPHYSICS () is carried out with an excellent agreement.     

几何精确NURBS有限元中边界条件施加方式对精度影响的三维计算分析

非均匀有理B样条(NURBS)有限元法把计算机辅助几何设计(CAGD)中的NURBS几何构形方法与有限元方法有机结合起来,有效消除了有限元离散模型的几何误差,提高了计算精度。但是由于NURBS基函数不是插值函数,直接在控制节点上施加位移边界条件会引起较大误差。本文详细讨论了NURBS基函数的插值特性,在NURBS有限元分析中采用罚函数法施加位移边界条件,提高了收敛率和计算精度。结合典型三维弹性力学问题,对两种施加位移边界条件的方法进行了对比和分析。计算结果表明,直接施加位移边界条件会导致收敛率和精度的明显降低,而基于罚函数法的NURBS有限元分析则能达到最优收敛率,并具有更高的精度。     

New spatial curved beam and cylindrical shell elements of gradient-deficient Absolute Nodal Coordinate Formulation

Based on previous studies, a new spatial curved slender-beam finite element and a new cylindrical shell finite element are proposed in the frame of gradient-deficient Absolute Nodal Coordinate Formulation (ANCF). The strain energy of the beam element is derived by using the definition of the Green?CLagrange strain tensor in continuum mechanics so that the assumption on small strain can be relaxed. By using the differential geometry and the continuum mechanics, the angle between two base vectors of a defined local coordinate frame of the cylindrical shell element is introduced into the strain energy formulations. Therefore, the new shell element can be used to model parallelogram shells. The analytical formulations of elastic forces and their Jacobian for the above two finite elements of gradient-deficient ANCF are also derived via the skills of tensor analysis. The generalized-alpha method is used to solve the huge set of system equations. Finally, four case studies including both static and dynamic problems are given to validate the proposed beam and cylindrical shell elements of gradient-deficient ANCF.     

Development of ANCF tetrahedral finite elements for the nonlinear dynamics of flexible structures

Nonlinear Dynamics - In this paper, methods for developing isoparametric tetrahedral finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are presented. The proposed ANCF...     

ANCF/CRBF平面梁闭锁问题及闭锁缓解研究

本文系统地研究了基于一致旋转场列式的绝对节点坐标 (ANCF consistentrotation-based formulation, ANCF/CRBF)平面梁单元的泊松闭锁问题及闭锁缓解技术.为了全面理解该类型单元的闭锁特性及明确单元的应用范围,文中首先开发了两种新的ANCF/CRBF刚性截面梁单元, 新单元在ANCF全参数梁的基础上,对梯度向量施加正交矩阵约束, 得到梯度与转角对时间导数之间的速度转换矩阵,从而引入转角参数. 新单元节点处完全消除了泊松闭锁和剪切效应,这是与传统ANCF/CRBF刚性截面梁单元的不同之处. 然后,对比分析了这三种ANCF/CRBF刚性截面梁单元泊松闭锁的特点.发现该类型单元对节点的横向梯度施加了运动学约束, 导致节点处截面不能变形,无法捕捉泊松效应, 但是单元内部能完全捕捉,这种不连续情况会加重单元整体的泊松闭锁问题. 并且发现对单元梯度约束的越多,闭锁问题越严重. 随后, 分别采用两种闭锁缓解技术, 弹性线方法和应变分解方法,进一步研究了单元的收敛性. 最终,通过多种静力学和动力学测试研究了泊松闭锁对ANCF/CRBF平面梁单元计算精度的影响及闭锁缓解技术在该类型单元上的缓解效果.