An improved dynamic model for a silicone material beam with large deformation

Dynamic modeling for incompressible hyperelastic materials with large deformation is an important issue in biomimetic applications. The previously proposed lower-order fully parameterized absolute nodal coordinate formulation (ANCF) beam element employs cubic interpolation in the longitudinal direction and linear interpolation in the transverse direction, whereas it cannot accurately describe the large bending deformation. On this account, a novel modeling method for studying the dynamic behavior of nonlinear materials is proposed in this paper. In this formulation, a higher-order beam element characterized by quadratic interpolation in the transverse directions is used in this investigation. Based on the Yeoh model and volumetric energy penalty function, the nonlinear elastic force matrices are derived within the ANCF framework. The feasibility and availability of the Yeoh model are verified through static experiment of nonlinear incompressible materials. Furthermore, dynamic simulation of a silicone cantilever beam under the gravity force is implemented to validate the superiority of the higher-order beam element. The simulation results obtained based on the Yeoh model by employing three different ANCF beam elements are compared with the result achieved from a commercial finite element package as the reference result. It is found that the results acquired utilizing a higher-order beam element are in good agreement with the reference results, while the results obtained using a lower-order beam element are different from the reference results. In addition, the stiffening problem caused by volumetric locking can be resolved effectively by applying a higher-order beam element. It is concluded that the proposed higher-order beam element formulation has satisfying accuracy in simulating dynamic motion process of the silicone beam.     

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.     

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.      

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.     

两类基于局部标架梁单元的闭锁缓解方法

对于大转动、大变形柔性体的刚柔耦合动力学问题,基于李群SE(3)局部标架(local frame formulation,LFF)的建模方法能够规避刚体运动带来的几何非线性问题,离散数值模型中广义质量矩阵与切线刚度矩阵满足刚体变换的不变性,可明显地提高柔性多体系统动力学问题的计算效率.有限元方法中,闭锁问题是导致单元收...     

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.     

A finite element beam model including cross-section distortion in the absolute nodal coordinate formulation

A new class of beam finite elements is proposed in a three-dimensional fully parameterized absolute nodal coordinate formulation, in which the distortion of the beam cross section can be characterized. The linear, second-order, third-order, and fourth-order models of beam cross section are proposed based on the Pascal triangle polynomials. It is shown that Poisson locking can be eliminated with the proposed higher-order beam models, and the warping displacement of a square beam is well described in the fourth-order beam model. The accuracy of the proposed beam elements and the influence of cross-section distortion on structure deformation and dynamics are examined through several numerical examples. We find that the proposed higher-order models can capture more accurately the structure deformation such as cross-section distortion including warping, compared to the existing beam models in the absolute nodal coordinate formulation.     

A NEW SPATIAL THIN-WALLED BEAM ELEMENT INCLUDING TRANSVERSE AND TORSIONAL SHEAR DEFORMATION

基于Timoshenko梁及Benscoter薄壁杆件理论,建立了考虑剪切变形、弯扭耦合以及翘曲剪应力影响的空间任意开闭口薄壁截面梁单元. 通过引入单元内部结点,对弯曲转角和翘曲角采用三节点Lagrange独立插值的方法,考虑了剪切变形和翘曲剪应力的影响并避免了横向剪切锁死问题;借助载荷作用下薄壁梁的截面运动分析,在位移和应变方程中考虑了弯扭耦合的影响. 通过数值算例将该单元的计算结果与理论解以及商用有限元软件和其他文献中的数值解进行对比和验证,结果对比表明该薄壁梁单元具有良好的精度和收敛性.     

基于绝对节点坐标法的弹性线方法研究

相比于浮动坐标系法, 绝对节点坐标法(absolute nodal coordinateformulation, ANCF)在处理柔性体非线性大变形问题上具有显著优势,ANCF将单元节点坐标定义在全局坐标系下,采用斜率矢量代替节点转角坐标, 具有常数质量阵,不存在科氏离心力等优点, 然而弹性力阵为非线性项,其求解将比较耗时且占用资源. 据此, 在弹性力求解方法中,引入弹性线方法(elastic line method, ELM),该方法将格林--拉格朗日应变张量定义在中心线上,采用曲率公式来定义弯曲应变, 转角公式来定义扭转应变.同时采用有限元法对三维柔性梁位移场进行离散,求解梁单元常数质量阵、广义刚度阵、广义力阵,进而得到单元的动力学方程, 通过转换矩阵得到三维梁的动力学方程.接着从理论上指出连续介质力学方法(continuum mechanics method,CMM)和弹性线方法在求解弹性力上的不同点, 并编制动力学仿真软件.最后分别采用连续介质力学方法和弹性线方法对柔性单摆以及履带式车辆的动力学问题进行仿真分析,结果表明:弹性线方法能在保证精度的前提下有效提高计算效率.      

ANCF索梁单元应变耦合问题与模型解耦

索粱结构在土木工程、航空航天等领域有着广泛的应用.在各类索梁动力学建模方法中,由于绝对节点坐标方法(absolute nodal coordinate formulation,ANCF)能够描述柔性体的大变形和大转动问题,因此非常适合大变形索梁结构的动力学建模.对绝对节点坐标索梁单元的应变进行分析可知,弯曲变形会引起单元内部轴向应变的不均匀分布,即单元轴向应变与弯曲应变相互耦合.这种应变耦合效应使单元产生伪应变能,导致单元刚度增大,造成单元失真.分析不同弯曲角下的单元应变及应变能可知,弯曲变形越大,单元失真越严重.通过构造等效一维杆单元重新描述轴向应变,实现了轴向应变与弯曲应变解耦.在此基础上推导广义弹性力,得到了绝对节点坐标索梁单元的应变解耦模型.对解耦前后的两种梁模型进行静力学和动力学仿真,结果表明;解耦模型消除了单元伪应变,相比原模型表现出更好的收敛性和曲率连续性,在相同单元数目下具有更高的精度.同时由于解耦模型降低了单元刚度,因此相比原模型,速度曲线中不再有高频振动.     

Boundary-type finite element method for wave propagation analysis

The boundary-type finite element method has been investigated and applied to the Helmholz and mild-slope equations. Four types of interpolation function are examined based on trigonometric function series. Three-node triangular, four-node quadrilateral, six-node triangular and eight-node quadrilateral elements are tested; these are all non-conforming elements. Three types of numerical example show that the three-node triangular and four-node quadrilateral elements are useful for practical analysis.     

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.

     

Experiments and theory in strain gradient elasticity

Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, b_h. To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam's thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, b_h, is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam's bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.     

复合材料厚梁精化锯齿理论及有限元分析

由于具有预先满足层间应力连续的优点,锯齿理论被广泛研究和应用。然而,至今锯齿理论仍然存在如下难题：基于锯齿理论构造单元时,需使用满足单元间C1连续的插值函数,难于构造多节点高阶单元,而且精度较低。如果这些问题不被重视和解决,应用此类理论分析复合材料力学问题可能得出不恰当的结论。通过发展高精度的考虑横法向应变的C0型锯齿理论,本文将克服已有锯齿理论遇到的上述难题。基于发展的锯齿理论,构造三节点梁单元验证发展理论模型的性能。     

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.     

基于绝对节点坐标法的平面梁有限变形下变形重构

现有的对有限变形条件下柔性结构变形重构的研究往往单纯基于曲率与应变间的几何关系, 同时忽略了被测体的纵向变形及其与弯曲变形的耦合效应. 为得到一种更加精确且能借助现有的力学工具进行应用方向扩展的变形重构方法, 以平面梁为对象, 借鉴变形重构逆有限元法的思想, 将平面梁的变形重构问题视作一类最优化问题. 首先, 通过引入绝对节点坐标法(absolute nodal coordinate formulation, ANCF)对柔性结构大变形下非线性的平面梁应变?位移关系进行精确描述, 构造了一种逆梯度缩减ANCF平面索梁单元. 然后, 对此逆ANCF单元进行改进, 在简化节点自由度的同时通过引入罚函数确保单元节点处的曲率连续性, 既保证了本问题的适定性, 也提升了最终解的精确性. 最后, 基于该单元利用Newton法构造了平面梁有限变形下变形重构问题的两种求解算法, 即逐单元算法和多单元整体算法, 以实现不同需求下的稳定求解. 数值仿真结果表明, 本方法在大变形条件下的变形重构误差小于1%, 而且在测点较少的情况下依然保持较高的精度, 同时验证了本方法的收敛性与计算效率.      

On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory： equilibrium, governing equation and static deflection

This paper has successfully addressed three critical but overlooked issues in nonlocal elastic stress field theory for nanobeams： （i） why does the presence of increasing nonlocal effects induce reduced nanostructural stiffness in many, but not consistently for all, cases of study, i.e., increasing static deflection, decreasing natural frequency and decreasing buckling load, although physical intuition according to the nonlocal elasticity field theory first established by Eringen tells otherwise？ （ii） the intriguing conclusion that nanoscale effects are missing in the solutions in many exemplary cases of study, e.g., bending deflection of a cantilever nanobeam with a point load at its tip; and （iii） the non-existence of additional higher-order boundary conditions for a higher-order governing differential equation. Applying the nonlocal elasticity field theory in nanomechanics and an exact variational principal approach, we derive the new equilibrium conditions, do- main governing differential equation and boundary conditions for bending of nanobeams. These equations and conditions involve essential higher-order differential terms which are opposite in sign with respect to the previously studies in the statics and dynamics of nonlocal nano-structures. The difference in higher-order terms results in reverse trends of nanoscale effects with respect to the conclusion of this paper. Effectively, this paper reports new equilibrium conditions, governing differential equation and boundary condi- tions and the true basic static responses for bending of nanobeams. It is also concluded that the widely accepted equilibrium conditions of nonlocal nanostructures are in fact not in equilibrium, but they can be made perfect should the nonlocal bending moment be replaced by an effective nonlocal bending moment. These conclusions are substantiated, in a general sense, by other approaches in nanostructural models such as strain gradient theory, modified couple stress models and experiments.     

Nonlinear dynamics of three-dimensional belt drives using the finite-element method

In this paper, new nonlinear dynamic formulations for belt drives based on the three-dimensional absolute nodal coordinate formulation are developed. Two large deformation three-dimensional finite elements are used to develop two different belt-drive models that have different numbers of degrees of freedom and different modes of deformation. Both three-dimensional finite elements are based on a nonlinear elasticity theory that accounts for geometric nonlinearities due to large deformation and rotations. The first element is a thin-plate element that is based on the Kirchhoff plate assumptions and captures both membrane and bending stiffness effects. The other three-dimensional element used in this investigation is a cable element obtained from a more general three-dimensional beam element by eliminating degrees of freedom which are not significant in some cable and belt applications. Both finite elements used in this investigation allow for systematic inclusion or exclusion of the bending stiffness, thereby enabling systematic examination of the effect of bending on the nonlinear dynamics of belt drives. The finite-element formulations developed in this paper are implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing more detailed models of mechanical systems that include belt drives subject to general loading conditions, nonlinear algebraic constraints, and arbitrary large displacements. The use of the formulations developed in this investigation is demonstrated using two-roller belt-drive system. The results obtained using the two finite-element formulations are compared and the convergence of the two finite-element solutions is examined.     

Daubechies条件小波有限元法研究

为拓展小波理论在结构工程中的应用,提高结构计算精度,提出了以Daubechies条件小波Ritz法为基础的Daubechies条件小波有限元法。该法结合广义变分原理和拉格朗日乘子法构造修正泛函,根据修正泛函的驻值条件得到全域法求解方程矩阵。根据构件的边界条件,按左右边界对求解矩阵进行相应拆分,构建条件小波单元刚度矩阵,并依据公共节点位移相等原则形成总体刚度矩阵,由此解得各单元的小波基待定系数,即可进一步求解位移场函数、内力分布函数及荷载集度函数。以工程中常见的弹性拉压杆及平面弯曲梁为例,详细阐述了该方法的构造过程。并通过典型算例将Daubechies条件小波有限元法计算值与理论解进行了对比,结果表明:在弹性拉压杆算例中,位移、应力、载荷集度的相对误差均在1.22×10-3%以内;在平面弯曲梁算例中,挠度、弯矩、载荷集度的相对误差均在8.91×10-2%以内。     

Analysis of Thin Beams and Cables Using the Absolute Nodal Co-ordinate Formulation

The purpose of this paper is to present formulations for beam elements based on the absolute nodal co-ordinate formulation that can be effectively and efficiently used in the case of thin structural applications. The numerically stiff behaviour resulting from shear terms in existing absolute nodal co-ordinate formulation beam elements that employ the continuum mechanics approach to formulate the elastic forces and the resulting locking phenomenon make these elements less attractive for slender stiff structures. In this investigation, additional shape functions are introduced for an existing spatial absolute nodal co-ordinate formulation beam element in order to obtain higher accuracy when the continuum mechanics approach is used to formulate the elastic forces. For thin structures where bending stiffness can be important in some applications, a lower order cable element is introduced and the performance of this cable element is evaluated by comparing it with existing formulations using several examples. Cables that experience low tension or catenary systems where bending stiffness has an effect on the wave propagation are examples in which the low order cable element can be used. The cable element, which does not have torsional stiffness, can be effectively used in many problems such as in the formulation of the sliding joints in applications such as the spatial pantograph/catenary systems. The numerical study presented in this paper shows that the use of existing implicit time integration methods enables the simulation of multibody systems with a moderate number of thin and stiff finite elements in reasonable CPU time.