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.     

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.     

完备化ANCF薄板单元及在钢板弹簧动力学建模中的应用

绝对节点坐标方法已在多体系统动力学研究中广泛应用, 但常用来描述板壳类结构的薄板单元, 由于梯度不完备而无法直接用于带有初始弯曲参考构型的柔性体变形描述. 为避免全参数板单元建立车辆钢板弹簧模型时存在的严重截面闭锁问题, 拟采用薄板单元用于板簧建模. 为此, 探索了将现有绝对节点坐标薄板单元纳入一般连续介质力学弹性力表达的方法, 采用中面上单位法向量作为单元厚度方向的梯度向量, 从而得到了完备化的薄板单元及其描述初始弯曲构型时消除初应变的方法. 在此基础上通过定义簧片的未变形构型, 在钢板弹簧中引入可控的预应力, 实现对钢板弹簧装配过程的准确模拟. 通过数值算例验证了本方法的正确性. 建立了车辆钢板弹簧模型, 通过建立在簧片上的局部坐标系实现接触点的跨单元搜索, 并采用惩罚函数法和平滑化的库伦摩擦模型施加簧片间的接触力. 引入参考节点的概念建立了整合车身与吊耳及其机构运动关系的刚柔耦合模型.}}      

Behavior of thin rectangular ANCF shell elements in various mesh configurations

The absolute nodal coordinate formulation (ANCF) is characterized by being developed specifically for dynamic analysis of large deformation problems. The objective of the study is to investigate how the shape of the initial mesh configuration influences the obtained numerical solution. After a thorough review of three available formulations, they are used in three different convergence studies. Initially a reference study is conducted to determine how the ANCF performs in an uniform and rectangular mesh. Subsequently, the ANCF methods sensitivity to irregular mesh is investigated and finally, the ability of the ANCF method to describe curved structures is evaluated. This study concludes that thin ANCF shell elements are sensitive to both the initial shape and their loading condition. Furthermore, both the initial configuration and the loading condition affect how the ANCF-based models converge. It is suggested that models containing thin ANCF shell elements are subjected to extensive validation studies, before they are used in a design process.     

Dynamic simulation of frictional contacts of thin beams during large overall motions via absolute nodal coordinate formulation

The aim of this study is to develop an approach of simulating the frictional contact dynamics of thin beams with large deformations and continuous contact zones of large size during their large overall motions. For this purpose, the thin beams are meshed via initially straight and gradient deficient thin beam elements of the absolute nodal coordinate formulation (ANCF) degenerated from a curved beam element of ANCF. A detection strategy for contact zone is proposed based on the combination of the minimal distance criterion and master-slave approach. By making use of the minimal distance criterion, the closest points of two thin beams can be found efficiently. The master-slave approach is employed to determine the continuous contact zone. The generalized frictional contact forces and their Jacobians are derived based on the principle of virtual work. Gauss integration is used to integrate the contact forces over the continuous contact zone. The generalized-alpha method is used to solve the dynamic equations of contacting beams. Numerical simulations of four static and dynamic contact problems, including those with continuous contact zones of large size, are completed to validate the high performance of the approach.     

Two new triangular thin plate/shell elements based on the absolute nodal coordinate formulation

Nonlinear Dynamics - In this paper, two three-node triangular thin plate/shell elements are proposed based on the absolute nodal coordinate formulation. As the gradient deficient element, the thin...     

Contact dynamics of elasto-plastic thin beams simulated via absolute nodal coordinate formulation

Under the frame of multibody dynamics, the contact dynamics of elasto-plastic spatial thin beams is numerically studied by using the spatial thin beam elements of absolute nodal coordinate formulation (ANCF). The inter-nal force of the elasto-plastic spatial thin beam element is derived under the assumption that the plastic strain of the beam element depends only on its longitudinal deformation. A new body-fixed local coordinate system is introduced into the spatial thin beam element of ANCF for efficient con-tact detection in the contact dynamics simulation. The linear isotropic hardening constitutive law is used to describe the elasto-plastic deformation of beam material, and the classical return mapping algorithm is adopted to evaluate the plastic strains. A multi-zone contact approach of thin beams previ-ously proposed by the authors is also introduced to detect the multiple contact zones of beams accurately, and the penalty method is used to compute the normal contact force of thin beams in contact. Four numerical examples are given to demonstrate the applicability and effectiveness of the pro-posed elasto-plastic spatial thin beam element of ANCF for flexible multibody system dynamics.     

一种薄壳厚壳通用的轴对称曲壳单元

有限元法分析轴对称壳体时，常需区分薄壳和厚壳并选用不同的单元，给计算带来不便．为此，通过对现有的轴对称壳体单元的研究，基于加权残值法，将广义协调条件引入剪应变场，构造了一种挠度和转角各自独立的新型轴对称曲壳单元．算例结果表明，新单元具有很高的精度，既可用于厚壳，也可用于薄壳．该单元可用于轴对称壳体结构的计算分析．     

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.     

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.