Bézier和B-spline曲线的绝对结点坐标列式有限元离散方法

Using Absolute Nodal Coordinate Formulation Elements to Model Bézier and B-spline Curve for Finite Element Analysis

CAD系统中广泛采用Bézier,B-spline描述复杂的几何形状,而传统有限元则采用完全不同的插值函数来描述分析模型,对几何形状的不同描述导致由CAD模型转换为有限元模型非常困难,且带来模型上的不一致。研究Bézier和B-spline曲线离散为绝对结点坐标列式(Absolute nodal coordinate formulation,ANCF)有限单元的方法,建立Bézier和B-spline曲线与ANCF索单元之间的线性转换关系,实现了两者之间的自动转换,从而为整合CAD和CAA系统提供一种方法,表明ANCF单元可完全涵盖Bézier曲线的特性,Bézier表达其实是ANCF表达的特例,采用ANCF单元可以精确表达Bézier和B-spline曲线。同时研究了B-spline曲线的连续性与ANCF单元结点之间的连续性关系,B-spline采用节点重复度控制所定义图形的连续性,ANCF单元因为采用位置和位置导数作为结点坐标可以自动保证C1连续,C2连续性要求实际上是在单元之间添加曲率相等的约束方程,从而实现消除一个位置导数坐标;对于3阶曲线,证明C3连续性要求下任意两个ANCF单元均可以合并为一个大的单元。进而得出在实际转换和分析中不必通过添加约束方程得到不同的连续性,可以直接由B-spline表达得到不同连续性要求的ANCF单元网格。以悬臂梁在冲击作用下的动态响应为例说明了不同的连续性要求对有限元分析的影响。

In CAD system,Bézier,B-spline representations are widely used to describe complex geometry,while traditional finite element analysis uses totally different method to describe the geometry of model.Such an incompatibility makes it difficult to transfer geometry created in CAD systems into CAA model,and leads to difference geometry between these two systems.The method of conversion from the Bézier and B-spline curve to absolute nodal coordinate formulation（ANCF） elements is focused on.The linear relationship between Bézier,B-spline and ANCF cable elements is established in order to facilitate the automatic transformation to each other.That makes it a possible way for integration of the CAD and CAA system.It shows that ANCF description has the character of Bézier description,and Bézier description is actuary a special case of ANCF description.Bézier and B-spline curve can be represented by ANCF elements exactly.The continuity of B-spline and ANCF elements are studied then.The concept of knot multiplicity is used to control the continuity of B-spline,while ANCF description can automatically ensure C1 continuity because it uses the position and gradient vectors as nodal coordinates.Linear algebraic constraint equations based on equal curvature will eliminate gradient coordinate,leading to C 2 continuity.For cubic curve,C 3 continuity will convert two elements to a single bigger element,even though the two elements are different.It shows also that ANCF elements with different continuity can be created directly by B-spline representation during conversion and analysis,no constraint equation is actuary used.The dynamic respond of a cantilever under the acting of impact force is taken as an example to demonstrate the effect of continuity on finite element analysis.