一种O（n）算法复杂度的递推绝对节点坐标法研究

摘要

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(articulated-body 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.%Compared with the tradition floating frame of reference formulation, the absolute nodal coordinate formulation (ANCF) has a natural advantage in solving nonlinear large deformation problems. However, the mathematic model es-tablished by ANCF is always converted to differential algebraic equation (DAE) based on analytical mechanics methods, which leads to an O(n2) or O(n3) algorithm complexity and position or speed constraint violation during the solution pro-cedure. In order to solve these problems, this paper proposes a recursive absolute nodal coordinate formulation (RANCF) with O(n) algorithm complexity. Firstly, the flexible bodies are described by RANCF. Secondly, a kinematic and dynamic recursive relationship between adjacent elements in the flexible multibody system is established based on the articulated-body algorithm (ABA). The equation obtained by RANCF is an ordinary differential equation (ODE), and the system generalized mass matrix is a tridiagonal block matrix. Thus, a recursive solution of the equation by element could be obtained through an appropriate matrix processing. On this basis, a particular algorithm flow of RANCF is provided with the eu000eciency of each step analyzed in detail, which proves the RANCF is an O(n) complexity algorithm. The RANCF maintains the advantage of ANCF that can accurately solve large deformation multibody problem, and vastly improves the computational eu000eciency of ANCF. In addition, because the ANCF avoids the constraint violation problems of DAE, it also has a higher algorithm accuracy. Finally, the validity and eu000eciently of this method is verified by the MSC.ADAMS software, the energy conservation test and the DOF-CPU time test.