约束动力系统的分析结构力学积分

INTEGRATION OF CONSTRAINED DYNAMICAL SYSTEM VIA ANALYTICAL STRUCTURAL MECHANICS

约束保守系统导出了微分代数方程,其数值求解总是采用差分法.微分-代数方程的约束带来Lagrange参变函数转变而得的微分方程,有其指标问题,扩大了求解的规模.虽然已经注意差分的保辛,但沿切面积分再投影,仍带来许多问题.本文运用分析结构力学的方法,以节点处的独立位移为未知数且严格满足节点的约束条件,再将有限元近似用于区段作用量函数,在区段内部用简单插值求解.则按分析结构力学的理论,不但达到了积分的保辛且区段内部的约束条件也可在变分原理的意义下近似满足.数值结果满意.

The conserved constrained dynamical system derives to differential-algebraic equation. Solving the constrained differential-algebraic equation via introducing the Lagrange parametric differential equation to treat the constraint, there is the differential index problem, which enlarges the variables to be solved. The existing solution methodology is always via the FDM. The symplectic preservation of the finite difference scheme is considered, however, the projection operation onto the constraint manifold still brings problems. The present paper applies the methodology of analytical structural mechanics. The independent displacements at the integration points are treated as primary variables to be solved, and the constraint conditions are satisfied strictly at the integration points. The time-domain finite element linear interpolation function is applied to approximate the orbit to generate the action function of the time-interval. According to the theory of analytical structural mechanics, not only the symplectic preservation of the integration scheme is reached, but also the constraint conditions are satisfied in the sense of variational approach approximately. Numerical examples demonstrate satisfactory results.