Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics
Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics
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DOI：
10.1016/0045-7825(92)90115-z
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发表时间：
1992-10
期刊：
影响因子：
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通讯作者：
J. Simo;N. Tarnow;K. Wong
中科院分区：
文献类型：
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作者：
J. Simo;N. Tarnow;K. Wong
结果表明，对于包括经典刚体动力学、非线性弹性动力学、非线性杆和非线性壳在内的非线性哈密顿系统，广泛使用的隐式格式，特别是经典的Newmark算法及其变种，一般不能守恒总角动量。对于线性哈密顿系统，众所周知，只有Crank-Nicholson格式精确地保持了系统的总能量。这种守恒性通常在非线性系统中消失。提出了一类隐式时间推进算法，它精确地保持了一般对称哈密顿系统中的守恒定律，特别是总角动量和总能量。值得注意的是，这类算法的实际实现可以通过一个简单的两步求解方案有效地完成，与标准隐式方法相比，这种方案基本上不会增加计算成本。对这些算法和相关的一类称为辛积分器的格式进行了完整的分析。通过构成非线性弹性动力学、非线性杆和非线性壳的三个典型模型问题的数值算例，验证了该方法的良好性能。
It is shown that widely used implicit schemes, in particular the classical Newmark family of algorithms and its variants, generally fail to conserve total angular momentum for nonlinear Hamiltonian systems including classical rigid body dynamics, nonlinear elastodynamics, nonlinear rods and nonlinear shells. For linear Hamiltonian systems, it is well known that only the Crank-Nicholson scheme exactly preserves the total energy of the system. This conservation property is typically lost in the nonlinear regime. A general class of implicit time-stepping algorithms is presented which preserves exactly the conservation laws present in a general Hamiltonian system with symmetry, in particular the total angular momentum and the total energy. Remarkably, the actual implementation of this class of algorithms can be effectively accomplished by means of a simple two-step solution scheme which results in essentially no added computational cost relative to standard implicit methods. A complete analysis of these algorithms and a related class of schemes referred to as symplectic integrators is given. The good performance of the proposed methodology is demonstrated by means of three numerical examples which constitute representative model problems of nonlinear elastodynamics, nonlinear rods and nonlinear shells.