线性阻尼声学系统仿真的结构保持算法
Structure preserving algorithms for simulation of linearly damped acoustic systems
摘要：
构造时间步进算法的能量方法在非线性问题的应用中越来越受到关注，因为数值稳定性可以从系统能量守恒中推断出来。或者，可以构造保持系统的辛形式的辛积分器。这种方法已经建立了哈密顿系统，在工程问题中有许多应用。本文将这种方法推广到非保守声学系统。离散守恒律，相当于能量守恒计划，推导出线性阻尼系统，包括外力的作用。进一步分析了连续和离散情形下辛结构的演化。现有的方法进行检查和新的方法设计使用集总振荡器作为一个元素模型。所提出的方法扩展到分布式系统的情况下，并举例说明通过一个案例研究的振动字符串反弹对刚性障碍物。
英文摘要：
Energy methods for constructing time-stepping algorithms are of increased interest in application to nonlinear problems, since numerical stability can be inferred from the conservation of the system energy. Alternatively, symplectic integrators may be constructed that preserve the symplectic form of the system. This methodology has been established for Hamiltonian systems, with numerous applications in engineering problems. In this paper an extension of such methods to non-conservative acoustic systems is presented. Discrete conservation laws, equivalent to that of energy-conserving schemes, are derived for systems with linear damping, incorporating the action of external forces. Furthermore the evolution of the symplectic structure is analysed in the continuous and the discrete case. Existing methods are examined and novel methods are designed using a lumped oscillator as an elemental model. The proposed methodology is extended to the case of distributed systems and exemplified through a case study of a vibrating string bouncing against a rigid obstacle.
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