不可压饱和多孔弹性杆动力响应的多辛方法

研究了不可压饱和多孔弹性杆的一维动力响应问题.基于多孔介质理论，在流相和固相微观不可压、固相骨架小变形的假定下，建立了不可压流体饱和多孔弹性杆一维轴向动力响应的数学模型.利用Hamilton空间体系的多辛理论，构造了不可压饱和多孔弹性杆轴向振动方程的多辛形式及其多种局部守恒律.采用中点Box离散方法得到轴向振动方程的多辛离散格式和局部能量守恒律以及局部动量守恒律的离散格式；数值模拟了不可压饱和多孔弹性杆的轴向振动过程，记录了每一时间步的局部能量数值误差和局部动量数值误差.结果表明，已构造的多辛离散格式具有很高的精确性和较长时间的数值稳定性，这为解决饱和多孔介质的动力响应问题提供了新的途径.

A Multi-Symplectic Method for Dynamic Responses of Incompressible Saturated Poroelastic Rods

Dynamic responses of incompressible saturated poroelastic rods were investigated. Based on the theory of porous media, the 1D axial vibration equation for a fluid saturated elastic porous rod was established, in which the saturated porous material was modeled as a 2-phase system composed of an incompressible solid phase and an incompressible fluid phase. Then a 1st-order multi-symplectic form for the axial vibration equation and several local conservation laws for the saturated poroelastic rod were derived with the multi-symplectic method. Moreover, the midpoint Box multi-symplectic scheme for the axial vibration equation, and the discrete schemes for the local energy conservation law and local momentum conservation law were constructed with the midpoint method. Finally, the axial vibration process of the incompressible saturated poroelastic rod was simulated numerically and numerical errors of the local energy conservation law and local momentum conservation law were also discussed by means of the numerical results of each time step and each time-space step, respectively. The results show that the proposed multi-symplectic scheme has advantages of high accuracy, long-time numerical stability and good conservation properties, and this method provides a new way to solve the dynamic responses of saturated porous media.