LIU Xue-mei, DENG Zi-chen, HU Wei-peng. A Multi-Symplectic Method for Dynamic Responses of Incompressible Saturated Poroelastic Rods[J]. Applied Mathematics and Mechanics, 2015, 36(3): 242-251. doi: 10.3879/j.issn.1000-0887.2015.03.002
 Citation: LIU Xue-mei, DENG Zi-chen, HU Wei-peng. A Multi-Symplectic Method for Dynamic Responses of Incompressible Saturated Poroelastic Rods[J]. Applied Mathematics and Mechanics, 2015, 36(3): 242-251.

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

##### doi: 10.3879/j.issn.1000-0887.2015.03.002
Funds:  The National Natural Science Foundation of China(11372252；11172239；11372253)
• Rev Recd Date: 2014-12-26
• Publish Date: 2015-03-15
• 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.
•  [1] Theodorakopoulos D D, Niskos D E. Flexural vibrations of poroelastic plate[J].Acta Mech,1994,103(1/4): 191-203. [2] Anke B, Martin S, Heinz A. A poroelastic Mindlin-plate[J].Proc Appl Math Mech,2003,3(1): 260-261. [3] 杨骁, 李丽. 不可压饱和多孔弹性梁、杆动力响应的数学模型[J]. 固体力学学报, 2006,27(2): 159-166.(YANG Xiao, LI Li. Mathematical model for dynamics of incompressible saturated poroelastic beam and rod[J].Acta Mechanica Solida Sinica,2006,27(2): 159-166.(in Chinese)) [4] FENG Kang. On difference schemes and symplectic geometry[C]// Proceeding of the 1984 Beijing Symposium on D D.Beijing: Science Press, 1984: 42-58. [5] Marsden J E, Patriek G P, Shkoller S. Multisymplectic geometry, variational integrators, and nonlinear PDEs[J].Comm Math Phys,1998,199(2): 351-395. [6] Marsden J E, Patriek G P, Shkoller S. Variational methods, multisymplectic geometry and mechanics[J].J Geom Phys,2001,38(2): 253-284. [7] Bridges T J. Multi-symplectic structures and wave propagation[J].Math Proc Cambridge Philos Soc, 1997,121(1): 147-190. [8] Bridges T J, Reich S. Multi-symplectic integrator: numerical schemes for Hamiltonian PDE that conserve symplecticity[J].Physics Letters A,2001,284(4/5): 184-193. [9] Reich S. Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations[J]. J Comput Phys,1999,157(2): 473-499. [10] 胡伟鹏, 邓子辰, 李文成. 膜自由振动的多辛方法[J]. 应用数学和力学, 2008,28(9): 1054-1062.(HU Wei-peng, DENG Zi-chen, LI Wen-cheng. Multi-symplectic methods for membrane free vibration equation[J].Applied Mathematics and Mechanics,2008,28(9): 1054-1062.(in Chinese)) [11] 杨骁, 李丽. 轴向扩散下简支饱和多孔弹性梁的大挠度分析[J]. 固体力学学报, 2007,28(3): 313-317.(YANG Xiao, LI Li. Larger deflection analysis of simply supported saturated poroelastic beam[J].Acta Mechanica Solida Sinica,2007,28(3): 313-317.(in Chinese)) [12] 周凤玺, 米海珍. 弹性地基上不可压含液饱和多孔弹性梁的自由振动[J]. 兰州理工大学学报, 2014,40(2): 118-122.(ZHOU Feng-xi, MI Hai-zhen. Free vibration of poroelastic beam with incompressible saturated liquid on elastic foundation[J].Journal of Lanzhou University of Technology,2014,40(2): 118-122.(in Chinese)) [13] 欧阳煜, 张雅男. 集中荷载作用下饱和多孔Timoshenko简支梁的动力学响应[J]. 工程力学, 2012,29(11): 325-331.（OUYANG Yu, ZHANG Ya-nan. Dynamical behavior of simply-supported saturated poroelastic Timoshenko beam under a concentrated load[J].Engineering Mechanics,2012,29(11): 325-331.(in Chinese)） [14] YANG Xiao. Gurtin-type variational principles for dynamics of a non-local thermal equilibrium saturated porous medium[J].Acta Mech Solida Sin,2005, 18(1): 37-45. [15] 杨骁, 程昌钧. 流体饱和多孔介质的动力学Gurtin型变分原理和有限元模拟[J]. 固体力学学报, 2003,24(3): 267-276.(YANG Xiao, CHENG Chang-jun. Gurtin variational principle and finite element simulation for dynamical problems of fluid-saturated porous media[J].Acta Mechanica Solida Sinica,2003,24(3): 267-276.(in Chinese))

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