不可压饱和多孔弹性杆动力响应的多辛方法

刘雪梅; 邓子辰; 胡伟鹏

doi:10.3879/j.issn.1000-0887.2015.03.002

不可压饱和多孔弹性杆动力响应的多辛方法

doi: 10.3879/j.issn.1000-0887.2015.03.002

¹西北工业大学力学与土木建筑学院,西安 710072;²长安大学工程力学系,西安 710064

基金项目: 国家自然科学基金(11372252；11172239；11372253)；中央高校基金（2014G1121096）

详细信息

作者简介:
刘雪梅（1980—），女，陕西榆林人，讲师，硕士（通讯作者. E-mail: liumei112@163.com）.

中图分类号: O343
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出版历程
- 收稿日期: 2014-12-02
- 修回日期: 2014-12-26
- 刊出日期: 2015-03-15

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

¹Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, P.R.China;²Department of Engineering Mechanics, Chang’an University, Xi’an 710064, P.R.China

Funds: The National Natural Science Foundation of China(11372252；11172239；11372253)

摘要

摘要: 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.
- 饱和多孔弹性杆 /
- 多辛方法 /
- 动力响应 /
- 离散
Abstract: 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.
- saturated poroelastic rod /
- multi-symplectic method /
- dynamic response /
- discrete

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参考文献(15)

[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))