Hamiltonian System Approach Simulation For Seismic Wave

Hamilton system approach is one method which develops based on symplecticgeometric algorithm. It，s canonical transformation is actually one kind of symplectictransformation in which it guarantees that the system’s energy does not change. So, it is avery good algorithm from this point. Taking into account symplectic geometric algorithmof Hamilton system which Professor Feng Kang have been proposed, Cao Yu etc. haveproposed Hamilton system approach for describes sound wave propagation and the elasticwave propagation. Based on this, this article expands this method to the three dimensionalsituation, and proposes Hamilton system approach which describes three dimensionalseismic wave propagation.First, this paper introduces the development of Hamilton system method and itsscientific significance and application prospect, and also presents the the basic theory ofsymplectic algorithm of Hamilton system, which makes theoretical preparations for 3 dmedium expansion.Second, the paper puts forward three dimensional Hamilton system method of thesound waves which adopts the same interaction model with the elastic wave and presentsinteraction coefficient suitable for sound waves and the relationship of interactioncoefficient with wave velocity, the mass density and spacing between discrete lattice point.The paper also deduces the discrete equations of second order symplectic differenceschemes which describes sound wave propagation in 3-D medium, and carrys outnumerical simulation for the given medium. The results show that Hamiltonian systemscan effectively achieve the propagation of sound waves in the three-dimensional medium.Third, this paper proposes 3-D Hamilton system method of elastic wave, whichadopts the interaction model of the center quasi-particle with six most neighbors particlesand twelve nearer particles. It presents interaction coefficient suitable for elastic wave andthe relationship of interaction coefficient with p wave velocity、mass density and Poisson’sratio. And the paper also deduces the discrete equations of symplectic difference schemeswhich describes elastic wave propagation, and carrys out numerical simulation for the given medium model. The simulation results show that the Hamiltonian system method iseffective for the achievement of the elastic wave propagation in three-dimensional media.Finally, for sound waves propagation in the 2-D media, the Hamilton systemapproach is put forward based on five interaction models. The method adopts fiveinteraction models , deduces the discrete equations of second order symplectic differenceschemes which describes sound wave propagation in 2-D medium, and carrys outnumerical simulation for the given medium. At the same time, we compare them withHamilton system approach based on nine interaction models in order to get morereasonable model parameters. Numerical simulation results not only verify the feasibilityof Hamiltonian systems based on the five-point interaction model, but also find areasonable model parameters.