Research On Numerical Solutions Of Some Time-domain Elastic Waves

The time-domain elastic wave is widely used in scientific and engineering fields,such as medical imaging,petroleum rocks,seismology.In this thesis,we consider the time-domain acoustic-elastic wave interaction problem and wave propagation in poroelastic media.The time-domain acoustic-elastic wave interaction problem is also called the time-domain fluid-solid interaction(FSI)problem and is mathematically formulated as a time-dependent transmission problem.In a time-domain FSI problem,an incident acoustic wave is scattered by a bounded elastic obstacle immersed in a homogeneous,compressible and inviscid fluid.Wave propagation in poroelastic medium can be mathematically considered as Stokes equations for the pore fluid and linear elastodynamics for the solid with no-slip conditions on the fluid-solid interface.This dissertation consists of three parts: the first part is an interior penalty discontinuous Galerkin(DG)method for the time-domain FSI problem,the second part is a coupling boundary integral equation-discontinuous Galerkin method for the time-domain FSI problem,the third part is a DG method for wave propagation in orthotropic poroelastic media.The main challenges of studying the time-domain acoustic-elastic wave interaction problem are dealing with the unboundedness and time dependance.In this thesis,we focus on numerical methods for the time-domain acoustic-elastic wave interaction problem,we use artificial boundary and boundary integral equation methods to transform the original transmission problem into a problem in a bounded region.In the second chapter,we first define the artificial boundary condition by using a new artificial boundary condition,and then use the DG method to analyse and solve the reduced problem.In the third chapter,we use boundary integral equation to solve the acoustic wave equation in the unbounded domian,and then employ the coupled BIE-DG method to solve the reduced problem.We give some analysis including stability analysis,convergence analysis and error estimate of the numerical solution.There are two frequency regimes for wave propagation in poroelastic medium.In low frequency,the fluid within the pores is assumed to be of Poiseuille type,and the viscous effect is a linear function of the relative velocity,the stiffness of the Biot equation remains the major challenge in the simulation procedure.In high frequency,the viscous effect is related to the square root of the frequency which results in a memory term in the time domain.Modeling task in high frequency will face more challenges with discretizing the memory term and storing the solution history.In low frequency,we use the direct method(unsplitting)to handle the viscous term with time step small than the characteristic decay time and to obtain higher order results.In high frequency,we use the two sided residue interpolation method to approximate the tortuosity based on the properties that the tortuosity is a Stieltjes function in frequency domain,and then obtain the augmented Biot-JKD model in the time domain by using the inverse Laplace transform.Also,we prove the property of energy decay of the augmented Biot-JKD model.By introducing the auxiliary variables,the augmented Biot-JKD model simplified the original problem and reduced the computational cost.Finally,numerical examples are presented to show the accuracy of the model and proposed method.