Ph.D Thesis

Ph.D StudentBaffet Daniel
SubjectAbsorbing Boundary Conditions for Elastic Waves
DepartmentDepartment of Applied Mathematics
Supervisor PROF. Dan Givoli


High-order local Absorbing Boundary Conditions (ABCs) are developed for elastodynamics and acoustics. Two high-order ABCs for elastodynamics are implemented by numerical schemes and tested. These seem to be the first high-order local ABCs for elastodynamics which are long-time stable. ABCs are a tool for numerical approximation of wave problems that are posed in unbounded domains. Such problems are common in fields of research such as solid-earth geophysics, oceanography and underwater acoustics. The unbounded domain is truncated by the introduction of an artificial boundary on which an ABC is enforced. The purpose of the ABC is to reduce spurious reflections of outgoing waves impinging on the artificial boundary.

Three ABCs are proposed for elastodynamics. The first ABC is a generalization of the Bamberger et al ABC for the displacement formulation. The stability of this ABC is analyzed theoretically. The second ABC is an extension of Hagstrom and Warburton's Complete Radiation Boundary Condition (CRBC) to the velocity-stress formulation of elastodynamics. For this ABC a uniform in time estimate is produced as proof of its stability. The ABC is also implemented by a finite difference (FD) scheme which is then tested for stability and accuracy. The third ABC is an extension of the recently developed Double Absorbing Boundary (DAB) method to elastodynamics. This ABC is also implemented by a FD scheme and tested for stability and accuracy. Numerical tests show that the scheme is stable and that it is capable of good accuracy, although it shows sensitivity to the choice of the ABC computational parameters. Our results indicate that it is possible to overcome this sensitivity by using a specific choice of parameters proposed by HW for the CRBC for acoustics. Our tests show that when using this choice of parameters, the error decreases consistently (and exponentially fast) as the order increases.

In the context of acoustics, the main results of this work concern the stability of ABCs. The first result relates the P-order Higdon ABC for acoustics to rational approximation of the symbol of the Dirichlet-Neumann operator. We then use a result by Ha-Duong and Joly regarding the problem in two dimensions to deduce a uniform in time bound on all the P-order derivatives of the solution. The second result concerns the DAB method stability. We prove uniform in time estimates on the DAB solution and auxiliary functions, whenever they are unique.