Analysis Of Time-domain Scattering Problems In Biperiodic Structures

The problem of periodic structure scattering(also known as diffraction grating)was first pro-posed by Rayleigh in 1907.Scattering theory in periodic structures have many important appli-cations in geophysics,micro-optics,earthquakes,which including photonic crystals,sensors,non-destructive testing.The time-domain scattering problems have received ever-increasing attention due to their capability of capturing wide-band signals and modeling more general material and non-linearity.In fact,dynamic impulsive data is usually easier to obtain and the information content of a temporal signal is usually much greater than that of the data available with a few discrete fre-quencies.However,compared with the time-harmonic scattering problems,mathematical studies are much less done for the time-domain counterparts due to the challenge of the temporal depen-dence.Due to our limited knowledge,we have found only a few mathematical works on medium scattering problems with bi-periodic structures for the time-dependent elastic wave equations.An-other challenge is how to truncate an unbounded domain into a bounded one when doing analysis or numerical calculation.It should be pointed out that when considering the finite time interval,we can't use Laplace transform.Moreover,TBC and PML are more complicated than in the case of time harmonics.This thesis is devoted to the time-domain analysis of the elastic and electromagnetic scat-tering problem in a biperiodic structure by using a different method.For the three-dimensional time-domain Navier's equations and Maxwell's equations in unbounded region,we propose a com-pressed coordinate transformation to reduce the problems equivalently into an initial-boundary value problem in a bounded domain and a finite time interval.The reduced problem can be for-mulated into a much smaller domain.This method is different from the Transparent Boundary Condition(TBC)or Perfect Matching Layer(PML)method in dealing with unbounded domain truncation,and has certain research application value.Then the Galerkin method is used to prove the uniqueness of the weak solutions of the two scattering problems,and the energy estimation method is used to establish the stability estimation.Furthermore,we obtain a priori estimates with explicit dependence on the time.