Frequency-domain Spectral Element Method And Domain Decomposition For Numerical Calculations Of Elastic And Acoustic Waves

The forward modeling of elastic wave and acoustic wave is an important foundation of seismic exploration and seismology.It plays an important role in oil and gas exploration,natural disaster prediction and other fields.In general,forward methods can be divided into two methods:time domain and frequency domain.In the simulation of seismic wave and acoustic wave,the time domain method has been widely studied by many scholars at home and abroad,and applied to practical problems.Unfortunately,the current research results of frequency domain algorithms are rare.In some applications,however,frequency-domain techniques can be advantageous over the time-domain approach when narrow band results are desired,especially if multiple sources can be handled more conveniently in the frequency domain.Moreover,the medium attenuation effects can be more accurately and conveniently modeled in the frequency domain.In this study,we present a spectral-element method(SEM)in frequency domain to simulate elastic and acoustic waves in anisotropic,heterogeneous,and lossy media.The SEM is based upon the finite-element framework and has exponential convergence because of the use of GLL basis functions.The SEM error decreases exponentially with the increase of the interpolation order,thus has high-precision.The anisotropic perfectly matched layer(PML)is employed to truncate the boundary for unbounded problems.Numerical examples show that PML we developed has good efficiency and performance.Compared with the conventional finite-element method,the number of unknowns in the SEM is significantly reduced,and higher order accuracy is obtained due to its spectral accuracy.Therefore,the spectral element method has great advantages compared with the conventional finite element method.Especially,when dealing with some anisotropic media problems,the traditional finite element method converges very slowly,and can even lead to erroneous results.On the other hand,the proposed SEM can still maintain the characteristics of fast convergence and high accuracy.To account for the acoustic-solid interaction,the domain decomposition method(DDM)based upon the discontinuous Galerkin spectral-element method is proposed.Numerical experiments show the proposed method can be an efficient alternative for accurate calculation of elastic and acoustic waves in frequency domain.Secondly,the spectral element method is developed for the first time for the determination of band structures of 3D isotropic/anisotropic phononic crystals(PCs).Generally,there are several methods,mostly numerical ones,for the computation of band structures in PCs,which include the plane-wave expansion method(PWE),the multiple-scattering theory method(MST),the finite difference time domain method(FDTD),and the finite element method(FEM).However,these methods have low precision,or the application scope is limited.Based on the Bloch theorem,we present a novel,intuitive discretization formulation for Navier equation in the SEM scheme for periodic media.By virtue of using the orthogonal Legendre polynomials,the generalized eigenvalue problem is converted to a regular one in our SEM implementation to improve the efficiency.Besides,according to the specific geometry structure,8-node and 27-node hexahedral elements as well as an analytic mesh have been used to accurately capture curved PC models in our SEM scheme.To verify its accuracy and efficiency,this study analyses the phononic-crystal plates with square and triangular lattice arrangements,and the 3D cubic phononic crystals consisting of simple cubic(SC),bulk central cubic(BCC)and faced central cubic(FCC)lattices with isotropic or anisotropic scatters.All the numerical results considered demonstrate that SEM is superior to the conventional FEM and can be an efficient alternative method for accurate determination of band structures of 3D phononic crystals.The third work of this thesis is that we propose a new domain decomposition algorithm based on the spectral element method.Spectral element method has obvious advantages over other ones in frequency domain.However,the direct inversion of the system for large scale problem often involves excessive demand on computational resources.In this study,we propose a new domain decomposition technique based on spectral element method(DDM-SEM)for elastic wave calculations in frequency domain.The proposed DDM-SEM combines the high accuracy of a spectral element method and the high degree of parallelism of a domain decomposition technique,which makes this method suitable for the accurate and efficient simulations of large scale problems in elastodynamics.Basically,the original problem is divided into a number of well designed subdomains.We use the spectral element method independently for each subsystem,and the neighboring subsystems are connected by a frequency-domain version of Riemann solver for elastic waves.For the proposed method,we can use the non-conforming meshes and employ different interpolation orders in different subdomains to to maximize the efficiency.By separating the internal and boundary unknowns of each subdomain,an efficient and naturally parallelizable block LDU direct solver is developed to solve the final system matrix.Numerical experiments verify the correctness,accuracy and efficiency and show that the proposed DDM-SEM can be a promising numerical tool for accurately and effectively solving for large scale problems of elastic waves.