Stable Algorithms With High-accuracy For Scattering And Dispersion Of Elastic Waves In Layered Media

The propagation problem of elastic waves in layered media is an important research issue,which has wide applications and development prospects in the fields of geophysics,engineering exploration,nondestructive detection and composite materials.Dispersion and scattering are two important characteristics for the research of elastic waves.This paper focuses the attention on these two aspects.Compared to isotropic media,the solution of wave propagation problem in anisotropic media is more difficult.These are reflected in two aspects:(1)for a generally anisotropic solid,there are 21 independent elastic constants and all of the displacement components are coupled;(2)the direction of energy flow does not coincide with the wave vector,so that the upward and downward waves in the medium must be separated by the direction of energy flow.The numerical methods for wave propagation in layered anisotropic media can be broadly classified into two groups.groups.One group is those methods based on continuous models,such as the transfer matrix method(TMM)and the stiffness matrix method(SMM),but they may cause numerical ill-conditioning problem under a certain condition,e.g.,the TMM for a large frequency-thickness product occurs the loss of precision;the SMM for a small frequency-thickness product causes a larger round-off error even instability.Additionally,the use of the continuous methods to solve dispersion relation is reduced to solve a transcendental eigenvalue problem related to the wavenumber and the frequency.Its solution is usually obtained by using an intensive search technique and a complicated iteration procedure.The loss of roots is a common phenomenon.The other group is those methods based on discrete models,such as the finite difference method(FDM)and the finite element method(FEM).This group of methods overcomes the difficulty of seeking the roots from the transcendental eigenequation by solving a quadratic eigenvalue problem.However,due to the need of the discretization,the accuracy of results depends on the size of element,and especially for high frequency cases,a fine mesh is usually adopted to achieve high accuracy,which gives rise to more degrees of freedom and thereby reduce the computational efficiency.The current methods are difficult to achieve three objectives of high accuracy,efficiency and stability simultaneously.Therefore,developing an algorithm for solving scattering and dispersion of waves in layered media,which has the high accuracy,efficiency,stability and universality,is of great significance.In a symplectic dual system framework,using the mixed energy matrix method(MEMM),the precise integration method(PIM),the Wittrick-Williams(W-W)algorithm and the symplectic method,this doctoral dissertation establishes methods with high accuracy,efficiency and stability for solving the dispersion and scattering of waves in layered media.The main research contents are summarized as follows:1.Based on the eigenvalue theory of the Hamiltonian matrix and the mixed energy matrix,an accurate and stable method,termed the mixed energy matrix method(MEMM),for the scattering of elastic waves in layered anisotropic structures is proposed.Using the symplectic structure of the Hamiltonian matrix,the relationship between the symplectic inner product and the Poynting vector are established;thus,the upward and downward waves for an arbitrary layer can be accurately separated.On this basis,an accurate and stable MEMM is proposed to accurately calculate the reflection and transmission coefficients of waves in an arbitrarily layered anisotropic media.Compared to the TMM and the SMM,a theoretical analysis shows that the TMM and the SMM are unstable for large and small frequency-thickness products,respectively,but the MEMM is stable for an arbitrary frequency-thickness product.The MEMM has the universality,and can be used to accurately solve the reflection and transmission coefficients of waves in layered anisotropic piezoelectric media.2.By combining the PIM with the W-W algorithm,a high-accuracy method is proposed for solving the dispersion of waves in layered media.Using the PIM and the W-W algorithm,three methods respectively based on the dynamic stiffness matrix,the symplectic transfer matrix and the mixed energy matrix are proposed for solving the dispersion relations of guided waves in layered finite media,surface waves in layered semi-infinite spaces,and Stoneley waves in layered infinite spaces.A theoretical analysis shows that,the three proposed methods are theoretically interconnected but the method based on the mixed energy matrix is more numerically stable than the other two methods.The performance of the PIM and the W-W algorithm based on the mixed energy matrix for solving wave propagation problems in layered media is demonstated by comparing with the semi-analytical finite element method.The proposed method has the following four advantages:(1)the use of the mixed energy matrix ensures that the method is stable for an arbitrary frequency-thickness product;(2)the application of the PIM ensures that the computation is highly accurate and efficient;(3)using the concept of the eigenvalue count in the W-W algorithm,the required eigenfrequencies for an arbitrarily specified range of frequencies can be computed and no eigenfrequency is missed;(4)it has the universality for solving wave propagation problems in layered media,and does not restrict layer number,layer thickness and properties of elastic materials of the structure.3.Based on the symplectic structure of the Hamiltonian matrix,by extensively applying the PIM and the W-W algorithm,a high-accuracy method for solving the dispersion of waves in layered piezoelectric-piezomagnetic(PM-PE)media is proposed.For PM-PE media,a submatrix of the Hamiltonian matrix is not positive definite due to the form of wave equation,so that the eigenvalue count of the sublayer is not zero when each layer is divided into sufficient number of sublayers,which makes that the W-W algorithm cannot be straightforward applied to solve wave propagation problems in layered PM-PE media.To overcome this difficulty,based on the symplectic structure of the Hamiltonian matrix,a symplectic transformation is introduced for the Hamiltonian matrix.A strictly theoretical analysis shows that after performing the symplectic transformation the eigenvalue count of the sublayer is zero when each layer is divided into sufficient number of sublayers,so that the W-W algorithm can be applied to PM-PE media.Then,by extensively applying the PIM and the W-W algorithm based on the MEMM,a high-accuracy method for solving the eigenfrequencies of waves in layered PM-PE media.4.Based on the symplectic geometry method and the W-W algorithm,an efficient method for solving the dispersion of surface waves in a semi-infinite periodically layered structure(PLS)with coating layers is proposed.Using the property of the symplectic transfer matrix,the eigenvalue problem for SH surface waves in a semi-infinite PLS is transformed into that of a unit cell,which can significantly improve the computational efficiency.By analyzing the relationship between the eigenvalues of a finite PLS within the stopbands and those of surface waves in the semi-infinite PLS,the eigenvalue problem of surface waves in an anisotropic semi-infinite PLS is transformed to solve few eigenvalues of a finite PLS within the stopbands.By taking advantages of the PIM and the W-W algorithm,a robust method for surface waves in an anisotropic semi-infinite PLS is developed.On this basic,by using the definition of the eigenvalue count in the W-W algorithm,an algorithm with high accuracy and efficient for solving the eigenfrequencies of surface waves in a semi-infinite PLS with coating layers.