基于广义Lippmann-Schwinger方程的地震波场模拟算法与应用

In the last several decades, due to the fast development of computer, numerical simulation has been an indispensable tool in scientific research. Numerical simulation methods which based on partial difference operators such as Finite Difference Method (FDM) and Finite Element Method (FEM) have been widely used. However, in the realm of seismology and seismic prospecting, one usually meets with geological models which have piece-wise heterogeneous structures as well as volume heterogeneities between layers, the continuity of displacement and stress across the irregular layers and seismic wave scattering induced by the perturbation of the volume usually bring in error when using conventional methods based on difference operators. The method discussed in this paper is based on elastic theory and integral theory. Seismic wave equation in the frequency domain is transformed into a generalized Lippmann-Schwinger equation, in which the seismic wavefield contributed by the background is expressed by the boundary integral equation and the scattering by the volume heterogeneities is considered. Boundary element-volume integral method based on this equation has advantages of Boundary Element Method (BEM), such as reducing one dimension of the model, explicit use the displacement and stress continuity across irregular interfaces, high precision, satisfying the boundary at infinite, etc. Also, this method could accurately simulate the seismic scattering by the volume heterogeneities. In this paper, the concrete Lippmann-Schwinger equation is specifically given according to the real geological models. Also, the complete coefficients of the non-smooth point for the integral equation are introduced. Because Boundary Element-Volume integral equation method uses fundamental solutions which are singular when the source point and the field are very close，both in the two dimensional and the three dimensional case, the treatment of the singular kernel affects the precision of this method. The method based on integral transform and integration by parts could treat the points on the boundary and inside the domain. It could transform the singular integral into an analytical one both in two dimensional and in three dimensional cases and thus it could eliminate the singularity. In order to analyze the elastic seismic wave scattering due to regional irregular topographies, the analytical solution for problems of this type is discussed and the analytical solution of P waves by multiple canyons is given. For the boundary reflection, the method used here is infinite boundary element absorbing boundary developed by a pervious researcher. The comparison between the analytical solutions and concrete numerical examples validate the efficiency of this method. We thoroughly discussed the sampling frequency in elastic wave simulation and find that, for a general case, three elements per wavelength is sufficient, however, when the problem is too complex, more elements per wavelength are necessary. Also, the seismic response in the frequency domain of the canyons with different types of random heterogeneities is illustrated. We analyzed the model of the random media, the horizontal and vertical correlation length, the standard deviation, and the dimensionless frequency how to affect the seismic wave amplification on the ground, and thus provide a basis for the choice of the parameter of random media during numerical simulation.

中能核反应中的统计发射过程及核温度参量

在过去的几年里，利用兰州重离子加速器（HIRFL）提供的束流，以及在OUVERTURE合作研究中，利用意大利国家核物理研究院南方实验室（INFN-LNS）超导回旋提供的束流，进行了多次中能区重离子核反应实验研究工作。如，最初的46.7MeV/u ~(12)C+~(58)Ni，~(115)In，~(197)Au的实验及30MeV/u ~(40)Ar+~(58)Ni，~(64)Ni~(115)In和30MeV/u Ni轰击Ni，Au，Al在MULTICS+MEDEA：4π装置上进行的实验工作。此外，本人还从事过一些理论研究工作，包括多粒子散射形式理论和相关数学物理问题研究，量子分子动力学和量子统计模型计算。本文是从事这些核物理研究工作的积累，主要侧重于实验结果的物理内容分析，而不强调实验技术，数据处理的技巧。主要的物理内容有以下几个方面：1.对于利用双同位素产额比提取同位素核温度的方法进行研究推广，使得对于实验中碰到的仅有部分能谱可以实现很好同位素分辩的情况，即使不能得到总的同位素产额，仅仅通过一段能区的同位素产额也可提取核温度。用于具体的实验研究工作中后，对于46.7MeV/u ~(12)C+~(58)Ni，~(115)In，~(197)Au核反应过程，同一体系利用这种方法得到的同位素核温度和利用粒子非稳态布居提取的核温度一致。2.围绕核反应过程中核温度的参量的提取，对于双同位素产额比与核温度的刻度关系进行了分析研究，通过计算考虑中等质量碎片（IMF）内部激发能后的内部配分函数表明，中等质量碎片的内部激发对刻度关系有重要影响。零阶近似下区域密度近似的结果和Gemini模拟计算的结果反映了相同的情况。3.研究核反应机制，多个粒子散射的形式理论的必需的，对于两体散射，其形式理论已经比较成熟，但是对于多个粒子散射问题出现的严重的困难是多体Lippmann-Schwinger方程无唯一收敛的解。作为一种探索性的研究工作，开展了多体散射理论研究工作，发展了一些具有普遍意义的数学物理方法。在本项研究工作中，通过能基础数学中的约当引理的推广，发现一个特例：对非连接图，Lippmann-Schwingwer方程存在收敛的解，因此多体散射形式理论，有可能重新建立。由于核力和多体问题是当今核物理研究的两大难点，世界各国的科学家都在努力以图攻克它们，而且多体问题还是物理学的其它许多领域的难题，因而多体散射还是引起诸多研究学科广泛兴趣的课题。通过发展一些新的数学理论和方法，我们已得到一些有意义的结果。4.将量子分子动力学这种中高能量区域所用的理论分析方法扩展至较低能区，通过对相空间中初始位置和动量抽样增加限制条件。如结合能和实验值要求一致，平均核势，核内Pauli阻塞更强一些，在演化中能量和动量守恒等等。得到一个很稳定的初始基态。均方半径保持不弥散的时间可达1600fm/c，用于研究10.6MeV/u Ne~(20)+Al~(27)的实验分析过程中。另外，量子统计模型（QSM）主要描述中心核-核碰撞，将它和碎裂模型结合，作一些改进后，可以对核-核碰撞进行统一描述。5.在中能核反应研究中发现，核反应过程中有大量的中子，轻带电粒子以及中等质量碎片发射出来，可以将这些粒子发射机制大致分为两大类。其中一类可以归结为动力学发射过程的产物。另一类则可以归于统计发射的产物。在低能核反应中，其发射能谱的斜率的负倒数，可作为复合核的核温度。而在中能重离子核反应中，其发射能谱变得很复杂，不再具有Maxwell分布。通常的三源拟合所给出的温度参数，已不能反映物理实质。提出多阶矩分析方法用于分析中能核反应中统计发射规律及受动力学过程的影响。

Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier-Stokes equations is presented. The scheme is constructed on a staggered-mesh grid system. The convection terms are discretized with a fifth-order-accurate upwind compact difference approximation, the viscous terms are discretized with a sixth-order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth-order difference approximation on a cell-centered mesh. Time advancement uses a three-stage Runge-Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented.

Characteristics and Sub-Characteristics of Two-Dimensional Parabolized Stability Equations

特征分析表明：对原始扰动量的抛物化稳定性方程组(PSE),它在亚超音速区分别具有椭圆和抛物特性,给出PSE特征对马赫数的依赖关系,阐明PSE仅把信息对流-扩散传播特性抛物化,而保留了信息对流-扰动传播特性,因此PSE应称为扩散抛物化稳定性方程(DPSE)。

Perturbational Finite Volume Method For The Solution of 2-D Navier-Stokes Equations on Unstructured and Structured Colocated Meshes

Based on the first-order upwind and second-order central type of finite volume( UFV and CFV) scheme, upwind and central type of perturbation finite volume ( UPFV and CPFV) schemes of the Navier-Stokes equations were developed. In PFV method, the mass fluxes of across the cell faces of the control volume (CV) were expanded into power series of the grid spacing and the coefficients of the power series were determined by means of the conservation equation itself. The UPFV and CPFV scheme respectively uses the same nodes and expressions as those of the normal first-order upwind and second-order central scheme, which is apt to programming. The results of numerical experiments about the flow in a lid-driven cavity and the problem of transport of a scalar quantity in a known velocity field show that compared to the first-order UFV and second-order CFV schemes, upwind PFV scheme is higher accuracy and resolution, especially better robustness. The numerical computation to flow in a lid-driven cavity shows that the under-relaxation factor can be arbitrarily selected ranging from 0.3 to 0. 8 and convergence perform excellent with Reynolds number variation from 102 to 104.

A Lagrangian lattice Boltzmann method for Euler equations

A Lagrangian lattice Boltzmann method for solving Euler equations is proposed. The key step in formulating this method is the introduction of the displacement distribution function. The equilibrium distribution function consists of macroscopic Lagrangian variables at time steps n and n + 1. It is different from the standard lattice Boltzmann method. In this method the element, instead of each particle, is required to satisfy the basic law. The element is considered as one large particle, which results in simpler version than the corresponding Eulerian one, because the advection term disappears here. Our numerical examples successfully reproduce the classical results.

The application of B-P constitutive equations in finite element analysis of high velocity impact

We present in this paper the application of B-P constitutive equations in finite element analysis of high velocity impact. The impact process carries out in so quick time that the heat-conducting can be neglected and meanwhile, the functions of temperature in equations need to be replaced by functions of plastic work. The material constants in the revised equations can be determined by comparison of the one-dimensional calculations with the experiments of Hopkinson bar. It can be seen from the comparison of the calculation with the experiment of a tungsten alloy projectile impacting a three-layer plate that the B-P constitutive equations in that the functions of temperature were replaced by the functions of plastic work can be used to analysis of high velocity impact.

Governing Equations and Numerical Solutions of Tension Leg Platform with Finite Amplitude Motion

It is demonstrated that when tension leg platform (TLP) moves with finite amplitude in waves, the inertia force, the drag force and the buoyancy acting on the platform are nonlinear functions of the response of TLP. The tensions of the tethers are also nonlinear functions of the displacement of TLP. Then the displacement, the velocity and the acceleration of TLP should be taken into account when loads are calculated. In addition, equations of motions should be set up on the instantaneous position. A theoretical model for analyzing the nonlinear behavior of a TLP with finite displacement is developed, in which multifold nonlinearities are taken into account, i.e., finite displacement, coupling of the six degrees of freedom, instantaneous position, instantaneous wet surface, free surface effects and viscous drag force. Based on the theoretical model, the comprehensive nonlinear differential equations are deduced. Then the nonlinear dynamic analysis of ISSC TLP in regular waves is performed in the time domain. The degenerative linear solution of the proposed nonlinear model is verified with existing published one. Furthermore, numerical results are presented, which illustrate that nonlinearities exert a significant influence on the dynamic responses of the TLP.

A further discussion on basic equations for two-phase flow: On collision stress of solid phase

The following points are argued: (i) there are two independent kinds of interaction on interfaces, i.e. the interaction between phases and the collision interaction, and the jump relations on interfaces can accordingly be resolved; (ii) the stress in a particle can also be divided into background stress and collision stress corresponding to the two kinds of interaction on interfaces respectively; (iii) the collision stress, in fact, has no jump on interface, so the averaged value of its derivative is equal to the derivative of its averaged value; (iv) the stress of solid phase in the basic equations for two\|phase flow should include the collision stress, while the stress in the expression of the inter\|phase force contains the background one only. Based on the arguments, the strict method for deriving the equations for two\|phase flow developed by Drew, Ishii et al. is generalized to the dense two\|phase flow, which involves the effect of collision stress.

A Lagrangian for von Karman Equations of Large Deflection Problem of Thin Circular Plate

By the semi-inverse method proposed by He, a Lagrangian is established for the large deflection problem of thin circular plate. Ritz method is used to obtain an approximate analytical solution of the problem. First order approximate solution is obtained, which is similar to those in open literature. By Mathematica a more accurate solution can be deduced.

A Cip/Multi-Moment Finite Volume Method For Shallow Water Equations With Source Terms

A novel finite volume method has been presented to solve the shallow water equations. In addition to the volume-integrated average (VIA) for each mesh cell, the surface-integrated average (SIA) is also treated as the model variable and is independently predicted. The numerical reconstruction is conducted based on both the VIA and the SIA. Different approaches are used to update VIA and SIA separately. The SIA is updated by a semi-Lagrangian scheme in terms of the Riemann invariants of the shallow water equations, while the VIA is computed by a flux-based finite volume formulation and is thus exactly conserved. Numerical oscillation can be effectively avoided through the use of a non-oscillatory interpolation function. The numerical formulations for both SIA and VIA moments maintain exactly the balance between the fluxes and the source terms. 1D and 2D numerical formulations are validated with numerical experiments. Copyright (c) 2007 John Wiley & Sons, Ltd.

Analysis and application of ellipticity of stability equations on fluid mechanics

By using characteristic analysis of the linear and nonlinear parabolic stability equations (PSE), PSE of primitive disturbance variables are proved to be parabolic intotal. By using sub-characteristic analysis of PSE, the linear PSE are proved to be elliptical and hyperbolic-parabolic for velocity U, in subsonic and supersonic, respectively; the nonlinear PSE are proved to be elliptical and hyperbolic-parabolic for relocity U + u in subsonic and supersonic, respectively. The methods are gained that the remained ellipticity is removed from the PSE by characteristic and sub-characteristic theories, the results for the linear PSE are consistent with the known results, and the influence of the Mach number is also given out. At the same time, the methods of removing the remained ellipticity are further obtained from the nonlinear PSE.

Dynamic equations for curved submerged floating tunnel

In virtue of reference Cartesian coordinates, geometrical relations of spatial curved structure are presented in orthogonal curvilinear coordinates. Dynamic equations for helical girder are derived by Hamilton principle. These equations indicate that four generalized displacements are coupled with each other. When spatial structure degenerates into planar curvilinear structure, two generalized displacements in two perpendicular planes are coupled with each other. Dynamic equations for arbitrary curvilinear structure may be obtained by the method used in this paper.

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.

Compact finite difference - Fourier spectral method for three-dimensional incompressible Navier-Stokes equations

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.