Initial design with L2 Monge-Kantorovich theory for the Monge–Ampère equation method of freeform optics

The Monge–Ampère (MA) equation arising in illumination design is highly nonlinear so that the convergence of the MA method is strongly determined by the initial design. We address the initial design of the MA method in this paper with the L2 Monge-Kantorovich (LMK) theory, and introduce an efficient approach for finding the optimal mapping of the LMK problem. Three examples, including the beam shaping of collimated beam and point light source, are given to illustrate the potential benefits of the LMK theory in the initial design. The results show the MA method converges more stably and faster with the application of the LMK theory in the initial design.

The Monge-Ampére equation method in freeform optics design

The Monge-Ampére equation method could be the most advanced point source algorithm of freeform optics design. This paper introduces this method, and outlines two key issues that should be tackles to improve this method.

The finite range Wiener-Hopf integral equation and a boundary value problem in a waveguide /

"Contract No. AF33(616)-6079 Project No. 9-(13-6278) Task 40572. Sponsored by: Wright Air Development Center"

Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle

The problem considered is that of determining the shape of a planar acoustically sound-soft obstacle from knowledge of the far-field pattern for one time-harmonic incident field. Two methods, which are based on the solution of a pair of integral equations representing the incoming wave and the far-field pattern, respectively, are proposed and investigated for finding the unknown boundary. Numerical resultsare included which show that the methods give accurate numerical approximations in relatively few iterations.

An Abstract Second Kind Fredholm Integral Equation with Degenerated Kernel

Mathematics Subject Classification: 44A40, 45B05

Existence and Asymptotic Stability of Solutions of a Perturbed Quadratic Fractional Integral Equation

Mathematics Subject Classification: 45G10, 45M99, 47H09

Diffraction Of Elastic Waves By Two Parallel Rigid Strips Embedded In An Infinite Orthotropic Medium

The elastodynamic response of a pair of parallel rigid strips embedded in an infinite orthotropic medium due to elastic waves incident normally on the strips has been investigated. The mixed boundary value problem has been solved by the Integral Equation method. The normal stress and the vertical displacement have been derived in closed form. Numerical values of stress intensity factors at inner and outer edges of the strips and vertical displacement at points in the plane of the strips for several orthotropic materials have been calculated and plotted graphically to show the effect of material orthotropy.

On the radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas

The radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas of finite depth is investigated. The analytical expressions for the radiated and diffracted potentials are derived as infinite series by use of the method of separation of variables. The unknown coefficients in the series are determined by the eigenfunction expansion matching method. The expressions for wave forces, hydrodynamic coefficients and reflection and transmission coefficients are given and verified by the boundary element method. Using the present analytical solution, the hydrodynamic influences of the angle of incidence, the submergence, the width and the thickness of the structure on the wave forces, hydrodynamic coefficients, and reflection and transmission coefficients are discussed in detail.

地震波谱元法数值模拟和偏移研究

With the development of both seismic theory and computer technology, numerical modeling technology of seismic wave has achieved great advancement during the past half century. The current methods under development include finite differentiation method (FDM), finite element method (FEM), pseudospectral method (PSM), integral equation method (IEM) and spectral element method (SEM). They exert their very important roles in every corner of seismology and seismic prospecting. Large quantity of researches towards spectral element method in the end of last century bring this method to a new era, which results in perfect solution of many difficult problems. However, parts of posterior works such as seismic migration and inversion which base on spectral element method have never been studied widely at least up to the present whereas are of importance to seismic imaging and seismic wave propagation. Based on previous work, this paper uses spectral element method to investigate the characteristics and laws of the seismic wave propagation in isotropic and anisotropic media. By thoroughly studying this high-accuracy method, we implement a kind of reverse-time pre- and post-stack migration based on SEM. In order to verify the validity of the SEM method, we have simulated the propagation of seismic wave in several different models. The simulation results show that: (1) spectral element method can be used to model any complex models and the computational results are comparable with the expected results and the analytic results; (2) the optimum accuracy can be achieved when the rank is between 4 and 9. When it is below 4, the dispersion may occur; and when it is above 9, the time step-length will be changed accordingly with the reducing space step-length in order to keep the computation stability. This will exponentially increase the computation time and at the same time the memory even if simulating the same media. This paper also applies explosive reflection surface imaging technology, time constancy principle of wave-filed extrapolation and least travetime raytracing technology of surface source to SEM pre- and post-stack migration of isotropic and anisotropic media. All imaging results derived by the above methods agree well with the real geological models and the position of interface and inflexions can also return to their right location well. This indicates that the method proposed in this paper is a kind of technology with high accuracy and robust stability. It can serve as an alternative method in real seismic data processing. All these work can boost the development of high-accuracy seismic imaging, and therefore have significant inference value.

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.

A new boundary condition solved with B.I.E.M.

Among the classical operators of mathematical physics the Laplacian plays an important role due to the number of different situations that can be modelled by it. Because of this a great effort has been made by mathematicians as well as by engineers to master its properties till the point that nearly everything has been said about them from a qualitative viewpoint. Quantitative results have also been obtained through the use of the new numerical techniques sustained by the computer. Finite element methods and boundary techniques have been successfully applied to engineering problems as can be seen in the technical literature (for instance [ l ] , [2], [3] . Boundary techniques are especially advantageous in those cases in which the main interest is concentrated on what is happening at the boundary. This situation is very usual in potential problems due to the properties of harmonic functions. In this paper we intend to show how a boundary condition different from the classical, but physically sound, is introduced without any violence in the discretization frame of the Boundary Integral Equation Method. The idea will be developed in the context of heat conduction in axisymmetric problems but it is hoped that its extension to other situations is straightforward. After the presentation of the method several examples will show the capabilities of modelling a physical problem.

SERBA: a B.I.E. program with linear elements for 2-D elastostatics analysis

The great developments that have occurred during the last few years in the finite element method and its applications has kept hidden other options for computation. The boundary integral element method now appears as a valid alternative and, in certain cases, has significant advantages. This method deals only with the boundary of the domain, while the F.E.M. analyses the whole domain. This has the following advantages: the dimensions of the problem to be studied are reduced by one, consequently simplifying the system of equations and preparation of input data. It is also possible to analyse infinite domains without discretization errors. These simplifications have the drawbacks of having to solve a full and non-symmetric matrix and some difficulties are incurred in the imposition of boundary conditions when complicated variations of the function over the boundary are assumed. In this paper a practical treatment of these problems, in particular boundary conditions imposition, has been carried out using the computer program shown below. Program SERBA solves general elastostatics problems in 2-dimensional continua using the boundary integral equation method. The boundary of the domain is discretized by line or elements over which the functions are assumed to vary linearly. Data (stresses and/or displacements) are introduced in the local co-ordinate system (element co-ordinates). Resulting stresses are obtained in local co-ordinates and displacements in a general system. The program has been written in Fortran ASCII and implemented on a 1108 Univac Computer. For 100 elements the core requirements are about 40 Kwords. Also available is a Fortran IV version (3 segments)implemented on a 21 MX Hewlett-Packard computer,using 15 Kwords.

Improvements for B.E.M. implementation in micros

This paper presents a computer program developed to run in a micro I.B.M.-P.C. wich incorporates some features in order to optimize the number of operations needed to compute the solution of plane potential problems governed by Laplace's equation by using the Boundary Integral Equation Method (B.I.E.M.). Also incorporated is a routine to plot isolines inside the domain under study.