High order numerical methods to two dimensional delta function integrals in level set methods

In this paper we design and analyze a class of high order numerical methods to delta function integrals appearing in level set methods in two dimensional case. The methods comprise approximating the mesh cell restrictions of the delta function integral. In each mesh cell the two dimensional delta function integral can be rewritten as a one dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral, and thus is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals proposed in [X. Wen, High order numerical methods to a type of delta function integrals, J. Comput. Phys. 226 (2007) 1952–1967]. We establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub-algorithms and has better accuracy under an assumption on the zero level set of the level set function which holds generally. Numerical examples are presented showing that the second to fourth order methods implemented in this paper achieve or exceed the expected accuracy and demonstrating the advantage of using our high order numerical methods.     

一类任意m×n阶矩形网络的电特性

任意矩形电路网络中的电位分布问题一直是科学研究的难题.本研究发展了研究电阻网络的递推-变换(RT)理论使之能够用于计算任意m×n阶电路网络模型.研究了一类含有任意边界的m×n阶矩形网络的电位分布及等效电阻,这是一个之前一直没有解决的深刻问题,因为之前的研究依赖于规则的边界条件或一个含有零电阻的边界条件.其他方法如格林函数技术和拉普拉斯矩阵方法计算电位函数比较困难,研究含有任意边界的电阻网络也是不可能的.电位函数问题是自然科学和工程技术领域研究的一个重要内容,如拉普拉斯方程的求解问题就是其中之一.本文给出了含有一条任意边界的m×n矩形电阻网络的节点电位函数解析式,并且得到了任意两节点间的等效电阻公式,同时导出了一些特殊情形下的结果.在对不同结果的比较研究时,得到了一个新的数学分式恒等式.     

The role of chiral Lagrangians in strongly interactingW L W L signals atpp supercolliders

In this paper we apply a simple phenomenological model to describe the scattering amplitudes of strongly interacting longitudinal gauge bosons at the futurepp colliders, LHC and SSC. The model is based on the use of the well-known techniques from chiral effective Lagrangians and chiral perturbation theory, supplemented by a unitarization method for the scattering amplitudes. The generality of the approach allows one to deal with various physical situations in the unknown symmetry breaking sector of the Standard Model. In particular we mimic the two typical scenarios for the symmetry breaking sector: one of Higgs-type and the other of QCD-type. Two different unitarization methods have been implemented for comparison: the Padé approximants method and theK-matrix method. The first one permits the incorporation of the possibility of dynamical resonances as it can develop poles for the various different (I, J)-channels. A systematic study of strongly interacting signals for both LHC and SSC is presented, and their rates and efficiencies are compared.     

A MONTE CARLO RENORMALIZATION GROUP STUDY OF A 2-DIMENSIONAL TRIANGULAR LATTICE

The method developed by D.J.E.Callaway is applied to Ising model on a two-dimensional triangular lattice. A fixed point and critical exponent are found. The results are consistent with one of the exact theories very well. Obviously this method show superiority to that obtained by some other approximate methods. The method is also applied to Z₂ gauge theory on a 2-dimensional triangular lattice, no fixed points are found, in agreement with other methods.     

Unconventional application of the two-flux approximation for the calculation of the Ambartsumyan-Chandrasekhar function and the angular spectrum of the backward-scattered radiation for a semi-infinite isotropically scattering medium

It is commonly accepted that the Schwarzschild-Schuster two-flux approximation (1905, 1914) can be employed only for the calculation of the energy characteristics of the radiation field (energy density and energy flux density) and cannot be used to characterize the angular distribution of radiation field. However, such an inference is not valid. In several cases, one can calculate the radiation intensity inside matter and the reflected radiation with the aid of this simplest approximation in the transport theory. In this work, we use the results of the simplest one-parameter variant of the two-flux approximation to calculate the angular distribution (reflection function) of the radiation reflected by a semi-infinite isotropically scattering dissipative medium when a relatively broad beam is incident on the medium at an arbitrary angle relative to the surface. We do not employ the invariance principle and demonstrate that the reflection function exhibits the multiplicative property. It can be represented as a product of three functions: the reflection function corresponding to the single scattering and two identical h functions, which have the same physical meaning as the Ambartsumyan-Chandrasekhar function (H) has. This circumstance allows a relatively easy derivation of simple analytical expressions for the H function, total reflectance, and reflection function. We can easily determine the relative contribution of the true single scattering in the photon backscattering at an arbitrary probability of photon survival Λ. We compare all of the parameters of the backscattered radiation with the data resulting from the calculations using the exact theory of Ambartsumyan, Chandrasekhar, et al., which was developed decades after the two-flux approximation. Thus, we avoid the application of fine mathematical methods (the Wiener-Hopf method, the Case method of singular functions, etc.) and obtain simple analytical expressions for the parameters of the scattered radiation. Note that the simplicity of the expressions is supplemented with unexpectedly high accuracy. The results demonstrate the unknown possibilities offered by the two-flux approximation, which is the simplest approximate method to solve the equations of transport theory. We assume that the method can be employed in the calculations of the angular characteristics of the reflected radiation for media whose single scattering is described using complicated (in comparison with isotropic) laws.     

Chaotic Motion of the cc System and a New Mechanism of J/Ø Suppression

Based on random matrix theory and reduced BS equation, it is found that the regular motion of cc system can be expected at a small value of color screening mass but the chaotic motion at a large one. It is shown that the level mixing due to color screening serves as a new mechanism resulting in J/Ø suppression. Moreover, this kind of suppression can occur before the color screening mass reaches its critical value for J/Ø dissociation. In addition, it is. implied that a strong J/Ø suppression is possible in the absence of dissociation of J/Ø.     

Spectral distributions in nuclei: General principles and applications

The subject of spectral distribution methods where one derives and applies the locally smoothed forms of observables in nuclei is briefly reviewed. It is well understood that the local forms (with respect to energy) of the level density function, expectation values and strength densities are Gaussian, linear (or ratio of Gaussians) and a bivariate Gaussian respectively. To accomodate symmetries in the above forms, one has to deal with multivariate distributions in general; for example the angular-momentum (J) decomposition leads to a bivariate Gaussian form for the level density. These results extend to indefinitely large spaces by method of partitioning and they generate convolution forms. The origin of these remarkable spectral properties is discussed and shell model examples are given to substantiate their applicability to nuclear systems. Spectral distribution theory is a practical, usable theory because the smoothed forms are defined in terms of traces of low particle-rank operators, and the trace information propagates. Finally we discuss the application of the spectral methods for a wide range of nuclear problems; these include binding energies, orbit occupancies, electromagnetic andβ-decay sum rule quantities, analysis of operators, symmetry breaking, numerical level densities, and determination of bounds on time-reversal non-invariant part of nucleon-nucleon interaction.     

爱因斯坦引力场Arnowitt-Deser-Misner约束方程的两种推导方法

给出两种不同方法,分别导出爱因斯坦引力理论中著名的Arnowitt-Deser-Misner(ADM)约束方程.其一是在具有洛伦兹号差的时空中,构造一个单参数引力场作用量,由此导出单参数ADM约束方程.该参数取某特定值时对应的就是熟知的ADM约束方程.其二是将二重复函数理论运用于爱因斯坦引力场的哈密顿形式表述中,得到引力场ADM约束的二重化形式,从而也能将通常的ADM约束作为其特殊情况包含其中.此外,这两种方法还能统一地表述具有不同时空号差(洛伦兹号差和欧几里得号差)的洛伦兹引力理论和欧几里得引力理论 关键词： Arnowitt-Deser-Misner约束方程 哈密顿表述 时空号差 引力场作用量     

Finite volume evolution Galerkin method for hyperbolic conservation laws with spatially varying flux functions

We present a generalization of the finite volume evolution Galerkin scheme [M. Luká?ová-Medvid’ová, J. Saibertov’a, G. Warnecke, Finite volume evolution Galerkin methods for nonlinear hyperbolic systems, J. Comp. Phys. (2002) 183 533– 562; M. Luká?ová-Medvid’ová, K.W. Morton, G. Warnecke, Finite volume evolution Galerkin (FVEG) methods for hyperbolic problems, SIAM J. Sci. Comput. (2004) 26 1–30] for hyperbolic systems with spatially varying flux functions. Our goal is to develop a genuinely multi-dimensional numerical scheme for wave propagation problems in a heterogeneous media. We illustrate our methodology for acoustic waves in a heterogeneous medium but the results can be generalized to more complex systems. The finite volume evolution Galerkin (FVEG) method is a predictor–corrector method combining the finite volume corrector step with the evolutionary predictor step. In order to evolve fluxes along the cell interfaces we use multi-dimensional approximate evolution operator. The latter is constructed using the theory of bicharacteristics under the assumption of spatially dependent wave speeds. To approximate heterogeneous medium a staggered grid approach is used. Several numerical experiments for wave propagation with continuous as well as discontinuous wave speeds confirm the robustness and reliability of the new FVEG scheme.     

Effects of dispersion on the inference of metal texture from S0 plate mode measurements. Part I. Evaluation of dispersion correction methods.

Ultrasonic S0 waves (fundamental symmetric Lamb modes) are being considered in several laboratories for the nondestructive characterization of the texture (preferred grain orientation) and formability of metal sheets and plates. In a typical experimental setup, the velocities of the S0 waves are measured as a function of wave propagation angle with respect to the rolling direction of the plate. However, the S0 waves are known to be dispersive, and that dispersion must be considered in order to isolate the small, texture-induced shifts in the S0 wave velocity. Currently, there are two approximate dispersion correction methods, one proposed by Thompson et al. [Met. Trans. A 20, 2431 (1989)] and the other introduced by Hirao and Fukuoka [J. Acoust. Soc. Am. 85, 2311 (1989)]. In this paper, these two methods will be evaluated using an exact theory for wave propagation in orthotropic plates. Through the evaluation, the limits of the current texture measurement techniques are established. It is found that when plate thickness to wavelength ratio is less than 0.15, both Thompson's and Hirao's methods work satisfactorily. When the thickness to wavelength ratio exceeds 0.3, neither Thompson's nor Hirao's dispersion correction method provides adequate corrections for the current texture measurement techniques. Within the range of 0.15-0.3, Thompson's method is recommended for weakly anisotropic sheets and plates and Hirao's method may be more appropriate for some strongly anisotropic cases.     

Thermodynamical irreversible approach for the stellar pulsation problem

In this paper, the stellar pulsation theory is reformulated following an irreversible thermodynamic approach (Lavenda, Thermodynamics of Irreversible Processes, Denver, New York, 1993). A general stability criterion and a thermodynamic Langrangian for the pulsating star problem are obtained using a variational method (Sicardi and Ferro Fontán, Phys. Lett. 113A (1958) 263; Sicardi et al., J. Math. Phys. 32 (1991) 1350). This formulation, based on the calculation of a free energy excess function, is applied to the adiabatic, non-adiabatic, radial and non-radial cases and, as a result, the already known energy principles are obtained as particular cases of the general stability criterion mentioned above. Eigenvectors and eigenvalues can be calculated in a systematic way from the thermodynamic Lagrangian obtained for the general dissipative case (being independent of the adiabatic one). This allows a better determination of periods and oscillation frequencies.     

An improved particle correction procedure for the particle level set method

The particle level set method [D. Enright, R. Fedkiw, J. Ferziger, I. Mitchell, A hybrid particle level set method for improved interface capturing, J. Comput. Phys. 183 (2002) 83–116.] can substantially improve the mass conservation property of the level set method by using Lagrangian marker particles to correct the level set function in the under-resolved regions. In this study, the limitations of the particle level set method due to the errors introduced in the particle correction process are analyzed, and an improved particle correction procedure is developed based on a new interface reconstruction scheme. Moreover, the zero level set is “anchored” as the level set functions are reinitialized; hence the additional particle correction after the level set reinitialization is avoided. With this new scheme, a well-defined zero level set can be obtained and the disturbances to the interface are significantly reduced. Consequently, the particle reseeding operation will barely result in the loss of interface characteristics and can be applied as frequently as necessary. To demonstrate the accuracy and robustness of the proposed method, two extreme particle reseeding strategies, one without reseeding and the other with reseeding every time step, are applied in several benchmark advection tests and the results are compared with each other. Three interfacial flow cases, a 2D surface tension driven oscillating droplet, a 2D gas bubble rising in a quiescent liquid, and a 3D drop impact onto a liquid pool are simulated to illustrate the advantages of the current method over the level set and the original particle level set methods with regard to the smoothness of geometric properties and mass conservation in real physical applications.     

Variational mesh adaptation II: error estimates and monitor functions

The key to the success of a variational mesh adaptation method is to define a proper monitor function which controls mesh adaptation. In this paper we study the choice of the monitor function for the variational adaptive mesh method developed in the previous work [J. Comput. Phys. 174 (2001) 924]. Two types of monitor functions, scalar matrix and non-scalar matrix ones, are defined based on asymptotic estimates of interpolation error obtained using the interpolation theory of finite element methods. The choice of the adaptation intensity parameter is also discussed for each of these monitor functions. Asymptotic bounds on interpolation error are obtained for adaptive meshes that satisfy the regularity and equidistribution conditions. Two-dimensional numerical results are given to verify the theoretical findings.     

Aspects of Green's function methods versus self-consistent field theory

The basic methods of solving fully symmetric, nonlinear theories are stated. These are discussed in terms of Green's function methods and self-consistent field theory methods. The equivalence of many-body theory based on Green's functions with quantum field theory, on which the self-consistent field theory is based, is reviewed. A number of similarities, differences, and cautions involved with these methods are determined. In particular, since very often both methods are based upon use of the adiabatic theorem, which is typicallynot applicable to the models under consideration, a deviation in the self-consistent theory is discussed that avoids this problem. A similar idea is used for solution of models with the functional integral method. Ferromagnetic models are used at various places in illustrating some of the ideas. By contrasting these methods further insight may be gained into solving nonlinear, physical theories.     

半空间背景下目标电磁散射等效模型研究

在电磁场积分方程方法框架下,应用等效原理求解了半空间背景下目标的电磁散射问题,可针对目标仅处于半空间界面单侧及目标跨越半空间界面处于半空间两侧的几何相对位置。与传统的半空间格林函数计算方法不同的是,这一方法使用的是各自介质的自由空间格林函数,因此可以回避索墨菲积分并且可以很方便与快速算法(如多层快速多级子)相结合,并对求解临近半空间界面的目标获得较好的收敛性。在等效原理方法中需要用有限的界面等效无限大半空间界面,因此采用了锥形入射波以降低开放边界引起的边缘效应。不同于一般目标的远场雷达散射截面表达,在该模型下,半空间电磁散射的远场描述需要用差场雷达散射截面。给出的算例能与现有文献以及仿真软件很好的吻合,并可作为实用工具分析半空间背景对目标远场响应的影响。     

Double Structures and Double Symmetries for the Einstein-Maxwell-Dilaton-Axion Theory

By using the so-called double-complex function method, it is found that the stationary axisymmetric D = 4 Einstein-Maxwell-Dilaton-Axion system can be written in a double-complex matrix Ernst-like form. Then the double symmetry symplectic group Sp((4, R(J)) of the theory and its double-complex fractional linear realization are given. These results demonstrate that the theory considered possesses more and richer symmetry structures than previously expected. Moreover, as an application, an infinite chain of double-solutions of the problem is obtained, which shows that the double-complex method is more effective. Some of the results of this paper cannot be obtained by the usual (non-double) scheme.     

Delta function approximations in level set methods by distance function extension

In [A.-K. Tornberg, B. Engquist, Numerical approximations of singular source terms in differential equations, J. Comput. Phys. 200 (2004) 462–488], it was shown for simple examples that the then most common way to regularize delta functions in connection to level set methods produces inconsistent approximations with errors that are not reduced with grid refinement. Since then, several clever approximations have been derived to overcome this problem. However, the great appeal of the old method was its simplicity. In this paper it is shown that the old method – a one-dimensional delta function approximation extended to higher dimensions by a distance function – can be made accurate with a different class of one-dimensional delta function approximations. The prize to pay is a wider support of the resulting delta function approximations.     

A Hamiltonian method for finding broadband modal eigenvalues

For shallow water waveguides over a layered elastic bottom, modal eigenvalues can be determined by searching the locations in the complex plane of the horizontal wave number at which the complex phase function is a multiple of π [C. T. Tindle and N. R. Chapman, J. Acoust. Soc. Am. 96, 1777-1782 (1994)]. In this paper, a Hamiltonian method is introduced for tracing the path in the complex plane along which the phase function keeps real. The Hamiltonian method can also be extended to compute the broadband modal eigenvalues or the modal dispersion curves in the Pekeris waveguide with fluid/elastic bottoms. For each proper or leaky normal mode, a different Hamiltonian is constructed in the complex plane and used to trace automatically the complex dispersion curve with the eigenvalue in a reference frequency as the initial value. In contrast to the usual methods, the dispersion curve for each mode is determined individually. The Hamiltonian method shows good performance by comparing with KRAKEN.     

Invariant resolutions for several Fueter operators

A proper generalization of complex function theory to higher dimension is Clifford analysis and an analogue of holomorphic functions of several complex variables were recently described as the space of solutions of several Dirac equations. The four-dimensional case has special features and is closely connected to functions of quaternionic variables. In this paper we present an approach to the Dolbeault sequence for several quaternionic variables based on symmetries and representation theory. In particular we prove that the resolution of the Cauchy–Fueter system obtained algebraically, via Gröbner bases techniques, is equivalent to the one obtained by R.J. Baston (J. Geom. Phys. 1992).     

A generalised manifestly gauge invariant exact renormalisation group for SU(N) Yang–Mills

We take the manifestly gauge invariant exact renormalisation group previously used to compute the one-loop β function in SU(N) Yang–Mills without gauge fixing, and generalise it so that it can be renormalised straightforwardly at any loop order. The diagrammatic computational method is developed to cope with general group theory structures, and new methods are introduced to increase its power, so that much more can be done simply by manipulating diagrams. The new methods allow the standard two-loop β function coefficient for SU(N) Yang–Mills to be computed, for the first time without fixing the gauge or specifying the details of the regularisation scheme.