Multi-step processes in high-energy elastic and inelastic nuclear scattering

Multi-step processes in elastic and inelastic nuclear scattering at intermediate and high energies are investigated using a formulation whereby a finite number of channels are explicitly treated while all the other channels are approximately accounted for through a “second-order potential matrix”. Within the framework of the eikonal approximation the problem reduces to a finite system of first-order coupled integro-differential equations with non-local potentials which depend on the two-body density matrix of the target nucleus. The relationship of the above formulation to the DWIA, the close-coupling method, and the Glauber multiple scattering model is examined. This approach is applied to the elastic and inelastic (2⁺, 4.43 MeV) scattering of 1 GeV nucleons by ¹²C. The corrections to the DWIA are sizeable, and the inelastic scattering appears to be very sensitive to the multi-step contributions and the nuclear structure.     

Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings

A formulation of the modal method for multi-element dielectric lamellar diffraction gratings is presented. It combines the conventional semi-analytic Kronig–Penny formulation with a Fresnel scattering matrix approach to the solution of the diffraction problem. The theory is intuitive, applicable to complex geometries, and provides insight into diffraction grating physics. With the Fresnel matrix extensions the method potentially has greater generality than other formulations, and is applicable to non-trivial waveguide problems, such as characterizing the coupling between 1D photonic crystal waveguides. A numerically stable way of using the modal method for multilayer stacks of lamellar gratings is described.     

Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings

A formulation of the modal method for multi-element dielectric lamellar diffraction gratings is presented. It combines the conventional semi-analytic Kronig-Penny formulation with a Fresnel scattering matrix approach to the solution of the diffraction problem. The theory is intuitive, applicable to complex geometries, and provides insight into diffraction grating physics. With the Fresnel matrix extensions the method potentially has greater generality than other formulations, and is applicable to non-trivial waveguide problems, such as characterizing the coupling between 1D photonic crystal waveguides. A numerically stable way of using the modal method for multilayer stacks of lamellar gratings is described.     

Singularity expansions for cylinders of finite conductivity

A new rigorous differential formulation to compute the scattering matrix of an obliquely oriented cylinder of finite conductivity and arbitrary shape is given. The analytic continuation of the scattering operator is examined in the frequency domain as well as in the propagation constant domain. The corresponding computer program is able to compute the singularity expansion for such a cylinder, as well as the propagating mode along it. Numerical examples are given.     

Parallel Jacobian-free Newton Krylov solution of the discrete ordinates method with flux limiters for 3D radiative transfer

The present study introduces a parallel Jacobian-free Newton Krylov (JFNK) general minimal residual (GMRES) solution for the discretized radiative transfer equation (RTE) in 3D, absorbing, emitting and scattering media. For the angular and spatial discretization of the RTE, the discrete ordinates method (DOM) and the finite volume method (FVM) including flux limiters are employed, respectively. Instead of forming and storing a large Jacobian matrix, JFNK methods allow for large memory savings as the required Jacobian-vector products are rather approximated by semiexact and numerical formulations, for which convergence and computational times are presented. Parallelization of the GMRES solution is introduced in a combined memory-shared/memory-distributed formulation that takes advantage of the fact that only large vector arrays remain in the JFNK process. Results are presented for 3D test cases including a simple homogeneous, isotropic medium and a more complex non-homogeneous, non-isothermal, absorbing–emitting and anisotropic scattering medium with collimated intensities. Additionally, convergence and stability of Gram–Schmidt and Householder orthogonalizations for the Arnoldi process in the parallel GMRES algorithms are discussed and analyzed. Overall, the introduction of JFNK methods results in a parallel, yet scalable to the tested 2048 processors, and memory affordable solution to 3D radiative transfer problems without compromising the accuracy and convergence of a Newton-like solution.     

求解洞穴散射问题的带有吸收边界层的有限元方法

给出二维洞穴散射问题的带有吸收边界层的有限元方法.通过吸收边界层将无界域问题截断,得到一个有界域问题,求解此有界域问题来代替原问题.进一步分析收敛性并进行数值模拟.结果表明了此方法的有效性.     

Polarization imaging of multiply-scattered radiation based on integral-vector Monte Carlo method

A new integral-vector Monte Carlo method (IVMCM) is developed to analyze the transfer of polarized radiation in 3D multiple scattering particle-laden media. The method is based on a “successive order of scattering series” expression of the integral formulation of the vector radiative transfer equation (VRTE) for application of efficient statistical tools to improve convergence of Monte Carlo calculations of integrals. After validation against reference results in plane-parallel layer backscattering configurations, the model is applied to a cubic container filled with uniformly distributed monodispersed particles and irradiated by a monochromatic narrow collimated beam. 2D lateral images of effective Mueller matrix elements are calculated in the case of spherical and fractal aggregate particles. Detailed analysis of multiple scattering regimes, which are very similar for unpolarized radiation transfer, allows identifying the sensitivity of polarization imaging to size and morphology.     

Analysis of scattering by conducting objects covered with arbitrarily magnetised plasma using finite element-boundary integral-fast multipole algorithm

The electromagnetic (EM) scattering by conducting objects covered with arbitrarily magnetised plasma is computed using the hybrid vector finite element-boundary integral-fast multipole method (FE-BI-FMM). The relative permittivity tensor of arbitrarily magnetised plasma is described. The FE-BI-FMM formulation for scattering by three-dimensional (3D) conducting objects covered with arbitrarily magnetised plasma is derived in detail. The validity of FE-BI-FMM is verified by comparing numerical results with analytical values. The radar cross-sections (RCS) of conducting objects covered with magnetised plasmas are simulated and the effects of plasma parameters on RCS are analysed. Numerical results indicate that magnetised plasma cover layer is effective on reducing the back scattering if appropriate parameters of plasma are chosen.     

Prediction of radiative heat transfer in 3D complex geometries using the unstructured control volume finite element method

In this paper, a 3D algorithm for the treatment of radiative heat transfer in emitting, absorbing, and scattering media is developed. The numerical approach is based on the utilization of the unstructured control volume finite element method (CVFEM) which, to the knowledge of the authors, is applied for the first time to simulate radiative heat transfer in participated media confined in 3D complex geometries. This simulation makes simultaneously the use of the merits of both the finite element method and the control volume method. Unstructured 3D triangular element grids are employed in the spatial discretization and azimuthal discretization strategy is employed in the angular discretization. The general discretization equation is presented and solved by the conditioned conjugate gradient squared method (CCGS). In order to test the efficiency of the developed method, several 3D complex geometries including a hexahedral enclosure, a 3D equilateral triangular enclosure, a 3D L-shaped enclosure and 3D elliptical enclosure are examined. The results are compared with the exact solutions or published references and the accuracy obtained in each case is shown to be highly satisfactory. Moreover, this approach required a less CPU time and iterations compared with those of even parity formulation of the discrete ordinates method.     

Burton–Miller-type singular boundary method for acoustic radiation and scattering

This paper proposes the singular boundary method (SBM) in conjunction with Burton and Miller?s formulation for acoustic radiation and scattering. The SBM is a strong-form collocation boundary discretization technique using the singular fundamental solutions, which is mathematically simple, easy-to-program, meshless and introduces the concept of source intensity factors (SIFs) to eliminate the singularities of the fundamental solutions. Therefore, it avoids singular numerical integrals in the boundary element method (BEM) and circumvents the troublesome placement of the fictitious boundary in the method of fundamental solutions (MFS). In the present method, we derive the SIFs of exterior Helmholtz equation by means of the SIFs of exterior Laplace equation owing to the same order of singularities between the Laplace and Helmholtz fundamental solutions. In conjunction with the Burton–Miller formulation, the SBM enhances the quality of the solution, particularly in the vicinity of the corresponding interior eigenfrequencies. Numerical illustrations demonstrate efficiency and accuracy of the present scheme on some benchmark examples under 2D and 3D unbounded domains in comparison with the analytical solutions, the boundary element solutions and Dirichlet-to-Neumann finite element solutions.     

A direct matrix method for computing analytical Jacobians of discretized nonlinear integro-differential equations

In this article, we present a simple direct matrix method for analytically computing the Jacobian of nonlinear algebraic equations that arise from the discretization of nonlinear integro-differential equations. The method is based on a formulation of the discretized equations in vector form using only matrix-vector products and component-wise operations. By applying simple matrix-based differentiation rules, the matrix form of the analytical Jacobian can be calculated with little more difficulty than required to compute derivatives in single-variable calculus. After describing the direct matrix method, we present numerical experiments demonstrating the computational performance of the method, discuss its connection to the Newton–Kantorovich method and apply it to illustrative 1D and 2D example problems from electrochemical transport.     

An investigation of transmission coefficients for finite and semi-infinite coupled plate structures

This paper introduces a method for determining the transmission coefficient for finite coupled plates using an analytical waveguide model combined with a scattering matrix. In the scattering matrix method, the amplitudes of the structural waves impinging on a junction are separated into incident, reflected, and transmitted components. The energy flow due to each of these waves is obtained using a wave impedance method, which is subsequently used to determine the transmission coefficient. Transmission coefficients for semi-infinite and finite L-shaped plates are investigated for single and multiple point force excitations, and for controlled incident wave sources. It is shown that the transmission coefficients can also be calculated from details of the modal transmission coefficients and the modal composition of the energy incident on the junction. Results show that the modal transmission coefficients are largely independent of whether the plates have finite or semi-infinite boundary conditions, and are only dependent on the details of the coupling. Finally, frequency averaged transmission coefficients are compared for semi-infinite and finite structures. In the cases considered, it is found that the semi-infinite system is a good approximation for finite systems after frequency averaging, especially if the system is excited with multiple point force excitation.     

A General T-Matrix Approach Applied to Two-Body and Three-Body Problems in Cold Atomic Gases

We propose a systematic T-matrix approach to solve few-body problems with s-wave contact interactions in ultracold atomic gases. The problem is generally reduced to a matrix equation expanded by a set of orthogonal molecular states, describing external center-of-mass motions of pairs of interacting particles; while each matrix element is guaranteed to be finite by a proper renormalization for internal relative motions. This approach is able to incorporate various scattering problems and the calculations of related physical quantities in a single framework, and also provides a physically transparent way to understand the mechanism of resonance scattering. For applications, we study two-body effective scattering in 2D–3D mixed dimensions, where the resonance position and width are determined with high precision from only a few number of matrix elements. We also study three fermions in a (rotating) harmonic trap, where exotic scattering properties in terms of mass ratios and angular momenta are uniquely identified in the framework of T-matrix.     

A piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids

We develop a piecewise linear (PWL) Galerkin finite element spatial discretization for the multi-dimensional radiation diffusion equation. It uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal (2D) or polyhedral (3D) grids. We first demonstrate some analytical properties of the PWL method and perform a simple mode analysis to compare the PWL method with Palmer’s vertex-centered finite-volume method and with a bilinear continuous finite element method. We then show that this new PWL method gives solutions comparable to those from Palmer’s. However, since the PWL method produces a symmetric positive-definite coefficient matrix, it should be substantially more computationally efficient than Palmer’s method, which produces an asymmetric matrix. We conclude that the Galerkin PWL method is an attractive option for solving diffusion equations on unstructured grids.     

Least-squares finite element formulations for one-dimensional radiative transfer

We present least-squares-based finite element formulations for the numerical solution of the radiative transfer equation in its first-order primitive variable form. The use of least-squares principles leads to a variational unconstrained minimization problem in a setting of residual minimization. In addition, the resulting linear algebraic problem will always have a symmetric positive definite coefficient matrix, allowing the use of robust and fast iterative methods for its solution. We consider space-angle coupled and decoupled formulations. In the coupled formulation, the space-angle dependency is represented by two-dimensional finite element expansions and the least-squares functional minimized in the continuous space-angle domain. In the decoupled formulation the angular domain is represented by discrete ordinates, the spatial dependence represented by one-dimensional finite element expansions, and the least-squares functional minimized continuously in space domain and at discrete locations in the angle domain. Numerical examples are presented to demonstrate the merits of the formulations in slab geometry, for absorbing, emitting, anisotropically scattering mediums, allowing for spatially varying absorption and scattering coefficients. For smooth solutions in space-angle domain, exponentially fast decay of error measures is demonstrated as the p-level of the finite element expansions is increased. The formulations represent attractive alternatives to weak form Galerkin finite element formulations, typically applied to the more complicated second-order even- and odd-parity forms of the radiative transfer equation.     

Understanding quantum entanglement by thermo field dynamics

We propose a new method to understand quantum entanglement using the thermo field dynamics (TFD) described by a double Hilbert space. The entanglement states show a quantum-mechanically complicated behavior. Our new method using TFD makes it easy to understand the entanglement states, because the states in the tilde space in TFD play a role of tracer of the initial states. For our new treatment, we define an extended density matrix on the double Hilbert space. From this study, we make a general formulation of this extended density matrix and examine some simple cases using this formulation. Consequently, we have found that we can distinguish intrinsic quantum entanglement from the thermal fluctuations included in the definition of the ordinary quantum entanglement at finite temperatures. Through the above examination, our method using TFD can be applied not only to equilibrium states but also to non-equilibrium states. This is shown using some simple finite systems in the present paper.     

Der CDD-Pol in der Zweikanalstreuung

Comparing the inelastic one channel N/D method of Frye and Warnock, and the many channel matrix N/D method of Bjorken it is shown, that for a large class of many channel models the solution of the Bjorken method is a solution of the Frye Warnock method. We demonstrate at a simple model of the two channel scattering, that one dynamical CDD pole exists in the Frye Warnock method.     

Semi-analytical finite element analysis of elastic waveguides subjected to axial loads

Predicting the influence of axial loads on the wave propagation in structures such as rails requires numerical analysis. Conventional three-dimensional finite element analysis has previously been applied to this problem. The process is tedious as it requires that a number of different length models be solved and that the user identify the computed modes of propagation. In this paper, the more specialised semi-analytical finite element method is extended to account for the effect of axial load. The semi-analytical finite element method includes the wave propagation as a complex exponential in the element formulation and therefore only a two-dimensional mesh of the cross-section of the waveguide is required. It was found that the stiffness matrix required to describe the effect of axial load is proportional to the mass matrix, which makes the extension to existing software trivial. The method was verified by application to an aluminium rod, where after phase and group velocities of propagating waves in a rail were computed to demonstrate the method.     

The layer multiple-scattering method for calculating transmission coefficients of 2D phononic crystals

We extend the layer multiple-scattering theory (LMST) to elastic waves propagating in two-dimensional (2D) periodical composites. The formalism to calculate the reflection and transmission coefficients for elastic waves through finite slabs is presented. In this spirit, the crystal is viewed as a sequence of identical monolayer which has one-dimensional (1D) periodicity along a given direction. The reflection and transmission coefficients for a multilayer slab can be obtained by a double-layer scheme through the calculation of the scattering matrix of a monolayer. To demonstrate the application of this formalism, we calculate transmission coefficients for systems consisting of pure solid components or mixing (solid and fluid) components. The validity of this method is checked by both band structure calculations and transmission measurement of ultrasonic experiment.