A meshless Galerkin method with moving least square approximations for infinite elastic solids

Combining moving least square approximations and boundary integral equations, a meshless Galerkin method, which is the Galerkin boundary node method （GBNM）, for twoand three-dimensional infinite elastic solid mechanics problems with traction boundary conditions is discussed. In this numerical method, the resulting formulation inherits the symmetry and positive definiteness of variational problems, and boundary conditions can be applied directly and easily. A rigorous error analysis and convergence study for both displacement and stress is presented in Sobolev spaces. The capability of this method is illustrated and assessed by some numerical examples.     

Electrodynamics of Superconductors as a Consequence of Local Gauge Invariance

The electromagnetic properties of superconductors are studied in the framework of a quantum gauge-invariant theory. The formulation is developed in the generalized pair approximation which preserves the Ward-Takahashi identities. The macroscopic equations which regulate current and electromagnetic fields are derived by means of the boson transformation method. Comparison with previous works is reported.     

Planar theory with spin and tensor forces

We develop techniques for summing parquet diagrams for systems with two-body interactions with spin, tensor, and isospin components. For boson systems, we sum the same set of diagrams which have been used to derive the optimized hypernetted-chain variational theory when the force is independent of spin and isospin. The present derivation leads to unique local approximations in each of the six independent channels. Singularities in any of these channels at either low or high density correspond to instabilities of the physical system. In contrast to previous variational approaches, the need for commutator terms does not arise in this limit. We derive an energy functional from which the equations of motion may be obtained by functional differentiation. The equations of motion may be solved by a paired-phonon-analysis method which requires time proportional to the number of channels present. Although the primary emphasis of this theory is to suggest ways of including realistic nuclear forces in fermion theories, we present some model calculations which demonstrate the capabilities of the present approach.     

pn-QRPA and higher-order approximations

The proton-neutron Quasi-particle Random Phase Approximation (pn-QRPA) is reviewed and higher-order approximations discussed with reference to the beta decay physics. The approach is fully developed in a boson formalism. Working within a schematic model, we first illustrate a fermion-boson mapping procedure and apply it to construct boson images of the fermion Hamiltonian at different levels of approximation. The quality of these images is tested through a comparison between approximate and exact spectra. Standard QRPA equations are derived in correspondence with the quasi-boson limit of the boson Hamiltonian. The use of higher-order Hamiltonians is seen to improve considerably the stability of the approximate solutions. The mapping procedure is also applied to Fermi beta operators and transition amplitudes are discussed. The range of applicability of the QRPA formalism is examined. Presented at Workshop on calculation of double-beta-decay matrix elements (MEDEX’97), Prague, May 27–31, 1997.     

Elementary Excitations of a Higgs–Yukawa System

This work investigates the physics of elementary excitations for the so-called relativistic quantum scalar plasma system, also known as the Higgs–Yukawa system. Following the Nemes–Piza–Kerman–Lin many-body procedure, the random-phase approximation (RPA) equations were obtained for this model by linearizing the time-dependent Hartree–Fock–Bogoliubov equations of motion around equilibrium. The resulting equations have a closed solution, from which the spectrum of excitation modes are studied. We show that the RPA oscillatory modes give the one-boson and two-fermion states of the theory. The results indicate the existence of bound states in certain regions in the phase diagram. Applying these results to recent Large Hadron Collider observations concerning the mass of the Higgs boson, we determine limits for the intensity of the coupling constant g of the Higgs–Yukawa model, in the RPA mean-field approximation, for three decay channels of the Higgs boson. Finally, we verify that, within our approximations, only Higgs bosons with masses larger than 190 GeV/ $c^2$ can decay into top quarks.     

Orthogonality properties of the coupled equations for rearrangement collisions

The orthogonality properties required for a rigorous formulation of the coupled equations method for reactions involving particle exchanges and rearrangements are presented without the explicit use of multichannel projection operators.     

A Schur complement formulation for solving free-boundary,Stefan problems of phase change

A Schur complement formulation that utilizes a linear iterative solver is derived to solve a free-boundary, Stefan problem describing steady-state phase change via the Isotherm–Newton approach, which employs Newton’s method to simultaneously and efficiently solve for both interface and field equations. This formulation is tested alongside more traditional solution strategies that employ direct or iterative linear solvers on the entire Jacobian matrix for a two-dimensional sample problem that discretizes the field equations using a Galerkin finite-element method and employs a deforming-grid approach to represent the melt–solid interface. All methods demonstrate quadratic convergence for sufficiently accurate Newton solves, but the two approaches utilizing linear iterative solvers show better scaling of computational effort with problem size. Of these two approaches, the Schur formulation proves to be more robust, converging with significantly smaller Krylov subspaces than those required to solve the global system of equations. Further improvement of performance are made through approximations and preconditioning of the Schur complement problem. Hence, the new Schur formulation shows promise as an affordable, robust, and scalable method to solve free-boundary, Stefan problems. Such models are employed to study a wide array of applications, including casting, welding, glass forming, planetary mantle and glacier dynamics, thermal energy storage, food processing, cryosurgery, metallurgical solidification, and crystal growth.     

The source galerkin method for scalar field theory

This is part I of a two-part series on the Source Galerkin method. This approach is based on the differential formulation of quantum field theory. On a finite lattice, the functional differential equations for a theory in the presence of an external source becomes a set of coupled differential equations for the generating functional Z. Systematic approximations to these equations are found using the Galerkin method. Calculations are straightforward to perform, and are executed rapidly compared to Monte Carlo. The bulk of the computation involves a single matrix inversion. In addition, bosons and fermions are treated in a symmetric manner. In this paper, we consider power series solutions for scalar field theory in D = 2, 3,4. Propagators and mass gaps are calculated for a number of systems. The calculations in this paper were made on a work station of modest power using a fourth order polynomial expansion for lattices of size 8², 4³, 2⁴ in 2D 3D, and 4D. In part II we consider the fermionic formulation.     

Evolution of vibrational excitations in glassy systems

The equations of the mode-coupling theory (MCT) for ideal liquid-glass transitions are used for a discussion of the evolution of the density-fluctuation spectra of glass-forming systems for frequencies within the dynamical window between the band of high-frequency motion and the band of low-frequency-structural-relaxation processes. It is shown that the strong interaction between density fluctuations with microscopic wavelength and the arrested glass structure causes an anomalous-oscillation peak, which exhibits the properties of the so-called boson peak. It produces an elastic modulus which governs the hybridization of density fluctuations of mesoscopic wavelength with the boson-peak oscillations. This leads to the existence of high-frequency sound with properties as found by x-ray-scattering spectroscopy of glasses and glassy liquids. The results of the theory are demonstrated for a model of the hard-sphere system. It is also derived that certain schematic MCT models, whose spectra for the stiff-glass states can be expressed by elementary formulas, provide reasonable approximations for the solutions of the general MCT equations.     

Why and how to avoid standard kinetic equations I. Principles and band conduction problem

A review of ideas leading to full rejection of any finite or partially-infinite order kinetic equation linearized in external field is given on grounds of the time-convolution Generalized Master Equations (GME). By two examples (two-level and band conduction problem), it is shown how standard kinetic equations result from GME in the lowest order approximations which obscure, however, a direct correspondence with the Kubo linear response theory. Without approximations, on the other hand, the rigorous approach is shown to be fully equivalent with the Kubo results. It is argued and illustrated that usual technical simplicity and seeming physical lucidity of standard theories (connected with the presence of field-independent transfer or scattering rates in the fielddependent linearized theory) are just owing to structural features which are solely due to the lowest order approximations involved. These features (i.e. also the possibility of standard physical interpretation of kinetic phenomena) are proved to disappear completely as far as the theory goes properly to higher orders.     

Two-state system coupled to a boson mode: Quantum dynamics and classical approximations

The relation between classical and quantum mechanical integrability is investigated for a boson mode coupled to a two-level system. Different semi-classical approximations of this system are considered which are obtained by (i) factorization of expectation values of the two-state variable and the boson, (ii) making a WKB-type approximation, (iii) replacing the boson by a classical field of constant amplitude and fixed frequency and (iv) putting the boson into a self-consistent coherent state. The results vary considerably and include cases of non-integrable and integrable classical dynamics. Quantum mechanically the system is found to satisfy a criterion of quantum mechanical integrability, which we formulate, but the separated Hamiltonian of the boson alone does not have a well-defined classical limit. Numerical results for the energy spectrum and expectation values are obtained, which show a high degree of regularity but also display overlapping avoided crossings usually associated with non-integrable Hamiltonians. The exact dynamics of the occupation probabilities of the two levels is also analysed numerically. The dependence of quantum mechanical recurrence effects (in quantum optics known as revivals) on coupling strength, frequency detuning and initial conditions is studied. The revivals are found to disappear in the case of strong coupling. The Fourier spectra of the dynamical expectation values are also calculated     

Derivation and application of the boson method in superconductivity

This is a review article on the boson method in superconductivity. It covers derivations of the basic equations in the boson method and applications of these equations to the magnetic properties of type II superconductors and to the Josephson phenomena.     

Three-dimensional radiative transfer in an anisotropically scattering, plane-parallel medium: generalized reflection and transmission functions

The problem of spatially varying, collimated radiation incident on an anisotropically scattering, plane-parallel medium is considered. A very general phase function is allowed. An integral transform is used to reduce the three-dimensional radiative transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to derive nonlinear integral and integro-differential equations for the generalized reflection and transmission functions. The integration is over the polar and azimuthal angles—this formulation is referred to as the double-integral formulation. The integral equations are used to illustrate symmetry relationships and to obtain single- and double-scattering approximations. The generalized reflection and transmission functions are important in the construction of the solutions to many multidimensional problems. Coupled integral equations for the interior and emergent intensities are developed and, for the case of two identical homogeneous layers, used to formulate a doubling procedure. Results for an isotropic and Rayleigh scattering medium are presented to illustrate the computational characteristics of the formulation.     

Dyson boson mapping theory including ground-state correlation

Since the usual boson mapping methods are based on either Tamm-Dancoff phonons or bare fermion pairs, the ground-state correlation is not well taken into account by them. To improve this, we propose a new treatment of Dyson boson mapping based on a modification of the RPA phonon, which is used in the practical treatment of the self-consistent collective-coordinate (SCC) method. Applying this new method to a solvable SU(3) model and comparing the results with the exact solutions, we show that it is a good improvement of the usual Dyson boson mapping method. We also compare this method with the quantized first-order Dyson-type representation of the SCC method, which can be considered as another way to include the ground-state correlation in a boson mapping.     

Mean-field approximation and the collective transformation for Dyson boson hamiltonians

Hartree and TDA approximations are derived for non-hermitian boson hamiltonians such as the one obtained with the Dyson boson expansion. It is shown how to construct a collective transformation from the structure of Hartree and TDA bosons. This approach is applied to monopole pairing hamiltonian in the tin region.     

A space–time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains

We present the first space–time hybridizable discontinuous Galerkin (HDG) finite element method for the incompressible Navier–Stokes and Oseen equations. Major advantages of a space–time formulation are its excellent capabilities of dealing with moving and deforming domains and grids and its ability to achieve higher-order accurate approximations in both time and space by simply increasing the order of polynomial approximation in the space–time elements. Our formulation is related to the HDG formulation for incompressible flows introduced recently in, e.g., [N.C. Nguyen, J. Peraire, B. Cockburn, A hybridizable discontinuous Galerkin method for Stokes flow, Comput. Methods Appl. Mech. Eng. 199 (2010) 582–597]. However, ours is inspired in typical DG formulations for compressible flows which allow for a more straightforward implementation. Another difference is the use of polynomials of fixed total degree with space–time hexahedral and quadrilateral elements, instead of simplicial elements. We present numerical experiments in order to assess the quality of the performance of the methods on deforming domains and to experimentally investigate the behavior of the convergence rates of each component of the solution with respect to the polynomial degree of the approximations in both space and time.     

Test of the resonating group method in a three-particle problem

Elastic scattering of a boson from a bound pair of bosons is used as a numerical example to test some qualities of the resonating group method. It is shown explicitly that the method does not converge when approximations are used for the asymptotic wave function. Reasonable results can be obtained despite the fact of non-convergence when the choice of distortion functions is restricted in a prescribed way. The same choice of distortion functions leads to very accurate results in resonating group calculations with exact channel functions.     

Equations-of-motion approach to a quantum theory of large-amplitude collective motion

The equations-of-motion approach to large-amplitude collective motion is implemented both for systems of coupled bosons, also studied in a previous paper, and for systems of coupled fermions. For the fermion case, the underlying formulation is that provided by the generalized Hartree-Fock approximation (or generalized density matrix method). To obtain results valid in the semi-classical limit, as in most previous work, we compute the Wigner transform of quantum matrices in the representation in which collective coordinates are diagonal and keep only the leading contributions. Higher-order contributions can be retained, however, and, in any case, there is no ambiguity of requantization. The semi-classical limit is seen to comprise the dynamics of time-dependent Hartree-Fock theory (TDHF) and a classical canonicity condition. By utilizing a well-known parametrization of the manifold of Slater determinants in terms of classical canonical variables, we are able to derive and understand the equations of the adiabatic limit in full parallelism with the boson case. As in the previous paper, we can thus show: (i) to zero and first order in the adiabatic limit the physics is contained in Villars' equations; (ii) to second order there is consistency and no new conditions. The structure of the solution space (discussed thoroughly in the previous paper) is summarized. A discussion of associated variational principles is given. A form of the theory equivalent to self-consistent cranking is described. A method of solution is illustrated by working out several elementary examples. The relationship to previous work, especially that of Zeievinsky and Marumori and coworkers is discussed briefly. Three appendices deal respectively with the equations-of-motion method, with useful properties of Slater determinants, and with some technical details associated with the fermion equations of motion.