Discrete wavelet structure and discrete energy of classical plane light waves

We find by the wavelet transform that the classical plane light wave of linear polarization can be decomposed into a series of discrete Morlet wavelets.In the theoretical frame,the energy of the classical light wave becomes discrete;interestingly,the discretization is consistent with the energy division of P portions in Planck radiation theory,where P is an integer.It is shown that the changeable energy of a basic plane light wave packet or wave train is H_(0k)=nP0 kω(n=1,2,3,...;k=|k|),with discrete wavelet structure parameter n,wave vector k and idler frequency ω,and a constant p0 k.The wave-particle duality from the Mach-Zehnder interference of single photons is simulated by using random basic plane light wave packets.

ON DISCRETE ENERGY DISSIPATION OF MAXWELL’S EQUATIONS IN A COLE-COLE DISPERSIVE MEDIUM

A simple criterion is studied for the first time for identifying the discrete energy dissipation of the Crank-Nicolson scheme for Maxwell’s equations in a Cole-Cole dispersive medium.Several numerical formulas that approximate the time fractional derivatives are investigated based on this criterion,including the L1 formula,the fractional BDF-2,and the shifted fractional trapezoidal rule(SFTR).Detailed error analysis is provided within the framework of time domain mixed finite element methods for smooth solutions.The convergence results and discrete energy dissipation law are confirmed by numerical tests.For nonsmooth solutions,the method SFTR can still maintain the optimal convergence order at a positive time on uniform meshes.Authors believe this is the first appearance that a second-order time-stepping method can restore the optimal convergence rate for Maxwell’s equations in a Cole-Cole dispersive medium regardless of the initial singularity of the solution.

Spherical parametrization of genus-zero meshes by minimizing discrete harmonic energy

The problem of spherical parametrization is that of mapping a genus-zero mesh onto a spherical surface. For a given mesh, different parametrizations can be obtained by different methods. And for a certain application, some parametrization results might behave better than others. In this paper, we will propose a method to parametrize a genus-zero mesh so that a surface fitting algorithm with PHT-splines can generate good result. Here the parametrization results are obtained by minimizing discrete har- monic energy subject to spherical constraints. Then some applications are given to illustrate the advantages of our results. Based on PHT-splines, parametric surfaces can be constructed efficiently and adaptively to fit genus-zero meshes after their spherical parametrization has been obtained.

Phase escape of current-biased Josephson junctions

Switching current distributions of an Nb/Al-AlO2/Nb Josephson junction are measured in a temperature range from 25 mK to 800 mK. We analyse the phase escape properties by using the theory of Larkin and Ovchinnikov （LO） which takes discrete energy levels into account. Our results show that the phase escape can be well described by the LO approach for temperatures near and below the crossover from thermal activation to macroscopic quantum tunneling. These results are helpful for further study of macroscopic quantum phenomena in Josephson junctions where discrete energy levels need to be considered.

INVARIANTS-PRESERVING DU FORT-FRANKEL SCHEMES AND THEIR ANALYSES FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH WAVE OPERATOR

Du Fort-Frankel finite difference method(FDM)was firstly proposed for linear diffusion equations with periodic boundary conditions by Du Fort and Frankel in 1953.It is an explicit and unconditionally von Neumann stable scheme.However,there has been no research work on numerical solutions of nonlinear Schrödinger equations with wave operator by using Du Fort-Frankel-type finite difference methods(FDMs).In this study,a class of invariants-preserving Du Fort-Frankel-type FDMs are firstly proposed for one-dimensional(1D)and two-dimensional(2D)nonlinear Schrödinger equations with wave operator.By using the discrete energy method,it is shown that their solutions possess the discrete energy and mass conservative laws,and conditionally converge to exact solutions with an order of for ofο(T^(2)+h_(x)^(2)+(T/h_(x))^(2))1D problem and an order ofο(T^(2)+h_(x)^(2)+h_(Y)^(2)+(T/h_(X))^(2)+(T/h_(y))^(2))for 2D problem in H1-norm.Here,τdenotes time-step size,while,hx and hy represent spatial meshsizes in x-and y-directions,respectively.Then,by introducing a stabilized term,a type of stabilized invariants-preserving Du Fort-Frankel-type FDMs are devised.They not only preserve the discrete energies and masses,but also own much better stability than original schemes.Finally,numerical results demonstrate the theoretical analyses.

Energy law preserving continuous finite element schemes for a gas metal arc welding system

In this paper a modifed continuous energy law was explored to investigate transport behavior in a gas metal arc welding(GMAW)system.The energy law equality at a discrete level for the GMAW system was derived by using the finite element scheme.The mass conservation and current density continuous equation with the penalty scheme was applied 10 improve the stability.According to the phase-field model coupled with the energy law preserving method,the GMAW model was discretized and a metal transfer process with a pulse current was simulated.It was found that the numerical solution agrees well with the data of the metal transfer process obtained by high-speed photography.Compared with the numerical solution of the volume of fuid model,which was widely studied in the GMAW system based on the finite element method Euler scheme,the energy law preserving method can provide better accuracy in predicting the shape evolution of the droplet and with a greater computing efficiency.

Effect of particle polydispersity on micromechanical properties and energy dissipation in granular mixtures

A series of numerical tests was conducted to study the micromechanical properties and energy dissipation in polydisperse assemblies of spherical particles subjected to uniaxial compression. In general, distributed particle size assemblies with standard deviations ranging from 0% to 80% of the particle mean diameter were examined. The microscale analyses included the trace of the fabric tensor, magnitude and orien- tation of the contact forces, trace of stress, number of contacts and degree of mobilization of friction in contacts between particles. In polydisperse samples, the average coordination numbers were lower than in monodisperse assemblies, and the mobilization of friction was higher than in monodisperse assemblies due to the non-uniform spatial rearrangement of spheres in the samples and the smaller displacements of the particles. The effect of particle size heterogeneity on both the energy density and energy dissipation in systems was also investigated.

An erosion model for the discrete element method

A shear impact energy model （SIEM） of erosion suitable for both dilute and dense particle flows is pro- posed based on the shear impact energy of particles in discrete element method （DEM） simulations. A number of DEM simulations are performed to determine the relationship between the shear impact energy predicted by the DEM model and the theoretical erosion energy. Simulation results show that nearly one-quarter of the shear impact energy will be converted to erosion during an impingement. According to the ratio of the shear impact energy to the erosion energy, it is feasible to predict erosion from the shear impact energy, which can be accumulated at each time step for each impingement during the DEM simulation. The total erosion of the target surface can be obtained by summing the volume of material removed from each impingement. The proposed erosion model is validated against experiment and results show that the SIEM combined with DEM accurately predicts abrasive erosions.

Structure-Preserving Finite-Element Schemes for the Euler-Poisson Equations

We discuss structure-preserving numerical discretizations for repulsive and attractive Euler-Poisson equations that find applications in fluid-plasma and selfgravitation modeling.The scheme is fully discrete and structure preserving in the sense that it maintains a discrete energy law,as well as hyperbolic invariant domain properties,such as positivity of the density and a minimum principle of the specific entropy.A detailed discussion of algorithmic details is given,as well as proofs of the claimed properties.We present computational experiments corroborating our analytical findings and demonstrating the computational capabilities of the scheme.

Corrected explicit-implicit domain decomposition algorithms for two-dimensional semilinear parabolic equations

Corrected explicit-implicit domain decomposition(CEIDD) algorithms are studied for parallel approximation of semilinear parabolic problems on distributed memory processors. It is natural to divide the spatial domain into some smaller parallel strips and cells using the simplest straightline interface(SI) . By using the Leray-Schauder fixed-point theorem and the discrete energy method,it is shown that the resulting CEIDD-SI algorithm is uniquely solvable,unconditionally stable and convergent. The CEIDD-SI method always suffers from the globalization of data communication when interior boundaries cross into each other inside the domain. To overcome this disadvantage,a composite interface(CI) that consists of straight segments and zigzag fractions is suggested. The corresponding CEIDD-CI algorithm is proven to be solvable,stable and convergent. Numerical experiments are presented to support the theoretical results.

A FINITE DIFFERENCE SCHEME FOR THE GENERALIZED NONLINEAR SCHRDINGER EQUATION WITH VARIABLE COEFFICIENTS

A finite difference scheme for the generalized nonlinear Schr dinger equation with variable coefficients is developed. The scheme is shown to satisfy two conservation laws. Numerical results show that the scheme is accurate and efficient.

A 2D DEM-LBM study on soil behaviour due to locally injected fluid

Leakage from underground pipes could result in foundations being undermined and cause damage to adjacent infrastructure. Soil particles surrounding the leaking area could be mobilised, displaced, and even washed out of the soil matrix by the leaking fluid, generating a void or cavity. A two-dimensional simulation using a coupled discrete element method and lattice Boltzmann method （DEM-LBM） has been used to investigate the behaviour of a soil bed subject to a locally injected fluid, which represents a leak in a pipe. Various values of inter-particle surface energy were also adopted to model the mechanical effects of cohesive particles. The results suggest that the inter-particle surface energy greatly influences the bed response with respect to the leaking fluid, including the excess pressure initiating the cavity, the cavity shape and its evolution rate.