Diffusion via splitting and remeshing via merging in vortex methods

The technique of splitting a fat vortex element (with a core width larger than some threshold) into some thin ones in order to fix the convergence problem of the core‐spreading vortex methods is convenient and efficient. In particular, it keeps the method purely Lagrangian. In the present investigation, the splitting process is further viewed as part of the physical diffusion process. A new splitting method in which several weaker child vortices surround a thinned but still strong parent vortex is proposed. It is found that because of the survival of the parent vortex, the error arising from the splitting events can be largely reduced. The computational amount on the other hand is kept reasonably large by merging similar and close‐by vortices. The merging scheme designed herein not only involves fewer restrictions but also allows merging vortices of opposite rotations through the viewpoint of remeshing. The validity and accuracy of these techniques, proposed particularly for simulations undergoing lots of splitting and merging events, are verified by successfully simulating the interactions between two Burgers vortices under an external straining field. Copyright © 2005 John Wiley & Sons, Ltd.     

On the use of the Galerkin method for 3D numerical modelling of the general circulation: the South Atlantic eperiment

A linear three‐dimensional hydrodynamical numerical model, with the application of the Galerkin Method for the vertical dependence, is here presented. The spherical coordinate system is used, in order to allow large‐scale simulations. The equations and mathematical development of the model are shown in detail, together with the boundary and initial conditions, and the sequence of equations' solution. The model is applied to the South Atlantic Ocean, for estimating typical seasonal circulations, and the results are summarized in maps of currents at surface and 1000 m depth, and in transport values of the Brazil Current between 30°S and 40°S. Copyright © 2002 John Wiley & Sons, Ltd.     

Least‐squares finite element method on adaptive grid for PDEs with shocks

We apply the least‐squares finite element method with adaptive grid to nonlinear time‐dependent PDEs with shocks. The least‐squares finite element method is also used in applying the deformation method to generate the adaptive moving grids. The effectiveness of this method is demonstrated by solving a Burgers' equation with shocks. Computational results on uniform grids and adaptive grids are compared for the purpose of evaluation. The results show that the adaptive grids can capture the shock more sharply with significantly less computational time. For moving shock, the adaptive grid moves correctly with the shock. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Meshless simulation of equilibrium swelling/deswelling of pH‐sensitive hydrogels

Hydrogels have been widely used in microelectromechanical systems (MEMS) and Bio‐MEMS devices. In this article, the equilibrium swelling/deswelling of the pH‐stimulus cylindrical hydrogel in the microchannel is studied and simulated by the meshless method. The multi‐field coupling model, called multi‐effect‐coupling pH‐stimulus (MECpH) model, is presented and used to describe the chemical field, electric field, and the mechanical field involved in the problem. The partial differential equations (PDEs) describing these three fields are either nonlinear or coupled together. This multi‐field coupling and high nonlinear characteristics produce difficulties for the conventional numerical methods (e.g., the finite element method or the finite difference method), so an alternative—meshless method is developed to discretize the PDEs, and the efficient iteration technique is adopted to solve the nonlinear problem. The computational results for the swelling/deswelling diameter of the hydrogel under the different pH values are firstly compared with experimental results, and they have a good agreement. The influences of other parameters on the mechanical properties of the hydrogel are also investigated in detail. It is shown that the multi‐field coupling model and the developed meshless method are efficient, stable, and accurate for simulation of the properties of the stimuli‐sensitive hydrogel. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 326–337, 2006     

Interval parameter perturbation method for evaluating the bounds on natural frequencies of structures with interval parameters

For structural parameters with uncertainties, interval mathematics can, in the case where the probabilistic distribution density of uncertain variables is unavailable, deal with the influence of uncertainties in structural parameters on the response of structures. In order to evaluate the region containing natural frequencies of structures with interval parameters, the interval parameter perturbation method is presented in this paper. The advantage of the present method is its computational efficiency in evaluating the region containing natural frequencies. A numerical example is used to illustrate the efficiency of the method proposed. The project is supported by National Youth Natural Science Foundation of China and National Outstanding Youth Science Foundation of China.     

低空大气对连续波干涉仪系统相位稳定性的影响

针对三站连续波干涉仪系统，根据实时测得的系统场区大气折射率漂移数据，应用几何光学方法，从理论上估计了连续波干波仪系统两条微波基线传输的相位稳定性。     

A method for selection and use of numerical integration points for molecules with cylindrical symmetry

Boys and Handy [1] have discussed the solution of the bivariational equations with restricted numerical integration. One of the weaknesses of the method was that in the numerical summations over points, some points arose with r _ij= 0 and non-zero weights. This makes the method quite impractical for the Schrodinger Hamiltonian (because of the singularity at r _ij= 0), and it cannot be advantageous for the transcorrelated Hamiltonian C^–1HC because there will be some discontinuous higher derivatives at r _ij=0. Here it is shown how the symmetry of cylindrically symmetric molecules can be used to eliminate such points, without losing any of the advantages of the overall method, such as the convergence of the eigensolutions. It is also shown how the primary numerical integration points (z _i, r_i) may be chosen in any calculation such that each is associated with an equal amount of one-electron density. The choice of the angular coordinates are governed by the removal of the r _ij=0 points and maintaining the natural orthogonality between orbitals of different symmetry types. The method has been programmed and found to be practical, although no new molecular calculations have yet been performed. It is to be hoped that these points will give a basis for new transcorrelated calculations on diatomic molecules.This paper was presented during the session on numerical integration methods for molecules of the 1970 Quantum Theory Conference in Nottingham. It has been revised in the light of the interesting discussion which followed.     

On the average length of lexicographic paths

A bound is given for the average length of a lexicographic path, a definition that is motivated by degeneracies encountered when using the randomized simplex method.     

Heat conductivity of three periodic Hard Disks via nonequilibrium molecular dynamics

We use a driving field, of the type first suggested by Evans, to generate a steady heat current in the simplest possible system, a two-dimensional periodic fluid of three hard disks. Hard-disk motion equations can be conveniently derived from repulsive constant-force or linear-force potentials by considering the infinitely repulsive limit of these potentials. We show that the isoenergetic and isokinetic forms of the nonequilibrium equations of motion generate steady-state heat conductivities differing by terms of order 1/N, whereN is the number of particles. The resulting conductivities appear to vary as the logarithm of the driving field strength. Even at low fields, the three-body periodic-system results lie well below Enskog's infinite-system prediction.     

The spherical-model limit in a random field

The spherical-model limitn of then-vector model in a random field, with either a statistically independent distribution or with long-range correlated random fields, is studied to demonstrate the correctness of the replica method in which then and replica limits limits are interchanged, provided the replica and thermodynamic limits are taken in the right order, in the case of long-range correlated random fields. A scaling form for the two-point correlation function relevant to the first-order phase transition below the lower critical dimensionality of the random system is also obtained.