A Cartesian‐grid collocation method based on radial‐basis‐function networks for solving PDEs in irregular domains

This paper reports a new Cartesian‐grid collocation method based on radial‐basis‐function networks (RBFNs) for numerically solving elliptic partial differential equations in irregular domains. The domain of interest is embedded in a Cartesian grid, and the governing equation is discretized by using a collocation approach. The new features here are (a) one‐dimensional integrated RBFNs are employed to represent the variable along each line of the grid, resulting in a significant improvement of computational efficiency, (b) the present method does not require complicated interpolation techniques for the treatment of Dirichlet boundary conditions in order to achieve a high level of accuracy, and (c) normal derivative boundary conditions are imposed by means of integration constants. The method is verified through the solution of second‐ and fourth‐order PDEs; accurate results and fast convergence rates are obtained. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A reference solution for phase change with convection

A reference solutions for phase change involving convection in the melt is currently missing. In the present study, we focus on the problem of melting of pure tin in a square cavity heated from the side, which is used as a benchmark test problem. The mathematical model used for the simulations is based on the enthalpy formulation. Extensive numerical computations are performed with grids as fine as 800 × 800. The convergence of the numerical solution is demonstrated and its level assessed. Data values and plots are provided for use as a reference solution. Copyright © 2005 John Wiley & Sons, Ltd.     

The Legnaro Francium Magneto-Optical Trap

Hyperfine Interactions - Laser cooling and trapping of radioactive atoms represent the new frontier in atomic physics and a new powerful tool in nuclear physics. We are setting up at the...     

Determination of k-factors by thermal neutron activation technique

The k_{o, Au}-factors of 10 elements have been determined by irradating the samples with thermal neutrons in the reactor thermal column. The proposed method has the advantage of eliminating the errors induced by the uncertainties of the neutron spectrum parameters and the nuclear data. The results obtained are in good agreement with the literature data.     

Numerical solution of Navier–Stokes equations using multiquadric radial basis function networks

A numerical method based on radial basis function networks (RBFNs) for solving steady incompressible viscous flow problems (including Boussinesq materials) is presented in this paper. The method uses a ‘universal approximator’ based on neural network methodology to represent the solutions. The method is easy to implement and does not require any kind of ‘finite element‐type’ discretization of the domain and its boundary. Instead, two sets of random points distributed throughout the domain and on the boundary are required. The first set defines the centres of the RBFNs and the second defines the collocation points. The two sets of points can be different; however, experience shows that if the two sets are the same better results are obtained. In this work the two sets are identical and hence commonly referred to as the set of centres. Planar Poiseuille, driven cavity and natural convection flows are simulated to verify the method. The numerical solutions obtained using only relatively low densities of centres are in good agreement with analytical and benchmark solutions available in the literature. With uniformly distributed centres, the method achieves Reynolds number Re = 100 000 for the Poiseuille flow (assuming that laminar flow can be maintained) using the density of , Re = 400 for the driven cavity flow with a density of and Rayleigh number Ra = 1 000 000 for the natural convection flow with a density of . Copyright © 2001 John Wiley & Sons, Ltd.     

Reversing the Stereoselectivity of a Palladium‐Catalyzed O‐Glycosylation through an Inner‐Sphere or Outer‐Sphere Pathway

An efficient and concise method for the construction of various O‐glycosidic bonds by a palladium‐catalyzed reaction with a 3‐O‐picoloyl glucal has been developed. The stereochemistry of the anomeric center derives from either an inner‐sphere or outer‐sphere pathway. Harder nucleophiles, such as aliphatic alcohols and sodium phenoxides give β‐products, and α products result from using softer nucleophiles, such as phenol.     

C‐Selective and Diastereoselective Alkyl Addition to β,γ‐Alkynyl‐α‐imino Esters with Zinc(II)ate Complexes

Since umpolung α‐imino esters contain three electrophilic centers, regioselective alkyl addition with traditional organometallic reagents has been a serious problem in the practical synthesis of versatile chiral α‐amino acid derivatives. An unusual C‐alkyl addition to α‐imino esters using a Grignard reagent (RMgX)‐derived zinc(II)ate was developed. Zinc(II)ate complexes consist of a Lewis acidic [MgX]⁺ moiety, a nucleophilic [R₃Zn]^? moiety, and 2 [MgX₂]. Therefore, the ionically separated [R₃Zn]^? selectively attacks the imino carbon atom ,which is most strongly activated by chelation of [MgX]⁺. In particular, chiral β,γ‐alkynyl‐α‐imino esters can strongly promote highly regio‐ and diastereoselective C‐alkylation because of structural considerations, and the corresponding optically active α‐quaternary amino acid derivatives are obtained within 5 minutes in high to excellent yields.