Questioning numerical integration methods for microsphere (and microplane) constitutive equations

In the last few years, more and more complex microsphere models have been proposed to predict the mechanical response of various polymers. Similarly than for microplane models, they consist in deriving a one-dimensional force vs. stretch equation and to integrate it over the unit sphere to obtain a three-dimensional constitutive equation. In this context, the focus of authors is laid on the physics of the one-dimensional relationship, but in most of the case the influence of the integration method on the prediction is not investigated.Here we compare three numerical integration schemes: a classical Gaussian scheme, a method based on a regular geometric meshing of the sphere, and an approach based on spherical harmonics. Depending on the method, the number of integration points may vary from 4 to 983,040! Considering simple quantities, i.e. principal (large) strain invariants, it is shown that the integration method must be carefully chosen. Depending on the quantities retained to described the one-dimensional equation and the required error, the performances of the three methods are discussed. Consequences on stress–strain prediction are illustrated with a directional version of the classical Mooney–Rivlin hyperelastic model. Finally, the paper closes with some advices for the development of new microsphere constitutive equations.     

Separation of hematite from banded hematite jasper (BHJ) by magnetic coating

The separation of iron oxide from banded hematite jasper(BHJ) assaying 47.8% Fe, 25.6% Si O2 and 2.30%Al2O3 using selective magnetic coating was studied. Characterization studies of the low grade ore indicate that besides hematite and goethite,jasper, a microcrystalline form of quartzite, is the major impurity associated with this ore. Beneficiation by conventional magnetic separation technique could yield a magnetic concentrate containing 60.8% Fe with 51% Fe recovery. In order to enhance the recovery of the iron oxide minerals, fine magnetite, colloidal magnetite and oleate colloidal magnetite were used as the coating material. When subjected to magnetic separation, the coated ore produces an iron concentrate containing 60.2% Fe with an enhanced recovery of56%. The AFM studies indicate that the coagulation of hematite particles with the oleate colloidal magnetite facilitates the higher recovery of iron particles from the low grade BHJ iron ore under appropriate conditions.     

Enhancement of optical and electrical properties of SILAR deposited ZnO thin films through fluorine doping and vacuum annealing for photovoltaic applications

Undoped and fluorine doped ZnO thin films were deposited onto glass substrates using successive ionic layer adsorption and reaction (SILAR) technique and then annealed at 350 °C in vacuum ambience. The F doping level was varied from 0 to 15 at% in steps of 5 at%. The XRD analysis showed that all the films are polycrystalline with hexagonal wurtzite structure and preferentially oriented along the (002) plane. Crystallite sizes were found to increase when 5 at% of F is doped and then decreased with further doping. It was seen from the SEM images that the doping causes remarkable changes in the surface morphology and the annealing treatment results in well-defined grains with an improvement in the grain size irrespective of doping level. All the films exhibit good transparency (>70%) after vacuum annealing. Electrical resistivity of the film was found to be minimum (1.32×10⁻³ Ω cm) when the fluorine doping level was 5 at%.     

Optimization of Pathogen Capture in Flowing Fluids with Magnetic Nanoparticles

Magnetic nanoparticles have been employed to capture pathogens for many biological applications; however, optimal particle sizes have been determined empirically in specific capturing protocols. Here, a theoretical model that simulates capture of bacteria is described and used to calculate bacterial collision frequencies and magnetophoretic properties for a range of particle sizes. The model predicts that particles with a diameter of 460 nm should produce optimal separation of bacteria in buffer flowing at 1 L h⁻¹. Validating the predictive power of the model, Staphylococcus aureus is separated from buffer and blood flowing through magnetic capture devices using six different sizes of magnetic particles. Experimental magnetic separation in buffer conditions confirms that particles with a diameter closest to the predicted optimal particle size provide the most effective capture. Modeling the capturing process in plasma and blood by introducing empirical constants (c_e), which integrate the interfering effects of biological components on the binding kinetics of magnetic beads to bacteria, smaller beads with 50 nm diameters are predicted that exhibit maximum magnetic separation of bacteria from blood and experimentally validated this trend. The predictive power of the model suggests its utility for the future design of magnetic separation for diagnostic and therapeutic applications.     

AlSb晶体内二维磁极化子的磁场与温度效应

在考虑声子之间相互作用的同时，本文应用线性组合算符和微扰法研究了电子自旋对弱耦合二维磁极化子特性的影响。对AlSb晶体所作的数值计算结果表明，随着磁场的加强，磁极化子平均数减少；随着温度的增加，磁极化子平均数也增加；随着温度的增加，电子自旋作用以及声子之间相互作用都加强；随着磁场的加强，电子自旋作用增加而声子之间相互作用基本不变。     

移动通信系统中功率控制研究

在阐述功率控制在移动通信系统中的发展过程的基础上，着重研究了WCDMA系统的功率控制，最后对功率控制未来的研究方向做了简要的分析。     

Synthesis of amphiphilic magnetic microspheres by dispersion copolymerization of styrene and poly(ethylene oxide) macromonomer

Amphiphilic magnetic microspheres ranging in diameter from 5 to 100 µm were prepared by dispersion copolymerization of styrene and poly(ethylene oxide) vinylbenzyl (PEO‐VB) macromonomer (MPEO) in the presence of Fe₃O₄ magnetic fluid. The effects of various polymerization parameters on the average particle size were systematically investigated. The average particle size was found to increase with increasing styrene concentration and initiator concentration. It also increased with decreasing stabilizer concentration and molecular weight of MPEO. The content of the hydroxyl groups localized in the microspheres ranged from 0.01 to 0.2 mmol g^?1. © 2003 Society of Chemical Industry     

A comparison of transformation methods for evaluating two‐dimensional weakly singular integrals

Accurate numerical evaluation of integrals arising in the boundary element method is fundamental to achieving useful results via this solution technique. In this paper, a number of techniques are considered to evaluate the weakly singular integrals which arise in the solution of Laplace's equation in three dimensions and Poisson's equation in two dimensions. Both are two‐dimensional weakly singular integrals and are evaluated using (in a product fashion) methods which have recently been used for evaluating one‐dimensional weakly singular integrals arising in the boundary element method. The methods used are based on various polynomial transformations of conventional Gaussian quadrature points where the transformation polynomial has zero Jacobian at the singular point. Methods which split the region of integration into sub‐regions are considered as well as non‐splitting methods. In particular, the newly introduced and highly accurate generalized composite subtraction of singularity and non‐linear transformation approach (GSSNT) is applied to various two‐dimensional weakly singular integrals. A study of the different methods reveals complex relationships between transformation orders, position of the singular point, integration kernel and basis function. It is concluded that the GSSNT method gives the best overall results for the two‐dimensional weakly singular integrals studied. Copyright © 2002 John Wiley & Sons, Ltd.