Application of Lobatto polynomials in atomic physics

The approximation properties of Lobatto polynomials are analyzed and then applied to approximate the atomic Kohn–Sham eigenfunction. In the first part of this article the approximation algorithm based on the Galerkin finite element method is derived. To obtain the approximation of the function, based on the presented algorithm, the linear set of equations must be solved. The matrix of the equation set is very sparse and its elements can be evaluated analytically. In the second part of this article, the algorithm is applied to evaluate adaptive polynomial approximation of selected Kohn–Sham eigenstates of indium (In) atom. The proposed r‐adaptive algorithm evaluates the minimum number of subintervals needed to represent the eigenfunction with required accuracy. Based on the r‐adaptive algorithm, the approximations of 4d, 5s 5p In eigenfunctions were calculated. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

四喷嘴对置式气化炉停留时间分布的随机模型

运用连续时间马尔可夫链及矩阵相关理论，建立了停留时间分布随机模型。根据对四喷嘴对置式气化炉流场的测试，将气化炉划分为若干区域，并且对各个区域体积进行了估算，组成马尔可夫链状态转移图。模拟计算表明，当射流回流区和撞击流回流区回流比为1，管流区为平推流模式,其他区域按全混流模式处理时，模拟值与实验值较接近。用优化的模型计算了不同条件的冷模气化炉的停留时间分布，结果表明，随着气体流量的增大，平均停留时间减小，无因次方差增大；随着炉体高度的增加，平均停留时间增大，无因次方差减小。对工业气化炉停留时间分布进行了预测，炉内流型总体上趋近于全混流，有利于炉内的气化反应。     

Combined genetic algorithm and multiple linear regression (GA-MLR) optimizer: Application to multi-exponential fluorescence decay surface

The optimization approach based on the genetic algorithm (GA) combined with multiple linear regression (MLR) method, is discussed. The GA-MLR optimizer is designed for the nonlinear least-squares problems in which the model functions are linear combinations of nonlinear functions. GA optimizes the nonlinear parameters, and the linear parameters are calculated from MLR. GA-MLR is an intuitive optimization approach and it exploits all advantages of the genetic algorithm technique. This optimization method results from an appropriate combination of two well-known optimization methods. The MLR method is embedded in the GA optimizer and linear and nonlinear model parameters are optimized in parallel. The MLR method is the only one strictly mathematical "tool" involved in GA-MLR. The GA-MLR approach simplifies and accelerates considerably the optimization process because the linear parameters are not the fitted ones. Its properties are exemplified by the analysis of the kinetic biexponential fluorescence decay surface corresponding to a two-excited-state interconversion process. A short discussion of the variable projection (VP) algorithm, designed for the same class of the optimization problems, is presented. VP is a very advanced mathematical formalism that involves the methods of nonlinear functionals, algebra of linear projectors, and the formalism of Fréchet derivatives and pseudo-inverses. Additional explanatory comments are added on the application of recently introduced the GA-NR optimizer to simultaneous recovery of linear and weakly nonlinear parameters occurring in the same optimization problem together with nonlinear parameters. The GA-NR optimizer combines the GA method with the NR method, in which the minimum-value condition for the quadratic approximation to chi(2), obtained from the Taylor series expansion of chi(2), is recovered by means of the Newton-Raphson algorithm. The application of the GA-NR optimizer to model functions which are multi-linear combinations of nonlinear functions, is indicated. The VP algorithm does not distinguish the weakly nonlinear parameters from the nonlinear ones and it does not apply to the model functions which are multi-linear combinations of nonlinear functions.     

Nonlinear analysis of a classical system: The double‐layer capacitor

The double‐layer capacitance (C_dl) of a heterogeneous interface dominates the high frequency band in impedance spectroscopy and is commonly handled as a pure capacitor in practical voltammetries. However, the results of nonlinear electrochemical impedance spectroscopy (NLEIS) performed on a classical redox couple system show significant harmonic peaks at the high frequency band. Numerical simulations based on the Helmholtz model and the Stern model were carried out to interpret experimental data. Good agreement between the nonlinear Stern model and the experimental data was found, implying that although C_dl is dominated by the linear Helmholtz capacitance, the nonlinear responses are inevitably introduced by the Gouy–Chapman capacitance and result in significant distortions to the linear approximation.     

Rapid fermentation of beer using an immobilized yeast multistage bioreactor system

A multistage bioreactor system for rapid beer fermentation was developed. The main fermentation process, which conventionally requires 7 d, could be shortened to 2 d by this system. The concentration of esters and higher alcohols are major factors in brewery fermentation, their production being closely related to the yeast growth phase. Yeast metabolism was successfully subdivided into a growth and a restricted phase through a combination of a continuous stirredtank reactor (CSTR) and an immobilized yeast packed-bed reactor (PBR). Production of higher alcohols was high in the CSTR because of its association with the level of biosynthetic activityde novo. A small amount was also produced in the PBR, however, possibly a result of an overflow in carbohydrate metabolism. Ester formation mainly occurred in the PBR, a linear increase in the level of ester being observed with flow through the PBR. The reactor system control strategy was to maximize the level of both higher alcohol and ester formation. The CSTR/PBR control range, based on extract consumption, was varied between 1:1 and 1:2. A ratio of 1:1 tended to create a flat beer, whereas a ratio 1:2 gave a beer of richer quality. Amino acid uptake by the yeast directly contributed to a reduction in the wort pH, whereas no relation could be observed between the level of organic acid production and pH.     

Modeling the adsorption of U(VI) onto animal chitin using coupled mass transfer and surface complexation

This paper reports the development of a treatment system, using animal chitin as a passive biosorbent, for removing U(VI) from aqueous waste streams. An integral part of this system is a model that provides for the optimization of the treatment system through simulation of U(VI) removal efficiency based on the characteristics of the influent waste stream. The model accounts for changing solution matrix conditions through the coupling of surface complexation and mass transfer models. Complexation of U(VI) by chitin surface sites was modeled using FITEQL. Application of FITEQL in the “forward” mode provided the sorbed and aqueous phase concentrations needed for the mass transfer model. The mass transfer model was derived for both batch and continuously stirred tank reactor (CSTR) configurations using Fick's Law, reactor mass balances and rate law expressions. The coupled model was successfully validated using CSTR data at pH 6.5 and rate constants determined from batch sorption experiments. The CSTR configuration yields a steady-state, eighty percent U(VI) removal for 1 μM influent U(VI) with a solution-phase pH of 6.5 and 3.9 g l⁻¹ chitin.     

Design of Nonlinear Model‐Based Control Using Bifurcation Analysis for Solution Polymerizations Carried Out in Lumped‐Distributed Reactors

In order to implement nonlinear control, nonlinear system identification must be performed, however, there are open questions concerning this field of process control, for example, experimental planning, model structure selection, parameter estimation, and validation. Therefore, the study of nonlinear model identification is a relevant unsolved problem that needs to be handled for nonlinear control synthesis. This paper presents the use of bifurcation theory, dynamic and stability analysis for nonlinear identification, and control of polymerization reactors. Peroxide‐initiated styrene‐solution polymerization reactors (lumped‐distributed) are investigated: batch, continuous stirred‐tank reactor (CSTR), and tubular reactors. Open and closed loop analyses are carried out using jacket temperature and weight average molecular weight setpoints as the bifurcation parameters. Phenomenological mathematical models, neural network nonlinear models, and an experimental data from a polymerization unit are employed for validating the proposed methodology in order to implement confident nonlinear controllers.     

使用微观动力学模型优化浆态床合成二甲醚

以合成气为原料，采用Cu-Zn-A l和γ-Al2O3构成的双功能催化剂，对使用浆态床一步法合成二甲醚的系统进行模拟研究。液固两相为全混流，气相为活塞流，组分方程使用四阶精度的Runge-Kutta法并结合C++编程进行求解，通过模拟计算，讨论了压力、温度、催化剂浓度对反应转化率、收率以及DME的选择性的影响，从而寻找合适的反应参数，为工业反应器放大设计和优化操作提供依据。     

Comparative analysis of linear and nonlinear models for ion transport problem in chronoamperometry

Ion transport problem related to controlled potential experiments in electrochemistry is studied. The problem is assumed to be superposition of diffusion and migration under the influence of an electric field. The comparative analysis are presented for three well-known models—pure diffusive (Cottrell’s), linear diffusion-migration, and nonlinear diffusion-migration (Cohn’s) models. The nonlinear model is derived by the identification problem for a nonlinear parabolic equation with nonlocal additional condition. This problem reduced to an initial-boundary value problem for nonlinear parabolic equation. The nonlinear finite difference approximation of this problem, with an appropriate iteration algorithm is derived. The comparative numerical analysis for all three models shows an influence of the nonlinear migration term, the valences of oxidized and reduced oxidized species, also diffusivity to the value of the total charge. The obtained results permits one to estimate bounds of linear and nonlinear ion transport models.     

Adaptive approach for nonlinear sensitivity analysis of reaction kinetics

We present a unified approach for linear and nonlinear sensitivity analysis for models of reaction kinetics that are stated in terms of systems of ordinary differential equations (ODEs). The approach is based on the reformulation of the ODE problem as a density transport problem described by a Fokker-Planck equation. The resulting multidimensional partial differential equation is herein solved by extending the TRAIL algorithm originally introduced by Horenko and Weiser in the context of molecular dynamics (J. Comp. Chem. 2003, 24, 1921) and discussed it in comparison with Monte Carlo techniques. The extended TRAIL approach is fully adaptive and easily allows to study the influence of nonlinear dynamical effects. We illustrate the scheme in application to an enzyme-substrate model problem for sensitivity analysis w.r.t. to initial concentrations and parameter values.     

Investigating the Time Dependent Behavior of Thermoplastic Polymers under Tensile Load

The relationship between the results of the tensile and the stress relaxation tests of polypropylene specimens were analyzed and an attempt was made to find a way to estimate the former from the latter based on the measurements and the theory of linear viscoelasticity. The mechanical response of real polymers are basically of nonlinear character, therefore their behavior patterns do not meet the idealized (linear) ones. Experiments were performed on poly(propylene) (PP) as a test material and the stress relaxation behavior, as well as the linear elastic and linear viscoelastic approximation of the tensile load-time curve were analyzed. To demonstrate the applicability of our idea and to perform the numerical calculations we have chosen a flexible function with three parameters to realize the nonlinear behavior.     

Hybrid neural network model of an industrial ethanol fermentation process considering the effect of temperature

In this work a procedure for the development of a robust mathematical model for an industrial alcoholic fermentation process was evaluated. The proposed model is a hybrid neural model, which combines mass and energy balance equations with functional link networks to describe the kinetics. These networks have been shown to have a good nonlinear approximation capability, although the estimation of its weights is linear. The proposed model considers the effect of temperature on the kinetics and has the neural network weights reestimated always so that a change in operational conditions occurs. This allow to follow the system behavior when changes in operating conditions occur.     

高碘酸盐-硫脲-硫酸反应体系的非线性动力学

研究了KIO4-SC(NH2)2-H2SO4反应体系在封闭、半封闭以及开放系统中的非线性动力学行为.发现在封闭体系中体系的吸光度、铂电极电位和pH值呈现单峰或准振荡现象;在半封闭和开放系统中体系的铂电极电位和碘电极电位均呈振荡现象.封闭、半封闭及开放系统的动力学曲线受体系酸度和初始浓度比值犤KIO4犦0/犤SC(NH2)2犦0的影响.对照其在封闭、半封闭和开放系统中的动力学行为,以碘单质产生和消耗驱使的反应动力学可解释该反应体系的复杂现象.     

Adaptive multilevel finite element solution of the Poisson–Boltzmann equation I. Algorithms and examples

This article is the first of two articles on the adaptive multilevel finite element treatment of the nonlinear Poisson–Boltzmann equation (PBE), a nonlinear eliptic equation arising in biomolecular modeling. Fast and accurate numerical solution of the PBE is usually difficult to accomplish, due to the presence of discontinuous coefficients, delta functions, three spatial dimensions, unbounded domain, and rapid (exponential) nonlinearity. In this first article, we explain how adaptive multilevel finite element methods can be used to obtain extremely accurate solutions to the PBE with very modest computational resources, and we present some illustrative examples using two well‐known test problems. The PBE is first discretized with piece‐wise linear finite elements over a very coarse simplex triangulation of the domain. The resulting nonlinear algebraic equations are solved with global inexact Newton methods, which we have described in an article appearing previously in this journal. A posteriori error estimates are then computed from this discrete solution, which then drives a simplex subdivision algorithm for performing adaptive mesh refinement. The discretize–solve–estimate–refine procedure is then repeated, until a nearly uniform solution quality is obtained. The sequence of unstructured meshes is used to apply multilevel methods in conjunction with global inexact Newton methods, so that the cost of solving the nonlinear algebraic equations at each step approaches optimal O(N) linear complexity. All of the numerical procedures are implemented in MANIFOLD CODE (MC), a computer program designed and built by the first author over several years at Caltech and UC San Diego. MC is designed to solve a very general class of nonlinear elliptic equations on complicated domains in two and three dimensions. We describe some of the key features of MC, and give a detailed analysis of its performance for two model PBE problems, with comparisons to the alternative methods. It is shown that the best available uniform mesh‐based finite difference or box‐method algorithms, including multilevel methods, require substantially more time to reach a target PBE solution accuracy than the adaptive multilevel methods in MC. In the second article, we develop an error estimator based on geometric solvent accessibility, and present a series of detailed numerical experiments for several complex biomolecules. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1319–1342, 2000     

pH探针在亚氯酸盐-硫代硫酸钠非线性反应体系研究 中的应用

当pH探针应用于ClO2^--S2O3^2-非线性反应体系动力学研究时,发现了封闭体系的pH准振荡和CSTR中的pH稳态振荡现象,在CSTR中振荡的pH与铂电位同相位,且与闭系中小峰性质相同,反应机理包括氯氧化合物自身反应,氯氧化合物-硫化合物反应,硫-硫反应。以硫价态变化驱使的动力学模型模拟出闭系准振荡和CSTR中pH稳态振荡。该模型也为解释反应-扩散体系中的pH前沿波提供模型机理。