Train Students to Solve the Practical Problem by Building Mathematics Models

It focuses on that students must be developed the ability to solve the practical problem by building the mathematics models and the ability to combine the theory with the practice. It also states that students must be improved the learning interests and practical experience.

Mathematical Models of Tire-Lateral Road Adhesion for Use in Road Vehicle Dynamics Studies

Mathematical models of tire-lateral mad adhesion for use in mad vehicle dynamics studies are set up to express the relations of adhesion coefficients with slip ratio in lateral direction.The models of tire-lateral mad adhesion revolutionize the Pacejka's model in concept and therefore make it possible for applications in vehicle dynamics studies by the expression of lateral adhesion coefficient as a function of wheel slip ratio,instead of the wheel slip angle,taking into account in the mean time the influences of mad surface condition, vehicle velocity,vertical load,tire slip angle,and wheel camber angle.

Flow field simulation and establishment for mathematical models of flow area of spool valve with sloping U-shape notch machined by different methods

Precise function expression of the flow area for the sloping U-shape notch orifice versus the spool stroke was derived. The computational fluid dynamics was used to analyze the flow features of the sloping U-shape notch on the spool, such as mass flow rates, flow coefficients, effiux angles and steady state flow forces under different operating conditions. At last, the reliability of the mathematical model of the flow area for the sloping U-shape notch orifice on the spool was demonstrated by the comparison between the orifice area curve derived and the corresponding experimental data provided by the test. It is presented that the bottom arc of sloping U-shape notch （ABU） should not be omitted when it is required to accurately calculate the orifice area of ABU. Although the theoretical flow area of plain bottom sloping U-shape notch （PBU） is larger than that of ABU at the same opening, the simulated mass flow and experimental flow area of ABU are both larger than these of PBU at the same opening, while the simulated flow force of PBU is larger than that of ABU at the same opening. Therefore, it should be prior to adapt the ABU when designing the spool with proportional character.

Study on Strength and Life of Anisotropic Single Crystal Blade - Part I: Crystallographic Constitutive Models and Applications

The single crystal blade is one of the key technologies for improving the performance, durability and reliability of aero-engines and ground gas-turbine engines. However, the anisotropic mechanical properties of the single crystal material makes a great deal of difficulties on the development and the application of the single crystal blade, which is a challenge for the engineering application of the single crystal superalloy and the theoretic bases of the application. Some researches on the strength analysis and the life prediction of the anisotropic single crystal blade were carried out by the authors' research team. They are as follows. The crystallographic constitutive models for the plastic and the creep behaviors and the method of the rupture life prediction were established and verified. The tensile or the creep experiments for DD3 single crystal alloy with different orientations under different temperatures and different tensile rates or under different temperatures and different stress levels were carried out. The experimental data and the anisotropic properties at intermediate and high temperatures revealed by the experiments are significant for the application of the single crystal alloy. In addition, the experimental research for a kind of single crystal blade was also made. As the application of the researches the strength analysis and the life prediction were carried out for the single crystal blade of a certain aero-engine. In this part, the constitutive models and their applications are described, and the experimental research work will be described in part II.

Mathematical models and expert system for grate-kiln process of iron ore oxide pellet production.Part Ⅱ:Rotary kiln process control

Rotary kiln process for iron ore oxide pellet production is hard to detect and control.Construction of one-dimensional model of temperature field in rotary kiln was described.And the results lay a solid foundation for online control.Establishment of kiln process control expert system was presented,with maximum temperature of pellet and gas temperature at the feed end as control cores,and interval estimate as control strategy.Software was developed and put into application in a pellet plant.The results show that control guidance of this system is accurate and effective.After production application for nearly one year,the compressive strength and first grade rate of pellet are increased by 86 N and 2.54%,respectively,while FeO content is 0.05% lowered.This system can reveal detailed information of real time kiln process,and provide a powerful tool for online control of pellet production.

Mathematical models and expert system for grate-kiln process of iron ore oxide pellet production(Part Ⅰ):Mathematical models of grate process

Grate process is an important step in grate-kiln pellet production.However,as a relatively closed system,the process on grate is inaccessible to direct detection,therefore,it is hard to control.As a result,mathematical models of temperature distribution,moisture distribution and oxidation degree distribution in pellet bed,with good universality,computation speed and calculation accuracy,are presented based on analysis of heat transfer and physical-chemical reactions during grate process.And real-time visualization of temperature,moisture and oxidation degree distribution in pellet bed during grate process is realized.Model validation is displayed,and the similarity of 91% is proved.The results can reveal real time status on grate,and provide a solid foundation for the subsequent study of artificial intelligence control system of pellet production.

On models for landscape connectivity: a case study of the new-born wetland of the Yellow River Delta

The models for landscape connectivity are distinguished into models for line connectivity, vertex connectivity, network connectivity and patch connectivity separately. Because the models for line connectivity, for vertex connectivity, and for network connectivity have long been studied and have become ripe, the model for patch connectivity is paid special attention in this paper. The patch connectivity is defined as the average movement efficiency (minimizing movement distance) of animal migrants or plant propagules in patches of a region under consideration. According to this definition, a model for landscape connectivity is mathematically deduced to apply to GIS data. The application of model for patch connectivity in the new-born wetland of the Yellow River Delta shows patch connectivity has a negative interrelation with human impact intensity and landscape diversity.

Mathematical models for foam-diverted acidizing and their applications

Foam diversion can effectively solve the problem of uneven distribution of acid in layers of different permeabilities during matrix acidizing. Based on gas trapping theory and the mass conservation equation, mathematical models were developed for foam-diverted acidizing, which can be achieved by a foam slug followed by acid injection or by continuous injection of foamed acid. The design method for foam-diverted acidizing was also given. The mathematical models were solved by a computer program. Computed results show that the total formation skin factor, wellhead pressure and bottomhole pressure increase with foam injection, but decrease with acid injection. Volume flow rate in a highpermeability layer decreases, while that in a low-permeability layer increases, thus diverting acid to the low-permeability layer from the high-permeability layer. Under the same formation conditions, for foamed acid treatment the operation was longer, and wellhead and bottomhole pressures are higher. Field application shows that foam slug can effectively block high permeability layers, and improve intake profile noticeably.

Progress in Pressure Swing Adsorption Models During the Recent 30 Years

The pressure swing adsorption (PSA) models discussed here are divided into three categories: partialdifferential equation model, electrical analogue model and neural network model. The partial differential equationmodel, including equilibrium and kinetic models, has provided an elementary viewpoint for PSA processes. Usingthe simplest equilibrium models, some influential factors, such as pressurization with product, incomplete purge,beds with dead volume and heat effects, are discussed respectively. With several approximate assumptions i.e.,concentration profile in adsorbent, 'frozen' column, symmetry and heat effects of bed wall, the more complexkinetic models can be simplified to a certain degree at the expense of a limited application. It has also been foundthat the electrical analogue model has great flexibility to handle more realistic PSA processes without any additionalhypothesis.

Mathematical models for properties of mortars with admixtures and recycled fine aggregates from demolished concretes

In order to expand the engineering application of recycle aggregate mortars （RAM） with aggregates from demolished concretes, the models for the properties of RAM and the replacement rate of these recycled fine aggregates were proposed. First, different kinds of mathematical models for the basic properties （compressive strength, water retention rate, and consistency loss） of RAM with two kinds of admixtures, thickening powders （TP） and self-made powdery admixtures （SSCT） designed for RAM, and the replacement rates were established, while the average relative errors and relative standard errors of these models were calculated. Additionally, the models and their error analyses for the curves of drying shrinkage and curing time of RAM ＋ SSCT at different replacement rates were put forward. The results show that polynomial functions should be used to calculate the basic properties of RAM ＋ TP and RAM ＋ SSCT at different replacement rates. In addition, polynonfial functions are the most optimal models for the sharp shrinkage sections in the curves of drying shrinkage-curing time of RAM ＋ SSCT, while exponential functions should be used as the models for the slow shrinkage sections and steady shrinkage sections.

Foundation of three—dimensional mathematical models for glass furnace regenerator

This paper presents a practical three dimensional mathematical model of circulation and heat transfer in generator of glass melting furnaces. The model was based on the heat transfer between the smoke flow and the lattice units, and between the air flow and the lattice units. This model not only bypassed the difficulty of complicated computation of the heat transfer process in the regenerator of glass furnaces, but also avoided the irrationality of fixing the temperature distribution on the surfaces. Use of the model yielded very important data and also the method for the design of the regenerator of glass furnaces in practical production.

Research on viral dynamic models of hepatitis B virus infection

A mathematical model with cytotoxic cells of hepatitis B virus (HBV)infection is set up based on a basic model of virus dynamics without cytotoxic cells andexperimental observation of anti-viral drag therapy for HBV infection patients. A quantitativeanalysis of dynamic behaviors shows that the model has three kinds of equilibrium points, whichrepresent the patient's complete recovery without immune ability, complete recovery with immuneability, and HBV persistent infection at the end of the treatment with drag lamivudine,respectively. Our model may provide possible quantitative interpretations for the treatments ofchronic HBV infections with the drag lamivudine, in particularly explain why the plasma virus ofNowak et al. 's patients turnover the original level after stopping the lamivudine treatment.

Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler’s equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic assumptions can be derived using the conservation and balance laws of classical or non-classical continuum mechanics. The mathematical models based on Kirchhoff hypothesis (classical plate theory, CPT) and first order shear deformation theory (FSDT) that are representative of most mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models.

Consistency and Validity of the Mathematical Models and the Solution Methods for BVPs and IVPs Based on Energy Methods and Principle of Virtual Work for Homogeneous Isotropic and Non-Homogeneous Non-Isotropic Solid Continua

Energy methods and the principle of virtual work are commonly used for obtaining solutions of boundary value problems (BVPs) and initial value problems (IVPs) associated with homogeneous, isotropic and non-homogeneous, non-isotropic matter without using (or in the absence of) the mathematical models of the BVPs and the IVPs. These methods are also used for deriving mathematical models for BVPs and IVPs associated with isotropic, homogeneous as well as non-homogeneous, non-isotropic continuous matter. In energy methods when applied to IVPs, one constructs energy functional (I) consisting of kinetic energy, strain energy and the potential energy of loads. The first variation of this energy functional (δI) set to zero is a necessary condition for an extremum of I. In this approach one could use δI = 0 directly in constructing computational processes such as the finite element method or could derive Euler’s equations (differential or partial differential equations) from δI = 0, which is also satisfied by a solution obtained from δI = 0. The Euler’s equations obtained from δI = 0 indeed are the mathematical model associated with the energy functional I. In case of BVPs we follow the same approach except in this case, the energy functional I consists of strain energy and the potential energy of loads. In using the principle of virtual work for BVPs and the IVPs, we can also accomplish the same as described above using energy methods. In this paper we investigate consistency and validity of the mathematical models for isotropic, homogeneous and non-isotropic, non-homogeneous continuous matter for BVPs that are derived using energy functional consisting of strain energy and the potential energy of loads. Similar investigation is also presented for IVPs using energy functional consisting of kinetic energy, strain energy and the potential energy of loads. The computational approaches for BVPs and the IVPs designed using energy functional and principle of virtual work, their consistency and validity are also investigated. Classical continuum mechanics (CCM) principles i.e. conservation and balance laws of CCM with consistent constitutive theories and the elements of calculus of variations are employed in the investigations presented in this paper.

Mathematical Models for Vacuum Drying Characteristics of Pomegranate Aril

The aim of this study was to investigate and determine the best mathematical models in describing vacuum drying characteristic of pomegranate arils in the range of 55-75 ℃. The vacuum batch dryer used in the evaluation was successful in drying a thin layer of pomegranate arils from the initial moisture content of 464.02% （d.b.） to 6.18% （d.b.） within 6.5 to 13.5 hr of continuous drying at the above mentioned temperature range. The drying rates increased with an increase in temperature and drying time. Five of the well known semi-theoretical and empirical models were fitted to the vacuum drying of pomegranate arils. The semi-empirical Midilli model has shown an excellent fit to predict drying behavior of the pomegranate arils because this model gave the highest coefficient of determination （RE）, the least chi-square （X2）, and the lowest root mean square error （RMSE）. The total drying occurs during falling period, signifying the influence of moisture diffusion during the drying. The effective diffusivity varied from 1.25× 10^10 to 2.91 × 10^10 m^2/s over the temperature range. Temperature dependence of the diffusivity was well documented by Arrhenius models. The activation energy of moisture diffusion during drying was found to be 40.46 kJ/mol.

Geographic Information Systems Used as a Practical Way of Teaching Mathematics

This paper presents reflections on proposals to merge Mathematics and Geography when teaching undergraduates from inter- and trans-disciplinary spatial models using mathematical modeling to enable spatial analysis and visualization through GIS—Geographic Information Systems. It brings forward an approach to Geography with new technologies integrating it with Statistics and Computer Science among other subjects. Thus, it describes some practical cases that are examples of geographical problems solved mathematically. The results show that GIS can be an educational tool for teaching both Geography and Mathematics.

Mathematical Models for a Social Partitioning Problem

In this paper we develop modeling techniques for a social partitioning problem. Different social interaction regulations are imposed during pandemics to prevent the spread of diseases. We suggest partitioning a set of company employees as an effective way to curb the spread, and use integer programming techniques to model it. The goal of the model is to maximize the number of direct interactions between employees who are essential for company’s work subject to the constraint that all employees should be partitioned into components of no more than a certain size implied by the regulations. Then we further develop the basic model to take into account different restrictions and provisions. We also give heuristics for solving the problem. Our computational results include sensitivity analysis on some of the models and analysis of the heuristic performance.

The calculation and optimal allocation of transmission capacity in natural gas networks with MINLP models

The transmission capacity of gas pipeline networks should be calculated and allocated to deal with the capacity booking with shippers. Technical capacities, which depend on the gas flow distribution at routes or interchange points, are calculated with a multiobjective optimization model and form a Pareto solution set in the entry/exit or point-to-point regime. Then, the commercial capacities, which can be directly applied in capacity booking, are calculated with single-objective optimization models that are transformed from the above multiobjective model based on three allocation rules and the demand of shippers.Next, peak-shaving capacities, which are daily oversupply or overdelivery amounts at inlets or deliveries,are calculated with two-stage transient optimization models. Considering the hydraulic process of a pipeline network and operating schemes of compressor stations, all the above models are mixed-integer nonlinear programming problems. Finally, a case study is made to demonstrate the ability of the models.

Some Equivalence Transform Methods for Simulating Models in Control Systems

In this paper two classes of equivalence transform methods for solving ordinary differential equations are proposed. One class of method is the equivalence integral transform method for special differential algebraic problems. The advantage of this class of method is such that the amount of work calculating one integration with parameters becomes that of two interpolations, when the system of nonlinear equations is solved on the right hand side function. The other class of method is the equivalence substitution method for avoiding calculating derivative on the right hand side function. In order to avoid calculation derivatives, two equivalence substitution methods are proposed here. The application instances of some special effect of the equivalence substitution methods are given.

Decoding of Factorial Experimental Design Models Implemented in Production Process

The paper deals with factorial experimental design models decoding.For the ease of calculation of the experimental mathematical models,it is convenient first to code the independent variables.When selecting independent variables,it is necessary to take into account the range covered by each.A wide range of choices of different variables is presented in this paper.After calculating the regression model,its variables must be returned to their original values for the model to be easy recognized and represented.In the paper,the procedures of simple first order models,with interactions and with second order models,are presented,which could be a very complicated process.Models without and with the mutual influence of independent variables differ.The encoding and decoding procedure on a model with two independent first-order parameters is presented in details.Also,the procedure of model decoding is presented in the experimental surface roughness parameters models’determination,in the face milling machining process,using the first and second order model central compositional experimental design.The simple calculation procedure is recommended in the case study.Also,a large number of examples using mathematical models obtained on the basis of the presented methodology are presented throughout the paper.