Introducing Modeling Transition Diagrams as a Tool to Connect Mathematical Modeling to Mathematical Thinking

This study contributes a methodological tool to reconstruct the cognitive processes and mathematical activities carried out by mathematical modelers. Represented as Modeling Transition Diagrams (MTDs), individual modeling routes were constructed for four engineering undergraduate students. Findings stress the importance and limitations of using micro-analysis to examine modeling processes. The findings and the MTDs were used to critically question the implications of using modeling cycles as a theory of mathematical modeling processes.     

Modeling of flow separation of assist gas as applied to laser cutting of thick sheet metal

The present paper describes the results of mathematical modeling of supersonic flows of a viscous compressible gas, obtained by numerically solving three-dimensional full Navier–Stokes equations, and also the results of experiments with visualization of gas jet flows in channels geometrically similar to the laser cut. Separation of the gas flow from the cut front is predicted numerically and then validated by experiments on a model setup. The gas flow structure arising in a narrow channel behind a sonic (conical) or supersonic nozzle is described. Specific features of originating in the flow separation on a smooth surface in a narrow channel are examined, and mechanisms controlling the separation are proposed. Flow separation directly affects the changes in the shape and structure of striations and is the one of main reason for the worse quality of the laser cut surface. It is shown that the changes in the structures of striations over the thickness of the sheet being cut are closely related to aerodynamic features of jet flows of the assisting gas in the cut channel.     

Modeling Complex Points up to Isotopy

In this paper, we examine the structure of complex points of real 4-manifolds embedded into complex 3-manifolds up to isotopy. We show that there are only two types of complex points up to isotopy and, as a consequence, show that any such embedding can be deformed by isotopy to a manifold having a 2-complete neighborhood basis.     

Modeling of strip mine excavations as dislocations in elastic rock

We propose a new model of excavation in rock based on Volterra dislocations. We develop a software package for determining the stressdeformed state in modeling excavations as dislocations, on the basis of which we study the qualitative properties of the vertical stress field of a solid problem of elasticity theory. Four figures.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 85–91.     

Modeling the hydrogen material interaction as parametric instability of the continuum

The hydrogen is contained in any material. Its concentration inside the materials leads to mechanical properties degradation. The two-continuum model of solid allows one to describe the influence of small concentration of hydrogen on the mechanical properties of materials in terms of changing the bonding energy of the second continuum, the latter being responsible for the hydrogen concentration. The application of this model to fatigue task give the hydrogen concentration that are critical for material destruction. Such fatigue destruction has a nature of parametric instability during the cyclic redistribution of the hydrogen under the cyclic loading. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling Stochastic Dominance as Infinite-Dimensional Constraint Systems via the Strassen Theorem

We use the Strassen theorem to solve stochastic optimization problems with stochastic dominance constraints. First, we show that a dominance-constrained problem on general probability spaces can be expressed as an infinite-dimensional optimization problem with a convenient representation of the dominance constraints provided by the Strassen theorem. This result generalizes earlier work which was limited to finite probability spaces. Second, we derive optimality conditions and a duality theory to gain insight into this optimization problem. Finally, we present a computational scheme for constructing finite approximations along with a convergence rate analysis on the approximation quality.     

An Approach to Modeling Artificial Gene Networks

We propose a new mathematical model of a repressilator, i.e., the simplest gene ring network consisting of three elements. The studied model is a three-dimensional system of ordinary differential equations depending on a single parameter. We study the existence and stability problems for relaxation periodic motion in this system.     

Modeling sourcing strategies to mitigate part obsolescence

Part obsolescence is a common problem across industries, from avionics and military sectors to most original equipment manufacturers serving industrial markets. When a part supplier announces that a part will become obsolete, the OEM can choose from a number of sourcing options. In practice, the three most commonly adopted mitigation strategies are: (1) a lifetime, or life-of-type (LOT), buy from the original supplier; (2) part substitution, which finds a suitable alternative; and (3) line redesign, which modifies the production line to accommodate a new part. We first develop a framework incorporating fixed cost, variable cost, leadtime, demand uncertainty and the discount rate to directly compare and characterize these three sourcing strategies in a static context. We next formulate an integrated sourcing approach that starts with a bridge buy and may continue with part substitution or line redesign when the originals parts are depleted. Through numerical studies, we identify the joint impact of the problem parameters on the static and integrated sourcing strategies and the optimal choice among them. While the integrated sourcing approach outperforms the static ones in many cases it is not a dominant strategy.     

基于Riemann-Hilbert问题建模求解孤立子解

基于矩阵谱问题构造了一种实用的方法来对一类实轴上的可积方程的Riemann-Hilbert问题进行建模。当跳跃矩阵是单位矩阵时,孤立子解通过特殊约化的Riemann-Hilbert问题显性表示。作为一个范例,对于具有任意阶矩阵谱问题的多分量非线性薛定谔方程,给出了该方法的具体应用。     

Phase Field Approach to Multiphase Flow Modeling

We review the phase field (otherwise called diffuse interface) model for fluid flows, where all quantities, such as density and composition, are assumed to vary continuously in space. This approach is the natural extension of van der Waals?? theory of critical phenomena both for one-component, two-phase fluids and for partially miscible liquid mixtures. The equations of motion are derived, assuming a simple expression for the pairwise interaction potential. In particular, we see that a non-equilibrium, reversible body force appears in the Navier-Stokes equation, that is proportional to the gradient of the generalized chemical potential. This, so called Korteweg, force is responsible for the convection that is observed in otherwise quiescent systems during phase change. In addition, in binary mixtures, the diffusive flux is modeled using a Cahn-Hilliard constitutive law with a composition-dependent diffusivity, showing that it reduces to Fick??s law in the dilute limit case. Finally, the results of several numerical simulations are described, modeling, in particular, a) mixing, b) spinodal decomposition, c) nucleation, d) enhanced heat transport, e) liquid-vapor phase separation.     

The Behavioral Approach to Systems and Modeling

Passive and active limited-slip differentials are used in high-performance cars to optimize the torque distribution on the driving wheels for traction maximization, driving comfort, stability and active safety of the vehicle. In this paper, detailed and reduced dynamic models for the simulation of four kinds of differential are presented. The models refer to the limited-slip steering differential with two clutches. The model of the conventional differential, of the mechanical limited-slip differential and of the controlled limited-slip differential can be obtained by simplification. The detailed model allows the simulation of the internal phenomena that influence the differential dynamics. The reduced model focuses only on the main dynamic behaviour of the differential. Some simulations show the use of the reduced model to compare the effects of the four differentials on the vehicle dynamics.     

计及多重阻塞的电动汽车移动储能特性建模

为实现城市交通电力耦合系统在城市道路、充电设施、输电线路阻塞环境下的优化运行,提出了计及多重阻塞的动态交通电力流联合优化方法。首先,基于时空网络模型,提出了计及电动汽车移动、静止、充电、排队模式的队列时空网络模型,构建了适用于电动汽车的车辆调度模型,进而形成动态交通分配模型,以减少交通出行损失。其次,通过优化发电机组、储能等的出力和备用计划,计及城市电网安全、备用约束,构建了安全约束动态经济调度模型,以降低碳排放及发电成本。随后,形成多目标动态优化模型,并将其转换为混合整数凸二次规划问题。最后,在耦合IEEE-30、Sioux Falls系统中验证了所提模型的有效性。     

Modeling phase diagrams as stochastic processes with application in vehicular traffic flow

Motivated by the similarity between the fundamental diagram of vehicular traffic and the Maxwell–Boltzmann distribution of ideal gases, this paper proposed a methodology to model the fundamental diagram as a stochastic process which also applies to other real-world systems with similar nature. A concrete example is provided to illustrate the application of the methodology where the fundamental diagram of vehicular traffic is modeled as a stochastic process to capture the scattering effect in flow–density relationship. A verification study was conducted on the model using empirical data and the statistical analysis shows that the overall quality of the fitted stochastic process is acceptable. Related existing efforts are referenced to the proposed stochastic fundamental diagram where their similarities and differences are elaborated. Further discussion is carried out on the significance of the stochastic fundamental diagram as well as the proposed methodology with an additional real-world example to illustrate its applications.     

Modeling and mitigating supply chain disruptions as a bilevel network flow problem

Years of globalization, outsourcing and cost cutting have increased supply chain vulnerability calling for more effective risk mitigation strategies. In our research, we analyze supply chain disruptions in a production setting. Using a bilevel optimization framework, we minimize the total production cost for a manufacturer interested in finding optimal disruption mitigation strategies. The problem constitutes a convex network flow program under a chance constraint bounding the manufacturer’s regrets in disrupted scenarios. Thus, in contrast to standard bilevel optimization schemes with two decision-makers, a leader and a follower, our model searches for the optimal production plan of a manufacturer in view of a reduction in the sequence of his own scenario-specific regrets. Defined as the difference in costs of a reactive plan, which considers the disruption as unknown until it occurs, and a benchmark anticipative plan, which predicts the disruption in the beginning of the planning horizon, the regrets allow measurement of the impact of scenario-specific production strategies on the manufacturer’s total cost. For an efficient solution of the problem, we employ generalized Benders decomposition and develop customized feasibility cuts. In the managerial section, we discuss the implications for the risk-adjusted production and observe that the regrets of long disruptions are reduced in our mitigation strategy at the cost of shorter disruptions, whose regrets typically stay far below the risk threshold. This allows a decrease of the production cost under rare but high-impact disruption scenarios.

     

Modeling myopia: Application to non-renewable resource extraction

We develop a parsimonious model of myopia with an infinitesimal period of commitment as an extension to a standard dynamic optimization in a continuous-time environment. We clearly distinguish the processes of planning future controls and choosing the current control, which makes the model both analytically and numerically convenient. In its application to a simple non-renewable resource extraction problem, we show that whether the terminal time is free or fixed determines the appropriateness of the approximation to myopic agents by constant discounting. We also show that the expiry of extraction permits may be useful in the presence of myopia.     

第十二届国际数学教育大会上与数学建模相关的活动

介绍了第十二届国际数学教育大会的基本情况,特别是会议上与数学建模相关的一些活动。