Showing papers on "Mathematical model published in 1999"

The nature of mathematical modeling

Abstract: Preface 1. Introduction Part I. Analytical Models: 2. Ordinary differential and difference equations 3. Partial differential equations 4. Variational principles 5. Random systems Part II. Numerical Models: 6. Finite differences: ordinary difference equations 7. Finite differences: partial differential equations 8. Finite elements 9. Cellular automata and lattice gases Part III. Observational Models: 10. Function fitting 11. Transforms 12. Architectures 13. Optimization and search 14. Clustering and density estimation 15. Filtering and state estimation 16. Linear and nonlinear time series Appendix 1. Graphical and mathematical software Appendix 2. Network programming Appendix 3. Benchmarking Appendix 4. Problem solutions Bibliography.

A mathematical theory of traffic hysteresis

Abstract: This paper presents a mathematical theory for modeling the hysteresis phenomenon observed in traffic flow. It proposes that acceleration, deceleration and equilibrium flow should be distinguished in obtaining speed-concentration and/or occupancy relationships, such that the phase transitions from one phase to another can be correctly identified. The analysis shows that the speed-concentration curves obtained following this approach are hysteresis loops, as predicted by the theory. The paper also gives a discussion of the general properties of the proposed modeling equations and examines the relationship between traffic hysteresis and stop-start waves observed in traffic flow.

Quasi-2d model for unsteady flow in pipe networks

Abstract: A quasi-two-dimensional model for unsteady-flow analysis in pipes and pipe networks is presented. The turbulence model is based on the mixing length hypothesis in the turbulent zone and on Newton’s law in the viscous sublayer. An expression of the mixing length in terms of the Reynolds number and an expression of the parameter of logarithmic law of the wall in terms of the friction Reynolds number are found from Nikuradse’s experimental data. An implicit numerical scheme for the integration of the equations is proposed to overcome the limitations of the explicit schemes. Uniqueness of the head and continuity of discharge are considered at the junctions. The results of both a quasi-steady 1D model and a quasi-2D model are compared with results from a laboratory network. For these experimental runs, the comparisons show that the average relative errors on the maximum head oscillations are 19.1% with the 1D model and 8.6% with the quasi-2D model; those on the minimum oscillations are 19.2% with the 1D model and 5.3% with the quasi-2D model. The latter model is in better agreement because it takes into account the velocity profile, thus allowing for a more accurate evaluation of the shear stress.

Optimal Use of Viscoelastic Dampers in Building Frames for Seismic Force

Abstract: Control of the seismic response of multistory building frames using optimally placed viscoelastic dampers (VEDs) is investigated. Responses are obtained in frequency domain using spectral analysis for narrow and broad band stationary random ground motions. Optimal locations of passive VEDs are found with the help of a controllability index, which is obtained with the help of the root-mean-square value of the interstory drift. To highlight the effect of modeling and optimal location of VEDs on the response reduction, three different mathematical models of the VEDs and three alternative schemes of placement of VEDs are considered. The response of the 20-story shear-frame model is controlled by the proposed strategy. It is shown that the scheme of the optimal placement of VEDs provides more reduction in response compared with other schemes of placement of VEDs considered in the study. Furthermore, the optimal placement of dampers is sensitive to the nature of excitation force, total quantity of viscoelastic ...

Development and comparison of models for light-pulse transport through scattering–absorbing media

Abstract: We examine the transport of short light pulses through scattering–absorbing media through different approximate mathematical models. It is demonstrated that the predicted optical signal characteristics are significantly influenced by the various models considered, such as PN expansion, two-flux, and discrete ordinates. The effective propagation speed of the scattered radiation, the predicted magnitudes of the transmitted and backscattered fluxes, and the temporal shape and spread of the optical signals are functions of the models used to represent the intensity distributions. A computationally intensive direct numerical integration scheme that does not utilize approximations is also implemented for comparison. Results of some of the models asymptotically approach those of direct numerical simulation if the order of approximation is increased. In this study therefore we identify the importance of model selection in analyzing short-pulse laser applications such as optical tomography and remote sensing and highlight the parameters, such as wave speed, that must be examined before a model is adopted for analysis.

Super–sensitivity to structure in biological models

Abstract: Applied scientific disciplines use mathematical models to make predictions. In the majority of cases these models are constructed using plausible mathematical characterizations of various component processes of the modelled system, rather than being based entirely on exact mathematical descriptions of proven mechanisms. We use general arguments and a specific example from applied ecology to demonstrate that model predictions can show alarming sensitivity to apparently tiny changes in model specification, in a manner that is counterintuitive and entirely invisible to conventional model sensitivity analysis. This result has serious implications for practical prediction using biological models.

Statistical Validation of Engineering and Scientific Models: Background

Abstract: A tutorial is presented discussing the basic issues associated with propagation of uncertainty analysis and statistical validation of engineering and scientific models. The propagation of uncertainty tutorial illustrates the use of the sensitivity method and the Monte Carlo method to evaluate the uncertainty in predictions for linear and nonlinear models. Four example applications are presented; a linear model, a model for the behavior of a damped spring-mass system, a transient thermal conduction model, and a nonlinear transient convective-diffusive model based on Burger's equation. Correlated and uncorrelated model input parameters are considered. The model validation tutorial builds on the material presented in the propagation of uncertainty tutoriaI and uses the damp spring-mass system as the example application. The validation tutorial illustrates several concepts associated with the application of statistical inference to test model predictions against experimental observations. Several validation methods are presented including error band based, multivariate, sum of squares of residuals, and optimization methods. After completion of the tutorial, a survey of statistical model validation literature is presented and recommendations for future work are made.

Formulation of Fuzzy Finite-Element Methods for Solid Mechanics Problems

Abstract: Accounting for uncertainties in mechanics problems has been accomplished previously by probabilistic methods that may require highly repetitive and time-consuming computations to analyze the behavior of mathematical models. In addition to the repetitions, knowledge of the probability distribution of state variables is often incomplete. This article introduces a new treatment of uncertainties in continuum mechanics based on fuzzy set theory. Uncertainties or fuzzy numbers herein are viewed through the concept of presumption level of the uncertainty (α-cut), α E [0, 1], which gives an interval of confidence A α = [a 1 (α) , a 2 (α) ]. This treatment is included in a new fuzzy finte-element formulation. The fuzzy approach to treating uncertainties in continuum mechanics is applied to load, geometric, and material uncertainties in a number of examples. Results demonstrate sharp inclusion of the fuzzy solutions in comparison with the exact solutions.

Reduced-Order Models Based on Linear and Nonlinear Aerodynamic Impulse Responses

Abstract: This paper discusses a method for the identification and application of reduced-order models based on linear and nonlinear aerodynamic impulse responses. The Volterra theory of nonlinear systems and an appropriate kernel identification technique are described. Insight into the nature of kernels is provided by applying the method to the nonlinear Riccati equation in a non-aerodynamic application. The method is then applied to a nonlinear aerodynamic model of an RAE 2822 supercritical airfoil undergoing plunge motions using the CFL3D Navier-Stokes flow solver with the Spalart-Allmaras turbulence model. Results demonstrate the computational efficiency of the technique.

Closed-loop Identification : Methods, Theory, and Applications

Abstract: System identification deals with constructing mathematical models of dynamical systems from measured data. Such models have important applications in many technical and nontechnical areas, such as ...

The effects of wind and human movement on the heat and vapour transfer properties of clothing

Abstract: This paper integrates the research presented in the papers in this special issue of Holmer et al. and Havenith et al. [Holmer, I., Nilsson, H., Havenith, G., Parsons, K. C. (1999) Clothing convective heat exchange: proposal for improved prediction in standards and models. Annals of Occupational Hygiene, in press; Havenith, G., Holmer, I., den Hartog, E. and Parsons, K. C. (1999) Clothing evaporative heat resistance: proposal for improved representation in standards and models. Annals of Occupational Hygiene, in press] to provide a practical suggestion for improving existing clothing models so that they can account for the effects of wind and human movement. The proposed method is presented and described in the form of a BASIC computer program. Analytical methods (for example ISO 7933) for the assessment of the thermal strain caused by human exposure to hot environments require a mathematical quantification of the thermal properties of clothing. These effects are usually considered in terms of 'dry' thermal insulation and vapour resistance. This simple 'model' of clothing can account for the insulation properties of clothing which reduce heat loss (or gain) between the body and the environment and, for example, the resistance to the transfer of evaporated sweat from the skin, which is important for cooling the body in a hot environment. When a clothed person is exposed to wind, however, and when the person is active, there is a potentially significant limitation in the simple model of clothing presented above. Heat and mass transfer can take place between the microclimate (within clothing and next to the skin surface) and the external environment. The method described in this paper 'corrects' static values of clothing properties to provide dynamic values that take account of wind and human movement. It therefore allows a more complete representation of the effects of clothing on the heat strain of workers.

Effect of fiber strength and fiber-matrix interface on crack bridging in cement composites

Abstract: This article proposes a new theory for predicting the crack-bridging performance of random short fibers involved in cementitious composites. The current theoretical model for estimating crack bridging performance of random short fiber reinforced cement composites under tension is limited to specific constituent properties: friction-dominant fiber-matrix interface and complete fiber pull-out from matrix without rupture. The new theory extends this model by accounting for two often-encountered features in practice: fiber strength reduction and rupture in composites, and chemical bond–dominant fiber-matrix interface. The new theory was verified to capture important characteristics in bridging performance in comparison with composite tensile test data. As a result, the new theory forms an important foundation for developing high-performance random short fiber reinforced cement composites.

Mathematical Models of Catalytic Combustors

Abstract: The various concepts of catalytic combustion for the production of energy are presented first. Then, the different configurations of catalytic combustion systems for gas turbine applications are described. The following aspects of the mathematical modeling of the catalyst section are addressed: (1) relevant physical and chemical phenomena; (2) comparison of mathematical models; (3) estimation of interphase transfer coefficients; (4) acquisition of kinetic data; (5) continuous versus discrete multichannel models. Selected examples of application of the models are then presented. Finally, the mathematical modeling of the homogeneous section is briefly discussed.

Theoretical Solution of Dam-Break Shock Wave

Abstract: A mathematical model of dam-break shock waves, or flood waves, in channels of trapezoidal cross section is presented. When the model is applied to channels of rectangular cross section, a new theoretical solution expressed using one independent multinominal algebraic equation is derived. In the past, the solution had to be described using three interrelated equations. The new equation indicates that the hydraulic parameters of a shock wave—such as depth and velocity of flow, and velocity of discontinuity—are determined only by the ratio of initial downstream depth to initial upstream depth. The model shows that results from the new equation are completely equivalent to those of the set of old equations. In addition, the flood hydrograph produced by a dam break at any time or any site can be described by a single curve in terms of dimensionless variables.

On the role and use of mathematical models in engineering design

Abstract: Mathematical models encompassing virtually all aspects of engineered products are in widespread use Indeed, entire industries exist in support of some models: finite element models, computational fluid dynamics models, electric power grid load-flow models, process models, simulation models, and so on Nonetheless, the purpose, construction, and use of models is not adequately understood One major issue is that no model perfectly represents reality Therefore models always diverge from reality somewhere, and they are always inaccurate by some amount Estimation of model error and the use of model-generated results in support of engineering design decision making are key to the successful use of models

Effect of the suspension structure on equivalent suspension parameters

Abstract: This paper examines the uncertainties in modelling a real suspension system that are due to the effect of suspension linkage layout (or structure) on the equivalent suspension parameters of a corresponding mathematical model. In most research on active suspension systems, a quarter-car model of two masses is very often used. However, without considering the influence of the suspension kinematic structure, the simple model may not be as effective as might be expected because of the uncertainties in the suspension parameters. Two sets of identified parameters for different suspension systems are compared to show the effect of suspension structure on the equivalent parameters. The relationships between specific parameters and changes in certain suspension linkage layouts are also investigated. The benefits of the parameter identification are demonstrated in the process of designing two active systems (one using a sky-hook control law and the other using a sliding mode control technique). The results show that suspension structure has a strong effect on the equivalent suspension parameters and this relationship becomes more important as the structure of suspension increases in complexity. The advantage of the identification process is crucial in designing both linear and non-linear active suspension systems.

Coupled thermal-electromagnetic model for microwave heating of temperature-dependent dielectric media

Abstract: Microwave heating processes involve electromagnetic and thermal effects coupled together through the local temperature dependence of the material dielectric properties. This paper presents a one dimensional model for the coupled electromagnetic-thermal process and demonstrates its solutions for typical problems. The local temperature dependence of the lossy dielectric medium is taken into account in two different time scales. One is the heat-generation time scale due the microwave radiation, and the other is the temperature diffusion time scale. The two time-scale approach minimizes the computation time and provides an efficient simulation tool for the analysis of various phenomena. The two-scale model presented in this paper is benchmarked by a comparison of its numerical results with other models published in the literature. Several examples of microwave heating processes in various materials are simulated. Effects of heat-wave propagation in matter are predicted by the model. The results show the temporal and spatial evolution of the temperature and power-dissipation profiles. Variations in the (microwave) impedance profile in the medium due to the heating are computed. A further development of this model, including more complicated geometries and various loss mechanisms, may yield useful numerical tools for the synthesis and design of microwave heaters in which the heated material acts as a nonlinear load in the microwave circuit.

Modeling Sediment Transport Using Depth-Averaged and Moment Equations

Abstract: A 1D mathematical model to calculate bed variations in alluvial channels is presented The model is based on the depth-averaged and moment equations for unsteady flow and sediment transport in open channels Particularly, the moment equation for suspended sediment transport is originally derived by the assumption of a simple vertical distribution for suspended sediment concentration By introducing sediment-carrying capacity, suspended sediment concentration can be solved directly from sediment transport and its moment equations Differential equations are then solved by using the control-volume formulation, which has been proven to have good convergence Numerical experiments are performed to test the sensitivity of the calibrated coefficients α and k in the modeling of the bed deposition and erosion Finally, the computed results are compared with available experimental data obtained in laboratory flumes Comparisons of this model with HEC-6 and other numerical models are also presented Good agreement

A comparison of models for characterizing the distribution of radionuclides with depth in soils

Abstract: Summary It is common practice to fit mathematical models to radionuclide activity–depth profiles in soils in order to quantify rates of vertical migration through the soil profile. We have fitted six such models to 21 different activity–depth profiles of radiocaesium (137Cs) derived from Chernobyl and determined relations between the models and the values of their parameters. The advection and dispersion parameters obtained using four solutions to the advection–dispersion equation (each based on different initial and boundary conditions or different simplifications) are in good agreement. We further develop a relation between parameter values obtained using the advection–dispersion models and those determined by a simpler exponential function of the form Aexp(–Bt) where t is the time and A and B are parameters to be estimated. One of the advection–dispersion models proved to be significantly better than the others in terms of goodness-of-fit, versatility and ease of use. A simple model, using calculations based on measured characteristics of the activity–depth profile, was shown to accord well with parameters derived from more complex models based on statistical curve fitting. We have also evaluated the ‘residence time’ or ‘compartmental’ model approach to characterizing radionuclide activity–depth profiles. We relate such models to a numerical solution of a simple advection equation, and we show that apparent dispersion in compartmental models is an artefact of numerical dispersion, which can be quantified by the Courant condition. For activity profiles that have a significant advection component, using solutions to the advection–dispersion equation, we have observed a strong positive correlation between advection and dispersion in the profile.

Speed of wave-front solutions to hyperbolic reaction-diffusion equations.

Abstract: The asymptotic speed problem of front solutions to hyperbolic reaction-diffusion (HRD) equations is studied in detail. We perform linear and variational analyses to obtain bounds for the speed. In contrast to what has been done in previous work, here we derive upper bounds in addition to lower ones in such a way that we can obtain improved bounds. For some functions it is possible to determine the speed without any uncertainty. This is also achieved for some systems of HRD (i.e., time-delayed Lotka-Volterra) equations that take into account the interaction among different species. An analytical analysis is performed for several systems of biological interest, and we find good agreement with the results of numerical simulations as well as with available observations for a system discussed recently.

Fast numerical simulation for full bore rupture of pressurized pipelines

Abstract: An efficient numerical simulation (CNGS-MOC), based on the method of characteristics for simulating full bore rupture of long pipelines containing two-phase hydrocarbons, was developed. The use of curved characteristics, in conjunction with a compound nested grid system, as well as a fast mathematical algorithm, lead to a significant reduction of CPU time, while improving accuracy. The model is validated extensively against field data including those obtained during the Piper Alpha tragedy, as well as the Isle of Grain depressurization tests. Its predictions are compared with those based on other mathematical models including PLAC, META-HEM, MSM-CS, as well as BLOWDOWN. Both CNGS-MOC and META-HEM produce reasonably accurate predictions with the remaining models assessed performing relatively poorly.

Mathematical models of diffusion-limited gas bubble dynamics in tissue

Abstract: Mathematical models of bubble evolution in tissue have recently been incorporated into risk functions for predicting the incidence of decompression sickness (DCS) in human subjects after diving and/or flying exposures. Bubble dynamics models suitable for these applications assume the bubble to be either contained in an unstirred tissue (two-region model) or surrounded by a boundary layer within a well-stirred tissue (three-region model). The contrasting premises regarding the bubble-tissue system lead to different expressions for bubble dynamics described in terms of ordinary differential equations. However, the expressions are shown to be structurally similar with differences only in the definitions of certain parameters that can be transformed to make the models equivalent at large tissue volumes. It is also shown that the two-region model is applicable only to bubble evolution in tissues of infinite extent and cannot be readily applied to bubble evolution in finite tissue volumes to simulate how such evolution is influenced by interactions among multiple bubbles in a given tissue. Two-region models that are incorrectly applied in such cases yield results that may be reinterpreted in terms of their three-region model equivalents but only if the parameters in the two-region model transform into consistent values in the three-region model. When such transforms yield inconsistent parameter values for the three-region model, results may be qualitatively correct but are in substantial quantitative error. Obviation of these errors through appropriate use of the different models may improve performance of probabilistic models of DCS occurrence that express DCS risk in terms of simulated in vivo gas and bubble dynamics.

Moment-rotation hysteresis behavior of top and seat angle steel frame connections

Abstract: This paper presents an approach toward formulating analytical models to predict the moment-rotation hysteresis behavior of top and seat angle connections. Experimental results obtained from 12 top and seat angle connection specimens are used to obtain the prediction equations for the parameters defining the moment rotation hysteresis loops of a typical top and seat angle connection. These parameters include the initial stiffness, ultimate moment capacity, ultimate rotation, the transition moment, characteristic moment, and rigidity parameter. Regression analysis results and comparisons with test results are presented to demonstrate the acceptability of these prediction equations. The prediction equations obtained for these parameters are used to develop four different moment rotation hysteresis models for the connection: the bilinear, elastoplastic, Ramberg-Osgood, and modified bilinear models. The results of the study show that the top and seat angle connection behaves as a semirigid connection. A wide range of initial stiffnesses and ultimate moment capacities are possible to achieve by altering the connection geometry related variables within a practical range. For certain geometric configurations of the connection, significant transfer of moment from the beam to the column can occur before the connection fails. Also, it is possible to design a connection with low stiffness and small moment transfer capability, so that it behaves in a manner such that it is close to being classified as a pin connection. The prediction equations developed for the parameters characterizing the four hysteresis models give acceptable results when compared to experimental results. The degree to which the models idealized the actual behavior varies with the elastoplastic model being the least conservative and the modified bilinear modeling being the best. The Ramberg-Osgood model is the most accurate in just modeling the nonpinching moment-rotation loops.