Showing papers on "Mathematical model published in 1995"

COMPARISON OF DIFFERENT k-ε MODELS FOR INDOOR AIR FLOW COMPUTATIONS

Abstract: In this article, Jive k-e, two-equation models are studied: the standard k-e model, a low-Reynolds-number k-e model, a two-layer k-e model, a two-scale k-e model, and a renormalization group (RNG) k-e model. They are evaluated for their performance in predicting natural convection, forced convection, and mixed convection in rooms, as well as an impinging jet flow. Corresponding experimental data from the literature are used for validation. It is found that the prediction of the mean velocity is more accurate than that of the turbulent velocity. These models are neither able to predict anisotropic turbulence correctly nor to pick up the secondary recirculation of indoor air flow; otherwise the performance of the standard k-e model is good. The RNG k-e model is slightly better than the standard k-e model and is therefore recommended for simulations of indoor airflow. The performance of the other models is not stable.

Time-Dependent Numerical Code for Extended Boussinesq Equations

Abstract: The extended Boussinesq equations derived by Nwogu (1993) significantly improve the linear dispersive properties of long-wave models in intermediate water depths, making it suitable to simulate wave propagation from relatively deep to shallow water. In this study, a numerical code based on Nwogu's equations is developed. The model uses a fourth-order predictor-corrector method to advance in time, and discretizes first-order spatial derivatives to fourth-order accuracy, thus reducing all truncation errors to a level smaller than the dispersive terms retained by the model. The basic numerical scheme and associated boundary conditions are described. The model is applied to several examples of wave propagation in variable depth, and computed solutions are compared with experimental data. These initial results indicate that the model is capable of simulating wave transformation from relatively deep water to shallow water, giving accurate predictions of the height and shape of shoaled waves in both regular and irregular wave experiments.

Frequency‐domain modelling of airborne electromagnetic responses using staggered finite differences

Abstract: A 3D frequency-domain EM modelling code has been implemented for helicopter electromagnetic (HEM) simulations. A vector Helmholtz equation for the electric fields is employed to avoid convergence problems associated with the first-order Maxwell's equations when air is present. Additional stability is introduced by formulating the problem in terms of the scattered electric fields. With this formulation the impressed dipole source is replaced with an equivalent source, which for the airborne configuration possesses a smoother spatial dependence and is easier to model. In order to compute this equivalent source, a primary field arising from dipole sources of either a whole space or a layered half-space must be calculated at locations where the conductivity is different from that of the background. The Helmholtz equation is approximated using finite differences on a staggered grid. After finite-differencing, a complex-symmetric matrix system of equations is assembled and preconditioned using Jacobi scaling before it is solved using the quasi-minimum residual (QMR) method. The modelling code has been compared with other 1D and 3D numerical models and is found to produce results in good agreement. We have used the solution to simulate novel HEM responses that are computationally intractable using integral equation (IE) solutions. These simulations include a 2D conductor residing at a fault contact with and without topography. Our simulations show that the quadrature response is a very good indicator of the faulted background, while the in-phase response indicates the presence of the conductor. However when interpreting the in-phase response, it is possible erroneously to infer a dipping conductor due to the contribution of the faulted background.

Simple mathematical model of a rough fracture

Abstract: A simple mathematical model of rough-walled fractures in rock is described which requires the specification of only three main parameters: the fractal dimension, the rms roughness at a reference length scale, and a length scale describing the degree of mismatch between the two fracture surfaces Fractured samples, collected from natural joints and laboratory specimens, have been profiled to determine the range of these three parameters in nature It is shown how this surface roughness model can be implemented on a computer, allowing future detailed study of the mechanical and transport properties of single fractures and the scale dependence of these properties

Nonlinear demographic dynamics: mathematical models, statistical methods, and biological experiments'

Abstract: Our approach to testing nonlinear population theory is to connect rigorously mathematical models with data by means of statistical methods for nonlinear time series. We begin by deriving a biologically based demographic model. The mathematical analysis identifies boundaries in parameter space where stable equilibria bifurcate to periodic 2-cy- cles and aperiodic motion on invariant loops. The statistical analysis, based on a stochastic version of the demographic model, provides procedures for parameter estimation, hypothesis testing, and model evaluation. Experiments using the flour beetle Tribolium yield the time series data. A three-dimensional map of larval, pupal, and adult numbers forecasts four possible population behaviors: extinction, equilibria, periodicities, and aperiodic motion including chaos. This study documents the nonlinear prediction of periodic 2-cycles in laboratory cultures of Tribolium and represents a new interdisciplinary approach to un- derstanding nonlinear ecological dynamics.

A description of the solar wind-magnetosphere coupling based on nonlinear filters

Abstract: A nonlinear filtering method is introduced for the study of the solar wind -- magnetosphere coupling and related to earlier linear techniques. The filters are derived from the magnetospheric state, a representation of the magnetospheric conditions in terms of a few global variables, here the auroral electrojet indices. The filters also couple to the input, a representation of the solar wind variables, here the rectified electric field. Filter-based iterative prediction of the indices has been obtained for up to 20 hours. The prediction is stable with respect to perturbations in the initial magnetospheric state; these decrease exponentially at the rate of 30/min. The performance of the method is examined for a wide range of parameters and is superior to that of other linear and nonlinear techniques. In the magnetospheric state representation the coupling is modeled as a small number of nonlinear equations under a time-dependent input.

A Mathematical Model of a Solid Oxide Fuel Cell

Abstract: A mathematical model of a tubular solid oxide fuel cell is presented. The complete electrochemical and thermal factors are accounted for in a rigorous manner. All required parameters are determined from independent sources; none are fit from performance data. To verify the accuracy of the model predictions, comparison is made with single cell test data from Westinghouse. Agreement with electrochemical and thermal results are within 5%, and for most points, much better. Predictions are shown for power-voltage, irreversibilities, and temperature and current distributions under various conditions.

Depth-averaged open-channel flow model

Abstract: A general mathematical model is developed to solve unsteady, depth-averaged equations. The model uses boundary-fitted coordinates, includes effective stresses, and may be used to analyze sub- and super critical flows. The time differencing is accomplished using a second-order accurate Beam and Warming approximation, while the spatial derivatives are approximated by second-order accurate central differencing. The equations are solved on a nonstaggered grid using an alternative-direction-implicit scheme. To enhance applicability, the equations are solved in transformed computational coordinates. The effective stresses are modeled by incorporating a constant eddy-viscosity turbulence model to approximate the turbulent Reynolds stresses. As is customary, the stresses due to depth-averaging are neglected. Excluding recirculating flows, it is observed that in most cases the effective stresses do not significantly affect the converged solution. The model is used to analyze a wide variety of hydraulics problems including flow in a channel with a hydraulic jump, flow in a channel contraction, flow near a spur-dike, flow in a 180° channel bend, and a dam-break simulation. For each of these cases, the computed results are compared with experimental data. The agreement between the computed and experimental results is satisfactory.

Application of Fractal Geometry to the Study of Networks of Fractures and Their Pressure Transient

Abstract: Typical models for the representation of naturally fractured systems generally rely on the double-porosity Warren-Root model or on random arrays of fractures. However, field observations have demonstrated the existence of multiple length scales in a variety of naturally fractured media. Present models fail to capture this important property of self-similarity. We first use concepts from the theory of fragmentation and from fractal geometry to construct numerically a network of fractures that exhibits self-similar behavior over a range of scales. The method is a combination of fragmentation concepts and the iterated function system approach and allows for great flexibility in the development of patterns. Next, numerical simulation of unsteady single-phase flow in such networks is described. It is found that the pressure transient response of finite fractals behaves according to the analytical predictions of Chang and Yortsos (1990) provided that there exists a power law in the mass-radius relationship around the test well location. Finite size effects can become significant and interfere with the identification of the fractal structure. The paper concludes by providing examples from actual well tests in fractured systems which are analyzed using fractal pressure transient theory.

Coupled turbulent flow, heat, and solute transport in continuous casting processes

Abstract: A fully coupled fluid flow, heat, and solute transport model was developed to analyze turbulent flow, solidification, and evolution of macrosegregation in a continuous billet caster. Transport equations of total mass, momentum, energy, and species for a binary iron-carbon alloy system were solved using a continuum model, wherein the equations are valid for the solid, liquid, and mushy zones in the casting. A modified version of the low-Reynolds numberk-e model was adopted to incorporate turbulence effects on transport processes in the system. A control-volume-based finite-difference procedure was employed to solve the conservation equations associated with appropriate boundary conditions. Because of high nonlinearity in the system of equations, a number of techniques were used to accelerate the convergence process. The effects of the parameters such as casting speed, steel grade, nozzle configuration on flow pattern, solidification profile, and carbon segregation were investigated. From the computed flow pattern, the trajectory of inclusion particles, as well as the density distribution of the particles, was calculated. Some of the computed results were compared with available experimental measurements, and reasonable agreements were obtained.

Mathematical studies in rigorous grating theory

Abstract: We consider the diffraction of a time-harmonic wave incident upon a grating (or periodic) structure. We study mathematical issues that arise in the direct modeling, inverse, and optimal design problems. Particular attention is paid to the variational approach and to finite-element methods. For the direct problem various results on existence, uniqueness, and numerical approximations of solutions are presented. Convergence properties of the variational method and sensitivity to TM polarization are examined. Our recent research on inverse diffraction problems and optimal design problems is also discussed.

A mostly linear model of transition to turbulence

Abstract: A simple model in three real dimensions is proposed, illustrating a possible mechanism of transition to turbulence. The linear part of the model is stable but highly non‐normal, so that certain inputs experience a great deal of growth before they eventually decay. The nonlinear terms of the model contribute no energy growth, but recycle some of the linear outputs into inputs, closing a feedback loop and allowing initially small solutions to ‘‘bootstrap’’ to a much larger amplitude. Although different choices of parameters in the nonlinearity lead to a variety of long‐term behaviors, the bootstrapping process is essentially independent of the details of the nonlinearity and varies predictably with the Reynolds number. The bootstrapping scenario demonstrated by this model is the basis of some recent explanations for the failure of classical hydrodynamic stability analysis to predict the onset of turbulence in certain flow configurations.

Modal and temporal analysis of head mathematical models

Abstract: The basic hypotheses used during these investigations were based on the vibration analysis of the head, which demonstrated that the head is not a solid nondeformable body, but a complex structure including deformable elements. Laboratoire des Systemes Biomecanique (LSBM) has recently proposed three mathematical models: a lumped model, a finite element model of the head in its sagittal plane, and a three-dimensional finite element model. These models were validated by their modal behavior and enabled the lesion mechanisms to be distinguished as a function of the spectral characteristics of the shock. The objective of this study is to complete these modal results by temporal analysis of the models by calculating the evolution of the intracranian mechanical parameters under shock conditions. To describe the head's dynamic behavior in the temporal domain, constant energy shocks of variable duration were simulated to evaluate their influence on different quantities as the intracerebral stresses in ter...

Critical Evaluation of Models and Their Parameters

Abstract: Some form of critical evaluatory procedure for models is needed to maintain the integrity of modeling and to ensure that the increasingly widespread use of models does not result in the propagation of misleading information. The term validation must be used with the clear understanding that no model can be validated in the sense that it has been unequivocally justified. All that can be achieved is to show how small the probability is that the model has been refuted. Whether this probability is acceptable is a subjective decision. The type of statistical test that is appropriate depends on the quality of the data against which the model is tested. Using a procedure that compares the sums of squares that result from the model not fitting the data with the sums of squares due to error in the data, gives a stringent test, but it requires the replication of measurements. Rigor is as important in evaluating parameters as it is in testing models. Direct measurement is the best option, but where a parameter has to be obtained by fitting, the statistical procedures used for validation are appropriate. In general, the further the data used for parameterization are removed from the data to be simulated the better. Problems can arise in both parameterization and validation if the model is nonlinear regarding its parameters, and the latter have appreciable variances. Parameterization and validation become more difficult as the complexity of the model or the scale at which it is used increase.

Modelling mathematical methods and scientific computation

Abstract: Preface Mathematical Modelling Introduction Definition of Mathematical Modelling Classification of Mathematical Modelling Modelling Methods Validation of Mathematical Models Mathematical Modelling as a Science Discrete Models Plan of Chapter 2 About Mathematical Modelling Mathematical Formulation of Problems On Existence, Uniqueness and Continuity Linear Systems Stability and Linearization From Bifurcation to Chaos Numerical Methods for Initial Value Problems Scientific Programs Continuous Models Introduction Mathematical Modelling Equilibrium Equation for the Vibration of an Elastic String Mathematical Models of Continuum Mechanics Mathematical Models of Electromagnetism Direct Simulation Models in Biology Classification and Characteristics Mathematical Formulation of Problems Finite Difference Methods The Collocation Method Decomposition of Domains Applications and Scientific Programs Inverse and Stochastic Problems Inverse Problems and Stochastic Models Classification of Inverse Problems Solution by Decomposition of Domains Solution by Minimization Techniques Mathematical Modelling and Stochasticity Classification of Discrete Stochastic Models Classification of Continuous Stochastic Models Modelling and Solution of Problems Stochastic Aspects and Inverse Problems Kinetic Models Application Discussion and Developments Scientific Programs Appendix 1. Functional Spaces and Fixed Point Theorems Appendix 2. Interpolation and Approximation Appendix 3. Random Variables References Subjects Index

Mathematical Models for Vehicular Traffic

Abstract: This survey contains a description of different types of mathematical models used for the simulation of vehicular traffic. It includes models based on ordinary differential equations, fluid dynamic equations and on equations of kinetic type. Connections between the different types of models are mentioned. Particular emphasis is put on kinetic models and on simulation methods for these models.

A two-factor saturation model for synchronous machines with multiple rotor circuits

Abstract: It is generally felt that no major accuracy breakthrough in predicting the steady-state and transient performance of synchronous machines could be achieved without taking proper account of the iron saturation effects as well as eddy-current losses. Although the two issues were often treated separately in the past, this paper attempts to unite them by developing a general model covering both the main-path magnetic saturation and frequency effects in the dynamic equations. Mathematical analysis in the d-q space pinpoints cross-saturation coupling which, a priori, does not seem to be symmetrical for salient-pole machines. Yet the model is theoretically sound, since it fulfils at least the physical constraints using energy balance principles. Some test points from a 555-MVA turbine-generator are used for an initial assessment the model's capability to predict the field current and internal angle for various loading conditions.

Mathematical Modeling of Groundwater Pollution

Abstract: Contents: Introduction.- Hydrodynamic Dispersion in Porous Media.- Analytical Solutions of Hydrodynamic Dispersion Equations.- Finite Difference Methods and the Method of Characteristics for Hydrodynamic Dispersion Equations.- Finite Element Methods for Solving Hydrodynamic Dispersion Equations.- Numerical Solutions of Advection-Dominated Problems.- Mathematical Models of Groundwater Quality.- Applications of Groundwater Quality Models.- Conclusions.- Appendix A: The Related Parameters in the Modeling of Mass Transport in Porous Media.- Appendix B: A FORTRAN Program for Simultaneously Simulating Groundwater Flow and Quality.- References.

Comparison of coupled radiative flow solutions with Project Fire II flight data

Abstract: A nonequilibrium, axisymmetric, Navier-Stokes flow solver with coupled radiation has been developed for use in the design or thermal protection systems for vehicles where radiation effects are important. The present method has been compared with an existing now and radiation solver and with the Project Fire 2 experimental data. Good agreement has been obtained over the entire Fire 2 trajectory with the experimentally determined values of the stagnation radiation intensity in the 0.2-6.2 eV range and with the total stagnation heating. The effects of a number of flow models are examined to determine which combination of physical models produces the best agreement with the experimental data. These models include radiation coupling, multitemperature thermal models, and finite rate chemistry. Finally, the computational efficiency of the present model is evaluated. The radiation properties model developed for this study is shown to offer significant computational savings compared to existing codes.

Finite-Element Model for High-Velocity Channels

Abstract: Numerical modelers of high-velocity channels are faced with supercritical transitions and the difficulty in capturing discontinuities in the flow field, known as hydraulic jumps. The implied smoothness of a numerical scheme can produce fictitious oscillations near these jump locations and can lead to instability. It is also important that the discrete numerical operations preserve the Rankine-Hugoniot conditions and accurately model jump speed and location. The geometric complexity of high-velocity channels with bridge piers and service ramps are easily represented using an unstructured model. A two-dimensional finite-element model that utilizes a characteristic based Petrov-Galerkin method and a shock-detection mechanism, which relies on elemental energy variation results in a robust system to model high-velocity channels. Comparisons are made between analytic shock-speed results, published laboratory data of a lateral contraction, and with a more general physical model.

A system identification approach to the detection of changes in both linear and non‐linear structural parameters

Abstract: This paper explores the potential of a new time domain identification procedure to detect changes in structural dynamic characteristics on the basis of measurements. This procedure is verified using mathematical models simulated on the computer. The experiments involve two eight-storey steel structures with and without energy devices, and a 47-storey building at San Francisco during the Loma Prieta earthquake. The recursive instrumental variable method and extended Kalman filter algorithm are used as identification algorithms. An exploratory investigation is made of the usefulness of various indices, such as mode shape and storey drift, that can be extracted accurately from identification to quantify changes in the characteristics of the physical system. It is concluded that the change of storey drift is the key information to the detection of changes in structural parameters, from which the proposed system identification algorithm can be applied with an appropriate inelastic model to simulate the dynamic behaviour of real structures undergoing strong ground motion excitations.

A mathematical model for a dissolving polymer

Abstract: In certain polymer-penetrant ,systems, nonlinear viscoelastic effects dominate those of Fickian diffusion. This behavior is often embodied in a memory integral incorporating nonlocal time effects into the dynamics; this integral can be derived from an augmented chemical potential. The mathematical framework presented is a moving boundary-value problem. The boundary separates the polymer into two distinct states: glassy and rubbery, where different physical processes dominate. The moving boundary condition that results is not solvable by similarity solutions, but can be solved by perturbation and integral equation techniques. Asymptotic solutions are obtained where sharp fronts move with constant speed. The resultant profiles are quite similar to experimental results in a dissolving polymer. It is then demonstrated that such a model has a limit on the allowable front speed and a self-regulating mass uptake.

Linear inversion of gravity data for 3-D density distributions

Abstract: We have developed an improved LevenburgMarquart technique to rapidly invert Bouguer gravity data for a 3-D density distribution as a source of the observed field. This technique is designed to replace tedious forward modeling with an automatic solver that determines density models constrained by geologic information supplied by the user. Where such information is not available, objective models are generated. The technique estimates the density distribution within the source volume using a least-squares inverse solution that is obtained iteratively by singular value decomposition using orthogonal decomposition of matrices with sequential Householder transformations. The source volume is subdivided into a series of right rectangular prisms of specified size but of unknown density. This discretization allows the construction of a system of linear equations relating the observed gravity field to the unknown density distribution.Convergence of the solution to the system is tightly controlled by a damping parameter which may be varied at each iteration. The associated algorithm generates statistical measures of solution quality not available with most forward methods. Along with the ability to handle large data sets within reasonable time constraints, the advantages of this approach are: (1) the ease with which pre-existing geological information can be included to constrain the solution, (2) its minimization of subjective user input, (3) the avoidance of difficulties encountered during wavenumber domain transformations, and (4) the objective nature of the solution.Application to a gravity data set from Hamilton County, Indiana, has yielded a geologically reasonable result that agrees with published models derived from interpretation of gravity, magnetic, seismic, and drilling data.

Solving the Estimation‐Identification Problem in Two‐Phase Flow Modeling

Abstract: In this paper a procedure is presented to solve the estimation-identification problem in two-phase flow modeling. Given discrete observations made on the system response, an optimum parameter set is derived for an appropriate conceptual model by solving the inverse problem using standard optimization techniques. Subsequently, a detailed error analysis is performed, and nonlinearity effects are considered. We discuss the iterative process of model identification and parameter estimation for a ventilation test performed at the Grimsel Rock Laboratory, Switzerland. A numerical model of the ventilation drift and the surrounding crystalline rock matrix is developed. Evaporation of moisture at the drift surface and the propagation of the unsaturated zone into the formation are simulated. A sensitivity analysis is performed to identify the parameters to be estimated. Absolute permeability and two parameters of van Genuchten`s characteristic curves are subsequently determined based on measurements of negative water potentials, evaporation rates, and gas pressure data. The performance of the minimization algorithm and the system behavior for the optimum parameter set are discussed. The study shows that a field experiment conducted under two-phase flow conditions can be successfully reproduced by taking into account a variety of physical processes and that it is possible to reliably determinemore » the two-phase hydraulic properties that are related to the given conceptual model. 32 refs., 8 figs., 7 tabs.« less

Computational Mechanics of Reinforced Concrete Structures

Abstract: Materials - Phenomena and Experiments - Mathematical Models - The Finite-Element-Method for RC and PC Structures Application to Engineering Problems

Neural Network Prediction of Solar Activity

Abstract: The neural network technique is used to analyze the time series of solar activity, as measured through the relative Wolf number. Firstly, the embedding dimension of the time-series characteristic attractor is obtained. Secondly, after describing the design and training of the net, the performance of the present approach in forecasting yearly mean sunspot numbers is favorably compared to that of conventional statistical methods. Finally, predictions for the remaining part of the 22th and the whole 23th cycle are presented.

Automated modeling of physical systems

Abstract: Effective reasoning about complex physical systems requires the use of models that are adequate for the task. Constructing such adequate models is often difficult. In this dissertation, we address this difficulty by developing efficient techniques for automatically selecting adequate models of physical systems. We focus on the important task of generating parsimonious causal explanations for phenomena of interest. Formally, we propose answers to the following: (a) what is a model and what is the space of possible models; (b) what is an adequate model; and (c) how do we find adequate models. We define a model as a set of model fragments, where a model fragment is a set of independent equations that partially describes some physical phenomenon. The space of possible models is defined implicitly by the set of applicable model fragments: different subsets of this set correspond to different models. An adequate model is defined as a simplest model that can explain the phenomenon of interest, and that satisfies any domain-independent and domain-dependent constraints on the structure and behavior of the physical system. We show that, in general, finding an adequate model is intractable (NP-hard). We address this intractability, by introducing a set of restrictions, and use these restrictions to develop an efficient algorithm for finding adequate models. The most significant restriction is that all the approximation relations between model fragments are required to be causal approximations. In practice this is not a serious restriction because most commonly used approximations are causal approximations. We also develop a novel order of magnitude reasoning technique, which strikes a balance between purely qualitative and purely quantitative methods. The order of magnitude of a parameter is defined on a logarithmic scale, and a set of rules propagate orders of magnitudes through equations. A novel feature of these rules is that they effectively handle non-linear simultaneous equations, using linear programming in conjunction with backtracking. The techniques described in this dissertation have been implemented and have been tested on a variety of electromechanical devices. These tests provide empirical evidence for the theoretical claims of the dissertation.