Showing papers on "Mathematical model published in 1983"

Efficient Three-Dimensional Global Models for Climate Studies: Models I and II

Abstract: Climate modeling based on numerical solution of the fundamental equations for atmospheric structure and motion permits the explicit modeling of physical processes in the climate system and the natural treatment of interactions and feedbacks among parts of the system. The main difficulty concerning this approach is related to the computational requirements. The present investigation is concerned with the development of a grid-point model which is programmed so that both horizontal and vertical resolutions can easily be changed. Attention is given to a description of Model I, the performance of sensitivity experiments by varying parameters, the definition of an improved Model II, and a study of the dependence of climate simulation on resolution with Model II. It is shown that the major features of global climate can be simulated reasonably well with a horizontal resolution as coarse as 1000 km. Such a resolution allows the possibility of long-range climate studies with moderate computer resources.

Sensitivity Analysis in Chemical Kinetics

Abstract: Complex mathematical models are increasingly being used as predictive tools and as aids for understanding the processes underlying observed chemical phenomena. The parameters appearing in these models, which may include rate constants, activation energies, thermodynamic constants, transport coefficients, initial conditions, and operating conditions, are seldom known to high precision. Thus, the predictions or conclusions of modeling endeavors are usually subject to uncertainty. Furthermore, regardless of uncertainty questions, there is always the overriding matter of which parameters control laboratory observations. Quantification of the role of the parameters in the model predictions is the traditional realm of sensitivity analysis. A significant amount of current research is directed at conceptualization and implementation of numerical techniques for determining parametric sensitivities for algebraic, differential, and partial differential equation models including those with stochastic character and nonconstant parameters. Recent studies have also served to extend the range of the conventional parametric analysis to address new questions, relevant to the process of model building and interpretation. This review attempts to present a full range of results and applications of sensitivity analysis, relevant to chemical kinetic modeling. We exclude a large class of literature that deals with system sensitivities from a control theory perspective (1). We further limit discussion of related subjects such as parameter identification (estimation of best parameter values for fitting data) and optimization, in which parametric sensitivities play only

A new set of spatial-interaction models: the theory of competing destinations.

Abstract: Members of the family of spatial-interaction models commonly referred to as gravity models are shown to be misspecified. One result of this misspecification is the occurrence of an undesirable "spatial-structure effect" in estimated distance-decay parameters and this effect is examined in detail. An alternative set of spatial-interaction models is formulated from which more accurate predictions of interactions and more accurate parameter estimates can be obtained. These new interaction models are termed competing destinations models, and estimated distance-decay parameters obtained in their calibration are shown to have a purely behavioural interpretation. The implications of gravity-model misspecification are discussed. (Author/TRRL)

Chaos in neurobiology

Abstract: Deterministic mathematical models of neural systems can give rise to complex aperiodic (`chaotic') dynamics in the absence of stochastic fluctuations (`noise') in the variables or parameters of the model or in the inputs to the system. The authors show that chaotic dynamics are expected in nonlinear feedback systems possessing time delays such as are found in recurrent inhibition and from the periodic forcing of neural oscillators. The implications of the possible occurrence of chaotic dynamics for experimental work and mathematical modeling of normal and abnormal functions neurophysiology are mentioned.

Strange Attractors and Chaos in Nonlinear Mechanics

Abstract: We review several examples of nonlinear mechanical and electrical systems and related mathematical models that display chaotic dynamics or strange attractors. Some simple mathematical models — iterated piecewise linear mappings — are introduced to explain and illustrate the concepts of sensitive dependence on initial conditions and chaos. In particular, we describe the role of homoclinic orbits and the horseshoe map in the generation of chaos, and indicate how the existence of such features can be detected in specific nonlinear differential equations.

Mathematical Analysis of a Zn / NiOOH Cell

Abstract: A mathematical model has been developed to predict the time dependent behavior of a Zn/NiOOH cell. The model uses experimentally determined polarization expressions to describe the losses between the positive and the negative electrodes. The electronic losses in the plane of the electrode are simulated by a network of resistors. The potential distribution, the current distribution, the cell voltage, the power capability, and the energy of a cell can be predicted. The mathematical model provides an analytical tool to evaluate, for example, the trade-offs between power capability and current collector mass, needed to design an electric vehicle battery.

A soil-tool model based on limit equilibrium analysis

Abstract: A mathematical model for predicting the behavior of narrow tillage tools in soils is based on a limit equilibrium analysis. Pertinent soil, tool and interface parameters influencing the tool performance have been identified and incorporated in the model. A comparison of predicted and experimental results is also included. Mathematical models based on emperical as well as semi-emperical methods have been developed to describe the soil-tillage tool interaction (Payne, 1956; Hettiaratchi and Reece, 1967; Hettiaratchi et al., 1966; Osman, 1964; Godwin and Spoor, 1977; McKyes, 1978; Desai et al., 1981). Even though the soil-tool interaction problem is three dimensional in nature, a majority of the models available are based on two-dimensional consideration (Hettiaratchi et al., 1966; Osman, 1964; Payne, 1956). In recent years some progress has been made toward the development of three dimensional models (Hettiaratchi and Reece, 1967; Godwin and Spoor, 1977; McKyes, 1978). However, most of these models are complex, and a sound mathematical background is essential to utilize them. Thus the need exists for more general and less complex models capable of predicting tillage-tool behavior in soils. Unlike costly experimental procedures, availability of such models would permit designers as well as researchers to develop with minimum effort a clear understanding of soil-tool interaction through parametric studies. Therefore, the overall objective of this study was to develop a generalized mathematical model and to examine its validity for predicting the tillage tool performance in soils..

Physical Interpretation of the Adjoint Functions for Sensitivity Analysis of Atmospheric Models

Abstract: The adjoint functions for an atmospheric model are the solution to a system of equations derived from a differential form of the model's equations. The adjoint functions can be used to calculate efficiently the sensitivity of one of the model's results to variations in any of the model's parameters. This paper shows that the adjoint functions themselves can be interpreted, as the sensitivity of a result to instantaneous perturbations of the model's dependent variables. This interpretation is illustrated for a radiative convective model, although the interpretation holds equally well for general circulation models. The adjoint functions are used to reveal the three time scales associated with 1) convective adjustment, 2) heat transfer between the atmosphere and space and 3) heat transfer between the ground and atmosphere. Calculating the eigenvalues and eigenvectors of the matrix of derivatives occurring in the set of adjoint equations reveals similar physical information without actually solving ...

A Comparison of Solutions of Two Model Equations for Long Waves.

Abstract: : This paper is concerned with mathematical models representing the unidirectional propagation of weakly nonlinear dispersive waves. Interest will be directed toward two particular models that are originally studied in the context of surface-wave phenomena in open-channel flows. The purpose of the present paper is to make a quantitative comparison between the solutions to the initial-value problem for each of these models. The basic conclusion of the study is that, on a long time scale T naturally related to the underlying physical situation, the equations predict the same outcome to within their implied order of accuracy. In this case the choice of one of these models over the other to describe a physical problem is apparently immaterial, with factors of incidental convenience probably providing the main criteria in a given situation. (Author)

Model equations of state

Abstract: Model equations of state determining the thermodynamic characteristics of matter in various states of aggregation are reviewed. Methods for describing the thermodynamics of gases, liquids, and plasmas are described. Quantum-mechanical models for solids and a quasiclassical model of matter are discussed. Models which can be used to study melting, evaporation, structural and electronic phase transitions in solids, and phase transitions in nonideal plasmas are also discussed. The ranges of applicability of the various methods are determined. The results of model-based calculations are compared with the experimental data available.

Reduced-order steady-state and dynamic models for separation processes. Part I. Development of the model reduction procedure

Abstract: One of the major difficulties with mathematical models of staged separation systems is the large dimensionality of the process model. This paper is concerned with simple (reduced-order) steady-state and dynamic models for processes such as distillation, absorption and extraction. The model reduction procedure is based on approximating the composition and flow profiles in the column using polynomials rather than as discrete functions of the stages. The number of equations required to describe the system is thus drastically reduced. The method is developed using a simple absorber system. In the second part of this paper, the application of the method to nonlinear multicomponent separation systems is demonstrated.

Models of the static and dynamic behavior of stripe geometry lasers

Abstract: The rate equations used to find carrier and photon densities in semiconductor lasers are extended to include a spatial variation of the carrier density in the junction plane and combined with a field equation for calculation of the intensity distribution. The equations are solved for both static and dynamic cases, and the model is able to account for nonlinear light current characteristics, near field displacements, and self-sustained pulsations in lasers without a built-in guiding mechanism. The results are compared to both experimental results and predictions from other models.

Probabilistic dynamic security assessment of power systems-I: Basic model

Abstract: A comprehensive framework for power system security assessment which incorporates probabilistic aspects of disturbances and system dynamic responses to disturbances is presented. Standard mathematical models for power system (steady-state) power flow analysis and transient stability (dynamic) analysis are used. A linear vector differential equation is derived whose solution gives the probability distribution of the time to insecurity. The coefficients of the differential equation contain the transition rates of system structural changes and a set of transition probabilities defined in terms of the steady-state and the dynamic security regions. These regions are defined in the space of power injections. Upper and lower bounds on the time to insecurity distribution are obtained.

Robustness of the independent modal-space control method

Abstract: The effect of parameter uncertainties on the control system performance of distributed-p arameter systems is examined. Because in general, the parameters contained in the equations of motion of the actual distributed system are, not known accurately, control forces designed on the basis of a postulated model will not control the actual distributed system effectively. In this paper it is shown by means of a stability theorem that, when the independent modal-space control (IMSC) method is used in conjunction with modal filters, any errors in the system parameters cannot lead to instability of the closed-loop system, so that the control system is very robust. A perturbation analysis is proposed for the computation of the closed-loop poles of large-order systems in the presence of parameter changes. I. Introduction T HE motion of a distributed-parameter system is governed generally by a set of simultaneous partial differential equations of motion. 1 The parameters contained in the equations of motion are, in general, continuous functions of the spatial variables. For flexible structures, these parameters represent mass, stiffness, and damping distributions. To control the distributed system, one must construct a mathematical model of the distributed system. The control forces then are designed on the basis of the mathematical model. A common approach to modeling is to convert the partial differential equations of the distributed system into an infinite set of ordinary differential equations.1'3 Then, a limited number of modes (generally the lowest) are retained for control. In designing the control system, one assumes that the eigensolution associated with the controlled modes is known with sufficient accuracy, which, in turn, assumes that the system parameters are known accurately. Errors in the eigensolution produce errors in the design and implementation of the controls. Hence, the question arises whether the control system designed on the basis of system parameters that are in error can control the actual system effectively; i.e., whether the control system is robust. The answer clearly depends on the degree of inaccuracy in the estimated state of the distributed system. For cases when this error is not very large, one intuitively expects very small deviations from the control system performance. In general, one should make some allowance in the control system design for parameter uncertainties. For cases when the parameters contained in the equations of motion, such as the mass and stiffness distributions, are known to within a multiplicative constant, only the system eigenvalues change and the eigenfunctions retain the same shape.4 A sensitivity study, based on a perturbation analysis treating the parameter errors as perturbations reveals that if the independent modal-space control (IMSC) method is used in conjunction with modal filters, the control system is relatively insensitive to parameter errors.4 When the spatial distributions of the system parameters are not known, however, both the estimated eigenvalues and eigenfunctions tend to differ from their actual values, so that the sensitivity analysis of Ref. 4 is not applicable.

Optimal control for xenon spatial oscillations in load follow of a nuclear reactor

Abstract: A simple core control model is developed for the control of xenon spatial oscillations in load following operations of a current-design nuclear pressurized water reactor. The model is formulated as a linear-quadratic tracking problem in the context of modern optimal control theory, and the resulting two-point boundary problem is solved directly by the techniques of initial value methods. The system of state equations is composed of the one-group diffusion equation with temperature and xenon feedbacks, the iodine-xenon dynamics equations, and an energy balance relation for the core. Control is via full-length and part-length control rod banks, boron, and coolant inlet temperature. The system equations are linearized around an equilibrium state, which is an eigensolution of the nonlinear static equations with feedback. The nonlinear eigenvalue problem is shown to have a unique positive solution under certain conditions by using the bifurcation theory, the solution being obtained by an iteration based on the use of monotone operators. A modal expansion reduces the linearized equations to a lumped parameter system.

Note on the R2 measure of goodness of fit for nonlinear models

Abstract: The coefficient of determination R2 is a standard measure of goodness of fit for mathematical models fitted to empirical data by means of least squares regression. However, for the case of nonlinear models, such as power models and exponential models frequently used in the behavioral sciences, the R2 measure is often subject to incorrect calculations and misinterpretations, producing potentially misleading results. This paper discusses these R2-related issues and presents the proper method of calculation. A fictitious example is used.

Two-dimensional plasma model for the arc-driven rail gun

Abstract: A previously developed one‐dimensional model for studying the fluid‐mechanical electrodynamical properties of the plasma in an arc‐driven rail gun is extended to two dimensions. The analysis includes deriving a set of general, time‐dependent equations, the solution of which yields the associated properties of the arc. These equations are then solved under the assumptions that the flow variables are steady in a frame of reference which accelerates with the arc, and that the effect of the arc’s acceleration upon these variables can be neglected. Numerical calculations are carried out to analyze arcs in recent experiments. In addition to the numerical calculations, some approximate analytic solutions, which are applicable under certain limiting conditions, are also worked out. These limiting‐case solutions are then used to derive a set of scaling relations which indicate how the arc properties vary with gun size, projectile mass, and acceleration characteristics. Considerable discussion of the assumptions and the results is given emphasizing particularly the physical reasons for the differences with previous one‐dimensional calculations.

Numerical fluid dynamics

Abstract: By 1755, Euler was explicitly envisaging analytical fluid dynamics as a mathematical science, that of integrating the Euler–Lagrange equations. But by 1915, at least seven different analytical models were being used to explain the behavior of real fluids. From 1945 on, von Neumann was trying to “arithmetize” the approximate solution of the equations associated with these models, with the help of large-scale, high-speed computers. Thus he was envisaging numerical fluid dynamics as a mathematical science.After briefly reviewing the gradual disintegration of Euler’s concept of analytical fluid dynamics as a mathematical science, this survey article tries to assess the present status and future prospects of numerical fluid dynamics, through a series of case studies.

Applicability of Dam‐Break Flood Wave Models

Abstract: Mathematical models based on the Saint-Venant equations for open channel flow often encounter serious difficulties when applied to natural channels. On the other hand, the approximate flood routing models used in practice may yield results which are in gross error. In this work five mathematical models are constructed based on equations ranging from the complete dynamic system to a simple normal-depth kinematic wave equation. The results of the models are compared between themselves and with experimental data, in the form of free-surface profiles and stage hydrographs. The models are then converted to dimensionless form, which reduces the number of independent parameters controlling their solution. The results of these calculations are presented in the form of dimensionless plots of maximum flood depth and time versus distance along the channel, for various levels of truncation of the open-channel flow equations. Estimates for the permissible range of application of the simplified routing models are given, and recommendations are made for their judicious application.

Nuclear level densities and partition functions with interactions

Abstract: A general theory, based on central limit theorems and unitary-group decompositions of the microscopic H, is given for the nuclear level density. The density appears in terms of convolutions of noninteracting-particle densities with easily calculable interaction functions given explicitly in terms of the Hamiltonian matrix elements.

Simulation of salt water–fresh water interface motion

Abstract: A mathematical model is presented which describes the salt water–fresh water motion with a sharp interface, assuming the validity of the Dupuit approximation. This model is used as a base to derive a numerical model (finite difference method) which is unconditionally convergent and stable. A method for solving the equations is selected together with a convergence accelerating procedure. The treatment of the boundary conditions in the interface is discussed, and a general and automatic solution for that problem is presented. Several tests with analytical solutions have been performed with good results.

Covariance Structures Under Polynomial Constraints: Applications to Correlation and Alpha-Type Structural Models

Abstract: This paper provides methods for the estimation of covariance structure models under polynomial constraints. Estimation is based on maximum likelihood principles under constraints, and the test statistics, parameter estimates, and standard errors are based on a statistical theory that takes into account the constraints. The approach is illustrated by obtaining statistics for the squared multiple correlation, for predictors in a standardized metric, and in the analysis of longitudinal data via old and new models having constraints that cannot be obtained by standard methods.

MADYMO Pedestrian Simulations

Abstract: In this paper four pedestrian models will be presented: three 2-dimensional models with 2, 5 and 7-segments respectively and one 3-dimensional model with 15 segments. All these models were formulated with the general Crash Victim Simulation package MADYMO. Model results will be compared with the experimental results of a Part 572 dummy impacted lateral at two velocities (30 and 40 km/h). The reliability of the models with respect to their complexity will be discussed. Special attention will be given to the mathematical representation of the contact between the pedestrian and sharp vehicle edges and the visualization of the complex 3-dimensional pedestrian motions with a recently developed 3D-Graphics Package.

Inviscid fluid flows

Abstract: Applied Mathematics is the art of constructing mathematical models of observed phenomena so that both qualitative and quantitative results can be predicted by the use of analytical and numerical methods. Theoretical Mechanics is concerned with the study of those phenomena which can be ob- served in everyday life in the physical world around us. It is often characterised by the macroscopic approach which allows the concept of an element or particle of material, small compared to the dimensions of the phenomena being modelled, yet large compared to the molecular size of the material. Then atomic and molecular phenomena appear only as quantities averaged over many molecules. It is therefore natural that the mathemati- cal models derived are in terms of functions which are continuous and well behaved, and that the analytical and numerical methods required for their development are strongly dependent on the theory of partial and ordinary differential equations. Much pure research in Mathematics has been stimu- lated by the need to develop models of real situations, and experimental observations have often led to important conjectures and theorems in Analysis. It is therefore important to present a careful account of both the physical or experimental observations and the mathematical analysis used. The authors believe that Fluid Mechanics offers a rich field for il- lustrating the art of mathematical modelling, the power of mathematical analysis and the stimulus of applications to readily observed phenomena.