Showing papers on "Mathematical model published in 2011"

Numerical Methods for Two-phase Incompressible Flows

Abstract: Introduction.- Part I One-phase incompressible flows.- Mathematical models.- Finite element discretization.- Time integration.-

Modelling of suspension components in a rail vehicle dynamics context

Abstract: Suspension components play key roles in the running behaviour of rail vehicles, and therefore, mathematical models of suspension components are essential ingredients of railway vehicle multi-body models. The aims of this paper are to review existing models for railway vehicle suspension components and their use for railway vehicle dynamics multi-body simulations, to describe how model parameters can be defined and to discuss the required level of detail of component models in view of the accuracy expected from the overall simulation model. This paper also addresses track models in use for railway vehicle dynamics simulations, recognising their relevance as an indispensable component of the system simulation model. Finally, this paper reviews methods presently in use for the checking and validation of the simulation model.

Simplified Modeling of a DFIG for Transient Studies in Wind Power Applications

Abstract: Improving the fault ride-through (FRT) capability of doubly fed induction generators (DFIGs) in wind power applications is a very important challenge for the wind power industry. The mathematical models of such generators enable us to analyze their response under generic conditions. However, their mathematical complexity does not contribute to simplifying the analysis of the system under transient conditions and hence does not help in finding straightforward solutions for enhancing their FRT. This paper presents a simplified model of the DFIG, which has been extracted from the classical fifth-order model, which can accurately estimate the behavior of the system while significantly reducing its complexity. In this paper, the mathematical deduction of this model will be presented, and simulations and experimental results will be shown to demonstrate the accuracy and reliability of the proposed algorithm.

Introduction to Finite Element Analysis: Formulation, Verification and Validation

Abstract: About the Authors. Series Preface. Preface. 1 Introduction. 1.1 Numerical simulation. 1.2 Why is numerical accuracy important? 1.3 Chapter summary. 2 An outline of the finite element method. 2.1 Mathematical models in one dimension. 2.2 Approximate solution. 2.3 Generalized formulation in one dimension. 2.4 Finite element approximations. 2.5 FEM in one dimension. 2.6 Properties of the generalized formulation. 2.7 Error estimation based on extrapolation. 2.8 Extraction methods. 2.9 Laboratory exercises. 2.10 Chapter summary. 3 Formulation of mathematical models. 3.1 Notation. 3.2 Heat conduction. 3.3 The scalar elliptic boundary value problem. 3.4 Linear elasticity. 3.5 Incompressible elastic materials. 3.6 Stokes' flow. 3.7 The hierarchic view of mathematical models. 3.8 Chapter summary. 4 Generalized formulations. 4.1 The scalar elliptic problem. 4.2 The principle of virtual work. 4.3 Elastostatic problems. 4.4 Elastodynamic models. 4.5 Incompressible materials. 4.6 Chapter summary. 5 Finite element spaces. 5.1 Standard elements in two dimensions. 5.2 Standard polynomial spaces. 5.3 Shape functions. 5.4 Mapping functions in two dimensions. 5.5 Elements in three dimensions. 5.6 Integration and differentiation. 5.7 Stiffness matrices and load vectors. 5.8 Chapter summary. 6 Regularity and rates of convergence. 6.1 Regularity. 6.2 Classification. 6.3 The neighborhood of singular points. 6.4 Rates of convergence. 6.5 Chapter summary. 7 Computation and verification of data. 7.1 Computation of the solution and its first derivatives. 7.2 Nodal forces. 7.3 Verification of computed data. 7.4 Flux and stress intensity factors. 7.5 Chapter summary. 8 What should be computed and why? 8.1 Basic assumptions. 8.2 Conceptualization: drivers of damage accumulation. 8.3 Classical models of metal fatigue. 8.4 Linear elastic fracture mechanics. 8.5 On the existence of a critical distance. 8.6 Driving forces for damage accumulation. 8.7 Cycle counting. 8.8 Validation. 8.9 Chapter summary. 9 Beams, plates and shells. 9.1 Beams. 9.2 Plates. 9.3 Shells. 9.4 The Oak Ridge experiments. 9.5 Chapter summary. 10 Nonlinear models. 10.1 Heat conduction. 10.2 Solid mechanics. 10.3 Chapter summary. A Definitions. A.1 Norms and seminorms. A.2 Normed linear spaces. A.3 Linear functionals. A.4 Bilinear forms. A.5 Convergence. A.6 Legendre polynomials. A.7 Analytic functions. A.8 The Schwarz inequality for integrals. B Numerical quadrature. B.1 Gaussian quadrature. B.2 Gauss Lobatto quadrature. C Properties of the stress tensor. C.1 The traction vector. C.2 Principal stresses. C.3 Transformation of vectors. C.4 Transformation of stresses. D Computation of stress intensity factors. D.1 The contour integral method. D.2 The energy release rate. E Saint-Venant's principle. E.1 Green's function for the Laplace equation. E.2 Model problem. F Solutions for selected exercises. Bibliography. Index.

Logistic modeling of the equilibrium speed–density relationship

Abstract: The fundamental diagram, as the graphical representation of the relationships among traffic flow, speed, and density, has been the foundation of traffic flow theory and transportation engineering. Seventy-five years after the seminal Greenshields model, a variety of models have been proposed to mathematically represent the speed-density relationship which underlies the fundamental diagram. Observed in these models was a clear path toward two competing goals: mathematical elegance and empirical accuracy. As the latest development of such a pursuit, this paper presents a family of speed-density models with varying numbers of parameters. All of these models perform satisfactorily and have physically meaningful parameters. In addition, speed variation with traffic density is accounted for; this enables statistical approaches to traffic flow analysis. The results of this paper not only improve our understanding of traffic flow but also provide a sound basis for transportation engineering studies.

On the Modeling of Elastic Contact between Rough Surfaces

Abstract: The contact force and the real contact area between rough surfaces are important in the prediction of friction, wear, adhesion, and electrical and thermal contact resistance. Over the last four decades various mathematical models have been developed. Built on very different assumptions and underlying mathematical frameworks, model agreement or effectiveness has never been thoroughly investigated. This work uses several measured profiles of real surfaces having vastly different roughness characteristics to predict contact areas and forces from various elastic contact models and contrast them to a deterministic fast Fourier transform (FFT)-based contact model. The latter is considered “exact” because surfaces are analyzed as they are measured, accounting for all peaks and valleys without compromise. Though measurement uncertainties and resolution issues prevail, the same surfaces are kept constant (i.e., are identical) for all models considered. Nonetheless, the effect of the data resolution of measured sur...

Practical limits for reverse engineering of dynamical systems: a statistical analysis of sensitivity and parameter inferability in systems biology models.

Abstract: The size and complexity of cellular systems make building predictive models an extremely difficult task. In principle dynamical time-course data can be used to elucidate the structure of the underlying molecular mechanisms, but a central and recurring problem is that many and very different models can be fitted to experimental data, especially when the latter are limited and subject to noise. Even given a model, estimating its parameters remains challenging in real-world systems. Here we present a comprehensive analysis of 180 systems biology models, which allows us to classify the parameters with respect to their contribution to the overall dynamical behaviour of the different systems. Our results reveal candidate elements of control in biochemical pathways that differentially contribute to dynamics. We introduce sensitivity profiles that concisely characterize parameter sensitivity and demonstrate how this can be connected to variability in data. Systematically linking data and model sloppiness allows us to extract features of dynamical systems that determine how well parameters can be estimated from time-course measurements, and associates the extent of data required for parameter inference with the model structure, and also with the global dynamical state of the system. The comprehensive analysis of so many systems biology models reaffirms the inability to estimate precisely most model or kinetic parameters as a generic feature of dynamical systems, and provides safe guidelines for performing better inferences and model predictions in the context of reverse engineering of mathematical models for biological systems.

Lattice models of nonequilibrium bacterial dynamics

Abstract: We study a model of self-propelled particles exhibiting run-and-tumble dynamics on a lattice. This non-Brownian diffusion is characterized by a random walk with a finite persistence length between changes of direction and is inspired by the motion of bacteria such as E. coli. By defining a class of models with multiple species of particles and transmutation between species we can recreate such dynamics. These models admit exact analytical results whilst also forming a counterpart to previous continuum models of run-and-tumble dynamics. We solve the externally driven non-interacting and zero-range versions of the model exactly and utilize a field-theoretic approach to derive the continuum fluctuating hydrodynamics for more general interactions. We make contact with prior approaches to run-and-tumble dynamics off lattice and determine the steady state and linear stability for a class of crowding interactions, where the jump rate decreases as density increases. In addition to its interest from the perspective of nonequilibrium statistical mechanics, this lattice model constitutes an efficient tool to simulate a class of interacting run-and-tumble models relevant to bacterial motion, so long as certain conditions (that we derive) are met.

The Mechanics of Solids and Structures - Hierarchical Modeling and the Finite Element Solution

Abstract: Mathematical Models in the Finite Element Solution.- Fundamental Steps in Structural Mechanics.- The Linear 3-D Elasticity Mathematical Model.- Mathematical Models used in Engineering Structural Analysis.- The Principle of Virtual Work.- The Finite Element Process of Solution.- Hierarchical Modeling Examples.- Modeling for Nonlinear Analysis.

Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology

Abstract: Newtonian and generalized Newtonian mathematical models for blood flow are compared in two different reconstructions of an anatomically realistic geometry of a saccular aneurysm, obtained from rotational CTA and differing to within image resolution. The sensitivity of the flow field is sought with respect to geometry reconstruction procedure and mathematical model choice in numerical simulations. Taking as example a patient specific intracranial aneurysm located on an outer bend under steady state simulations, it is found that the sensitivity to geometry variability is greater, but comparable, to the one of the rheological model. These sensitivities are not quantifiable a priori. The flow field exhibits a wide range of shear stresses and slow recirculation regions that emphasize the need for careful choice of constitutive models for the blood. On the other hand, the complex geometrical shape of the vessels is found to be sensitive to small scale perturbations within medical imaging resolution. The sensitivity to mathematical modeling and geometry definition are important when performing numerical simulations from in vivo data, and should be taken into account when discussing patient specific studies since differences in wall shear stress range from 3% to 18%.

A Bayesian Framework for Parameter Estimation in Dynamical Models

Abstract: Mathematical models in biology are powerful tools for the study and exploration of complex dynamics. Nevertheless, bringing theoretical results to an agreement with experimental observations involves acknowledging a great deal of uncertainty intrinsic to our theoretical representation of a real system. Proper handling of such uncertainties is key to the successful usage of models to predict experimental or field observations. This problem has been addressed over the years by many tools for model calibration and parameter estimation. In this article we present a general framework for uncertainty analysis and parameter estimation that is designed to handle uncertainties associated with the modeling of dynamic biological systems while remaining agnostic as to the type of model used. We apply the framework to fit an SIR-like influenza transmission model to 7 years of incidence data in three European countries: Belgium, the Netherlands and Portugal.

A bottom-up solution for the multi-facility optimal pavement resurfacing problem

Abstract: Transportation infrastructure management has been a subject of growing economic importance in recent years due to the magnitude of agency expenditures. Increasingly sophisticated methods have been developed to model pavement deterioration and solve for optimal management strategies. However, it is unclear whether these more complex methods are providing more useful results. This paper presents a simple approach for optimizing the frequency and intensity of resurfacing for multiple highway facilities. It builds upon existing optimization methods for the single-facility, continuous-state, continuous-time problem and corresponding results, which include a threshold structure for optimal solutions. This threshold structure allows for mathematical simplifications and for a straightforward optimization approach to be applied to the multi-facility case. The approach is bottom-up rather than top-down, preserving facility-specific features to develop informative budget allocation results. Application of the approach in a case study indicates that solutions are likely to be robust to deterioration model uncertainty, which is consistent with previous facility-level findings. In addition, the methodology is shown to be robust to the form of the deterioration model.

Waves and Structures in Nonlinear Nondispersive Media: General Theory and Applications to Nonlinear Acoustics

Abstract: Part I. Foundations of The Theory of Waves in Nondispersive. Nonlinear Equations of The First Order.- Generalized Solutions of Nonlinear Equation.- Nonlinear Equations of The Second Order.- Field Evolution Governed by Burgers Equation.- Evolution of A Noise Field Governed.- Multi-dimensional Nonlinear Equations.- Part II. Mathematical Models and Physical Phenomena in Nonlinear Acoustics. Model Equations and Methods Of Finding of Their Exact Solutions.- Types of Acoustic Nonlinearities and Methods of Nonlinear Acoustic Diagnostics.- Nonlinear Sawtooth Waves.- Self-Action of Spatially Bounded Waves Containing Shock Fronts.- Nonlinear Standing Waves, Resonance Phenomena and Frequency Characteristics of Distributed Systems.

Application of Expert Evaluation Method to Determine the Importance of Operating Asphalt Mixing Plant Quality Criteria and Rank Correlation

Abstract: Road asphalt concrete pavement is usually laid of hot-mix asphalt (HMA) mixture. HMA mixture is produced in an asphalt mixing plant (AMP) according to the technology applied in its structure. AMP shall meet certain requirements set in the norms not only to the quality of HMA mix produced in it, but to environmental protection as well. A possibility to produce an HMA mixture of all required types and marks in it shall be available through the use of various materials and additives, including reclaimed asphalt pavement. HMA mixture production costs shall be as low as possible. AMP shall be of appropriate technical condition, and with proper equipment, which mostly influence the technological, ecological, and economic parameters. Its actual productivity shall meet the required scope (HMA mixture amount) of road construction works carried out in the serviced region. The article presents 9 criteria of operating AMP quality, mathematical models of determining their significance through the application of an expert research method as well as expert opinion correlation values. Ranks of AMP quality criteria are replaced by their weight indices through the application of 2 different methodologies. Quality criteria weight indices may be used according to an additive model through the calculation of operating AMP quality multi-criteria index. A numerical sample is presented at the end of the article.

Matter Density and Relativistic Models of Wave Function Collapse

Abstract: Mathematical models for the stochastic evolution of wave functions that combine the unitary evolution according to the Schroedinger equation and the collapse postulate of quantum theory are well understood for non-relativistic quantum mechanics. Recently, there has been progress in making these models relativistic. But even with a fully relativistic law for the wave function evolution, a problem with relativity remains: Different Lorentz frames may yield conflicting values for the matter density at a space-time point. We propose here a relativistic law for the matter density function. According to our proposal, the matter density function at a space-time point x is obtained from the wave function psi on the past light cone of x by setting the i-th particle position in |psi|^2 equal to x, integrating over the other particle positions, and averaging over i. We show that the predictions that follow from this proposal agree with all known experimental facts.

Toward real-time simulation of cardiac dynamics

Abstract: We show that through careful and model-specific optimizations of their GPU implementations, simulations of realistic, detailed cardiac-cell models now can be performed in 2D and 3D in times that are close to real time using a desktop computer. Previously, large-scale simulations of detailed mathematical models of cardiac cells were possible only using supercomputers. In our study, we consider five different models of cardiac electrophysiology that span a broad range of computational complexity: the two-variable Karma model, the four-variable Bueno-Orovio-Cherry-Fenton model, the eight-variable Beeler-Reuter model, the 19-variable Ten Tusscher-Panfilov model, and the 67-variable Iyer-Mazhari-Winslow model. For each of these models, we treat both their single- and double-precision versions and demonstrate linear or even sub-linear growth in simulation times with an increase in the size of the grid used to model cardiac tissue. We also show that our GPU implementations of these models can increase simulation speeds to near real-time for simulations of complex spatial patterns indicative of cardiac arrhythmic disorders, including spiral waves and spiral wave breakup. The achievement of real-time applications without the need for supercomputers may, in the near term, facilitate the adoption of modeling-based clinical diagnostics and treatment planning, including patient-specific electrophysiological studies.

Hydrologic procedures of storm event watershed models: a comprehensive review and comparison

Abstract: Currently, many watershed models are available that have various complexities, strengths, and weaknesses. The basic mathematical foundations of these mathematical models are often overlooked due to high demands on convenient applications with graphical user interfaces. Although this and other factors are important while selecting a model, the mathematical foundation should also be taken into account, as performance or efficiency and accuracy of a model depend on its simplicity or complexity. A comprehensive review of 14 storm event watershed models was conducted. Hydrologic procedures (rainfall excess, flow routing, and subsurface flow) of the models are presented and compiled. Among the procedures, flow routing has the most influence on model performances (speed and accuracy). Overland and channel flow routing procedures using different flow-governing equations, having various approximations and solved by different methods, are compared based on their relative levels of physical bases, complexities, and expected accuracies in simulating the dynamics of water flow. Models using more mathematical terms in the flow-governing equations are more physically based and expected to be more accurate than models using approximations, however, are more complex due to more intensive but approximate numerical schemes (inefficient). Models using approximate equations with analytical solutions may provide a balance between complexity and accuracy. The review and comparisons are useful to modellers, water resources managers, and researchers in understanding the basic foundations of the models and making informed selections for practical applications or further developments. Other factors such as data intensiveness, user friendliness, and resource requirements are also important considerations. Copyright © 2011 John Wiley & Sons, Ltd.

Mathematical Models and Numerical Schemes for the Simulation of Human Phonation

Abstract: Acoustic data has long been harvested in fundamental voice investigations since it is easily obtained using a microphone. However, acoustic signals alone do not reveal much about the complex interplay between sound waves, structural surface waves, mechanical vibrations, and fluid flow involved in phonation. Available high speed imaging techniques have over the past ten years provided a wealth of information about the mechanical deformation of the superior surface of the larynx during phonation. Time-resolved images of the inner structure of the deformable soft tissues are not yet feasible because of low temporal resolution (MRI and ultrasound) and x-ray dose-related hazards (CT and standard x- ray). One possible approach to circumvent these challenges is to use mathematical models that reproduce observable behavior such as phonation frequency, closed quotient, onset pressure, jitter, shimmer, radiated sound pressure, and airflow. Mathematical models of phonation range in complexity from systems with relatively small degrees of freedom (multi-mass models) to models based on partial differential equations (PDEs) mostly solved by finite element (FE) methods resulting in millions of degrees-of-freedom. We will provide an overview about the current state of mathematical models for the human phonation process, since they have served as valuable tools for providing insight into the basic mechanisms of phonation and may eventually be of sufficient detail and accuracy to allow surgical planning, diagnostics, and rehabilitation evaluations on an individual basis. Furthermore, we will also critically discuss these models w.r.t. the used geometry, boundary conditions, material properties, their verification, and reproducibility.

Experimental and Numerical Analysis of Microwave Heating of Water and Oil Using a Rectangular Wave Guide: Influence of Sample Sizes, Positions, and Microwave Power

Abstract: The heating process of water and oil using microwave oven with rectangular wave guide is investigated numerically and experimentally. The numerical model is validated with an experimental study. The transient Maxwell’s equations are solved by using the finite difference time domain method to describe the electromagnetic field in the wave guide and sample. The temperature profiles and velocity field within sample are determined by the solution of the momentum, energy, and Maxwell’s equations. In this study, the effects of physical parameters, e.g., microwave power, the position of sample in wave guide, size, and thickness of sample, are studied. The results of distribution of electric field, temperature profiles, and velocity field are presented in details. The results show that the mathematical models are in agreement with the experimental data. Conclusively, the mathematical model presented in this study correctly explains the phenomena of microwave heating within the liquid layer.

Discretised link travel time models based on cumulative flows: Formulations and properties

Abstract: In the research area of dynamic traffic assignment, link travel times can be derived from link cumulative inflow and outflow curves which are generated by dynamic network loading. In this paper, the profiles of cumulative flows are piecewise linearized. Both the step function (SF) and linear interpolation (LI) are used to approximate cumulative flows over time. New formulations of the SF-type and LI-type link travel time models are developed. We prove that these two types of link travel time models ensure first-in-first-out (FIFO) and continuity of travel times with respect to flows, and have other desirable properties. Since the LI-type link travel time model does not satisfy the causality property, a modified LI-type (MLI-type) link travel time model is proposed in this paper. We prove that the MLI-type link travel time model ensures causality, strong FIFO and travel time continuity, and that the MLI-type link travel time function is strictly monotone under the condition that the travel time of each vehicle on a link is greater than the free flow travel time on that link. Numerical examples are set up to illustrate the properties and accuracy of the three models.