Showing papers on "Mathematical model published in 2019"

Advances in pantographic structures: design, manufacturing, models, experiments and image analyses

Abstract: In the last decade, the exotic properties of pantographic metamaterials have been investigated and different mathematical models (both discrete or continuous) have been introduced. In a previous publication, a large part of the already existing literature about pantographic metamaterials has been presented. In this paper, we give some details about the next generation of research in this field. We present an organic scheme of the whole process of design, fabrication, experiments, models and image analyses.

Assessing parameter identifiability in compartmental dynamic models using a computational approach: application to infectious disease transmission models.

Abstract: Mathematical modeling is now frequently used in outbreak investigations to understand underlying mechanisms of infectious disease dynamics, assess patterns in epidemiological data, and forecast the trajectory of epidemics. However, the successful application of mathematical models to guide public health interventions lies in the ability to reliably estimate model parameters and their corresponding uncertainty. Here, we present and illustrate a simple computational method for assessing parameter identifiability in compartmental epidemic models. We describe a parametric bootstrap approach to generate simulated data from dynamical systems to quantify parameter uncertainty and identifiability. We calculate confidence intervals and mean squared error of estimated parameter distributions to assess parameter identifiability. To demonstrate this approach, we begin with a low-complexity SEIR model and work through examples of increasingly more complex compartmental models that correspond with applications to pandemic influenza, Ebola, and Zika. Overall, parameter identifiability issues are more likely to arise with more complex models (based on number of equations/states and parameters). As the number of parameters being jointly estimated increases, the uncertainty surrounding estimated parameters tends to increase, on average, as well. We found that, in most cases, R0 is often robust to parameter identifiability issues affecting individual parameters in the model. Despite large confidence intervals and higher mean squared error of other individual model parameters, R0 can still be estimated with precision and accuracy. Because public health policies can be influenced by results of mathematical modeling studies, it is important to conduct parameter identifiability analyses prior to fitting the models to available data and to report parameter estimates with quantified uncertainty. The method described is helpful in these regards and enhances the essential toolkit for conducting model-based inferences using compartmental dynamic models.

Real-time simulation of large-scale HTS systems: multi-scale and homogeneous models using the T-A formulation

Abstract: The emergence of second-generation high temperature superconducting (HTS) tapes has favored the development of large-scale superconductor systems. The mathematical models capable of estimating electromagnetic quantities in superconductors have evolved from simple analytical models to complex numerical models. The available analytical models are limited to the analysis of single wires or infinite arrays that, in general, do not represent actual devices in real applications. The numerical models based on the finite element method using the H formulation of Maxwell's equations are useful for the analysis of medium-size systems, but their application in large-scale systems is problematic due to the excessive computational cost in terms of memory and computation time. Therefore it is necessary to devise new strategies to make the computation more efficient. The homogenization and the multi-scale methods have successfully simplified the description of the systems allowing the study of large-scale systems. Also, efficient calculations have been recently achieved using the T–A formulation. In the present work, we propose a series of adaptations to the multi-scale and homogenization methods so that they can be efficiently used in conjunction with the T–A formulation to compute the distribution of current density and hysteresis losses in the superconducting layer of superconducting tapes. The computation time and the amount of memory are substantially reduced up to a point that it is possible to achieve real-time simulations of large-scale HTS systems under slow ramping cycles of practical importance on personal computers.

Physics-Informed Neural Networks for Power Systems.

Abstract: This paper introduces for the first time, to our knowledge, a framework for physics-informed neural networks in power system applications. Exploiting the underlying physical laws governing power systems, and inspired by recent developments in the field of machine learning, this paper proposes a neural network training procedure that can make use of the wide range of mathematical models describing power system behavior, both in steady-state and in dynamics. Physics-informed neural networks require substantially less training data and can result in simpler neural network structures, while achieving high accuracy. This work unlocks a range of opportunities in power systems, being able to determine dynamic states, such as rotor angles and frequency, and uncertain parameters such as inertia and damping at a fraction of the computational time required by conventional methods. This paper focuses on introducing the framework and showcases its potential using a single-machine infinite bus system as a guiding example. Physics-informed neural networks are shown to accurately determine rotor angle and frequency up to 87 times faster than conventional methods.

A Comparative Study of Control-Oriented Thermal Models for Cylindrical Li-Ion Batteries

Abstract: An accurate control-oriented thermal model is of extreme importance for temperature monitoring and thermal management of lithium (Li)-ion batteries in automotive and grid applications. This article, for the first time, presents a comprehensively comparative study of seven representative control-oriented thermal models for cylindrical Li-ion batteries. These models were selected from the state-of-the-art simplified models reported in the existing literature. All these models are introduced in detail. The model assumptions of physical structure, heat generation, and heat conduction are analyzed. Particle swarm optimization (PSO) algorithm is utilized to identify model parameters. Modeling fidelity is evaluated and compared using both simulation campaigns and experimental data sets of cylindrical Li-iron phosphate batteries. The sensitivity of all 1-D models to convective heat transfer coefficient (or convective heat resistance) is quantitatively analyzed, and the computational complexity of the models is compared under a real drive cycle. All the comparisons are thoroughly discussed, and useful insights are provided.

A Review of Physical and Numerical Approaches for the Study of Gas Stirring in Ladle Metallurgy

Abstract: This article presents a review of the research into gas stirring in ladle metallurgy carried out over the past few decades. Herein, the physical modeling experiments are divided into four major areas: (1) mixing and homogenization in the ladle; (2) gas bubble formation, transformation, and interactions in the plume zone; (3) inclusion behavior at the steel–slag interface and in the molten steel; and (4) open eye formation. Several industrial trials have also been carried out to optimize gas stirring and open eye formation. Approaches for selecting criteria for scaling to guarantee flow similarity between industrial trials and physical modeling experiments are discussed. To describe the bubble behavior and two-phase plume structure, four main mathematical models have been used in different research fields: (1) the quasi-single-phase model, (2) the volume of fluid (VOF) model, (3) the Eulerian multiphase (E–E) model, and (4) the Eulerian–Lagrangian (E–L) model. In recent years, the E–E model has been used to predict gas stirring conditions in the ladle, and specific models in commercial packages, as well as research codes, have been developed gradually to describe the complex physical and chemical phenomena. Furthermore, the coupling of turbulence models with multiphase models is also discussed. For physical modeling, some general empirical rules have not been analyzed sufficiently. Based on a comparison with the available experimental results, it is found that the mathematical models focusing on the mass transfer phenomenon and inclusion behaviors at the steel-slag interface, vacuum degassing at the gas–liquid interface, dissolution rate of the solid alloy at the liquid–solid interface, and the combination of fluid dynamics and thermodynamics need to be improved further. To describe industrial conditions using mathematical methods and improve numerical modeling, the results of physical modeling experiments and industrial trials must offer satisfactory validations for the improvement of numerical modeling.

Gradient-Based Iterative Parameter Estimation Algorithms for Dynamical Systems from Observation Data

Abstract: It is well-known that mathematical models are the basis for system analysis and controller design. This paper considers the parameter identification problems of stochastic systems by the controlled autoregressive model. A gradient-based iterative algorithm is derived from observation data by using the gradient search. By using the multi-innovation identification theory, we propose a multi-innovation gradient-based iterative algorithm to improve the performance of the algorithm. Finally, a numerical simulation example is given to demonstrate the effectiveness of the proposed algorithms.

Day-Ahead Optimal Operation for Multi-Energy Residential Systems With Renewables

Abstract: The intermittency and stochasticity nature of distributed renewable energy sources has introduced great challenges to the efficiency and security of energy distribution system operations. To address the negative impacts of intermittent renewable energy sources, this paper proposes a day-ahead optimal operation strategy utilizing distributed energy resources based on the framework of the interconnected multi-energy system. First, a framework and mathematical models of multi-energy residential systems (MERS) are proposed. Based on the characteristics of residential energy distribution networks, the complex MERS models are reformulated to relieve the computational burden. Furthermore, the uncertainty factors such as renewable energy generation fluctuations and demand variations are handled by a reformulated chance constrained programing technique. The feasibility and effectiveness of the proposed method are validated through a combined electric power and natural gas test system. Compared to similar models and methods in the existing literature, the proposed method performs better in terms of solution time, model scalability, and robustness in handling uncertainties.

Constitutive relations for compressible granular flow in the inertial regime

Abstract: Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the μ(I),Φ(I)-rheology, which postulates that the bulk friction coefficient μ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction ϕ are functions of the inertial number I only. Although the μ(I),Φ(I)-rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the μ(I),Φ(I)-rheology that does not suffer from such defects is proposed. In the framework of compressible I-dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established μ(I) and Φ(I) relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.

Optimal Design of Experiments for a Lithium-Ion Cell: Parameters Identification of an Isothermal Single Particle Model with Electrolyte Dynamics

Abstract: Advanced battery management systems rely on mathematical models to guarantee optimal functioning of Lithium-ion batteries. The pseudo-two dimensional (P2D) model is a very detailed electrochemical model suitable for simulations. On the other side, its complexity prevents its usage in control and state estimation. Therefore, the use of simplified electrochemical models such as the Single Particle Model with electrolyte dynamics (SPMe) is more appropriate, which exhibits good adherence to real data when suitably calibrated. This work focuses on a Fisher-based optimal experimental design for identifying the SPMe parameters assuming an isothermal setup. The proposed approach relies on an iterative nonlinear optimization scheme to minimize the covariance parameters matrix. At first, the parameters are estimated by considering the SPMe as the real plant. Subsequently, a more realistic scenario is considered where the isothermal P2D model is used to reproduce a real battery behavior. Results show the effectivene...

Polynomial Dynamical Systems, Reaction Networks, and Toric Differential Inclusions

Abstract: Some of the most common mathematical models in biology, chemistry, physics, and engineering are polynomial dynamical systems, i.e., systems of differential equations with polynomial right-hand side...

A topological approach to selecting models of biological experiments.

Abstract: We use topological data analysis as a tool to analyze the fit of mathematical models to experimental data. This study is built on data obtained from motion tracking groups of aphids in [Nilsen et al., PLOS One, 2013] and two random walk models that were proposed to describe the data. One model incorporates social interactions between the insects via a functional dependence on an aphid’s distance to its nearest neighbor. The second model is a control model that ignores this dependence. We compare data from each model to data from experiment by performing statistical tests based on three different sets of measures. First, we use time series of order parameters commonly used in collective motion studies. These order parameters measure the overall polarization and angular momentum of the group, and do not rely on a priori knowledge of the models that produced the data. Second, we use order parameter time series that do rely on a priori knowledge, namely average distance to nearest neighbor and percentage of aphids moving. Third, we use computational persistent homology to calculate topological signatures of the data. Analysis of the a priori order parameters indicates that the interactive model better describes the experimental data than the control model does. The topological approach performs as well as these a priori order parameters and better than the other order parameters, suggesting the utility of the topological approach in the absence of specific knowledge of mechanisms underlying the data.

Mathematical Models for Air Traffic Conflict and Collision Probability Estimation

Abstract: Increasing traffic demands and technological developments provide novel design opportunities for future air traffic management (ATM). In order to evaluate current air traffic operations and future designs, over the past decades, several mathematical models have been proposed for air traffic conflict and collision probability estimation. However, few comparative evaluations of these models with respect to their mathematical core exist. Such comparative evaluations are particularly difficult since different authors employ different model definitions, notations, and assumptions, even when using the same modeling techniques. The aim of this paper is: 1) to present the mathematical core of the existing approaches for air traffic conflict and collision probability estimation using the same body of notations and definitions; 2) to outline the advances in estimating the probability of air traffic conflict and collision using a unified mathematical framework; 3) to various air traffic applications and their use of directed mathematical models for air traffic conflict and collision probability estimation; and 4) to provide insight into the capabilities and restrictions of the mathematical models in the evaluation of future ATM designs.

Classical and contemporary fractional operators for modeling diarrhea transmission dynamics under real statistical data

Abstract: This paper is all about the development and analysis of an epidemiological model related to the disease of diarrhea that occurred in Ghana during 2008–2015. Using real statistical data, three new fractional-order mathematical models have been developed on the basis of having information about existing classical model . The new models are formulated with Caputo, Caputo–Fabrizio–Caputo and the Atangana–Baleanu–Caputo fractional-order approaches while taking care of the dimensional analysis during the process of fractionalization. Besides, existence and uniqueness for the solutions of the fractional-order models under each case are proved with the help of fixed point theory whereas positivity and boundedness of models’ solution are also investigated. Steady-states (disease-free and endemic equilibria) points of the model and sensitivity of the basic reproductive number ( R 0 ) are also explored. While many of the model’s parameters are fixed, the transmission rate ( β ) of the disease has been estimated and so is the case with orders of the fractional models. Using minimum distance approach, it has been found that the diarrhea model under investigation estimates the real statistical data well enough when considered with the Atangana–Baleanu–Caputo fractional order operator which has non-local and non-singular kernel. Thus, this fractional-order operator of Atangana–Baleanu in the present research study for the diarrhea model outperforms those having index law, power law and stretched exponential kernels.

First order phase transitions and the thermodynamic limit

Abstract: We consider simple mean field continuum models for first order liquid-liquid demixing and solid-liquid phase transitions and show how the Maxwell construction at phase coexistence emerges on going from finite-size closed systems to the thermodynamic limit. The theories considered are the Cahn-Hilliard model of phase separation, which is also a model for the liquid-gas transition, and the phase field crystal model of the solid-liquid transition. Our results show that states comprising the Maxwell line depend strongly on the mean density with spatially localized structures playing a key role in the approach to the thermodynamic limit.

Dimensionless Analysis for Investigating the Quality Characteristics of Aluminium Matrix Composites Prepared through Fused Deposition Modelling Assisted Investment Casting

Abstract: The aluminium matrix composites (AMCs) have become a tough competitor for various categories of metallic alloys, especially ferrous materials, owing to their tremendous servicing in the diversified application. In this work, additional efforts have been made to formulate a mathematical model, by using dimensionless analysis, able to predict the mechanical characteristics of the AMCs that have already been optimized and characterized by the authors. Here, the experimental and statistical data obtained from the Taguchi L18 orthogonal array and analysis of variance (ANOVA) have been used. They permit collection of the output responses and allow the identification of significant process parameters, respectively, which thereafter were used to design the mathematical model. Second order polynomial equations have been obtained from the specific output response and the relevant input parameter were incorporated with the highest level of contribution. The obtained quadratic equations indicate the regression values (R2) equal to unity, hence, proving the performances of the fit. The results demonstrate that the developed mathematical models present very high accuracy for predicting the output responses.

In-process identification of modal parameters using dimensionless relationships in milling chatter

Abstract: Machining parameters needed for stable, high-performance high-speed machining could be found using mathematical models that need accurate measurements of modal parameters of the machining system. In-process modal parameters, however, can slightly differ from those measured offline and this can limit the applicability of simple measurement methods such as impact hammer tests. To study and extract the in-process modal parameters, mathematical models are used to define two key dimensionless parameters and establish their relationships with each other and the modal parameters. Based on these relationships, the modal parameters are extracted using two analytical methods, the two-point method (TPM), and the regression method (RM). As shown with experimental studies, the RM extracts the modal parameters successfully and while being much faster than the existing iteration-based methods, it provides stability lobe predictions that match well the experimental results. Furthermore, it is noted that the natural frequency parameter is estimated with much better relative precision compared to the damping ratio and the modal stiffness parameters.

Numerical Investigation of the Time Fractional Mobile-Immobile Advection-Dispersion Model Arising from Solute Transport in Porous Media

Abstract: Evolution equations containing fractional derivatives can offer efficient mathematical models for determination of anomalous diffusion and transport dynamics in multi-faceted systems that cannot be precisely modeled by using normal integer order equations. In recent times, researches have found out that lots of physical processes illustrate fractional order characteristics that alters with time or space. The continuum of order in the fractional calculus permits the order of the fractional operator be accounted for as a variable. In the current research work, radial basis functions (RBFs) approximation is utilized for solving fractional mobile-immobile advection-dispersion (TF-MIM-AD) model in a bounded domain which is applied for explaining solute transport in both porous and fractured media. In this approach, firstly, the discretization process of the aforesaid equation with of convergence order $$\mathcal {O}(\delta t^{})$$ in the t-direction is described via the finite difference scheme for $$ 0< \alpha <1$$ . Afterwards, by help of the meshless methods based on RBFs, we will illustrate how to obtain the approximated solution. The stability and convergence of time-discretized scheme are also theoretically discussed in detail throughout the paper. Finally, two numerical instances are included to clarify effectiveness and accuracy of our proposed concepts which is investigated in the current research work.