Numerical Simulation

Nowadays mathematical modeling and numerical simulations play an important role in life and natural science. Numerous researchers are working in developing different methods and techniques to help understand the behavior of very complex systems, from the brain activity with real importance in medicine to the turbulent flows with important applications in physics and engineering. This book presents an overview of some models, methods, and numerical computations that are useful for the applied research scientists and mathematicians, fluid tech engineers, and postgraduate students

Um segundo estudo de invasão populacional dinâmica a partir da equação do telégrafo reativo e formulação de elementos de contorno - Um ensaio sobre o crescimento tumoral in vitro

This paper is a continuation of a study already carried out on the use of the reactive-telegraph equation to analyse problems of population dynamics based on a formulation of the boundary element method (BEM). In this paper, the numerical model simulates the evolution of a tumour as a problem of population density of cancer cells from different reactive terms coupled to the reactive-telegraph equation to describe the growth and distribution of the population, similar to the two-dimensional in vitro tumour growth experiment. The mathematical model developed is called D-BEM, uses a time independent fundamental solution and the finite difference method is combined with BEM to approximate the time derivative terms and the Gaussian quadrature is used to calculate the domain integrals. The solution of the system nonlinear of equations is based on the Gaussian elimination method and the stability of the proposed formulation was verified. As the telegraph equation has a wave behaviour, a phase change phenomenon that can lead to the appearance of negative population density may occur, an algorithm was developed to guarantee the solution's positivity. Important results were obtained and demonstrate the effect of the delay parameter on tumour growth. In one of the tested cases, the results indicated an oscillatory behaviour in the tumour growth when the delay parameter assumed increasing values. The results of numerical simulations that sought to represent tumour growth, as well as the entire formulation of the boundary elements are presented below.Este artigo é a continuação de um estudo já realizado sobre o uso da equação do telégrafo reativo para analisar problemas de dinâmica populacional a partir de uma formulação do método dos elementos de contorno (BEM). Neste artigo, o modelo numérico simula a evolução de um tumor como um problema de densidade populacional de células cancerosas a partir de diferentes termos reativos acoplados à equação do telégrafo reativo para descrever o crescimento e distribuição da população, semelhante ao experimento de crescimento do tumor in vitro. O modelo matemático desenvolvido é denominado D-BEM, usa uma solução fundamental independente do tempo e o método das diferenças finitas é combinado com o BEM para aproximar os termos de tempo derivativos e a quadratura Gaussiana é usada para calcular as integrais de domínio. A solução do sistema de equações é baseada no método de eliminação gaussiana e foi verificada a estabilidade da formulação proposta. Como a equação do telégrafo possui comportamento ondulatório, pode ocorrer um fenômeno de mudança de fase que pode levar ao aparecimento de densidade populacional negativa, para tanto, foi desenvolvido um algoritmo que garantir a positividade da solução. Resultados importantes foram obtidos e demonstram o efeito do parâmetro de atraso no crescimento do tumor. Em um dos casos testados, os resultados indicaram um comportamento oscilatório no crescimento tumoral quando o parâmetro de retardo assumiu valores crescentes. O importante resultado dessa antítese para o crescimento do tumor, bem como toda a formulação dos elementos de contorno são apresentados a seguir

A Fractional Analog of Crank–Nicholson Method for the Two Sided Space Fractional Partial Equation with Functional Delay

For two sided space fractional diffusion equation with time functional after-effect, an implicit numerical method is constructed and the order of its convergence is obtained. The method is a fractional analogue of the Crank–Nicholson method, and also uses interpolation and extrapolation of the prehistory of model with respect to time.This work was supported by Government of the Russian Federation program 02.A03.21.0006on 27.08.2013 and by Russian Science Foundation 14-35-00005

Numerical evaluation of corona discharge as a means of boundary layer control and drag reduction

Problems of viscous drag reduction and boundary layer control have been and continue to be objectives of research for their economic impact and the enhancement of the flight characteristics of flying vehicles. Corona discharge is a new technique in this regard;The study of this technique requires consideration of the electrostatics and fluid mechanics. In order to effectively evaluate the technique with minimum complications, the geometry chosen was a dc positive corona discharge on a flat plate of zero thickness at zero angle of attack with the fluid flow as steady state, two-dimensional, incompressible viscous one. Five coupled partial differential equations govern this model requiring the simultaneous solution of these equations. A finite difference method has been employed to approximate these equations through an appropriate scheme for each equation. A clustered grid is used in the vertical direction to handle the high velocity gradient inside the boundary layer. The insufficient boundary conditions necessary for the numerical solution of Poisson\u27s equation is compensated by making the numerical model find the appropriate computational domain which leads to a unique solution. Stability conditions of the five-point scheme approximating Poisson\u27s equation has been determined computationally;Results obtained using the numerical model are presented. As a result of this research the corona discharge near a surface of finite conductivity is now better understood as an electrostatic phenomena. The corona discharge between wire-to-wire electrodes occurs if the electric potential difference between the electrodes is raised to a value higher than the corona onset voltage. The corona current is proportional to the potential difference between the electrodes and inversely proportional to the corona wire diameter. At the same time it does not significantly respond to the gap length between the electrodes until the corona wire diameter becomes large, then it varies inversely as the gap length;The corona discharge can be applied to reduce drag on bodies when the Reynolds number is relatively small. The drag reduction achieved by corona discharge inside a boundary layer is a function of many parameters. The drag reduction is proportional to both the electric potential difference and the gap length between the two electrodes and inversely proportional to the free stream velocity. It is also proportional to the location of the corona electrodes as measured from the plate leading edge and inversely proportional to the corona wire diameter. The increased effect of corona discharge at low flow speeds confirms its ability to significantly enhance the cooling rate of a hot body by boosting the convection of the flow around that body. The quantitative analysis of electrostatic cooling is the natural extension of this study

Quantitative non-destructive testing

The work undertaken during this period included two primary efforts. The first is a continuation of theoretical development from the previous year of models and data analyses for NDE using the Optical Thermal Infra-Red Measurement System (OPTITHIRMS) system, which involves heat injection with a laser and observation of the resulting thermal pattern with an infrared imaging system. The second is an investigation into the use of the thermoelastic effect as an effective tool for NDE. As in the past, the effort is aimed towards NDE techniques applicable to composite materials in structural applications. The theoretical development described produced several models of temperature patterns over several geometries and material types. Agreement between model data and temperature observations was obtained. A model study with one of these models investigated some fundamental difficulties with the proposed method (the primitive equation method) for obtaining diffusivity values in plates of thickness and supplied guidelines for avoiding these difficulties. A wide range of computing speeds was found among the various models, with a one-dimensional model based on Laplace's integral solution being both very fast and very accurate

Investigation and development of implicit numerical methods for building energy simulation

A variety of building energy analysis and simulation tools are increasingly used to determine peak heating and cooling loads, size thermal plant, anticipate annual energy consumption and analyse thermal comfort. Numerical solution techniques are considered the most flexible for building energy simulation. When applied to the differential equations modelling energy flows in buildings, they give rise to a system of non-linear algebraic (difference) equations. In order to evaluate numerical methods for building energy simulation, the problem has been characterized mathematically and comprehensive test problems (equation sets) with these characteristics have been prepared. The principal attribute of the problem was found to be a stifiess ratio of the order of lo4. Candidate methods have been programmed and their outputs compared, in numerical experiments, with highly accurate (converged) solutions for the test problems. The accepted validation methods, empirical validation, analytical verification and inter-modal comparison were considered inappropriate. The first estimates total and not just numerical error, the second is too confined and the third lacks an absolute standard. The main evaluation parameter used was computational efficiency which is defined as accuracy attained per unit (computational) effort expended. An improved difference equation solver has been proposed and compared with the one used in the European reference model (ESP) and elsewhere. It was found to produce 27% less error than the currently used method. A fundamental method for estimating the pre-conditioning period of a building has been put forward in this part of the work. The trapezoidal rule (TR) is currently used in a number of building energy simulation packages including ESP. A known instability associated with the method is described and an implicit member of the Runge-Kutta family, possessing the necessary strong stability, has been shown, using the test problems, to be more efficient than TR by a factor of 4.27

The starting transient of solid propellant rocket motors with high internal gas velocities

A comprehensive analytical model which considers time and space development of the flow field in solid propellant rocket motors with high volumetric loading density is described. The gas dynamics in the motor chamber is governed by a set of hyperbolic partial differential equations, that are coupled with the ignition and flame spreading events, and with the axial variation of mass addition. The flame spreading rate is calculated by successive heating-to-ignition along the propellant surface. Experimental diagnostic studies have been performed with a rectangular window motor (50 cm grain length, 5 cm burning perimeter and 1 cm hydraulic port diameter), using a controllable head-end gaseous igniter. Tests were conducted with AP composite propellant at port-to-throat area ratios of 2.0, 1.5, 1.2, and 1.06, and head-end pressures from 35 to 70 atm. Calculated pressure transients and flame spreading rates are in very good agreement with those measured in the experimental system

Numerical Method for Solving the Nonlinear Superdiffusion Equation with Functional Delay

For a space-fractional diffusion equation with a nonlinear superdiffusion coefficient and with the presence of a delay effect, the grid numerical method is constructed. Interpolation and extrapolation procedures are used to account for the functional delay. At each time step, the algorithm reduces to solving a linear system with a main matrix that has diagonal dominance. The convergence of the method in the maximum norm is proved. The results of numerical experiments with constant and variable delays are presented. © 2023 by the authors.Russian Science Foundation, RSF: 22-21-00075This research was funded by the Russian Science Foundation grant number 22-21-00075