<div>Multiscale and multiphysics have been two major challenges in analyzing and designing new emerging engineering devices, materials, circuits, and systems. When simulating a multiscale problem, numerical methods have to overcome the challenges in both space and time to account for the scales spanning many orders of magnitude difference. In the finite-difference time-domain (FDTD) method, subgridding techniques have been developed to address the multiscale challenge. However, the accuracy and stability in existing subgridding algorithms have always been two competing factors. In terms of the analysis of a multiphysics problem, it involves the solution of multiple partial differential equations. Existing partial differential equation solvers require solving a system matrix when handling inhomogeneous materials and irregular geometries discretized into unstructured meshes. When the problem size, and hence the matrix size, is large, existing methods become highly inefficient.</div><div><br></div><div>In this work, a symmetric positive semi-definite FDTD subgridding algorithm in both space and time is developed for fast transient simulations of multiscale problems. This algorithm is stable and accurate by construction. Moreover, the method is further made unconditionally stable, by analytically finding unstable modes, and subsequently deducting them from the system matrix. To address the multiphysics simulation challenge, we develop a matrix-free time domain method for solving thermal diffusion equation, and the combined Maxwell-thermal equations, in arbitrary unstructured meshes. The counterpart of the method in frequency domain is also developed for fast frequency-domain analysis. In addition, a generic time marching scheme is proposed for simulating unsymmetrical systems to guarantee their stability in time domain. </div>

Este trabalho apresenta um método numérico para simular escoamentos viscoelásticos bidimensionais governados pela equação constitutiva Giesekus [Schleiniger e Weinacht 1991]. As equações governantes são resolvidas pelo método de diferenças finitas numa malha deslocada. A superfície livre do fluido é modelada por partículas marcadoras possibilitando assim a sua visualização e localização. O cálculo da velocidade é efetuado por um método implícito enquanto a pressão é calculada por um método explícito. A equação constitutiva de Giesekus é resolvida pelo método de Euler modificado explícito. O método numérico desenvolvido nesse trabalho é verificado comparando-se a solução numérica com a solução analítica para o escoamento de um fluido Giesekus em um canal. Resultados de convergência são obtidos pelo uso de refinamento de malha. Os resultados alcançados incluem um estudo da aplicação do modelo Giesekus para simular o escoamento numa contração planar 4:1 e o problema de um jato incidindo sobre uma placa rígida, em que o fenômeno jet buckling é simulado. / This work presents a numerical method to simulate two-dimensional viscoelastic flows governed by the Giesekus constitutive equation [Schleiniger e Weinacht 1991]. The governing equations are solved by the finite difference method on a staggered grid. The free surface of the fluid is modeled by tracer particles thus enabling its visualization and location. The calculation of the velocity is performed by an implicit method while pressure is calculated by an explicit method. The Giesekus constitutive equation is resolved by the explicit modified Euler method. The numerical method developed in this work is verified by comparing the numerical solution with the analytical solution for the flow of a Giesekus fluid in a channel. Convergence results are obtained by the use of mesh refinement. Results obtained include a study of the application of the Giesekus model to simulate the flow through a 4:1 contraction and the problem of a jet flowing onto a rigid plate where the phenomenon of jet buckling is simulated.

Este trabalho apresenta parte de um sistema de simulação integrado para escoamento de fluido incompressível bidimensional em malhas não estruturadas denominado UmFlow-2D. O sistema consiste de três módulos: um módulo modelador, um módulo simulador e um módulo visualizador. A parte do sistema apresentado neste trabalho é o módulo simulador. Este módulo, implementa as equações de Navier-Stokes. As equações governantes são discretizadas pelo método de diferenças finitas generalizadas e os termos convectivos pelo método semi-lagrangeano. Um método de projeção é empregado para desacoplar as componentes da velocidade e pressão. O gerenciamento da malha, não estruturada é feito pela estrutura de dados SHE. Os resultados numéricos obtidos pelo UmFlow-2D são comparados com soluções analíticas e soluções numéricas de outros trabalhos. / This work presents an integratc simulation system, called UmFlow-2D, wich aims a,t simulating two-dimensional íncompressible fluid flow using unstructed mesh. The system is divided three modules: modeling module, simulation module and visualization module. In this work we present the simulation module. The simulation module implements the Navier-Stokes equation. The governing equations are discretized by a generalized flnite dillerence method and the convective terms by semi-lagrangean method. A projection method is employed to uncouple the velocity componentes and pressure. The management at the unstructed mesh is ready using a data structure called SHE. The numérica! results are compared with analytical solutions and numerical simulations of other works.

Submitted by Maria Cristina (library@lncc.br) on 2017-08-14T19:07:20Z No. of bitstreams: 1 Tese-AlanAmad.pdf: 11294057 bytes, checksum: cadab8a6da3988a5a62791507562b196 (MD5) / Approved for entry into archive by Maria Cristina (library@lncc.br) on 2017-08-14T19:07:34Z (GMT) No. of bitstreams: 1 Tese-AlanAmad.pdf: 11294057 bytes, checksum: cadab8a6da3988a5a62791507562b196 (MD5) / Made available in DSpace on 2017-08-14T19:07:45Z (GMT). No. of bitstreams: 1 Tese-AlanAmad.pdf: 11294057 bytes, checksum: cadab8a6da3988a5a62791507562b196 (MD5) Previous issue date: 2016-02-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (Capes) / Wave propagation modeling is a challenging problem with many important practical applications in engineering and applied sciences. These applications include the modeling in acoustic, scattering, vibration, structural dynamic response, earthquake, seismic, electromagnetism, photonic, and so on. In fluid-structure modeling, the applications include, for example, simulations in aircraft, rockets, turbines, marine structures, storage tanks, dams, suspension bridges and noise reduction. Our interest is the development of numerical methods to accurately solving time-harmonic wave problems. In this thesis, we propose finite difference and finite element methods to solve the acoustic and elastic problems and a coupled acoustic fluid-structure problem. We also develop a numerical model to simulate hyperthermia therapy, based on topological derivatives and on a stabilized hybrid method. / Modelagem em propagação de ondas é um problema desafiador, com muitas aplicações práticas importantes em engenharia e ciências aplicadas. Estas aplicações incluem a modelagem em acústica, dispersão, vibração, resposta dinâmica estrutural, terremoto, sísmica, eletromagnetismo, fotônica, e assim por diante. Em modelagem de fluido-estrutura, as aplicações incluem simulações em aviões, foguetes, turbinas, estruturas marítimas, tanques de armazenamento, barragens e pontes suspensas, redução de ruído, por exemplo. Nosso interesse é o desenvolvimento de métodos numéricos para resolver precisamente problemas de ondas harmônicas no tempo. Nesta tese consideramos métodos de diferenças finitas e elementos finitos para resolver problemas acústicos e elásticos e um problema acoplado de fluido-estrutura acústica. Também desenvolvemos um modelo numérico para simular terapia por hipertermia, baseado em derivadas topológicas e um método híbrido estabilizado.

Este trabalho tem como objetivo o desenvolvimento de um método numérico para simular escoamentos viscoelásticos tridimensionais com superfícies livres governados pelo modelo constitutivo Giesekus. As equações governantes são resolvidas pelo método de diferenças finitas numa malha deslocada. A superfície livre do fluido é modelada por partículas marcadoras, possibilitando assim a visualização e localização da superfície livre do fluido. A equação constitutiva de Giesekus é resolvida utilizando as seguintes formulações: método de Runge-Kutta de segunda ordem (também conhecido como método de Euler modificado) e transformação logarítmica do tensor conformação. O método numérico apresentado é verificado comparando-se os resultados obtidos por meio de refinamento de malha para os escoamentos em um tubo e de um jato incidindo em uma placa plana. Resultados de convergência foram obtidos por meio de refinamento de malha do escoamento totalmente desenvolvido em um tubo. Os resultados numéricos obtidos incluem a simulação de um jato incidindo em uma caixa vazia e a simulação do inchamento do extrudado (dieswell) para vários números de Weissenberg utilizando diferentes valores do fator de mobilidade do fluido. Resultados adicionais incluem simulações do fenômeno delayed dieswell para altos números de Weissenberg e altos valores do número de Reynolds. Uma comparação qualitativa com resultados experimentais é apresentada. / In this work, a numerical method for simulating viscoelastic free surface flows governed by the Giesekus constitutive equation is developed. The governing equations are solved by the finite difference method on a staggered grid. The fluid free surface is approximated by marker particles which enables the visualization and location of the free surface fluid. The Giesekus constitutive equation is solved by the following techniques: second-order Runge-Kutta, conformation tensor and logarithmic transformation of the conformation tensor. The numerical method is verified by comparing the numerical solutions obtained on a series of embedding meshes of the flow in a tube and by the flow produced by a jet flowing onto a planar surface. Additional verification and convergence results are obtained by solving tube flow employing several meshes. Results obtained include the simulation of a jet flowing into a three dimensional container and the simulation of extrudate swell using several values of the Reynolds and Weissenberg numbers and different values of the mobility parameter a. Furthermore, we present results from the simulation of the phenomenon know as delayed dieswell using highWeissenberg and Reynolds numbers. Comparisons with experimental results are given.

Este trabalho propõe um esquema de diferenças finitas compacta de alta ordem para resolver problemas elípticos com coeficientes variáveis em malhas composta. São apresentados a formulação matemática e a dedução do método compacto de quarta ordem aplicado à problemas elípticos bidimensionais, em malha regular e composta. Foi adotado o uso da biblioteca PETSc com os seus pré-condicionadores e métodos numéricos para resolver os sistemas lineares resultantes da discretização do problema. Por fim, testes visando verificar o código foram feitos, utilizando o método de soluções manufaturadas, para mostrar alta eficiência e acurácia do método desenvolvido. / This paper proposes a scheme of compact finite difference higher order for solve elliptic problems with variable coeficients in composite meshes. we present the mathematical formulation and the deduction of the compact method of fourth order applied to two-dimensional elliptic problems in regular and composite mesh . It was adopted using the PETSc library with its pre- conditioners and numerical methods for solving linear systems resulting from discretization of the problem. Finally , tests to verify the code were made using the method of manufactured solutions to show high eficiency and accuracy of the method developed .

"Material property estimation method is developed with 1-D heat conduction model and bounding exercise for Fire Dynamics Simulator (FDS) analysis. The purpose of this study is to develop an unsophisticated tool to convert small scale cone calorimeter data into input data that can be used in computational fluid dynamics (CFD) models to predict flame spread. Specific interests of input data for FDS in this study include thermal conductivity, specific heat, pre exponential factor, activation energy, heat of vaporization. The tool consists of two objects; 1-D model and bounding exercise. Main structure of the model is based on one of the thermal boundary conditions in the FDS, named as “Pyrolysis Model, Thermally-Thick Solid”, in which pyrolysis flux occurs on the surface of the object under radiant heat flux. This boundary condition is adopted because it has the best characteristics in the dynamics of modeling which are subject to our interests. The structure of the model is simple and concise. For engineering point of view, a practical model ought to have such simplicity that saves time and effort. Pyrolysis model in FDS meets this requirement. It is also a part of reason that this study is to develop a computational model which converts a set of data from the cone calorimeter test to a set of input data for FDS. A pyrolysis term on a surface of an object in this boundary condition will be playing an important role regarding a surface temperature and a mass loss rate of the object. Bounding exercise is introduced to guide proper outcome out of the modeling. Prediction of the material properties from the simulation is confirmed by the experimental data in terms of surface temperature history and mass loss rate under the bounding exercise procedure. For the cone calorimeter, thirteen different materials are tested. Test materials vary with their material composition such as thermoplastics, fiber reinforced plastics (FRP), and a wood. Throughout the modeling fed by a set of the cone calorimeter test data, estimated material properties are provided. So called “Bounding Exercise” is introduced here to draw the estimated material properties. Bounding exercise is a tool in order to guide the material property estimation procedure. Three sets of properties (Upper, Standard and Lower) are derived from the boundary exercise as recommended material properties. From the modeling results, PMMA shows the best agreement regarding the estimated material properties compared with already known results from the references. Wood indicates, however, somewhat different results, in which the mass loss rate takes a peak around the ignition and decreases sharply. This burning behavior can not be predicted using the “Pyrolysis Model”. The model in this study does not account so called “Charring Behavior” that a charring layer toward a surface or difference between a charred density in a charring layer and a normal density in a virgin layer of a wood. These factors result in a discrepancy of the estimated material properties with the reference data. Unlike PMMA and wood, FRP materials show a unique ignition characteristic. Mass loss rate history from some FRP materials indicate more a thermoplastic burning behavior and other materials tend to char. In addition there are few known material property data for theses materials and it is difficult to verify the results from this study with pre-existing data. Some plastic samples also indicate difficulties of the modeling. Because some samples melt and disfigure during the test, one dimensional heat transfer boundary condition is no longer applicable. Each bounding exercise results are fully examined and analyze in Chapter 6. Some of limitations contain model’s structural limitation, in which the model is too simple for certain cases, as well as limitations of bounding exercise. Finally, recommendations are made for future work including upgraded model accountable for the pyrolysis of charring material and FRP materials, data comparison with FDS results, and improved bounding exercise method."

The Finite Difference Time Domain (FDTD) Method has been a powerful tool in numerical simulation of electromagnetic (EM) problems for decades. In recent years, it has also been applied to biomedical research to investigate the interaction between EM waves and biological tissues. In Wireless Body Area Networks (WBANs) studies, to better understand the localization problem within the body, an accurate source/receiver model must be investigated. However, the traditional source models in FDTD involve effective volume and may cause error in near field arbitrary direction. This thesis reviews the basic mathematical and numerical foundation of the Finite Difference Time Domain method and the material properties needed when modeling a human body in FDTD. Then Coincident Phase Centers (CPCs) point sources models have been introduced which provide nearly the same accuracy at the distances as small as 3 unit cells from the phase center. Simultaneously, this model outperforms the usual sources in the near field when an arbitrary direction of the electric or magnetic dipole moment is required.

Technological advancement has thrust MEMS design and fabrication into the forefront of modern technologies. It has become sufficiently self-sustained to allow mass production. The limiting factor which is stalling commercialization of MEMS is the packaging and device reliability. The challenging issues with MEMS packaging are application specific. The function of the package is to give the MEMS device mechanical support, protection from the environment, and electrical connection to other devices in the system. The current state of the art in MEMS packaging transcends the various packaging techniques available in the integrated circuit (IC) industry. At present the packaging of MEMS includes hermetic ceramic packaging and metal packaging with hermetic seals. For example the ADXL202 accelerometer from the Analog Devices. Study of the packaging methods and costs show that both of these methods of packaging are expensive and not needed for majority of MEMS applications. Due to this the cost of current MEMS packaging is relatively high, as much as 90% of the finished product. Reducing the cost is therefore of the prime concern. This Thesis explores the possibility of an inexpensive plastic package for MEMS sensors like accelerometers, optical MEMS, blood pressure sensors etc. Due to their cost effective techniques, plastic packaging already dominates the IC industry. They cost less, weigh less, and their size is small. However, porous nature of molding materials allows penetration of moisture into the package. The Thesis includes an extensive study of the plastic packaging and characterization of three different plastic package samples. Polymeric materials warp upon absorbing moisture, generating hygroscopic stresses. Hygroscopic stresses in the package add to the thermal stress due to high reflow temperature. Despite this, hygroscopic characteristics of the plastic package have been largely ignored. To facilitate understanding of the moisture absorption, an analytical model is presented in this Thesis. Also, an empirical model presents, in this Thesis, the parameters affecting moisture ingress. This information is important to determine the moisture content at a specific time, which would help in assessing reliability of the package. Moisture absorption is modeled using the single phase absorption theory, which assumes that moisture diffusion occurs freely without any bonding with the resin. This theory is based on the Fick's Law of diffusion, which considers that the driving force of diffusion is the water concentration gradient. A finite difference simulation of one-dimensional moisture diffusion using the Crank-Nicolson implicit formula is presented. Moisture retention causes swelling of compounds which, in turn, leads to warpage. The warpage induces hygroscopic stresses. These stresses can further limit the performance of the MEMS sensors. This Thesis also presents a non invasive methodology to characterize a plastic package. The warpage deformations of the package are measured using Optoelectronic holography (OEH) methodology. The OEH methodology is noninvasive, remote, and provides results in full-field-of-view. Using the quantitative results of OEH measurements of deformations of a plastic package, pressure build up can be calculated and employed to assess the reliability of the package.

This thesis introduces models for pricing credit default swaps (CDS) and evaluating the counterparty risk when buying a CDS in the over-the-counter (OTC) market from a counterpart subjected to default risk. Rather than assuming that the default of the referencing firm of the CDS is independent of the trading parties in the CDS, this thesis proposes models that capture the default correlation amongst the three parties involved in the trade, namely the referencing firm, the buyer and the seller. We investigate how the counterparty risk that CDS buyers face can be affected by default correlation and how their balance sheet could be influenced by the changes in counterparty risk. The correlation of corporate default events has been frequently observed in credit markets due to the close business relationships of certain firms in the economy. One of the many mathematical approaches to model that correlation is default contagion. We propose an innovative model of default contagion which provides more flexibility by allowing the affected firm to recover from a default contagion event. We give a detailed derivation of the partial differential equations (PDE) for valuing both the CDS and the credit value adjustment (CVA). Numerical techniques are exploited to solve these PDEs. We compare our model against other models from the literature when measuring the CVA of an OTC CDS when the default risk of the referencing firm and the CDS seller is correlated. Further, the model is extended to incorporate economy-wide events that will damage all firms' credit at the same time-this is another kind of default correlation. Advanced numerical techniques are proposed to solve the resulting partial-integro differential equations (PIDE). We focus on investigating the different role of default contagion and economy-wide events have in terms of shaping the default correlation and counterparty risk. We complete the study by extending the model to include bilateral counterparty risk, which considers the default of the buyer and the correlation among the three parties. Again, our extension leads to a higher-dimensional problem that we must tackle with hybrid numerical schemes. The CVA and debit value adjustment (DVA) are analysed in detail and we are able to value the profit and loss to the investor's balance sheet due to CVA and DVA profit and loss under different market circumstances including default contagion.