Using an explicit integration method in physically based animations has many advantages including conceptual and computational simplicity, however, it re- quires small time steps to ensure low numerical instability. Simulations with large numbers of individually interacting components such as cloth, hair, and fluid models, are limited by the sections of particles most susceptible to error. This results in the need for smaller time steps than required for the majority of the system. These sections can be diverse and dynamic, quickly changing in size and location based on forces in the system. Identifying and handling these trou- blesome sections could allow for a larger time step to be selected, while preventing a breakdown in the simulation. This thesis presents Smarticles (smart particles), a method of individually de- tecting particles exhibiting signs of instability and stabilizing them with minimal adverse effects to visual accuracy. As a result, higher levels of error introduced from large time steps can be tolerated with minimal overhead. Two separate approaches to Smarticles were implemented. They attempt to find oscillating particles by analyzing a particle’s (1) past behavior and (2) behavior with re- spect to its neighbors along a strand. Both versions of Smarticles attempt to correct unstable particles using velocity dampening. Smarticles was applied to a two dimensional hair simulation modeled as a continuum using smooth particle hydrodynamic. Hair strands are formed by linking particles together using one of two methods: position based dynamics or mass-spring forces. Both versions of Smarticles, as well as a control of normal particles, were directly compared and evaluated based on stability and visual fluidity. Hair particles were exposed to various forms of external forces under increasing time step lengths. Testing showed that both versions of Smarticles working together allowed an average increase of 18.62% in the time step length for hair linked with position based dynamics. In addition, Smarticles was able to significantly reduce visible instability at even larger time steps. While these results suggest Smarticles is successful, the method used to correct particle instability may jeopardize other important aspects of the simulation. A more accurate correction method would likely need to be developed to make Smarticles an advantageous method.

[pt] Zika Vírus (ZIKV) é um vírus transmitido pelos mosquitos Aedes aegypti (mesmo transmissor da dengue e da febre chikungunya) e o Aedes albopictus. O contágio principal pelo ZIKV se dá pela picada do mosquito que, após se alimentar do sangue de alguém contaminado, pode transportar o ZIKV durante toda a sua vida, transmitindo a doença para uma população que não possui anticorpos contra ele. Também pode ser transmitido através de relação sexual de uma pessoa com Zika para os seus parceiros ou parceiras, mesmo que a pessoa infectada não apresente os sintomas da doença.Neste trabalho, apresentamos dois modelos matemáticos para a epidemia do ZIKV usando (1) equações diferenciais ordinárias e (2) equações diferenciais ordinárias com atraso temporal, que é o tempo que os mosquitos levam para desenvolver o vírus. Fazemos uma comparação entre as duas variantes de modelagem e, para facilitar o trabalho com os modelos, fornecemos uma interface gráfica com o usuário. Simulações computacionais são realizadas para o Suriname e El Salvador, que são países propensos a desenvolver a epidemia de maneira endêmica. Para estudar a difusão espacial do ZIKV, propomos um modelo baseado em equações de advecção-difusão e criamos um esquema numérico com elementos finitos e diferenças finitas para resolvê-lo. / [en] The Zika Virus (ZIKV) is a virus transmitted by Aedes aegypti mosquitoes (same as the one transmitting dengue and chikungunya fever) and Aedes albopictus. The main way of contagion by the ZIKV is caused by the bite of a mosquito that, after feeding from someone contaminated, can transport the ZIKV throughout its life, transmitting the disease to a population that does not have the immunity. It can also be transmitted through a person s sexual relationship with ZIKV to their partners, even if the infected person does not have the symptoms of the disease. In this work, we present two mathematical models for the ZIKV epidemic by using (1) ordinary differential equations and, (2) ordinary differential equations with temporal delay, which is the time it takes mosquitoes to develop the virus. We make a comparison between the two modeling variants and, to facilitate the work with the models, we provide a graphical user interface. Computational simulations are performed for Suriname and El Salvador, which are countries that are prone to develop the epidemic in an endemic manner. In order to study the spatial diffusion of ZIKV, we propose a model based on advection-diffusion equations and create a numerical scheme with finite elements and finite differences to resolve it.

Cardiovascular diseases (CVDs) are a leading cause of death worldwide, emphasizing the need for advanced and effective intervention and treatment measures. Hypertension, a significant risk factor for CVDs, is characterized by reduced vascular compliance in arterial vessels. There is a significant rise in interest in exploring the viscoelastic properties of arteries in the last few years, for the treatment of these diseases. This study aims to develop an experimental setup using 3D Printing Technology to test viscoelastic arterial models for the validation of a diagnostic device for cardiovascular diseases. The research investigates the selection of polymer-based materials that closely mimic the viscoelastic properties of arterial vessels. An experimental setup is designed and fabricated to perform mechanical tests on 3D-printed specimens. The study utilizes a mathematical model to describe the viscoelastic behavior of the materials. The model's predictions are validated using experimental data obtained from the mechanical tests. This study demonstrates the potential of 3D printing technology in fabricating specimens using elastic and flexible resin materials. These specimens closely replicate the mechanical properties of native arteries, offering a tangible platform for controlled mechanical testing. Stress relaxation tests on the3D printed specimens highlight the viscoelastic properties of fabricated materials, shedding light on their behavior under strain. The study goes further to model the mechanics of these materials, utilizing the Fractional Voigt model to capture the intricate balance between elastic and resistive behaviors under varying deformation levels. The results highlight the successful fitting of the Fractional Voigt model to the experimental data, confirming the viscoelastic behavior of the specimens. The obtained values of α and RMSE indicate a good representation of arterial mechanical properties within the viscoelastic arterial model, under different loading conditions. This research contributes to improving cardiovascular device validation and offers a practical and reliable alternative to invasive experiments. Future works include exploring different materials and conditions for arterial modeling and enhancing the precision and scope of the viscoelastic model. Overall, this study advances the understanding of cardiovascular biomechanics, contributing to the development of more effective diagnostic devices for cardiovascular diseases.

Modelling and simulation of complex and multi-domain mechanical systems has become of major importance in the last few years to address energy and fuel consumption performance evaluation. The goal is to unify the available modelling languages aiming to improve scalability and easiness of handling complex multi-domain models. Modelica Modelling Language was born in 1997. It has three main features: object-oriented, equation based with non-causal design structure and multi-domain environment. This thesis aims to give an overview of using Modelica on Modelon Impact software to model and simulate a complete 3D wheel loader dynamic system. The project wants to show how the model has been developed focusing on each sub-system implementation. The 3D wheel loader model is designed following the top-down and bottom-up design approaches and focusing on the powertrain sub-system with the engine, transmission and driveline blocks. The combination of the two logics is used to smooth the modelling path and exploit all the benefits. For the simulation experiments, test rig models are implemented to verify the dynamics of individual sub-systems. The model is simulated giving a set of input signals and solving the dynamic equations using different numerical solvers and comparing the elapsed simulation time. The simulation results show that the Radau5ODE explicit solver achieves faster simulation with stable solution given by the variable step size parameter. However, more studies and specific background are needed to update the complexity of the model and compare it with the already existing one. / Modellering och simulering av komplexa mekaniska system med flera domäner har fått stor betydelse under de senaste åren för att utvärdera energi och bränsleförbrukning. Målet är att förena de tillgängliga modelleringsspråken för att förbättra skalbarheten och underlätta hanteringen av komplexa modeller med flera områden. Modelica-modelleringsspråket föddes 1997. Det har tre huvudfunktioner: objektorienterat, ekvationsbaserat med icke-kausal designstruktur och en miljö med flera områden. Syftet med denna avhandling är att ge en översikt över användningen av Modelica i programvaran Modelon Impact för att modellera och simulera ett komplett dynamiskt 3D-system för hjullastare. Projektet vill visa hur modellen har utvecklats med fokus på varje delsystems genomförande. 3D-modellen för hjullastaren har utformats enligt top-down och bottom-up principerna och fokuserar på delsystemet drivlina med motor, transmission och drivlina. Kombinationen av de två logikerna används för att jämna ut modelleringsvägen och utnyttja alla fördelar. För simuleringsförsöken har testriggmodeller införts för att kontrollera dynamiken hos enskilda delsystem. Modellen simuleras med en uppsättning insignaler och de dynamiska ekvationerna löses med hjälp av olika numeriska lösare, varefter den förflutna simuleringstiden jämförs. Simuleringsresultaten visar att den explicita lösaren Radau5ODE ger en snabbare simulering med en stabil lösning som ges av parametern variabel stegstorlek. Det behövs dock fler studier och mer specifik bakgrund för att uppdatera modellens komplexitet och jämföra den med redan existerande modeller.

Bifurcations in Huang's chaotic chemical reactor system leading from simple dynamics into chaotic regimes are considered. Following the linear stability analysis, the periodic orbit resulting from a Hopf bifurcation of any of the six fixed points is constructed analytically by the method of multiple scales across successively slower time scales, and its stability is then determined by the resulting final secularity condition. Furthermore, we run numerical simulations of our chemical reactor at a particular fixed point of interest, alongside a set of parameter values that forces our system to undergo Hopf bifurcation. These numerical simulations then verify our analysis of the normal form.

<p>We consider existence and stability of breather solutions to discrete nonlinear Schrodinger (dNLS) and discrete Klein-Gordon (dKG) equations near the anti-continuum limit, the limit of the zero coupling constant. For sufficiently small coupling, discrete breathers can be uniquely extended from the anti-continuum limit where they consist of periodic oscillations on excited sites separated by "holes" (sites at rest).</p> <p>In the anti-continuum limit, the dNLS equation linearized about its discrete breather has a spectrum consisting of the zero eigenvalue of finite multiplicity and purely imaginary eigenvalues of infinite multiplicities. Splitting of the zero eigenvalue into stable and unstable eigenvalues near the anti-continuum limit was examined in the literature earlier. The eigenvalues of infinite multiplicity split into bands of continuous spectrum, which, as observed in numerical experiments, may in turn produce internal modes, additional eigenvalues on the imaginary axis. Using resolvent analysis and perturbation methods, we prove that no internal modes bifurcate from the continuous spectrum of the dNLS equation with small coupling.</p> <p>Linear stability of small-amplitude discrete breathers in the weakly-coupled KG lattice was considered in a number of papers. Most of these papers, however, do not consider stability of discrete breathers which have "holes" in the anti-continuum limit. We use perturbation methods for Floquet multipliers and analysis of tail-to-tail interactions between excited sites to develop a general criterion on linear stability of multi-site breathers in the KG lattice near the anti-continuum limit. Our criterion is not restricted to small-amplitude oscillations and it allows discrete breathers to have "holes" in the anti-continuum limit.</p> / Doctor of Philosophy (PhD)

The direct and adjoint methods are to linearize the time-averaged solution of bounded dynamical systems about one or more design parameters. Hence, such methods are one way to obtain the gradient necessary in locally optimizing a dynamical system’s time-averaged behavior over those design parameters. However, when analyzing nonlinear systems whose solutions exhibit chaos, standard direct and adjoint sensitivity methods yield meaningless results due to time-local instability of the system. The present work proposes a new method of solving the direct and adjoint linear systems in time, then tests that method’s ability to solve instances of the Lorenz system that exhibit chaotic behavior. Promising results emerge and are presented in the form of a regression analysis across a parametric study of the Lorenz system.

In vitro tissue engineering is a method for developing living and functional tissues external to the body, often within a device called a bioreactor to control the chemical and mechanical environment. However, the quality of bone tissue engineered products is currently inadequate for clinical use as the implant cannot bear weight. In an effort to improve the quality of the construct, hydrostatic pressure, the pressure in a fluid at equilibrium that is required to balance the force exerted by the weight of the fluid above, has been investigated as a mechanical stimulus for promoting extracellular matrix deposition and mineralisation within bone tissue. Thus far, little research has been performed into understanding the response of bone tissue cells to mechanical stimulation. In this thesis we investigate an in vitro bone tissue engineering experimental setup, whereby human mesenchymal stem cells are seeded within a collagen gel and cultured in a hydrostatic pressure bioreactor. In collaboration with experimentalists a suite of mathematical models of increasing complexity is developed and appropriate numerical methods are used to simulate these models. Each of the models investigates different aspects of the experimental setup, from focusing on global quantities of interest through to investigating their detailed local spatial distribution. The aim of this work is to increase understanding of the underlying physical processes which drive the growth and development of the construct, and identify which factors contribute to the highly heterogeneous spatial distribution of the mineralised extracellular matrix seen experimentally. The first model considered is a purely temporal model, where the evolution of cells, solid substrate, which accounts for the initial collagen scaffold and deposited extracellular matrix along with attendant mineralisation, and fluid in response to the applied pressure are examined. We demonstrate that including the history of the mechanical loading of cells is important in determining the quantity of deposited substrate. The second and third models extend this non-spatial model, and examine biochemically and biomechanically-induced spatial patterning separately. The first of these spatial models demonstrates that nutrient diffusion along with nutrient-dependent mass transfer terms qualitatively reproduces the heterogeneous spatial effects seen experimentally. The second multiphase model is used to investigate whether the magnitude of the shear stresses generated by fluid flow, can qualitatively explain the heterogeneous mineralisation seen in the experiments. Numerical simulations reveal that the spatial distribution of the fluid shear stress magnitude is highly heterogeneous, which could be related to the spatial heterogeneity in the mineralisation seen experimentally.

Introduction: Antimalarial resistance, particularly artemisinin resistance, is a major threat to P. falciparum malaria elimination efforts worldwide. Urgent intervention is required to tackle artemisinin resistance but field data on which to base planning of strategies are limited. The aims were to collect available field data and develop population level mathematical models of P. falciparum malaria treatment and artemisinin resistance in order to determine the optimal strategies for elimination of artemisinin resistant malaria in Cambodia and treatment of pre-hospital and severe malaria in Cambodia and Bangladesh. Methods: Malaria incidence and parasite clearance data from Cambodia and Bangladesh were collected and analysed and modelling parameters derived. Population dynamic mathematical models of P. falciparum malaria were produced. Results: The modelling demonstrated that elimination of artemisinin resistant P. falciparum malaria would be achievable in Cambodia in the context of artemisinin resistance using high coverages with ACT treatment, ideally combined with LLITNs and adjunctive single dose primaquine. Sustained efforts would be necessary to achieve elimination and effective surveillance is essential, both to identify the baseline malaria burden and to monitor parasite prevalence as interventions are implemented. A modelled policy change to rectal and intravenous artesunate in the context of pre-existing artemisinin resistance would not compromise the efficacy of ACT for malaria elimination. Conclusions: By being developed rapidly in response to specific questions the models presented here are helping to inform planning efforts to combat artemisinin resistance. As further field data become available, their planned on-going development will produce increasingly realistic and informative models which can be expected to play a central role in planning efforts for years to come.

The heart is one the most important organs in the human body and many other live creatures. The electrical activity in the heart controls the heart function, and many heart diseases are linked to the abnormalities in the electrical activity in the heart. Mathematical equations and computer simulation can be used to model the electrical activity in the heart. The heart models are challenging to solve because of the complexity of the models and the huge size of the problems. Several cell models have been proposed to model the electrical activity in a single heart cell. These models must be coupled with a heart model to model the electrical activity in the entire heart. The bidomain model is a popular model to simulate the propagation of electricity in myocardial tissue. It is a continuum-based model consisting of non-linear ordinary differential equations (ODEs) describing the electrical activity at the cellular scale and a system of partial differential equations (PDEs) describing propagation of electricity at the tissue scale. Because of this multi-scale, ODE/PDE structure of the model, splitting methods that treat the ODEs and PDEs in separate steps are natural candidates as numerical methods. First, we need to solve the problem at the cellular scale using ODE solvers. One of the most popular methods to solve the ODEs is known as the Rush-Larsen (RL) method. Its popularity stems from its improved stability over integrators such as the forward Euler (FE) method along with its easy implementation. The RL method partitions the ODEs into two sets: one for the gating variables, which are treated by an exponential integrator, and another for the remaining equations, which are treated by the FE method. The success of the RL method can be understood in terms of its relatively good stability when treating the gating variables. However, this feature would not be expected to be of benefit on cell models for which the stiffness is not captured by the gating equations. We demonstrate that this is indeed the case on a number of stiff cell models. We further propose a new partitioned method based on the combination of a first-order generalization of the RL method with the FE method. This new method leads to simulations of stiff cell models that are often one or two orders of magnitude faster than the original RL method. After solving the ODEs, we need to use bidomain solvers to solve the bidomain model. Two well-known, first-order time-integration methods for solving the bidomain model are the semi-implicit method and the Godunov operator-splitting method. Both methods decouple the numerical procedure at the cellular scale from that at the tissue scale but in slightly different ways. The methods are analyzed in terms of their accuracy, and their relative performance is compared on one-, two-, and three-dimensional test cases. As suggested by the analysis, the test cases show that the Godunov method is significantly faster than the semi-implicit method for the same level of accuracy, specifically, between 5 and 15 times in the cases presented. Second-order bidomain solvers can generally be expected to be more effective than first-order bidomain solvers under normal accuracy requirements. However, the simplest and the most commonly applied second-order method for the PDE step, the Crank-Nicolson (CN) method, may generate unphysical oscillations. We investigate the performance of a two-stage, L-stable singly diagonally implicit Runge-Kutta method for solving the PDEs of the bidomain model and present a stability analysis. Numerical experiments show that the enhanced stability property of this method leads to more physically realistic numerical simulations compared to both the CN and Backward Euler (BE) methods.