This thesis discusses a novel method for modelling facial action units. It presents facial action units model based on boundary value problems for accurate representation of human facial expression in three-dimensions. In particular, a solution to a fourth order elliptic Partial Differential Equation (PDE) subject to suitable boundary conditions is utilized, where the chosen boundary curves are based on muscles movement defined by Facial Action Coding System (FACS). This study involved three stages: modelling faces, manipulating faces and application to simple facial animation. In the first stage, PDE method is used in modelling and generating a smooth 3D face. The PDE formulation using small sets of parameters contributes to the efficiency of human face representation. In the manipulation stage, a generic PDE face of neutral expression is manipulated to a face with expression using PDE descriptors that uniquely represents an action unit. A combination of the PDE descriptor results in a generic PDE face having an expression, which successfully modelled four basic expressions: happy, sad, fear and disgust. An example of application is given using simple animation technique called blendshapes. This technique uses generic PDE face in animating basic expressions.

Neste trabalho, estudamos a existˆencia e multiplicidade de solu¸c˜oes de certos problemas p-sublineares envolvendo o operador p-laplaciano usando teoria de Morse. / The purpose of this text is to provide a didactic exposition of the paper “Solutions of p-sublinear p-Laplacian equation via Morse theory” by Yuxia Guo and Jiaquan Liu [8]. This paper addresses the existence and multiplicity of solutions for the problem where is a smooth, bounded domain of RN, p is the p-Laplacian operator and f satisfies certain conditions, in particular f is p-sublinear at 0. Morse theory is used to infer the existence of critical points of a functional associated to this problem. In Chapter 2, we introduce the necessary Morse theoretic concepts, assuming basic knowledge of singular homology theory. In Chapter 3, we introduce basic properties of the p-Laplacian operator, assuming knowledge of Sobolev spaces, including imbedding and compactness results. Finally, in Chapter 4, we follow Guo and Liu’s paper itself.

In this thesis we study boundary value problems for the Laplace equation on do mains with smooth boundary. Central to our analysis is a relation, known as the global relation, that couples the boundary data for a given BVP. Previously, the global re lation has primarily been applied to elliptic PDEs defined on polygonal domains. In this thesis we extend the use of the global relation to domains with smooth boundary. This is done by introducing a new transform, denoted by F_p, that is an analogue of the Fourier transform on smooth convex curves. We show that the F_p-transform is a bounded and invertible integral operator. Following this, we show that the F_p-transform naturally arises in the global relation for the Laplace equation on domains with smooth boundary. Using properties of the F_p-transform, we show that the global relation defines a continuously invertible map between the Dirichlet and Neumann data for a given BVP for the Laplace equation. Following this, we construct a numerical method that uses the global relation to find the Neumann data, given the Dirichlet data, for a given BVP for the Laplace equation on a domain with smooth boundary.

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Motivated by applications in tissue engineering, this thesis is concerned with the flow through and around a free-moving porous tissue construct (TC) within a high-aspect-ratio vessel (HARV) bioreactor. We formalise and extend various results for flow within a Hele-Shaw cell containing a porous obstacle. We also consider the impact of the flow on related nutrient transport problems. The HARV bioreactor is a cylinder with circular cross-section which rotates about its axis at a constant rate, and is filled with a nutrient-rich culture medium. The porous TC is modelled as a rigid porous cylinder with circular cross-section and is fully saturated with the fluid. We formulate the flow problem for a porous TC (governed by Darcy's equations) within a HARV bioreactor (governed by the Navier-Stokes equations). We couple the two regions via appropriate interfacial conditions which are derived by consideration of the intricate boundary-layer structure close to the TC surface. By exploiting various small parameters, we simplify the system of equations by performing an asymptotic analysis, and investigate the resulting system for the flow due to a prescribed TC motion. The motion of the TC is determined by analysis of the force and torque acting upon it, and the resulting equations of motion (which are coupled to the flow) are investigated. The short-time TC behaviour is periodic, but we are able to study the long-time drift from this periodic solution by considering the effect of inertia using a multiple-scale analysis. We find that, contrary to received wisdom, inertia affects TC drift on a similar timescale to tissue growth. Finally, we consider the advection of nutrient through the bioreactor and TC, and investigate the problem of nutrient advection-diffusion for a simplified model involving nutrient uptake.

Neste trabalho, estudamos a existˆencia e multiplicidade de solu¸c˜oes de certos problemas p-sublineares envolvendo o operador p-laplaciano usando teoria de Morse. / The purpose of this text is to provide a didactic exposition of the paper “Solutions of p-sublinear p-Laplacian equation via Morse theory” by Yuxia Guo and Jiaquan Liu [8]. This paper addresses the existence and multiplicity of solutions for the problem where is a smooth, bounded domain of RN, p is the p-Laplacian operator and f satisfies certain conditions, in particular f is p-sublinear at 0. Morse theory is used to infer the existence of critical points of a functional associated to this problem. In Chapter 2, we introduce the necessary Morse theoretic concepts, assuming basic knowledge of singular homology theory. In Chapter 3, we introduce basic properties of the p-Laplacian operator, assuming knowledge of Sobolev spaces, including imbedding and compactness results. Finally, in Chapter 4, we follow Guo and Liu’s paper itself.

abstract: This thesis considers two problems in the control of robotic swarms. Firstly, it addresses a trajectory planning and task allocation problem for a swarm of resource-constrained robots that cannot localize or communicate with each other and that exhibit stochasticity in their motion and task switching policies. We model the population dynamics of the robotic swarm as a set of advection-diffusion- reaction (ADR) partial differential equations (PDEs). Specifically, we consider a linear parabolic PDE model that is bilinear in the robots' velocity and task-switching rates. These parameters constitute a set of time-dependent control variables that can be optimized and transmitted to the robots prior to their deployment or broadcasted in real time. The planning and allocation problem can then be formulated as a PDE-constrained optimization problem, which we solve using techniques from optimal control. Simulations of a commercial pollination scenario validate the ability of our control approach to drive a robotic swarm to achieve predefined spatial distributions of activity over a closed domain, which may contain obstacles. Secondly, we consider a mapping problem wherein a robotic swarm is deployed over a closed domain and it is necessary to reconstruct the unknown spatial distribution of a feature of interest. The ADR-based primitives result in a coefficient identification problem for the corresponding system of PDEs. To deal with the inherent ill-posedness of the problem, we frame it as an optimization problem. We validate our approach through simulations and show that reconstruction of the spatially-dependent coefficient can be achieved with considerable accuracy using temporal information alone. / Dissertation/Thesis / Masters Thesis Mechanical Engineering 2014

Orientador: Jose Luiz Boldrini / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T14:04:36Z (GMT). No. of bitstreams: 1 Araujo_AndersonLuisAlbuquerquede_M.pdf: 835300 bytes, checksum: 26817c8db069e82f056a44e0e6ebd1d5 (MD5) Previous issue date: 2008 / Resumo: Neste trabalho, consideramos um problema de controle ótimo governado por uma equação diferencial parcial parabólica, que modela o crescimento e a difusão de uma população de mosquitos em uma certa região do plano. Para este modelo relativamente simples, mostramos a existência de uma trajetória ótima a ser seguida por uma unidade volante de pulverização de inseticida, no sentido de minimizar um certo funcional que leva em conta a população total de mosquitos bem como os custos da operação. Caracterizamos também tais trajetórias (controles) ótimas pela derivação de suas respectivas condições de otimalidade de primeira ordem. Para isso, usamos o formalismo de Dubovitskii e Milyutin, o qual está baseado na separação de certos cones associados ao funcional a ser minimizado e ás restrições do problema, incluindo a equação. Também analisamos o problema do ponto de vista do método de penalização / Abstract: In this work, we consider an optimal control problem governed by a parabolic partial differential equation, which models the growth and diffusion of a mosquito population in a certain region of the Euclidean plane. For this relatively simple model, we show the existence of an optimal trajectory to be followed by a insecticide spraying device, in the sense of minimizing a certain functional that takes in consideration both the the total mosquito population and the operational costs. We also characterize such optimal trajectories (controls) by deriving their respective first order optimal conditions. For this, we use the Dubovitskii and Milyutin formalism, which is based on the separation of certain cones associated to the functional to be minimized, and to the restrictions of the problem, including the equation. We also analyze the problem from the point of view of the penalization method / Mestrado / Analise Matematica / Mestre em Matemática

Orientadores: Djairo Guedes de Figueiredo, João Marcos Bezerra do Ó / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-16T16:52:57Z (GMT). No. of bitstreams: 1 Ribeiro_BrunoHenriqueCarvalho_D.pdf: 1676664 bytes, checksum: 8517caa733a0141397500732b70a6ae6 (MD5) Previous issue date: 2010 / Resumo: Estudamos problemas do tipo Ambrosetti-Prodi para classes de sistemas elípticos gradientes com não-linearidades em crescimento crítico unilateral de Sobolev e de Trudinger-Moser. Com uso de métodos variacionais, provamos multiplicidade de solução para problemas homogêneos sem ressonância na parte linear e existência de solução não-trivial para problemas homogêneos com ressonância / Abstract: We study Ambrosetti-Prodi problems for classes of gradient elliptic systems with nonlinearities in the critical growth range of Sobolev and Trudinger-Moser types. Using variational methods, we prove multiplicity of solutions for nonhomogeneous problems without resonance in the linear part and homogeneous problems involving resonance / Doutorado / Analise / Doutor em Matemática