Deforming a complex generic shape into a representation of another complex shape is investigated. An initial study is done on the effect of cranial shape variation on masticatory induced stress. A finite element analysis is performed on two different skull geometries. One skull geometry has a prognathic shape, characterised by jaws protruding forward, while the other has a non-prognathic form. Comparing the results of the initial nite element analyses, the effect of an undesired variation in shape and topology on the resulting stress field is observed. This variation in shape and topology can not be attributed to the cranial shape variation that is investigated. This means that the variation in the masticatory induced stress field that is due to the relative degree in prognathism can not be quantified effectively. To best compare results, it would be beneficial to have a computational domain for the different skull geometries that have one-to-one correspondence. An approach to obtain a computational domain that represents various geometries with the exact same mesh size and connectivity between them does exist. This approach involves deforming a generic mesh to represent different target shapes. This report covers an introductory study to register and deform a generic mesh to approximately represent a complex target geometry. Various procedures are investigated, implemented and combined to specifically accommodate complex geometries like that of the human skull. A surface registration procedure is implemented and combined with a feature registration procedure. Feature lines are extracted from the surface representation of each skull as well as the generic shape. These features are compared and an initial deformation is applied to the generic shape to better represent the corresponding features on the target. Selective feature preserved elastic surface registration is performed after the initial feature based registration. Only the registration to surfaces of featureless areas and matched feature areas are allowed along with user selected areas during surface registration. The implemented procedures have various aspects that still require improvement before the desired study regarding prognathism's effect on masticatory induced stress could truly be approached pragmatically. Focus is only given to the use of existing procedures while the additional required improvements could be addressed in future work. It is however required that the resulting discretised domain obtained in this initial study be of sufficient quality to be used in a finite element analysis (FEA). The implemented procedure is illustrated using the two original skull geometries. Symmetric versions of these geometries are generated with a one-to-one correspondence map between them. The skull representations are then used in a finite element analysis to illustrate the appeal of having computational domains with a consistent mapping between them. The variation in the masticatory induced stress field due to the variation in cranial shape is illustrated using the consistent mapping between the geometries as part of this example. / Dissertation (MEng)--University of Pretoria, 2011. / Mechanical and Aeronautical Engineering / unrestricted

In 2014, Astala, Päivärinta, Reyes, and Siltanen conducted numerical experiments reconstructing a piecewise continuous conductivity. The algorithm of the shortcut method is based on the reconstruction algorithm due to Nachman, which assumes a priori that the conductivity is Hölder continuous. In this dissertation, we prove that, in the presence of infinite-precision data, this shortcut procedure accurately recovers the scattering transform of an essentially bounded conductivity, provided it is constant in a neighborhood of the boundary. In this setting, Nachman’s integral equations have a meaning and are still uniquely solvable. To regularize the reconstruction, Astala et al. employ a high frequency cutoff of the scattering transform. We show that such scattering transforms correspond to Beltrami coefficients that are not compactly supported, but exhibit certain decay at infinity. For this class of Beltrami coefficients, we establish that the complex geometric optics solutions to the Beltrami equation exist and exhibit the same subexponential decay as described in the 2006 work of Astala and Päivärinta. This is a first step toward extending the inverse scattering map of Astala and Päivärinta to non-compactly supported conductivities.

[pt] Uma metodologia eficiente e segura é proposta para prever a propagação de trincas de fadiga sob carregamento complexo em estruturas bidimensionais com geometria genérica. Primeiro, o caminho da trinca (em geral curvo) e os fatores de intensidade de tensão KI(a) e KII(a) ao longo do comprimento da trinca a são calculados num programa de elementos finitos especialmente desenvolvido para este fim, o Quebra2D. Estes cálculos são feitos usando pequenos incrementos especificáveis no tamanho da trinca e técnicas de remalhamento automatizadas. Os valores de KI(a) são usados como dados de entrada num programa de previsão de vida à fadiga, o ViDa. Esse programa foi desenvolvido para prever a iniciação e a propagação de trincas 1D e 2D sob carregamento complexo por todos os métodos clássicos, incluindo SN, eN e IIW (estruturas soldadas) para a iniciação da trinca, e o método da/dN para a propagação. Em particular, o módulo que propaga a trinca aceita qualquer expressão de KI(a) e qualquer regra da/dN, e usa o método DKrms ou CCC (crescimento ciclo-a-ciclo) para prever a propagação de trincas uni e bidimensionais sob carregamento complexo. A análise numérica proposta foi verificada através de vários experimentos representativos, cuja metodologia experimental é discutida em detalhes. / [en] A reliable and cost effective two-phase methodology is proposed to predict fatigue crack propagation in generic two-dimensional structural components under complex loading. First, the fatigue crack path and its stress intensity factor are calculated in a specialized finite- element software, using small crack increments. Numerical methods are used to calculate the crack propagation path, based on the computation of the crack incremental direction, and the stress-intensity factors KI, from the finite element response. Then, an analytical expression is adjusted to the calculated KI(a) values, where a is the length along the crack path. This KI(a) expression is used as an input to a powerful general purpose fatigue design software based on the local approach, developed to predict both initiation and propagation fatigue lives under complex loading by all classical design methods, including the SN, the eN and the IIW (for welded structures) to deal with crack initiation, and the da/dN to treat propagation problems. In particular, its crack propagation module accepts any KI expression and any da/dN rule, using the DKrms or the cycle-by-cycle propagation methods to deal with one and twodimensional crack propagation under complex loading. If requested, this latter method may include overload-induced crack retardation effects. This two-phase methodology is experimentally validated by fatigue tests on compact tension and bending single edge notch specimens, modified with holes positioned to attract or to deflect the cracks.

This thesis is concerned with the inverse problem of determining a unitary connection $A$ on a Hermitian vector bundle $E$ of rank $m$ over a compact Riemannian manifold $(M, g)$ from the Dirichlet-to-Neumann (DN) map $\Lambda_A$ of the associated connection Laplacian $d_A^*d_A$. The connection is to be determined up to a unitary gauge equivalence equal to the identity at the boundary. In our first approach to the problem, we restrict our attention to conformally transversally anisotropic (cylindrical) manifolds $M \Subset \mathbb{R}\times M_0$. Our strategy can be described as follows: we construct the special Complex Geometric Optics solutions oscillating in the vertical direction, that concentrate near geodesics and use their density in an integral identity to reduce the problem to a suitable $X$-ray transform on $M_0$. The construction is based on our proof of existence of Gaussian Beams on $M_0$, which are a family of smooth approximate solutions to $d_A^*d_Au = 0$ depending on a parameter $\tau \in \mathbb{R}$, bounded in $L^2$ norm and concentrating in measure along geodesics when $\tau \to \infty$, whereas the small remainder (that makes the solution exact) can be shown to exist by using suitable Carleman estimates. In the case $m = 1$, we prove the recovery of the connection given the injectivity of the $X$-ray transform on $0$ and $1$-forms on $M_0$. For $m > 1$ and $M_0$ simple we reduce the problem to a certain two dimensional $\textit{new non-abelian ray transform}$. In our second approach, we assume that the connection $A$ is a $\textit{Yang-Mills connection}$ and no additional assumption on $M$. We construct a global gauge for $A$ (possibly singular at some points) that ties well with the DN map and in which the Yang-Mills equations become elliptic. By using the unique continuation property for elliptic systems and the fact that the singular set is suitably small, we are able to propagate the gauges globally. For the case $m = 1$ we are able to reconstruct the connection, whereas for $m > 1$ we are forced to make the technical assumption that $(M, g)$ is analytic in order to prove the recovery. Finally, in both approaches we are using the vital fact that is proved in this work: $\Lambda_A$ is a pseudodifferential operator of order $1$ acting on sections of $E|_{\partial M}$, whose full symbol determines the full Taylor expansion of $A$ at the boundary.