This thesis develops an algorithm to decrease high-curvature sections in tool paths for complex parts to achieve shorter machining times resulting in higher production rates. In the research sample cases, the algorithm decreased machining times by 1% to 9% for design-induced sections of high curvature and by 16% to 75% for CAM induced ripples using high path tolerances. High-curvature sections in tool paths are caused by complex part geometry, noise, and discontinuities in the model. The curvature is decreased by deviating the tool path within an allowable path tolerance. The feedrate along the tool path is directly related to the curvature of the tool path. High-curvature sections cause the NC machine to reduce the feedrate along the tool path due to acceleration and jerk limits. These lower feedrates increase machining time and slow production rates. This new algorithm decreases curvature, which increases feedrates and decreases machining times, thereby increasing production rates for manufacturing companies. The tool paths are represented by cubic B-splines. The algorithm is based on the basic principle that the curvature of a B-spline directly relates to the geometry of its control polygon. If the control polygon's geometry has many tight corners then the B-spline will have high curvature. If the control polygon's geometry is a straight line then the B-spline will be a straight line with zero curvature. The algorithm deviates the control polygon's points so that they move towards forming a straight line. The control polygon will rarely form a straight line because the spline is limited by the path tolerance. However, as the control polygon moves towards forming a straight line, the curvature decreases, which allows the feedrate to increase. Six sample cases are explored in which the machining time is decreased. Three of the cases are tool paths that contain curvature sections with a range of unnecessary curvature from low to high. One sample is the tool path for the complex geometry in a snow tire mold. Another sample tool path contains ripples caused by noise in the CAD model. The last tool path contains ripples caused by tangency discontinuities in the CAD model. The percent of time saved directly relates to the severity of the curvature in the part. This thesis provides a quick and efficient means to reduce curvature in complex parts, resulting in decreased machining times and increased production rates.

Optical testing in adverse environments, ophthalmology and applications where characterization by curvature is leveraged all have a common goal: accurately estimate wavefront shape. This dissertation investigates wavefront sensing techniques as applied to optical testing based on gradient and curvature measurements. Wavefront sensing involves the ability to accurately estimate shape over any aperture geometry, which requires establishing a sampling grid and estimation scheme, quantifying estimation errors caused by measurement noise propagation, and designing an instrument with sufficient accuracy and sensitivity for the application. Starting with gradient-based wavefront sensing, a zonal least-squares wavefront estimation algorithm for any irregular pupil shape and size is presented, for which the normal matrix equation sets share a pre-defined matrix. A Gerchberg–Saxton iterative method is employed to reduce the deviation errors in the estimated wavefront caused by the pre-defined matrix across discontinuous boundary. The results show that the RMS deviation error of the estimated wavefront from the original wavefront can be less than λ/130~ λ/150 (for λ equals 632.8nm) after about twelve iterations and less than λ/100 after as few as four iterations. The presented approach to handling irregular pupil shapes applies equally well to wavefront estimation from curvature data. A defining characteristic for a wavefront estimation algorithm is its error propagation behavior. The error propagation coefficient can be formulated as a function of the eigenvalues of the wavefront estimation-related matrices, and such functions are established for each of the basic estimation geometries (i.e. Fried, Hudgin and Southwell) with a serial numbering scheme, where a square sampling grid array is sequentially indexed row by row. The results show that with the wavefront piston-value fixed, the odd-number grid sizes yield lower error propagation than the even-number grid sizes for all geometries. The Fried geometry either allows sub-sized wavefront estimations within the testing domain or yields a two-rank deficient estimation matrix over the full aperture; but the latter usually suffers from high error propagation and the waffle mode problem. Hudgin geometry offers an error propagator between those of the Southwell and the Fried geometries. For both wavefront gradient-based and wavefront difference-based estimations, the Southwell geometry is shown to offer the lowest error propagation with the minimum-norm least-squares solution. Noll’s theoretical result, which was extensively used as a reference in the previous literature for error propagation estimate, corresponds to the Southwell geometry with an odd-number grid size. For curvature-based wavefront sensing, a concept for a differential Shack-Hartmann (DSH) curvature sensor is proposed. This curvature sensor is derived from the basic Shack-Hartmann sensor with the collimated beam split into three output channels, along each of which a lenslet array is located. Three Hartmann grid arrays are generated by three lenslet arrays. Two of the lenslets shear in two perpendicular directions relative to the third one. By quantitatively comparing the Shack-Hartmann grid coordinates of the three channels, the differentials of the wavefront slope at each Shack-Hartmann grid point can be obtained, so the Laplacian curvatures and twist terms will be available. The acquisition of the twist terms using a Hartmann-based sensor allows us to uniquely determine the principal curvatures and directions more accurately than prior methods. Measurement of local curvatures as opposed to slopes is unique because curvature is intrinsic to the wavefront under test, and it is an absolute as opposed to a relative measurement. A zonal least-squares-based wavefront estimation algorithm was developed to estimate the wavefront shape from the Laplacian curvature data, and validated. An implementation of the DSH curvature sensor is proposed and an experimental system for this implementation was initiated. The DSH curvature sensor shares the important features of both the Shack-Hartmann slope sensor and Roddier’s curvature sensor. It is a two-dimensional parallel curvature sensor. Because it is a curvature sensor, it provides absolute measurements which are thus insensitive to vibrations, tip/tilts, and whole body movements. Because it is a two-dimensional sensor, it does not suffer from other sources of errors, such as scanning noise. Combined with sufficient sampling and a zonal wavefront estimation algorithm, both low and mid frequencies of the wavefront may be recovered. Notice that the DSH curvature sensor operates at the pupil of the system under test, therefore the difficulty associated with operation close to the caustic zone is avoided. Finally, the DSH-curvature-sensor-based wavefront estimation does not suffer from the 2π-ambiguity problem, so potentially both small and large aberrations may be measured.

No description available.

Master of Science / Department of Civil Engineering / Asadollah Esmaeily / Realistic strength assessment of reinforced concrete structural elements, especially columns in bridges and tall buildings is a critical need not only at design time, but also when an accurate evaluation of the strength is needed for decisions such as retrofit or replacement of an existing structure. Assessment of the flexural strength of a column under a specific axial load level is usually done by constructing the axial force-bending moment interaction response curve of the section. This assessment can be done using the code procedure. However, the code does not consider the confinement effect, and is based on the “stress block” assumption for a pre-assumed failure strain for concrete. It has been shown by various experimental and analytical studies that the performance of a reinforced concrete section is affected by different factors such loading history and material behavior. A realistic performance assessment should consider not only proper models for the monotonic and cyclic response of the material, but also analytical methods and procedures that can capture the effects of loading pattern and provide realistic predictions of the section capacity. Accuracy of the analytical methods in strength assessment of reinforced concrete sections was explored in a comparative study. These methods were compared and validated against the existing experimental data. The factors considered in these analytical procedures, included the effect of confinement, and the method employed in assessment of the axial-force-bending moment interaction response of a column section. The experimental data were collected from tests conducted on circular and rectangular columns under a constant axial load. It has been shown that the axial force-bending moment interaction curve, constructed based on the moment-curvature response of a section using a more detailed analytical method such as fiber-model, considering the confining effect of the lateral reinforcement, represents the most realistic and optimal response of a cross section.

The main objective of this thesis is to obtain critical insight on the role of crystalline incompatibilities in strain and curvature, induced in presence of line defects i.e. dislocations and disclinations, on the energy and geometry of specific features of the local microstructure, and on the bulk mechanical response of nanocrystalline/ultra-fine grained materials. To that end, studies are performed at the (1) inter-atomic and fine scale, and (2) at the mesoscale. The modelling approach is based on the field dislocation and disclination mechanics theory of continuously representated dislocations and disclinations. New, thermodynamically rigorous, multi-scale elastic constitutive laws based on the couple stress theory are developed to capture the effect of strain and curvature incompatibilities on the Cauchy and couple stresses. A new meso-scale elasto-viscoplastic constitutive model of defect incompatibilities based on a fast Fourier transform technique is developed. The desired scale transitioning is achieved via novel phenomenological defect density transport equations and the newly developed elastic constitutive laws. At the fine scale, the model is applied to study energetic interactions between strain and curvature incompatibilities associated with grain boundaries and their influence on triple line energies. Results reveal that incompatible lattice strains have the most significant contribution to the energy. Incompatible lattice curvatures have negligible energetic contributions but are necessary to characterize the geometry of grain boundaries. Finally, both incompatible lattice strains and curvatures are necessary to capture the structure sensitive mechanical behavior of grain boundaries. At the mesoscale, deformation of nanocrystalline aggregates characterized by residual curvatures is studied to identify the impact of the latter's presence on the local and bulk mechanical response of the aggregate. Relaxation of local stresses generated from residual curvatures reproduces the effect of GB dislocation emission. Uniaxial tensile loading of nanocrystalline microstructures containing residual curvatures reveals a softening in the yield stress which could explain the breakdown in Hall-Petch law in the nanocrystalline regime. Next, the possibility of characterizing incompatibilities using X-ray or neutron diffraction techniques is tested. Results reveal that only strains and their gradients contribute to the broadening of diffraction peaks; curvatures and their gradients have no contribution. This study leads to the development of a new multi-scale averaged strain based Fourier technique for generating virtual diffraction peaks.

The purpose of this research was to enhance the methods towards the development of an efficient three dimensional face recognition system. More specifically, one of our aims was to investigate how the use of curvature of the diagonal profiles, extracted from 3D facial geometry models can help the neutral face recognition processes. Another aim was to use a gender classifier employed on 3D facial geometry in order to reduce the search space of the database on which facial recognition is performed. 3D facial geometry with facial expression possesses considerable challenges when it comes face recognition as identified by the communities involved in face recognition research. Thus, one aim of this study was to investigate the effects of the curvature-based method in face recognition under expression variations. Another aim was to develop techniques that can discriminate both expression-sensitive and expression-insensitive regions for ii face recognition based on non-neutral face geometry models. In the case of neutral face recognition, we developed a gender classification method using support vector machines based on the measurements of area and volume of selected regions of the face. This method reduced the search range of a database initially for a given image and hence reduces the computational time. Subsequently, in the characterisation of the face images, a minimum feature set of diagonal profiles, which we call T shape profiles, containing diacritic information were determined and extracted to characterise face models. We then used a method based on computing curvatures of selected facial regions to describe this feature set. In addition to the neutral face recognition, to solve the problem arising from data with facial expressions, initially, the curvature-based T shape profiles were employed and investigated for this purpose. For this purpose, the feature sets of the expression-invariant and expression-variant regions were determined respectively and described by geodesic distances and Euclidean distances. By using regression models the correlations between expressions and neutral feature sets were identified. This enabled us to discriminate expression-variant features and there was a gain in face recognition rate. The results of the study have indicated that our proposed curvature-based recognition, 3D gender classification of facial geometry and analysis of facial expressions, was capable of undertaking face recognition using a minimum set of features improving efficiency and computation.

Differential geometry is about space (a manifold) and a geometric structure on that space. In Riemann’s lecture (see [17]), he stated that “Thus arises the problem, to discover the matters of fact from which the measure-relations of space may be determined...”. It is key then to understand how manifolds differ from one another geometrically. The results of this dissertation concern how the geometry of a manifold changes when we alter metrical connections. We investigate how diverse geodesics are in different metrical connections. From this, we investigate a new class of metrical connections which are dependent on the class of smooth functions. Specifically, we fix a Riemannian metric and investigate the geometry of the manifold when we change the metrical connections associated with the fixed Riemannian metric. We measure the change in the Riemannian curvatures associated with this new class of metrical connections, and then give uniqueness and existence criterion for curvature of compact 2-manifolds. These results depend on the use of Hodge Theory and ultimately on the function f we choose to define a metrical connection.

Berry curvature appears in the semi-classical theory of Bloch electrons already to first order in electromagnetic fields, resulting in profound modification of the carrier velocity and phase space density of states. Here we derive the equations of motion for the physical position and crystal momentum to second order in the fields. The dynamics still has a Hamiltonian structure, albeit with noncanonical Poisson brackets between the physical variables. We are able to expand both the carrier energy and the Poisson brackets to second order in the fields with terms of clear physical meaning. To demonstrate the utility of our theory, we obtain with much ease the electromagnetic response and orbital magnetic susceptibility. / text

This thesis focuses on a certain type of nonlinear parabolic partial differential equations, i.e. PME and FDE. Chapter 1 consists of a survey on results related to PME and FDE, and a short review on some works about deriving gradient estimates in probabilistic ways. In Chapter 2 we estimate gradient on space variables of solutions to the heat equation on Euclidean space. The main idea is to construct two semimartingales by letting the solution and its gradient running backward on the path space of a diffusion process. Estimates derived from decompositions of those two semimartingales are then combined to give rise to an upper bound on gradient that only involves the maximum of the initial data and time variable. In particular, it is independent of the dimension. In Chapter 3 we carry the idea in Chapter 2 onto the study of positive solutions to PME or FDE, and obtained a similar type of bound on |∇u| for local solutions to PME or FDE on Euclidean space. In existing literature there have always been constraints on m. By considering a more general form of transformation on u and introducing a family of equivalent measures on path space, we add more flexibility to our method. Thus our result is valid for a larger range of m. For global solutions, when m violates our constraint, we need two-sided bound on u to control |∇u|. In Chapter 4 we utilize maximum principle to derive Li-Yau type gradient estimate for PME on a compact Riemannian manifold with Ricci curvature bounded from below. Our result is able to yield a Harnack inequality possessing the right order in time variable when the lower bound of Ricci curvature is negative.

In this work we study two problems about parabolic partial differential equations on Riemannian manifolds with conical singularities. The first problem we are concerned with is the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. By introducing so called weighted Hölder and Sobolev spaces with discrete asymptotics, we provide a complete existence and regularity theory for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. The second problem we study is the short time existence problem for the generalized Lagrangian mean curvature flow in almost Calabi-Yau manifolds, when the initial Lagrangian submanifold has isolated conical singularities that are modelled on stable special Lagrangian cones. First we use Lagrangian neighbourhood theorems for Lagrangian submanifolds with conical singularities to integrate the generalized Lagrangian mean curvature flow to a nonlinear parabolic equation of functions, and then, using the existence and regularity theory for the heat equation, we prove short time existence of the generalized Lagrangian mean curvature flow with isolated conical singularities by letting the conical singularities move around in the ambient space and the model cones to rotate by unitary transformations.