Approximating the Derivative of Manifold-valued Functions

We consider the approximation of manifold-valued functions by embedding the manifold into a higher dimensional space, applying a vector-valued approximation operator and projecting the resulting vector back to the manifold. It is well known that the approximation error for manifold-valued functions is close to the approximation error for vector-valued functions. This is not true anymore if we consider the derivatives of such functions. In our paper we give pre-asymptotic error bounds for the approximation of the derivative of manifold-valued function. In particular, we provide explicit constants that depend on the reach of the embedded manifold.Comment: 25 pages, 5 figure

An Optimal Ansatz Space for Moving Least Squares Approximation on Spheres

We revisit the moving least squares (MLS) approximation scheme on the sphere $\mathbb S^{d-1} \subset \mathbb R^d$, where $d>1$. It is well known that using the spherical harmonics up to degree $L \in \mathbb N$ as ansatz space yields for functions in $\mathcal C^{L+1}(\mathbb S^{d-1})$ the approximation order $\mathcal O \left( h^{L+1} \right)$, where $h$ denotes the fill distance of the sampling nodes. In this paper we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degree up to $L$, while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as $h \to 0$. Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space as ansatz space.Comment: 21 pages, 4 figures. Will be submitted to 'Advances in Computational Mathematics

Habit plane determination from reconstructed parent phase orientation maps

This study details the development and validation of a new algorithm that determines the dominant habit plane of a transformed child phase from orientation maps of a single planar cross-section. The method describes the habit plane in terms of its five-parameter grain boundary character and couples it to the specific orientation relationship of the identified orientation variant. The symmetry operations associated with the specific orientation relationship of the variants are applied to transform habit plane traces as determined in the specimen-fixed reference frame into the parent or child reference frame, allowing for the fitting of the habit plane. Our algorithm stands out by its robustness, computational efficiency, automation and ability to operate on fully transformed microstructures. Four automated methods for habit plane trace determination are proposed and compared. Detailed sensitivity analysis reveals that the proposed algorithm is exceptionally robust against poor accuracy in the measured traces and distortions in the orientation map, but more sensitive to inaccuracies propagated from parent grain reconstruction. Validation on a synthetic microstructure with a known habit plane and returned consistent results when applied to high and low carbon steels with different prior austenite grain sizes and orientation map resolutions. The habit planes were not significantly affected by the austenite grain sizes. The habit plane of the steel with 0.35 wt.% C was close to (111){\gamma} whereas the habit plane of steel with 0.71 wt.% C was closer to (575){\gamma}, in close agreement with previous work using two-surface stereological analysis and transmission electron microscopy-based trace analysis

Установка первичной переработки нефти с разработкой клапанной ректификационной колонны

В дипломном проекте рассмотрена установка первичной переработки нефти с разработкой клапанной ректификационной колонны. В дипломном проекте рассмотрены различные конструкции контактных устройств. Приведены расчеты основных технологических и конструктивных показателей аппарата. Приведены расчеты основных технико-экономических показателей, выполнен годовой график Планово-предупредительных работ, по которым видно, что работа установки первичной переработки нефти является экономически эффективной. Дипломный проект содержит расчетно-пояснительную записку из 136 страниц текста, 15 таблиц, 24 рисунка, 27 литературных источника и графическую часть из 3 листов формата А1.The thesis project covers installation of primary oil processing with the development of the valve rectification column. Different designs of contact devices are examined in the thesis project. The calculations of basic technological and structural indicators of the device are shown in it. The calculations of the main technical and economic indicators are also shown in the thesis project, the annual schedule of preventive work is made, which shows that the work of installation of primary oil refining is cost-effective. The thesis project includes the settlement and explanatory note from the 136 pages of the text, 15 tables, 24 figures, 27 references and the graphic part from the 3 sheets of A1 format

EBSD orientation analysis based on experimental Kikuchi reference patterns

Orientation determination does not necessarily require complete knowledge of the local atomic arrangement in a crystalline phase. We present a method for microstructural phase discrimination and orientation analysis of phases for which there is only limited crystallographic information available. In this method, experimental Kikuchi diffraction patterns are utilized to generate a self-consistent master reference for use in the technique of Electron Backscatter Diffraction (EBSD). The experimentally derived master data serves as an application-specific reference in EBSD pattern matching approaches. As application examples, we map the locally varying orientations in samples of icosahedral quasicrystals observed in a Ti40Zr40Ni20 alloy, and we analyse AlNiCo decagonal quasicrystals