Mathematical And Computational Methods For Freeform Optical Shape Description

Slow-servo single-point diamond turning as well as advances in computer controlled small lap polishing enable the fabrication of freeform optics, specifically, optical surfaces for imaging applications that are not rotationally symmetric. Freeform optical elements will have a profound importance in the future of optical technology. Orthogonal polynomials added onto conic sections have been extensively used to describe optical surface shapes. The optical testing industry has chosen to represent the departure of a wavefront under test from a reference sphere in terms of orthogonal φ-polynomials, specifically Zernike polynomials. Various forms of polynomials for describing freeform optical surfaces may be considered, however, both in optical design and in support of fabrication. More recently, radial basis functions were also investigated for optical shape description. In the application of orthogonal φ-polynomials to optical freeform shape description, there are important limitations, such as the number of terms required as well as edge-ringing and ill-conditioning in representing the surface with the accuracy demanded by most stringent optics applications. The first part of this dissertation focuses upon describing freeform optical surfaces with φ-polynomials and shows their limitations when including higher orders together with possible remedies. We show that a possible remedy is to use edge-clusteredfitting grids. Provided different grid types, we furthermore compared the efficacy of using different types of φ-polynomials, namely Zernike and gradient orthogonal Q-polynomials. In the second part of this thesis, a local, efficient and accurate hybrid method is developed in order to greatly reduce the order of polynomial terms required to achieve higher level of accuracy in freeform shape description that were shown to require thousands of terms including many higher order terms under prior art. This comes at the expense of multiple sub-apertures, and as such iv computational methods may leverage parallel processing. This new method combines the assets of both radial basis functions and orthogonal phi-polynomials for freeform shape description and is uniquely applicable across any aperture shape due to its locality and stitching principles. Finally in this thesis, in order to comprehend the possible advantages of parallel computing for optical surface descriptions, the benefits of making an effective use of impressive computational power offered by multi-core platforms for the computation of φ-polynomials are investigated. The φ-polynomials, specifically Zernike and gradient orthogonal Q-polynomials, are implemented with a set of recurrence based parallel algorithms on Graphics Processing Units (GPUs). The results show that more than an order of magnitude speedup is possible in the computation of φ-polynomials over a sequential implementation if the recurrence based parallel algorithms are adopted

Differential-Algebraic Equations

Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed

Application of Graph Neural Networks and graph descriptors for graph classification

Graph classification is an important area in both modern research and industry. Multiple applications, especially in chemistry and novel drug discovery, encourage rapid development of machine learning models in this area. To keep up with the pace of new research, proper experimental design, fair evaluation, and independent benchmarks are essential. Design of strong baselines is an indispensable element of such works. In this thesis, we explore multiple approaches to graph classification. We focus on Graph Neural Networks (GNNs), which emerged as a de facto standard deep learning technique for graph representation learning. Classical approaches, such as graph descriptors and molecular fingerprints, are also addressed. We design fair evaluation experimental protocol and choose proper datasets collection. This allows us to perform numerous experiments and rigorously analyze modern approaches. We arrive to many conclusions, which shed new light on performance and quality of novel algorithms. We investigate application of Jumping Knowledge GNN architecture to graph classification, which proves to be an efficient tool for improving base graph neural network architectures. Multiple improvements to baseline models are also proposed and experimentally verified, which constitutes an important contribution to the field of fair model comparison.Comment: Master's thesis submitted at AGH University of Science and Technolog

Design and applications of a graphics package for the HP1000 computer.

The objective of this thesis is to develop the FORTRAN subroutine PLOTER which is a general-purpose plotting tool to plot charts on a Hewlett Packard plotter. The programs RESP and INVLAP which can plot the frequency and time responses of system functions are modified to adopt the PLOTER subroutine and are stored of the HP1000-A900 minicomputer whose software, the GRAPHICS/1000, supports the graphics ability of PLOTER. This thesis describes the theories, functions, software techniques and operations of the PLOTER subroutine and the application programs RESP and the INVLAP. It also provides program listings and example plots

Full-field analysis of the dynamic behaviour of thermally stressed panels

This thesis details the research conducted over the course of three years under funding from the European Office of the United States Air Force (EOARD) and the Engineering and Physical Sciences Research Council (EPSRC) as a part of a long-standing effort to collect high-quality experimental data which can be used in the development and validation of predictive computational mechanics models. The focus of this study is on the acquisition of full-field displacement and temperature data when thermally and thermo-mechanically loading aerospace grade material panels as a means to study the effect of non-uniform temperature distributions on their dynamic behaviour at a component level (macroscale). The inclusion of this data in the development of a robust predictive model has also been investigated. To that end, a review of the existing literature is provided which highlights the current knowledge gaps in the modelling and experiments on the thermal and thermo-vibratory loading of panels, as well as the state-of-the-art in full-field data analysis. Initially, a finite element (FE) model was developed and compared to predictive and experimental data available in literature. This allowed for an investigation into the best practices to adopt in the development of a computational mechanics model with temperature-dependent material properties. It was found that a successful representation of experimental conditions strongly depends on the effective depiction of the thermal load and initial shape of the component. Then, a thin plate with free edges and constrained about its centre was heated using quartz lamps arranged in two different configurations and mechanically loaded using a shaker. Experimental modal analysis was used to acquire the resonant frequencies and mode shapes of the plate. Mode shapes were studied by exciting the plate to its first eleven resonant frequencies and acquiring displacement data using a Pulsed Laser Digital Image Correlation method (PL-DIC). Infra-red imaging was used to acquire temperature maps across the specimen. Experimentally-acquired temperature maps and measurements of the plate’s initial shape were included in a temperature-dependent FE model, developed according to the findings in the preliminary study, previously described. For the first time, experimental results showed the resonant response of the plate to strongly depend on the temperature distribution across the structure, correlating well with past predictive work in the literature. This was supported by the results from the finite element model, which were validated against experimental data and found to yield reliable predictions. The influence of temperature distribution in the deformation of panels was further investigated using a 1 mm plate with reinforced edges. The geometry was designed to emulate an aircraft’s skin with the reinforced edges performing the function of stringers and ribs. High temperatures were achieved using quartz lamps arranged in various configurations with controllable power output. PL-DIC was used to measure surface displacements and a commercially-available micro bolometer mapped the temperature distribution across the plate. Deflection results for the reinforced plate showed it to behave as a dynamic system that buckles out-of-plane when heated before relaxing to a steady state. It was demonstrated that the out-of-plane displacement experienced by the plate is strongly influenced by the in-plane spatial distribution of temperature