Blending techniques in Curve and Surface constructions

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Ship Hull Representation by Non-Uniform Rational B-Spline Surface Patches

The purpose of this work is to propose a new method for representing the ship hull shape with mathematic surfaces so that geometric data can be generated for any point on the hull where required to assist the production process. An extensive survey of previous work is presented covering both the use of parametric curves and surfaces to model the ship hull and also the most relevant software systems developed for that purpose. The main methods and algorithms available for the generation and edition of curves and surfaces are presented and compared taking into consideration the intended application. From the analysis of the formulations available it was concluded that the most adequate one, which however had not yet been extensively used to model ship hulls was the Non-Uniform Rational B-Splines (NURBS), due to the potential of their capability to represent exactly conic curves and surfaces. Therefore these surfaces were selected as the basis of the method developed in this thesis. A procedure is proposed for the representation of a given hull form in a two step approach, creating first a wireframe model over which the surface patches are generated. Both curves and surfaces are based on the NURBS formulation. To create the wireframe model, first a set of longitudinal boundary lines is selected, dividing the surface into areas of similar shape. Then, these lines are fitted by curves and faired to some extent. Next, transverse sections are defined and split by the boundary lines. Surface patches are then generated over the transverse section curves within the limits of each patch. Finally, to obtain the traditional representation of the ship surface by transverse sections, buttocks and waterlines, contour lines are generated for constant values of x, y and z coordinates. A computer system has been developed incorporating an interface that allows the visualization of the curves and surfaces being modeled. The system incorporates several algorithms for generation and edition of curves and surfaces, in addition to the main contribution of this thesis which is the use of NURBS to represent the ship hull surface. The system also incorporates curve and surface analysis tools and some basic fairing algorithms so that during the several steps of the creation of the model, the fairness of the curves and surfaces can be evaluated and improved to some extent. The procedure is tested and compared with an existing commercial system through some application examples, of a complete hull and in more detail in the bow region, showing that good results can be obtained with the system presented here

Modelling the term structure of interest rates for three European countries

It is extremely important to understand the inner workings of debt markets, since they involve interesting and complex financial concepts. Moreover, the concepts related to government debt issued in form of bonds have a series of characteristics that might impact the average person reality, for which it should be completely understood. Having the previous motivations in mind, in this thesis, concepts related to the correct estimation of the zero-coupon yield curves (or term structure of interest rates) will be deeply analysed. In order to do so, several relevant models, from the one of Nelson and Siegel (which was the basis for several other more complete models), to the widely currently applied Adjusted Svensson model proposed by De Pooter will be discussed and applied. In the end of all the applications of the different exposed models, a comparison between the criteria resulting from such applications is performed, in order to assess which model fits in a more accurate way into the dataset used in this thesis. Finally, the relationship between risk and return will also be approached in a bond market context, as well as the connection between this relationship and the different risk profiles that an investor may present.É extremamente importante compreender o funcionamento dos mercados de dívida, uma vez que esses mercados envolvem conceitos financeiros interessantes e complexos. Para além disso, os conceitos associados à dívida pública emitida sob a forma de obrigações compreendem um conjunto de caraterísticas que podem ter impacto na realidade de cada pessoa no geral, pelo o qual devem ser totalmente compreendidos. Tendo estas motivações em mente, no decorrer desta tese, conceitos relacionados com a estimação correta da estrutura temporal das taxas de juro serão profundamente analisados, através da apresentação e aplicação de um conjunto de modelos relevantes para o efeito, desde o modelo proposto por Nelson and Siegel (sendo este a base para uma série de outros modelos mais completos), até ao modelo ajustado de Svensson, proposto por De Pooter, que é vastamente aplicado na atualidade. Após a aplicação dos diferentes modelos expostos, é feita uma comparação entre os resultados de cada aplicação, com vista a determinar qual o modelo que se adequa de uma forma mais precisa nos dados utilizados. Por fim, a relação entre risco e retorno será também abordada num contexto de mercado de obrigações, assim como a ligação entre essa relação com os diferentes perfis de risco que um investidor pode apresentar

Representation and application of spline-based finite elements

Isogeometric analysis, as a generalization of the finite element method, employs spline methods to achieve the same representation for both geometric modeling and analysis purpose. Being one of possible tool in application to the isogeometric analysis, blending techniques provide strict locality and smoothness between elements. Motivated by these features, this thesis is devoted to the design and implementation of this alternative type of finite elements. This thesis combines topics in geometry, computer science and engineering. The research is mainly focused on the algorithmic aspects of the usage of the spline-based finite elements in the context of developing generalized methods for solving different model problems. The ability for conversion between different representations is significant for the modeling purpose. Methods for conversion between local and global representations are presented

ESTIMATION OF THE FISHER EFFECT ON THE TERM STRUCTURE OF INTEREST RATES EMPLOYING A TERM STRUCTURE OF INFLATIONARY EXPECTATIONS

This study deals with the estimation of the effect of a term structure of inflationary expectations on the term structure of interest rates. By estimating the Fisher effect over the entire term structure, this analysis captures the associational effects between values of the interest rates along a term structure and values of inflationary expectations along a term structure. The study contains a discussion of the Fisher hypothesis as it was developed and tested by Irving Fisher. Models of inflationary expectations and previous applications of these to test the Fisher effect are discussed. A method of constructing a term structure of inflationary expectations is developed. Inflationary expectations are estimated and proof is offered to demonstrate that these expectations are statistically rational. A model for estimating nominal interest rates from yields on Treasury notes and Bank discounts on Treasury bills is developed. The technique is applied to obtain estimates of the term structure of nominal interest rates, monthly, for the period of January 1970 through November 1982. In order to summarize the term structures of inflationary expectations and the term structures of nominal interest rates as functions, this study estimates empirical term structures using cubic exponential spline functions. There is a discussion of spline methodology from a modeling perspective and from an econometric perspective. The Fisher effect of the term structure of inflationary expectations on the term structure of interest rates is estimated by pooling the cross-section data described by the coefficients of the cubic exponential splines, and the time-series data. The evidence does not reject the Fisher hypothesis of a complete pass-through of inflationary expectations to nominal interest rates in a world of taxes. The evidence also suggests that associational effects along the term structures are present

Smooth and Time-Optimal Trajectory Generation for High Speed Machine Tools

In machining complex dies, molds, aerospace and automotive parts, or biomedical components, it is crucial to minimize the cycle time, which reduces costs, while preserving the quality and tolerance integrity of the part being produced. To meet the demands for high quality finishes and low production costs in machining parts with complex geometry, computer numerical control (CNC) machine tools must be equipped with spline interpolation, feedrate modulation, and feedrate optimization capabilities. This thesis presents the development of novel trajectory generation algorithms for Non Uniform Rational B-Spline (NURBS) toolpaths that can be implemented on new low-cost CNC's, as well as, in conjunction with existing CNC's. In order to minimize feedrate fluctuations during the interpolation of NURBS toolpaths, the concept of the feed correction polynomial is applied. Feedrate fluctuations are reduced from around 40 % for natural interpolation to 0.1 % for interpolation with feed correction. Excessive acceleration and jerk in the axes are also avoided. To generate jerk-limited feed motion profiles for long segmented toolpaths, a generalized framework for feedrate modulation, based on the S-curve function, is presented. Kinematic compatibility conditions are derived to ensure that the position, velocity, and acceleration profiles are continuous and that the jerk is limited in all axes. This framework serves as the foundation for the proposed heuristic feedrate optimization strategy in this thesis. Using analytically derived kinematic compatibility equations and an efficient bisection search algorithm, the command feedrate for each segment is maximized. Feasible solutions must satisfy the optimization constraints on the velocity, control signal (i.e. actuation torque), and jerk in each axis throughout the trajectory. The maximized feedrates are used to generate near-optimal feed profiles that have shorter cycle times, approximately 13-26% faster than the feed profiles obtained using the worst-case curvature approach, which is widely used in industrial CNC interpolators. The effectiveness of the NURBS interpolation, feedrate modulation and feedrate optimization techniques has been verified in 3-axis machining experiments of a biomedical implant

Implementation of B-splines in a Conventional Finite Element Framework

The use of B-spline interpolation functions in the finite element method (FEM) is not a new subject. B-splines have been utilized in finite elements for many reasons. One reason is the higher continuity of derivatives and smoothness of B-splines. Another reason is the possibility of reducing the required number of degrees of freedom compared to a conventional finite element analysis. Furthermore, if B-splines are utilized to represent the geometry of a finite element model, interfacing a finite element analysis program with existing computer aided design programs (which make extensive use of B-splines) is possible. While B-splines have been used in finite element analysis due to the aforementioned goals, it is difficult to find resources that describe the process of implementing B-splines into an existing finite element framework. Therefore, it is necessary to document this methodology. This implementation should conform to the structure of conventional finite elements and only require exceptions in methodology where absolutely necessary. One goal is to implement B-spline interpolation functions in a finite element framework such that it appears very similar to conventional finite elements and is easily understandable by those with a finite element background. The use of B-spline functions in finite element analysis has been studied for advantages and disadvantages. Two-dimensional B-spline and standard FEM have been compared. This comparison has addressed the accuracy as well as the computational efficiency of B-spline FEM. Results show that for a given number of degrees of freedom, B-spline FEM can produce solutions with lower error than standard FEM. Furthermore, for a given solution time and total analysis time B-spline FEM will typically produce solutions with lower error than standard FEM. However, due to a more coupled system of equations and larger elemental stiffness matrix, B-spline FEM will take longer per degree of freedom for solution and assembly times than standard FEM. Three-dimensional B-spline FEM has also been validated by the comparison of a three-dimensional model with plane-strain boundary conditions to an equivalent two-dimensional model using plane strain conditions