Visualization Of Curve And Surface Data Using Rational Cubic Ball Functions

This study considered the problem of shape preserving interpolation through regular data using rational cubic Ball which is an alternative scheme for rational Bézier functions. A rational Ball function with shape parameters is easy to implement because of its less degree terms at the end polynomial compared to rational Bézier functions. In order to understand the behavior of shape parameters (weights), we need to discuss shape control analysis which can be used to modify the shape of a curve, locally and globally. This issue has been discovered and brought to the study of conversion between Ball and Bézier curve

Constrained Interpolation And Shape Preserving Approximation By Space Curves [QA297.6. K82 2006 f rb].

Dua jenis masalah rekabentuk lengkung telah ipertimbangkan. Terlebih dahulu kami mempertimbangkan interpolasi satu set titik data ruang yang bertertib dengan satu lengkung licin tertakluk kepada satu set satah kekangan yang berbentuk terhingga atau tak terhingga di mana garis cebis demi cebis yang menyambung titik data secara berturutan tidak bersilang dengan satah kekangan. Two types of curve designing problem have been considered. We first consider the interpolation of a given set of ordered spatial data points by a smooth curve in the presence of a set of finite or infinite constraint planes, where the polyline joining consecutive data points does not intersect with the constraint planes

Splainidega lahendamine

Dissertatsioonis on käsitletud kolme probleemi splainide teooriast. Esiteks vaadeldakse kuupsplainidega histopoleerimist. Antud on suvaliselt paiknevate sõlmedega ja suvaliste kõrgustega histogramm. Näidatakse, et alati on olemas histopoleeriv kuupsplain üldka-su¬tatavate rajatingimuste korral. Splaini leidmisel kasutatakse erinevaid esitusi. Teiseks uuritakse suvalise võrgu korral histopoleeriva polünomiaalse perioodilise splaini ole¬mas-olu. Saadud tulemustest järelduvad kirjandusest varem teadaolevad tulemused ühtla¬se võrgu korral. Kolmas probleem, mida dissertatsioonis lahendatakse, on ruut/lineaar ratsionaalsplainidega histopoleerimine. Taolised splainid säilitavad lähteandmete kume-ru¬se, sest nad on ise alati selle omadusega. Et vabadus on siin splaini sõlmede valikus, on loomulik küsida, kas suvalise rangelt kumera histogrammi korral on võimalik valida splaini sõlmed nii, et eksisteeriks histopoleeriv ruut/lineaar ratsionaalsplain. Vastus on siin eitav. Uurimisel leitakse splaini parameetreid määrav baasvõrrandite süsteem ning splaini eksisteerimine on samaväärne selle mittelineaarse süsteemi lahendi olemasoluga. Probleemi lahendamiseks on leitud splaini sobiv esitus.The dissertation treats three kinds of problems from the theory of splines. Firstly, a par¬ti-cular interpolation problem about the cubic spline histopolation with arbitrary place¬ment of histogram knots and spline knots between them is discussed. A cubic spline is studied provided that its integral on a prescribed interval equals the area of the corres¬pon¬ding histogram rectangle. It is considered the most common boundary value con¬di¬tions like given values of the spline and its first and second derivatives in endpoints of given interval and then solved the problem of existence and uniqueness of the solution for such histopolation problem. Secondly, the periodic polynomial spline histopolation problem with the arbitrary placement of histogram knots and coinciding histogram knots is considered. Several results about the existence and uniqueness of solution are obtained and they imply known results in the case of uniform grid. In the last problem, the rational spline histopolation of convex data is studied. For the concern about the con¬vexity, an appropriate tool is interpolation or histopolation with quadratic/linear rational splines because these splines keep the sign of its second derivative on the whole interval. For this reason the given histogram is assumed to be strictly convex. The main task is at the study of existence of solution for a nonlinear system of basic equations to determine the values of second derivatives in spline knots. The other parameters in the representation of spline are determined from a linear system with regular matrix. It is shown that there is a strictly convex histogram without the solution of histopolation problem for any choice of spline knots.https://www.ester.ee/record=b524283

Preserving Positivity And Monotonicity Of Real Data Using Bézier-Ball Function And Radial Basis Function

In this thesis, a rational cubic Bézier-Ball function which refers to a rational cubic Bézier function expressed in terms of Ball control points and weights are used to preserve positivity and monotonicity of real data sets. Four shape parameters are proposed to preserve the characteristics of the data. A rational Bi-Cubic Bézier-Ball function is introduced to preserve the positivity of surface generated from real data set and from known functions. Eight shape parameters proposed can be modified to preserve the positivity of the surface. Interpolating 2D and 3D real data using radial basis function (RBF) is proposed as an alternative method to preserve the positivity of the data. Two types of RBF which are Multiquadric (MQ) function and Gaussian function, which contains a shape parameter are used. The boundaries (lower and upper limit) of the shape parameter which preserves the positivity of real data are proposed. Comparisons are made using the root-mean-square (RMS) error between the proposed interpolation methods with existing works in literature. It was found that MQ function and rational cubic Bézier-Ball is comparable with existing literature in preserving positivity for both curves and surfaces. For preserving monotonicity, the rational cubic Bézier-Ball is comparable but the MQ quasi-interpolation introduced can only linearly interpolate the curve and the RMS values are big. Gaussian function is able to preserve positivity of curves and surfaces but with unwanted oscillations which result to unsmooth curves

Controlling the interpolation of NURBS curves and surfaces

The primary focus of this thesis is to determine the best methods for controlling the interpolation of NURBS curves and surfaces. The various factors that affect the quality of the interpolant are described, and existing methods for controlling them are reviewed. Improved methods are presented for calculating the parameter values, derivative magnitudes, data point spacing and twist vectors, with the aim of producing high quality interpolants with minimal data requirements. A new technique for obtaining the parameter values and derivative magnitudes is evaluated, which constructs a C$^1$ cubic spline with orthogonal first and second derivatives at specified parametric locations. When this data is used to create a C$^2$ spline, the resulting interpolant is superior to those constructed using existing parameterisation and derivative magnitude estimation methods. Consideration is given to the spacing of data points, which has a significant impact on the quality of the interpolant. Existing methods are shown to produce poor results with curves that are not circles. Three new methods are proposed that significantly reduce the positional error between the interpolant and original geometry. For constrained surface interpolation, twist vectors must be estimated. A method is proposed that builds on the Adini method, and is shown to have improved error characteristics. In numerical tests, the new method consistently outperforms Adini. Interpolated surfaces are often required to join together smoothly along their boundaries. The constraints for joining surfaces with parametric and geometric continuity are discussed, and the problem of joining $N$ patches to form an $N$-sided region is considered. It is shown that regions with odd $N$ can be joined with G$^1$ continuity, but those with even $N$ or requiring G$^2$ continuity can only be obtained for specific geometries

Essays in Power System Economics

In the first chapter, we propose a new method for modeling competition in electricity spot markets, namely, by approximating the supply functions of the competitors with cubic splines. We argue that this method is preferable to approximation by linear or piecewise-affine functions, which have been the main approaches to date. We apply our method to the firms competing in the Texas market. We also show that, more often than not, we will observe that the marginal revenue functions of the firms will have increasing segments which may lead to multiple profit-maximizing optima for a firm. In the second chapter, we model the effects of forward contracting on power prices in wholesale electricity markets. In contrast to most of the previous literature, we explicitly model power retailers, and introduce risk aversion. As expected, increasing the number of players have pro-competitive effects on the spot price of electricity. We also find that as the generators bid more competitively, spot and forward prices converge. Our model also captures the effects of level and variability of power demand on the players' contracting decisions. In the final chapter, we depart from equilibrium approach and utilizing agent-based modeling, analyze the effects of increased power demand price sensitivity on the level and volatility of power prices. We find that as the price sensitivity increases at the demand side, power price as well as its volatility decrease significantly. We also argue that the celebrated Herfindahl-Hirschman Index to measure market concentration is not a suitable metric for power markets

Adaptive methods for linear dynamic systems in the frequency domain with application to global optimization

Designers often seek to improve their designs by considering several discrete modifications. These modifications may require changes in materials and geometry, as well as the addition or removal of individual components. In general, if the modifications are applied one at a time, none of them may sufficiently improve the performance. Also, the total number of modifications that may be included in the final design is often limited due to cost or other constraints. The designer must therefore determine the optimal combination of modifications in order to complete the design. While this design challenge arises fairly commonly in practice, very little research has studied it in its full generality. This work assumes that the mathematical description of the design and its modifications are frequency dependent matrices. Such matrices typically arise due to finite element analysis as well as other modeling techniques. Computing performance metrics related to steady-state forced response, also known as performing a frequency sweep, involves factorizing these matrices many times. Additionally, determining the globally optimum design in this case involves an exhaustive search of the combinations of modifications. These factors lead to prohibitively long run times particularly as the size of the system grows. The research presented here seeks to reduce these costs, making such a search feasible. Several innovative techniques have been developed and tested over the course of the research, focused in two primary areas: adaptive frequency sweeps and efficient combinatorial optimization. The frequency sweep methods rely on an adaptive bisection of the frequency range and either a subspace approximation based on implicit interpolatory model order reduction or an elementwise approximation using piecewise multi-point Padé interpolants. Additionally, a strategy for augmenting the adaptive methods with the system's modal information is presented. For combinatorial optimization, an approximation algorithm is developed that capitalizes on any presence of dynamic uncoupling between modifications. The net effect of this work is to allow designers and researchers to develop new dynamic systems and perform analyses faster and more efficiently than ever before

ShapeWright--finite element based free-form shape design

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1990.Includes bibliographical references (p. 179-192).by George Celniker.Ph.D