Approximation with interval B-splines for robust reverse engineering

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1997.Includes bibliographical references (leaves 132-136).by Guoling Shen.M.S

MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

This paper studies the problem of finding the smallest $n$-simplex enclosing a given $n^{\text{th}}$-degree polynomial curve. Although the Bernstein and B-Spline polynomial bases provide feasible solutions to this problem, the simplexes obtained by these bases are not the smallest possible, which leads to undesirably conservative results in many applications. We first prove that the polynomial basis that solves this problem (MINVO basis) also solves for the $n^\text{th}$-degree polynomial curve with largest convex hull enclosed in a given $n$-simplex. Then, we present a formulation that is \emph{independent} of the $n$-simplex or $n^{\text{th}}$-degree polynomial curve given. By using Sum-Of-Squares (SOS) programming, branch and bound, and moment relaxations, we obtain high-quality feasible solutions for any $n\in\mathbb{N}$ and prove numerical global optimality for $n=1,2,3$. The results obtained for $n=3$ show that, for any given $3^{\text{rd}}$-degree polynomial curve, the MINVO basis is able to obtain an enclosing simplex whose volume is $2.36$ and $254.9$ times smaller than the ones obtained by the Bernstein and B-Spline bases, respectively. When $n=7$, these ratios increase to $902.7$ and $2.997\cdot10^{21}$, respectively.Comment: 25 pages, 16 figure

Analysis and design of developable surfaces for shipbuilding

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1997, and Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1997.Includes bibliographical references (p. 97-99).by Julie Steele Chalfant.M.S

Applications and Modeling Techniques of Wind Turbine Power Curve for Wind Farms - A Review

In the wind energy industry, the power curve represents the relationship between the “wind speed” at the hub height and the corresponding “active power” to be generated. It is the most versatile condition indicator and of vital importance in several key applications, such as wind turbine selection, capacity factor estimation, wind energy assessment and forecasting, and condition monitoring, among others. Ensuring an effective implementation of the aforementioned applications mostly requires a modeling technique that best approximates the normal properties of an optimal wind turbines operation in a particular wind farm. This challenge has drawn the attention of wind farm operators and researchers towards the “state of the art” in wind energy technology. This paper provides an exhaustive and updated review on power curve based applications, the most common anomaly and fault types including their root-causes, along with data preprocessing and correction schemes (i.e., filtering, clustering, isolation, and others), and modeling techniques (i.e., parametric and non-parametric) which cover a wide range of algorithms. More than 100 references, for the most part selected from recently published journal articles, were carefully compiled to properly assess the past, present, and future research directions in this active domain

Shape-Invariant Models for Non-Independent Functional Data

Non-independent functional data frequently arise in evolutionary and biological studies. It is important to possess models that incorporate correlations between subjects and appropriately describe the relationships between response and covariates. The variation in the response curves is usually a mixture of amplitude and phase variation, both of which should be explicitly modeled for efficient statistical inference. In this dissertation we propose a shape-invariant model that explicitly addresses amplitude and phase variability. We incorporate genetic and environmental random effects for the parameters, and use the additive genetic information matrix in the representation of the covariance matrices to make the unobservable genetic components mathematically identifiable. We derive the asymptotic properties of the maximum likelihood estimators and study their finite sample behavior by simulation. Then we apply the new method to the analysis of growth curves of flour beetles

Trajectory definition with high relative accuracy (HRA) by parametric representation of curves in nano-positioning systems

Nanotechnology applications demand high accuracy positioning systems. Therefore, in order to achieve sub-micrometer accuracy, positioning uncertainty contributions must be minimized by implementing precision positioning control strategies. The positioning control system accuracy must be analyzed and optimized, especially when the system is required to follow a predefined trajectory. In this line of research, this work studies the contribution of the trajectory definition errors to the final positioning uncertainty of a large-range 2D nanopositioning stage. The curve trajectory is defined by curve fitting using two methods: traditional CAD/CAM systems and novel algorithms for accurate curve fitting. This novel method has an interest in computer-aided geometric design and approximation theory, and allows high relative accuracy (HRA) in the computation of the representations of parametric curves while minimizing the numerical errors. It is verified that the HRA method offers better positioning accuracy than commonly used CAD/CAM methods when defining a trajectory by curve fitting: When fitting a curve by interpolation with the HRA method, fewer data points are required to achieve the precision requirements. Similarly, when fitting a curve by a least-squares approximation, for the same set of given data points, the HRA method is capable of obtaining an accurate approximation curve with fewer control points

Numerical proper reparametrization of parametric plane curves

We present an algorithm for reparametrizing algebraic plane curves from a numerical point of view. More precisely, given a tolerance ϵ>0 and a rational parametrization P of a plane curve C with perturbed float coefficients, we present an algorithm that computes a parametrization Q of a new plane curve D such that Q is an ϵ –proper reparametrization of D. In addition, the error bound is carefully discussed and we present a formula that measures the “closeness” between the input curve C and the output curve D