On multi-degree splines

Multi-degree splines are piecewise polynomial functions having sections of different degrees. For these splines, we discuss the construction of a B-spline basis by means of integral recurrence relations, extending the class of multi-degree splines that can be derived by existing approaches. We then propose a new alternative method for constructing and evaluating the B-spline basis, based on the use of so-called transition functions. Using the transition functions we develop general algorithms for knot-insertion, degree elevation and conversion to B\'ezier form, essential tools for applications in geometric modeling. We present numerical examples and briefly discuss how the same idea can be used in order to construct geometrically continuous multi-degree splines

Constructing an overall dynamical model for a system with changing design parameter properties

This study considers the identification problem for a class of non-linear parameter-varying systems associated with the following scenario: the system behaviour depends on some specifically prescribed parameter properties, which are adjustable. To understand the effect of the varying parameters, several different experiments, corresponding to different parameter properties, are carried out and different data sets are collected. The objective is to find, from the available data sets, a common parameter-dependent model structure that best fits the adjustable parameter properties for the underlying system. An efficient Common Model Structure Selection (CMSS) algorithm, called the Extended Forward Orthogonal Regression (EFOR) algorithm, is proposed to select such a common model structure. Two examples are presented to illustrate the application and the effectiveness of the new identification approach

Matrix representations for multi-degree B-splines

The paper is concerned with computing the B-spline basis of a multi-degree spline space, namely a space of piecewise functions comprised of polynomial segments of different degrees. To this aim, we provide a general method to work out a matrix representation relating the sought basis with another one easier to compute. This will allow us, for example, to calculate a multi degree B-spline basis starting from local Bernstein bases of different degrees or from the B-spline basis of a spline space where all sections have the same degree. This change of basis can be translated into a conceptually simple and computationally efficient algorithm for the evaluation of multi-degree B-splines

Identification of Nonlinear Parameter-Dependent Common-Structured models to accommodate varying experimental conditions and design parameter properties

This study considers the identification problem for a class of nonlinear parameter-varying systems associated with the following scenario: the system behaviour depends on some specifically prescribed parameter properties, which are adjustable. To understand the effect of the varying parameters, several different experiments, corresponding to different parameter properties, are carried out and different data sets are collected. The objective is to find, from the available data sets, a common parameter-dependent model structure that best fits the adjustable parameter properties for the underlying system. An efficient common model structure selection (CMSS) algorithm, called the extended forward orthogonal regression (EFOR) algorithm, is proposed to select such a common model structure. Several examples are presented to illustrate the application and the effectiveness of the new identification approach

Interactive display of isosurfaces with global illumination

Journal ArticleAbstract-In many applications, volumetric data sets are examined by displaying isosurfaces, surfaces where the data, or some function of the data, takes on a given value. Interactive applications typically use local lighting models to render such surfaces. This work introduces a method to precompute or lazily compute global illumination to improve interactive isosurface renderings. The precomputed illumination resides in a separate volume and includes direct light, shadows, and interreflections. Using this volume, interactive globally illuminated renderings of isosurfaces become feasible while still allowing dynamic manipulation of lighting, viewpoint and isovalue

Evolutionary structural pptimisation based on boundary element representation of b-spline geometry

Evolutionary Structural Optimisation (ESO) has become a well-established technique for determining the optimum shape and topology of a structure given a set of loads and constraints. The basic ESO concept that the optimum topology design evolves by slow removal and addition of material has matured over the last ten years. Nevertheless, the development of the method has almost exclusively considered finite elements (FE) as the approach for providing stress solutions. This thesis presents an ESO approach based on the boundary element method. Non-uniform rational B-splines (NURBS) are used to define the geometry of the component and, since the shape of these splines is governed by a set of control points, use can be made of the locations of these control points as design variables. The developed algorithm creates internal cavities to accomplish topology changes. Cavities are also described by NURBS and so they have similar behaviour to the outside boundary. Therefore, both outside and inside are optimised at the same time. The optimum topologies evolve allowing cavities to merge between each other and to their closest outer boundary. Two-dimensional structural optimisation is investigated in detail exploring multi-load case and multi-criteria optimisation. The algorithm is also extended to three-dimensional optimisation, in which promising preliminary results are obtained. It is shown that this approach overcomes some of the drawbacks inherent in traditional FE-based approaches, and naturally provides accurate stress solutions on smooth boundary representations at each iteration

Progettazione e Sviluppo di una Web App per curve MD-spline

La tesi si occupa della progettazione e sviluppo di una Web App scritta in HTML5 e canvas e JavaScript che ha l'obiettivo di permettere di sperimentare le potenzialità di una nuova classe di curve 2D utili per il disegno denominata C1 MD-spline

Exchangeability and sets of desirable gambles

Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We give a general discussion of such models and their rationality criteria. We study exchangeability assessments for them, and prove counterparts of de Finetti's finite and infinite representation theorems. We show that the finite representation in terms of count vectors has a very nice geometrical interpretation, and that the representation in terms of frequency vectors is tied up with multivariate Bernstein (basis) polynomials. We also lay bare the relationships between the representations of updated exchangeable models, and discuss conservative inference (natural extension) under exchangeability and the extension of exchangeable sequences.Comment: 40 page

Statistically optimal continuous free energy surfaces from biased simulations and multistate reweighting

Free energies as a function of a selected set of collective variables are commonly computed in molecular simulation and of significant value in understanding and engineering molecular behavior. These free energy surfaces are most commonly estimated using variants of histogramming techniques, but such approaches obscure two important facets of these functions. First, the empirical observations along the collective variable are defined by an ensemble of discrete observations and the coarsening of these observations into a histogram bins incurs unnecessary loss of information. Second, the free energy surface is itself almost always a continuous function, and its representation by a histogram introduces inherent approximations due to the discretization. In this study, we relate the observed discrete observations from biased simulations to the inferred underlying continuous probability distribution over the collective variables and derive histogram-free techniques for estimating this free energy surface. We reformulate free energy surface estimation as minimization of a Kullback-Leibler divergence between a continuous trial function and the discrete empirical distribution and show that this is equivalent to likelihood maximization of a trial function given a set of sampled data. We then present a fully Bayesian treatment of this formalism, which enables the incorporation of powerful Bayesian tools such as the inclusion of regularizing priors, uncertainty quantification, and model selection techniques. We demonstrate this new formalism in the analysis of umbrella sampling simulations for the $\chi$ torsion of a valine sidechain in the L99A mutant of T4 lysozyme with benzene bound in the cavity.Comment: 24 pages, 5 figure