Würzburg, Univ., Diss., 2008

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / In this thesis we study a standard example of refinable functions, that is, functions which can be reproduced by the integer shifts of their own dilations. Using the cardinal B-spline as an introductory example, we prove some of its properties, thereby building a basis for a later extension to the vector setting. Defining a subdivision scheme associated to the B-spline refinement mask, we then present the proof of a well-known convergence result. Subdivision is a powerful tool used in computer-aided geometric design (CAGD) for the generation of curves and surfaces. The basic step of a subdivision algorithm consists of starting with a given set of points, called the initial control points, and creating new points as a linear combination of the previous ones, thereby generating new control points. Under certain conditions, repeated applications of this procedure yields a continuous limit curve. One important goal of this thesis is to study a particular extension of scalar subdivision to matrix subdivision ...

We consider the problem of reconstructing a surface from noisy samples by approximating the point set with non-uniform rational B-spline surfaces. We focus on the fact that the knot sequences should also be part of the unknown variables that include the control points and the weights in order to find their optimal positions. We show how to set up the free knot problem such that constrained nonlinear optimization can be applied efficiently. We describe in detail a parallel implementation of our approach that give almost linear speedup. Finally, we provide numerical results obtained on the Chemnitzer Linux Cluster supercomputer.

In this work we first introduce the basics of surface study and than we get acquainted with three basic types of surfaces of computer graphics, Bézier, rational Bézier and B-spline surfaces. This is to prepare the reader for the study of the fourth type of surfaces - NURBS. All types of introduced surfaces are accompanied by examples and algorithms with programs implemented in MatLab environment. When studying the NURBS surfaces properties we also introduce the associated important geometric tools. With this base of knowledge we concentrate on examples of common surfaces represented as NURBS, that are again accompanied with representation in programs. i

The Wavelet Finite Element Method (WFEM) involves combining the versatile wavelet analysis with the classical Finite Element Method (FEM) by utilizing the wavelet scaling functions as interpolating functions; providing an alternative to the conventional polynomial interpolation functions used in classical FEM. Wavelet analysis as a tool applied in WFEM has grown in popularity over the past decade and a half and the WFEM has demonstrated potential prowess to overcome some difficulties and limitations of FEM. This is particular for problems with regions of the solution domain where the gradient of the field variables are expected to vary fast or suddenly, leading to higher computational costs and/or inaccurate results. The properties of some of the various wavelet families such as compact support, multiresolution analysis (MRA), vanishing moments and the “two-scale” relations, make the use of wavelets in WFEM advantageous, particularly in the analysis of problems with strong nonlinearities, singularities and material property variations present. The wavelet based finite elements (WFEs) presented in this study, conceptually based on previous works, are constructed using the Daubechies and B-spline wavelet on the interval (BSWI) wavelet families. These two wavelet families possess the desired properties of multiresolution, compact support, the “two scale” relations and vanishing moments. The rod, beam and planar bar WFEs are used to study structural static and dynamic problems (moving load) via numerical examples. The dynamic analysis of functionally graded materials (FGMs) is further carried out through a new modified wavelet based finite element formulation using the Daubechies and BSWI wavelets, tailored for such classes of composite materials that have their properties varying spatially. Consequently, a modified algorithm of the multiscale Daubechies connection coefficients used in the formulation of the FGM elemental matrices and load vectors in wavelet space is presented and implemented in the formulation of the WFEs. The approach allows for the computation of the integral of the products of the Daubechies functions, and/or their derivatives, for different Daubechies function orders. The effects of varying the material distribution of a functionally graded (FG) beam on the natural frequency and dynamic response when subjected to a moving load for different velocity profiles are analysed. The dynamic responses of a FG beam resting on a viscoelastic foundation are also analysed for different material distributions, velocity and viscous damping profiles. The approximate solutions of the WFEM converge to the exact solution when the order and/or multiresolution scale of the WFE are increased. The results demonstrate that the Daubechies and B-spline based WFE solutions are highly accurate and require less number of elements than FEM due to the multiresolution property of WFEM. Furthermore, the applied moving load velocities and viscous damping influence the effects of varying the material distribution of FG beams on the dynamic response. Additional aspects of WFEM such as, the effect of altering the layout of the WFE and selection of the order of wavelet families to analyse static problems, are also presented in this study.

This paper is concerned with approximating noisy samples by non-uniform rational B-spline curves with special emphasis on free knots. We show how to set up the problem such that nonlinear optimization methods can be applied efficiently. This involves the introduction of penalizing terms in order to avoid undesired knot positions. We report on our implementation of the nonlinear optimization and we show a way to implement the program in parallel. Parallel performance results are described. Our experiments show that our program has a linear speedup and an efficiency value close to unity. Runtime results on a parallel computer are displayed.

No description available.

B-spline surfaces have been widely used in aircraft design to represent different types of components in a uniform format. Unlike the visual trimming of B-spline surfaces, which hides unwanted portions in rendering, the geometric trimming approach provides a mathematically clean representation. This dissertation focuses on the geometric trimming of fuselage and wing components represented by B-spline surfaces. To trim two intersecting surfaces requires finding their intersections effectively. Most of the existing algorithms focus on providing intersections suitable for rendering. In this dissertation, an intersection algorithm suitable for geometric trimming of B-spline surfaces is presented. The number of intersection points depends on the number of isoparametric curves selected, and thus is controllable and independent of the error bound of intersection points. Trimming curves are classified and a new scheme for trimming by a closed trimming curve is provided to improve the accuracy. The surface trimmed by a closed trimming curve is subdivided into four patches and the trimming curve is converted into two open trimming curves. Two surface patches are created by knot insertion, which match the original surface exactly. The other two surface patches are trimmed by the converted open trimming curves. Factors affecting the trimming process are discussed and metrics are provided to measure trimming errors. Exact trimming is precluded due to the high degree of intersections. The process may lead to significant deviation from the corresponding portion on the original surface. Optimizations are employed to minimize approximation errors and obtain higher accuracy. The hybrid Parallel Tempering and Simulated Annealing optimization method, which is an effective algorithm to overcome the slow convergence waiting dilemma and initial value sensitivity, is applied for the minimization of B-spline surface representation errors. The results confirm that trimming errors are successfully reduced. / Ph. D.

近年來剖面資料的監控在統計製程控制中有很大範圍的應用。在這篇論文裡，我們針對監控無母數剖面資料提出一個實務上的操作方法。這個操作方法有下列這些重要的特色：(1)使用一個靈活且有計算效率的無母數模型B-spline來描述反應變數與解釋變數的關係；(2)一般迴歸模型中之殘差結構假設是不需要的；(3)允許剖面資料內之觀測值間具有相關性之結構。最後，我們利用一個無線偵測器的實際資料來評估所提出方法的效率。 / Profile monitoring has received increasingly attention in a wide range of applications in statistical process control (SPC). In this work, we propose a practical proposed guide which has the following important features: (i) a flexible and computationally efficient smoothing technique, called the B-spline, is employed to describe the relationship between the response variable and the explanatory variable(s); (ii) the usual structural assumptions on the residuals are not require; and (iii) the dependence structure for the within-profile observations is appropriately accommodated. Finally, a real data set from a wireless sensor is used to evaluate the efficiency of our proposed method.

En esta tesis se presenta un método para analizar antenas montadas sobre estructuras arbitrarias. La Optica Geométrica (GO) y la Teoría Uniforme de la difraccion (UTD), han sidoempleadas para analizar los efectos que la estructura produce sobre el diagrama de radiación de la antena emisora. Para la descripción geométrica de la estructura, han sido utilizados parches NURBS (Non Uniform Rational B-Spline), por lo que el método presentado, es compatible con la mayoría de los programas gráficos disponibles en el mercado.EL tratamiento de geometrías arbitrarias requiere un código eficiente en el análisis de tres dimensiones.Por otro lado, para obtener resultados satisfactorios, la descripción de la superficie de la estructura, debe ser muy próxima al modelo real, sin embargo, esto complica el tratamiento computacional. Aquí la estructura es modelada mediante un conjunto de parches NURBS, que unidos entre sí, describen el modelo completo. Esta descripción permite manipular superficies arbitrarias con un bajo numero de parches, lo que significa un volumen de información reducido.La descripción inicial por NURBS del modelo, es acompañada con información complemetaria como por ejemplo: la tipología de las superficies, las curvas frontera, el tipo de material, etc. Esto es imprescindible para la aplicación de criterios de selección dedicados a la aceleración del proceso.El método tras leer la descripción del modelo, descompone los parches NURBS en superficies racionales de Bezier. Un parche de Bezier es también una superficie paramétrica definida en términos de una combinación lineal de los polinomios de Bernstein.Las antenas son modeladas usando modelos numéricos simples basados en agrupaciones de dipolos infinitesimales eléctricos y magnéticos. Esta caracterización de la antena es muyventajosa ya que con un numero reducido de datos de entrada, la fuente queda definida para cualquier dirección del espacio y el valor del campo radiado puede ser calculado fácilmente.El análisis electromagnético de los efectos que contribuyen al campo dispersado por la geometría comienza con una selección rigurosa de la geometría iluminada desde la fuente.Unicamente los parches de Bezier iluminados serán almacenados por el ordenador durante el análisis. La filosofía de este proceso es descartar aquella parte de la geometría que no contribuye a los efectos de dispersión.El campo total calculado es la superposición de los siguientes efectos pertenecientes a la GO y a la UTD: campo directo procedente de la fuente, campo reflejado por los parches de Bezier, campo difractado por las aristas del modelo definidas como curvas de Bezier, ondas de superficie, dobles reflexiones, reflexione-difraccion y difraccion-reflexión. El método ha sido diseñado para analizar campo cercano y lejano. El mayor gasto computacional se debe a la búsqueda de los puntos de dispersión, por lo que antes de emplear los algoritmos de intersección es necesario aplicar un conjunto de criterios rápidos dependientes de la dirección de observación.El principio de Fermat en combinación con el Gradiente Conjugado (CGM) es usado para obtener de manera eficiente los puntos de dispersión sobre la estructura. Para cada efecto, laposible ocultación de la trayectoria completa del rayo es examinada, por ello, si el rayo corta alguno de los parches de Bezier su contribución será descartada. Los dobles efectos son tratados como una generalización de los simples efectos.El método desarrollado es eficiente ya que precisa de un numero reducido de superficies para modelar objetos complejos lo que se traduce en bajos requerimientos de memoria y reducidos tiempos de calculo. / In this thesis a method to analyze antennas on board of complex bodies is presented. The Geometrical Optics (GO) and Uniform Theory of Diffraction (UTD) have been used to analyze the effect of the structure in the radiation pattern of the antennas. The bodies are geometrically modelled by using NURBS (Non Uniform Rational B-Spline) surfaces. In addition to be accurate and efficient, the method is compatible with most of the modern CAGD (Computer Aided Geometric Design) available programs.The treatment of arbitrary geometries requires a code which can carry out an efficient 3D analysis. To obtain accurate results the description of the surface must be close to the real model, however this complicates the computational procedure. Here the structure is modeled by a collection of individual N.U.R.B.S. surface patches joined to form a complete description of the surface model. The NURBS description is able to manipulate free form surfaces with a low number of patches, and therefore, with a low amount of information. The initial description of the model by NURBS surfaces is accompanied with other complementary data for example : the topology of the surfaces, the boundary curves, the types of material and other inputs. It is very interesting to apply criteria to make the complete analysis faster.The method reads the NURBS description of the model and transforms the NURBS into the rational BEZIER surfaces. A rational BEZIER patch is also a parametric surface defined in terms of a linear combination of Bernstein polynomials.The antennas are modelled using simple numerical models based on arrays of electric and magnetic infinitesimal dipoles. This antenna modelization is very advantageous because with a little input data, the source is defined in any direction and the field value is readily accessible.The electromagnetic analysis of the contributive effects to the scattering field by the geometry, starts with the rigorous selection of the geometry illuminated from the source. Only the Bezier patches illuminated will be in memory of the computer during the analysis. The philosophy of this previous process is to discard in the process the part of the geometry which does not contribute to the scattering effects.The total field is the superposition of the following GO and UTD field components: direct field from the source, reflected fields from the Bezier patches of the model, diffracted fields from the arbitrary edges defined as a Bezier curves, creeping waves, double reflected field and diffracted-reflected and reflected-diffracted fields. The search of specular and diffraction points are the most CPU time consuming, thus before using the intersection algorithms it is necessary to apply a set of fast selection criteria which depend on the observation direction.The Fermat principle in conjunction with the Conjugate Gradient Method (CGM) is used for obtaining efficiently the reflection points and diffraction points on the structure. For each effect the complete ray path is examined to see whether or not it is interrupted by any Bezier patch of the model, in this case the field component is not computed. The double effects are treated using a generalization of the single effects algorithms. The method has been developed to analyze the near and far field cases for different frequencies.The developed method is quite efficient because it makes use of a small number of surfaces to model complex bodies, so it requires few memory and low computing time.