A new class of trigonometric B-Spline Curves

We construct one-frequency trigonometric spline curves with a de Boor-like algorithm for evaluation and analyze their shape-preserving properties. The convergence to quadratic B-spline curves is also analyzed. A fundamental tool is the concept of the normalized B-basis, which has optimal shape-preserving properties and good symmetric properties

Numerical Solution of a Class of Nonlinear System of Second-Order Boundary-Value Problems: a Fourth-Order Cubic Spline Approach

A cubic B-spline collocation approach is described and presented for the numerical solution of an extended system of linear and nonlinear second-order boundary-value problems. The system, whether regular or singularly perturbed, is tackled using a spline collocation approach constructed over uniform or non-uniform meshes. The rate of convergence is discussed theoretically and verified numerically to be of fourth-order. The efficiency and applicability of the technique are demonstrated by applying the scheme to a number of linear and nonlinear examples. The numerical solutions are contrasted with both analytical and other existing numerical solutions that exist in the literature. The numerical results demonstrate that this method is superior as it yields more accurate solutions

Quadrature methods for 2D and 3D problems

AbstractIn this paper we give an overview on well-known stability and convergence results for simple quadrature methods based on low-order composite quadrature rules and applied to the numerical solution of integral equations over smooth manifolds. First, we explain the methods for the case of second-kind equations. Then we discuss what is known for the analysis of pseudodifferential equations. We explain why these simple methods are not recommended for integral equations over domains with dimension higher than one. Finally, for the solution of a two-dimensional singular integral equation, we prove a new result on a quadrature method based on product rules

NURBS-SEM: A hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces

Non Uniform Rational B-spline (NURBS) patches are a standard way to describe complex geometries in Computer Aided Design tools, and have gained a lot of popularity in recent years also for the approximation of partial differential equations, via the Isogeometric Analysis (IGA) paradigm. However, spectral accuracy in IGA is limited to relatively small NURBS patch degrees (roughly p 648), since local condition numbers grow very rapidly for higher degrees. On the other hand, traditional Spectral Element Methods (SEM) guarantee spectral accuracy but often require complex and expensive meshing techniques, like transfinite mapping, that result anyway in inexact geometries. In this work we propose a hybrid NURBS-SEM approximation method that achieves spectral accuracy and maintains exact geometry representation by combining the advantages of IGA and SEM. As a prototypical problem on non trivial geometries, we consider the Laplace\u2013Beltrami and Allen\u2013Cahn equations on a surface. On these problems, we present a comparison of several instances of NURBS-SEM with the standard Galerkin and Collocation Isogeometric Analysis (IGA)

Doctor of Philosophy

dissertationVolumetric parameterization is an emerging field in computer graphics, where volumetric representations that have a semi-regular tensor-product structure are desired in applications such as three-dimensional (3D) texture mapping and physically-based simulation. At the same time, volumetric parameterization is also needed in the Isogeometric Analysis (IA) paradigm, which uses the same parametric space for representing geometry, simulation attributes and solutions. One of the main advantages of the IA framework is that the user gets feedback directly as attributes of the NURBS model representation, which can represent geometry exactly, avoiding both the need to generate a finite element mesh and the need to reverse engineer the simulation results from the finite element mesh back into the model. Research in this area has largely been concerned with issues of the quality of the analysis and simulation results assuming the existence of a high quality volumetric NURBS model that is appropriate for simulation. However, there are currently no generally applicable approaches to generating such a model or visualizing the higher order smooth isosurfaces of the simulation attributes, either as a part of current Computer Aided Design or Reverse Engineering systems and methodologies. Furthermore, even though the mesh generation pipeline is circumvented in the concept of IA, the quality of the model still significantly influences the analysis result. This work presents a pipeline to create, analyze and visualize NURBS geometries. Based on the concept of analysis-aware modeling, this work focusses in particular on methodologies to decompose a volumetric domain into simpler pieces based on appropriate midstructures by respecting other relevant interior material attributes. The domain is decomposed such that a tensor-product style parameterization can be established on the subvolumes, where the parameterization matches along subvolume boundaries. The volumetric parameterization is optimized using gradient-based nonlinear optimization algorithms and datafitting methods are introduced to fit trivariate B-splines to the parameterized subvolumes with guaranteed order of accuracy. Then, a visualization method is proposed allowing to directly inspect isosurfaces of attributes, such as the results of analysis, embedded in the NURBS geometry. Finally, the various methodologies proposed in this work are demonstrated on complex representations arising in practice and research

Unequally spaced knot techniques for the numerical solution of partial differential equations.

Cubic spline approximations to time dependent partial differential equations, having both constant and variable coefficients, are developed in which the knot points may be chosen to be unequally spaced. Four methods are presented for obtaining 'optimal' knot positions, these being chosen so at to produce an increase in accuracy compared with methods based on equally spaced knots. Three of the procedures described produce knot partitions which are fixed throughout time. The fourth procedure yields differently placed 'optimal' knots on each time line, thus enabling us to better approximate the varying time nature characteristic of many partial differential equations. Truncation errors and stability criteria are derived and full numerical implementation procedures are given. Five case studies are presented to enable comparisons to be drawn between the knot placement methods and results found using equally spaced knots. Possible extensions of the work of this thesis are given

Summary of research in applied mathematics, numerical analysis, and computer sciences

The major categories of current ICASE research programs addressed include: numerical methods, with particular emphasis on the development and analysis of basic numerical algorithms; control and parameter identification problems, with emphasis on effective numerical methods; computational problems in engineering and physical sciences, particularly fluid dynamics, acoustics, and structural analysis; and computer systems and software, especially vector and parallel computers