Applications of Isogeometric Analysis Coupled with Finite Volume Method

In this thesis, a combination of Isogeometric Analysis (IGA) and Finite Volume Method (FVM) on geometries parameterized by Non-Uniform Rational Basis Splines (NURBS) is explored with applications in fluid flow, heat transfer, and shape optimization. An IGA framework supplemented with FVM is created in MATLAB® to solve problems defined over single patch domains with mesh refinement by node insertion. Additionally, a second-order finite difference method is developed using non-orthogonal curvilinear coordinates and a numerical Jacobian of the NURBS geometry. The examples include fully developed laminar flow through ducts, potential flow around a tilted ellipse, transient heat conduction, linear advection-diffusion, and a basic shape optimization example using a particle swarm technique. The numerical results are compared among the methods and verified with available analytical solutions

Constructing Wavelet Frames and Orthogonal Wavelet Bases on the Sphere

Non

Doctor of Philosophy

dissertationWhile boundary representations, such as nonuniform rational B-spline (NURBS) surfaces, have traditionally well served the needs of the modeling community, they have not seen widespread adoption among the wider engineering discipline. There is a common perception that NURBS are slow to evaluate and complex to implement. Whereas computer-aided design commonly deals with surfaces, the engineering community must deal with materials that have thickness. Traditional visualization techniques have avoided NURBS, and there has been little cross-talk between the rich spline approximation community and the larger engineering field. Recently there has been a strong desire to marry the modeling and analysis phases of the iterative design cycle, be it in car design, turbulent flow simulation around an airfoil, or lighting design. Research has demonstrated that employing a single representation throughout the cycle has key advantages. Furthermore, novel manufacturing techniques employing heterogeneous materials require the introduction of volumetric modeling representations. There is little question that fields such as scientific visualization and mechanical engineering could benefit from the powerful approximation properties of splines. In this dissertation, we remove several hurdles to the application of NURBS to problems in engineering and demonstrate how their unique properties can be leveraged to solve problems of interest

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

Isogeometric Analysis for Electromagnetism

The combination of numerical analysis with the scanning technology has been seeing increased use in many research areas. There is an emerging need for high-fidelity geometric modeling and meshing for practical applications. The Isogeometric Analysis (IGA) is a comprehensive computational framework, which integrates geometric modeling and meshing with analysis. Different from other existing numerical methods, the IGA can generate analysis ready models without loss of geometrical accuracy. In IGA, the continuity and the quality of a solution can be conveniently controlled and refined. These features enable IGA to integrate modeling, analysis, and design in a unified framework, the root idea of IGA. The IGA for electromagmetics is studied here for steady and transient electromagnetics as well as electromagnetic scattering. The solution procedure and the associated Matlab codes are developed to simulate the electromagnetic radiation on a biological tissues. The scattered and the total electrical fields are computed over the complex geometry of a brain section with realistic material properties. A perfectly matched layer (PML) is developed to model the far field boundary condition. The IGA platform developed here offers a reliable simulation within an accurate representation of the geometry. The results of this research can be used both in evaluating the potential health and safety risks of electromagnetic radiations and in optimizing the design of radiating devices used in non-invasive diagnostics and therapies

Proceedings of the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), 10-11 July 2017, Nottingham Conference Centre, Nottingham Trent University

This book contains the abstracts and papers presented at the Eleventh UK Conference on Boundary Integral Methods (UKBIM 11), held at Nottingham Trent University in July 2017. The work presented at the conference, and published in this volume, demonstrates the wide range of work that is being carried out in the UK, as well as from further afield

Point-set manifold processing for computational mechanics: thin shells, reduced order modeling, cell motility and molecular conformations

In many applications, one would like to perform calculations on smooth manifolds of dimension d embedded in a high-dimensional space of dimension D. Often, a continuous description of such manifold is not known, and instead it is sampled by a set of scattered points in high dimensions. This poses a serious challenge. In this thesis, we approximate the point-set manifold as an overlapping set of smooth parametric descriptions, whose geometric structure is revealed by statistical learning methods, and then parametrized by meshfree methods. This approach avoids any global parameterization, and hence is applicable to manifolds of any genus and complex geometry. It combines four ingredients: (1) partitioning of the point set into subregions of trivial topology, (2) the automatic detection of the local geometric structure of the manifold by nonlinear dimensionality reduction techniques, (3) the local parameterization of the manifold using smooth meshfree (here local maximum-entropy) approximants, and (4) patching together the local representations by means of a partition of unity. In this thesis we show the generality, flexibility, and accuracy of the method in four different problems. First, we exercise it in the context of Kirchhoff-Love thin shells, (d=2, D=3). We test our methodology against classical linear and non linear benchmarks in thin-shell analysis, and highlight its ability to handle point-set surfaces of complex topology and geometry. We then tackle problems of much higher dimensionality. We perform reduced order modeling in the context of finite deformation elastodynamics, considering a nonlinear reduced configuration space, in contrast with classical linear approaches based on Principal Component Analysis (d=2, D=10000's). We further quantitatively unveil the geometric structure of the motility strategy of a family of micro-organisms called Euglenids from experimental videos (d=1, D~30000's). Finally, in the context of enhanced sampling in molecular dynamics, we automatically construct collective variables for the molecular conformational dynamics (d=1...6, D~30,1000's)