Smooth representation of thin shells and volume structures for isogeometric analysis

The purpose of this study is to develop self-contained methods for obtaining smooth meshes which are compatible with isogeometric analysis (IGA). The study contains three main parts. We start by developing a better understanding of shapes and splines through the study of an image-related problem. Then we proceed towards obtaining smooth volumetric meshes of the given voxel-based images. Finally, we treat the smoothness issue on the multi-patch domains with C1 coupling. Following are the highlights of each part. First, we present a B-spline convolution method for boundary representation of voxel-based images. We adopt the filtering technique to compute the B-spline coefficients and gradients of the images effectively. We then implement the B-spline convolution for developing a non-rigid images registration method. The proposed method is in some sense of “isoparametric”, for which all the computation is done within the B-splines framework. Particularly, updating the images by using B-spline composition promote smooth transformation map between the images. We show the possible medical applications of our method by applying it for registration of brain images. Secondly, we develop a self-contained volumetric parametrization method based on the B-splines boundary representation. We aim to convert a given voxel-based data to a matching C1 representation with hierarchical cubic splines. The concept of the osculating circle is employed to enhance the geometric approximation, where it is done by a single template and linear transformations (scaling, translations, and rotations) without the need for solving an optimization problem. Moreover, we use the Laplacian smoothing and refinement techniques to avoid irregular meshes and to improve mesh quality. We show with several examples that the method is capable of handling complex 2D and 3D configurations. In particular, we parametrize the 3D Stanford bunny which contains irregular shapes and voids. Finally, we propose the B´ezier ordinates approach and splines approach for C1 coupling. In the first approach, the new basis functions are defined in terms of the B´ezier Bernstein polynomials. For the second approach, the new basis is defined as a linear combination of C0 basis functions. The methods are not limited to planar or bilinear mappings. They allow the modeling of solutions to fourth order partial differential equations (PDEs) on complex geometric domains, provided that the given patches are G1 continuous. Both methods have their advantages. In particular, the B´ezier approach offer more degree of freedoms, while the spline approach is more computationally efficient. In addition, we proposed partial degree elevation to overcome the C1-locking issue caused by the over constraining of the solution space. We demonstrate the potential of the resulting C1 basis functions for application in IGA which involve fourth order PDEs such as those appearing in Kirchhoff-Love shell models, Cahn-Hilliard phase field application, and biharmonic problems

Learning to Segment and Represent Motion Primitives from Driving Data for Motion Planning Applications

Developing an intelligent vehicle which can perform human-like actions requires the ability to learn basic driving skills from a large amount of naturalistic driving data. The algorithms will become efficient if we could decompose the complex driving tasks into motion primitives which represent the elementary compositions of driving skills. Therefore, the purpose of this paper is to segment unlabeled trajectory data into a library of motion primitives. By applying a probabilistic inference based on an iterative Expectation-Maximization algorithm, our method segments the collected trajectories while learning a set of motion primitives represented by the dynamic movement primitives. The proposed method utilizes the mutual dependencies between the segmentation and representation of motion primitives and the driving-specific based initial segmentation. By utilizing this mutual dependency and the initial condition, this paper presents how we can enhance the performance of both the segmentation and the motion primitive library establishment. We also evaluate the applicability of the primitive representation method to imitation learning and motion planning algorithms. The model is trained and validated by using the driving data collected from the Beijing Institute of Technology intelligent vehicle platform. The results show that the proposed approach can find the proper segmentation and establish the motion primitive library simultaneously

The Construction of Optimized High-Order Surface Meshes by Energy-Minimization

Despite the increasing popularity of high-order methods in computational fluid dynamics, their application to practical problems still remains challenging. In order to exploit the advantages of high-order methods with geometrically complex computational domains, coarse curved meshes are necessary, i.e. high-order representations of the geometry. This dissertation presents a strategy for the generation of curved high-order surface meshes. The mesh generation method combines least-squares fitting with energy functionals, which approximate physical bending and stretching energies, in an incremental energy-minimizing fitting strategy. Since the energy weighting is reduced in each increment, the resulting surface representation features high accuracy. Nevertheless, the beneficial influence of the energy-minimization is retained. The presented method aims at enabling the utilization of the superior convergence properties of high-order methods by facilitating the construction of coarser meshes, while ensuring accuracy by allowing an arbitrary choice of geometric approximation order. Results show surface meshes of remarkable quality, even for very coarse meshes representing complex domains, e.g. blood vessels

Constrained Interpolation By Parametric Rational Cubic Splines

Interpolasi terkekang adalah berguna dalam masalah seperti mereka bentuk sebuah Iengkung yang perlu dihadkan dalam suatu kawasan tertentu. Dalam disertasi ini, kami membincangkan interpolasi terkekang dengan menggunakan splin kubik nisbah yang diperkenalkan dalam (Goodman et aI, 1991). Terdapat dua kaedah pengubahsuaian lengkung disarankan, kaedah yang melibatkan modifikasi pemberat a,p berkaitan dengan titik hujung segmen lengkung dibincangkan dalam disertasi ini. Skim ini memperoleh sebuah G2 lengkung interpolasi yang terletak di sebelah garis-garis yang diberikan seperti data yang diberikan. Sebagai perkembangan daripada kertas ini, kami akan memperoleh satu skim interpolasi terkekang altematif dengan menggunakan lengkung kubik nisbah. Pemberat n, e yang berkaitan dengan titik kawalan dalaman diubah suai untuk memperoleh sebuah G1 lengkung interpolasi yang terletak di sebelah garis-garis yang diberikan seperti data yang diberikan. Constrained interpolation could be useful in problem like designing a curve that must be restricted within a specified region. In this dissertation, we discuss constrained interpolation using rational cubic splines introduced in (Goodman et aI, 1991). There are two curve modification methods suggested and the one which involves modification of the weights a ,fJ associated with the end points of the curve segments is discussed in this dissertation. This scheme obtains a G2 interpolating curve which lies on one side of the given lines as the given data. Extension from this paper, we will derive an alternative constrained interpolation scheme using rational cubic curve. The weights Q , e associated with the inner control points are modified to obtain a G1 interpolating curve which lies on one side of the given lines as the given data

Approximation and geometric modeling with simplex B-splines associated with irregular triangles

Bivariate quadratic simplical B-splines defined by their corresponding set of knots derived from a (suboptimal) constrained Delaunay triangulation of the domain are employed to obtain a C1-smooth surface. The generation of triangle vertices is adjusted to the areal distribution of the data in the domain. We emphasize here that the vertices of the triangles initially define the knots of the B-splines and do generally not coincide with the abscissae of the data. Thus, this approach is well suited to process scattered data.\ud \ud With each vertex of a given triangle we associate two additional points which give rise to six configurations of five knots defining six linearly independent bivariate quadratic B-splines supported on the convex hull of the corresponding five knots.\ud \ud If we consider the vertices of the triangulation as threefold knots, the bivariate quadratic B-splines turn into the well known bivariate quadratic Bernstein-Bézier-form polynomials on triangles. Thus we might be led to think of B-splines as of smoothed versions of Bernstein-Bézier polynomials with respect to the entire domain. From the degenerate Bernstein-Bézier situation we deduce rules how to locate the additional points associated with each vertex to establish knot configurations that allow the modeling of discontinuities of the function itself or any of its directional derivatives. We find that four collinear knots out of the set of five defining an individual quadratic B-spline generate a discontinuity in the surface along the line they constitute, and that analogously three collinear knots generate a discontinuity in a first derivative.\ud Finally, the coefficients of the linear combinations of normalized simplicial B-splines are visualized as geometric control points satisfying the convex hull property.\ud Thus, bivariate quadratic B-splines associated with irregular triangles provide a great flexibility to approximate and model fast changing or even functions with any given discontinuities from scattered data.\ud An example for least squares approximation with simplex splines is presented

The Surface Topography of a Magnetic Fluid -- a Quantitative Comparison between Experiment and Numerical Simulation

The normal field instability in magnetic liquids is investigated experimentally by means of a radioscopic technique which allows a precise measurement of the surface topography. The dependence of the topography on the magnetic field is compared to results obtained by numerical simulations via the finite element method. Quantitative agreement has been found for the critical field of the instability, the scaling of the pattern amplitude and the detailed shape of the magnetic spikes. The fundamental Fourier mode approximates the shape to within 10% accuracy for a range of up to 40% of the bifurcation parameter of this subcritical bifurcation. The measured control parameter dependence of the wavenumber differs qualitatively from analytical predictions obtained by minimization of the free energy.Comment: 21 pages, 16 figures; corrected typos, added reference to Kuznetsov and Spector(1976), S.J. Fortune(1995) and Harkins&Jordan (1930). Figures revise