Complete Subdivision Algorithms, II: Isotopic Meshing of Singular Algebraic Curves

Given a real valued function f(X,Y), a box region B_0 in R^2 and a positive epsilon, we want to compute an epsilon-isotopic polygonal approximation to the restriction of the curve S=f^{-1}(0)={p in R^2: f(p)=0} to B_0. We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga and Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities

Grid generation for the solution of partial differential equations

A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given

Representation of conformal maps by rational functions

The traditional view in numerical conformal mapping is that once the boundary correspondence function has been found, the map and its inverse can be evaluated by contour integrals. We propose that it is much simpler, and 10-1000 times faster, to represent the maps by rational functions computed by the AAA algorithm. To justify this claim, first we prove a theorem establishing root-exponential convergence of rational approximations near corners in a conformal map, generalizing a result of D. J. Newman in 1964. This leads to the new algorithm for approximating conformal maps of polygons. Then we turn to smooth domains and prove a sequence of four theorems establishing that in any conformal map of the unit circle onto a region with a long and slender part, there must be a singularity or loss of univalence exponentially close to the boundary, and polynomial approximations cannot be accurate unless of exponentially high degree. This motivates the application of the new algorithm to smooth domains, where it is again found to be highly effective

Different approaches on the implementation of implicit polynomials in visual tracking /

Visual tracking has emerged as an important component of systems in several application areas including vision-based control, human-computer interfaces, surveillance, agricultural automation, medical imaging and visual reconstruction. The central challenge in visual tracking is to keep track of the pose and location of one or more objects through a sequence of frames. Implicit algebraic 2D curves and 3D surfaces are among the most powerful representations and have proven very useful in many model-based applications in the past two decades. With this approach, objects in 2D images are described by their silhouettes and then represented by 2D implicit polynomial curves. In our work, we tried different approaches in order to efficiently apply the powerful implicit algebraic 2D curve representation to the phenomenon of visual tracking. Through the proposed concepts and algorithms, we tried to reduce the computational burden of fitting algorithms. Besides showing the usage of this representation on boundary data simulations, use of the implicit polynomial as a representative of the target region is also experimented on real videos

A review on feature-mapping methods for structural optimization

Acknowledgments We thank Dr. Lukas Pflug from the Department of Mathematics at the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Germany, for fruitful discussion and support. The initiative for this review goes back to critical yet constructive comments by Prof. Kurt Maute, from the University of Colorado Boulder, USA. We also thank Prof. Horea Ilies from the University of Connecticut, USA, for guidance and insight into some of the geometric aspects of this work. The first author acknowledges support by Deutsche Forschungsgemeinschaft (DFG) in the framework of the collaborative research center CRC 814 (subproject C2). The third author thanks the support of the US National Science Foundation, award CMMI-1634563.Peer reviewedPreprintPostprin

Improvements to the APBS biomolecular solvation software suite

The Adaptive Poisson-Boltzmann Solver (APBS) software was developed to solve the equations of continuum electrostatics for large biomolecular assemblages that has provided impact in the study of a broad range of chemical, biological, and biomedical applications. APBS addresses three key technology challenges for understanding solvation and electrostatics in biomedical applications: accurate and efficient models for biomolecular solvation and electrostatics, robust and scalable software for applying those theories to biomolecular systems, and mechanisms for sharing and analyzing biomolecular electrostatics data in the scientific community. To address new research applications and advancing computational capabilities, we have continually updated APBS and its suite of accompanying software since its release in 2001. In this manuscript, we discuss the models and capabilities that have recently been implemented within the APBS software package including: a Poisson-Boltzmann analytical and a semi-analytical solver, an optimized boundary element solver, a geometry-based geometric flow solvation model, a graph theory based algorithm for determining p$K_a$ values, and an improved web-based visualization tool for viewing electrostatics

Discontinuous Galerkin discretised level set methods with applications to topology optimisation

This thesis presents research concerning level set methods discretised using discontinuous Galerkin (DG) methods. Whilst the context of this work is level set based topology optimisation, the main outcomes of the research concern advancements which are agnostic of application. The first of these outcomes are the development of two novel DG discretised PDE based level set reinitialisation techniques, the so called Elliptic and Parabolic reinitialisation methods, which are shown through experiment to be robust and satisfy theoretical optimal rates of convergence. A novel Runge-Kutta DG discretisation of a simplified level set evolution equation is presented which is shown through experiment to be high-order accurate for smooth problems (optimal error estimates do not yet exist in the literature based on the knowledge of the author). Narrow band level set methods are investigated, and a novel method for extending the level set function outside of the narrow band, based on the proposed Elliptic Reinitialisation method, is presented. Finally, a novel hp-adaptive scheme is developed for the DG discretised level set method driven by the degree with which the level set function can locally satisfy the Eikonal equation defining the level set reinitialisation problem. These component parts are thus combined to form a proposed DG discretised level set methodology, the efficacy of which is evaluated through the solution of numerous example problems. The thesis is concluded with a brief exploration of the proposed method for a minimum compliance design problem