High order discretizations for spatial dependent SIR models

In this paper, an SIR model with spatial dependence is studied and results regarding its stability and numerical approximation are presented. We consider a generalization of the original Kermack and McKendrick model in which the size of the populations differs in space. The use of local spatial dependence yields a system of integro-differential equations. The uniqueness and qualitative properties of the continuous model are analyzed. Furthermore, different choices of spatial and temporal discretizations are employed, and step-size restrictions for population conservation, positivity, and monotonicity preservation of the discrete model are investigated. We provide sufficient conditions under which high order numerical schemes preserve the discrete properties of the model. Computational experiments verify the convergence and accuracy of the numerical methods.Comment: 33 pages, 5 figures, 3 table

Skeleton Based Parametric 2D Region Representation: Disk B-Spline Curves

The skeleton, or medial axis, is an important attribute of 2D shapes. The disk B-spline curve (DBSC) is a skeleton-based parametric freeform 2D region representation, which is defined in B-spline form. The DBSC describes not only a 2D region, which is suitable for describing heterogonous materials in the region, but also the center curve (skeleton) of the region explicitly, which is suitable for animation, simulation and recognition. In addition to being useful for error estimation of the B-spline curve, the DBSC can be used in designing and animating freeform 2D regions. Despite increasing DBSC applications, its theory and fundamentals have not been thoroughly investigated. In this paper, we discuss several fundamental properties and algorithms, such as the de Boor algorithm for DBSCs. We first derive the explicit evaluation and derivatives formulas at arbitrary points of a 2D region (interior and boundary) represented by a DBSC and then provide heterogeneous object representation. We also introduce modeling and interactive heterogeneous object design methods for a DBSC, which consolidates DBSC theory and supports its further applications

ISOGEOMETRIC ANALYSIS AND PATCHWISE REPRODUCING POLYNOMIAL PARTICLE METHOD FOR PLATES

Isogeometric analysis (IGA) ([8, 16, 27]) is designed to combine two tasks, design by Computer Aided Design (CAD) and Finite Element Analysis (FEA), so that it drastically reduces the error in the representation of the computational domain and the re-meshing by the use of “exact” CAD geometry directed at the coarsest level of discretization. This is achieved by using B-splines or non-uniform rational B-splines (NURBS) for the description of geometries as well as for the representation of unknown solution fields. In order to handle the singularities arising in the PDEs, Babu?ska and Oh [7] introduced mapping techniques, called the Method of Auxiliary Mapping (MAM), into conventional p-version of Finite Element Methods (FEM). In a similar spirit to MAM, it is possible to construct a novel NURBS geometrical mapping that generates singular functions resembling the singularities. The proposed mapping technique is concerned with constructions of unconventional novel geometrical mappings by which push-forward of B-spline functions defined on the parameter space generates singular functions in a physical domain that resemble the given point singularities. In other words, the pull-back of the singularity into the parameter space by the non standard NURBS mapping becomes highly smooth. However, the mapping technique is not able to handle in the framework of IGA. Thus, we consider how to use the proposed mapping method in IGA of elliptic prob- lems and elasticity containing singularities without changing the design mapping. For this end, we embed the mapping method into the standard IGA that uses NURBS basis functions for which h - p - k-refinements are applicable for improved computational solution. In other words, the mapping method will be used to enrich NURBS basis functions around neighborhood of singularities so that they can capture singular behaviors of the solution to be approximated. Finally, Reproducing Polynomial Particle Method (RPPM) is one of meshless methods that use meshes minimally or do not use meshes at all. In this disserta- tion, the RPPM is employed for free vibration and buckling of the first order shear deformation model (FSDT), called the Reissner-Mindlin plate, and for analysis of boundary layer of the Reissner-Mindlin plate. For numerical implementation, we use flat-top partition of unity functions, introduced by Oh et al, and patchwise RPPM in which approximation functions have high order polynomial reproducing property and shape functions satisfying the Kronecker delta property. Also, we demonstrate that our method is more effective than other existing methods in dealing with Reissner- Mindlin plates with various material properties and boundary conditions

ARMANDO, a SPH code for CERN

The Smoothed Particle Hydrodynamics methodologies may be a useful numerical tool for the simulation of particle beam interaction with liquid targets and obstacles. ARMANDO code is a state of the art SPH code interfaced with FLUKA and capable to solve these problems. This report presents the basic theoretical elements behind the method, describes the most important aspects of the implementation and shows some simple examples

IGA-based Multi-Index Stochastic Collocation for random PDEs on arbitrary domains

This paper proposes an extension of the Multi-Index Stochastic Collocation (MISC) method for forward uncertainty quantification (UQ) problems in computational domains of shape other than a square or cube, by exploiting isogeometric analysis (IGA) techniques. Introducing IGA solvers to the MISC algorithm is very natural since they are tensor-based PDE solvers, which are precisely what is required by the MISC machinery. Moreover, the combination-technique formulation of MISC allows the straight-forward reuse of existing implementations of IGA solvers. We present numerical results to showcase the effectiveness of the proposed approach.Comment: version 3, version after revisio

ARMANDO, a SPH code for CERN Some theory, a short tutorial, the code description and some examples

The Smoothed Particle Hydrodynamics methodologies may be a useful numerical tool for the simulation of particle beam interaction with liquid targets and obstacles. ARMANDO code is a state of the art SPH code interfaced with FLUKA and capable to solve these problems. This report presents the basic theoretical elements behind the method, describes the most important aspects of the implementation and shows some simple examples

Coronal loop detection from solar images and extraction of salient contour groups from cluttered images.

This dissertation addresses two different problems: 1) coronal loop detection from solar images: and 2) salient contour group extraction from cluttered images. In the first part, we propose two different solutions to the coronal loop detection problem. The first solution is a block-based coronal loop mining method that detects coronal loops from solar images by dividing the solar image into fixed sized blocks, labeling the blocks as Loop or Non-Loop , extracting features from the labeled blocks, and finally training classifiers to generate learning models that can classify new image blocks. The block-based approach achieves 64% accuracy in IO-fold cross validation experiments. To improve the accuracy and scalability, we propose a contour-based coronal loop detection method that extracts contours from cluttered regions, then labels the contours as Loop and Non-Loop , and extracts geometric features from the labeled contours. The contour-based approach achieves 85% accuracy in IO-fold cross validation experiments, which is a 20% increase compared to the block-based approach. In the second part, we propose a method to extract semi-elliptical open curves from cluttered regions. Our method consists of the following steps: obtaining individual smooth contours along with their saliency measures; then starting from the most salient contour, searching for possible grouping options for each contour; and continuing the grouping until an optimum solution is reached. Our work involved the design and development of a complete system for coronal loop mining in solar images, which required the formulation of new Gestalt perceptual rules and a systematic methodology to select and combine them in a fully automated judicious manner using machine learning techniques that eliminate the need to manually set various weight and threshold values to define an effective cost function. After finding salient contour groups, we close the gaps within the contours in each group and perform B-spline fitting to obtain smooth curves. Our methods were successfully applied on cluttered solar images from TRACE and STEREO/SECCHI to discern coronal loops. Aerial road images were also used to demonstrate the applicability of our grouping techniques to other contour-types in other real applications