Streamline integration as a method for twodimensional elliptic grid generation
Abstract
We propose a new numerical algorithm to construct a structured numerical elliptic grid of a doubly connected domain. Our method is applicable to domains with boundaries defined by two contour lines of a twodimensional function. Furthermore, we can adapt any analytically given boundary aligned structured grid, which specifically includes polar and Cartesian grids. The resulting coordinate lines are orthogonal to the boundary. Grid points as well as the elements of the Jacobian matrix can be computed efficiently and up to machine precision. In the simplest case we construct conformal grids, yet with the help of weight functions and monitor metrics we can control the distribution of cells across the domain. Our algorithm is parallelizable and easy to implement with elementary numerical methods. We assess the quality of grids by considering both the distribution of cell sizes and the accuracy of the solution to elliptic problems. Among the tested grids these key properties are best fulfilled by the grid constructed with the monitor metric approach.  Graphical abstract:  Highlights: • Construct structured, elliptic numerical grids with elementary numerical methods. • Align coordinate lines with or make them orthogonal to the domain boundary. • Compute grid points and metric elements upmore »
 Authors:
 Institute for Ion Physics and Applied Physics, Universität Innsbruck, A6020 Innsbruck (Austria)
 Numerical Analysis group, Universität Innsbruck, A6020 Innsbruck (Austria)
 Publication Date:
 OSTI Identifier:
 22622303
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 340; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICAL METHODS AND COMPUTING; ACCURACY; ALGORITHMS; COMPUTERIZED SIMULATION; CONTROL; COORDINATES; DISTRIBUTION; MATHEMATICAL SOLUTIONS; NUMERICAL ANALYSIS; TWODIMENSIONAL CALCULATIONS; TWODIMENSIONAL SYSTEMS; WEIGHT
Citation Formats
Wiesenberger, M., Email: Matthias.Wiesenberger@uibk.ac.at, Held, M., and Einkemmer, L.. Streamline integration as a method for twodimensional elliptic grid generation. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.03.056.
Wiesenberger, M., Email: Matthias.Wiesenberger@uibk.ac.at, Held, M., & Einkemmer, L.. Streamline integration as a method for twodimensional elliptic grid generation. United States. doi:10.1016/J.JCP.2017.03.056.
Wiesenberger, M., Email: Matthias.Wiesenberger@uibk.ac.at, Held, M., and Einkemmer, L.. Sat .
"Streamline integration as a method for twodimensional elliptic grid generation". United States.
doi:10.1016/J.JCP.2017.03.056.
@article{osti_22622303,
title = {Streamline integration as a method for twodimensional elliptic grid generation},
author = {Wiesenberger, M., Email: Matthias.Wiesenberger@uibk.ac.at and Held, M. and Einkemmer, L.},
abstractNote = {We propose a new numerical algorithm to construct a structured numerical elliptic grid of a doubly connected domain. Our method is applicable to domains with boundaries defined by two contour lines of a twodimensional function. Furthermore, we can adapt any analytically given boundary aligned structured grid, which specifically includes polar and Cartesian grids. The resulting coordinate lines are orthogonal to the boundary. Grid points as well as the elements of the Jacobian matrix can be computed efficiently and up to machine precision. In the simplest case we construct conformal grids, yet with the help of weight functions and monitor metrics we can control the distribution of cells across the domain. Our algorithm is parallelizable and easy to implement with elementary numerical methods. We assess the quality of grids by considering both the distribution of cell sizes and the accuracy of the solution to elliptic problems. Among the tested grids these key properties are best fulfilled by the grid constructed with the monitor metric approach.  Graphical abstract:  Highlights: • Construct structured, elliptic numerical grids with elementary numerical methods. • Align coordinate lines with or make them orthogonal to the domain boundary. • Compute grid points and metric elements up to machine precision. • Control cell distribution by adaption functions or monitor metrics.},
doi = {10.1016/J.JCP.2017.03.056},
journal = {Journal of Computational Physics},
number = ,
volume = 340,
place = {United States},
year = {Sat Jul 01 00:00:00 EDT 2017},
month = {Sat Jul 01 00:00:00 EDT 2017}
}

The purpose of this paper is to prove a more precise result which yields the generation of essentially infinitely many solutsion of (1) as p {yields} {infinity}. Our proof makes use of the argument. We are not sure whether the method of energy estimate employed in such literatures and is applicable to this situation.

Adaptive grid generation by elliptic equations with grid control at all of the boundaries
A method that combines Anderson's grid method and a gridpoint control scheme is developed in order to solve elliptic equations in a manner that simultaneously controls grid spacing and orthogonality on all of the boundaries. Both the finitedifference and finitevolume methods have become increasingly important in the solving of partial differential equations. These numerical methods rely on discretization of the domain of definition, most frequently employing numerical grid generation. In order to accurately solve a problem numerically, the proper location of the nodal points of the computational domain and the orthogonal grid around the boundaries are of prime importance. Whilemore » 
Elliptic grid generation based on Laplace equations and algebraic transformations
An elliptic grid generation method is presented to generate boundary conforming grids in domains in 2D and 3D physical space and on minimal surfaces and parametrized surfaces in 3D physical space. The elliptic grid generation method is based on the use of a composite mapping. This composite mapping consists of a nonlinear transfinite algebraic transformation and an elliptic transformation. The elliptic transformation is based on the Laplace equations for domains, or on the LaplaceBeltrami equations for surfaces. The algebraic transformation maps the computational space one toone onto a parameter space. The elliptic transformation maps the parameter space onetoone onto themore » 
Numerical study of the turbulent heat transfer in a motorized engine utilizing a twoboundary methodgrid generation technique
An algorithm for a multidimensional numerical solution was developed to predict the turbulent flow field and heat flux int eh cylinder of a reciprocating engine. The twoboundary methodgrid generation technique associated with cubic interpolation was used to map the complex physical domain onto an ideal rectangle for every time step. Hence, the metrics of the coordinate transformation could be obtained by direct analytic differentiation, and the rapid solution time was another advantage. The heat and flow patterns in the cylinder were vividly visualized by means of the contour maps of velocity vectors and isotherm. The effects of piston crown shapemore » 
Convergence of a shockcapturing streamline diffusion finite element method for a scalar conservation law in two space dimensions
A convergence result for a shockcapturing streamline diffusion finite element method applied to a timedependent scalar nonlinear hyperbolic conservation law in two space dimensions is demonstrated. The proof is based on a uniqueness result for measurevalued solutions by DiPerna. An almost optimal error estimate for a linearized conservation law having a smooth exact solution is also demonstrated. 11 refs.