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On Consistent Boundary Conditions for the Yee Scheme in 3D
Department of Mathematics and Institute for Computational Engineering and Sciences, The University of Texas at Austin, USA.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.ORCID iD: 0000-0002-6321-8619
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The standard staircase approximation of curved boundaries in the Yee scheme is inconsistent. Consistency can however be achieved by modifying the algorithm close to the boundary.  We consider a technique to consistently model curved boundaries where the coefficients of the update stencil is modified, thus preserving the Yee structure.  The method has previously been successfully applied to acoustics in two and three dimension, as well as electromagnetics in two dimensions.  In this paper we generalize to electromagnetics in three dimensions.  Unlike in previous cases there is a non-zero divergence growth along the boundary that needs to be projected away.  We study the convergence and provide numerical examples that demonstrates the improved accuracy.

Keyword [en]
FDTD, Yee, Staircasing
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-95504OAI: oai:DiVA.org:kth-95504DiVA: diva2:528690
Note
QC 20120530Available from: 2012-07-30 Created: 2012-05-28 Last updated: 2012-07-30Bibliographically approved
In thesis
1. Modified Stencils for Boundaries and Subgrid Scales in the Finite-Difference Time-Domain Method
Open this publication in new window or tab >>Modified Stencils for Boundaries and Subgrid Scales in the Finite-Difference Time-Domain Method
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis centers on modified stencils for the Finite-Difference Time-Domain method (FDTD), or Yee scheme, when modelling curved boundaries, obstacles and holes smaller than the discretization length.  The goal is to increase the accuracy while keeping the structure of the standard method, enabling improvements to existing implementations with minimal effort.

We present an extension of a previously developed technique for consistent boundary approximation in the Yee scheme.  We consider both Maxwell's equations and the acoustic equations in three dimensions, which require separate treatment, unlike in two dimensions.

The stability properties of coefficient modifications are essential for practical usability.  We present an analysis of the requirements for time-stable modifications, which we use to construct a simple and effective method for boundary approximations. The method starts from a predetermined staircase discretization of the boundary, requiring no further data on the underlying geometry that is being approximated.

Not only is the standard staircasing of curved boundaries a poor approximation, it is inconsistent, giving rise to errors that do not disappear in the limit of small grid lengths. We analyze the standard staircase approximation by deriving exact solutions of the difference equations, including the staircase boundary. This facilitates a detailed error analysis, showing how staircasing affects amplitude, phase, frequency and attenuation of waves.

To model obstacles and holes of smaller size than the grid length, we develop a numerical subgrid method based on locally modified stencils, where a highly resolved micro problem is used to generate effective coefficients for the Yee scheme at the macro scale.

The implementations and analysis of the developed methods are validated through systematic numerical tests.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2012. xi, 34 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2012:07
Keyword
FDTD, Yee, Staircasing
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-95510 (URN)978-91-7501-417-3 (ISBN)
Public defence
2012-06-15, F3, Lindstedtsvägen 26, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish e‐Science Research Center
Note

QC 20120530

Available from: 2012-05-30 Created: 2012-05-28 Last updated: 2013-04-09Bibliographically approved

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Häggblad, JonRunborg, OlofTornberg, Anna-Karin
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