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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, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.ORCID-id: 0000-0002-6321-8619
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA.
(Engelska)Manuskript (preprint) (Övrigt vetenskapligt)
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.

Nyckelord [en]
FDTD, Yee, Staircasing
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
URN: urn:nbn:se:kth:diva-95504OAI: oai:DiVA.org:kth-95504DiVA, id: diva2:528690
Anmärkning
QC 20120530Tillgänglig från: 2012-07-30 Skapad: 2012-05-28 Senast uppdaterad: 2012-07-30Bibliografiskt granskad
Ingår i avhandling
1. Modified Stencils for Boundaries and Subgrid Scales in the Finite-Difference Time-Domain Method
Öppna denna publikation i ny flik eller fönster >>Modified Stencils for Boundaries and Subgrid Scales in the Finite-Difference Time-Domain Method
2012 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Stockholm: KTH Royal Institute of Technology, 2012. s. xi, 34
Serie
Trita-CSC-A, ISSN 1653-5723 ; 2012:07
Nyckelord
FDTD, Yee, Staircasing
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
urn:nbn:se:kth:diva-95510 (URN)978-91-7501-417-3 (ISBN)
Disputation
2012-06-15, F3, Lindstedtsvägen 26, KTH, Stockholm, 10:00 (Engelska)
Opponent
Handledare
Forskningsfinansiär
Swedish e‐Science Research Center
Anmärkning

QC 20120530

Tillgänglig från: 2012-05-30 Skapad: 2012-05-28 Senast uppdaterad: 2013-04-09Bibliografiskt granskad

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Runborg, Olof

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Häggblad, JonRunborg, OlofTornberg, Anna-Karin
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