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Consistent modeling of boundaries in acoustic finite-difference time-domain simulations
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
University of Texas.
2012 (English)In: Journal of the Acoustical Society of America, ISSN 0001-4966, E-ISSN 1520-8524, Vol. 132, no 3, 1303-1310 p.Article in journal (Refereed) Published
Abstract [en]

The finite-difference time-domain method is one of the most popular for wave propagation in the time domain. One of its advantages is the use of a structured staggered grid, which makes it simple and efficient on modern computer architectures. A drawback however is the difficulty in approximating oblique boundaries, having to resort to staircase approximations.  In many scattering problems this means that the grid resolution required to obtain an accurate solution is much higher than what is dictated by propagation in a homogeneous material.  In this paper zero boundary data is considered, first for the velocity and then the pressure. These two forms of boundary conditions model perfectly rigid and pressure-release boundaries, respectively.  A simple and efficient method to consistently model curved rigid boundaries in two dimensions was developed in [A.-K. Tornberg and B. Engquist, J. Comput. Phys. 227, 6922--6943 (2008)].  Here this treatment is generalized to three dimensions.  Based on the approach of this method, a technique to model pressure-release surfaces with second order accuracy and without additional restriction on the timestep is also introduced.  The structure of the standard method is preserved, making it easy to use in existing solvers.  The effectiveness is demonstrated in several numerical tests.

Place, publisher, year, edition, pages
Acoustical Society of America (ASA), 2012. Vol. 132, no 3, 1303-1310 p.
Keyword [en]
Computer architecture, Finite difference time domain method
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-63593DOI: 10.1121/1.4740490ISI: 000309155000021Scopus ID: 2-s2.0-84866306609OAI: oai:DiVA.org:kth-63593DiVA: diva2:482429
Funder
Swedish e‐Science Research Center
Note

QC 20121031

Available from: 2012-01-23 Created: 2012-01-23 Last updated: 2017-12-08Bibliographically 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, JonEngquist, Björn
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