2,252 research outputs found
A General Perfectly Matched Layer Model for Hyperbolic-Parabolic Systems
This paper describes a very general absorbing layer model for hyperbolic-parabolic systems of partial differential equations. For linear systems with constant coefficients it is shown that the model possesses the perfect matching property, i.e., it is a perfectly matched layer (PML). The model is applied to two linear systems: a linear wave equation with a viscous damping term and the linearized Navier–Stokes equations. The resulting perfectly matched layer for the viscous wave equation is proved to be stable. The paper also presents how the model can be used to construct an absorbing layer for the full compressible Navier–Stokes equations. For all three applications, numerical experiments are presented. Especially for the linear problems, the results are very promising. In one experiment, where the performance of a “hyperbolic PML” and the new hyperbolic-parabolic PML is compared for a hyperbolic-parabolic system, an improvement of six orders of magnitude is observed. For the compressible Navier–Stokes equations results obtained with the presented layer are competitive with existing methods
An axisymmetric time-domain spectral-element method for full-wave simulations: Application to ocean acoustics
The numerical simulation of acoustic waves in complex 3D media is a key topic
in many branches of science, from exploration geophysics to non-destructive
testing and medical imaging. With the drastic increase in computing
capabilities this field has dramatically grown in the last twenty years.
However many 3D computations, especially at high frequency and/or long range,
are still far beyond current reach and force researchers to resort to
approximations, for example by working in 2D (plane strain) or by using a
paraxial approximation. This article presents and validates a numerical
technique based on an axisymmetric formulation of a spectral finite-element
method in the time domain for heterogeneous fluid-solid media. Taking advantage
of axisymmetry enables the study of relevant 3D configurations at a very
moderate computational cost. The axisymmetric spectral-element formulation is
first introduced, and validation tests are then performed. A typical
application of interest in ocean acoustics showing upslope propagation above a
dipping viscoelastic ocean bottom is then presented. The method correctly
models backscattered waves and explains the transmission losses discrepancies
pointed out in Jensen et al. (2007). Finally, a realistic application to a
double seamount problem is considered.Comment: Added a reference, and fixed a typo (cylindrical versus spherical
Space-time Methods for Time-dependent Partial Differential Equations
Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space.
Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations
Three-dimensional sound propagation models using the parabolic-equation approximation and the split-step Fourier method
Author Posting. © IMACS, 2012. This article is posted here by permission of World Scientific Publishing for personal use, not for redistribution. The definitive version was published in Journal of Computational Acoustics 21 (2013): 1250018, doi:10.1142/S0218396X1250018X.The split-step Fourier method is used in three-dimensional parabolic-equation (PE) models to compute underwater sound propagation in one direction (i.e. forward). The method is implemented in both Cartesian (x, y, z) and cylindrical (r, θ, z) coordinate systems, with forward defined as along x and radial coordinate r, respectively. The Cartesian model has uniform resolution throughout the domain, and has errors that increase with azimuthal angle from the x axis. The cylindrical model has consistent validity in each azimuthal direction, but a fixed cylindrical grid of radials cannot produce uniform resolution. Two different methods to achieve more uniform resolution in the cylindrical PE model are presented. One of the methods is to increase the grid points in azimuth, as a function of r, according to nonaliased sampling theory. The other is to make use of a fixed arc-length grid. In addition, a point-source starter is derived for the three-dimensional Cartesian PE model. Results from idealized seamount and slope calculations are shown to compare and verify the performance of the three methods.This work was sponsored by the Office of Naval Research under the grants N00014-10-1-0040
and N00014-11-1-0701
Development of Stresses in Cohesionless Poured Sand
The pressure distribution beneath a conical sandpile, created by pouring sand
from a point source onto a rough rigid support, shows a pronounced minimum
below the apex (`the dip'). Recent work of the authors has attempted to explain
this phenomenon by invoking local rules for stress propagation that depend on
the local geometry, and hence on the construction history, of the medium. We
discuss the fundamental difference between such approaches, which lead to
hyperbolic differential equations, and elastoplastic models, for which the
equations are elliptic within any elastic zones present .... This displacement
field appears to be either ill-defined, or defined relative to a reference
state whose physical existence is in doubt. Insofar as their predictions depend
on physical factors unknown and outside experimental control, such
elastoplastic models predict that the observations should be intrinsically
irreproducible .... Our hyperbolic models are based instead on a physical
picture of the material, in which (a) the load is supported by a skeletal
network of force chains ("stress paths") whose geometry depends on construction
history; (b) this network is `fragile' or marginally stable, in a sense that we
define. .... We point out that our hyperbolic models can nonetheless be
reconciled with elastoplastic ideas by taking the limit of an extremely
anisotropic yield condition.Comment: 25 pages, latex RS.tex with rspublic.sty, 7 figures in Rsfig.ps.
Philosophical Transactions A, Royal Society, submitted 02/9
Perfectly Matched Layers for the heat and advection-diffusion equations
We design a perfectly matched layer for the advection-diffusion equation. We show that the reflection coefficient is exponentially small with respect to the damping parameter and the width of the PML and this independently of the advection and of the viscosity. Numerical tests assess the efficiency of the approach
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