4,100 research outputs found
An efficient method for the incompressible Navier-Stokes equations on irregular domains with no-slip boundary conditions, high order up to the boundary
Common efficient schemes for the incompressible Navier-Stokes equations, such
as projection or fractional step methods, have limited temporal accuracy as a
result of matrix splitting errors, or introduce errors near the domain
boundaries (which destroy uniform convergence to the solution). In this paper
we recast the incompressible (constant density) Navier-Stokes equations (with
the velocity prescribed at the boundary) as an equivalent system, for the
primary variables velocity and pressure. We do this in the usual way away from
the boundaries, by replacing the incompressibility condition on the velocity by
a Poisson equation for the pressure. The key difference from the usual
approaches occurs at the boundaries, where we use boundary conditions that
unequivocally allow the pressure to be recovered from knowledge of the velocity
at any fixed time. This avoids the common difficulty of an, apparently,
over-determined Poisson problem. Since in this alternative formulation the
pressure can be accurately and efficiently recovered from the velocity, the
recast equations are ideal for numerical marching methods. The new system can
be discretized using a variety of methods, in principle to any desired order of
accuracy. In this work we illustrate the approach with a 2-D second order
finite difference scheme on a Cartesian grid, and devise an algorithm to solve
the equations on domains with curved (non-conforming) boundaries, including a
case with a non-trivial topology (a circular obstruction inside the domain).
This algorithm achieves second order accuracy (in L-infinity), for both the
velocity and the pressure. The scheme has a natural extension to 3-D.Comment: 50 pages, 14 figure
The effective conductivity of arrays of squares: large random unit cells and extreme contrast ratios
An integral equation based scheme is presented for the fast and accurate
computation of effective conductivities of two-component checkerboard-like
composites with complicated unit cells at very high contrast ratios. The scheme
extends recent work on multi-component checkerboards at medium contrast ratios.
General improvement include the simplification of a long-range preconditioner,
the use of a banded solver, and a more efficient placement of quadrature
points. This, together with a reduction in the number of unknowns, allows for a
substantial increase in achievable accuracy as well as in tractable system
size. Results, accurate to at least nine digits, are obtained for random
checkerboards with over a million squares in the unit cell at contrast ratio
10^6. Furthermore, the scheme is flexible enough to handle complex valued
conductivities and, using a homotopy method, purely negative contrast ratios.
Examples of the accurate computation of resonant spectra are given.Comment: 28 pages, 11 figures, submitted to J. Comput. Phy
Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves
A method is proposed for accurately describing arbitrary-shaped free
boundaries in single-grid finite-difference schemes for elastodynamics, in a
time-domain velocity-stress framework. The basic idea is as follows: fictitious
values of the solution are built in vacuum, and injected into the numerical
integration scheme near boundaries. The most original feature of this method is
the way in which these fictitious values are calculated. They are based on
boundary conditions and compatibility conditions satisfied by the successive
spatial derivatives of the solution, up to a given order that depends on the
spatial accuracy of the integration scheme adopted. Since the work is mostly
done during the preprocessing step, the extra computational cost is negligible.
Stress-free conditions can be designed at any arbitrary order without any
numerical instability, as numerically checked. Using 10 grid nodes per minimal
S-wavelength with a propagation distance of 50 wavelengths yields highly
accurate results. With 5 grid nodes per minimal S-wavelength, the solution is
less accurate but still acceptable. A subcell resolution of the boundary inside
the Cartesian meshing is obtained, and the spurious diffractions induced by
staircase descriptions of boundaries are avoided. Contrary to what occurs with
the vacuum method, the quality of the numerical solution obtained with this
method is almost independent of the angle between the free boundary and the
Cartesian meshing.Comment: accepted and to be published in Geophys. J. In
Computation and visualization of Casimir forces in arbitrary geometries: non-monotonic lateral forces and failure of proximity-force approximations
We present a method of computing Casimir forces for arbitrary geometries,
with any desired accuracy, that can directly exploit the efficiency of standard
numerical-electromagnetism techniques. Using the simplest possible
finite-difference implementation of this approach, we obtain both agreement
with past results for cylinder-plate geometries, and also present results for
new geometries. In particular, we examine a piston-like problem involving two
dielectric and metallic squares sliding between two metallic walls, in two and
three dimensions, respectively, and demonstrate non-additive and non-monotonic
changes in the force due to these lateral walls.Comment: Accepted for publication in Physical Review Letters. (Expected
publication: Vol. 99 (8) 2007
Spectral methods for partial differential equations
Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed. Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems. Both collocation and tau methods are considered. These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type. Fluid dynamical applications are emphasized
PyFR: An Open Source Framework for Solving Advection-Diffusion Type Problems on Streaming Architectures using the Flux Reconstruction Approach
High-order numerical methods for unstructured grids combine the superior
accuracy of high-order spectral or finite difference methods with the geometric
flexibility of low-order finite volume or finite element schemes. The Flux
Reconstruction (FR) approach unifies various high-order schemes for
unstructured grids within a single framework. Additionally, the FR approach
exhibits a significant degree of element locality, and is thus able to run
efficiently on modern streaming architectures, such as Graphical Processing
Units (GPUs). The aforementioned properties of FR mean it offers a promising
route to performing affordable, and hence industrially relevant,
scale-resolving simulations of hitherto intractable unsteady flows within the
vicinity of real-world engineering geometries. In this paper we present PyFR,
an open-source Python based framework for solving advection-diffusion type
problems on streaming architectures using the FR approach. The framework is
designed to solve a range of governing systems on mixed unstructured grids
containing various element types. It is also designed to target a range of
hardware platforms via use of an in-built domain specific language based on the
Mako templating engine. The current release of PyFR is able to solve the
compressible Euler and Navier-Stokes equations on grids of quadrilateral and
triangular elements in two dimensions, and hexahedral elements in three
dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented
for various benchmark flow problems, single-node performance is discussed, and
scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The
software is freely available under a 3-Clause New Style BSD license (see
www.pyfr.org)
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