Prsentation PowerPoint

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Some numerical issues in flow simulations using
particles G.-H. Cottet, Grenoble

• Field calculations
• subgrid-scale modeling vs artificial viscosity
  models
• Regridding
• Variable-size particles

Illustrations on vortex flows
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Vorticity conservation for incompressible flows
Incompressible Navier-Stokes equations in
velocity-pressure formulation
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Other types of particles vorticity contours and
filaments
Vorticity filament a curve that concentrates a
vortex tube
Circulation of vortex tube
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Velocity must be computed in self-consistent way
from vorticity

• Two classical approaches
• a completely grid-free approach, based on
  integral representation formulas
• an approach using grid-based Poisson solvers
• First approach uses Biot-Savart law

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When w replaced by a set of particles, velocity
on each particle is expressed as

• Two remarks
• Kernel is singular need to mollify to avoid
  large values when particles
• Approach
• N-body problem complexity in O(N2) if Nnumber
  of particles

Mollification is performed by convolution with
cut-off with radial symmetry and core size e -gt
explicit algebraic formulas High order formulas
obtained by imposing that cut-off shares as many
moments as possible with Dirac function Example
of formula at order 4
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Complete numerical method can be rephrased as a
set of coupled differential equations Some
natural conservation properties result from this
formulation
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O(N2) complexity can be reduce to O(NLogN) by
using Fast Summation Algorithms idea is to
replace kernel by algebraic expansions
(Greengard-Rocklin, for logarithmic kernel)
with precise estimates
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Typical tree-code Divide recursively into boxes
containing about the same number of
particles Upward pass form mulipole expansions,
from finer to coarser level (using shifts of
previously computed expansions) Downward pass
accumulate contributions of well-separated
boxes, from coarser to finer level At finest
level, complete with direct summation of nearby
particles
Still many improvements to come in 3D
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• Other approach for field calculations use an
  underlying Eulerian grid and grid-based Poisson
  solvers (Particle-In-Cell/Vortex-In-Cell
  methods)
• Project particle strength on grid points
• Use a Poisson solver on that grid, and
  differentiate on the grid to get grid field
  values
• Interpolate back fields on particles

• Drawbacks
• against Lagrangian features of particles (and
  possible loss of information in grid-particle
  interpolations)
• require far-field artificial boundary conditions
• Advantage
• Cheap (for relatively simple geometries)

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Comparison of CPU times for velocity evaluations
in 3D (Krasny tree-code vs VIC with Fishpack and
64 points interpolation formulas)
VIC1 cartesian grid with 100 particles VIC2
polar grid with 65 particles VIC3 polar grid
with 25 particles
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Conclusion choice of solver depends on how
localized vorticity is in the computational box
needed
Flow past a sphere grid-free calculation
(Ploumehans et al.,JCP02)
Flow past a cylinder VIC calculation
(C.-Poncet, JCP03)
About 600,000 particles, Total cpu is 200 hours
on 32 HP processors
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Possible to combine both approaches (grid-free
and particle-in-cell) through domain
decomposition approaches far-field by grid-free,
boundary layer by PIC.
Other issue related to velocity evaluations
time-stepping to push particles and update
strengths In general RK2 or RK4 Linear stability
only requires particle not to cross each
other (no conventional CFL type condition)
CPU savings depend on particular flow
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Particle resampling schemes for diffusion
Based on rewriting diffusion as an integral
operator
Where h satisfies moment properties
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Resulting particle scheme
Here, we distinguish local values and volumes
that make the particles strengths
Not constrained with time-stepping, can be high
order Slightly more expensive than random walk
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Implicit subgrid-scale models in particle methods
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Mollified particles (blobs) thus satisfy
This is an averaged Euler equations This means
that the particle method is achieving some
subgrid scale (implicit) model It thus allows
for backscatter energy. Same remark applies to
particle methods for compressible flows
potential increase in enstrophy (resp energy)
must be compensated by diffusion term.
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In 1D optimal form of artificial viscosity can
be derived rigorously from energy principle
Equivalent equation for particles
Modified non energy-increasing equation
Can be interprated as a diffusion model to
correct for energy increasing part of truncation
error in PM
Resulting particle AV scheme
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Remark reminiscent to LES models for turbulent
flows
Gradient model
Integral approximations of diffusion tensors
allows to identify Positive contributions to
dissipation. For incompressible flows

• Where z is a cut-off function
• z non-increasing function of radius gives a 3d
  version of previous
• AV model

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In all particle simulations, accuracy requires
frequent and accurate regridding
 classical  interpolation formulas
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Typical example showing importance of regridding
circular patch with high strain
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1d Burgers equation comparison of PM (including
AV and regridding) with 3rd order WENO schemes
Steady shock
Moving shock
Black circles PM White circles WENO E-O White
sqaures WENO L-F
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Gas dynamics for compressible isentropic viscous
flows
Conservation of density, momentum and energy
Comparaison of PIC method with TVD schemes for
a 2D shock boundary layer interaction
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Re200, t1
N1000
N500
PIC
TVD (Mac-Cormack with 3rd order limiter)
Tenaud-Daru
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• Regridding can also be used as a way to adapt
  particles
• to the flow topology -gt variable-size particles
• three different approaches
• Regridding into variable-sized particles via
  global mappings
• Regridding via local mappings
• Regridding onto piecewise uniform particles

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Physical space
Mapped space
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1) Mollified particles (for field evaluation)
with blob size
2) Diffusion in mapped coordinates
Can be written in divergence form, using the fact
that
/
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Using particle diffusion formulas for anisotropic
diffusion (Degond-Mas-Gallic, 1989) we obtain
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Remark conservative in physical variables
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Example rebound of a dipole with exponentially
stretched particle distribution (C-Koumoutsakos,
JCP 00)
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Flow around 2D cylinder - Variable-size particles
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• Variable-size particles can be used in an
  adaptive fashion
• Two possible strategies (Bergdorf,C.,Koumoutsakos,
  SIAM MMS)
• Adaptively build global mappings
• Adaptively define zones with piecewise constant
  volumes
• (AMR approach)

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First approach define global mappings through
finite-dimensional maps
Previous strategy would require to invert the
mapping. Alternative approach move particles in
the mapped space
Example 1D convection-diffusion equation
Mapping jacobian
Mapping velocity
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With
Easy to derive motion equations in mapped space
And to discretize on particles
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Method of adaptation relies on r-adaptive FEM
map Velcoity given by
Where M monitors the regularity of the solution
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Second approach particles have piecewise
constant volumes Inside each population,
particles dynamics is straightforward Remeshing
allows to transfer information between
populations through a buffer zone
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Illustration case of an elliptical vortex
Adaptation based on vorticity gradients Fine and
coarse zone in the AMR approach defined using a
package designed for FD
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Global mapping approach Particle grid
Vorticity contours
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Measure of accuracy/cost enstrophy profiles and
number of particles compared to uniform size
particles
Dotted line uniform particle distribution
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AMR approach
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• Conclusion
• About AV more to learn about specifics of PM in
  small-scale parametrizations
• About adaptive methods
• AMR approach more straightforward but accuracy
• strongly dependent on definition of fine and
  coarse grids (in progress gas dynamics coupled
  with radiative transfer for inertial fusion)
• Global mapping approach more subtle to implement
  but optimally accurate