Boltzmann Solver with Adaptive Mesh in
Phase Space
Vladimir Kolobov, Robert Arslanbekov CFD Research Corporation, Huntsville, AL, USA
Anna Frolova Dorodnitsyn Computing Center of RAS, Moscow, Russia
Topical Workshop
“Issues in Solving the Boltzmann Equation for Aerospace Applications”
The Institute for Computational and Experimental Research in Mathematics
Brown University
Providence, RI, USA
June, 2013
Agenda
• Unified Flow Solver
• Tree-based Adaptive Cartesian Mesh
• Direct Boltzmann Solver
• Examples of UFS Applications
• Adaptive Mesh in Phase Space: Three-of-Trees Structure
• Boltzmann Solver with AMPS
• Importance Sampling
• Multi-Point Projection Method
• Deviation Method
• Testing & Demonstration
• Relaxation Problem
• Shock Wave Structure
• Hypersonic Flow Around Square
• Lorentz Gas: Kinetics of Light Particles
• Acknowledgements & References
Kinetic and Hydrodynamic Approach
Particles are described by five
characteristics:
1. Density
2. Mean directed velocity,
3. Temperature,
They depend on 4 scalar arguments –
3 spatial coordinates and time.
,n tr
Hydrodynamic Kinetic
, tv r
,T tr
The only characteristic is the
velocity distribution function
(VDF) – the particle density in
phase space
It depends on 7 scalar arguments
– 3 spatial coordinates, 3 velocity
components and time.
• Kinetic description is much more detailed and much more
expensive computationally
• Smart solver should select appropriate model based on global
conditions and local properties of transport phenomena
, ,f tξ r
Unified Flow Solver
Coupling
Algorithm
Boltzmann
Equation
Domain
Decomposition
Continuum Kinetic
Schemes
Direct Simulation
Monte-Carlo
(DSMC)
Direct Numerical
Solution
Direct Numerical Solution
of the Boltzmann equation
is preferable to DSMC for
coupling kinetic and
hydrodynamic models
Self-Aware Physics and Adaptive Numerics
• Dynamic adaptation of computational (Cartesian)
mesh to solution and geometry
• Automatic switching between kinetic and fluid
models based on continuum breakdown criteria
• Efficient Parallel Execution
recently added
Kolobov et al, J. Comp. Phys 223, (2007) 589
Kolobov & Arslanbekov, J. Comp. Phys 231, (2012) 839
Density Profile
Continuum Breakdown Criterion
Mesh Refinement Criterion:
0.1
2 221
0.1p u
Knp T x y
Illustration of UFS methodology
Steady Flow Around Cylinder for different Kn numbers: M=3
Kinetic Flag
Kn=0.01
Kn=0.1
Kn=0.3
Kn=1
Tree-based Adaptive Cartesian Mesh
• The computational grid is generated by subsequent division of square boxes to half
of the initial dimension. The procedure of grid generation is represented by a tree.
• To perform computations for a part of the domain (shown by blue color), one
introduces a flag for each leaf. The procedure of cell traversing is modified in such a
way as to visit only the cells belonging to the selected sub-domain (connected by
solid lines). To specify boundary conditions ghost cells are used (dashed lines on the
graph).
• SFC allows complete flexibility for a fine-grained domain decomposition with highly efficient
dynamic load balancing (DLB) among processors.
• During sequential traversing of cells, the physical space is filled with curves in N-order (Morton
ordering), and all cells are numbered a one-dimensional array.
• A weight is assigned to each cell, proportional to CPU time required for computations in this
cell. The array modified with corresponding weights, is subdivided into sub-arrays equal to the
number of processors.
• Coarse-grained domain decomposition is obtained by using multiple octrees (a “Forest of
Octrees”) connected through their common boundaries. Graph partitioning algorithms are used
for domain decomposition and DLB.
Space Filling Curves & Forest of Octrees
Coarse-grained and fine-grained parallelism
Highly Scalable Computational Framework
Automatic grid generation and Domain
Decomposition between processors for the
Space Shuttle Orbiter. Each processor ID is
shown by a different color.
Dynamic Load
Balancing between
processors for flow
around cylinder
Gas Density obtained
with a continuum (NS)
solver
Discrete Velocity Method (DVM)
Introduce Cartesian mesh in velocity space
with a cell size and nodes
- differential collision cross-section,
( ) ( )f
f f ft
r ξξ a
3 2( ) ( ) ( )
R Sg gf f d d ξ ξ ξ
3 2( ) ( )
R Sg gf d d ξ ξ
is a unit vector on the sphere / Ω ξ
2 3 1S R
relative velocity g
ξ ξ
Collisional Sphere
in Velocity Space
' '
1 1
2 22 2 ' '
1
,
1
ξ ξ ξ ξ
ξ ξ ξ ξ
conservation of momentum and energy
ξiξ
Direct Boltzmann Solver with DVM
( ) ( , )ii i i i
ff I f f
t
r ξ
Using discrete velocity mesh, the Boltzmann equation is reduced to a set of linear hyperbolic
transport equations in physical space with a nonlinear source term
Introducing a computational grid in physical space, we split the solution into two stages: free
streaming and relaxation. For the streaming part, we use an explicit finite volume numerical
scheme (index j denotes cell number in physical space)
* 1
1
, 0
k k
ij ij k
i i face facefaceface
f fV f S
t
ξ n
For calculation of the face values of the distribution function, we use standard interpolation
schemes of the first and second order. A tree-based dynamically adaptive isotropic Cartesian
grid with a local mesh size h is automatically generated in a computational domain.
The relaxation stage
Using explicit scheme poses a restriction for the time step:
*
* * *
k k
ij ij k k k
ij ij ij
f ff
t
max max
1min( , )
ht
Calculation of Boltzmann Collision Integral
Here, , b and are the impact parameters.
is a sphere with the center and the radius determined by the
characteristic parameters of the problem.
Symmetric form in eight-dimensional space 3 3 2 mb
2
* * * *
1 1 1 1
0 0
1( ) [ ( ) ( ) ( ) ( )] ( , )
2
mb
I d d d i gbdb
*ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ
1 1( , ) ( ) ( )i f fξ ξ ξ ξ
3
• straightforward numerical integration is very expensive (~ , where is the
number of points inside the sphere ).
• post-collision velocities do not fall into cell centers – interpolation needed to
satisfy conservation laws
• integral must be zero for any Maxwellian distribution
3
( ) ( , ) 0I f f d ξ ξ2( ) (1, , ) ξ ξ
2
vNvN
NtN method
2 3 3
1 1 1 1
1(ζ) ω ( (ξ - ζ) (ξ - ζ) (ξ - ζ) (ξ - ζ)) (ξ) (ξ ) ( ,ω) ξ ξ
2S R R
I d f f g g d d
The NtN method takes into account only those post-collisional velocities that fall exactly into the nodes of the
velocity grid. Therefore all properties (i-iii) are satisfied automatically (*)
(*) Z.Tan and P.L. Varghese , The method for the Boltzmann equation, J. Comp. Phys. 110, 327 (1994)
2
6( )( ) ( )
2
N N
i j i j ij ij
i j S
I f f d g
ζ ω ω
where ( ) ( )i i i ξ ζ ξ ζ , ( ) ( )j j j ξ ζ ξ ζ ,
and N is the total number of nodes in velocity space
0 ( ) / 2i j ξ ξ ξ
Collision sphere in velocity space. This sphere with
center and radius |g|/2 is wrapped
around pre- and post collision velocities
2
( )ijM
i j ij ijl i j
lS
d w ω ω
where ijM is the total number of such nodes (for each pair of pre-collisional nodes i and j)
and ijlw are their weights. For the VHS-like models with isotropic scattering, the number
and position of nodes on the collisional sphere can be determined a priory and the
calculation of weights is simple, 1/ijl ijw M . For more general potentials, the angle
between direct and inverse collisions is a function of relative velocity and these
calculations become cumbersome.
To evaluate the integral over the unit sphere, the NtN method takes into
account only those post-collisional velocity nodes that lie on the
collisional sphere
Cheremisin’s method
To generalize the NtN method for more complex models of molecular collisions it is necessary to take into account
inverse collisions that do not fall exactly into the nodes of the velocity grid
F.G.Tcheremissine, Solution to the Boltzmann kinetic equation for high-speed flows, Comp. Math. Math. Phys. 46 (2006) 315
Selection of post-collision nodes for Cheremisin’s
method
3 3
2
1 1 1
0 0
1( ) [ ( ) ( ) ( ) ( )] ( , )
4
mb
R R
I d db i bgd d
1ζ ξ ζ ξ ζ ξ ζ ξ ζ ξ ξ ξ ξ
1 1 1( , ) ( ) ( ) ( ) ( )i f f f f ξ ξ ξ ξ ξ ξ
On the discrete velocity grid, the velocities before collision ( , )i jξ ξ are selected at integer
nodes ,i j of the grid and the post collision velocities ( , ) ξ ξ do not necessarily fall into
the nodes. To obtain the mass, impulse, and energy conservation in each collision and
satisfy the condition ( ) ( ) ( ) ( )i j ξ ξ ξ ξ , the value of ( ) ( ) ξ ξ is
interpolated to the nearest integer nodes ,k mξ ξ using the following interpolation:
1 1( ) ( ) (1 )( ( ) ( )) ( ( ) ( ))m m k kr r ξ ξ ξ ξ ξ ξ
the nodes 1 1,k mξ ξ are selected symmetrically with respect to the nodes ,k mξ ξ
(see Figure)
On a uniform grid, it is possible to perform this interpolation with one coefficient for five
scalar invariant functions of vector , since conservation of mass and impulse in this
case is satisfied automatically due to the symmetric position of the nodes. The coefficient
r is found from the equation
2 2 2 2 2 2
1 1(1 ) m m k kr r
Korobov Sequences
Straightforward summation over all velocity nodes requires operations, where is
the number of points used for integration over impact parameters. To reduce the number of operations,
one can select collision events with a Monte Carlo method or use special sequences such as Korobov
nodes.
In the general case, Korobov’s points inside an s-dimensional hypercube are defined as
* N.M.Korobov, Exponential Sums and Their Applications, Springer, (2001), 232p;
/ , 1,2,..., , 1,2,..., 1p
r rx a p r s p
where p is a prime number, p
ra are pre-calculated integer coefficients, and the brace
denotes the reminder on dividing an integer by an integer. The velocity grid points closest
to the selected Korobov’s points are taken as the velocity grid points. The accuracy of
this procedure is estimated as ((ln ) / )as a
c cO N N , where cN is the number of quasi-
random trials, and the exponent 1a depends on the smoothness of the integrated
function (for a piecewise-constant function, 1a
). The above error is less than estimated
error of 1/ 2( )cO N
for Monte Carlo methods of calculating multi-dimensional integrals.
The typical value of cN in our simulations was equal to 34000. We have accounted only
for those collisions inside a sphere (with the center and radius defined by the
characteristic parameters of the problem), for which inverse collisions also fall inside this
sphere. Depending on the value of cN , and the number of cells in velocity space,
different Korobov’s sequences were selected
2
0( )aO n N1/3
0~ ( )an O N
Boundary Conditions
Cut-cell and IBM
approaches. Cut
cells are yellow,
ghost cells are
blue.
• The main challenge of Cartesian grids is the implementation
of surface boundary conditions.
• In the cut-cell method, boundary conditions are specified by
extrapolating solution within the fluid domain; cut-cells of
small size are merged with larger neighboring cells.
• In IBM approach, ghost cells are introduced inside solids;
boundary conditions on a solid surface are obtained using
these ghost cell outside the flow domain.
VDF of reflected particles from a boundary moving with
velocity uw (Maxwell’s approximation):
( , , ) 1 ( , 2 , ) ( , )w M w wf t f t F T r ξ r ξ ξ u n u
0w ξ u n
n is a unit normal vector to the boundary pointing to the gas;
is accommodation coefficient;
FM is a Maxwellian VDF providing zero normal mass flux at
the wall.
Example of UFS Simulations
Shock Wave
Penetration into
Micro Channel
Transient Problems
M=3
Kn=0.2
Computational grid
Kinetic &
Fluid Domains
Gas Density
V.I. Kolobov, et al, Unified Flow Solver for Transient
Rarefied-Continuum Flows, 28th Symposium on Rarefied
Gas Dynamics, AIP Conf. Proceedings 1501 (2012) 414
• Two species with masses of 3.2 and 1.6
(reference mass = 10), with no chemical
reactions
• The HS model for the Boltzmann solver.
• For the domain decomposition, continuum
breakdown criterion
is used with the total density
UFS for Gas Mixtures
Computational mesh and kinetic/continuum
domains for binary mixture of monatomic
gases at M=2, for Kn = 0.125, 0.025, and 0.0125
1
S Kn
/t i i im m
The temperatures of species become
different in the kinetic domains while they
are equal in the continuum domains.
V.I.Kolobov, et al, “Unified Solver for Rarefied and
Continuum Flows in Multi-Component Gas Mixtures”,
Rarefied Gas Dynamics: 25 Int. Symposium, (2006)
Euler-Boltzmann coupling for gas mixtures
The computational time increases
sharply with increasing the mass ratio
Internal Degrees of Freedom
Rotational spectrum for 25 levels at several
points along the shock for Ma = 13. The
center of shock is located at x = 0.
Rotational equilibrium inside the
shock does not exists at high
Mach numbers
F.G. Tcheremissine, V.I.Kolobov and R.R.Arslanbekov, “Simulation of Shock Wave Structure in Nitrogen
with Realistic Rotational Spectrum and Molecular Interaction Potential”, Rarefied Gas Dynamics: 25 Int.
Symposium, St. Petersburg, Russia (2006)
Adaptive Mesh in Phase Space
To generate adaptive velocity grid with different -grids in r-space cells (as
opposed to previously used global velocity grid across all r-space cells),
we introduce a new structure-within-a-structure method.
We “grow” tree-based -meshes in each r-space cell.
Tree-of-Trees
Key Features of the Tree-of-Trees Structure
• -simulation objects are controlled by separate scripts, one for the solver
in r-space and another for the solver in -space
• Adapting both r-space and -space grids can be done independently, or
r-space adaptation can control -space adaptation and vice versa
• It is possible to have different topology -grids in each r-space cells:
different number of boxes (building blocks) and boxes of different sizes
and (center) positions
• However, for efficient implementation of the advection operator, it is
desirable to have similar topology -grids so that each -space cell in
r-space cell can find a corresponding leaf, parent or children cell in
neighboring r-cells
• Such implementation allows improved conservation of (at least to a
second degree of accuracy) when advecting VDF from one r-space cell to
another
Mapping between Velocity Space Grids
• Examples of coarse-fine (left) and coarse-coarse (right) mappings
between -grids in 2 neighboring r-cells.
• Coarse-fine mapping involves calculation of (Van Leer limiter) cell
center gradient of VDF
• Coarse-coarse mapping involves summation over all leaf cells (total
7 in this case) of the non-leaf -cell in cell r2.
VDFs are stored in r- and -cell centers. Advection in r-space requires
calculating normal fluxes across cell faces of neighboring r-cells
Operator splitting and time advancing
• finite-volume discretization of advection in both the r- and -spaces
• explicit time stepping
• Second-order accuracy in r- and -spaces is achieved by using cell-
centered gradients with minmod and Van Leer limiters to ensure VDF
monotonicity.
• Second-order accuracy in time is achieved by using the Hancock, two-
step predictor-corrector technique Eq. 1 and a higher order Godunov’s
method in Eq. 2 (advection in -space), where the temporal derivative is
replaced by spatial derivatives.
• Time steps are estimated from the CFL criterion both in r- and -spaces,
and using time step limitation from collisions. For steady-state problems,
local time stepping can be used.
*
* * *
*
div ( , )
(0, , ) ( , )n
ff I f f
t
f f
r ξ
r ξ r ξ
**
**
** *
div 0
(0, , ) ( , , )
ff
t
f f t
ξ a
r ξ r ξ
Kinetic equation is solved in two steps
(1) (2)
Importance Sampling
Random sampling of velocity cells based on a given distribution
Choose velocity cells based on distribution and a uniform
distribution of the collision parameters ( ):
( ) ξ
,b
1/ if || || / 2( )
0 otherwise
l l
l
V h
( )( )
f
N
ξξ
1
( )vN
i i
i
N f V
, where is gas density
11 1
1 1
( ) ( )cN
m l ll l l l l l
lc l l
b f fI b g
N
ξ
|| || max{| |,| |,| |}x y z
Importance sampling is superior with respect to the Korobov’s method,
especially for sharply localized distribution functions
Multipoint Projection Method
• To satisfy conservation laws, one
has to redistribute (project)
contributions of post-collisional
velocities into neighbor cells.
• Adapt 7-point projection method [*]
to non-uniform unstructured grids in
velocity space:
Selection procedure for adaptive
Cartesian mesh in velocity space
7
1
i i i
i
X
2 2 2{1, , , , }t
xi yi zi xi yi ziX
2 2 2{1, , , , }t
x y z x y z
To satisfy five conservation laws (density, momentum, end energy) with
seven coefficients, additional conditions are used: 4 5 6
[*] A.B. Morris, P.L. Varghese, D.B. Goldstein, J. Comput. Phys. 230 (2011) 1265
Deviation Method
• Collision integral must vanish for a Maxwellian VDF:
• For static mesh: use symmetric form of the collision integral and log
interpolation of VDF (see above)
• For adaptive velocity mesh, assume , after [*]: ( ) ( ) ( )M df F f ξ ξ ξ
1 1( ) ( ) ( ) ( )f f f f ξ ξ ξ ξ
2
* * * * *
1 1 1 1
0 0
1( ) [ ( ) ( ) ( ) ( )](2 )
2
mb
M d d
S S
I F f f gbdbd d d
ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ
Sample pre-collision velocities from the distributions
2
1 1 1
1
( ) (2 ) ( )( )c
M d d
Mm F f f
Ml dl d l l l l l l l
lc
bI sign F f sign f b g
M
ξ
2( ) | 2 | /M dM d F fF f N
1( ) | | /dd ff N
Typical number of collisions 2 / (| | )M d dc F f f m vN N N g t f V
selected minimal value of the VDF | |f
an average volume of the velocity cell 3 / (6 )v m vV g N
[*] L. Baker and N. Hadjiconstantinou, Physics of Fluids 17 (2005) 051703
adapted mesh and VDF contours for Ma = 10 (left, time 0.32)
and Ma=20 (right, time 0.15).
Homogeneous Relaxation
Both computation time and
memory are ~10 times
smaller using adaptive grid
compared to uniform grid
Dimensionless variables.
velocity space of size 40 (ξ-grid
[-20,20]×[-20,20]×[-20,20]) for
Ma = 10 and 80 for Ma = 20.
grid adaptation on gradients of
VDF (L3-6 ξ-grid) for Ma = 10
(L3-7 ξ-grid for Ma = 20
0x
0x
0D3V problem: Initial VDF
consists of two parts:
pre-shock conditions at
post-shock conditions at
pressure components (left) and 4th moment components (right)
on static (L6 and L7) and adaptive (L3-7) ξ-grid
Normalized gas density and temperature
as functions of distance. Also shown are
results of the BGK model
Shock Wave Structure (Ma=10)
Comparison of solvers with adapted and uniform grids for 1D3V problem
Heat flux and number of collisions
as functions of distance.
• velocity grid adapted on gradients of VDF for an L4-6 ξ-grid: minimum level of 4
(corresponding uniform mesh 16×16×16) and maximum level of 6 (corresponding
uniform mesh of size 64×64×64).
• The number of collisions (varied from ~ 3,000 to ~20,000) is lower compared to
that used with Korobov sequences.
Shock Wave Structure
Adapted velocity grids and VDFs, , at different locations inside
the shock: pre-shock (left), transition (middle) and post-shock (right).
( , , 0)x y zf
• The adapted L4-6 ξ-grid allows properly capturing VDF details
• When a uniform ξ-grid is used for this problem, similar accuracy could only be
achieved by using L6 ξ-grids. Computations with such ξ-grids in all r-cells are
very expensive.
• Both CPU time and memory requirements are higher by a factor of ~40 when
using a uniform L6 ξ-grid
Hypersonic flow around a square
Spatial distributions of gas temperature and adapted r-grid for
Ma = 10 (left) and Ma = 30 (right) for hypersonic flow over a
cold (Twall = 1) square at Kn = 0.1.
2D2V, BGK model
r-grid is adapted on
gradients of density, mean
velocity and temperature
-grid adapted (in each r-
cell) on gradients of VDF
with a threshold value
reduced near the wall to
resolve reflected part of
VDF for proper simulation
of reflection.
Gas density, mean velocity and temperature along
stagnation lines for Ma = 20 (left) and Ma = 30 (right) at Kn =
0.1. (temperature scaled down for clarity).
magnitudes of jumps
across the shock wave
are predicted correctly
Hypersonic flow around a square
Adapted velocity mesh and VDF contours for Ma = 20 (top) and Ma =30 (bottom), Kn = 0.1
at different locations: free stream (left), inside shock wave (middle) and near the wall (right).
Velocity grid adaptation is highly beneficial inside the shock and near the wall. VDF
discontinuity near the surface is being smeared by collisions in the Knudsen layer.
Hypersonic flow around a square
Comparison of AMPS-Boltzmann results with
UFS-DSMC results for gas macro-parameters
along stagnation line at Ma =10 and Kn = 0.1
DSMS results were obtained with the
UFS-DSMC solver using tree-based
mesh and the HS collision model.
Despite the use of different collision
models in the AMPS-Boltzmann and
UFS-DSMC, we observed surprisingly
good agreement for the gas density,
mean velocity and temperature profiles
along the stagnation line, except some
region in front of the shock.
The difference in the temperature profiles
in front of the shock can be attributed to
the BGK model and to the first-order
accuracy scheme in configuration space
used in AMPS-Boltzmann.
R.R. Arslanbekov, V.I. Kolobov, J. Burt, E. Josyula, Direct Simulation Monte Carlo with Octree
Cartesian Mesh, AIAA paper 2012-2990.
Discussion: AMPS Advantages and Drawbacks
• The r-space and -space grid adaptations are currently carried out
independently.
• We use - grid adaptation on gradients of VDF with a given threshold
parameter scaled to local peak value of VDF (in r-space and in time). This
allows fine tuning of the - grid adaptation in some regions of r-space
(such as near a wall)
• We have found that grid level variation of 3–4 levels (e.g., from 4 to 7 or
from 5 to 9) is adequate to obtain acceptable accuracy during VDF
advection & mapping.
• Using adapted -grid with 3–4 levels of refinement only in regions where
VDF is significant allows obtaining a CPU time and memory gain factor of
43 – 44 = 64 – 256 in 2V and 83 – 84 = 512 – 4096 in 3V compared to
uniform -grids.
• There is however an overhead related to dealing with unstructured and
spatially varying -grids. This overhead will be reduced in a future work by
storing mapping data between grid adaptation events.
Discussion (Continued)
• In the current implementation, VDFs are stored only on leaf r-cells (as
opposed to VDFs in -space, which are stored on all cells, as well as
other quantities, such as density, velocity and temperature in r-space).
• Grid adaptation in r-space requires that VDFs are dynamically created
during r-cell refinement and destroyed during r-cell coarsening. During
coarsening of an r-cell, a new -simulation object is created in a parent
cell with a VDF formed as an average over its 4 (in 2D) children cells
whose -simulation objects are then destroyed.
• During refinement of an r-cell, 4 (in 2D) new VDFs (or -simulation
objects) are created which are all clones of the parent VDF (-simulation
object), which thus assumes first-order cell refinement algorithm
• Second order refinement can be implemented as it is done in -space,
which will require calculation of cell centered gradients of VDF in the
parent cell based on neighboring cells with different -grids.
Outlook
• Advection of VDF at a given velocity across faces of neighboring r-cells requires
that in each given r-cell, -space trees are traversed and searched in all neighbors
of this r-cell. For fine-coarse mapping, gradients of VDF need also be calculated in
the coarse -cells in a neighboring r-cell. For coarse-fine mapping, all leaf children
need to be visited in the corresponding non-leaf cells of a neighboring r-cell. Thus,
roughly, in 2D, each r-cell involves (in 2D) traverses/searches of -grids (with ~
1000–10000 number of -cells for each r-cell).
• Although traversing octrees is done in an efficient manner, this searching
represents a significant overhead compared to codes where structured -grids are
used and no searching/mapping of --grids is involved. To reduce this overhead, it
is possible to store the mapping data in each --cell of a given r-cell. This mapping
data will include --cells involved in mapping in each r-cell’s neighbor. One can
then update these data only when grid adaptation takes place in --space of a
current or neighboring r-cell or this r-cell gets refined or coarsened.
• Storage of this mapping data will only involve lists to existing data structures (--
cells) and thus will only require minimal additional memory allocation. With this
approach being implemented, we expect to get significant speed ups (factor of 10
to 100 in 2D).
Lorentz Gas
Linear Boltzmann-Lorentz collision operator
2
' '( , ) ( ) ( )
S
I N f f d Ω Ω Ω
Elastic collisions of light particles with heavy species (leading term at ) M
/Ω ξ ξ
Inelastic collisions of light particles
2
2
2( , ') ( , ) ( , ) ( , ) 'k k
k S
I N f f d
Ω Ω Ω
after inelastic collision the particle finds itself on a smaller sphere of radius
distributed according to scattering law with a weighting factor of
modify the direction of the particle velocity but conserve its kinetic energy.
• Generate Np uniformly distributed points on the sphere using the Marsaglia
method: http://mathworld.wolfram.com/SpherePointPicking.html
• Numerical experiments showed that Np ~20–50 points were sufficient for most
of the studied problems and conditions
2 2
0
2 2 2
0
Beam penetration through a thin film
VDF slices at =0 and adapted computational mesh in ξ-space at x = L/40 (left)
and x=L (right) for isotropic scattering at , and iso-surfaces of VDF in
3V space at some representative values.
2 2 2
0 0
0
exp [( ) ] / , 0, 0( , )
( , ) 0, , 0
x y z x
x
u T xf x C
f x x L
ξξ 0 max
max
( )0
/ 1/ 6L
1D3V Problem. Boundary Conditions
0 9u 0 1T
Scattering law
grid adaptation allows
capturing fine details of
the angular distributions
at different locations
AMR drastically reduces computational cost compared to uniformly refined grids
used in prior works.
Beam penetration through a thin film
Same for anisotropic scattering
Predicted dependences closely resemble those
obtained using the Boltzmann-Lorentz and
Fokker-Plank operators (for gliding collisions)
Angular distributions at exit location (x = L) for
isotropic and anisotropic scattering at .
For clarity, VDF obtained using isotropic scattering is
clipped to 0.01 (actual maximum value ~ 0.1).
/ 1/ 6L
Electron Streaming in Semiconductors
Computational mesh and VDF contours on slices for
and different values of : 0 (left), 1 (center), and 10 (right). White
circles denote inelastic collisional sphere ( ).
0 E 0 /E m eE
0 0 0/E m
0z 0/ 0.25E E
/ E
0/ 1
Phased motion of hot electrons in configuration and momentum spaces causes
spatial modulation of the electron concentration and mean electron velocity with
the period , which has been observed in structures of finite size.
In gas discharges, similar phenomena result in plasma stratification and
propagation of ionization waves (moving striations).
0 /E eE
Conclusions
• An Adaptive Mesh in Phase Space (AMPS) methodology has been
developed for solving Boltzmann equation via discrete velocity method.
• Adaptive Cartesian mesh is generated in both configuration (r) and
velocity () spaces using a Tree-of -Trees (ToT) data structure.
• A tree-based Cartesian mesh in configuration space is automatically
generated around complex boundaries and can dynamically adapt to
moving boundaries and local flow properties.
• A tree-based Cartesian mesh in velocity space is dynamically created for
each leaf cell in configuration space to minimize the number of cells in
velocity space.
• Mapping procedures have been developed for streaming operator in
configuration space on locally adapted velocity grids.
Conclusions (Continued)
• For the first time, we have computed the full Boltzmann collisional
operator (for HS model) on adaptive velocity meshes by implementing
importance sampling, multi-point projection, and deviation methods.
• We have implemented efficient algorithms for the linear Boltzmann-
Lorentz collision integral for both elastic and inelastic collisions of light
particles in a Lorentz gas.
• AMPS methodology has been demonstrated for several problems:
• Rarefied Gas Dynamics: shock wave structure (1D3V) and
hypersonic flows (2D2V);
• Electron Kinetics in a Lorentz gas (semiconductors and plasmas)
• We have demonstrated that AMPS allows minimizing the number of
cells in phase space to reduce computational cost and memory usage
for solving challenging problems.
• Future developments: (i) improve mapping procedure; (ii) develop
strategies for synchronized mesh adaptation in configuration and
velocity space; (iii) moving bodies; (iv) parallel (GPU).
Acknowledgements
Collaborators and co-authors: • Drs V.V. Aristov and S.A. Zabelok (RAS, Moscow, Russia)
• Dr F.G. Tcheremissine (RAS, Moscow, Russia)
• Mr Eswar Josyula (AFRL)
• Dr Jonathan Burt (AFRL)
Financial support provided by DoD and NASA
through SBIR/STTR Projects
References for Further Details
v • V.I.Kolobov, R.R.Arslanbekov, V.V.Aristov, A.A.Frolova, S.A.Zabelok, “Unified
solver for rarefied and continuum flows with adaptive mesh and algorithm
refinement”, J. Comput. Phys. 223 (2007) 589
• V.I.Kolobov and R.R.Arslanbekov, “Towards Adaptive Kinetic-Fluid Simulations
of Weakly Ionized Plasmas”, J. Comput. Phys. 231 (2012) 839
• V.V. Aristov, A.A. Frolova, S.A. Zabelok, R.R. Arslanbekov, V.I. Kolobov,
“Simulations of pressure-driven flows through channels and pipes with unified
flow solver”, Vacuum 86 (2012) 1717
• R.R. Arslanbekov, V.I. Kolobov, J. Burt, E. Josyula, “Direct Simulation Monte
Carlo with Octree Cartesian Mesh”, 43rd AIAA Thermophysics Conference 25 -
28 June 2012, New Orleans, Louisiana, AIAA 2012-2990
• R.R. Arslanbekov, V.I. Kolobov, and A.A. Frolova, “Kinetic Solvers with
Adaptive Mesh in Phase Space”, arXiv:1304.3330 [physics.comp-ph]
