A Hybrid Continuum–Atomistic Scheme for Viscous,
Incompressible Gaseous Flow
Wijesinghe, H.S. and Hadjiconstantinou, N.G.†
†
Department of Aeronautics/Astronautics, Massachusetts Institute of Technology
Department of Mechanical Engineering, Massachusetts Institute of Technology,
77 Massachusetts Ave., Cambridge, MA 02139, USA.
Abstract. We present a hybrid continuum–atomistic computational scheme for viscous, incompressible, gaseous flows. The
continuum domain is described by the steady Navier–Stokes equations, in this case solved using a finite element formulation.
The atomistic domain is described by the dilute gas model solved by direct simulation Monte Carlo (DSMC). The coupling
procedure is derived from a domain decomposition method known as the Schwarz alternating method [1]. Test results from
a two–dimensional driven cavity problem show convergence within 10 iterations for flow Reynolds numbers of order 1.
Extension of the scheme to unsteady flows is also discussed.
INTRODUCTION AND BACKGROUND
Existing continuum–atomistic formulations [2, 3, 4, 5] have been limited to compressible flows where time–dependent
flux–based schemes are typically used for solution of the continuum equations of motion. Explicit and synchronized
time marching of the continuum and atomistic sub–domains in these schemes can become prohibitively expensive
when the characteristic timescales of the sub–domains are very different. The hybrid scheme proposed in the present
work provides solutions to steady problems in an implicit sense, thus achieving timescale decoupling and superior
computational efficiency compared to explicit time marching algorithms. Our approach also caters to low speed (incompressible) flows where the compressible numerical formulation fails. The proposed scheme can also be extended
to unsteady flows without time–step based stability restrictions. The hybrid coupling scheme will be described first
with a brief summary of the continuum and atomistic sub–domain solvers. Results from applying the scheme to a
driven cavity test problem will be presented next together with an analysis of factors affecting convergence. Extension
of the scheme to unsteady flows is then presented with reference to an impulsive flat plate test problem.
SCHWARZ COUPLING
The hybrid coupling scheme proposed is based on the Schwarz alternating method [1]. The basic features of Schwarz
coupling can be illustrated with respect to a one–dimensional example as shown in Figure 1. Within this coupling
framework, the overlap region facilitates information exchange between the continuum and atomistic sub–domains
in the form of Dirichlet boundary conditions. A steady state continuum solution is first obtained using boundary
conditions taken from the atomistic sub–domain solution. At the first iteration this latter solution can be a guess. A
steady state atomistic solution is then found using boundary conditions taken from the continuum sub–domain. This
exchange of boundary conditions corresponds to a single Schwarz iteration. Successive Schwarz iterations are repeated
until convergence, i.e. until the solutions in the two sub–domains are identical in the overlap region. The exchange of
boundary conditions in this manner has three major advantages; 1) Dirichlet boundary conditions are arguably easier
to impose on molecular simulations compared to flux boundary conditions, 2) fluxes are matched without the need
to be explicitly imposed, and 3) timescales can be decoupled [8] since only steady state solutions are required from
each sub–domain (this is of great importance in multiscale applications). The Schwarz procedure is guaranteed to
converge for elliptic problems [1], and has recently been shown to converge for finite but sufficiently small Reynolds
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
907
Boundary for
atomistic solution
Boundary for
continuum solution
Continuum solution
Atomistic solution
Overlap region
FIGURE 1.
Illustration of the Schwarz coupling method in one–dimension.
numbers [6]. The method has been used by Hadjiconstantinou and Patera [7] and Hadjiconstantinou [8] to couple a
molecular dynamics description of a dense fluid with a Navier–Stokes continuum flow solver. In gaseous flows the
Schwarz method was used by Aktas and Aluru [9] to couple a DSMC based atomistic solver with a Stokes continuum
flow solver. Their implementation was effectively limited to one–dimensional flows in the overlap region and utilized a
Maxwell Boltzmann distribution to set boundary conditions on the molecular simulation. The objective of the present
paper is to develop a rigorous coupling for general gaseous flow cases (both two–dimensional and three–dimensional)
where the boundary conditions on the molecular sub–domains are imposed using a Chapman Enskog distribution.
Chapman Enskog distributions allow for the correct imposition of continuum boundary conditions on the atomistic
sub–domain in the presence of finite gradients in the continuum flow field. As will be shown, this results in superior
accuracy over the equilibrium Maxwell Boltzmann distribution as suggested in [8].
Extension of Schwarz coupling to two–dimensions is illustrated in Figure 2. In the present work the continuum
sub–domain is solved using a finite element solver based on the steady, incompressible Navier–Stokes equations.
The continuum solution is obtained in an implicit sense and does not require time marching to steady state. The
element equations are derived using a Bubnov–Galerkin approach described by Gartling and Becker [10]. To apply
the Galerkin method the nonlinear convective terms of the Navier–Stokes equations are linearized as shown below,
∂u
∂x
∂v
ρ un
∂x
ρ un
∂u
∂y
∂v
vn
∂y
∂u ∂v
∂x ∂y
2µ
∂u
∂x
, σy
2µ
∂v
∂y
, τxy
(2)
0
(3)
∂u
∂y
µ
(1)
where, σx
∂ τxy
∂ σx P
∂x
∂y
∂ τxy ∂ σy P
∂x
∂y
vn
∂v
∂x
, µ is the viscosity and P is the pressure.
The variables (un vn ) are initially set to an approximate solution of the flow (e.g., a Stokes solution). The velocity
and pressure fields are interpolated across Taylor Hood triangle elements using quadratic and linear shape functions
respectively. The resulting matrix equations are solved using direct inversion for the unknown velocities and pressure.
The inversion is repeated with updated values for (un vn ) until convergence.
The flow in the atomistic sub–domain is solved using the direct simulation Monte Carlo (DSMC) algorithm described by Bird [11]. Argon gas (molecular mass m 6 63 10 26kg and hard sphere diameter σ 3 66 10 10m)
908
−6
1
x 10
0.9
Continuum only
0.8
Atomistic only
0.7
Y (m)
0.6
0.5
0.4
0.3
0.2
h
Overlap width
0.1
0
0
0.2
0.4
0.6
X (m)
FIGURE 2.
0.8
1
−6
x 10
Continuum and atomistic sub-domains for Schwarz coupling in two–dimensions.
was used for all simulations.
The imposition of continuum sub–domain boundary conditions on the atomistic sub–domain is facilitated by a
particle reservoir in the overlap region as shown in Figure 3. Particles are created at locations x y within the reservoir
with velocities u v drawn from a Chapman-Enskog velocity distribution [2]. The mean and gradient of velocities
from the continuum solution are used as input to generate the Chapman Enskog distribution as detailed by Garcia
and Alder [12]. The continuum velocities at x y required to generate the Chapman Enskog velocity distribution are
interpolated from cell nodes using the same quadratic interpolation functions used by the finite element solver. The
number and spatial distribution of particles in the reservoir are chosen according to the overlying continuum cell
mean density and density gradients. After particles are created in the reservoir they are convected for a single DSMC
timestep. Particles that enter DSMC cells are incorporated into the standard convection/collision routines of the
DSMC algorithm. Particles that remain in the reservoir are discarded. Particles that leave the DSMC domain are also
deleted from the computation.
The velocity boundary condition originating from the atomistic sub–domain can be imposed on the continuum sub–
domain more directly. As shown in Figure 3, the centers of the DSMC cells are aligned along the nodes of the finite
element cells. While alignment of finite element and DSMC nodes is not a requirement for the scheme, this helps avoid
interpolation errors which may be substantial due to fluctuations in DSMC cell velocities. The DSMC cell velocities
obtained by time averaging particle velocities can be specified directly as Dirichlet conditions on the corresponding
finite element nodes. A correction to nodal velocities to ensure mass conservation is also made [8].
RESULTS
The hybrid coupling scheme was applied to a driven cavity flow in the domain outlined in Figure 2. The hybrid
solution is expected to recover the fully continuum solution (which will now be referred to as the exact solution)
since the atomistic sub–domain is far from solid boundaries and from regions of large velocity gradients. This test
therefore provides a consistency check for the scheme. Standard Dirichlet velocity boundary conditions for a driven
cavity problem were applied on the continuum sub–domain; the u velocity component on the left, right and lower
909
−7
5
x 10
Chapman Enskog
distribution
generated by
interpolating
velocity solution
from FE triangle
nodes
4.5
DSMC Cells
centered on FE
nodes
Y (m)
4
3.5
Particle
created in
reservoir
Reservoir region
3
2.5
2.5
3
3.5
4
4.5
X (m)
FIGURE 3.
5
−7
x 10
Particle reservoir in the overlap region.
TABLE 1. Baseline simulation parameters for the driven cavity test problem.
Property
Value
Total domain width Lx
Total domain height Ly
Finite element nodes in Lx
Finite element nodes in Ly
Imposed flow velocity at y Ly
Reynolds number based on Lx
Atomistic domain width lx
Atomistic domain height ly
Atomistic domain origin lxo
Atomistic domain origin lyo
1
1
6
10 m
10 6 m
41
41
50 m/s
43
2 25 10 7 m
2 25 10 7 m
3 875 10 7 m
3 875 10 7 m
Property
Value
DSMC cells in lx
DSMC cells in ly
Number of particles per cell
Overlap region width h
Reservoir region width
DSMC timestep ∆τ
Argon mean free path λ
Mean temperature T
DSMC timesteps per Schwarz iteration
No. of timesteps before averaging
9 (2.5 cells/λ )
9 (2.5 cells/λ )
50
1 0 10 7 m
8 75 10 8 m
3 7 10 11 s
6 258 10 8 m
273K
500000
50000
walls were held at zero while the upper wall u velocity was set to 50 m/s, the v velocity component on all boundaries
was set to zero. Despite the high velocity, the flow is essentially incompressible and isothermal. Runs at lower
velocities produced the same results. The pressure level is set by setting the middle node on the lower boundary at
atmospheric pressure (1 013 105 ) Pa. Additional parameters used in the simulation are listed in Table 1. A zero
velocity solution in the atomistic sub–domain was used as an initial guess. The DSMC simulations were advanced for
a total of 500000 (18 5µ s) time steps per Schwarz iteration with averaging beginning after 50000 (1 85 µ s) time steps.
The convergence of the u velocity along the y 0 425 10 6m plane as a function of Schwarz iterations is plotted
in Figure 4. Good comparison is achieved between the fully continuum numerical solution and the coupled hybrid
solution. The continuum numerical solution is reached to within 1m s at the 3rd Schwarz iteration and to within
0 2m s at the 10th Schwarz iteration. Similar convergence of the v velocity field is also observed. For a more global
indication of convergence the L2 norm of the velocity and pressure variables at each Schwarz iteration is plotted
in Figure 5. The velocity L2 norm shows rapid decay and convergence. The pressure L2 norm also shows a general decay but indicates that further iterations are required for convergence. This effect is currently under investigation.
910
0
U = f (X,Y=0.425× 10−6m)
−2
−4
−6
−8
Exact
Iterate=1
Iterate=2
Iterate=3
Iterate=5
Iterate=10
−10
−12
0
FIGURE 4.
0.1
0.2
0.3
0.4
0.5
X (m)
0.6
0.7
0.8
0.9
1
−6
x 10
Convergence of the u velocity component with successive Schwarz iterations.
While the above test problem was not selected to demonstrate computational savings, substantial savings are
expected in practical applications where the reduction in cost achieved by the use of the continuum description significantly outweighs the increase in cost due to the small number (O 10 ) of Schwarz iterations required. Additional
contributions to computational efficiency include the drastically reduced time to which the molecular sub–domain
needs to be simulated before it reaches a steady state, and the improved computer performance for calculations with
small memory requirements [9].
Factors Governing Convergence
A range of tests were conducted to assess the effect of overlap region widths (h 0 8λ and h 1 6λ ), Maxwell
Boltzmann based equilibrium distribution particle reservoirs (i.e., equivalent to ignoring gradient information in the
overlying continuum solution) and number of DSMC time steps per Schwarz iteration on the convergence of the
coupling scheme. Results from these tests are plotted in Figure 5 and Figure 6. The following observations can be
made from these plots:
1. Convergence of the velocity field is only weakly coupled to the overlap region width h in the range considered. A
similar conclusion was reached by Aktas and Aluru [9]. The pressure field shows greater sensitivity to h but no
significant differences in convergence can be seen.
2. The velocity and pressure error norms are one order of magnitude larger when a Maxwell Boltzmann reservoir is
used.
3. Halving the number of DSMC time steps per Schwarz iteration (via advancing the DSMC solution to 275000
time steps and averaging after 50000) has an insignificant effect on the convergence of the velocity L2 norm.
The pressure L2 norm tracks the baseline solution for 3 Schwarz iterations but then fluctuates above the baseline
solution.
911
6
2
log10(sqrt((Exact−Iterate) )
5
4
3
2
Velocity, h=1.6 λ
Pressure, h=1.6 λ
Velocity, h=0.8 λ
Pressure, h=0.8 λ
1
0
−1
1
2
3
4
5
6
7
No. of Schwarz iterates Ns
8
9
10
FIGURE 5. Comparison of the convergence of the velocity and pressure fields with Schwarz iterations. The velocity norm is
constructed from the sum of both u and v velocities.
6
2
log10(sqrt((Exact−Iterate) )
5
4
3
2
1
0
−1
1
Velocity: Baseline
Pressure: Baseline
Velocity: Maxwell Boltzmann reservoirs
Pressure: Maxwell Boltzmann reservoirs
Velocity: Coupling at 275000 time steps
Pressure: Coupling at 275000 time steps
2
3
4
5
6
7
No. of Schwarz iterates N
8
9
10
s
FIGURE 6. Comparison of the convergence of the velocity and pressure fields with Schwarz iterations. The velocity norm is
constructed using the sum of both u and v velocities.
912
TABLE 2.
Unsteady schwarz simulation parameters for the impulsive flat plate test problem.
Property
Value
Total domain width Lx
Continuum domain width Lc
Continuum nodes in Lx
Continuum time step ∆t
Atomistic domain width La
DSMC cells in La
No. of particles in each cell
Flat plate velocity at x Lx
6
1 10 m
0 75625 10 6 m
51
1 0 10 10 s
0 24375 10 6 m
20
2000
30 m/s
Property
Value
Overlap region width h
Reservoir region width
DSMC timestep ∆τ
Argon mean free path λ
DSMC timesteps per ensemble
No. of ensembles
Schwarz iterations at each time level t n
Total no. of time steps
4 0 10 8 m
3 375 10 8 m
1 0 10 11 s
6 258 10 8 m
100
1000
5
4
EXTENSION TO UNSTEADY FLOWS
The Schwarz alternating method can be extended to couple unsteady hybrid flows. Similar to steady flow coupling, an
overlap region between the subdomains facilitates information exchange in the form of Dirichlet boundary conditions.
Unlike the steady flow case however, successive Schwarz iterations are used to converge the solution to a given time
t n . The converged solution at t n forms the initial condition for subsequent Schwarz iterations to advance the solution
to time level t n 1 . The unsteady Schwarz scheme still allows for time–scale decoupling; each subdomain can be
advanced at the local most favourable time step, while the choice of t n 1 is arbitrary. The computational cost of performing multiple Schwarz iterations per time level is partially offset by the ability to implicitly advance to the time of
interest without the need for explicit coupling at previous times. Further work in accelerating this scheme is in progress.
This scheme has been implemented to calculate the unsteady one–dimensional flow generated by an impulsive flat
plate. To help reduce noise in the DSMC solution, ensemble averaging was also performed using simulations initiated
from a different random number seed. The continuum solution was obtained by solving an unsteady one–dimensional
equation for momentum diffusion using an implicit backward difference scheme. The imposition of continuum
boundary conditions on the atomistic sub–domain is facilitated using particle reservoirs as described for the steady
flow case. Imposition of the atomistic boundary conditions on the continuum sub–domain also follows the use of
overlapping continuum nodes and DSMC cell centers as described for the steady flow case. The parameters used in
the unsteady simulations are listed in Table 2.
Figure 7 compares the hybrid solution obtained at times 1 10 9s through 4 10 9s with the fully DSMC atomistic
solution. The hybrid solution shows good comparison and captures the unsteady velocity slip at the wall.
CONCLUSIONS
A hybrid continuum–atomistic scheme has been developed to couple a Navier–Stokes description of a continuum
field with a DSMC description of a dilute gas. Coupling of the continuum/atomistic sub–domains is achieved by
exchange of boundary conditions via a Schwarz alternating method. Continuum sub–domain boundary conditions are
imposed on the atomistic sub–domain using particle reservoirs based on the Chapman Enskog velocity distribution.
The atomistic sub–domain boundary conditions are imposed on the continuum sub–domain via simple averaging. The
following conclusions have been reached in this study:
1. The Schwarz coupling scheme applied to a two–dimensional driven cavity flow at Reynolds number 4.3 converges
to within 0 2 m/s of the fully continuum solution after 10 Schwarz iterations.
2. Convergence of the scheme is found to be weakly dependent on the overlap region width in the range h 0 8λ
to h 1 6λ .
3. The use of Chapman Enskog distributions significantly improves the accuracy of the solution compared to
Maxwell Boltzmann equilibrium distributions.
4. Schwarz coupling extends naturally to unsteady flows. Solution of a one–dimensional flat plate problem shows
good comparison with a fully DSMC solution.
913
30
25
Hybrid: Continuum
Hybrid: Atomistic
Atomistic
Atomistic + Reservoir
20
V (m/s)
Continuum
−8
Overlap h=4 ×10
15
m
10
τ = 4 × 10−9 s
5
0
−5
0
−9
τ = 1 × 10
0.1
0.2
0.3
0.4
0.5
X (m)
s
0.6
0.7
0.8
0.9
1
−6
x 10
FIGURE 7. Comparison of the unsteady hybrid scheme with the fully DSMC atomistic solution. 5 Schwarz iterations were
required to converge the solution at each time level.
ACKNOWLEDGMENTS
The authors wish to thank Prof. Alejandro Garcia, Prof. Anthony Patera and Prof. Berni Alder for useful discussions
and suggestions. This work has been supported by the Lawrence Livemore National Laboratory.
REFERENCES
1.
Lions, P., “On the Schwarz Alternating Method, I,” in First International Symposium on Domain Decomposition Methods for
Partial Differential Equations, edited by G. M. R. Glowinski, G. Golub and J. Periaux, SIAM, Philadelphia, 1988, p. 1.
2. Garcia, A., Bell, J., Crutchfield, W., and Alder, B., Journal of Computational Physics, 154, 134–155 (1999).
3. Wadsworth, D., and Erwin, D., AIAA Paper 90-1690 (1990).
4. Eggers, J., and Beylich, A., Prog. Astro. Aero., 159, 166 (1994).
5. Hash, D., and Hassan, H., AIAA Paper 95-0410 (1995).
6. Liu, S., SIAM J. Sci. Comput., 22, 1974–1986 (2001).
7. Hadjiconstantinou, N., and Patera, A., International Journal of Modern Physics C, 8, 967–976 (1997).
8. Hadjiconstantinou, N., Journal of Computational Physics, 154, 245–265 (1999).
9. Aktas, O., and Aluru, N., Journal of Computational Physics, 178, 342–372 (2000).
10. Gartling, D., and Becker, E., Comput. Methods Appl. Mech. Eng., 8, 51–60 (1960).
11. Bird, G., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, England, UK, 1994.
12. Garcia, A., and Alder, B., Journal of Computational Physics, 140 (1998).
914