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over a unit square, with
ʷ
a uniform random variable in [
0
.
05
,
0
.
05], phase-
separation can be observed. In Figure 2, snapshots of the evolution of an initially
mixed and immiscible binary fluid are shown. Notice that at steady state the
phases have separated, which is consistent with the thermodynamics of this pro-
cess. Given the adaptive time-stepping algorithm used, and the highly nonlinear
nature due to the logarithmic chemical potential of the problem, the number of
nonlinear iterations per (accepted) time step varies between 2 and 10 throughout
the simulation. More details on this particular method can be found in [8].
We present preliminary strong scaling results for this problem and show them
on table 5.1. The code was run on Shaheen, a BlueGene/P supercomputer at
King Abdullah University of Science and Technology, and the results show how
PetIGA is well suited for high-performance applications.
−
Table 1.
Scaling results for the two-dimensional Cahn-Hilliard equation. The compu-
tational mesh used consisted of 8192
2
C
1
-quadratic B-splines. The computational time
reported refers to the time taken for 10 time steps.
Cores (
N
) Time t (
s
) Speedup Eciency
512
2296
1.00
100%
1024
1207
1.90
95%
2048
578
3.97
99%
8192
147
15.62
98%
5.2 The Phase-Field Crystal Equation
The phase-field crystal equation is a sixth-order, nonlinear time-dependent par-
tial differential equation. Although initially developed to solve solidification
problems with both spatial and temporal scales orders of magnitude larger than
the ones available through molecular dynamics [13], it has since then been used
to tackle issues in crack propagation, dislocation dynamics, and formation of
foams among others [14]. In this equation, the order parameter
ˆ
represents an
atomistic density field, which is periodic in the solid state and uniform in the
liquid one. The free energy functional used in this model is given by
F
PFC
=
ʩ
ˆ
4
4
−
2
+(
ʔˆ
)
2
d
V
ˆ
2
2
2
ˆ
2
+
1
−
2
|∇
ˆ
|
(9)
The same procedure shown in section 5.1 to derive the partial differential equa-
tion is again applied, such that the evolution in time of the atomistic density
field is defined as
∂t
=
ʔ
(1 +
ʔ
)
2
ˆ
+
ˆ
3
ˆ
)
.
∂ˆ
−
(10)
