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# ------------------------------------------------------------------------
#
# Solid Fuel Ignition (SFI) problem. This problem is modeled by the
# partial differential equation
#
# -Laplacian(u) - lambda * exp(u) = 0, 0 < x,y,z < 1,
#
# with boundary conditions
#
# u = 0 for x = 0, x = 1, y = 0, y = 1, z = 0, z = 1
#
# A finite difference approximation with the usual 7-point stencil
# is used to discretize the boundary value problem to obtain a
# nonlinear system of equations. The problem is solved in a 3D
# rectangular domain, using distributed arrays (DAs) to partition
# the parallel grid.
#
# ------------------------------------------------------------------------
try: range = xrange
except: pass
import sys, petsc4py
petsc4py.init(sys.argv)
from numpy import exp, sqrt
from petsc4py import PETSc
class Bratu3D(object):
def __init__(self, da, lambda_):
assert da.getDim() == 3
self.da = da
self.lambda_ = lambda_
self.localX = da.createLocalVec()
def formInitGuess(self, snes, X):
#
x = self.da.getVecArray(X)
#
mx, my, mz = self.da.getSizes()
hx, hy, hz = [1.0/(m-1) for m in [mx, my, mz]]
lambda_ = self.lambda_
scale = lambda_/(lambda_ + 1.0)
#
(xs, xe), (ys, ye), (zs, ze) = self.da.getRanges()
for k in range(zs, ze):
min_k = min(k,mz-k-1)*hz
for j in range(ys, ye):
min_j = min(j,my-j-1)*hy
for i in range(xs, xe):
min_i = min(i,mx-i-1)*hx
if (i==0 or j==0 or k==0 or
i==mx-1 or j==my-1 or k==mz-1):
# boundary points
x[i, j, k] = 0.0
else:
# interior points
min_kij = min(min_i,min_j,min_k)
x[i, j, k] = scale*sqrt(min_kij)
def formFunction(self, snes, X, F):
#
self.da.globalToLocal(X, self.localX)
x = self.da.getVecArray(self.localX)
f = self.da.getVecArray(F)
#
mx, my, mz = self.da.getSizes()
hx, hy, hz = [1.0/m for m in [mx, my, mz]]
hxhyhz = hx*hy*hz
hxhzdhy = hx*hz/hy;
hyhzdhx = hy*hz/hx;
hxhydhz = hx*hy/hz;
lambda_ = self.lambda_
#
(xs, xe), (ys, ye), (zs, ze) = self.da.getRanges()
for k in range(zs, ze):
for j in range(ys, ye):
for i in range(xs, xe):
if (i==0 or j==0 or k==0 or
i==mx-1 or j==my-1 or k==mz-1):
f[i, j, k] = x[i, j, k] - 0
else:
u = x[ i , j , k ] # center
u_e = x[i+1 , j , k ] # east
u_w = x[i-1 , j , k ] # west
u_n = x[ i , j+1 , k ] # north
u_s = x[ i , j-1 , k ] # south
u_u = x[ i , j , k+1] # up
u_d = x[ i , j , k-1] # down
u_xx = (-u_e + 2*u - u_w)*hyhzdhx
u_yy = (-u_n + 2*u - u_s)*hxhzdhy
u_zz = (-u_u + 2*u - u_d)*hxhydhz
f[i, j, k] = u_xx + u_yy + u_zz \
- lambda_*exp(u)*hxhyhz
def formJacobian(self, snes, X, J, P):
#
self.da.globalToLocal(X, self.localX)
x = self.da.getVecArray(self.localX)
#
mx, my, mz = self.da.getSizes()
hx, hy, hz = [1.0/m for m in [mx, my, mz]]
hxhyhz = hx*hy*hz
hxhzdhy = hx*hz/hy;
hyhzdhx = hy*hz/hx;
hxhydhz = hx*hy/hz;
lambda_ = self.lambda_
#
P.zeroEntries()
row = PETSc.Mat.Stencil()
col = PETSc.Mat.Stencil()
#
(xs, xe), (ys, ye), (zs, ze) = self.da.getRanges()
for k in range(zs, ze):
for j in range(ys, ye):
for i in range(xs, xe):
row.index = (i,j,k)
row.field = 0
if (i==0 or j==0 or k==0 or
i==mx-1 or j==my-1 or k==mz-1):
P.setValueStencil(row, row, 1.0)
else:
u = x[i,j,k]
diag = (2*(hyhzdhx+hxhzdhy+hxhydhz)
- lambda_*exp(u)*hxhyhz)
for index, value in [
((i,j,k-1), -hxhydhz),
((i,j-1,k), -hxhzdhy),
((i-1,j,k), -hyhzdhx),
((i, j, k), diag),
((i+1,j,k), -hyhzdhx),
((i,j+1,k), -hxhzdhy),
((i,j,k+1), -hxhydhz),
]:
col.index = index
col.field = 0
P.setValueStencil(row, col, value)
P.assemble()
if J != P: J.assemble() # matrix-free operator
return PETSc.Mat.Structure.SAME_NONZERO_PATTERN
OptDB = PETSc.Options()
n = OptDB.getInt('n', 16)
nx = OptDB.getInt('nx', n)
ny = OptDB.getInt('ny', n)
nz = OptDB.getInt('nz', n)
lambda_ = OptDB.getReal('lambda', 6.0)
da = PETSc.DMDA().create([nx, ny, nz], stencil_width=1)
pde = Bratu3D(da, lambda_)
snes = PETSc.SNES().create()
F = da.createGlobalVec()
snes.setFunction(pde.formFunction, F)
fd = OptDB.getBool('fd', False)
mf = OptDB.getBool('mf', False)
if mf:
J = None
snes.setUseMF()
else:
J = da.createMat()
snes.setJacobian(pde.formJacobian, J)
if fd:
snes.setUseFD()
snes.getKSP().setType('cg')
snes.setFromOptions()
X = da.createGlobalVec()
pde.formInitGuess(snes, X)
snes.solve(None, X)
U = da.createNaturalVec()
da.globalToNatural(X, U)
if OptDB.getBool('plot_mpl', False):
def plot_mpl(da, U):
comm = da.getComm()
rank = comm.getRank()
scatter, U0 = PETSc.Scatter.toZero(U)
scatter.scatter(U, U0, False, PETSc.Scatter.Mode.FORWARD)
if rank == 0:
try:
from matplotlib import pylab
except ImportError:
PETSc.Sys.Print("matplotlib not available")
else:
from numpy import mgrid
nx, ny, nz = da.sizes
solution = U0[...].reshape(da.sizes, order='f')
xx, yy = mgrid[0:1:1j*nx,0:1:1j*ny]
pylab.contourf(xx, yy, solution[:, :, nz//2])
pylab.axis('equal')
pylab.xlabel('X')
pylab.ylabel('Y')
pylab.title('Z/2')
pylab.show()
comm.barrier()
plot_mpl(da, U)
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