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dumbbell.py
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dumbbell.py
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# Copyright (c) 2015-2017 Lester Hedges <[email protected]>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see <https://www.gnu.org/licenses/>.
"""
dumbbell.py
An example showing perimeter minimisation with a shape matching
constraint.
Here we construct a simple example system with two minima separated by
a free energy barrier. The matched shape is a narrow-necked dumbbell
constructed from two vertically offset, overlapping circles. The initial
configuration is a circle centred in the upper lobe of the dumbbell.
We construct two minima by making the perimeter (objective) sensitivities
a function of the y coordinate of each boundary point. Sensitivities
are linearly reduced between the centres of the upper and lower lobes,
hence within the lower lobe it's possible to form a circle with a smaller
perimeter at the same cost.
To reach the global minimum in the lower lobe the shape must pass through
the neck of the dumbbell. This transition requires a significant deformation
to the interface and a corresponding increase in the perimeter of the zero
contour. The pathway is not possible at zero temperature since it requires
a fluctuation that is uphill in free energy. As such the circle remains
trapped in the upper lobe.
The output file, "dumbbell_*.txt", contains the measured
perimeter and mismatch vs time data for the optmisation run. Level set
information for each sample interval is written to ParaView readable VTK
files, "level-set_*.vtk". Boundary segment data is written to
"boundary-segments_*.txt".
"""
import math
import pyslsm
import sys
# HELPER FUNCTIONS
# Constraint sensitivity function.
def computeConstraintSensitivity(coord, levelSet):
"""
Interpolate nodal signed distance mismatch to a boundary point using
inverse squared distance weighting. On length scales larger than a
grid spacing we are only concerned with the sign of the mismatch,
i.e. the direction that the boundary should move (out or in) in order
to reduce the mismatch.
"""
# Zero the mismatch.
mismatch = 0
# Zero the weighting factor.
weight = 0
# Find the node that is cloest to the boundary point.
node = levelSet.mesh.getClosestNode(coord)
# Loop over node and all of its neighbours.
for i in range(-1,3):
# Work out index of neighbour.
# First test the node itself.
if i < 0:
n = node
# Then its neighbours.
else:
n = levelSet.mesh.nodes[node].neighbours[i]
# Distance from the boundary point to the node in x & y direction.
dx = levelSet.mesh.nodes[n].coord.x - coord.x
dy = levelSet.mesh.nodes[n].coord.y - coord.y
# Squared distance.
rSqd = dx*dx + dy*dy
# If boundary point lies exactly on a node then use the sign of
# the mismatch at that node.
if rSqd < 1e-6:
# Calculate nodal mismatch.
m = levelSet.target[n] - levelSet.signedDistance[n]
# Smooth mismatch over a grid spacing.
if abs(m) < 1.0:
return m
# Return the sign of the mismatch.
if m < 0:
return -1.0
else:
return 1.0
# Otherwise, update the interpolation estimate.
else:
mismatch += (levelSet.target[n] - levelSet.signedDistance[n]) / rSqd
weight += 1.0 / rSqd
# Compute weighted mismatch.
mismatch /= weight
# Smooth mismatch over a grid spacing.
if abs(mismatch) < 1.0:
return mismatch
# Return the sign of the interpolated mismatch.
if mismatch < 0:
return -1.0
else:
return 1.0
# Mismatch function.
def computeMismatch(mesh, targetArea):
"""
Compute the total area mismatch between the level set domain
and its target value.
"""
# Zero the mismatch.
areaMismatch = 0
# Compute the total absolute area mismatch.
for i in range(0, mesh.nElements):
areaMismatch += abs(targetArea[i] - mesh.elements[i].area)
return areaMismatch
# Perimeter function.
def computePerimeter(points):
length = 0
# Compute the total weighted boundary length.
for i in range(0, len(points)):
length += 0.5*computePointLength(points[i])
return length
# Boundary point length function.
def computePointLength(point):
length = 0
# Store coordinates of the point.
x1 = point.coord.x
y1 = point.coord.y
# Sum the distance to each neighbour.
for i in range(0, point.nNeighbours):
# Store coordinates of the neighbouring point.
x2 = boundary.points[point.neighbours[i]].coord.x
y2 = boundary.points[point.neighbours[i]].coord.y
# Compute separation components.
dx = x2 - x1
dy = y2 - y1
# Compute segment length.
len = math.sqrt(dx*dx + dy*dy) / nDiscrete
# Perform discrete boundary (line) integral.
for j in range(0, nDiscrete):
# Compute y position along segment.
y = y1 + (j+0.5)*dy/nDiscrete
# Add weighted segment length.
length += computePerimeterWeight(y)*len
return length
# Perimeter weight function.
def computePerimeterWeight(y):
if y > upperLobeCentre: return 1.0
elif y < lowerLobeCentre: return reduce
else:
# Fractional distance from the centre of the upper dumbbell lobe.
dy = (upperLobeCentre - y) / lobeSeparation
return (1.0 - dy*(1.0 - reduce))
# MAIN PROGRAM
# Boundary integral discretisation factor.
nDiscrete = 10
# Sensitivity reduction factor.
reduce = 0.65
# Maximum displacement per iteration, in units of the mesh spacing.
# This is the CFL limit.
moveLimit = 0.05
# Default temperature of the thermal bath.
temperature = 0.02
# Override temperature if command-line argument is passed.
if (len(sys.argv) > 1):
temperature = float(sys.argv[1])
# Override sensitivity reduction factor if command-line argument is passed.
if (len(sys.argv) > 2):
reduce = float(sys.argv[2])
# Set maximum running time.
maxTime = 8000
# Set maximumum area mismatch.
maxMismatch = 0.2
# Set sampling interval.
sampleInterval = 40
# Set time of the next sample.
nextSample = 40
# Hole vectors.
initialHoles = pyslsm.VectorHole()
targetHoles = pyslsm.VectorHole()
# Create a dumbbell from two vertically offset holes.
targetHoles.append(pyslsm.Hole(50, 69, 20))
targetHoles.append(pyslsm.Hole(50, 31, 20))
# Store dumbbell data (needed for sensitivity callback).
upperLobeCentre = targetHoles[0].coord.y
lowerLobeCentre = targetHoles[1].coord.y
lobeSeparation = upperLobeCentre - lowerLobeCentre
# Initialise the system with a circle in the upper lobe.
initialHoles.append(pyslsm.Hole(50, 69, 15))
# Initialise a 100x100 level set domain.
levelSet = pyslsm.LevelSet(100, 100, initialHoles, targetHoles, moveLimit, 6, True)
# Store the mesh area.
meshArea = levelSet.mesh.width * levelSet.mesh.height
# Initialise io object.
io = pyslsm.InputOutput()
# Reinitialise the level set to a signed distance function.
levelSet.reinitialise()
# Initialise the boundary object.
boundary = pyslsm.Boundary()
# Initialise target area fraction vector.
targetArea = pyslsm.VectorDouble()
# Discretise the target structure.
boundary.discretise(levelSet, True)
io.saveBoundarySegmentsTXT(0, boundary)
# Compute the element area fractions.
levelSet.computeAreaFractions(boundary)
# Store the target area fractions.
for i in range(0, levelSet.mesh.nElements):
targetArea.append(levelSet.mesh.elements[i].area)
# Perform initial boundary discretisation.
boundary.discretise(levelSet)
# Compute the element area fractions.
levelSet.computeAreaFractions(boundary)
# Compute the initial boundary point normal vectors.
boundary.computeNormalVectors(levelSet)
# Initialise random number generator.
rng = pyslsm.MersenneTwister()
# Number of cycles since signed distance reinitialisation.
nReinit = 0
# Running time.
runningTime = 0
# Time measurements.
times = pyslsm.VectorDouble()
# Boundary length measurements (objective).
lengths = pyslsm.VectorDouble()
# Area mismatch measurements (constraint).
mismatches = pyslsm.VectorDouble()
# Lambda values for the optimiser.
lambdas = pyslsm.VectorDouble([0, 0])
print("Starting dumbbell demo...\n")
# Print output header.
print("--------------------------")
print("%6s %8s %10s" % ("Time", "Length", "Mismatch"))
print("--------------------------")
# Integrate until we exceed the maximum time.
while runningTime < maxTime:
# Initialise the sensitivity object.
sensitivity = pyslsm.Sensitivity()
# Initialise the sensitivity callback.
cb = pyslsm.Callback()
cb.callback = computePointLength
# Assign boundary point sensitivities.
for i in range(0, len(boundary.points)):
boundary.points[i].sensitivities[0] = \
sensitivity.computeSensitivity(boundary.points[i], cb.callback)
boundary.points[i].sensitivities[1] = \
computeConstraintSensitivity(boundary.points[i].coord, levelSet)
# Apply deterministic Ito correction.
sensitivity.itoCorrection(boundary, temperature)
# Time step associated with the iteration.
timeStep = pyslsm.MutableFloat()
# Constraint distance vector.
constraintDistances = pyslsm.VectorDouble()
# Current area mismatch.
mismatch = computeMismatch(levelSet.mesh, targetArea)
# Push current distance from constraint violation into vector.
constraintDistances.append(meshArea*maxMismatch - mismatch)
# Initialise the optimisation object.
optimise = pyslsm.Optimise(boundary.points, constraintDistances, \
lambdas, timeStep, levelSet.moveLimit)
# Perform the optimisation.
optimise.solve()
# Extend boundary point velocities to all narrow band nodes.
levelSet.computeVelocities(boundary.points, timeStep, temperature, rng)
# Compute gradient of the signed distance function within the narrow band.
levelSet.computeGradients()
# Update the level set function.
isReinitialised = levelSet.update(timeStep.value)
# Reinitialise the signed distance function, if necessary.
if not isReinitialised:
# Reinitialise at least every 20 iterations.
if nReinit == 20:
levelSet.reinitialise()
nReinit = 0
else:
nReinit = 0
# Increment the number of steps since reinitialisation.
nReinit += 1
# Compute the new discretised boundary.
boundary.discretise(levelSet)
# Compute the element area fractions.
levelSet.computeAreaFractions(boundary)
# Compute the boundary point normal vectors.
boundary.computeNormalVectors(levelSet)
# Increment the time.
runningTime += timeStep.value
# Check if the next sample time has been reached.
while runningTime >= nextSample:
# Compute the weighted boundary perimeter.
length = computePerimeter(boundary.points)
# Record the time, length, and mismatch area.
times.append(runningTime)
lengths.append(length)
mismatches.append(mismatch)
# Update the time of the next sample.
nextSample += sampleInterval
# Print statistics.
print("%6.1f %8.1f %10.4f" % (runningTime, length, mismatch / meshArea))
# Write level set and boundary segments to file.
io.saveLevelSetVTK(len(times), levelSet)
io.saveBoundarySegmentsTXT(len(times), boundary)
# Print results to file.
fileName = "dumbbell_%5.4f.txt" % (temperature)
file = open(fileName, "w")
for i in range(0, len(times)):
file.write("%lf %lf %lf\n" % (times[i] - times[0], lengths[i], mismatches[i] / meshArea))
file.close()
print("\nDone!")