This page was generated from unit-7-optimization/01_Shape_Derivative_Levelset.ipynb.
7.1 Shape optimization via the shape derivative¶
In this tutorial we discuss the implementation of shape optimization algorithms to solve
\[\min_{\Omega} J(\Omega) := \int_{\Omega} f(x)\;dx, \quad f\in C^1(\mathbf{R}^d).\]
The analytic solution to this problem is
\[\Omega^*:= \{x\in \mathbf{R}^d:\; f(x)\le 0\}.\]
This problem is solved by fixing an initial guess \(\Omega_0\subset \mathbf{R}^d\) and then considering the problem
\[\min_{X} J((\mbox{id}+X)(\Omega_0)),\]
where \(X:\mathbf{R}^d \to \mathbf{R}^d\) are (at least) Lipschitz vector fields. We approximate \(X\) by a finite element function.
Initial geometry and shape function¶
We choose \(f\) as
\[\begin{split}\begin{array}{rl}
f(x,y) =& \left(\sqrt{(x - a)^2 + b y^2} - 1 \right) \left(\sqrt{(x + a)^2 + b y^2} - 1 \right) \\
& \left(\sqrt{b x^2 + (y - a)^2} - 1 \right) \left(\sqrt{b x^2 + (y + a)^2} - 1 \right) - \varepsilon.
\end{array}\end{split}\]
where \(\varepsilon = 0.001, a = 4/5, b = 2\). The corresponding zero level sets of this function are as follows. The green area indicates where \(f\) is negative.
Let us set up the geometry.
[1]:
# Generate the geometry
from ngsolve import *
# load Netgen/NGSolve and start the gui
from ngsolve.webgui import Draw
from netgen.geom2d import SplineGeometry
geo = SplineGeometry()
geo.AddCircle((0,0), r = 2.5)
ngmesh = geo.GenerateMesh(maxh = 0.08)
mesh = Mesh(ngmesh)
mesh.Curve(2)
[1]:
<ngsolve.comp.Mesh at 0x7efbffd00170>
[2]:
#gfset.vec[:]=0
#gfset.Set((0.2*x,0))
#mesh.SetDeformation(gfset)
#scene = Draw(gfzero,mesh,"gfset", deformation = gfset)
[3]:
# define the function f and its gradient
a =4.0/5.0
b = 2
f = CoefficientFunction((sqrt((x - a)**2 + b * y**2) - 1) \
* (sqrt((x + a)**2 + b * y**2) - 1) \
* (sqrt(b * x**2 + (y - a)**2) - 1) \
* (sqrt(b * x**2 + (y + a)**2) - 1) - 0.001)
# gradient of f defined as vector valued coefficient function
grad_f = CoefficientFunction((f.Diff(x),f.Diff(y)))
[4]:
# vector space for shape gradient
VEC = H1(mesh, order=1, dim=2)
# grid function for deformation field
gfset = GridFunction(VEC)
gfX = GridFunction(VEC)
Draw(gfset)
[4]:
WebGLScene
Shape derivative¶
\[DJ(\Omega)(X) = \int_\Omega f \mbox{div}(X) + \nabla f\cdot X\;dx\]
[5]:
# Test and trial functions
PHI, PSI = VEC.TnT()
# shape derivative
dJOmega = LinearForm(VEC)
dJOmega += (div(PSI)*f + InnerProduct(grad_f, PSI) )*dx
Bilinear form¶
\[(\varphi,\psi) \mapsto b_\Omega(\varphi,\psi):= \int_\Omega (\nabla \varphi+\nabla \varphi^\top): \nabla\psi+\varphi\cdot \psi\; dx.\]
to compute the gradient \(X:= \mbox{grad}J(\Omega)\) by
\[b_\Omega(X,\psi) = DJ(\Omega)(\psi)\quad \text{ for all } \quad \psi \in H^1(\Omega)\]
[6]:
# bilinear form for H1 shape gradient; aX represents b_\Omega
aX = BilinearForm(VEC)
aX += InnerProduct(grad(PHI) + grad(PHI).trans, grad(PSI)) * dx
aX += InnerProduct(PHI, PSI) * dx
The first optimisation step¶
Fix an initial domain \(\Omega_0\) and define
\[\mathcal J_{\Omega_0}(X):= J((\mbox{id} + X)(\Omega_0)).\]
Then we have the following relation between the derivative of \(\mathcal J_{\Omega_0}\) and the shape derivative of \(J\):
\[D\mathcal J_{\Omega_0}(X_n)(X) = DJ((\mbox{id}+X_n)(\Omega_0))(X\circ(\mbox{id}+X_n)^{-1}).\]
Here
\((\mbox{id}+X_n)(\Omega_0)\) is current shape
Now we note that \(\varphi \mapsto \varphi\circ(\mbox{id}+X_n)^{-1}\) maps the finite element space on \((\mbox{id}+X_n)(\Omega_0)\) to the finite element space on \(\Omega_0\). Therefore the following are equivalent:
assembling \(\varphi \mapsto D\mathcal J_{\Omega_0}(X_n)(\varphi)\) on the fixed domain \(\Omega_0\)
assembling \(\varphi \mapsto DJ((\mbox{id}+X_n)(\Omega_0))(\varphi)\) on the deformed domain \((\mbox{id}+X_n)(\Omega_0)\).
[7]:
# deform current domain with gfset
mesh.SetDeformation(gfset)
# assemble on deformed domain
aX.Assemble()
dJOmega.Assemble()
mesh.UnsetDeformation()
# unset deformation
Now let’s make one optimization step with step size \(\alpha_1>0\):
\[\Omega_1 = (\mbox{id} - \alpha_1 X_0)(\Omega_0).\]
[8]:
# compute X_0
gfX.vec.data = aX.mat.Inverse(VEC.FreeDofs(), inverse="sparsecholesky") * dJOmega.vec
print("current cost ", Integrate(f*dx, mesh))
print("Gradient norm ", Norm(gfX.vec))
alpha = 20.0 / Norm(gfX.vec)
gfset.vec[:]=0
scene = Draw(gfset)
# input("A")
gfset.vec.data -= alpha * gfX.vec
mesh.SetDeformation(gfset)
#draw deformed shape
scene.Redraw()
# input("B")
print("cost after gradient step:", Integrate(f, mesh))
mesh.UnsetDeformation()
scene.Redraw()
current cost 63.58544055768985
Gradient norm 441.5929065866167
cost after gradient step: 5.023994315952169
Optimisation loop¶
Now we can set up an optimisation loop. In the second step we compute
\[\Omega_2 = (\mbox{id} - \alpha_0 X_0 - \alpha_1 X_1)(\Omega_0)\]
by the same procedure as above.
[9]:
import time
iter_max = 50
gfset.vec[:] = 0
mesh.SetDeformation(gfset)
scene = Draw(gfset,mesh,"gfset")
converged = False
alpha =0.11#100.0 / Norm(gfX.vec)
# input("A")
for k in range(iter_max):
mesh.SetDeformation(gfset)
scene.Redraw()
Jold = Integrate(f, mesh)
print("cost at iteration ", k, ': ', Jold)
# assemble bilinear form
aX.Assemble()
# assemble shape derivative
dJOmega.Assemble()
mesh.UnsetDeformation()
gfX.vec.data = aX.mat.Inverse(VEC.FreeDofs(), inverse="sparsecholesky") * dJOmega.vec
# step size control
gfset_old = gfset.vec.CreateVector()
gfset_old.data = gfset.vec
Jnew = Jold + 1
while Jnew > Jold:
gfset.vec.data = gfset_old
gfset.vec.data -= alpha * gfX.vec
mesh.SetDeformation(gfset)
Jnew = Integrate(f, mesh)
mesh.UnsetDeformation()
if Jnew > Jold:
print("reducing step size")
alpha = 0.9*alpha
else:
print("linesearch step accepted")
alpha = alpha*1.5
break
print("step size: ", alpha, '\n')
time.sleep(0.1)
Jold = Jnew
cost at iteration 0 : 63.58544055768985
linesearch step accepted
step size: 0.165
cost at iteration 1 : -0.738254540552506
linesearch step accepted
step size: 0.2475
cost at iteration 2 : -0.8176867041657307
linesearch step accepted
step size: 0.37124999999999997
cost at iteration 3 : -0.9325775036547923
linesearch step accepted
step size: 0.556875
cost at iteration 4 : -1.07016091722071
linesearch step accepted
step size: 0.8353125
cost at iteration 5 : -1.1521932981049592
linesearch step accepted
step size: 1.25296875
cost at iteration 6 : -1.154748206242885
reducing step size
reducing step size
linesearch step accepted
step size: 1.52235703125
cost at iteration 7 : -1.154955030640749
reducing step size
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linesearch step accepted
step size: 1.4982276723046877
cost at iteration 8 : -1.1550471182642865
reducing step size
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linesearch step accepted
step size: 1.3270326873287925
cost at iteration 9 : -1.1554534107715186
reducing step size
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linesearch step accepted
step size: 1.4511102435940346
cost at iteration 10 : -1.1555975937049026
reducing step size
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linesearch step accepted
step size: 1.4281101462330694
cost at iteration 11 : -1.1556252951639108
reducing step size
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linesearch step accepted
step size: 1.4054746004152756
cost at iteration 12 : -1.1558537093254497
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linesearch step accepted
step size: 1.3831978279986936
cost at iteration 13 : -1.155956915499012
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linesearch step accepted
step size: 1.3612741424249144
cost at iteration 14 : -1.1561527931843583
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linesearch step accepted
step size: 1.3396979472674797
cost at iteration 15 : -1.1561633003430372
reducing step size
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linesearch step accepted
step size: 1.1866173613229614
cost at iteration 16 : -1.1563067482063283
reducing step size
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linesearch step accepted
step size: 1.0510285285313932
cost at iteration 17 : -1.1563736701659089
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linesearch step accepted
step size: 1.034369726354171
cost at iteration 18 : -1.1564003185514078
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linesearch step accepted
step size: 1.0179749661914572
cost at iteration 19 : -1.1564107875935878
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linesearch step accepted
step size: 1.0018400629773228
cost at iteration 20 : -1.1564509045111546
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linesearch step accepted
step size: 0.9859608979791321
cost at iteration 21 : -1.1565107679749391
reducing step size
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linesearch step accepted
step size: 0.9703334177461629
cost at iteration 22 : -1.1565852374271406
reducing step size
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linesearch step accepted
step size: 0.9549536330748863
cost at iteration 23 : -1.1566653246131884
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linesearch step accepted
step size: 0.9398176179906492
cost at iteration 24 : -1.1567405554672066
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linesearch step accepted
step size: 1.027690565272775
cost at iteration 25 : -1.1567416778332902
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linesearch step accepted
step size: 1.0114016698132016
cost at iteration 26 : -1.1567526503917527
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linesearch step accepted
step size: 0.9953709533466624
cost at iteration 27 : -1.156774772484465
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linesearch step accepted
step size: 0.9795943237361178
cost at iteration 28 : -1.156804369104932
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linesearch step accepted
step size: 0.9640677537049005
cost at iteration 29 : -1.1568391174358499
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linesearch step accepted
step size: 0.9487872798086778
cost at iteration 30 : -1.1568738713243505
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linesearch step accepted
step size: 0.9337490014237104
cost at iteration 31 : -1.1569061928764737
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linesearch step accepted
step size: 1.0210545330568275
cost at iteration 32 : -1.1569106526567583
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linesearch step accepted
step size: 1.0048708187078768
cost at iteration 33 : -1.156919026053408
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linesearch step accepted
step size: 0.988943616231357
cost at iteration 34 : -1.1569303603266774
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step size: 0.97326885991409
cost at iteration 35 : -1.156944276358052
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step size: 0.9578425484844518
cost at iteration 36 : -1.1569588543052922
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linesearch step accepted
step size: 0.9426607440909731
cost at iteration 37 : -1.1569732380754987
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step size: 1.0307995236634793
cost at iteration 38 : -1.1569759057953024
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linesearch step accepted
step size: 1.0144613512134133
cost at iteration 39 : -1.1569802651385417
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step size: 0.9983821387966807
cost at iteration 40 : -1.1569860038671813
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step size: 0.9825577818967535
cost at iteration 41 : -1.1569932294443086
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step size: 0.96698424105369
cost at iteration 42 : -1.1570010425039097
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step size: 1.05739726759221
cost at iteration 43 : -1.1570011999144414
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step size: 1.0406375209008738
cost at iteration 44 : -1.157001977135632
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step size: 1.0241434161945948
cost at iteration 45 : -1.1570042248609775
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step size: 1.0079107430479106
cost at iteration 46 : -1.1570078095931227
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step size: 0.9919353577706012
cost at iteration 47 : -1.1570130549871247
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step size: 0.9762131823499371
cost at iteration 48 : -1.157019211552103
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step size: 0.9607402034096908
cost at iteration 49 : -1.15702607786857
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linesearch step accepted
step size: 1.050569412428497
Using SciPy optimize toolbox¶
We use the package scipy.optimize and wrap the shape functions around it.
We first setup the initial geometry as a circle of radius r = 2.5 (as before)
[10]:
# The code in this cell is the same as in the example above.
from ngsolve import *
from ngsolve.webgui import Draw
from netgen.geom2d import SplineGeometry
import numpy as np
ngsglobals.msg_level = 1
geo = SplineGeometry()
geo.AddCircle((0,0), r = 2.5)
ngmesh = geo.GenerateMesh(maxh = 0.08)
mesh = Mesh(ngmesh)
mesh.Curve(2)
# define the function f
a =4.0/5.0
b = 2
f = CoefficientFunction((sqrt((x - a)**2 + b * y**2) - 1) \
* (sqrt((x + a)**2 + b * y**2) - 1) \
* (sqrt(b * x**2 + (y - a)**2) - 1) \
* (sqrt(b * x**2 + (y + a)**2) - 1) - 0.001)
# Now we define the finite element space VEC in which we compute the shape gradient
# element order
order = 1
VEC = H1(mesh, order=order, dim=2)
# define test and trial functions
PHI = VEC.TrialFunction()
PSI = VEC.TestFunction()
# define grid function for deformation of mesh
gfset = GridFunction(VEC)
gfset.Set((0,0))
# only for new gui
#scene = Draw(gfset, mesh, "gfset")
#if scene:
# scene.setDeformation(True)
# plot the mesh and visualise deformation
#Draw(gfset,mesh,"gfset")
#SetVisualization (deformation=True)
Generate Mesh from spline geometry
Curve elements, order = 2
Shape derivative¶
\[DJ(\Omega)(X) = \int_\Omega f \mbox{div}(X) + \nabla f\cdot X\;dx\]
[11]:
fX = LinearForm(VEC)
# analytic shape derivative
fX += f*div(PSI)*dx + (f.Diff(x,PSI[0])) *dx + (f.Diff(y,PSI[1])) *dx
Bilinear form for shape gradient¶
Define the bilinear form
\[(\varphi,\psi) \mapsto b_\Omega(\varphi,\psi):= \int_\Omega (\nabla \varphi+\nabla \varphi^\top): \nabla\psi + \varphi\cdot\psi\; dx\]
to compute the gradient \(X =: \mbox{grad}J\) by
\[b_\Omega(X,\psi) = DJ(\Omega)(\psi)\quad \text{ for all } \psi\in H^1(\Omega)\]
[12]:
# Cauchy-Riemann descent CR
CR = False
# bilinear form for gradient
aX = BilinearForm(VEC)
aX += InnerProduct(grad(PHI)+grad(PHI).trans, grad(PSI))*dx + InnerProduct(PHI,PSI)*dx
## Cauchy-Riemann regularisation
if CR == True:
gamma = 100
aX += gamma * (PHI.Deriv()[0,0] - PHI.Deriv()[1,1])*(PSI.Deriv()[0,0] - PSI.Deriv()[1,1]) *dx
aX += gamma * (PHI.Deriv()[1,0] + PHI.Deriv()[0,1])*(PSI.Deriv()[1,0] + PSI.Deriv()[0,1]) *dx
aX.Assemble()
invaX = aX.mat.Inverse(VEC.FreeDofs(), inverse="sparsecholesky")
gfX = GridFunction(VEC)
Wrapping the shape function and its gradient¶
Now we define the shape function \(J\) and its gradient grad(J) use the shape derivative \(DJ\). The arguments of \(J\) and grad(J) are vector in \(\mathbf{R}^d\), where \(d\) is the dimension of the finite element space.
[13]:
def J(x_):
gfset.Set((0,0))
# x_ NumPy vector
gfset.vec.FV().NumPy()[:] += x_
mesh.SetDeformation(gfset)
cost_c = Integrate (f, mesh)
mesh.UnsetDeformation()
Redraw(blocking=True)
return cost_c
def gradJ(x_, euclid = False):
gfset.Set((0,0))
# x_ NumPy vector
gfset.vec.FV().NumPy()[:] += x_
mesh.SetDeformation(gfset)
fX.Assemble()
mesh.UnsetDeformation()
if euclid == True:
gfX.vec.data = fX.vec
else:
gfX.vec.data = invaX * fX.vec
return gfX.vec.FV().NumPy().copy()
Gradient descent and Armijo rule¶
We use the functions \(J\) and grad J to define a steepest descent method.We use scipy.optimize.line_search_armijo to compute the step size in each iteration. The Arjmijo rule reads
\[J((\mbox{Id}+\alpha_k X_k)(\Omega_k)) \le J(\Omega_k) - c_1 \alpha_k \|\nabla J(\Omega_k)\|^2,\]
where \(c_1:= 1e-4\) and \(\alpha_k\) is the step size.
[14]:
# import scipy linesearch method
# from scipy.optimize.linesearch import line_search_armijo
# try:
# from scipy.optimize._linesearch import line_search_armijo # SciPy >= 1.8
# except ImportError:
# from scipy.optimize.linesearch import line_search_armijo # older SciPy
def line_search_armijo(f, xk, pk, gfk, old_fval, c1=1e-4, alpha0=1.0):
phi0 = f(xk) if old_fval is None else old_fval
derphi0 = np.dot(gfk, pk)
alpha, nfev = alpha0, 0
for _ in range(30):
fval = f(xk + alpha * pk); nfev += 1
if fval <= phi0 + c1 * alpha * derphi0:
return alpha, nfev, fval
alpha *= 0.5
return None, nfev, phi0
def gradient_descent(x0, J_, gradJ_):
xk_ = np.copy(x0)
# maximal iteration
it_max = 50
# count number of function evals
nfval_total = 0
print('\n')
for i in range(1, it_max):
# Compute a step size using a line_search to satisfy the Wolf
# compute shape gradient grad J
grad_xk = gradJ_(xk_,euclid = False)
# compute descent direction
pk = -gradJ_(xk_)
# eval cost function
fold = J_(xk_)
# perform armijo stepsize
if CR == True:
alpha0 = 0.15
else:
alpha0 = 0.11
step, nfval, b = line_search_armijo(J_, xk_, pk = pk, gfk = grad_xk, old_fval = fold, c1=1e-4, alpha0 = alpha0)
nfval_total += nfval
# update the shape and print cost and gradient norm
xk_ = xk_ - step * grad_xk
print('Iteration ', i, '| Cost ', fold, '| grad norm', np.linalg.norm(grad_xk))
mesh.SetDeformation(gfset)
scene.Redraw()
mesh.UnsetDeformation()
if np.linalg.norm(gradJ_(xk_)) < 1e-4:
#print('#######################################')
print('\n'+'{:<20} {:<12} '.format("##################################", ""))
print('{:<20} {:<12} '.format("### success - accuracy reached ###", ""))
print('{:<20} {:<12} '.format("##################################", ""))
print('{:<20} {:<12} '.format("gradient norm: ", np.linalg.norm(gradJ_(xk_))))
print('{:<20} {:<12} '.format("n evals f: ", nfval_total))
print('{:<20} {:<12} '.format("f val: ", fold) + '\n')
break
elif i == it_max-1:
#print('######################################')
print('\n'+'{:<20} {:<12} '.format("#######################", ""))
print('{:<20} {:<12} '.format("### maxiter reached ###", ""))
print('{:<20} {:<12} '.format("#######################", ""))
print('{:<20} {:<12} '.format("gradient norm: ", np.linalg.norm(gradJ_(xk_))))
print('{:<20} {:<12} '.format("n evals f: ", nfval_total))
print('{:<20} {:<12} '.format("f val: ", fold) + '\n')
Now we are ready to run our first optimisation algorithm:¶
[15]:
gfset.vec[:]=0
x0 = gfset.vec.FV().NumPy() # initial guess = 0
scene = Draw(gfset, mesh, "gfset")
gradient_descent(x0, J, gradJ)
Iteration 1 | Cost 63.58544055768985 | grad norm 441.5929065866169
Iteration 2 | Cost -0.738254540552506 | grad norm 4.795270097704415
Iteration 3 | Cost -0.785772763146643 | grad norm 4.810253357264034
Iteration 4 | Cost -0.834010074162516 | grad norm 4.778142295071014
Iteration 5 | Cost -0.8819828810623977 | grad norm 4.690302282301073
Iteration 6 | Cost -0.9285213589747617 | grad norm 4.5397982631673575
Iteration 7 | Cost -0.9723556245428674 | grad norm 4.322976857862519
Iteration 8 | Cost -1.0122531682835856 | grad norm 4.04105181526849
Iteration 9 | Cost -1.0471884111239973 | grad norm 3.7012245498532246
Iteration 10 | Cost -1.0765044830550639 | grad norm 3.316885832891531
Iteration 11 | Cost -1.1000163330677026 | grad norm 2.906202755156897
Iteration 12 | Cost -1.1180151853481513 | grad norm 2.489788164133312
Iteration 13 | Cost -1.131171998485402 | grad norm 2.0875804970132315
Iteration 14 | Cost -1.14037467642678 | grad norm 1.7159371235340144
Iteration 15 | Cost -1.1465561839174976 | grad norm 1.385718678918182
Iteration 16 | Cost -1.1505622106495064 | grad norm 1.1024720780232362
Iteration 17 | Cost -1.1530807107883256 | grad norm 0.8664390301856512
Iteration 18 | Cost -1.1546253818026362 | grad norm 0.6744733131571095
Iteration 19 | Cost -1.1555550069868308 | grad norm 0.5214023587336207
Iteration 20 | Cost -1.1561072041599014 | grad norm 0.40125724712232974
Iteration 21 | Cost -1.1564329219351202 | grad norm 0.30812647847843966
Iteration 22 | Cost -1.1566250019098752 | grad norm 0.2366460842080783
Iteration 23 | Cost -1.1567391511183982 | grad norm 0.18221681468365186
Iteration 24 | Cost -1.1568081782564767 | grad norm 0.1410442451728761
Iteration 25 | Cost -1.156851143672455 | grad norm 0.11008000257674284
Iteration 26 | Cost -1.1568790147384713 | grad norm 0.08691942912250199
Iteration 27 | Cost -1.1568980658952455 | grad norm 0.06968543512602497
Iteration 28 | Cost -1.1569118763196693 | grad norm 0.05692156240844297
Iteration 29 | Cost -1.1569224898912507 | grad norm 0.04750014325185898
Iteration 30 | Cost -1.1569310797324186 | grad norm 0.04055031381079077
Iteration 31 | Cost -1.1569383260293806 | grad norm 0.035404371147590444
Iteration 32 | Cost -1.1569446301476731 | grad norm 0.03155781707996613
Iteration 33 | Cost -1.1569502351939924 | grad norm 0.028637497719291145
Iteration 34 | Cost -1.1569552935463383 | grad norm 0.026373993717594742
Iteration 35 | Cost -1.1569599053139974 | grad norm 0.024577086183133326
Iteration 36 | Cost -1.1569641397932513 | grad norm 0.02311474669624772
Iteration 37 | Cost -1.1569680476107744 | grad norm 0.021896245450960323
Iteration 38 | Cost -1.156971667734236 | grad norm 0.020859390544323834
Iteration 39 | Cost -1.156975031503332 | grad norm 0.019961362418991987
Iteration 40 | Cost -1.1569781648802602 | grad norm 0.019172877193450764
Iteration 41 | Cost -1.1569810896434538 | grad norm 0.01847191228418589
Iteration 42 | Cost -1.1569838257845129 | grad norm 0.017845413165404424
Iteration 43 | Cost -1.1569863886485672 | grad norm 0.017278988193844106
Iteration 44 | Cost -1.1569887950477002 | grad norm 0.01676565064074146
Iteration 45 | Cost -1.1569910584232708 | grad norm 0.01629879591809281
Iteration 46 | Cost -1.1569931906328965 | grad norm 0.015873146903417634
Iteration 47 | Cost -1.1569952027417363 | grad norm 0.015485165897309285
Iteration 48 | Cost -1.1569971040368816 | grad norm 0.015129700903153891
Iteration 49 | Cost -1.1569989040745596 | grad norm 0.01480437146262209
#######################
### maxiter reached ###
#######################
gradient norm: 0.014506461247563517
n evals f: 49
f val: -1.1569989040745596
L-BFGS method¶
Now we use the L-BFGS method provided by SciPy. The BFGS method requires the shape function \(J\) and its gradient grad J. We can also specify additional arguments in options, such as maximal iterations and the gradient tolerance.
In the BFGS method we replace
\[b_\Omega(X,\varphi) = \nabla J(\Omega)(\varphi)\]
by
\[H_\Omega(X,\varphi) = \nabla J(\Omega)(\varphi),\]
where \(H_\Omega\) is an approximation of the second shape derivative at \(\Omega\). On the discrete level we solve
\[\begin{split}\begin{array}{rl}
\text{ solve }\quad B_nX_k & = \nabla J(\Omega_k) \\
& \\
\text{ update } \quad s_k & := -\alpha_k p_k \\
y_k & := \nabla J_h(\Omega_{k+1}) - \nabla J_h(\Omega_k) \\
B_{k+1} &:= B_k + \frac{y_k\otimes y_k}{y_k\cdot s_k} + \frac{B_ks_k\otimes B_ks_k}{B_ky_k\cdot B_ks_k}.
\end{array}\end{split}\]
[16]:
from scipy.optimize import minimize
x0 = gfset.vec.FV().NumPy()
# options for optimiser
options = {"maxiter":1000,
"gtol":1e-10}
# we use quasi-Newton method L-BFGS
minimize(J, x0, method='L-BFGS-B', jac=gradJ, options=options)
[16]:
message: CONVERGENCE: RELATIVE REDUCTION OF F <= FACTR*EPSMCH
success: True
status: 0
fun: -1.1571141910331146
x: [-5.215e-04 6.981e-01 ... -1.464e-03 4.577e-03]
nit: 16
jac: [-7.356e-08 -5.266e-04 ... 4.357e-06 -3.471e-06]
nfev: 116
njev: 116
hess_inv: <7760x7760 LbfgsInvHessProduct with dtype=float64>
Improving mesh quality via Cauchy-Riemann equations¶
In the previous section we computed the shape gradient grad J:= X of \(J\) at \(\Omega\) via
\[\int_{\Omega} \nabla X : \nabla \varphi + X\cdot \varphi\; dx = DJ(\Omega)(\varphi) \quad \forall \varphi \in H^1(\Omega)^d.\]
This may lead to overly stretched or even degenerated triangles. One way to improve this is to modify the above equation by
\[\int_{\Omega} \nabla X : \nabla \varphi + X\cdot \varphi + \color{red}\gamma \color{red}B\color{red}X\cdot \color{red}B\color{red}\varphi\;dx = DJ(\Omega)(\varphi) \; \forall \varphi \in H^1(\Omega)^d,\]
where
\[\begin{split}B := \begin{pmatrix}
-\partial_x & \partial_y \\
\partial_y & \partial_x
\end{pmatrix}\end{split}\]
The two equations \(BX = 0\) are precisely the Cauchy-Riemann equations \(-\partial_x X_1 + \partial_y X_2=0\) and \(\partial_y X_1 + \partial_x X_2\). So the bigger \(\gamma\) the more angle preserving is the gradient. So by adding the B term we enforce conformaty with strength \(\gamma\).
This only means we need to add the \(\alpha\) term to the above bilinear form aX:
[17]:
alpha = 100
aX += alpha * (PHI.Deriv()[0,0] - PHI.Deriv()[1,1])*(PSI.Deriv()[0,0] - PSI.Deriv()[1,1]) *dx
aX += alpha * (PHI.Deriv()[1,0] + PHI.Deriv()[0,1])*(PSI.Deriv()[1,0] + PSI.Deriv()[0,1]) *dx
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