This page was generated from unit-7-optimization/03_Shape_Derivative_Laplace_SemiAuto.ipynb.

7.3 PDE-Constrained Shape Optimization in NGSolve

We want to solve the PDE-constrained shape optimization problem :nbsphinx-math:`begin{equation}

underset{Omegasubset mathsf{D}}{mbox{min}} ; J(u) := int_Omega |u-u_d|^q ; dx, quad qge 2

end{equation}` subject to that \((\Omega,u)\) satisfy :nbsphinx-math:`begin{equation}

int_Omega nabla u cdot nabla v ; dx = int_Omega f v ; dx ; quad text{ for all } v in H_0^1(Omega),

end{equation}` where \(\Omega \subset \mathbb R^2\) for given \(u_d, f \in C^1(\mathbb R^2)\).

[1]:

import netgen.gui

from ngsolve import *
from netgen.geom2d import SplineGeometry

[2]:

geo = SplineGeometry()
geo.AddCircle(c=(0.5,0.5), r=0.5, bc = 'circle')
mesh = Mesh(geo.GenerateMesh(maxh = 0.08))

[3]:

#given data of our problem (chosen such that \Omega^* = [0,1]^2 is the optimal shape)
f = CoefficientFunction(2*y*(1-y)+2*x*(1-x))
ud = x*(1-x)*y*(1-y)

grad_f = CoefficientFunction( (f.Diff(x), f.Diff(y) ) )
grad_ud = CoefficientFunction( (ud.Diff(x), ud.Diff(y) ) )

State equation

[4]:

fes = H1(mesh, order=2, dirichlet=".*")
u, v = fes.TnT()
gfu = GridFunction(fes)
Draw (gfu, mesh, "state")

a = BilinearForm(fes, symmetric=True)
a += grad(u)*grad(v)*dx

fstate = LinearForm(fes)
fstate += f*v*dx

[5]:

def SolveStateEquation():
    rhs = gfu.vec.CreateVector()
    rhs.data = fstate.vec - a.mat * gfu.vec
    update = gfu.vec.CreateVector()
    update.data = a.mat.Inverse(fes.FreeDofs()) * rhs
    gfu.vec.data += update

[6]:

a.Assemble()
fstate.Assemble()
SolveStateEquation()

Redraw()

Adjoint equation

We set up the adjoint equation :nbsphinx-math:`begin{align*}

mbox{Find } p in H_0^1(Omega): int_Omega nabla w cdot nabla p , mbox dx = - partial_u J(u)(w) quad text{ for all } w in H_0^1(Omega)

end{align*}` where \(u\) is the solution to the state equation. For \(J(u) = \int_\Omega |u-u_d|^q \mbox dx\), we get :nbsphinx-math:`begin{align*}

partial_u J(u)(w) = q int_Omega (u-u_d)^{d-1}w ,mbox dx.

end{align*}` However, we can also use the Diff(…) command:

[7]:

q=6

def Cost(u):
    return (u-ud)**q*dx

p, w = fes.TnT()
gfp = GridFunction(fes)
Draw (gfp, mesh, "adjoint")

fadjoint = LinearForm(fes)
fadjoint += -1*Cost(gfu).Diff(gfu,w)

[8]:

def SolveAdjointEquation():
    rhs = gfp.vec.CreateVector()
    rhs.data = fadjoint.vec - a.mat.T * gfp.vec
    update = gfp.vec.CreateVector()
    update.data = a.mat.Inverse(fes.FreeDofs()).T * rhs
    gfp.vec.data += update

[9]:

fadjoint.Assemble()
SolveAdjointEquation()
Redraw()

Note that (for linear problems) the operator on the left hand side of the adjoint equation is just the transpose of the state operator.

Automatic Shape Differentiation

The formula for the shape derivative was derived as the partial derivative of the perturbed Lagrangian (brought back to the original domain): :nbsphinx-math:`begin{align*}
dmathcal J(Omega; X) = frac{partial}{partial t} left( int_Omega xi(t)|u - u_d^t|^q mbox{d} x

  • int_{Omega} (F_t^{-top}nabla u) cdot (F_t^{-top} nabla p) xi(t) , dx - int_{Omega} xi(t) f^t p ,dx right)|_{t=0}

end{align*}` where

  • \(T_t(x)=x+tX(x)=y\)

  • \(F_t = DT_t = I+t DX\)

  • \(\xi(t) = \mbox{det}(F_t)\)

  • \(u_d^t = u_d \circ T_t\)

  • \(f^t = f \circ T_t\)

The integrand depends on the parameter \(t\) only via \(\xi(t)\), \(F_t\) and \(T_t\). Denoting the integrand by \(G^{u,p}\), the derivative is given by :nbsphinx-math:`begin{align*}

dmathcal J(Omega; X) =& frac{d G^{u,p}}{d xi} frac{d xi}{d t} + frac{d G^{u,p}}{d F} frac{d F}{d t} + frac{d G^{u,p}}{dy} cdot frac{d T_t}{dt} \ =& frac{d G^{u,p}}{d xi} mbox{div}(X) + frac{d G^{u,p}}{d F} DX + frac{d G^{u,p}}{dy} cdot X

end{align*}`

[10]:

J = Parameter(1)
F = Id(2)
#Finv = 2*Id(2) - F    # consistent linearization at F = I (avoids calling Inv() for speedup)

def Equation(u,v):
    return ( (Inv(F.trans)*grad(u))*(Inv(F.trans)*grad(v))-f*v)*J*dx
#    return ( (Finv.trans*grad(u))*(Finv.trans*grad(v))-f*v)*J*dx

def CostAuto(u):
    return J*(u-ud)**q*dx

[11]:

VEC = H1(mesh, order=2, dim=2)
PHI, X = VEC.TnT()

Lagrangian = CostAuto(gfu) + Equation(gfu,gfp)
dJOmegaAuto = LinearForm(VEC)
dJOmegaAuto += Lagrangian.Diff(J, div(X))
dJOmegaAuto += Lagrangian.Diff(F, grad(X).trans)   # grad(X).trans is Jacobian
dJOmegaAuto += Lagrangian.Diff(x, X[0]) + Lagrangian.Diff(y, X[1])

[12]:

b = BilinearForm(VEC)
b += InnerProduct(grad(X),grad(PHI))*dx + InnerProduct(X,PHI)*dx

gfX = GridFunction(VEC)

# gfset denotes the deformation of the original domain and will be updated during the shape optimization
gfset = GridFunction(VEC)
gfset.Set((0,0))
Draw(gfset,mesh,"gfset")
SetVisualization (deformation=True)

[13]:

def SolveDeformationEquationAuto():
    rhs = gfX.vec.CreateVector()
    rhs.data = dJOmegaAuto.vec - b.mat * gfX.vec
    update = gfX.vec.CreateVector()
    update.data = b.mat.Inverse(VEC.FreeDofs()) * rhs
    gfX.vec.data += update

[14]:

b.Assemble()
dJOmegaAuto.Assemble()
SolveDeformationEquationAuto()
Draw(-gfX, mesh, "gfX")

[15]:

print('Cost at initial design', Integrate (CostAuto(gfu), mesh))
gfset.Set((0,0))
scale = 0.5 / Norm(gfX.vec)
gfset.vec.data -= scale * gfX.vec

Cost at initial design 6.14314805222293e-13

[16]:

mesh.SetDeformation(gfset)
a.Assemble()
fstate.Assemble()
SolveStateEquation()
print('Cost at new design', Integrate (CostAuto(gfu), mesh))

Cost at new design 5.1568007223315504e-14

Equation can also be used to define the bilinear form. The following defines the same bilinear form as above:

[17]:

aAuto = BilinearForm(fes, symmetric=True)
aAuto += Equation(u,v)

Thus, the user has to enter the PDE (in its transformed form) only once.

Finally, let us again run the full algorithm:

[18]:

#reset to and solve for initial configuration
gfset.Set((0,0))
mesh.SetDeformation(gfset)
a.Assemble()
fstate.Assemble()
SolveStateEquation()

LineSearch = False

# visoptions.vecfunction='gfset'
# visoptions.scalfunction='state'

iter_max = 600
Jold = Integrate(CostAuto(gfu), mesh)
converged = False
for k in range(iter_max):
    print('cost at iteration', k, ': ', Jold)
    mesh.SetDeformation(gfset)

    a.Assemble()
    fstate.Assemble()
    SolveStateEquation()

    fadjoint.Assemble()
    SolveAdjointEquation()

    b.Assemble()
    dJOmegaAuto.Assemble()
    SolveDeformationEquationAuto()

    scale = 0.01 / Norm(gfX.vec)
    gfsetOld = gfset
    gfset.vec.data -= scale * gfX.vec

    Jnew = Integrate(CostAuto(gfu), mesh)

    if LineSearch:
        while Jnew > Jold and scale > 1e-12:
            # input('a')
            scale = scale / 2

            if scale <= 1e-12:
                converged = True
                break

            gfset.vec.data = gfsetOld.vec - scale * gfX.vec
            mesh.SetDeformation(gfset)

            a.Assemble()
            fstate.Assemble()
            SolveStateEquation()
            Jnew = Integrate(CostAuto(gfu), mesh)

    if converged==True:
        break
    Jold = Jnew

    Redraw(blocking=True)

cost at iteration 0 :  6.143148052222901e-13
cost at iteration 1 :  5.748924421710597e-13
cost at iteration 2 :  5.207294625812289e-13
cost at iteration 3 :  4.712190017435656e-13
cost at iteration 4 :  4.2601114125018297e-13
cost at iteration 5 :  3.8477703747758833e-13
cost at iteration 6 :  3.4720812047686925e-13
cost at iteration 7 :  3.1301529239678015e-13
cost at iteration 8 :  2.819281271068429e-13
cost at iteration 9 :  2.5369407268049563e-13
cost at iteration 10 :  2.280776583897301e-13
cost at iteration 11 :  2.0485970785397564e-13
cost at iteration 12 :  1.8383655997753434e-13
cost at iteration 13 :  1.6481929930262426e-13
cost at iteration 14 :  1.4763299739883728e-13
cost at iteration 15 :  1.3211596690422567e-13
cost at iteration 16 :  1.1811902982603246e-13
cost at iteration 17 :  1.0550480169609736e-13
cost at iteration 18 :  9.414699314953546e-14
cost at iteration 19 :  8.392973044264107e-14
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