This tutorial is part of multiphenicsx.

Tutorial 06, case 2b: advection diffusion reaction problem with distributed control

In this tutorial we solve the optimal control problem

$$\min J(y, u) = \frac{1}{2} \int_{\Omega_1 \cup \Omega_2} (y - y_d)^2 dx + \frac{\alpha}{2} \int_{\Omega} u^2 dx$$ s.t. $$\begin{cases} - \epsilon \Delta y + \beta \cdot \nabla y + \sigma y = f + u & \text{in } \Omega\\ y = g & \text{on } \partial\Omega \end{cases}$$

where $$\begin{align*} & \Omega & \text{domain}\\ & u \in L^2(\Omega) & \text{control variable}\\ & y \in H^1_0(\Omega) & \text{state variable}\\ & \alpha > 0 & \text{penalization parameter}\\ & y_d & \text{desired state}\\ & \epsilon > 0 & \text{diffusion coefficient}\\ & \beta \in \mathbb{R}^2 & \text{advection field}\\ & \sigma > 0 & \text{reaction coefficient}\\ & f & \text{forcing term}\\ & g & \text{non homogeneous piecewise constant Dirichlet BC}\\ \end{align*}$$ using an adjoint formulation solved by a one shot approach.

The test case is from section 5.2 of

F. Negri, G. Rozza, A. Manzoni and A. Quarteroni. Reduced Basis Method for Parametrized Elliptic Optimal Control Problems. SIAM Journal on Scientific Computing, 35(5): A2316-A2340, 2013.

import typing

import dolfinx.fem
import dolfinx.fem.petsc
import dolfinx.io
import dolfinx.mesh
import gmsh
import mpi4py.MPI
import numpy as np
import numpy.typing
import petsc4py.PETSc
import ufl
import viskex

import multiphenicsx.fem
import multiphenicsx.fem.petsc

Geometrical parameters

mesh_size = 0.05

Mesh

class GenerateRectangleLines:
    """Generate a rectangle."""

    def __init__(self) -> None:
        self.points: dict[tuple[float, float], int] = dict()
        self.lines: dict[tuple[int, int], int] = dict()

    def add_point(self, x: float, y: float) -> int:
        """Add a point to gmsh, if not present already."""
        key = (x, y)
        try:
            return self.points[key]
        except KeyError:
            p = gmsh.model.geo.addPoint(x, y, 0.0, mesh_size)
            self.points[key] = p
            return p  # type: ignore[no-any-return]

    def add_line(self, p0: int, p1: int) -> int:
        """Add a line to gmsh, if not present already."""
        try:
            return self.lines[(p0, p1)]
        except KeyError:
            l01 = gmsh.model.geo.addLine(p0, p1)
            self.lines[(p0, p1)] = l01
            self.lines[(p1, p0)] = -l01
            return l01  # type: ignore[no-any-return]

    def __call__(
        self, x_min: float, x_max: float, y_min: float, y_max: typing.Union[float, list[float]]
    ) -> tuple[int, list[int], int, int]:
        """Add points and lines that define a rectangle with the provided coordinates."""
        p0 = self.add_point(x_min, y_min)
        p1 = self.add_point(x_max, y_min)
        if isinstance(y_max, list):
            p2 = [self.add_point(x_max, y) for y in y_max]
            p3 = self.add_point(x_min, y_max[-1])
        else:
            p2 = [self.add_point(x_max, y_max)]
            p3 = self.add_point(x_min, y_max)
        l0 = self.add_line(p0, p1)
        p1_p2 = [p1, *p2]
        l1 = [self.add_line(p1_p2[i], p1_p2[i + 1]) for i in range(len(p2))]
        l2 = self.add_line(p2[-1], p3)
        l3 = self.add_line(p3, p0)
        return (l0, l1, l2, l3)

gmsh.initialize()
gmsh.model.add("mesh")

generate_rectangle_lines = GenerateRectangleLines()
[l0, l1, l2, l3] = generate_rectangle_lines(0.0, 1.0, 0.0, 1.0)
[l4, l5, l6, _] = generate_rectangle_lines(1.0, 2.5, 0.0, [0.3, 0.7, 1.0])
[l7, l8, l9, l10] = generate_rectangle_lines(0.2, 0.8, 0.3, 0.7)
[l11, l12, l13, l14] = generate_rectangle_lines(1.2, 2.5, 0.3, 0.7)
line_loop_rectangle_outer_left = gmsh.model.geo.addCurveLoop([l0, l1[0], l2, l3])
line_loop_rectangle_outer_right = gmsh.model.geo.addCurveLoop([l4, l5[0], l5[1], l5[2], l6, -l1[0]])
line_loop_rectangle_inner_left = gmsh.model.geo.addCurveLoop([l7, l8[0], l9, l10])
line_loop_rectangle_inner_right = gmsh.model.geo.addCurveLoop([l11, l12[0], l13, l14])
rectangle_outer_left = gmsh.model.geo.addPlaneSurface(
    [line_loop_rectangle_outer_left, line_loop_rectangle_inner_left])
rectangle_outer_right = gmsh.model.geo.addPlaneSurface(
    [line_loop_rectangle_outer_right, line_loop_rectangle_inner_right])
rectangle_inner_left = gmsh.model.geo.addPlaneSurface([line_loop_rectangle_inner_left])
rectangle_inner_right = gmsh.model.geo.addPlaneSurface([line_loop_rectangle_inner_right])

gmsh.model.geo.synchronize()
gmsh.model.addPhysicalGroup(1, [l0, l2, l3], 1)
gmsh.model.addPhysicalGroup(1, [l4, l6], 2)
gmsh.model.addPhysicalGroup(1, [l5[0], l5[1], l5[2]], 3)
gmsh.model.addPhysicalGroup(2, [rectangle_outer_left], 3)
gmsh.model.addPhysicalGroup(2, [rectangle_outer_right], 4)
gmsh.model.addPhysicalGroup(2, [rectangle_inner_left], 1)
gmsh.model.addPhysicalGroup(2, [rectangle_inner_right], 2)
gmsh.model.mesh.generate(2)

Info    : Meshing 1D...
Info    : [  0%] Meshing curve 1 (Line)
Info    : [ 10%] Meshing curve 2 (Line)
Info    : [ 20%] Meshing curve 3 (Line)
Info    : [ 20%] Meshing curve 4 (Line)
Info    : [ 30%] Meshing curve 5 (Line)
Info    : [ 40%] Meshing curve 6 (Line)
Info    : [ 40%] Meshing curve 7 (Line)
Info    : [ 50%] Meshing curve 8 (Line)
Info    : [ 60%] Meshing curve 9 (Line)
Info    : [ 60%] Meshing curve 10 (Line)
Info    : [ 70%] Meshing curve 11 (Line)
Info    : [ 70%] Meshing curve 12 (Line)
Info    : [ 80%] Meshing curve 13 (Line)
Info    : [ 90%] Meshing curve 14 (Line)
Info    : [ 90%] Meshing curve 15 (Line)
Info    : [100%] Meshing curve 16 (Line)
Info    : Done meshing 1D (Wall 0.00136342s, CPU 0.001816s)
Info    : Meshing 2D...
Info    : [  0%] Meshing surface 1 (Plane, Frontal-Delaunay)
Info    : [ 30%] Meshing surface 2 (Plane, Frontal-Delaunay)
Info    : [ 60%] Meshing surface 3 (Plane, Frontal-Delaunay)
Info    : [ 80%] Meshing surface 4 (Plane, Frontal-Delaunay)
Info    : Done meshing 2D (Wall 0.0262892s, CPU 0.026351s)
Info    : 1301 nodes 2734 elements

mesh, subdomains, boundaries, *other_tags = dolfinx.io.gmshio.model_to_mesh(
    gmsh.model, comm=mpi4py.MPI.COMM_WORLD, rank=0, gdim=2)
gmsh.finalize()
assert subdomains is not None
assert boundaries is not None

# Create connectivities required by the rest of the code
mesh.topology.create_connectivity(mesh.topology.dim - 1, mesh.topology.dim)

# Define associated measures
dx = ufl.Measure("dx", subdomain_data=subdomains)

viskex.dolfinx.plot_mesh(mesh)

error: XDG_RUNTIME_DIR is invalid or not set in the environment.

viskex.dolfinx.plot_mesh_tags(mesh, subdomains, "subdomains")

error: XDG_RUNTIME_DIR is invalid or not set in the environment.

viskex.dolfinx.plot_mesh_tags(mesh, boundaries, "boundaries")

error: XDG_RUNTIME_DIR is invalid or not set in the environment.

Function spaces

Y = dolfinx.fem.functionspace(mesh, ("Lagrange", 1))
U = dolfinx.fem.functionspace(mesh, ("Lagrange", 1))
Q = Y.clone()

Trial and test functions

(y, u, p) = (ufl.TrialFunction(Y), ufl.TrialFunction(U), ufl.TrialFunction(Q))
(z, v, q) = (ufl.TestFunction(Y), ufl.TestFunction(U), ufl.TestFunction(Q))

Problem data

alpha = 0.01
y_d_1 = 0.6
y_d_2 = 1.8
epsilon = 1. / 15.
x = ufl.SpatialCoordinate(mesh)
beta = ufl.as_vector((x[1] * (1 - x[1]), 0))
sigma = dolfinx.fem.Constant(mesh, petsc4py.PETSc.ScalarType(0))
ff = dolfinx.fem.Constant(mesh, petsc4py.PETSc.ScalarType(0))

Optimality conditions

state_operator = (epsilon * ufl.inner(ufl.grad(y), ufl.grad(q)) * dx
                  + ufl.inner(ufl.dot(beta, ufl.grad(y)), q) * dx + sigma * ufl.inner(y, q) * dx)
adjoint_operator = (epsilon * ufl.inner(ufl.grad(p), ufl.grad(z)) * dx
                    - ufl.inner(ufl.dot(beta, ufl.grad(p)), z) * dx + sigma * ufl.inner(p, z) * dx)
a = [[ufl.inner(y, z) * (dx(1) + dx(2)), None, adjoint_operator],
     [None, alpha * ufl.inner(u, v) * dx, - ufl.inner(p, v) * dx],
     [state_operator, - ufl.inner(u, q) * dx, None]]
f = [ufl.inner(y_d_1, z) * dx(1) + ufl.inner(y_d_2, z) * dx(2),
     None,
     ufl.inner(ff, q) * dx]
a[0][0] += dolfinx.fem.Constant(mesh, petsc4py.PETSc.ScalarType(0)) * ufl.inner(y, z) * dx
a[2][2] = dolfinx.fem.Constant(mesh, petsc4py.PETSc.ScalarType(0)) * ufl.inner(p, q) * dx
f[1] = ufl.inner(dolfinx.fem.Constant(mesh, petsc4py.PETSc.ScalarType(0)), v) * dx
a_cpp = dolfinx.fem.form(a)
f_cpp = dolfinx.fem.form(f)


def bdofs_Y(idx: int) -> np.typing.NDArray[np.int32]:
    """Return DOFs of the space Y located on the boundary `idx`."""
    assert boundaries is not None
    return dolfinx.fem.locate_dofs_topological(
        Y, mesh.topology.dim - 1, boundaries.indices[boundaries.values == idx])


def bdofs_Q(idx: int) -> np.typing.NDArray[np.int32]:
    """Return DOFs of the space Q located on the boundary `idx`."""
    assert boundaries is not None
    return dolfinx.fem.locate_dofs_topological(
        Q, mesh.topology.dim - 1, boundaries.indices[boundaries.values == idx])


bc_state = [dolfinx.fem.dirichletbc(petsc4py.PETSc.ScalarType(idx), bdofs_Y(idx), Y) for idx in (1, 2)]
bc_adjoint = [dolfinx.fem.dirichletbc(petsc4py.PETSc.ScalarType(0), bdofs_Q(idx), Q) for idx in (1, 2)]

Solution

(y, u, p) = (dolfinx.fem.Function(Y), dolfinx.fem.Function(U), dolfinx.fem.Function(Q))

Cost functional

J = (0.5 * ufl.inner(y - y_d_1, y - y_d_1) * dx(1) + 0.5 * ufl.inner(y - y_d_2, y - y_d_2) * dx(2)
     + 0.5 * alpha * ufl.inner(u, u) * dx)
J_cpp = dolfinx.fem.form(J)

Uncontrolled functional value

# Extract state forms from the optimality conditions
a_state = ufl.replace(a[2][0], {q: z})
f_state = ufl.replace(f[2], {q: z})
a_state_cpp = dolfinx.fem.form(a_state)
f_state_cpp = dolfinx.fem.form(f_state)

# Assemble the linear system for the state
A_state = dolfinx.fem.petsc.assemble_matrix(a_state_cpp, bcs=bc_state)
A_state.assemble()
F_state = dolfinx.fem.petsc.assemble_vector(f_state_cpp)
dolfinx.fem.petsc.apply_lifting(F_state, [a_state_cpp], [bc_state])
F_state.ghostUpdate(addv=petsc4py.PETSc.InsertMode.ADD, mode=petsc4py.PETSc.ScatterMode.REVERSE)
dolfinx.fem.petsc.set_bc(F_state, bc_state)

# Solve
ksp = petsc4py.PETSc.KSP()
ksp.create(mesh.comm)
ksp.setOperators(A_state)
ksp.setType("preonly")
ksp.getPC().setType("lu")
ksp.getPC().setFactorSolverType("mumps")
ksp.setFromOptions()
ksp.solve(F_state, y.x.petsc_vec)
y.x.petsc_vec.ghostUpdate(addv=petsc4py.PETSc.InsertMode.INSERT, mode=petsc4py.PETSc.ScatterMode.FORWARD)
ksp.destroy()

<petsc4py.PETSc.KSP at 0x7f92a4066070>

J_uncontrolled = mesh.comm.allreduce(dolfinx.fem.assemble_scalar(J_cpp), op=mpi4py.MPI.SUM)
print("Uncontrolled J =", J_uncontrolled)
assert np.isclose(J_uncontrolled, 0.028096831)

Uncontrolled J = 0.028096638296093254

viskex.dolfinx.plot_scalar_field(y, "uncontrolled state")

error: XDG_RUNTIME_DIR is invalid or not set in the environment.

Optimal control

# Assemble the block linear system for the optimality conditions
bc = bc_state + bc_adjoint
A = dolfinx.fem.petsc.assemble_matrix_block(a_cpp, bcs=bc)
A.assemble()
F = dolfinx.fem.petsc.assemble_vector_block(f_cpp, a_cpp, bcs=bc)

# Solve
yup = dolfinx.fem.petsc.create_vector_block(f_cpp)
ksp = petsc4py.PETSc.KSP()
ksp.create(mesh.comm)
ksp.setOperators(A)
ksp.setType("preonly")
ksp.getPC().setType("lu")
ksp.getPC().setFactorSolverType("mumps")
ksp.setFromOptions()
ksp.solve(F, yup)
yup.ghostUpdate(addv=petsc4py.PETSc.InsertMode.INSERT, mode=petsc4py.PETSc.ScatterMode.FORWARD)
ksp.destroy()

<petsc4py.PETSc.KSP at 0x7f92a40656c0>

# Split the block solution in components
with multiphenicsx.fem.petsc.BlockVecSubVectorWrapper(yup, [Y.dofmap, U.dofmap, Q.dofmap]) as yup_wrapper:
    for yup_wrapper_local, component in zip(yup_wrapper, (y, u, p)):
        with component.x.petsc_vec.localForm() as component_local:
            component_local[:] = yup_wrapper_local

J_controlled = mesh.comm.allreduce(dolfinx.fem.assemble_scalar(J_cpp), op=mpi4py.MPI.SUM)
print("Optimal J =", J_controlled)
assert np.isclose(J_controlled, 0.001775304)

Optimal J = 0.0017753044854971769

viskex.dolfinx.plot_scalar_field(y, "state")

error: XDG_RUNTIME_DIR is invalid or not set in the environment.

viskex.dolfinx.plot_scalar_field(u, "control")

error: XDG_RUNTIME_DIR is invalid or not set in the environment.

viskex.dolfinx.plot_scalar_field(p, "adjoint")

error: XDG_RUNTIME_DIR is invalid or not set in the environment.