Minimal surface solution in Python

Question

I have a set of 3D points defining a 3D contour. What I want to do is to obtain the minimal surface representation corresponding to this contour (see Minimal Surfaces in Wikiped

Answer 1

You can use FEniCS:

from fenics import (
    UnitSquareMesh,
    FunctionSpace,
    Expression,
    interpolate,
    assemble,
    sqrt,
    inner,
    grad,
    dx,
    TrialFunction,
    TestFunction,
    Function,
    solve,
    DirichletBC,
    DomainBoundary,
    MPI,
    XDMFFile,
)

# Create mesh and define function space
mesh = UnitSquareMesh(100, 100)
V = FunctionSpace(mesh, "Lagrange", 2)

# initial guess (its boundary values specify the Dirichlet boundary conditions)
# (larger coefficient in front of the sin term makes the problem "more nonlinear")
u0 = Expression("a*sin(2.5*pi*x[1])*x[0]", a=0.2, degree=5)
u = interpolate(u0, V)
print(
    "initial surface area: {}".format(assemble(sqrt(1 + inner(grad(u), grad(u))) * dx))
)

# Define the linearized weak formulation for the Newton iteration
du = TrialFunction(V)
v = TestFunction(V)
q = (1 + inner(grad(u), grad(u))) ** (-0.5)
a = (
    q * inner(grad(du), grad(v)) * dx
    - q ** 3 * inner(grad(u), grad(du)) * inner(grad(u), grad(v)) * dx
)
L = -q * inner(grad(u), grad(v)) * dx
du = Function(V)

# Newton iteration
tol = 1.0e-5
maxiter = 30
for iter in range(maxiter):
    # compute the Newton increment by solving the linearized problem;
    # note that the increment has *homogeneous* Dirichlet boundary conditions
    solve(a == L, du, DirichletBC(V, 0.0, DomainBoundary()))
    u.vector()[:] += du.vector()  # update the solution
    eps = sqrt(
        abs(assemble(inner(grad(du), grad(du)) * dx))
    )  # check increment size as convergence test
    area = assemble(sqrt(1 + inner(grad(u), grad(u))) * dx)
    print(
        f"iteration{iter + 1:3d}  H1 seminorm of delta: {eps:10.2e}  area: {area:13.5e}"
    )
    if eps < tol:
        break

if eps > tol:
    print("no convergence after {} Newton iterations".format(iter + 1))
else:
    print("convergence after {} Newton iterations".format(iter + 1))

with XDMFFile(MPI.comm_world, "out.xdmf") as xdmf_file:
    xdmf_file.write(u)

(Modified from http://www-users.math.umn.edu/~arnold/8445/programs/minimalsurf-newton.py.)