Fem-fenics

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Package for solving Partial Differential Equations based on Fenics.

Tutorials

Poisson Equation

Here is a first example for the solution of the Poisson equation. The equation being solved is

Code: Define Poisson problem
 
pkg load fem-fenics msh
import_ufl_Problem ('Poisson')

# Create mesh and define function space
x = y = linspace (0, 1, 33);
mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4));

# File Poisson.ufl
# element = FiniteElement("Lagrange", triangle, 1)
V = FunctionSpace('Poisson', mesh);





# Define boundary condition

bc = DirichletBC(V, @(x, y) 0.0, [2;4]);

# Define variational problem
# File Poisson.ufl
# u = TrialFunction(element)
# v = TestFunction(element)
# f = Coefficient(element)
# g = Coefficient(element)

f = Expression ('f', 
                 @(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));
g = Expression ('g', @(x,y) sin (5.0 * x));

# File Poisson.ufl
# a = inner(grad(u), grad(v))*dx
# L = f*v*dx + g*v*ds

a = BilinearForm ('Poisson', V);
L = Poisson_LinearForm ('Poisson', V, f, g);

# Compute solution

[A, b] = assemble_system (a, L, bc);
sol = A \ b;
u = Function ('u', V, sol);

# Save solution in VTK format

save (u)

# Plot solution
plot (u);
Code: Define Poisson problem
 
from dolfin import *


# Create mesh and define function space

mesh = UnitSquareMesh(32, 32)



V = FunctionSpace(mesh, "Lagrange", 1)

# Define Dirichlet boundary (x = 0 or x = 1)
def boundary(x):
    return x[0] < DOLFIN_EPS or x[0] > 1.0 - DOLFIN_EPS

# Define boundary condition
u0 = Constant(0.0)
bc = DirichletBC(V, u0, boundary)

# Define variational problem

u = TrialFunction(V)
v = TestFunction(V)



f = Expression("10*exp(-(pow(x[0] - 0.5, 2) + pow(x[1] - 0.5, 2)) / 0.02)")

g = Expression("sin(5*x[0])")





a = inner(grad(u), grad(v))*dx
L = f*v*dx + g*v*ds

# Compute solution
u = Function(V)
(A, b) = assemble_system (a, L, bc);
solve(A, u.vector(), b, "gmres", "default")


# Save solution in VTK format
file = File("poisson.pvd")
file << u

# Plot solution
plot(u, interactive=True)