Fem-fenics
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Package for solving Partial Differential Equations based on Fenics.
Introduction
Tutorials
Poisson Equation
Here is a first example for the solution of the Poisson equation. The equation being solved is
A complete description of the problem is avilable on the Fenics website.
| Code: Define Poisson problem with fem-fenics |
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 = 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, 'poisson')
# Plot solution
plot (u);
|
| Code: Define Poisson problem with fenics python |
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)
© Copyright 2011, The FEniCS Project
|
Mixed Formulation for Poisson Equation
A complete description of the problem is avilable on the Fenics website.
| Code: Define MixedPoisson problem with fem-fenics |
pkg load fem-fenics msh
import_ufl_Problem ('MixedPoisson')
# Create mesh
x = y = linspace (0, 1, 33);
mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4));
# File MixedPoisson.ufl
# BDM = FiniteElement("BDM", triangle, 1)
# DG = FiniteElement("DG", triangle, 0)
# W = BDM * DG
V = FunctionSpace('MixedPoisson', mesh);
# Define trial and test function
# File MixedPoisson.ufl
# (sigma, u) = TrialFunctions(W)
# (tau, v) = TestFunctions(W)
# CG = FiniteElement("CG", triangle, 1)
# f = Coefficient(CG)
f = Expression ('f',
@(x,y) 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02));
# Define variational form
# File MixedPoisson.ufl
# a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx
# L = - f*v*dx
a = BilinearForm ('MixedPoisson', V);
L = LinearForm ('MixedPoisson', V, f);
# Define essential boundary
bc1 = DirichletBC (SubSpace (V, 1), @(x,y) [0; -sin(5.0*x)], 1);
bc2 = DirichletBC (SubSpace (V, 1), @(x,y) [0; sin(5.0*x)], 3);
# Compute solution
[A, b] = assemble_system (a, L, bc1, bc2);
sol = A \ b;
func = Function ('func', V, sol);
sigma = Function ('sigma', func, 1);
u = Function ('u', func, 2);
# Plot solution
plot (sigma);
plot (u);
|
| Code: Define MixedPoisson problem with fenics python |
from dolfin import *
# Create mesh
mesh = UnitSquareMesh(32, 32)
# Define function spaces and mixed (product) space
BDM = FunctionSpace(mesh, "BDM", 1)
DG = FunctionSpace(mesh, "DG", 0)
W = BDM * DG
# Define trial and test functions
(sigma, u) = TrialFunctions(W)
(tau, v) = TestFunctions(W)
f = Expression
("10*exp(-(pow(x[0] - 0.5, 2) + pow(x[1] - 0.5, 2)) / 0.02)")
# Define variational form
a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx
L = - f*v*dx
# Define function G such that G \cdot n = g
class BoundarySource(Expression):
def __init__(self, mesh):
self.mesh = mesh
def eval_cell(self, values, x, ufc_cell):
cell = Cell(self.mesh, ufc_cell.index)
n = cell.normal(ufc_cell.local_facet)
g = sin(5*x[0])
values[0] = g*n[0]
values[1] = g*n[1]
def value_shape(self):
return (2,)
G = BoundarySource(mesh)
# Define essential boundary
def boundary(x):
return x[1] < DOLFIN_EPS or x[1] > 1.0 - DOLFIN_EPS
bc = DirichletBC(W.sub(0), G, boundary)
# Compute solution
w = Function(W)
solve(a == L, w, bc)
(sigma, u) = w.split()
# Plot sigma and u
plot(sigma)
plot(u)
interactive()
© Copyright 2011, The FEniCS Project
|
Hyperelasticity
This time we compare the code with the c++ version of DOLFIN. A complete description of the problem is avilable on the Fenics website. The final solution will look like this
| Code: HyperElasticity Problem: the ufl file |
# Function spaces
element = VectorElement("Lagrange", tetrahedron, 1)
# Trial and test functions
du = TrialFunction(element) # Incremental displacement
v = TestFunction(element) # Test function
# Functions
u = Coefficient(element) # Displacement from previous iteration
B = Coefficient(element) # Body force per unit volume
T = Coefficient(element) # Traction force on the boundary
# Kinematics
I = Identity(element.cell().d) # Identity tensor
F = I + grad(u) # Deformation gradient
C = F.T*F # Right Cauchy-Green tensor
# Invariants of deformation tensors
Ic = tr(C)
J = det(F)
# Elasticity parameters
mu = Constant(tetrahedron)
lmbda = Constant(tetrahedron)
# Stored strain energy density (compressible neo-Hookean model)
psi = (mu/2)*(Ic - 3) - mu*ln(J) + (lmbda/2)*(ln(J))**2
# Total potential energy
Pi = psi*dx - inner(B, u)*dx - inner(T, u)*ds
# First variation of Pi (directional derivative about u in the direction of v)
F = derivative(Pi, u, v)
# Compute Jacobian of F
J = derivative(F, u, du)
© Copyright 2011, The FEniCS Project
|
| Code: Define HyperElasticity problem with fem-fenics |
pkg load fem-fenics msh
problem = 'HyperElasticity';
import_ufl_Problem (problem);
Rotation = @(x,y,z) ...
[0; ...
0.5*(0.5 + (y - 0.5)*cos(pi/3) - (z-0.5)*sin(pi/3) - y);...
0.5*(0.5 + (y - 0.5)*sin(pi/3) + (z-0.5)*cos(pi/3) - z)];
#Create mesh and define function space
x = y = z = linspace (0, 1, 17);
mshd = Mesh (msh3m_structured_mesh (x, y, z, 1, 1:6));
V = FunctionSpace (problem, mshd);
# Create Dirichlet boundary conditions
bcl = DirichletBC (V, @(x,y,z) [0; 0; 0], 1);
bcr = DirichletBC (V, Rotation, 2);
bcs = {bcl, bcr};
# Define source and boundary traction functions
B = Constant ('B', [0.0; -0.5; 0.0]);
T = Constant ('T', [0.1; 0.0; 0.0]);
# Set material parameters
E = 10.0;
nu = 0.3;
mu = Constant ('mu', E./(2*(1+nu)));
lmbda = Constant ('lmbda', E*nu./((1+nu)*(1-2*nu)));
u = Expression ('u', @(x,y,z) [0; 0; 0]);
# Create (linear) form defining (nonlinear) variational problem
L = ResidualForm (problem, V, mu, lmbda, B, T, u);
# Solve nonlinear variational problem F(u; v) = 0
u0 = assemble (L, bcs{:});
# Create function for the resolution of the NL problem
function [y, jac] = f (problem, xx, V, bc1, bc2, B, T, mu, lmbda)
u = Function ('u', V, xx);
a = JacobianForm (problem, V, mu, lmbda, u);
L = ResidualForm (problem, V, mu, lmbda, B, T, u);
if (nargout == 1)
[y, xx] = assemble (L, xx, bc1, bc2);
elseif (nargout == 2)
[jac, y, xx] = assemble_system (a, L, xx, bc1, bc2);
endif
endfunction
fs = @(xx) f (problem, xx, V, bcl, bcr, B, T, mu, lmbda);
[x, fval, info] = fsolve (fs, u0, optimset ("jacobian", "on"));
func = Function ('u', V, x);
# Save solution in VTK format
save (func, 'displacement');
# Plot solution
plot (func);
|
| Code: Define HyperElasticity problem with fenics c++ |
#include <dolfin.h>
#include "HyperElasticity.h"
using namespace dolfin;
// Sub domain for clamp at left end
class Left : public SubDomain
{
bool inside(const Array<double>& x, bool on_boundary) const
{
return (std::abs(x[0]) < DOLFIN_EPS) && on_boundary;
}
};
// Sub domain for rotation at right end
class Right : public SubDomain
{
bool inside(const Array<double>& x, bool on_boundary) const
{
return (std::abs(x[0] - 1.0) < DOLFIN_EPS) && on_boundary;
}
};
// Dirichlet boundary condition for clamp at left end
class Clamp : public Expression
{
public:
Clamp() : Expression(3) {}
void eval(Array<double>& values, const Array<double>& x) const
{
values[0] = 0.0;
values[1] = 0.0;
values[2] = 0.0;
}
};
// Dirichlet boundary condition for rotation at right end
class Rotation : public Expression
{
public:
Rotation() : Expression(3) {}
void eval(Array<double>& values, const Array<double>& x) const
{
const double scale = 0.5;
// Center of rotation
const double y0 = 0.5;
const double z0 = 0.5;
// Large angle of rotation (60 degrees)
double theta = 1.04719755;
// New coordinates
double y = y0 + (x[1]-y0)*cos(theta) - (x[2]-z0)*sin(theta);
double z = z0 + (x[1]-y0)*sin(theta) + (x[2]-z0)*cos(theta);
// Rotate at right end
values[0] = 0.0;
values[1] = scale*(y - x[1]);
values[2] = scale*(z - x[2]);
}
};
int main()
{
// Create mesh and define function space
UnitCubeMesh mesh (16, 16, 16);
HyperElasticity::FunctionSpace V(mesh);
// Define Dirichlet boundaries
Left left;
Right right;
// Define Dirichlet boundary functions
Clamp c;
Rotation r;
// Create Dirichlet boundary conditions
DirichletBC bcl(V, c, left);
DirichletBC bcr(V, r, right);
std::vector<const BoundaryCondition*> bcs;
bcs.push_back(&bcl); bcs.push_back(&bcr);
// Define source and boundary traction functions
Constant B(0.0, -0.5, 0.0);
Constant T(0.1, 0.0, 0.0);
// Define solution function
Function u(V);
// Set material parameters
const double E = 10.0;
const double nu = 0.3;
Constant mu(E/(2*(1 + nu)));
Constant lambda(E*nu/((1 + nu)*(1 - 2*nu)));
// Create (linear) form defining (nonlinear) variational problem
HyperElasticity::ResidualForm F(V);
F.mu = mu; F.lmbda = lambda; F.B = B; F.T = T; F.u = u;
// Create jacobian dF = F' (for use in nonlinear solver).
HyperElasticity::JacobianForm J(V, V);
J.mu = mu; J.lmbda = lambda; J.u = u;
// Solve nonlinear variational problem F(u; v) = 0
solve(F == 0, u, bcs, J);
// Save solution in VTK format
File file("displacement.pvd");
file << u;
// Plot solution
plot(u);
interactive();
return 0;
}
© Copyright 2011, The FEniCS Project
|
A complete description of the problem is avilable on the Fenics website.
| Code: Define HyperElasticity problem with fem-fenics |
pkg load fem-fenics msh
import_ufl_Problem ("TentativeVelocity");
import_ufl_Problem ("VelocityUpdate");
import_ufl_Problem ("PressureUpdate");
# We can either load the mesh from the file as in Dolfin but
# we can also use the msh pkg to generate the L-shape domain
name = [tmpnam ".geo"];
fid = fopen (name, "w");
fputs (fid,"Point (1) = {0, 0, 0, 0.1};\n");
fputs (fid,"Point (2) = {1, 0, 0, 0.1};\n");
fputs (fid,"Point (3) = {1, 0.5, 0, 0.1};\n");
fputs (fid,"Point (4) = {0.5, 0.5, 0, 0.1};\n");
fputs (fid,"Point (5) = {0.5, 1, 0, 0.1};\n");
fputs (fid,"Point (6) = {0, 1, 0,0.1};\n");
fputs (fid,"Line (1) = {5, 6};\n");
fputs (fid,"Line (2) = {2, 3};\n");
fputs (fid,"Line(3) = {6,1,2};\n");
fputs (fid,"Line(4) = {5,4,3};\n");
fputs (fid,"Line Loop(7) = {3,2,-4,1};\n");
fputs (fid,"Plane Surface(8) = {7};\n");
fclose (fid);
msho = msh2m_gmsh (canonicalize_file_name (name)(1:end-4),...
"scale", 1,"clscale", .2);
unlink (canonicalize_file_name (name));
mesh = Mesh (msho);
# Define function spaces (P2-P1). From ufl file
# V = VectorElement("CG", triangle, 2)
# Q = FiniteElement("CG", triangle, 1)
V = FunctionSpace ('VelocityUpdate', mesh);
Q = FunctionSpace ('PressureUpdate', mesh);
# Define trial and test functions. From ufl file
# u = TrialFunction(V)
# p = TrialFunction(Q)
# v = TestFunction(V)
# q = TestFunction(Q)
# Set parameter values. From ufl file
# nu = 0.01
dt = 0.01;
T = 3.;
# Define boundary conditions
noslip = DirichletBC (V, @(x,y) [0; 0], [3, 4]);
outflow = DirichletBC (Q, @(x,y) 0, 2);
# Create functions
u0 = Expression ('u0', @(x,y) [0; 0]);
# Define coefficients
k = Constant ('k', dt);
f = Constant ('f', [0; 0]);
# Tentative velocity step. From ufl file
# eq = (1/k)*inner(u - u0, v)*dx + inner(grad(u0)*u0, v)*dx \
# + nu*inner(grad(u), grad(v))*dx - inner(f, v)*dx
a1 = BilinearForm ('TentativeVelocity', V, k);
# Pressure update. From ufl file
# a = inner(grad(p), grad(q))*dx
# L = -(1/k)*div(u1)*q*dx
a2 = BilinearForm ('PressureUpdate', Q);
# Velocity update
# a = inner(u, v)*dx
# L = inner(u1, v)*dx - k*inner(grad(p1), v)*dx
a3 = BilinearForm ('VelocityUpdate', V);
# Assemble matrices
A1 = assemble (a1, noslip);
A3 = assemble (a3, noslip);
# Time-stepping
t = dt; i = 0;
while t < T
# Update pressure boundary condition
inflow = DirichletBC (Q, @(x,y) sin(3.0*t), 1);
# Compute tentative velocity step
"Computing tentative velocity"
L1 = LinearForm ('TentativeVelocity', V, k, u0, f);
b1 = assemble (L1, noslip);
utmp = A1 \ b1;
u1 = Function ('u1', V, utmp);
# Pressure correction
"Computing pressure correction"
L2 = LinearForm ('PressureUpdate', Q, u1, k);
[A2, b2] = assemble_system (a2, L2, inflow, outflow);
ptmp = A2 \ b2;
p1 = Function ('p1', Q, ptmp);
# Velocity correction
"Computing velocity correction"
L3 = LinearForm ('VelocityUpdate', V, k, u1, p1);
b3 = assemble (L3, noslip);
ut = A3 \ b3;
u1 = Function ('u0', V, ut);
# Plot solution
plot (p1);
plot (u1);
# Save to file
save (p1, sprintf ("p_%3.3d", ++i));
save (u1, sprintf ("u_%3.3d", i));
# Move to next time step
u0 = u1;
t += dt
end
|
| Code: Define NS problem with fenics python |
from dolfin import *
# Load mesh from file
mesh = Mesh("lshape.xml.gz")
# Define function spaces (P2-P1)
V = VectorFunctionSpace(mesh, "CG", 2)
Q = FunctionSpace(mesh, "CG", 1)
# Define trial and test functions
u = TrialFunction(V)
p = TrialFunction(Q)
v = TestFunction(V)
q = TestFunction(Q)
# Set parameter values
dt = 0.01
T = 3
nu = 0.01
# Define time-dependent pressure boundary condition
p_in = Expression("sin(3.0*t)", t=0.0)
# Define boundary conditions
noslip = DirichletBC(V, (0, 0),
"on_boundary && \
(x[0] < DOLFIN_EPS | x[1] < DOLFIN_EPS | \
(x[0] > 0.5 - DOLFIN_EPS && x[1] > 0.5 - DOLFIN_EPS))")
inflow = DirichletBC(Q, p_in, "x[1] > 1.0 - DOLFIN_EPS")
outflow = DirichletBC(Q, 0, "x[0] > 1.0 - DOLFIN_EPS")
bcu = [noslip]
bcp = [inflow, outflow]
# Create functions
u0 = Function(V)
u1 = Function(V)
p1 = Function(Q)
# Define coefficients
k = Constant(dt)
f = Constant((0, 0))
# Tentative velocity step
F1 = (1/k)*inner(u - u0, v)*dx + inner(grad(u0)*u0, v)*dx \
+ nu*inner(grad(u), grad(v))*dx - inner(f, v)*dx
a1 = lhs(F1)
L1 = rhs(F1)
# Pressure update
a2 = inner(grad(p), grad(q))*dx
L2 = -(1/k)*div(u1)*q*dx
# Velocity update
a3 = inner(u, v)*dx
L3 = inner(u1, v)*dx - k*inner(grad(p1), v)*dx
# Assemble matrices
A1 = assemble(a1)
A2 = assemble(a2)
A3 = assemble(a3)
# Use amg preconditioner if available
prec = "amg" if has_krylov_solver_preconditioner("amg")
else "default"
# Create files for storing solution
ufile = File("results/velocity.pvd")
pfile = File("results/pressure.pvd")
# Time-stepping
t = dt
while t < T + DOLFIN_EPS:
# Update pressure boundary condition
p_in.t = t
# Compute tentative velocity step
begin("Computing tentative velocity")
b1 = assemble(L1)
[bc.apply(A1, b1) for bc in bcu]
solve(A1, u1.vector(), b1, "gmres", "default")
end()
# Pressure correction
begin("Computing pressure correction")
b2 = assemble(L2)
[bc.apply(A2, b2) for bc in bcp]
solve(A2, p1.vector(), b2, "gmres", prec)
end()
# Velocity correction
begin("Computing velocity correction")
b3 = assemble(L3)
[bc.apply(A3, b3) for bc in bcu]
solve(A3, u1.vector(), b3, "gmres", "default")
end()
# Plot solution
plot(p1, title="Pressure", rescale=True)
plot(u1, title="Velocity", rescale=True)
# Save to file
ufile << u1
pfile << p1
# Move to next time step
u0.assign(u1)
t += dt
print "t =", t
# Hold plot
interactive()
© Copyright 2011, The FEniCS Project
|