Fem-fenics

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Package for solving Partial Differential Equations based on Fenics.

Introduction

Fem-Fenics Octave is a package for solving partial differential equations. Obviously, Fem-fenics is not the only extra package for Octave with this purpose. For example, Bim_package uses finite volumes to solve diffusion-advection-reaction equations, while secs1d/2d/3d [1] are suited for the resolution of the drift-diffusion system. Furthermore, to use profitably the software, you can integrate it with msh [2] for the generation of the mesh and with fpl [3] for the post-processing of data. The objective of Fem-fenics is to be a generic library of finite elements, thereby allowing the user to resolve any type of pde, choosing also the most appropriate Finite Element space for any specific problem.

As the name suggests, the Fem-fenics pkg is a wrapper for FEniCS [4] functions and classes. Thus, ideally the Fem-fenics final goal would be to be able to reproduce all the features available in FEniCS, simplifying them where it is possible or using the Octave function whenever available (like the "\" for the resolution of a linear system or the odepkg [5] for the resolution of a time dependent problem).

Tutorials

A generic problem has to be solved in two steps:

  • a file where the abstract problem is described: this file has to be written in Unified Form Language (UFL), which is a domain specific language for defining discrete variational forms and functionals in a notation close to pen-and-paper formulation. UFL is easy to learn, and in any case the User manual provides explanations and examples. [6]
  • a script file where the abstract problem is imported and a specific problem is implemented and solved: this is the script file where the fem-fenics functions are needed. Their sintax is as close as possible to the python interface, so that Fenics users are comfortable with that, but it is alos quite intuitive for beginners. The examples below show the equivalence between the different programming languages.


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

HyperElasticity.png

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


Incompressible Navier-Stokes equations

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