Fem-fenics

Package for solving Partial Differential Equations based on Fenics.

Introduction

Fem-Fenics 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

The solution of a problem can be logically divided in two steps. According to convenience or personal preference, they can be addressed with different files or just in one Octave script:

• the description of the abstract problem: this should be done via the 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] As mentioned before, the problem can be defined in a separate .ufl file or handled directly in an m-file using ufl blocks.
• the implementation of a specific problem, an instance of the abstract one: this is done in a script file (.m) where the fem-fenics functions are used and the problem is solved. Their syntax is as close as possible to the python interface, so that Fenics users should be comfortable with it, but it is also 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

$-\mathrm {div} \ (\nabla u(x,y)))=1\qquad {\mbox{ in }}\Omega$

$u(x,y)=0\qquad {\mbox{ on }}\Gamma _{D}$

$(\nabla u(x,y))\cdot \mathbf {n} =0\qquad {\mbox{ on }}\Gamma _{N}$

A complete description of the problem is avilable on the Fenics website.

Code: Define Poisson problem with fem-fenics

pkg load fem-fenics msh ufl start Poisson ufl element = FiniteElement("Lagrange", triangle, 1) ufl ufl u = TrialFunction(element) ufl v = TestFunction(element) ufl f = Coefficient(element) ufl g = Coefficient(element) ufl ufl a = inner(grad(u), grad(v))*dx ufl L = f*v*dx + g*v*ds ufl end # Create mesh and define function space x = y = linspace (0, 1, 33); mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4)); V = FunctionSpace('Poisson', mesh); # Define boundary condition bc = DirichletBC(V, @(x, y) 0.0, [2;4]); # Define variational problem # Operation performed in the ufl snippet above # 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)); # As in the ufl snippet above # a = inner(grad(u), grad(v))*dx # L = f*v*dx + g*v*ds a = BilinearForm ('Poisson', V, 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

In this example the Poisson equation is solved with a mixed approach: the stable FE space obtained using Brezzi-Douglas-Marini polynomial of order 1 and Discontinuos elements of order 0 is used.

$-\mathrm {div} \ (\mathbf {\sigma } (x,y)))=f(x,y)\qquad {\mbox{ in }}\Omega$

$\sigma (x,y)=\nabla u(x,y)\qquad {\mbox{ in }}\Omega$

$u(x,y)=0\qquad {\mbox{ on }}\Gamma _{D}$

$(\sigma (x,y))\cdot \mathbf {n} =\sin(5x)\qquad {\mbox{ on }}\Gamma _{N}$

A complete description of the problem is available on the Fenics website.

Code: Define MixedPoisson problem with fem-fenics

pkg load fem-fenics msh ufl start MixedPoisson ufl ufl BDM = FiniteElement("BDM", triangle, 1) ufl DG = FiniteElement("DG", triangle, 0) ufl W = BDM * DG ufl ufl "(sigma, u)" = TrialFunctions(W) ufl "(tau, v)" = TestFunctions(W) ufl ufl CG = FiniteElement("CG", triangle, 1) ufl f = Coefficient(CG) ufl ufl a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx ufl L = - f*v*dx ufl end # Create mesh x = y = linspace (0, 1, 33); mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4)); # ufl snippet above # BDM = FiniteElement("BDM", triangle, 1) # DG = FiniteElement("DG", triangle, 0) # W = BDM * DG V = FunctionSpace('MixedPoisson', mesh); # Define trial and test function # ufl snippet above # (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 # ufl snippet above # a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx # L = - f*v*dx a = BilinearForm ('MixedPoisson', V, 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. The problem for an elastic material can be expressed as a minimization problem

$\min _{u\in V}\Pi$

$\Pi =\int _{\Omega }\psi (u)\,{\rm {d}}x-\int _{\Omega }B\cdot u\,{\rm {d}}x-\int _{\partial \Omega }T\cdot u\,{\rm {d}}s$

where \Pi is the total potential energy, \psi is the elastic stored energy, \B is a body force and \T is a traction force.

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 ufl start HyperElasticity ufl # Function spaces ufl element = VectorElement("Lagrange", tetrahedron, 1) ufl ufl # Trial and test functions ufl du = TrialFunction(element) # Incremental displacement ufl v = TestFunction(element) # Test function ufl ufl # Functions ufl u = Coefficient(element) # Displacement from previous iteration ufl B = Coefficient(element) # Body force per unit volume ufl T = Coefficient(element) # Traction force on the boundary ufl ufl # Kinematics ufl I = Identity(element.cell().d) # Identity tensor ufl F = I + grad(u) # Deformation gradient ufl C = F.T*F # Right Cauchy-Green tensor ufl ufl # Invariants of deformation tensors ufl Ic = tr(C) ufl J = det(F) ufl ufl # Elasticity parameters ufl mu = Constant(tetrahedron) ufl lmbda = Constant(tetrahedron) ufl ufl # Stored strain energy density (compressible neo-Hookean model) ufl psi = (mu/2)*(Ic - 3) - mu*ln(J) + (lmbda/2)*(ln(J))**2 ufl ufl # Total potential energy ufl Pi = psi*dx - inner(B, u)*dx - inner(T, u)*ds ufl ufl # First variation of Pi (directional derivative about u in the direction of v) ufl F = derivative(Pi, u, v) ufl ufl # Compute Jacobian of F ufl J = derivative(F, u, du) ufl end problem = 'HyperElasticity'; 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

The incompressible Navier-Stokes equations are solved using the Chorin-Temam projection algorithm. [7]. A complete description of the specific 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, 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, Q); # Velocity update # a = inner(u, v)*dx # L = inner(u1, v)*dx - k*inner(grad(p1), v)*dx a3 = BilinearForm ('VelocityUpdate', V, 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

Obstacles in the Domain

$-\mathrm {div} (a\nabla u)=1\quad {\mbox{ in }}\Omega ,$

$u=5\quad {\mbox{ on }}\Gamma _{T},$

$u=0\quad {\mbox{ on }}\Gamma _{B},$

$\nabla u\cdot \mathbf {n} =-10e^{-(y-0.5)^{2}}\quad {\mbox{ on }}\Gamma _{L},$

$\nabla u\cdot \mathbf {n} =1\quad {\mbox{ on }}\Gamma _{R}$

The above example is a weighted Poisson problem on the unit square. The diffusion coefficient $a$ assumes value 0.01 on the obstacle $\Omega _{1}=[0.5,0.7]\times [0.2,1.0]$, whilst 1 outside on $\Omega _{0}=\Omega \setminus \Omega _{1}$. You can find a detailed explanation on the FEniCS website. On the side, you can see the solution.

Code: Define Poisson problem with obstacle with fem-fenics

pkg load fem-fenics msh # Create mesh x = y = linspace (0, 1, 65); [msh, facets] = Mesh (msh2m_structured_mesh (x, y, 0, 4:-1:1)); ufl start Subdomains ufl fe = FiniteElement "(""CG"", triangle, 2)" ufl u = TrialFunction (fe) ufl v = TestFunction (fe) ufl ufl a0 = Coefficient (fe) ufl a1 = Coefficient (fe) ufl g_L = Coefficient (fe) ufl g_R = Coefficient (fe) ufl f = Coefficient (fe) ufl ufl a= "inner(a0*grad(u),grad(v))*dx(0) + inner(a1*grad(u),grad(v))*dx(1)" ufl L = g_L*v*ds(1) + g_R*v*ds(3) + f*v*dx(0) + f*v*dx(1) ufl end V = FunctionSpace ("Subdomains", msh); # Define problem coefficients a0 = Constant ("a0", 1.0); a1 = Constant ("a1", 0.01); g_L = Expression ("g_L", @(x, y) - 10*exp(- (y - 0.5) ^ 2)); g_R = Constant ("g_R", 1.0); f = Constant ("f", 1.0); # Define subdomains domains = MeshFunction ("dx", msh, 2, 0); obstacle = SubDomain (@(x,y) (y >= 0.5) && (y <= 0.7) && ... (x >= 0.2) && (x <= 1.0), false); domains = mark (obstacle, domains, 1); # Define boundary conditions bc1 = DirichletBC (V, @(x, y) 5.0, facets, 2); bc2 = DirichletBC (V, @(x, y) 0.0, facets, 4); # Define variational form a = BilinearForm ("Subdomains", V, V, a0, a1, domains); L = LinearForm ("Subdomains", V, g_L, g_R, f, facets, domains); # Solve problem [A, b] = assemble_system (a, L, bc1, bc2); sol = A \ b; u = Function ("u", V, sol); # Plot solution [X, Y] = meshgrid (x, y); U = u (X, Y); surf (X, Y, U);

Code: Define Poisson problem with obstacle with FEniCS python

from dolfin import * # Define mesh mesh = UnitSquareMesh(64, 64) # Create classes for defining parts of the boundaries class Left(SubDomain): def inside(self, x, on_boundary): return near(x[0], 0.0) class Right(SubDomain): def inside(self, x, on_boundary): return near(x[0], 1.0) class Bottom(SubDomain): def inside(self, x, on_boundary): return near(x[1], 0.0) class Top(SubDomain): def inside(self, x, on_boundary): return near(x[1], 1.0) # Initialize sub-domain instances left = Left() top = Top() right = Right() bottom = Bottom() # Initialize mesh function for boundary domains boundaries = FacetFunction("size_t", mesh) boundaries.set_all(0) left.mark(boundaries, 1) top.mark(boundaries, 2) right.mark(boundaries, 3) bottom.mark(boundaries, 4) # Define function space and basis functions V = FunctionSpace(mesh, "CG", 2) u = TrialFunction(V) v = TestFunction(V) # Define input data a0 = Constant(1.0) a1 = Constant(0.01) g_L = Expression("- 10*exp(- pow(x[1] - 0.5, 2))") g_R = Constant("1.0") f = Constant(1.0) # Initialize mesh function for interior domains class Obstacle(SubDomain): def inside(self, x, on_boundary): return (between(x[1], (0.5, 0.7)) and between(x[0], (0.2, 1.0))) obstacle = Obstacle() domains = CellFunction("size_t", mesh) domains.set_all(0) obstacle.mark(domains, 1) # Define Dirichlet boundary conditions at top and bottom boundaries bcs = [DirichletBC(V, 5.0, boundaries, 2), DirichletBC(V, 0.0, boundaries, 4)] # Define new measures associated with the interior domains and # exterior boundaries dx = Measure("dx")[domains] ds = Measure("ds")[boundaries] # Define variational form F = (inner(a0*grad(u), grad(v))*dx(0) + inner(a1*grad(u), grad(v))*dx(1) - g_L*v*ds(1) - g_R*v*ds(3) - f*v*dx(0) - f*v*dx(1)) # Separate left and right hand sides of equation a, L = lhs(F), rhs(F) # Solve problem u = Function(V) solve(a == L, u, bcs) # Plot solution plot(u, title="u") interactive() © Copyright 2011, The FEniCS Project

Relevant Implementation Details

The relevant implementation details which the user should know are:

• All the objects are managed using boost::shared_ptr <>. It means that the same resource can be shared by more objects and useless copies should be avoided. For example, if we have two different functional spaces in the same problem, like with Navier-Stokes for the velocity and the pressure, the mesh is shared between them and no one has its own copy.

• The essential BC are imposed directly to the matrix with the command assemble(), which sets to zero all the off diagonal elements in the corresponding line, sets to 1 the diagonal element and sets to the exact value the rhs. This means that we could loose the symmetry of the matrix, if any. To avoid this problem and preserve the symmetry of the system it is available the assemble_system() command which builds at once the lhs and the rhs.

• The coefficient of the variational problem can be specified using either an Expression(), a Constant() or a Function(). They are different objects which behave in different ways: an Expression or a Constant object overloads the eval() method of the dolfin::Expression class and it is evaluated at run time using the octave function feval (). A Function instead doesn't need to be evaluated because it is assembled copying element-by-element the values contained in the input vector.

• Unfortunately the feature used in previous versions of the package to handle DirichletBC is not implemented yet in the FEniCS library for distributed execution. This means that prior to running code via MPI it should be adapted to use the newly introduced MeshFunction to mark border subsets, so that essential boundary conditions are correctly applied.

Known issues

• Fem-fenics needs both in the installation phase and in normal usage some includes that are not in standard system include directories, namely those related to MPI and the Eigen template library. If you experience an installation failure or errors when importing your UFL files, then you should properly set your CPPFLAGS environment variable before launching Octave, as per [8]. This can be done in sh compatible shells with the command below:

 export CPPFLAGS="-I/usr/include/eigen3 $(mpicxx -showme:compile)"

Notice that /usr/include/eigen3 has to be changed accordingly to the path in which Eigen are to be found on your system.
It should readily work in Ubuntu 13.10 and 14.04.

• There is a known bug with Openmpi, discussed on the maintainers list [9], that needs the preloading of a MPI shared library with the following command:

 export LD_PRELOAD=/usr/lib/openmpi/lib/libmpi.so

Again, notice that /usr/lib/openmpi/lib/libmpi.so has to be changed to fit to your system.

Additional functionality / TODOS

Obviously, Fem-fenics is not (yet) able to reproduce all the functionality available in Fenics. If there is any important features missing, please add it to the list below. (Or, better, you can directly submit your extension to the mercurial repository [10]).

• Normal: add the possibility to use a reserved keyword (normal ?) to be used with the DirichletBC.

   bc = DirichletBC (V, @(x, y, normal) [sin(x)*normal; 0], [3, 4]);
  

• @function/feval: the function should accept as input also an array of values. Show how it can be used in an example with odepkg.

External Links

• User manual [11].
• Repository [12]
• Functions reference [13]
• Presentation at MOX [14]