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== Tutorials == | == 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. [http://fenicsproject.org/documentation/ufl/1.2.0/user/user_manual.html#ufl-user-manual] | ||
* | * a script file ('''.m''') 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 used. 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. | ||
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A complete description of the problem is avilable on the [http://fenicsproject.org/documentation/dolfin/1.2.0/python/demo/pde/poisson/python/documentation.html Fenics website.] | A complete description of the problem is avilable on the [http://fenicsproject.org/documentation/dolfin/1.2.0/python/demo/pde/poisson/python/documentation.html Fenics website.] | ||
<div style="width: 100%;"> | <div style="width: 100%;"> | ||
<div style="float:left; width: 48%"> | <div style="float:left; width: 48%"> | ||
{{Code|Define Poisson problem with fem-fenics|<syntaxhighlight lang="octave" style="font-size:13px"> | {{Code|Define Poisson problem with fem-fenics|<syntaxhighlight lang="octave" style="font-size:13px"> | ||
pkg load fem-fenics msh | pkg load fem-fenics msh | ||
import_ufl_Problem ('Poisson') | |||
# Create mesh and define function space | # Create mesh and define function space | ||
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mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4)); | mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4)); | ||
# File Poisson.ufl | |||
# element = FiniteElement("Lagrange", triangle, 1) | |||
V = FunctionSpace('Poisson', mesh); | V = FunctionSpace('Poisson', mesh); | ||
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# Define variational problem | # Define variational problem | ||
# | # File Poisson.ufl | ||
# u = TrialFunction(element) | # u = TrialFunction(element) | ||
# v = TestFunction(element) | # v = TestFunction(element) | ||
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g = Expression ('g', @(x,y) sin (5.0 * x)); | g = Expression ('g', @(x,y) sin (5.0 * x)); | ||
# | # File Poisson.ufl | ||
# a = inner(grad(u), grad(v))*dx | # a = inner(grad(u), grad(v))*dx | ||
# L = f*v*dx + g*v*ds | # L = f*v*dx + g*v*ds | ||
a = BilinearForm ('Poisson' | a = BilinearForm ('Poisson', V); | ||
L = LinearForm ('Poisson', V, f, g); | L = LinearForm ('Poisson', V, f, g); | ||
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# Create mesh and define function space | |||
mesh = UnitSquareMesh(32, 32) | |||
V = FunctionSpace(mesh, "Lagrange", 1) | V = FunctionSpace(mesh, "Lagrange", 1) | ||
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=== Mixed Formulation for Poisson Equation === | === 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 | 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 Dicontinuos element of order 0 is used. | ||
<math>-\mathrm{div}\ ( \mathbf{\sigma} (x, y) ) ) = f (x, y) \qquad \mbox{ in } \Omega</math> | <math>-\mathrm{div}\ ( \mathbf{\sigma} (x, y) ) ) = f (x, y) \qquad \mbox{ in } \Omega</math> | ||
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<math>(\sigma (x, y) ) \cdot \mathbf{n} = \sin (5x) \qquad \mbox{ on } \Gamma_N</math> | <math>(\sigma (x, y) ) \cdot \mathbf{n} = \sin (5x) \qquad \mbox{ on } \Gamma_N</math> | ||
A complete description of the problem is | A complete description of the problem is avilable on the [http://fenicsproject.org/documentation/dolfin/1.2.0/python/demo/pde/mixed-poisson/python/documentation.html Fenics website.] | ||
<div style="width: 100%;"> | <div style="width: 100%;"> | ||
<div style="float:left; width: 48%"> | <div style="float:left; width: 48%"> | ||
{{Code|Define MixedPoisson problem with fem-fenics|<syntaxhighlight lang="octave" style="font-size:13px"> | {{Code|Define MixedPoisson problem with fem-fenics|<syntaxhighlight lang="octave" style="font-size:13px"> | ||
pkg load fem-fenics msh | pkg load fem-fenics msh | ||
import_ufl_Problem ('MixedPoisson') | |||
# Create mesh | # Create mesh | ||
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mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4)); | mesh = Mesh(msh2m_structured_mesh (x, y, 1, 1:4)); | ||
# ufl | # File MixedPoisson.ufl | ||
# BDM = FiniteElement("BDM", triangle, 1) | # BDM = FiniteElement("BDM", triangle, 1) | ||
# DG = FiniteElement("DG", triangle, 0) | # DG = FiniteElement("DG", triangle, 0) | ||
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# Define trial and test function | # Define trial and test function | ||
# ufl | # File MixedPoisson.ufl | ||
# (sigma, u) = TrialFunctions(W) | # (sigma, u) = TrialFunctions(W) | ||
# (tau, v) = TestFunctions(W) | # (tau, v) = TestFunctions(W) | ||
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# Define variational form | # Define variational form | ||
# ufl | # File MixedPoisson.ufl | ||
# a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx | # a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx | ||
# L = - f*v*dx | # L = - f*v*dx | ||
a = BilinearForm ('MixedPoisson' | a = BilinearForm ('MixedPoisson', V); | ||
L = LinearForm ('MixedPoisson', V, f); | L = LinearForm ('MixedPoisson', V, f); | ||
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{{Code|Define MixedPoisson problem with fenics python|<syntaxhighlight lang="python" style="font-size:13px"> | {{Code|Define MixedPoisson problem with fenics python|<syntaxhighlight lang="python" style="font-size:13px"> | ||
from dolfin import * | from dolfin import * | ||
# Create mesh | # Create mesh | ||
mesh = UnitSquareMesh(32, 32) | mesh = UnitSquareMesh(32, 32) | ||
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(sigma, u) = TrialFunctions(W) | (sigma, u) = TrialFunctions(W) | ||
(tau, v) = TestFunctions(W) | (tau, v) = TestFunctions(W) | ||
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=== Hyperelasticity === | === Hyperelasticity === | ||
This time we compare the code with the | 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 | ||
<math> \min_{u \in V} \Pi</math> | <math> \min_{u \in V} \Pi</math> | ||
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A complete description of the problem is avilable on the [http://fenicsproject.org/documentation/dolfin/1.2.0/cpp/demo/pde/hyperelasticity/cpp/documentation.html Fenics website.] The final solution will look like this | A complete description of the problem is avilable on the [http://fenicsproject.org/documentation/dolfin/1.2.0/cpp/demo/pde/hyperelasticity/cpp/documentation.html Fenics website.] The final solution will look like this | ||
[[File:HyperElasticity.png| | |||
[[File:HyperElasticity.png|Alignment = center]] | |||
{{Code|HyperElasticity Problem: the ufl file|<syntaxhighlight lang="octave" style="font-size:13px"> | {{Code|HyperElasticity Problem: the ufl file|<syntaxhighlight lang="octave" style="font-size:13px"> | ||
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{{Code|Define HyperElasticity problem with fem-fenics|<syntaxhighlight lang="octave" style="font-size:13px"> | {{Code|Define HyperElasticity problem with fem-fenics|<syntaxhighlight lang="octave" style="font-size:13px"> | ||
pkg load fem-fenics msh | pkg load fem-fenics msh | ||
problem = 'HyperElasticity'; | |||
import_ufl_Problem (problem); | |||
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using namespace dolfin; | using namespace dolfin; | ||
// Sub domain for clamp at left end | // Sub domain for clamp at left end | ||
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// Solve nonlinear variational problem F(u; v) = 0 | // Solve nonlinear variational problem F(u; v) = 0 | ||
solve(F == 0, u, bcs, J); | solve(F == 0, u, bcs, J); | ||
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=== Incompressible Navier-Stokes equations === | === Incompressible Navier-Stokes equations === | ||
A complete description of the problem is avilable on the [http://fenicsproject.org/documentation/dolfin/1.2.0/python/demo/pde/navier-stokes/python/documentation.html Fenics website.] | |||
A complete description of the | |||
<div style="width: 100%;"> | <div style="width: 100%;"> | ||
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# eq = (1/k)*inner(u - u0, v)*dx + inner(grad(u0)*u0, v)*dx \ | # 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 | # + nu*inner(grad(u), grad(v))*dx - inner(f, v)*dx | ||
a1 = BilinearForm ('TentativeVelocity' | a1 = BilinearForm ('TentativeVelocity', V, k); | ||
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# a = inner(grad(p), grad(q))*dx | # a = inner(grad(p), grad(q))*dx | ||
# L = -(1/k)*div(u1)*q*dx | # L = -(1/k)*div(u1)*q*dx | ||
a2 = BilinearForm ('PressureUpdate' | a2 = BilinearForm ('PressureUpdate', Q); | ||
# Velocity update | # Velocity update | ||
# a = inner(u, v)*dx | # a = inner(u, v)*dx | ||
# L = inner(u1, v)*dx - k*inner(grad(p1), v)*dx | # L = inner(u1, v)*dx - k*inner(grad(p1), v)*dx | ||
a3 = BilinearForm ('VelocityUpdate' | a3 = BilinearForm ('VelocityUpdate', V); | ||
# Assemble matrices | # Assemble matrices | ||
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</div> | </div> | ||
<div style="clear:both"></div> | <div style="clear:both"></div> | ||
== Relevant Implementation Details == | == Relevant Implementation Details == | ||
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The relevant implementation details which the user should know are: | 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 ''Expession'' 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. | |||
* | |||
== Additional functionality / TODOS == | == Additional functionality / TODOS == | ||
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* '''Norma'''l: add the possibility to use a reserved keyword (normal ?) to be used with the DirichletBC. <syntaxhighlight lang="octave" style="font-size:13px"> bc = DirichletBC (V, @(x, y, normal) [sin(x)*normal; 0], [3, 4]);</syntaxhighlight> | * '''Norma'''l: add the possibility to use a reserved keyword (normal ?) to be used with the DirichletBC. <syntaxhighlight lang="octave" style="font-size:13px"> bc = DirichletBC (V, @(x, y, normal) [sin(x)*normal; 0], [3, 4]);</syntaxhighlight> | ||
* | * @function/'''feval''': the function should accept as input also an array of values. Show how it can be used in an example with odepkg. | ||
[[Category: | [[Category:OctaveForge]] | ||
[[Category:Packages]] |