cell_problem

cell_problem ¶

Helper problems that are used to solve the cell problem in the HMM.

Classes:

Name	Description
`PeriodicLinearProblem`	linear problem class that automatically adds periodic boundary conditions on boxes.

hommx.cell_problem.PeriodicLinearProblem ¶

Bases: dolfinx_mpc.LinearProblem

Class for solving a linear variational problem with periodic boundary conditions with multi point constraints of the form \(a(u, v) = L(v)\) for all v using PETSc as a linear algebra backend.

The solution is only unique up to a constant. This is handled by telling the PETSc KSP solver about the nullspace.

Parameters:

Name	Description	Default
`a`	A bilinear UFL form, the left hand side of the variational problem.	required
`L`	A linear UFL form, the right hand side of the variational problem.	required
`mpc`	The multi point constraint.	required
`bcs`	A list of Dirichlet boundary conditions.	required
`u`	The solution function. It will be created if not provided. The function has to be based on the functionspace in the mpc, i.e. .. highlight:: python .. code-block:: python `u = dolfinx.fem.Function(mpc.function_space)`	required
`petsc_options`	Parameters that is passed to the linear algebra backend PETSc. #type: ignore For available choices for the 'petsc_options' kwarg, see the PETSc-documentation https://www.mcs.anl.gov/petsc/documentation/index.html.	required
`form_compiler_options`	Parameters used in FFCx compilation of this form. Run `ffcx --help` at the commandline to see all available options. Takes priority over all other parameter values, except for `scalar_type` which is determined by DOLFINx.	required
`jit_options`	Parameters used in CFFI JIT compilation of C code generated by FFCx. See https://github.com/FEniCS/dolfinx/blob/main/python/dolfinx/jit.py#L22-L37 for all available parameters. Takes priority over all other parameter values.	required

Examples: Example usage:

.. highlight:: python
.. code-block:: python

   problem = LinearProblem(
       a, L, mpc, [bc0, bc1], petsc_options={"ksp_type": "preonly", "pc_type": "lu"}
   )

solve ¶

solve() -> fem.Function

Solve the problem.

hommx.cell_problem.create_periodic_boundary_conditions ¶

create_periodic_boundary_conditions(function_space: fem.FunctionSpace, bcs: list[fem.DirichletBC] | None = None) -> dolfinx_mpc.MultiPointConstraint

Creates periodic boundary condition on the unit square or unit cube. For implementation details see _create_periodic_boundary_conditions_2d and _create_periodic_boundary_conditions_3d

hommx.cell_problem._create_periodic_boundary_conditions_2d ¶

_create_periodic_boundary_conditions_2d(msh: mesh.Mesh, function_space: fem.FunctionSpace, bcs: list[fem.DirichletBC] | None = None) -> dolfinx_mpc.MultiPointConstraint

Creates periodic boundary condition on the unit square using dolfinx_mpc.

This is done using create_periodic_constraints. We do this by forcing \(u(x, y_1) = u(x, y_0)\) and \(u(x_1, y) = u(x_0, y)\), on the unit square we would have \((x_0, x_1) = (y_0, y_1) = (0, 1)\) Internally we create slave DoFs at \(u(x, y_1)\) and \(u(x_1, y)\). Some attention need to be paid because the node at \((x_1,y_1)\) is constrained twice. We use a workaround adapted from: dolfinx_mpc

Parameters:

Name	Type	Description	Default
`msh`	`mesh.Mesh`	Mesh that the boundary conditions should be applied to	required
`function_space`	`fem.FunctionSpace`	Function Space on whicht the bc should be applied	required
`bcs`	`list[fem.DirichletBC] \| None`	Dirichlet Boundary Conditions, here the constraint is ignored	`None`

hommx.cell_problem._create_periodic_boundary_conditions_3d ¶

_create_periodic_boundary_conditions_3d(msh: mesh.Mesh, function_space: fem.FunctionSpace, bcs: list[fem.DirichletBC] | None = None) -> dolfinx_mpc.MultiPointConstraint

Creates periodic boundary condition on the unit cube using dolfinx_mpc.

This is done using create_periodic_constraints. We do this by forcing \(u(x, y_1, z) = u(x, y_0, z)\) and so on for all directions on the unit square we would have \((z_0, z_1) = (x_0, x_1) = (y_0, y_1) = (0, 1)\) Internally we create slave DoFs at \(u(x, y_1, z)\) and \(u(x_1, y, z)\) etc. Some attention need to be paid because the nodes at \((x_1, y_1, z)\) are constrained twice etc for other edges And \((x_1, y_1, z_1)\) is constrained three times. We use a workaround adapted from: dolfinx_mpc

Parameters:

Name	Type	Description	Default
`msh`	`mesh.Mesh`	Mesh that the boundary conditions should be applied to	required
`function_space`	`fem.FunctionSpace`	Function Space on which the bc should be applied	required
`bcs`	`list[fem.DirichletBC] \| None`	Dirichlet Boundary Conditions, here the constraint is ignored	`None`