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# Biharmonic equation

This demo is implemented in a single Python file,
{download}`demo_biharmonic.py`, which contains both the variational forms
and the solver. It illustrates how to:

- Solve a linear partial differential equation
- Use a discontinuous Galerkin method
- Solve a fourth-order differential equation

## Equation and problem definition

### Strong formulation

The biharmonic equation is a fourth-order elliptic equation.
On the domain $\Omega \subset \mathbb{R}^{d}$, $1 \le d \le 3$, it reads

$$
\nabla^{4} u = f \quad {\rm in} \ \Omega,
$$

where $\nabla^{4} \equiv \nabla^{2} \nabla^{2}$ is the biharmonic operator
and $f$ is a prescribed source term.
To formulate a complete boundary value problem, the biharmonic equation
must be complemented by suitable boundary conditions.

### Weak formulation

Multiplying the biharmonic equation by a test function and integrating
by parts twice leads to a problem of second-order derivatives, which would
require $H^{2}$ conforming (roughly $C^{1}$ continuous) basis functions.
To solve the biharmonic equation using Lagrange finite element basis
functions, the biharmonic equation can be split into two second-order
equations (see the Mixed Poisson demo for a mixed method for the Poisson
equation), or a variational formulation can be constructed that imposes
weak continuity of normal derivatives between finite element cells.
This demo uses a discontinuous Galerkin approach to impose continuity
of the normal derivative weakly.

Consider a triangulation $\mathcal{T}$ of the domain $\Omega$, where
the set of interior facets is denoted by $\mathcal{E}_h^{\rm int}$.
Functions evaluated on opposite sides of a facet are indicated by the
subscripts $+$ and $-$.
Using the standard continuous Lagrange finite element space

$$
V = \left\{v \in H^{1}_{0}(\Omega)\,:\, v \in P_{k}(K) \
\forall \ K \in \mathcal{T} \right\}
$$

and considering the boundary conditions

$$
u &= 0 \quad {\rm on} \ \partial\Omega, \\
\nabla^{2} u &= 0 \quad {\rm on} \ \partial\Omega,
$$

a weak formulation of the biharmonic problem reads: find $u \in V$ such that

$$
a(u,v)=L(v) \quad \forall \ v \in V,
$$

where the bilinear form is

$$
a(u, v) =
\sum_{K \in \mathcal{T}} \int_{K} \nabla^{2} u \nabla^{2} v \, {\rm d}x \
+\sum_{E \in \mathcal{E}_h^{\rm int}}\left(\int_{E} \frac{\alpha}{h_E}
[\!\![ \nabla u ]\!\!] [\!\![ \nabla v ]\!\!] \, {\rm d}s
- \int_{E} \left<\nabla^{2} u \right>[\!\![ \nabla v ]\!\!]  \, {\rm d}s
- \int_{E} [\!\![ \nabla u ]\!\!] \left<\nabla^{2} v \right> \,
{\rm d}s\right)
$$

and the linear form is

$$
L(v) = \int_{\Omega} fv \, {\rm d}x.
$$

Furthermore, $\left< u \right> = \frac{1}{2} (u_{+} + u_{-})$,
$[\!\![ w ]\!\!]  = w_{+} \cdot n_{+} + w_{-} \cdot n_{-}$,
$\alpha \ge 0$ is a penalty parameter and
$h_E$ is a measure of the cell size.

The input parameters for this demo are defined as follows:

- $\Omega = [0,1] \times [0,1]$ (a unit square)
- $\alpha = 8.0$ (penalty parameter)
- $f = 4.0 \pi^4\sin(\pi x)\sin(\pi y)$ (source term)

## Implementation

We first import the modules and functions that the program uses:

```python
import numpy as np

import ufl
from dolfinx import fem, io, mesh, plot
from dolfinx.fem.petsc import LinearProblem
from dolfinx.mesh import CellType, GhostMode
from ufl import (CellDiameter, FacetNormal, avg, div, dS, dx, grad, inner,
                 jump, pi, sin)

from mpi4py import MPI
from petsc4py.PETSc import ScalarType  # type: ignore
```

We begin by using {py:func}`create_rectangle
<dolfinx.mesh.create_rectangle>` to create a rectangular
{py:class}`Mesh <dolfinx.mesh.Mesh>` of the domain, and creating a
finite element {py:class}`FunctionSpaceBase <dolfinx.fem.FunctionSpaceBase>`
$V$ on the mesh.

```python
msh = mesh.create_rectangle(comm=MPI.COMM_WORLD,
                            points=((0.0, 0.0), (1.0, 1.0)), n=(32, 32),
                            cell_type=CellType.triangle,
                            ghost_mode=GhostMode.shared_facet)
V = fem.functionspace(msh, ("Lagrange", 2))
```

The second argument to {py:func}`functionspace
<dolfinx.fem.functionspace>` is a tuple consisting of `(family,
degree)`, where `family` is the finite element family, and `degree`
specifies the polynomial degree. in this case `V` consists of
second-order, continuous Lagrange finite element functions.

Next, we locate the mesh facets that lie on the boundary
$\Gamma_D = \partial\Omega$.
We do this using using {py:func}`locate_entities_boundary
<dolfinx.mesh.locate_entities_boundary>` and providing  a marker
function that returns `True` for points `x` on the boundary and
`False` otherwise.

```python
facets = mesh.locate_entities_boundary(msh, dim=1,
                                       marker=lambda x: np.logical_or.reduce((
                                           np.isclose(x[0], 0.0),
                                           np.isclose(x[0], 1.0),
                                           np.isclose(x[1], 0.0),
                                           np.isclose(x[1], 1.0))))
```

We now find the degrees-of-freedom that are associated with the
boundary facets using {py:func}`locate_dofs_topological
<dolfinx.fem.locate_dofs_topological>`

```python
dofs = fem.locate_dofs_topological(V=V, entity_dim=1, entities=facets)
```

and use {py:func}`dirichletbc <dolfinx.fem.dirichletbc>` to create a
{py:class}`DirichletBC <dolfinx.fem.DirichletBC>`
class that represents the boundary condition. In this case, we impose
Dirichlet boundary conditions with value $0$ on the entire boundary
$\partial\Omega$.

```python
bc = fem.dirichletbc(value=ScalarType(0), dofs=dofs, V=V)
```

Next, we express the variational problem using UFL.

First, the penalty parameter $\alpha$ is defined. In addition, we define a
variable `h` for the cell diameter $h_E$, a variable `n`for the
outward-facing normal vector $n$ and a variable `h_avg` for the
average size of cells sharing a facet
$\left< h \right> = \frac{1}{2} (h_{+} + h_{-})$. Here, the UFL syntax
`('+')` and `('-')` restricts a function to the `('+')` and `('-')`
sides of a facet.

```python
alpha = ScalarType(8.0)
h = CellDiameter(msh)
n = FacetNormal(msh)
h_avg = (h('+') + h('-')) / 2.0
```

After that, we can define the variational problem consisting of the bilinear
form $a$ and the linear form $L$. The source term is prescribed as
$f = 4.0 \pi^4\sin(\pi x)\sin(\pi y)$. Note that with `dS`, integration is
carried out over all the interior facets $\mathcal{E}_h^{\rm int}$, whereas
with `ds` it would be only the facets on the boundary of the domain, i.e.
$\partial\Omega$. The jump operator
$[\!\![ w ]\!\!] = w_{+} \cdot n_{+} + w_{-} \cdot n_{-}$ w.r.t. the
outward-facing normal vector $n$ is in UFL available as `jump(w, n)`.

```python
# Define variational problem
u = ufl.TrialFunction(V)
v = ufl.TestFunction(V)
x = ufl.SpatialCoordinate(msh)
f = 4.0 * pi**4 * sin(pi * x[0]) * sin(pi * x[1])

a = inner(div(grad(u)), div(grad(v))) * dx \
    - inner(avg(div(grad(u))), jump(grad(v), n)) * dS \
    - inner(jump(grad(u), n), avg(div(grad(v)))) * dS \
    + alpha / h_avg * inner(jump(grad(u), n), jump(grad(v), n)) * dS
L = inner(f, v) * dx
```

We create a {py:class}`LinearProblem <dolfinx.fem.petsc.LinearProblem>`
object that brings together the variational problem, the Dirichlet
boundary condition, and which specifies the linear solver. In this
case we use a direct (LU) solver. The {py:func}`solve
<dolfinx.fem.petsc.LinearProblem.solve>` will compute a solution.

```python
problem = LinearProblem(a, L, bcs=[bc], petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
uh = problem.solve()
```

The solution can be written to a  {py:class}`XDMFFile
<dolfinx.io.XDMFFile>` file visualization with ParaView or VisIt

```python
with io.XDMFFile(msh.comm, "out_biharmonic/biharmonic.xdmf", "w") as file:
    V1 = fem.functionspace(msh, ("Lagrange", 1))
    u1 = fem.Function(V1)
    u1.interpolate(uh)
    file.write_mesh(msh)
    file.write_function(u1)
```

and displayed using [pyvista](https://docs.pyvista.org/).

```python
try:
    import pyvista
    cells, types, x = plot.vtk_mesh(V)
    grid = pyvista.UnstructuredGrid(cells, types, x)
    grid.point_data["u"] = uh.x.array.real
    grid.set_active_scalars("u")
    plotter = pyvista.Plotter()
    plotter.add_mesh(grid, show_edges=True)
    warped = grid.warp_by_scalar()
    plotter.add_mesh(warped)
    if pyvista.OFF_SCREEN:
        pyvista.start_xvfb(wait=0.1)
        plotter.screenshot("uh_biharmonic.png")
    else:
        plotter.show()
except ModuleNotFoundError:
    print("'pyvista' is required to visualise the solution")
    print("Install 'pyvista' with pip: 'python3 -m pip install pyvista'")
```
