Biharmonic equation
This demo 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
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
and considering the boundary conditions
a weak formulation of the biharmonic problem reads: find \(u \in V\) such that
where the bilinear form is
and the linear form is
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
The implementation is in two files: a form file containing the definition of the variational forms expressed in UFL and a C++ file containing the actual solver.
Running this demo requires the files: demo_biharmonic/main.cpp,
demo_biharmonic/biharmonic.py and
demo_biharmonic/CMakeLists.txt.
UFL form file
The UFL file is implemented in demo_biharmonic/biharmonic.py.
UFL form implemented in python
The first step is to define the variational problem at hand. We define the variational problem in UFL terms in a separate form file. We begin by defining the finite element:
from basix.ufl import element
from ufl import (
CellDiameter,
Coefficient,
Constant,
FacetNormal,
FunctionSpace,
Mesh,
TestFunction,
TrialFunction,
avg,
div,
dS,
dx,
grad,
inner,
jump,
)
e = element("Lagrange", "triangle", 2)
The first argument to :py:class:FiniteElement is the finite element
family, the second argument specifies the domain, while the third
argument specifies the polynomial degree. Thus, in this case, our
element element consists of second-order, continuous Lagrange basis
functions on triangles (or in order words, continuous piecewise linear
polynomials on triangles).
Next, we use this element to initialize the trial and test functions (\(u\) and \(v\)) and the coefficient function \(f\):
coord_element = element("Lagrange", "triangle", 1, shape=(2,))
mesh = Mesh(coord_element)
V = FunctionSpace(mesh, e)
u = TrialFunction(V)
v = TestFunction(V)
f = Coefficient(V)
Next, the outward unit normal to cell boundaries and a measure of the
cell size are defined. The average size of cells sharing a facet will
be used (h_avg). The UFL syntax ('+') and ('-') restricts a
function to the ('+') and ('-') sides of a facet, respectively.
The penalty parameter alpha is made a :cpp:class:Constant so
that it can be changed in the program without regenerating the code.
# Normal component, mesh size and right-hand side
n = FacetNormal(mesh)
h = CellDiameter(mesh)
h_avg = (h("+") + h("-")) / 2
alpha = Constant(mesh)
Finally, we define the bilinear and linear forms according to the
variational formulation of the equations. Integrals over
internal facets are indicated by *dS.
# Bilinear form
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
)
# Linear form
L = inner(f, v) * dx
Note
TODO: explanation on how to run cmake and/or shell commands for ffcx.
To compile biharmonic.py using FFCx with an option
for PETSc scalar type float64 one would execute the command
ffcx biharmonic.py --scalar_type=float64
C++ program
The main solver is implemented in the demo_biharmonic/main.cpp
file.
At the top we include the DOLFINx header file and the generated header file “biharmonic.h” containing the variational forms for the Biharmonic equation, which are defined in the UFL form file. For convenience we also include the DOLFINx namespace.
#include "biharmonic.h"
#include <basix/finite-element.h>
#include <cmath>
#include <dolfinx.h>
#include <dolfinx/common/types.h>
#include <dolfinx/fem/Constant.h>
#include <dolfinx/fem/petsc.h>
#include <numbers>
#include <utility>
#include <vector>
using namespace dolfinx;
using T = PetscScalar;
using U = typename dolfinx::scalar_value_type_t<T>;
Inside the main function, we begin by defining a mesh of the
domain. As the unit square is a very standard domain, we can use a
built-in mesh provided by the UnitSquareMesh factory. In
order to create a mesh consisting of 32 x 32 squares with each square
divided into two triangles, and the finite element space (specified
in the form file) defined relative to this mesh, we do as follows
int main(int argc, char* argv[])
{
dolfinx::init_logging(argc, argv);
PetscInitialize(&argc, &argv, nullptr, nullptr);
{
// Create mesh
auto part = mesh::create_cell_partitioner(mesh::GhostMode::shared_facet);
auto mesh = std::make_shared<mesh::Mesh<U>>(
mesh::create_rectangle<U>(MPI_COMM_WORLD, {{{0.0, 0.0}, {1.0, 1.0}}},
{32, 32}, mesh::CellType::triangle, part));
// A function space object, which is defined in the generated code,
// is created:
auto element = basix::create_element<U>(
basix::element::family::P, basix::cell::type::triangle, 2,
basix::element::lagrange_variant::unset,
basix::element::dpc_variant::unset, false);
// Create function space
auto V = std::make_shared<fem::FunctionSpace<U>>(
fem::create_functionspace(mesh, element));
// The source function $f$ and the penalty term $\alpha$ are
// declared:
auto f = std::make_shared<fem::Function<T>>(V);
f->interpolate(
[](auto x) -> std::pair<std::vector<T>, std::vector<std::size_t>>
{
std::vector<T> f;
for (std::size_t p = 0; p < x.extent(1); ++p)
{
auto pi = std::numbers::pi;
f.push_back(4.0 * std::pow(pi, 4) * std::sin(pi * x(0, p))
* std::sin(pi * x(1, p)));
}
return {f, {f.size()}};
});
auto alpha = std::make_shared<fem::Constant<T>>(8.0);
// Define variational forms
auto a = std::make_shared<fem::Form<T>>(fem::create_form<T>(
*form_biharmonic_a, {V, V}, {}, {{"alpha", alpha}}, {}, {}));
auto L = std::make_shared<fem::Form<T>>(
fem::create_form<T>(*form_biharmonic_L, {V}, {{"f", f}}, {}, {}, {}));
// Now, the Dirichlet boundary condition ($u = 0$) can be
// created using the class {cpp:class}`DirichletBC`. A
// {cpp:class}`DirichletBC` takes two arguments: the value of the
// boundary condition, and the part of the boundary on which the
// condition applies. In our example, the value of the boundary
// condition (0.0) can represented using a {cpp:class}`Function`,
// and the Dirichlet boundary is defined by the indices of degrees
// of freedom to which the boundary condition applies. The
// definition of the Dirichlet boundary condition then looks as
// follows:
// Define boundary condition
auto facets = mesh::exterior_facet_indices(*mesh->topology());
const auto bdofs = fem::locate_dofs_topological(
*V->mesh()->topology_mutable(), *V->dofmap(), 1, facets);
auto bc = std::make_shared<const fem::DirichletBC<T>>(0.0, bdofs, V);
// Now, we have specified the variational forms and can consider
// the solution of the variational problem. First, we need to
// define a {cpp:class}`Function` `u` to store the solution. (Upon
// initialization, it is simply set to the zero function.) Next, we
// can call the `solve` function with the arguments `a == L`, `u`
// and `bc` as follows:
// Compute solution
fem::Function<T> u(V);
auto A = la::petsc::Matrix(fem::petsc::create_matrix(*a), false);
la::Vector<T> b(L->function_spaces()[0]->dofmap()->index_map,
L->function_spaces()[0]->dofmap()->index_map_bs());
MatZeroEntries(A.mat());
fem::assemble_matrix(la::petsc::Matrix::set_block_fn(A.mat(), ADD_VALUES),
*a, {bc});
MatAssemblyBegin(A.mat(), MAT_FLUSH_ASSEMBLY);
MatAssemblyEnd(A.mat(), MAT_FLUSH_ASSEMBLY);
fem::set_diagonal<T>(la::petsc::Matrix::set_fn(A.mat(), INSERT_VALUES), *V,
{bc});
MatAssemblyBegin(A.mat(), MAT_FINAL_ASSEMBLY);
MatAssemblyEnd(A.mat(), MAT_FINAL_ASSEMBLY);
b.set(0.0);
fem::assemble_vector(b.mutable_array(), *L);
fem::apply_lifting<T, U>(b.mutable_array(), {a}, {{bc}}, {}, T(1.0));
b.scatter_rev(std::plus<T>());
bc->set(b.mutable_array(), std::nullopt);
la::petsc::KrylovSolver lu(MPI_COMM_WORLD);
la::petsc::options::set("ksp_type", "preonly");
la::petsc::options::set("pc_type", "lu");
lu.set_from_options();
lu.set_operator(A.mat());
la::petsc::Vector _u(la::petsc::create_vector_wrap(*u.x()), false);
la::petsc::Vector _b(la::petsc::create_vector_wrap(b), false);
lu.solve(_u.vec(), _b.vec());
// Update ghost values before output
u.x()->scatter_fwd();
// The function `u` will be modified during the call to solve. A
// {cpp:class}`Function` can be saved to a file. Here, we output the
// solution to a `VTK` file (specified using the suffix `.pvd`) for
// visualisation in an external program such as Paraview.
// Save solution in VTK format
io::VTKFile file(MPI_COMM_WORLD, "u.pvd", "w");
file.write<T>({u}, 0.0);
}
PetscFinalize();
return 0;
}