laplace

Solving Laplace equation with ArcaneFEM

Here, we employ Arcane to solve the Laplace equation, which stands as one of the fundamental partial differential equations (PDEs). The provided code presents a straightforward implementation of a 2D unstructured mesh Galerkin finite element method (FEM) solver.

The Laplace equation, also known as the harmonic equation, appears in various scientific and engineering applications. It plays a crucial role in fields such as physics, engineering, and mathematics. By solving the Laplace equation, we can gain insights into phenomena such as steady-state heat conduction, electrostatics, fluid flow, potential fields, and more. The code showcased here provides a foundation for tackling problems governed by this fundamental equation.

Problem description

The 2D Laplace equation is solved for a closed meshed domain $\Omega^h$ in order to know the Laplace solution $u(x,y)$ within the domain. The equation reads

$$\frac{\partial}{\partial x}\left(\frac{\partial u}{\partial x} \right) + \frac{\partial}{\partial y}\left(\frac{\partial u}{\partial y} \right) = 0 \quad \forall (x,y)\in\Omega^h $$

or in a more compact forms

$$\nabla(\nabla u)= 0 \quad \forall (x,y)\in\Omega^h.$$ or

$$\nabla^2 u= 0 \quad \forall (x,y)\in\Omega^h.$$ or

$$\Delta^2 u= 0 \quad \forall (x,y)\in\Omega^h.$$ or

To complete the problem description, three first type (Dirichlet) boundary conditions are applied to this problem:

$u = 50.0 \quad \forall(x,y)\in\partial\Omega^h_{\text{inner}}\subset\partial \Omega^h,$

$u = 20.0 \quad \forall(x,y)\in\partial\Omega^h_{\text{outer}}\subset\partial \Omega^h,$ and

We work with approximation, $\lambda$ is homogeneous $\lambda : \Omega^h \in \mathbb{R}^{+}$, in this case the variational formulation in $H^1_{0}(\Omega) \subset H^1{\Omega}$ reads

search FEM trial function $u^h(x,y)$ satisfying

$$- \int_{\Omega^h}\lambda\nabla u^h \nabla v^h + \int_{\partial\Omega_N} (\overline{q} \cdot \mathbf{n}) v^h v^h = 0 \quad \forall v^h\in H^1_0(\Omega^h)$$

given

$u^h=50.0 \quad \forall (x,y)\in\partial\Omega^h_{\text{inner}}$,

$u^h=20.0 \quad \forall (x,y)\in\partial\Omega^h_{\text{outer}}$,

$\int_{\Omega^h_{{N}}}(\mathbf{q} \cdot \mathbf{n}) v^h=0$

The code

Mesh

The mesh plancher.msh is provided in the Test.ring.arc file

  <meshes>
    <mesh>
      <filename>ring.msh</filename>
    </mesh>
  </meshes>

Please not that use version 4.1 .msh file from Gmsh.

Boundary conditions

The Dirichlet (constant $u$) boundary conditions are provided in Test.ring.arc file

    <dirichlet-boundary-condition>
      <surface>inner</surface>
      <value>50.0</value>
    </dirichlet-boundary-condition>
    <dirichlet-boundary-condition>
      <surface>outer</surface>
      <value>20.0</value>
    </dirichlet-boundary-condition>

So in the snippet above, three Dirichlet conditions are applied ($50, 20.0$) on three borders ('inner', 'outer') for the loaded mes ring.msh.

The Neumann boundary conditions are absent but could be provided in such a way

    <neumann-boundary-condition>
      <surface>outer</surface>
      <value>16.0</value>
    </neumann-boundary-condition>

Post Process

For post processing the Mesh0.hdf file is outputted (in output/depouillement/vtkhdfv2 folder), which can be read by PARAVIS. The output is of the $\mathbb{P}_1$ FE order (on nodes).

Tests available in this module

The tests are present in the form of .arc files with a prefix Test.:

Name	Dimension	Boundary Condition	Solver	Comment
L-shape	3D	Dirichlet + Null flux	Default (PETSc)	- Serves as validation test
ring	2D	Dirichlet only	Default (PETSc)
PointDirichlet	2D	Point Dirichlet + Null flux	Sequential Direct LU	- Serves as validation test
PointDirichlet-refined	2D	Point Dirichlet + Null flux	Default (PETSc)	- Refined version of test

Point loading example