README.md

1D Example

One-dimensional planewave propagation in an elongated rectangle.

• Domain, $\Omega = {(0,L)\times(0,h)}$
  • Length, $L=1$
  • Width, $h$ is the mesh-width.
• Boundary conditions
  • Rigid boundary, $\boldsymbol{u} \cdot \boldsymbol{n} = 0$ at $y=0$ and $y=h$,
  • Pressure $p = 1.0$ at $x=0$,
  • Impedance $Z = 1.0$ at $x=L$.

The boundary conditions are expressly chosen such that the exact solution can be computed analytically and the errors can be computed.

The exact solution here is given as,

$$ \begin{aligned} \boldsymbol{u}_{\mathrm{ex}} (x,y) &= {\frac{{p}_{0}}{k^2}} \exp (\mathrm{i} k x) \boldsymbol{e}_{x}, \\ {p}_{\mathrm{ex}} (x,y) &= {p}_{0} \exp(\mathrm{i} k x). \end{aligned} $$

Errors are computed for a range of mesh resolutions for interpolated exact displacement field and FEM-solver computed displacement field. Errors are also computed for different projection techniques proposed for one and two orders of approximation.

Run

The following instructions run the example assuming you are in the current directory.

An automated script main.jl is provided to run scripts in a particular order,

julia +1.11.0 --project=.. main.jl

This should generate the meshes of various resolutions in the meshes folder. All output fields are written within .jld2 files and visualizations in .pdf files within the results folder.