11.4 Examples

11.4.1 L-shaped domain

The first example is the classic L-shaped domain, in which an exact solution is known, which makes it an ideal test case [20]. The domain is the polygon formed from the vertices

The exact solution for the displacements is known in polar co-ordinates (r,θ) as

ur(r,θ)	=
uθ(r,θ)	=

with λ and μ being the Lamé constants and ω = 3π∕4. Boundary conditions are set to be the exact solution and f = 0. The solution has a singularity at the origin, and so in order to test convergence h-refinement is required.

A simple example of how the linear elastic solver can be set up can be found in the Tests/L-shaped.xml session file in the linear elastic solver directory. A more refined domain with the obtained u solution is shown in figure 11.1. The solver can be run using the command:

The obtained solution L-domain.fld can be applied to the mesh to obtain a deformed XML file using the deform module in FieldConvert:

11.4.2 Boundary layer deformation

In this example we use the iterative elastic system to apply a large deformation to a triangular boundary layer mesh of a square mesh Ω = [0,1]². At the bottom edge, we apply a Dirichlet condition g = sin(πx) that is enforced by splitting it into N substeps, so that at each step we solve the system with the boundary condition gⁿ(x) = g(x)∕N. The process is depicted in figure 11.2.

The setup is very straightforward. The geometry can be found inside the file Examples/bl-mesh.xml and the conditions inside Examples/bl-conditions.xml. The solver can be set up using the following parameters, with NumSteps denoting N:

To identify the boundary that we intend to split up into substeps, we must assign the WALL tag to our boundary regions:

The solver can then be run using the command:

This will produce a series of meshes bl-mesh-%d.xml, where %d is an integer running between 0 and 100. If at any point the mesh becomes invalid, that is, a negative Jacobian is detected, execution will cease.