# Solid mechanics with finite differences: How to handle “corner nodes”?

I have a question concerning coding boundary conditions for solid mechanics (linear elasticity). In the special case I have to use finite differences (3D). I am very new to this topic, so perhaps some of the following questions may be very basic.

To lead to my specific problem, first of all I want to show what I already implemented (To keep it clear, I will only use 2D).

1.) I have the following discretization of $$div(\sigma) = 0$$, showing the first component of the divergence $$\frac{\partial\sigma_{xx}}{\partial x} + \frac{\partial\sigma_{xy}}{\partial y} = 0$$:

I use a non-staggered grid, so Ux and Uy are defined at the same place.

2.) The next step was to treat the boundaries, where I use "ghost nodes". According to $$\sigma \bullet n = t^*$$, where $$t^*$$ is the stress on the boundary.

a) Here I use $$(\lambda + 2\mu)\frac{\partial U_x}{\partial x} + \lambda \frac{\partial U_y}{\partial y} = \sigma_{xx}^*$$ to obtain Ux at the ghost point as all other values of Ux and Uy are given (inside the body). $$\sigma_{xx}^*$$ is the value of this stress on the boundary (normally zero).

b) Same procedure, only via $$\mu\frac{\partial U_x}{\partial y} + \mu \frac{\partial U_y}{\partial x} = \sigma_{xy}^*$$ I obtain Uy at the ghost point. Again $$\sigma_{xy}^*$$ is the value of this stress on the boundary (normally zero).

3.) I think until now all my steps seem to be logic, if not, please correct me. But now there are also the "corner nodes", where I don't have a clue how to handle them.

To keep my scheme for $$div(\sigma) = 0$$ working on the corner node, I need Ux and Uy at the node in the lower left. But here my previously procedure like in 2.) doesn't work, as the node is not orthogonal to the boundary. I already tried to extrapolate the displacements, but it seems that this will result in stability problems (I am solving the whole problem implicit with an iterative solver).

So my question is what's the correct way to handle these "corner nodes"? I am happy for every idea.