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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 $\frac{\partial\sigma_{xx}}{\partial x} + \frac{\partial\sigma_{xy}}{\partial y} = 0$:

disc

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.

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.

Corner

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.

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I have had similar problems with the corner boundary conditions, especially in solving structural plate problems with a uniformly applied transverse pressure. In particular if one is trying to obtain the Shear loads on the edges (including the corners). The shear loads are a function of the ∂^3 w/∂^2 x∂y. Using a central difference scheme this causes one to need the the "ghost" node that is diagonal to the corner node to determine this derivative. I don't believe that averaging based on adjacent nodes is appropriate. What I did was to use the twisting moment Mxy that I calculated at the corner node and equated it to the finite difference "molecule" for the twisting moment as a function of the displacements. Since I already knew the displacements of all other adjacent nodes (based on the boundary conditions along the edges of the plate) it was a simple matter to solve for this "tricky" corner node. Ii hope this helps.

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You might be trying to solve a system of equations that does not have a unique solution. Imagine you have a bunch of nodes connected by springs, floating in space, and you want to find the equilibrium position of each node. If the system is not anchored to something fixed (or no force is applied), there are many possible solutions. Any one solution can always be translated or rotated and it is still a solution. Have you tried fixing the displacements at one corner node to eliminate translation, and fixing one displacement at another corner to eliminate rotations?

I once tried this approach of fixing some nodes and adjusting normal forces at others, but it seemed to focus large amounts of force at individual boundary nodes, resulting in instability. What ended up working was to not try to anchor just a few of nodes, but to anchor all of the nodes relative to a homogeneous strain. Essentially you strain the entire system homogeneously, but then include the homogeneous component in the local definition of strain at each node, so it does not contribute any additional elastic energy. You can read more about it in this paper and the cited references: http://pubs.acs.org/doi/abs/10.1021/nn204177u.

This instability problem is probably a good reason to choose finite elements for mechanics problems when possible.

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