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I've been trying to get a simple solution to the 2D Navier-Lame equations using finite difference on a rectangular grid. I want to see the displacements, u and v, when a simple deformation is imposed - e.g. top boundary is displaced by 10%.

The equations are as follows:

\begin{eqnarray*}

(λ+2μ)\frac{∂^2u}{∂x^2} + (λ+μ)\frac{∂^2v}{∂x∂y} + μ\frac{∂^2u}{∂y^2} = 0

\\

\\μ\frac{∂^2v}{∂x^2} + (λ+μ)\frac{∂^2u}{∂x∂y} + (λ+2μ)\frac{∂^2v}{∂y^2} = 0

\end{eqnarray*}

I have tried using the Gauss-Siedel method but am not getting the expected results (which have been sovled using e.g. Mathematica/Abaqus)

Is there something else I have to consider since the two equations are coupled via the mixed partials?

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# A Finite Difference solver for 2D Elasticity equations

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