Finite Difference solver for 2D Elasticity equations

[smmr89]

Hi

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?

 

[the_wolfman]

It will help us if you tell us in detail how you are formulating the problem.

Gauss-Siedel is an iterative method that allows us to solve a system of linear algebraic equations. The details of how you convert you system of differential equations into a system of algebraic equations are important.

 

[Twigg]

Also, what are the wacky results? How big is your rectangular grid? What values of the Lame parameters have you tried? Can you tell us what differencing scheme you used?

 

[EndWord]

Hi Smmr98 ,
Have you sloved the 2D Navier equation?