# A Finite Difference solver for 2D Elasticity equations

1. Mar 17, 2016

### 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?

2. Mar 18, 2016

### 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.

3. Mar 25, 2016

### 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?

4. Dec 10, 2017

### EndWord

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