# Linear elasticity

For small deformations the equilibrium equation in the displacement formulation is [1]
(μ + λ)∇divu + μΔu + f = 0,     f = div(I Ta) = ∇Ta,
where u = (X, Y) is displacement of a point from equilibrium position, Ta represents the active stress in cardiac tissue [2], μ, λ are Lame parameters (we put μ = 1 ). Using 2-nd order finite differences we obtain the approximate equation
(2 + λ)(Xi+1,j + Xi-1,j - 2Xi,j ) + Xi,j+1 + Xi,j-1 - 2Xi,j + (Yi+1,j+1 + Yi-1,j-1 - Yi+1,j-1 - Yi-1,j+1 )(1 + λ)/4 + h2Fxi,j = 0,
Fxi,j = (Tai+1,j - Tai-1,j )/2h
.     (*)

## 2D weighted Jacobi relaxation algorithm on CPU

For weighted Jacobi relaxation [3] of (Xi,j , Yi,j ) on n×n grid we rewrite equations (*) as
2(3 + λ)Xi,j = (2 + λ)(Xi+1,j + Xi-1,j ) + Xi,j+1 + Xi,j-1 + (Yi+1,j+1 + Yi-1,j-1 - Yi+1,j-1 - Yi-1,j+1 )(1 + λ)/4 + h2Fxi,j
where Ta ~ exp[-(r/a)2]. 50 cross-sections Xi, n/2 and error log max |Xi,jnew - Xi,jold| (the red curve below) are ploted after every 2it iterations. Set it = 50 to get accurate displacement values.
n it×2 ω

You can check that relaxation is fast for small n (e.g. n = 32) but it is very slow for fine-grids.

[1] Linear elasticity (Wiki)
[2] M.P. Nasha, A.V. Panfilov   Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias
Progress in Biophysics & Molecular Biology 85 (2004) 501.
[3] William L. Briggs, Van Emden Henson, Steve F. McCormick   A Multigrid Tutorial (look at "Tutorial Slides")

Simulations on GPU     updated 18 July 2012