# Finite element convergence rate and possion's ratio

I am running simulations of a cantilever beam where it is fixed on one end and negative force applied to the other end. The first simulation is with 4-node linear quadrilateral elements and the other is simulation with 9-node quadratic quadrilaterial elements. At first I ran the simulations with a poisson's ratio of 0.3, then with a Poisson's ratio of 0.499 and calculated the convergence rate.

I got the convergence rate for the 4-node as 1.7 but it should be 2 and for the 9-node it is 3. My question is why they are different than each other is it because 4-node is linear and 9-node is quadratic.

In addition, what would be the reasons that cause the convergence rate to be 1.7 not 2? when I asked my professor he said your algorithm is right but you need to know why its not 2. When I changed the Poisson's ratio to 0.499. everything messed up it the 4node simulation but it in it is fine for the 9node, why is that? why the 9-node can handle 0.499 material and 4-node can'

• Regarding the poisson's ratio and the element type: quadratic elements will always perform better than linear elements when you have a high poisson's ratio. Quadratic simpy have 'more options to move' and thus to satisfy the constraint of nearby incompressibility. – P. G. Dec 2 '18 at 11:23
• Maybe you could add graphs/plots or tell us how you calculated your convergence rate. What's the difference between your numerical and the analytical solution? – P. G. Dec 2 '18 at 11:31
• I know the 9node perform better with higher poisson's ratio but why its better/ – Abdelrahman Alhammadi Dec 2 '18 at 19:53
• I will add a plot of the error and convergence rate of the 4node problem to show you , the answer should be 2 but I got 1.7 for some reason. – Abdelrahman Alhammadi Dec 2 '18 at 19:57
• how can I add a plot here? – Abdelrahman Alhammadi Dec 2 '18 at 20:00   