User:EML5526.S11.Team1.HS/HW7

=Problem 7.3 Static Solution for Unit Circle=

Given
Let $$ \Omega = $$ circle with unit radius.

$$ T = 4 $$

$$ f (x) = 1 in \Omega $$

$$ g = 0 on \Gamma_g = \partial \Omega to 10^{-6} $$ accuracy at center.

Find
For both quad and triangle elements.

sym $$ \Rightarrow $$ use $$~ \frac{1}{4} ~$$ of meshes.

Plot deformed shape in 3-D.

Quad)
Using SolidWorks software. A unit circle was created, with a mesh of 9 elements, all quadrilateral. It was from the data points that the mesh nodes were created by hand using the measurement tool. The picture below shows the mesh construction on SolidWorks.



After obtaining the coordinates from the unit circle and numbering the nodes from 1 to 16 since there are 9 elements, the boundary conditions were defined.

The boundary on the bottom is defined as

and for the left hand side of the circle, the boundary condition is as follows:

This means that these boundaries are "free" and the nodes are open to deflection. However the arc of the circle is not "free" these boundary conditions are locked and the nodes stay in one place.

After obtaining this information, a Matlab code was created to re-create the nodes and boundary conditions for the structure. The code is listed below.

The following code then calls the mesh file and makes a 3D plot of the unit circle.

Here is the result of the 3D plot. The arc of the circle does not deflect, but the free ends of the circle are clearly displaced.



Triangular)
The procedure is done the same way except with a different Matlab code, shown below for the mesh and plot.