Dave Grant
Homework 4
Problem 1 A
𝜕2𝑢 𝜕2𝑢
+
=0
𝜕𝑥 2 𝜕𝑦 2
Show that u(x,y)=2xy satisfies.
first derivative : 2y+2x=0
second derivative 0+0=0 checks.
Check BC: u(0,y)=u(x,0)=0,
By inspection substituting zero into either variable will result in zero, checks.
u(x,1)=2x, by inspection substituting y=1 into 2xy equals 2x, checks
u(1,y), by inspection substituting x=1 into 2xy equals 2y, checks.
Part C Abaqus
Fringe comparison
Figure 1Linear Mesh with ele size 0.25
Figure 2 Linear Elements with ele size 0.1
Figure 3 Quadratic with 0.25 ele size
As the graph shows in each plot at the left and bottom edges the u(x,y) is zero and is 2 times the x or y value on the top or right
side respectively. So at the top right corner the red indicates a value of 2. This agrees with the exact solution given.
Problem 2 b).
Figure 4Linear with 0.25 size
Figure 5Linear with 0.1ele size
Figure 6 quadratic with 0.25 ele size,
The problem gave no exact solution so only mesh extensions were compared for validity.
Problem 3
Abaqus
Figure 7 Linear with ele size 0.25
Figure 8 quadratic with 0.25 ele size
Figure 9 linear with 0.1 ele size