User:EML5526.S11.Team1.HS/HW6

=HW6=

Given
6.1)

Given vector $$ q_x = -y^2 ~,~ q_y = -2xy $$ on domain shown in Figure 6.2.

6.2)

Given a vector field $$ q_x = 3x^2 y +y^3 ~,~ q_y = 3x + y^3 $$ on the domain shown in Figure 6.9. The curved boundary of the domain is a parabola.

6.3) Given $$ \oint_{T} \mathbf{n} d\Gamma = 0 $$

Find
6.1)

Verify the divergence theorem.

6.2)

Verify the divergence theorem.

6.3)

Prove using divergence theorem

FB 6.1)


We have to show that:

Where $$ \Omega $$ is the domain, $$ \Gamma $$ is the boundary, n is the normal vector and the notation $$ q_{i,i} $$  means $$ \frac{ \partial q_i}{ \partial x_i} $$

The integration on the boundary is

FB 6.2)
To calculate the integral on the surface we see that part of the surface is the parabola and the other part is in the x axis; therefore the integral will be:

The first integral is on the parabola and the second part is on the x axis.

On the curved surface the, normal vector on a segment of surface ds is shown in the figure. From that figure we observe that:

Also, it is clear that on the x axis we have y=0. Replacing in the integral we obtain:

Taking the derivative on the equation of the parabola we obtain:

Replacing on the integral we have:

Changing the limits of integration in the first two terms and organizing we obtain:



The integral on the volume will be:

Performing first the integration with respect to y we have:

FB 6.3)
The theorem says that for any tensor $$ T_{i...j} $$ (of any order) inside a volume $$ \Omega $$ surrounded by a surface $$ \Gamma $$ with a normal vector n

In this case the tensor T is the constant 1; therefore, changing the integral of surface to an integral of volume we have

But as 1 is a constant, its derivatives with respect to any coordinate are zero and we have

=Problem 6.5: Solve G2DM10/D1 using 2DLIBF=

Given: 2DLIBF from p. 29-2
PDE

$$ K = I $$ (identity matrix)

$$ f = 0, \frac{\partial u}{\partial t} = 0 $$

$$ \frac{\partial}{\partial x_i} \left(k_{ij} \frac{\partial u}{\partial x_j} \right) = 0 $$

Essential B.C.

$$ g = 2 on  \partial \Omega $$

Natural B.C.

none

Find
Solve G2DM10/D1 for $$ u^h $$ until accuracy $$ 10^{-6} $$ at center $$ (x,y) = (0,0) $$

Solution
Using the weak form to be

and

2DLIBF

where

$$ I = i + \left( j-1 \right) m $$

$$ n = m = 2, 4, 6, 8... $$

Thus

Therefore by substituting equation 6.5.3 into equation 6.5.1 and integrating, the stiffness matrix can be solved for. This is done by using Matlab.

The solution yields