User:Eml5526.s11.team2.penultimate yay.cleveland

=Problem 6.8: ADD TITLE HERE!!!!!!!!!!!!=

2. Solve the Following:
The following problems can be found in Fish and Belytschko pg. 91-92:

Problem 4.6: Gauss Quadrature to Obtain Numerical Integration
Use the Gauss quadrature to obtain exact values for the following integrals. Also verify by analytic integration:

(a) Integrate the following:
$$ \displaystyle \int_{0}^{4}{\left(x^2+1\right)dx} $$

Using the two-point Gauss quadrature where $$\displaystyle n_{gp} = 2, a=0, b=4 $$ we can determine the roots $$\displaystyle \xi_1 $$ and $$ \displaystyle \xi_2 $$ along with the weights $$\displaystyle W_1 $$ and $$\displaystyle W_2 $$ as shown above or by using the following system of equations. Either method is valid and for completeness we have decided to demonstrate both methods.


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$$ \displaystyle \begin{align} \left[ {\begin{array}{*{20}{c}} 1&1\\ \xi_1 & \xi_2 \\ \xi_1^2 & \xi_2^2 \\ \xi_1^3 & \xi_2^3 \end{array}} \right]
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\left[ {\begin{array}{*{20}{c}} W_1\\ W_2 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2\\ 0\\ \frac{2}{3}\\ 0 \end{array}} \right] \end{align} $$     (6.8.1)
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$$ \displaystyle \therefore \quad W_1 = W_2 = 1 \quad and \quad \xi_1=-\frac{\sqrt{3}}{3} \quad, \quad \xi_2 = \frac{\sqrt{3}}{3} $$     (6.8.2)
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Next we must consider the interval that $$\displaystyle x $$ is defined on. In its current form the Gauss quadrature is defined for $$\displaystyle x \in [-1,1] $$. To correct for this we will define the following relation between $$\displaystyle \xi $$ and $$ x $$.


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$$ \displaystyle x= \frac{1}{2}(a+b)+\frac{1}{2} \xi (b-a) $$     (6.8.3)
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$$ \begin{align} \displaystyle \quad x &= \frac{1}{2}(0+4)+\frac{1}{2} \xi (4-0) \\ &= \frac{1}{2}(0+4)+\frac{1}{2} \xi (4-0) \\ &= 2+ {2} \xi \end{align} $$     (6.8.4)
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$$ \displaystyle \therefore \quad f(\xi)= (2+ {2} \xi)^2+1 $$     (6.8.5)
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$$ \begin{align} \displaystyle I &= \int_{0}^{4}{\left(x^2+1\right) \underbrace{dx}_{{\color{red}\frac{1}{2}(b-a)d \xi } }} \\ &= \frac{1}{2}(b-a) \int_{-1}^{1}{\left( 2+ {2} \xi)^2+1 \right) d\xi} \\ &= \frac{1}{2}(b-a) \left[ \cancelto{w_1}((2+ {2} \cancelto{\xi_1})^2+1) + \cancelto{w_2}((2+ {2} \cancelto{\xi_2})^2+1) \right] \\  &= \frac{76}{3} \end{align} $$      (6.8.6)
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(b) Integrate the following:
$$\displaystyle \int_{-1}^{1}{\left(\xi^4+2\xi^2\right)d\xi} $$

(c) Write MATLAB Code
That utilizes the function gauss.m and performs Gauss integration. Check the solutions in (a) and (b) against the MATLAB code.

Problem 4.7: Three-Point Gauss Quadrature to Obtain Numerical Integration
Use the three-point Gauss quadrature to evaluate the following integrals. Compare the results to the analytic integral.

(a) Integrate the following:
$$\displaystyle \int_{-1}^{1}{\frac{\xi}{\xi^2+1}d\xi} $$

(b) Integrate the following:
$$\displaystyle \int_{-1}^{1}{cos^2\pi\zeta d\zeta} $$

(c) Write MATLAB Code
That utilizes the function gauss.m and performs Gauss integration. Check the solutions in (a) and (b) against the MATLAB code

Problem 4.8: n-Point Gauss Quadrature Comparison for $$ n=1,2,3 $$
The integral $$ \displaystyle \int_{-1}^{1}{\left(3\xi^3+2\right)d\xi} $$ can be integrated exactly using the two-point Gauss quadrature. How is the accuracy effected if

(c) Use MATLAB Code
Check the solutions in (a) and (b) against the MATLAB code

=Problem 6.9: ADD TITLE HERE!!!!!!!!!!!!=

3. Solve the following heat conduction problem
Consider a chimney constructed of two isotropic materials: dense concrete $$ (k = 2.0 W^{\circ}C^{-1} ) $$ and bricks $$ (k = 0.9 W^{\circ}C^{-1} ) $$. The temperature of the hot gases on the inside surface of the chimney is $$140^{\circ}C$$, whereas the outside is exposed to the surrounding air, which is at $$ T= 10C^{\circ}C^{-1} $$. The dimensions of the chimney (in meters) are shown below. For the analysis, exploit the symmetry and consider 1/8 of the chimney cross-sectional area. Consider a mesh of eight elements as shown below. Determine the temperature and flux in the two materials.

Analyze the problem with $$2x2$$, $$4x4$$ and $$8x8$$ quadrilateral elements for 1/8 of the problem domain. $$A2x2$$ finite element mesh is shown in Figure 8.23. Symmetry implies insulated boundary conditions on edges AD and BC. Note that element boundaries have to coincide with the interface between the concrete and bricks.

Solution
Matlab Code 1: Heat2d.m

adfghasdf

Matlab Code 2:  Include_flags.m

qwrgasg

Matlab Code 3:   preprocessor.m

Matlab Code 4:  input_file_16ele.m

Matlab Code 5:  mesh2d.m

Matlab Code 6:   plotmesh.m

Matlab Code 7:   setup_ID_LM.m

Matlab Code 8:   heat2Delem.m

Matlab Code 9:   gauss.m

Matlab Code 10:   BmatHeat2D.m

Matlab Code 11:   assembly.m

Matlab Code 12:   src_and_flux.m

Matlab Code 13:   solvedr.m Matlab Code 14:   postprocessor.m

Matlab Code 15:   postprocessor.m

Here is a link to the pseudo code that explains the processes behind the above matlab codes,

1. Graphs for 8.3
The above figure corresponds to Figure8.9 on page 200 of Fish and Belytschko.

The above figure corresponds to Figure8.10 on page 201 of Fish and Belytschko.

The above figure corresponds to Figure8.12 on page 202 of Fish and Belytschko.