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Finite Element Method

100 points] The two-dimensional Laplace equation around a cylinder with a domain [0, 2π]×[1, 5] is given
by.


∂x2
+


∂y2
= 0 (1)
The analytical solution is given by
ψ(r, θ) = U∞

r −
R2
r

sin(θ) (2)
(3)
In here R = 1 and U∞ = 1. The Drichlet boundary conditions:
ψ(1, θ) = 0 (4)
ψ(5, θ) =
5 −
1
5

sin(θ) (5)
(6)
Use the second-order accurate finite element discretization with uniform 80 × 40, 160 × 80 and 320 × 160
quadrilateral elements to solve the above Laplace equation. For this purpose

  1. Implement the fully implicit finite element solution algorithm and use a direct solver (LU factorization) to solve.
  2. Plot Error function versus ∆r and ∆θ in a log-log scale. Compute the spatial convergence rate.
  3. Compare the error with the solutions of finite difference method.
    The error function is given by
    Error = kψi,j − ψanalytick2

    imaxjmax
    (7)
    [20 points] Solve the same problem with an unstructured quadrilateral/triangular elements.
    Several useful MATLAB commands:
    Crate a sparse matrix
    i=[];
    j=[];
    s=[];
    m=100;
    n=100;
    A=sparse(i, j, s, m, n);
    To solve a sparse linear system
    x = A\b ;

Sample Solution

Over the past century the field of criminology has changed drastically, with ideas like intersectionality quickly becoming more dominant when attempting to explain crime and the driving forces behind it. Gender has become increasingly important in the quest to understand criminal statistics and the disparities between the sexes. Gendered behaviors influence even street level crimes in more ways than the early criminologists would have ever believed. One important question is how does the re-construction of gender occur and influence offenders and how can examining crime through an intersectional framework help us understand it? I firmly think that the gendered behaviors, or the action of “doing gender” by offenders, plays an important role in crime and that the intersectional framework can provide serious opportunities to further understand how gender, race, and class intertwines with crime. It is important to first understand how men and women re-construct gender on the streets. Typically, men are the “inner circle” of the gang, and this immediately leads to gender stereotypes being reinforced. The men reinforce stereotypes they have absorbed from the wider society, like family or media. They then enforce these stereotypes on the rest of the gang, especially the women. In a study published in the International Journal for Crime, Justice and Social Democracy, titled Homegirls, Hoodrats and Hos: Co-constructing Gang Status through Discourse and Performance, Dr. Abigail Kolb and Dr. Ted Palys (2016) investigate this phenomenon in street gangs. Women who join a gang by sleeping with one or more members are not respected and are seen as “hoodrats”. They are not trusted with important matters and are seen as quick to snitch if caught. Women who “do masculinity”, or dress and act more masculine, are seen as much more trustworthy than hoodrats. Unfortunately, to keep their status they have to condone and often act out the bias of the men in the group by putting down the other women designated as hoodrats. In their study, Doing Gender, Dr. Candace West and Dr. Don H. Zimmerman (1987) further explain that by “doing gender” people are simply acting out a socially constructed “achieved property of situated conduct.” Drug dealers reconstruct this in how they treat women in gangs as well as how they interact with other males. Another way that these gender stereotypes are reconstructed are during dru
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