关键词 > EE575
EE575: HOMEWORK 2 (DUE 02/13/2020)
发布时间:2020-02-06
Problem 1. Simulate 1D metal rod of unit length. Use 1D assumptions (constant temperature across crossection etc). Initial temperature f(x,0) =
boundary conditions f(0,t) = f(L,t) = 0. Run the heat equation and plot the temperature as a function of time. (Hint: Refer to the handout posted on blackboard for some useful source code). (10pt)
Problem 2. Simulate 2D unit square plate using a 256x256 regular grid, with initial temperature f(x,y,t = 0) = sin(2πx)cos(2πy). Assume the Dirichlet boundary conditions of 0 temperature on the boundary. Run the heat equation and plot the temperature distribution as a function of time. Take any black and white 256x256 image and run the heat equation on the intensity values. Show the smoothing behavior of the heat equation. (Extra credit (2pt): Create a movie showing heat change over a period of time). Hint: Check delsq function in matlab. (10pt)
![](/Uploads/20200207/5e3cd2f736eee.png)
and show that
![](/Uploads/20200207/5e3cd6923672f.png)
that is, the curve of shortest length from α(a) to α(b) is the straight line joining these points.
Problem 4. (2pt) The trace of the parameterized curve (arbitrary parameter) α(t) = (t,cosht), t ∈ R, is called catenary. Show that the signed curvature of the catenary is
.
Problem 5. (4pt) Plot the Frenet frame for a regular parameterized curve for (tcos(6t),tsin(6t),t). You can write your own code or use the Matlab code that I wrote as a starting point available on blackboard.
![](/Uploads/20200207/5e3ccd5cebffa.png)
Problem 2. Simulate 2D unit square plate using a 256x256 regular grid, with initial temperature f(x,y,t = 0) = sin(2πx)cos(2πy). Assume the Dirichlet boundary conditions of 0 temperature on the boundary. Run the heat equation and plot the temperature distribution as a function of time. Take any black and white 256x256 image and run the heat equation on the intensity values. Show the smoothing behavior of the heat equation. (Extra credit (2pt): Create a movie showing heat change over a period of time). Hint: Check delsq function in matlab. (10pt)
Problem 3. (4pt) (Straight Lines as Shortest) Let α : I → R 3 be a parameterized curve. Let [a,b] ⊂ I and set α(a) = p, α(b) = q. a. (2pt) Show that, for any constant vector v, |v| = 1,
![](/Uploads/20200207/5e3cd2f736eee.png)
and show that
![](/Uploads/20200207/5e3cd6923672f.png)
that is, the curve of shortest length from α(a) to α(b) is the straight line joining these points.
Problem 4. (2pt) The trace of the parameterized curve (arbitrary parameter) α(t) = (t,cosht), t ∈ R, is called catenary. Show that the signed curvature of the catenary is
![](/Uploads/20200207/5e3cd6af45ca3.png)
Problem 5. (4pt) Plot the Frenet frame for a regular parameterized curve for (tcos(6t),tsin(6t),t). You can write your own code or use the Matlab code that I wrote as a starting point available on blackboard.