CVEN Computer Applications in Engineering and Construction. Programming Assignment #2 Random Number Generation and Particle Diffusion

Transcription

1 CVE 0-50 Computer Applications in Engineering and Construction Programming Assignment # Random umber Generation and Particle Diffusion Date distributed: 0/06/09 Date due: 0//09 by :59 pm (submit an electronic copy via turnitin.com) Submit your solution on turnitin.com by the due date and time shown above. Please show all your work and follow the rules outlined in the course syllabus. MATLAB has two built-in random number generators, rand and randn, which are frequently used in numerical simulations. rand generates uniformly distributed numbers (between 0 and ) while randn generates normally distributed numbers (mean = 0, standard deviation = ). We are going to test random number generator rand and apply it to solve a diffusion problem.. Random umber Generation and Diffusion Random number generation is an important task in many simulations which require input of random numbers. One of the most important methods relying on the use of random numbers in engineering and physics is called Monte-Carlo method. For certain studies when an exact result is infeasible or impossible to obtain, Monte-Carlo simulations may be performed with a large numbers of repeated computations to obtain statistical properties. It is frequently used in problems with significant uncertainty in inputs. One good example is random walk of molecules in a stagnant environment, also called molecular diffusion. It is useful in simulating the spreading of scalar substances (such as pollutant) in a zero-mean-current environment.. Assignment Use Uniformly Distributed Random umbers You will first need to use the MATLAB function rand to generate 0,000 random points in the (x, y), (x, y),, (x0000, y0000) format between x = [ ] and y = [ ]. Use rand to generate numbers between 0 and, and then rescale them to be between and. Use help rand to get information about this function and its implementation. After you successfully generated those 0,000 random points, plot them in a subplot (,,) in Figure () using red dots but no lines. The second step is to evenly divide the range of [ ] in x into 5 intervals so each interval is Use IF loops and FOR loops to count how many points you get in each interval. Plot the numbers of points versus their corresponding x locations (at the center of each interval) in a second subplot (,,) in Figure () using the bar command. The second subplot should plot 5 data points with values like this:

2 where i indicates the number of particles in the ith interval (If done correctly, i should be close to 400). Plot these 5 data points using the bar command. Use help bar for information about this command. Give appropriate labels and title to each subplot. The third step is to use the hist command to generate a histogram with 5 evenly spaced containers, and plot the result in a third subplot (,,) in Figure (). Use help hist for information about this command. Compare the distribution with what you got in step. Discuss whether the randomly generated numbers are evenly distributed and judge whether the random number generator rand generates randomly but uniformly distributed numbers. You may increase the total number of points (from example, from 0,000 to 00,000) before you make your conclusions. Organize your results using the memorandum template from the Assignments page of the course website. In the body of the memorandum, you should briefly describe the methods used to classify each point and include a copy of the final figure produced by your working computer code. You may also discuss any problems with your code and ideas you may have to fix them. Include a listing of your MATLAB program in an Appendix. Be sure to include descriptive text with the computer code or codes in the appendix. To help you get started, the following provides a skeleton for your completed program: PROGRAM random_walk.m This program generate random numbers and examine their distributions using MATLAB function rand. It then simulates random walk using the Monte-Carlo method. Define Variables: random_number -- generated random numbers x, y -- coordinates of random points ni -- number of intervals in x-direction xi -- x-coordinate of each interval threshold -- upper threshold of each interval ncount -- number of points in the interval corresponding to xi nstep -- total number of time steps in random walk step -- displacement for each step of random walk Your ame CVE 0-50, Spring 09

3 clear all; close all; n = 0000; (A) Use rand to generate uniformly distributed points: random_number = rand(n, ); x = random_number(:,)*-; y = random_number(:,)*-; Your results should look like Figure below: Figure. Random points generated using rand and their distribution

4 . Assignment - Random Walk and Particle Diffusion Based on the 0,000 points you generated using the rand function in Section, you may simulate the spread of oil spill based on the theory of random walk. You can consider each point as an oil particle. Assuming at each time step (e.g., with a unit of minutes) each particle can move a distance of 0. (with a unit of meters) in the positive or negative x direction randomly (but not in the y direction to make the problem easier), compute 00 time steps (i.e., 5 hours) and plot their locations and distributions in Figure () using red dots but no lines and discuss your observation. Use rand to generate a random number for each particle at each time step. Since the motion is random, each particle should be given an equal chance to move left or right. For example, you may move a particle to the right if the given random number is greater than 0.5 and vice versa. Make sure you cover enough range of such as [-8 8] in x in your plot since the points are diffusing over time. Similar to what you did in step in Section, evenly divide the range of [-8 8] in x into 40 intervals so each interval is 0.4. Use IF loops and FOR loops to count how many points you get in each interval. Plot numbers of points versus their corresponding x locations (at the center of each interval) in a second subplot in Figure (). Figure () therefore has two subplots - the first subplot should plot the 0,000 red dots, and the second subplot should plot 40 data points with values like this: where i indicates the number of particles in the ith interval. Plot these 40 data points using the bar command. Give appropriate labels and title to each subplot. MATLAB is a powerful tool for displaying results as both images and movies. After getting your program working successfully, you can plot the particle locations at every time step and use pause to pause a desired duration, such as 0.05 second, in between adjacent plots so you can see the particles in motion. The process discussed here is called diffusion. You may also assemble all image files together to produce a movie (avi) file and observe how particles spread during the 5-hour simulation period. Observe the distribution of the particles based on your Figure (). Is it related to the central limit theorem? Calculate and record the mean ( x ) and standard deviation (σ ) of x for the 0,000 particles at each time step (you may use MATLAB functions mean and std). Is x close to zero at all time? To answer the question, you may plot t versus x but do not include the plot in your memo. 4

5 d Diffusion theory states that the diffusion coefficient (E) can be expressed as ( E = dt σ ). Plot t versus σ as your Figure () and check if E is approximately constant (i.e., if the t versus σ line is straight?). If it is, estimate the value of E using second and centimeter as the units of t and σ, respectively. Make sure you include the units when reporting the E value. 5