New Mexico Tech Hyd 510

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1 Vectors vector - has magnitude and direction (e.g. velocity, specific discharge, hydraulic gradient) scalar - has magnitude only (e.g. porosity, specific yield, storage coefficient) unit vector - a unit vector is a vector with magnitude = 1 Vectors are usually identified by a lower case letter. To differentiate between scalar and vector variables, vectors are identified in boldface, such as a, or with an arrow a r above the vector name. When writing by hand they are often indicated by underline a or sometimes an overline a ; some printed papers/books also use these notations. In mechanics, i, j, k are unit vectors in the Cartesian x-, y-, and -directions, respectively. Other notation for Cartesian unit vectors includes n x, n y, n. Example: Suppose a is a 2 dimensional vector, in particular let a = 2 i + 3 j Then, a is a vector of magnitude 2 in the x-direction and 3 in the y-direction. In an x-y plot, it could be drawn as an arrow with its tail at the origin and its head at coordinate (2,3). But vectors aren t tied to an origin, so a could also be drawn as an arrow with is tail at (5,5) and its head at (7,8). In other words, the position of the tail is arbitrary, but the magnitude and direction of the vector is fixed. The vector a can also be written as (2,3). Vector Operations Define: where a = α 1 i + α 2 j + α 3 k b = β 1 i + β 2 j + β 3 k, α 1, α 2, α 3, β 1, β 2, and β 3 are scalars Scalar multiplication: c a = c α 1 i + c α 2 j + c α 3 k, where c is a scalar (result is a vector) Addition: a + b = (α 1 + β 1 ) i + (α 2 + β 2 ) j + (α 3 + β 3 ) k (result is a vector) Subtraction: a - b = (α 1 - β 1 ) i + (α 2 - β 2 ) j + (α 3 - β 3 ) k (result is a vector) Magnitude a = ( α α α 2 3 ) 1/2 (result is a scalar) Dot (Scalar) Product a. b = α 1 β 1 + α 2 β 2 + α 3 β 3 = a b cos θ where θ is the angle between the vectors (result is a scalar) -1.1-

2 Arrays An array is an ordered list of data, often referred to as a vector, as we and Matlab will do. We ve already seen how a vector in mechanics can be written as an array. The vector a in the example on the previous page can be written as a two-dimensional array (2,3), or if you prefer [2,3], which is used in Matlab. But other information can also be written as an array. Here are some examples. Example 1: Suppose we measure water level in a well at eleven different times. We can define a time array, t, that contains the 11 sampling times, and a head array, h, that contains the 11 water levels. So we might have: t = [0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0] (in hours) h = [10.0, 9.90, 9.81, 9.75, 9.70, 9.66, 9.64, 9.63, , 9.62] (in feet) Here we have two (11-dimensional) row vectors, t and h. Time is the independent variable and head in the dependent variable. For each time in the t vector, we have a corresponding water level in the h vector; therefore the order of the data in each vector is important. For example, at t=2.0 hr. the water level h=9.70 ft, corresponding to the fifth entry in each vector. Arrays can be organied as rows, as above, or in columns. If we take the transpose of t and h we get column vectors t = M ; h = M Example 2: Suppose that we examine a transect across the landscape and measure topographic elevation,, vegetation density, d, and soil moisture, θ, at N locations (i.e., N = the number of locations). Let x denote the horiontal position along the transect. Then all three metrics vary with x, that is, they are functions of position x. However, we don t have complete functions, only values at the N discrete locations. We then represent this data set by four N-dimensional arrays with corresponding data: x = [ x 1, x 2, x 3,, x N-1, x N ] = [ 1, 2, 3,, N-1, N ] d = [ d 1, d 2, d 3,, d N-1, d N ] θ = [θ 1, θ 2, θ 3,, θ N-1, θ N ] This data could be organied in the form of a matrix, with four rows representing, respectively, the location and the three metrics, and with N columns representing the data. x x = d d θ 1 1 x d 2 2 x d 3 3 L L L d N 1 N N 1 θ1 θ2 θ3 L θ N 1 x xn N d N θ N We describe matrix dimensions as the number of rows by the number of columns. Here we have a (4 N) matrix on the right hand side (RHS), and a (4 1) column vector (of vectors) on the LHS

3 Example 3: Suppose that you have a continuous mathematical function, c(x), representing solute concentration over space in flowing system, with space described by a Cartesian coordinate, x. You wish to plot this function, in order to make a presentation, but your plotting program, like all plotting programs, can t handle a continuous function. It requires you to translate your continuous function into discrete values at a discrete number of points. Thus you select a set of n spatial locations at which you determine the value of c, and feed this to the plotting program. The n spatial locations are given by the vector x = [ x 1, x 2, x 3,, x n-1, x n ], while the associated values of c are given by c = [ c 1, c 2, c 3,, c n-1, c n ]. The program then plots the values c i (x i ), i=1,2,, n, and interpolates between them. The most general plotting programs (especially in two or three dimensions) have two steps. The first is deciding where to locate the points x i. The second is to interpolate c between the points. The simplest location algorithm is to uniformly space the points x i. That is what we mostly do this semester. The simplest interpolation scheme is to draw a straight line between points, that is, to linearly interpolate. As long as the points are located very close to each other these simple steps are adequate. Adaptive schemes put more points where the action is, that is, where the function is changing rapidly (i.e., where the derivative is sufficient different from ero). More sophisticated interpolations use higher order polynominals, splines, or other methods. Suppose that c(x) = c erfc(x-5)], where x is in meters. Then sample the complementary error function between 0 and 10, say at 21 points. (Why did I pick 21 points instead of, say, 20 points?) The result, expressed as the ratio of c/c 0 is given by the table and graph: x (m) c/c e e e e e-13 Relative Concentration Flow Solute Plume Plume front Sampled value Distance, m Where did this come from? Matlab: > x=linspace(0,10,21); c = 0.5*erfc(x-5); plot(x,c,'o') -1.3-

4 BASIC MATLAB MATLAB command structure: > varname = command; where > denotes the command line prompt, varname is the name of the variable to be assigned, and command is the MATLAB command. If no varname is specified, the default is ans. If the semi-colon is omitted, the result is displayed. Some commands contain calls to functions that require input variables. For example, the function sqrt takes the square root of the number that is inputted to it. All input variables are enclosed in parentheses; order is important. Input variable can be numbers or variables. Comments can be added on separate lines, or at the end of a line, by preceeding it with the symbol %. Examples: > a = 7.2; > b=a+5.1; > c=b/4.6+(a-1)*8-(a*b)^2; > d=cos(1.57)-sqrt(c*b/a); > e = [1,2,3]; % this defines an array (row vector) e with elements 1, 2, and 3. > f = [3,2,1]; > g=e+f; % if two vectors have the same number of elements, they % can be added or subtracted using + and - >h= e ; % this transposes the row vector into a column vector Some Basic Commands: whos - prints a list of all variables and their sie help - prints a list of help categories help command - prints help for command (e.g. > help sqrt ) plot - makes an x-y plot of data (requires input) quit - exits Matlab sie - displays the sie of the variable (requires input) linspace - generates an array of evenly spaced numbers (requires input) title - adds a graph title (requires input) xlabel - adds a label to the x-axis of a plot (requires input) hold on - freees the plot so that additional data can be plotted on the same plot hold off - unfreees the plot % - comment line Script files: A sequence of Matlab commands and comments can be saved as a script file, called an m-file, and can be reused. In essence, an m-file is a simple program. Some advantages of using a script m-file instead of the command line are (1) the program can be run many times without re-entering the commands; (2) if you have errors, it is easier to edit and debug the program; and (3) the program is not removed from memory when you exit Matlab

5 MATLAB Exercise 1 For each of the following turn in an m-file. When asked for discussion or explanation, include it as a comment(s) in the m-file. You files should be named 1_#_lastname.m (the.m is added automatically by Matlab), where the 1 refers to this exercise number, # is the problem number (below, 1 to 6), and lastname is your last name. 1.1 Solving a vector problem Darcy s law states that the specific discharge [ms -1 ] vector, q, in a porous media is a linear function of the hydraulic gradient vector, h [-], or q = -K h where porous media hydraulic conductivity K [ms -1 ] is a scalar proportionality coefficient. Specific discharge is the rate (and direction) at which water moves through the porous media (discharge per unit area per unit time). Hydraulic conductivity describes the resistance to flow (resistance = K -1 ). The hydraulic gradient provides the driving force. If a non-reactive tracer or contaminant is introduced into the porous media it will move with the flow at the average linear (seepage) velocity, v = q /n, where n= effective porosity is a scalar. Suppose that there is two-dimensional flow with gradient h = 0.03 i j. The porous media has conductivity K = 1 ft/d and porosity n = Use Matlab for the following calculations. a. Calculate vector q b. Calculate the average linear (seepage) velocity vector, v. c. What is the magnitude of velocity, v? d. What is the direction of velocity (assume 0 deg is east, 90 deg is north)? 1.2. Plotting data Build the vectors t and h in example 1 (p. 1.2) and plot h as a function of t. Use a symbol for h. Do not connect the symbols with lines. Label the axes and title the graph Plotting multiple functions Build the vectors x,, d, and θ in example 2 (p. 1.2) and plot, d and θ as a function of x on the same graph. Use different symbols for each dependent variable. Connect with lines. Label the axes and title the graph. Note that example 2 does not assign actual numerical values, we ll have to do that here. Assume that the measurements are evenly spaced in x, over a distance of 1km. Use the matlab linspace command to make this vector. Then use functions of x for the other parameters. Make the elevation to be a linear function of x, so that = (x-x 0 ), where x 0 is the initial location and x is measured in meters. Make vegetation density to be a sinusoidal function of x with a wave length of 10m, a mean value of 0.4, and an amplitude of 0.2, so that d = sin (2π/10m). Finally, make moisture content a stochastic variable with mean value of 0.2 and a random component with a maximum range of 0.1, θ = ε. The random variable is uniformly distributed between -1 and +1 and spatially uncorrelated. Use the matlab rand command. Potentially useful commands: plot, xlim, ylim, hold on, text, xlabel, ylabel, title, legend. 1.4 Variable spacing Using the output from the following line, and if necessary the commands help linspace and help logspace, write a short description (in proper English) of Matlab commands linspace and logspace. x1=0:0.1:1, x2=linspace(0,1,11), x3=logspace(0,1,11) 1.5. Plotting with variable spacing. You will often find occasion to examine the output of computer codes that you or someone else develops. In some cases the data will consist of values of one or more variables as a function of time, t, or space, x. With time, and with one-dimensional spatial models, the results will be presented as graphs of one or more dependent variables (heads, concentrations, temperatures, etc) vs. an independent variable (usually time or distance). The data will often be evenly spaced as a function of the independent variable, but that will not always be the case. Data is usually in the form of a series of data pairs (triplets, etc), consisting of a dependent variable value and its location (say, t or x)

6 This problem allows you to learn how to plot these types of graphs using Matlab. The structure is similar in other math codes and in standard plotting packages. Consider two functions of space, x. h = cos( x 2 k = x sin( x); Plot these two functions in the domain 0 x 3;, using 31 points in x created with the linspace command. Represent each function by a different symbol. Do not connect the points. Provide a concise listing of the commands that you use to create the data file and plot it. Label the x and y axes appropriately. Include a plot title and legend. a. Use evenly spaced points in the interval (0,3) with x = 0.1 increments. b. Place points spaced more closely at the larger x, where there is more action, using locations given by x = i, with 0 i 9 and i = Compare to the plot in part a. Note that both a and b use 31 points, but b uses them more efficiently by employing uneven spacing of the independent variable. Here you specified the region where more points will be used. The most sophisticated plotting routines go one step further, adaptively computing where more points are needed. The symbolic math code Mathematica does this automatically. 1.6 Plotting with Subplot Recall that sin(x) = - sin(-x) and cos(x) = + cos(-x). In words these two equations state "sin is an odd function" and "cosine is an even function". Create an m-file which makes a plot using the subplot command. Then use fplot command to plot the functions. See the Matlab help page for these commands, or a reference book or web page. The plot should have four subplots, one of sin(x), one of sin(-x), one of cos(x) and one of cos(-x). The range of each plot should be from -2π to 2π. Label all axes and title each plot. Visually inspect the plots. Write a short written definition (proper English) of even and odd functions