MATLAB Computer Applications (MSE 360/METE 352, MSE 262/METE 258) Lecture Notes PDF

17-May-21 MSE 360/METE 352 MSE 262/METE 258 Computer Applications Prepared by: Dr. K. Mensah - Darkwa Lesson-01 Introduction Computer Applications  Lecturer: Dr. K. Mensah-Darkwa  Class Schedule: Tue 08....

17-May-21 MSE 360/METE 352 MSE 262/METE 258 Computer Applications Prepared by: Dr. K. Mensah - Darkwa Lesson-01 Introduction Computer Applications  Lecturer: Dr. K. Mensah-Darkwa  Class Schedule: Tue 08.00 – 10.00, 10.30 – 12.30; (MSE2, METE2) Thur 08.00 – 10.00, 10.30 – 12.30; (MSE3, METE3)  Teaching Assistant: Miss Nancy Boye Mensah/Miss Christabel Wireko Arthur 2 1  17-May-21 Course Features Presentation Tutorials and Class discussions In class Quizzes (Every weeks) Assignments Attendance Assessments –Mid-Semester (to be announced) –End of Semester (to be announced) 3 Class Policy Attendance is mandatory, strictly enforced, and will be recorded each time the class meets. Missing Class: Reason for absence, with proof, must be stated within 2 days of absence. Quizzes may be announced / unannounced, and/or given at the beginning (10 min)/middle/ or end of any class. Late classwork / homework / assignments will not be accepted 4 2  17-May-21 Grade Evaluation  Continuous Assessment (30%) Assignments + Quizzes + Group Project (20%) Mid-Semester (10%) Participation in Class (Bonus)  End of Semester (70%) Section A (40%) Section B (60%) The University Grading Scale 5 Quizzes/Assignments - Submission Checklist All Quizzes/Assignments are to be submitted on the Template. Make as much copies as you need. Each Quiz/Assignment problem must be on a separate sheet of paper. If you need more than one sheet you should staple them together. (do not submit lose papers) Submitted codes will be checked for copying. Any one found guilty will not have the chance for a defense. Both students shall score zero(0) score Turn assignment in before class starts on the due date. (No late work will be graded) 6 3  17-May-21 Check “The test of a good teacher is not how many questions he can ask his pupils that they will answer readily, but how many questions he inspires them to ask him which he finds it hard to answer” By: Alice Wellington Rollins The mediocre teacher tells. The good teacher explains. The superior teacher demonstrates. The great teacher inspires.” By: William Arthur Ward “Tell me and I'll forget; show me and I may remember; involve me and I'll understand.” Chinese Proverb 7 Check “If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.” By: Martin Luther King Jr. 8 4  17-May-21 Course contents In this course you will learn how to use Matlab as an effective tool in science and engineering. From the course content: The possibilities and limitations in Matlab, syntax and interactive computations, programming in Matlab, visualisation, optimizing code for fast computations. Syntax and interactive computations, programming in Matlab using scripts and functions, rudimentary algebra and analysis. One- and two-dimensional graphical presentations. Examples on engineering applications. 9 Intended learning outcomes The course will give the fundamental knowledge and practical abilities in MATLAB required to effectively utilize this tool in technical numerical computations and visualisation in other courses. After the course you will be Able to use Matlab for interactive computations. Familiar with memory and file management in Matlab. Able to generate plots and export this for use in reports and presentations. Able to program scripts and functions using the Matlab development environment. Able to use basic flow controls (if-else, for, while). Familiar with strings and matrices and their use. 10 5  17-May-21 Ethical approach All members of a group are responsible for the group's work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution. 11 6  MSE 360/METE 352 MSE 262/METE 258 Computer Applications Prepared by: Dr. K. Mensah - Darkwa Lesson-02 Overview of MATLAB® What is MATLAB?  MATrix LABoratory (MATLAB) was created in the late 1970s by Cleve Moler, the chairman of the computer science department at the University of New Mexico  MATLAB is a high-level language and interactive environment that enables you to perform computationally intensive tasks faster than with traditional programming languages such as C, C++, and Fortran.  Mathworks, Inc. (www.mathworks.com) 1  Accessing the MATLAB software Look for the MATLAB icon on desktop. Or search for the icon on the startup menu. Click the icon, once. MATLAB takes a few moments to load, especially the initial access. NOTE: Do not continue clicking the icon; this would cause multiple MATLAB windows to appear and could slow down access time. Interface: The default MATLAB Desktop 2  MATLAB File Extensions Extension.m MATLAB function, script.txt MATLAB user-generated text file.fig MATLAB Figure.mat MATLAB binary file for storing variables – do not use or send as assignment submission Typewriter font to represent MATLAB commands Code >> y = 6*x Syntax: for example Getting MATLAB Assistance  MATLAB has three levels of assistance – Find the square root of 9 using MATLAB Level 1: >> lookfor root use lookfor when the command is unknown >> lookfor a topic Level 2: >>help sqrt use help to discover the command syntax >> help correctly spelled command Level 3: >> doc sqrt a pop-up window gives examples of command use >> doc correctly spelled command 3  Common MATLAB Commands >> clc Clears the command window and sends the prompt to the top of the window, clc does not clear or delete the variables, only the window contents. >> clear Clears (or deletes) all current variables. If one wishes to delete some, not all, of the variables, he or she would use clear var1 var2. e.g. Type this at the command prompt >> x =15 >> y = 6*x Common MATLAB Commands >> who Requests a list of the variables in use during the current session >> whos Requests a list of the variables plus their size and type e.g. Type this at the command prompt >> x =15 >> y = 6*x try who and whos 4  Symbols Usage % Use to denote a non-executable line or portion of a line. MATLAB does not attempt to execute the line or part of line that follows the % symbol. Used for commenting ; Use the semicolon to suppress the display of the MATLAB line. It can also be used to separate commands on one line , Use the comma to separate several MATLAB commands on one line … The ellipsis is used to continue a MATLAB command onto the following line. Common MATLAB Commands 5  Common MATLAB Commands The format of numerical values, show in the command window, is automatically set to have 4 values to follow the decimal, even if trailing zeros must be written. >> pi >> format long >> pi % displays a total of 16 numerical digits >> format bank % displays the number in banking format Common MATLAB Commands: 6  MATLAB Variables  Variable name: A location used to store either the results of mathematical operations or to store data  Begins with a letter  Can contain numbers, other letters, and/or the underscore  Is case sensitive  Must be less than 33 characters (for purposes of this class)  Should be relevant to the value it represents  Avoid list of special variables (i, j, eps, etc.) - check next slide  If you do not specify a location, ans is the default variable name >> 0.25+0.5 ans = 0.7500 >> store_var1=0.25+0.5; >> store_var1 Special Variables Do not use these to name your variables! 7 Common Mathematics Functions Common Mathematics Functions 8  Mathematical Operations >> x=240 % for radian measures >> tan(x), sin(x), cos(x), atan(x), asin(x), acos(x) % the ‘d’ is used for degree measures >> tand(x), sind(x), cosd(x),atand(x), asind(x), acosd(x) Mathematical Operations: Complex values >> N=3+4i >> M=3+4j Here, N and M are equivalent complex values. (No multiplication symbol is required for the complex part; also, recall that i and j are reserved variables) When dealing with complex numbers, leave your format type as short Else display complexities will occur. 9  Matrix Manipulations A (3 x 3) matrix can be assigned as follows >> A = [1 2 3; 4 5 6; 7 8 9] A = 1 2 3 4 5 6 7 8 9 % Separating rows by a semicolon and column by space or comma % Square brackets are used Matrix Manipulations A row vector can be assigned as follows: >> a = [1 2 3 4 5] a = 1 2 3 4 5 Creating a matrix on row basis: >> x = [8 6 9]; >> y = [-5 8 1]; >> z = [4 8 2]; >> B = [x; y; z] %concatenated or joining 10  Matrix Manipulations There are several built-in functions that can be used to create matrices. For example, the ones and zeros functions create vectors or matrices filled with ones and zeros, respectively: >> E = zeros(2,3) >> F = ones(1,3) >> G = eye(1,3) >> H = rand(5) Matrix Manipulations >>size(F) Returns the size m(rows) and n(cols) of matrix F >> tril(A) Lower triangular part of matrix A >> triu(A) Upper triangular part of matrix A >> DI = diag(A) Diagonals of matrix A 11  Matrix Manipulations >> B = [-1 2 8; 6 5 -6; 7 0 -9] >> C = A+B >> A = C-B >> A*B Array or element-by-element operations >> A.*B Try who and whos Matrix Manipulations We could construct the matrix A by concatenating (i.e., joining) the vectors representing each column: >> A = [[1 4 7]' [2 5 8]' [3 6 9]'] For an array, A(m, n) selects the element in mth row and the nth column. >> A(2,3) ans = 6 Try: >> A(4) 12  Matrix Manipulations Selecting columns >> A(:,3) Selecting rows >> A(3,:) Matrix Manipulations >> A = [1 2 3; 4 5 6; 7 8 9] A = 1 2 3 4 5 6 7 8 9 >> A^2 ans = 30 36 42 66 81 96 102 126 150 >> A.^2 ans = 1 4 9 16 25 36 49 64 81 13  Matrix Manipulations 1 5 6 𝐴= 7 4 2 −3 6 7 The transpose of [A] can be obtained using the ' operator: >> A’ >> I=eye(3) Matrices can be augmented using: >> Aug=[A I] >> D=[A;I] Matrix Manipulations The matrix inverse can be computed with the inv function: >> AI = inv(A) The matrix determinant of a can be computed with the det function: >> D = det(A) 14  Matrix Manipulations  a11 a12 a13   x1   b1      [ A]{x}  {b}   a21 a22 a23   x2   b2   a31 a32 a33   x3  b3  −1 𝑥 = 𝐴 𝑏 Left-division 1 5 6 1 >> x = A\b 𝐴= 7 4 2 𝑏= 1 −3 6 7 1 Matrix inversion >> x = inv(A)*b Arrays, Vectors and Matrices The colon operator is used to create and manipulate arrays. If a colon is used to separate two numbers, MATLAB generates the numbers between them using an increment of one (1): Syntax: var = start : end >> t = 1:5 t = 1 2 3 4 5 15  Arrays, Vectors and Matrices If colons are used to separate three numbers, MATLAB generates the numbers between the first and third numbers using an increment equal to the second number: Syntax: var = start : increment : end >> t = 1:0.5:3 t = 1.0000 1.5000 2.0000 2.5000 3.0000 Try: >> t = 10:-1:5 Arrays, Vectors and Matrices The linspace function generates a row vector of equally spaced points. Syntax: linspace(x1, x2, n) This generates n points between x1 and x2 >> linspace(0,1,6) ans = 0 0.2000 0.4000 0.6000 0.8000 1.0000 If the n is omitted, the function automatically generates 100 points 16  Arrays, Vectors and Matrices The logspace function generates a row vector that is logarithmically equally spaced. Syntax: logspace(x1, x2, n) This generates n logarithmically equally spaced points between decades 10x1 and 10x2. >> logspace(-1,2,4) If the n is omitted, the function automatically generates 50 points Character Strings Aside from numbers, alphanumeric information or character strings can be represented by enclosing the strings within single quotation marks. For example, >> f = ‘Mete3'; >> s = ‘Mse3'; >> x = [f s] 17 Mathematical Operations: Operator precedence Mathematical Operations: Operator precedence 18  Mathematical Operations (PEMDAS) Mathematical Operations (PEMDAS) P Parentheses first E Exponents (i.e. Powers and Square Roots, etc.) MD Multiplication and Division (left-to-right) AS Addition and Subtraction (left-to-right) 19  Mathematical Operations: Operations on arrays MATLAB has four additional arithmetic operators, that work on corresponding elements of arrays with equal dimensions. They are sometimes called array or element-by- element operations because they are performed element by element. Mathematical Operations: Built in Functions >> E = [-1.6 -1.5 -1.4 1.4 1.5 1.6]; >> round(E) >> ceil(E) >> floor(E) %---------------------------------------- >> F = [3 5 4 6 1]; >> sum(F) >> min(F),max(F),mean(F),prod(F),sort(F) %---------------------------------------- >> t = [0:2:20]‘ >> length(t) 20  Programming in MATLAB: M - file An M-file consists of a series of statements that can be run all at once. Note that the nomenclature M-file comes from the fact that such files are stored with a.m extension Script Files A script file is merely a series of MATLAB commands that are saved on a file. The script can be executed by typing the file name in the command window or by invoking the menu selections in the edit window: Debug, Run. New, Script-file Programming in MATLAB: M - file Function Files Function files are M-files that start with the word function. In contrast to script files, they can accept input arguments and return outputs. The syntax for the function file can be represented generally as: function outvar = funcname(arglist) % helpcomments statements outvar = value; 21  Programming in MATLAB: M - file Anatomy of M-file Programming in MATLAB: M - file A script is a sequence of instructions that are executed in order. Variables in script files are shared with the working directory’s workspace. You can build your own functions and scripts in the MATLAB editor. 22  Programming in MATLAB: M - file A function is a self-contained set of instructions designed to do a specific task. A function has its own workspace for its variables. Information can be added to a function’s workspace only through a function’s input variables. Information can leave the function’s workspace only through a function’s output variables. You can assign functions to variables using function handles. Programming in MATLAB: Script M - file Script File: File, New, M-file clc clear all close all % specify Cylinder Parameters r=5; h=10; % Volume of Cylinder V=pi*r^2*h % Lateral surface area s=2*pi*r*h %Total surface area S=s+2*pi*r^2 23  Programming in MATLAB: Function M - file Function File: File, New, M-file function [mean, stdev] = stats(x) % this will calculate std n = length(x); mean = sum(x)/n; stdev = sqrt(sum((x-mean).^2/(n- 1))); Programming in MATLAB: Function M - file Function File: File, New, M-file @ The Command Prompt >> y = [8 5 10 12 6 7.5 4]; >> [m, s] = stats(y) 24  Programming in MATLAB: Input - Output Input Function This function allows you to prompt the user for values directly from the command window. Syntax: n = input('promptstring') m = input('Mass (kg):') When this line is executed, the user is prompted with the message Mass (kg): Programming in MATLAB: input - output disp Function This function provides a way to display a value. Syntax: disp (value) disp(the values or equation) disp(‘The Mass in kg:’) fprintf Function This function provides a way to display a set of values. 25  Input – Output Example function dropball % velocity of a dropping ball g = 9.81; % acceleration of gravity m = input('Mass (kg): '); cd = input('Drag cof(kg/m): '); t = input('Time (s): '); disp(' ') % display a blank line disp(‘The Vel (m/s):') % display the string disp(sqrt(g * m / cd)*tanh(sqrt(g * cd / m) * t)) end Programming in MATLAB: Example @ the command prompt type >> dropball Mass (kg): xx Drag cof (kg/m): xx Time (s): xx 26  Structured Programming in MATLAB The simplest of all M-files perform instructions sequentially. That is, the program statements are executed line by line starting at the top of the function and moving down to the end.  Decisions (or Selection) The branching of flow based on a decision.  Loops (or Repetition) The looping of flow to allow statements to be repeated. Programming in MATLAB: Decisions Flow Control in Matlab This structure allows you to execute a set of statements if a logical condition is true. The basic tool for flow control is the if construct 1. Plain if 2.if...else 3.if...elseif 27  Programming in MATLAB: Decisions The if Structure. The if construct is used to isolate a block (or blocks) of code that is executed only if a condition is true.. Syntax: if condition statements end Programming in MATLAB: Decisions The if Structure G = 10; if G x = fzero(@(x) x^2-9,0) >> x = fzero(@(x) x^2-9,[0 4]) >> x = fzero(@(x) x^2-9,[-4 4]) >> options = optimset('display','iter'); >> [x,fx] = fzero(@(x) x^10-1,0.5,options) MATLAB Funtions: roots To determine all the roots, both real and complex The roots function has the syntax f(x) = x5− 3.5x4+ 2.75x3+ 2.125x2− 3.875x+ 1.25 >> a = [1 -3.5 2.75 2.125 -3.875 1.25]; >> polyval(a,1) >> x = roots(a) >> a = poly(x) 41  MATLAB Funtions: lu MATLAB has a built-in function lu that generates the LU factorization. It has the general syntax: [L,U] = lu(X) 3 −0.1 −.02 𝑥1 7.85 0.1 7 −0.3 𝑥2 = −19.3 0.3 −0.2 10 𝑥3 71.4 MATLAB Funtions: chol MATLAB has a built-in function chol that generates the Cholesky factorization. It has the general syntax, U = chol(X) 42  16-Jun-21 MSE 360/METE 352 Computer Applications Prepared by: Dr. K. Mensah - Darkwa Lesson-03 Plotting Plotting with MATLAB  Simple Plotting Syntax: plot (x, y) Open a new Script: Example_plot1 Plot the function y = 5sin 2x for x from 0 to 7 x = 0:0.01:7; y = 5*sin(2*x); plot(x,y),xlabel('x'),ylabel('y') 2 MSE 360_~Dr. K. Mensah-Darkwa 1  16-Jun-21 Plotting with MATLAB  Multiple Plotting Syntax: plot(x, y, x, z) Open a new Script: Example_plot2 Plot the functions 𝑦 = 2√𝑥 and z = 4𝑠𝑖𝑛3𝑥 For x from 0 to 5 on the same plot x = 0:0.01:5; y = 2*sqrt(x); z = 4*sin(3*x); plot(x,y,x,z),xlabel('x'),gtext('y') ,gtext('z') 3 Plotting with MatLAB  legend Syntax: legend('string1','string2',...) Open a new Script: Example_plot3 x = -pi:pi/20:pi; plot(x,cos(x),'-ro',x,sin(x),'-.b') legend('cos(x)','sin(x)'); grid on; 4 MSE 360_~Dr. K. Mensah-Darkwa 2  16-Jun-21 Plotting Options Example_plot4 t = [0:2:20]'; g=9.81; cd=0.8; m=100; v =sqrt(g*m/cd)*tanh(sqrt(g*cd/m)*t) plot(t, v); title('Plot of v versus t') xlabel('Values of t') ylabel('Values of v') grid on 5 Plotting Options % Replace the plot line with… plot(t,v,'--dc','LineWidth',2,... 'MarkerSize',10,... 'MarkerEdgeColor','k',... 'MarkerFaceColor','m') 6 MSE 360_~Dr. K. Mensah-Darkwa 3  16-Jun-21 Creating Multiple Figures figure(1) x = 0:0.05:2; y1 = exp(x); plot(x,y1); title(' exp(x)'); grid on; figure(2) y2 = exp(-x); plot (x,y2); title(' exp(-x)'); grid on; 7 Plotting Options - Hold on/off Open a new Script: Example_plot5 x1 = 1:0.1:3.1; y1 = sin(x1); x2 = 1:0.3:3.1; y2 = sin(-x2+pi/3); hold on grid on plot(x1,y1,'o'); plot(x2,y2,'h--g') hold off 8 MSE 360_~Dr. K. Mensah-Darkwa 4  16-Jun-21 Plotting Options – Hold on/off Open a new Script: Example_plot6 x = 0:0.1:20; y1 = exp(-x/10).*sin(x); y2 = exp(-x/5).*sin(x); plot(x,y1,'b'), grid on hold on plot(x,y2,'r') xlabel('x'); ylabel('f(x) = e^{-x/10} sin(x)'); title('A simple plot'); 9 Plotting Options 10 MSE 360_~Dr. K. Mensah-Darkwa 5  16-Jun-21 Plotting Options - Subplot  subplot Syntax: subplot(m, n, p) This command breaks the graph window into an m-by-n matrix of small axes, and selects the p-th axes for the current plot. 11 Plotting Options - Subplot Open a new Script: Example_plot6a t = 0:pi/50:10*pi; subplot(2,2,1);plot(sin(t)) title('sin t') subplot(2,2,2);plot(cos(t)) title('cos t') subplot(2,2,3);plot(sin(2*t)) title('sin 2t') subplot(2,2,4);plot(cos(5*t)) title('sin 5t') 12 MSE 360_~Dr. K. Mensah-Darkwa 6  16-Jun-21 Plotting Options - Subplot Open a new Script: Example_plot6b x = linspace(0,2*pi,50); subplot(2,2,1), plot(x,sin(x)), xlabel('x'), ylabel('sin(x)'); subplot(2,2,2), plot(x,cos(x)), xlabel('x'), ylabel('cos(x)'); subplot(2,2,3), plot(x,sin(2*x)), xlabel('x'), ylabel('sin(2x)'); subplot(2,2,4), plot(x,cos(2*x)), xlabel('x'), ylabel('cos(2x)'); 13 Plotting Options - Subplot Open a new Script: Example_plot7 x = 1:0.1:4; y1 = sin(3*x); y2 = cos(5*x); y3 = sin(3*x).*cos(5*x); subplot(1,3,1); plot(x,y1,'m-'); title('sin(3x)'); subplot(1,3,2); plot(x,y2,'g-'); title('cos(5x)'); subplot(1,3,3); plot(x,y3,'k-'); title('sin(3x)cos(5x)') MSE 360_~Dr. K. Mensah-Darkwa 7  16-Jun-21 3D-Plotting Options Use the plot3D command to plot the functions below. t vary from t=0 to t=10𝜋 (use divisions of 𝜋/50) 𝑥 = 𝑒 −0.05𝑡 sin 𝑡 𝑦 = 𝑒 −0.05𝑡 cos 𝑡 z=𝑡 3D-Plotting Options Open a new Script: Example_plot8 t=0:pi/50:10*pi; x=exp(-0.05*t).*sin(t); y=exp(-0.05*t).*cos(t); z=t; plot3(x,y,z,'>'); grid on xlabel('x'),ylabel('y'),… zlabel('z') MSE 360_~Dr. K. Mensah-Darkwa 8  16-Jun-21 3D-Plotting Options - Mesh and surface plots >> x=-1:3; >> y=1:4; >> [X,Y]=meshgrid(x,y) 3D-Plotting Options Open a new Script: Example_plot9 [X,Y] = meshgrid(-1:.05:1, 0:.05:2); Z = sin(5*X).* cos(2*Y); xlabel('x');ylabel('y');zlabel('z') MSE 360_~Dr. K. Mensah-Darkwa 9  16-Jun-21 3D-Plotting Options mesh(X,Y,Z) meshz(X,Y,Z) meshc(X,Y,Z) surf(X,Y,Z) surfc(X,Y,Z) waterfall(X,Y,Z) contour3(X,Y,Z,15) MSE 360_~Dr. K. Mensah-Darkwa 10  16-Jun-21 3D-Plotting Options x = linspace(-1,1,50); y = x; a = 3; c = 0.5; [xx, yy] = meshgrid(x,y); z = c*sin(2*pi*a*sqrt(xx.^2+yy.^2)); surf(xx,yy,z), colorbar, xlabel('x'), ylabel('y'), zlabel('z'),... title('f(x,y)=csin(2\pia\surd(x^2+y^2))') figure; mesh(xx,yy,z*0.5), colorbar, xlabel('x'), ylabel('y'), zlabel('z'),... title('f(x,y)=csin(2\pia\surd(x^2+y^2))') Logarithmic Scales x = 0:0.2:100; y = 2 * x.^2; % For the linear / linear case plot(x,y); title('Linear / linear Plot'); xlabel('x'); ylabel('y'); grid on; MSE 360_~Dr. K. Mensah-Darkwa 11  16-Jun-21 Logarithmic Scales % For the log / linear case semilogx(x,y); title('Log / linear Plot'); % For the linear / log case semilogy(x,y); title('Log / linear Plot'); % For the log / log case loglog(x,y); title('Log / log Plot'); Controlling x- and y-axis Plotting Limits x = -2*pi:pi/20:2*pi; y = sin(x); plot(x,y); % axis([0 pi 0 1]) title ('Plot of sin(x) vs x'); grid on; MSE 360_~Dr. K. Mensah-Darkwa 12  16-Jun-21 Controlling x- and y-axis Plotting Limits MSE 360_~Dr. K. Mensah-Darkwa 13  16-Jun-21 Plotting Options Plot the polynomial f(x) = x4+√30x−1 between x = −2 and x = 2 Use Command Prompt >>f2=@(x) x^4+sqrt(30*x)-1 >>fplot(f2,[-2 2]) MSE 360_~Dr. K. Mensah-Darkwa 14  16-Jun-21 Group Home Work 5.1 5.15 5.17 5.28 Group09_51_mse2 https://compengknust- my.sharepoint.com/:f:/g/personal/kmdarkwa_coe _knust_edu_gh/Ej_GiQgeez5IkDveJ- dsHUEBuULHTX6aCJFRixC7dgfLaw Agenda for next meeting Discuss the governing equations Input and output perimeters What more? How more? MSE 360_~Dr. K. Mensah-Darkwa 15