Computational Methods for Engineers Programming in Engineering Problem Solving

Transcription

1 Computational Methods for Engineers Programming in Engineering Problem Solving Abu Hasan Abdullah January 6, 2009 Abu Hasan Abdullah 2009

2 An Engineering Problem Problem Statement: The length of a belt in an open-belt drive, L, is given by L = q 4c 2 (D d) `DθD + dθ d 2 (1) where «D d θ D = π + 2 sin 1 2c «D d θ d = π 2 sin 1 2c c is the centre distance, D is the diameter of the larger pulley, d is the diameter of the smaller pulley, θ D is the angle of contact of the belt with the larger pulley, and θ d is the angle of contact of the belt with the smaller pulley, see Figure-2.8 of Rao (2002). If a belt having a length 11 m is used to connect the two pulleys with diameters 0.4 m and 0.2 m, determine the centre distance between the pulleys. Abu Hasan Abdullah

3 An Engineering Problem Figure 1: Open belt drive. Abu Hasan Abdullah

4 Analysis of Engineering Problem 1. Problem Statement: Recognise and understand the problem (what is it that needed to be solved?). 2. Governing Equations or Mathematical Models: Identify parameters affecting the problem, make the necessary assumptions, develop mathematical model or governing equations (based on theories from Engineering Mathematics and other Engineering Subjects). 3. Solution: Solution of the governing equations may make use of the computer programming (why?). 4. Verification: Verify and interpret the solution (right/wrong?). Abu Hasan Abdullah

5 Analysis of Engineering Problem Problem Statement The length of a belt in an open-belt drive, L, is given by q L = 4c 2 (D d) `DθD + dθ d 2 (2) where «D d θ D = π + 2 sin 1 2c «D d θ d = π 2 sin 1 2c c is the centre distance, D is the diameter of the larger pulley, d is the diameter of the smaller pulley, θ D is the angle of contact of the belt with the larger pulley, and θ d is the angle of contact of the belt with the smaller pulley, see Figure-2.8 of Rao (2002). If a belt having a length 11 m is used to connect the two pulleys with diameters 0.4 m and 0.2 m, determine the centre distance between the pulleys. Abu Hasan Abdullah

6 Analysis of Engineering Problem Problem Statement Figure 2: Open belt drive. Abu Hasan Abdullah

7 Analysis of Engineering Problem Mathematical Model Defined as a formulation or equation that expresses the essential features of a physical system or process in mathematical terms. Its simplest form can be represented as a functional relationship thus where Dependent variable = f(independent variables, parameters, forcing functions) dependent variable: a characteristic that reflects the behaviour/state of system independent variables: dimensions (time, space, mass) along which the system s behaviour that is being determined parameters: reflective of system s properties or composition forcing functions: external influences acting on the system Abu Hasan Abdullah

8 Analysis of Engineering Problem Mathematical Model Mathematical model ranges from a simple algebraic relationship to large complicated set of DE. Mathematical models (a.k.a. governing equations) are derived by applying physical laws such as Equilibrium Equation Newton s Law of Motion Conservation Laws: Mass, Momentum, Energy Equation of State Abu Hasan Abdullah

9 Analysis of Engineering Problem Solution Solution of the governing equation or mathematical model may appear as Transcendental Functions Linear or Nonlinear Algebraic Equations Homogeneous Equations leading to an Eigenvalue Problem Ordinary or Partial Differential Equations Equations involving Integrals or Derivatives which are either closed-form or open-ended. Abu Hasan Abdullah

10 Analysis of Engineering Problem Solution: Closed-form Closed-form mathematical expression I 1 = b a xe x2 dx = = 1 2 e b e a2 = 1 2 [ 12 e x2] b a (e a2 e b2) leads to analytical solution Abu Hasan Abdullah

11 Analysis of Engineering Problem Solution: Open-ended Open-ended mathematical expressions, e.g. I 1 = Z b a need to be approximated numerically f(x)dx = Z b a e x2 dx Nowadays, approximated numerical solutions are done by developing a computer program. Because numerical methods deal extensively with approximations connected with the manipulation of numbers, accuracy, precision and error feature prominently in programming the solution. We shall cover these later! Abu Hasan Abdullah

12 Analysis of Engineering Problem Solution: Computer Program Steps in computer program development: Algorithm Design: Listing down of the sequence of steps to define the problem at hand. Techniques available: algorithm, flowchart, pseudocode Program Coding: Writing these steps in a computer language. Debugging: Testing the program to ensure that it is error-free and reliable. Documentation: Making the program easy to understand and use through manual or guide. See SME 1013 Programming for Engineers for details. Abu Hasan Abdullah

13 Analysis of Engineering Problem Solution: Computer Program Algorithm Algorithm: A general sequence of the logical steps in solving a specific problem. Flowchart: A graphical representation of the algorithm. Better suited for visualizing complex algorithms. Pseudocode: Uses code-like statements in place of the graphical symbols of flowchart. Easier to develop a program with it than with a flowchart. Elements of good algorithm Each step must be deterministic i.e. not ambiguous. The process must end after a finite number of steps. The algorithm must be general enough to deal with any contingency. Abu Hasan Abdullah

14 Analysis of Engineering Problem Solution: Computer Program Algorithm Find roots of equation ax 2 + bx + c = 0 using the quadratic formula x = b ± b 2 4ac 2a 1. Start 2. Read coefficients a, b and c 3. Implement quadratic formula (Avoid division by zero, allow for complex roots) 4. Display solution i.e. values of x 5. Stop Abu Hasan Abdullah

15 Analysis of Engineering Problem Solution: Computer Program Pseudocode Find roots of equation ax 2 + bx + c = 0 using the quadratic formula x = b ± b 2 4ac 2a DO READ a, b, c root1 = (-b + SQRT(b^2-4ac)/(2a) root2 = (-b - SQRT(b^2-4ac)/(2a) PRINT root1, root2 PRINT Try again? Answer yes or no READ response IF response = no EXIT ENDDO Abu Hasan Abdullah

16 Analysis of Engineering Problem Solution: Computer Program Flowchart Start/Stop Process Input/Output Refers to separate flowchart Decision Connector Off-page Connector Preparation (for Loop, etc.) Abu Hasan Abdullah

17 Analysis of Engineering Problem Solution: Computer Program Coding A program is a sequence of instructions to the computer for it to solve a particular problem. A set of programs is called code. Programs are written in some programming language, e.g. Fortran, Matlab, Basic, C++, Pascal, Java. Programs are stored in files which are a sequence of bytes which is given a name and stored on a disk. Abu Hasan Abdullah

18 Analysis of Engineering Problem Solution: Computer Program Coding A program is a file containing a sequence of statements, each of which tells the computer to do a specific action. Once a program is run or executed the commands are followed and actions occur in a sequential manner. If the program is designed to interact with the outside world, then it must have input and output. Abu Hasan Abdullah

19 Analysis of Engineering Problem Solution: Computer Program Coding A program is said to have a bug if it contains a mistake or it does not function in the way it is intended to. Bugs can happen both in the logic of the program, as well as in the commands. In order that the program perform the exact actions it is intended to do, before the actual program is written an algorithm for solving the problem must first be outlined. Abu Hasan Abdullah

20 Example Problem 1 Problem Statement: Assuming that the thrust T of a screw propeller is dependent upon diameter D, speed of advance v, fluid density ρ, rotational speed of propeller N and coefficient of viscosity µ, derive and expression that relates all the parameters involved and solve for T. Governing Equation: Through dimensional analysis ( µ T = ρv 2 D 2 f ρvd, ND ) v Abu Hasan Abdullah

21 Example Problem 2 Problem Statement: Given temperature in degrees Fahrenheit, the temperature in degrees Kelvin is to be computed and shown. Governing Equation: From Physics, these two temperature scales are related through ( ) TF 32 T k = and the parameters involved in this problem are T K and T F Abu Hasan Abdullah

22 Example Problem 2 Algorithm: 1. Start 2. Get the temperature in Fahrenheit 3. Compute the temperature in Kelvin using the formula: T k = ( ) TF Show the temperature in Kelvin 5. Stop Abu Hasan Abdullah

23 Example Problem 2 Pseudocode: Start Read TF TK = (TF-32)/ Print TK Stop Abu Hasan Abdullah

24 Example Problem 3 Problem Statement: Determine the mass of the bungee jumper with a drag coefficient of 0.25 kg/m to have a velocity of 36 m/s after 4 s of free fall. Note: The acceleration of gravity is 9.81 m/s. Governing Equation: From Physics and Mechanics the fall velocity as function of time is ( ) gm gcd v(t) = tanh c d m t or, on re-arranging in terms of a function of mass, f(m), f(m) = ( ) gm gcd tanh c d m t v(t) = 0 Abu Hasan Abdullah

25 Example Problem 3 Algorithm: The governing equation indicates a root-finding problem. 1. Choose lower m_l and upper m_u guesses for the root such that the function changes sign over the interval. This can be checked by ensuring that f(m_l) x f(m_u) < An estimate of the root m_r is determined by m R = m L + m U 2 3. Make the following evaluations to determine in which subinterval the root lies: If f(m_l) x f(m_u) < 0, root lies in lower subinterval; repeat step 2. If f(m_l) x f(m_u) > 0, root lies in upper subinterval; repeat step 2. If f(m_l) x f(m_u) = 0, root equals m_r; terminate computation. Abu Hasan Abdullah

26 Example Problem 3 Computer Program: Fortran code for root finding C*********************************************************************** C BISECTION METHOD * C*********************************************************************** C TO FIND A SOLUTION TO F(X)=0 GIVEN THE CONTINOUS FUNCTION C F ON THE INTERVAL <A,B>, WHERE F(A) AND F(B) HAVE C OPPOSITE SIGNS: C C INPUT: ENDPOINTS A,B; TOLERANCE TOL; C MAXIMUM INTERATIONS N0. C OUTPUT: APPROXIMATE SOLUTION P OR A C MESSAGE THAT THE ALGORITHM FAILS. C CHARACTER NAME*14,NAME1*14,AA*1 INTEGER INP,OUP,FLAG LOGICAL OK REAL A,B,FA,FB,X,TOL INTEGER N0 Abu Hasan Abdullah

27 C*********************************************************************** C DEFINE YOUR FUNCTION HERE * C*********************************************************************** C DEFINE F C! F(X)=X**3+4*X**2-10 c c OPEN(UNIT=5,FILE= CON,ACCESS= SEQUENTIAL ) OPEN(UNIT=6,FILE= CON,ACCESS= SEQUENTIAL ) WRITE(6,*) This is the Bisection Method. WRITE(6,*) Has the function F been created in the program? WRITE(6,*) Enter Y or N WRITE(6,*) READ(5,*) AA IF(( AA.EQ. Y ).OR. ( AA.EQ. y )) THEN OK =.FALSE. 10 IF (OK) GOTO 11 WRITE(6,*) Input endpoints A < B separated by blank WRITE(6,*) READ(5,*) A, B IF (A.GT.B) THEN X = A A = B Abu Hasan Abdullah

28 B = X ENDIF IF (A.EQ.B) THEN WRITE(6,*) A cannot equal B WRITE(6,*) ELSE FA = F( A ) FB = F( B ) IF ( FA * FB.GT. 0.0 ) THEN WRITE(6,*) F(A) and F(B) have same sign WRITE(6,*) ELSE OK =.TRUE. ENDIF ENDIF GOTO OK =.FALSE. 12 IF (OK) GOTO 13 WRITE(6,*) Input tolerance WRITE(6,*) READ(5,*) TOL Abu Hasan Abdullah

29 IF (TOL.LE.0.0) THEN WRITE(6,*) Tolerance must be positive WRITE(6,*) ELSE OK =.TRUE. ENDIF GOTO OK =.FALSE. 14 IF (OK) GOTO 15 WRITE(6,*) Input maximum number of iterations WRITE(6,*) - no decimal point WRITE(6,*) READ(5,*) N0 IF ( N0.LE. 0 ) THEN WRITE(6,*) Must be positive integer WRITE(6,*) ELSE OK =.TRUE. ENDIF GOTO CONTINUE Abu Hasan Abdullah

30 ELSE WRITE(6,*) The program will end so that the function F WRITE(6,*) can be created OK =.FALSE. ENDIF IF (.NOT.OK) GOTO 40 WRITE(6,*) Select output destinations: WRITE(6,*) 1. Screen WRITE(6,*) 2. Text file WRITE(6,*) Enter 1 or 2 WRITE(6,*) READ(5,*) FLAG IF ( FLAG.EQ. 2 ) THEN WRITE(6,*) Input the file name in the form - WRITE(6,*) drive:name.ext WRITE(6,*) with the name contained within quotes WRITE(6,*) as example: A:OUTPUT.DTA WRITE(6,*) READ(5,*) NAME1 OUP = 3 OPEN(UNIT=OUP,FILE=NAME1,STATUS= NEW ) Abu Hasan Abdullah

31 ELSE OUP = 6 ENDIF WRITE(6,*) Select amount of output WRITE(6,*) 1. Answer only WRITE(6,*) 2. All intermediate approximations WRITE(6,*) Enter 1 or 2 WRITE(6,*) READ(5,*) FLAG WRITE(OUP,*) BISECTION METHOD IF (FLAG.EQ.2) THEN WRITE(OUP,004) 004 FORMAT(3X, I,15X, P,12X, F(P) ) ENDIF C STEP 1 I=1 C STEP IF (I.GT.N0) GOTO 020 C STEP 3 C COMPUTE P(I) P=A+(B-A)/2 Abu Hasan Abdullah

32 FP=F(P) IF (FLAG.EQ.2) THEN WRITE(OUP,005) I,P,FP 005 FORMAT(1X,I3,2X,E15.8,2X,E15.8) ENDIF C STEP 4 C C STEP 5 I=I+1 C STEP 6 C IF( ABS(FP).LE.1.0E-20.OR. (B-A)/2.LT. TOL) THEN PROCEDURE COMPLETED SUCCESSFULLY WRITE(OUP,002) P, I, TOL GOTO 040 ENDIF COMPUTE A(I) AND B(I) IF( F(A)*FP.GT. 0) THEN A=P ELSE B=P ENDIF GOTO 016 Abu Hasan Abdullah

33 020 CONTINUE C STEP 7 C PROCEDURE COMPLETED UNSUCCESSFULLY WRITE(OUP,003) N0 IF(OUP.NE.6) WRITE(6,003) N0 040 CLOSE(UNIT=5) CLOSE(UNIT=OUP) IF (OUP.NE.6) CLOSE(UNIT=6) STOP 002 FORMAT(1X, THE APPROXIMATE SOLUTION IS,1X *,E15.8,1X, AFTER,1X,I2,1X, ITERATIONS, WITH TOLERANCE *,1X,E15.8) 003 FORMAT(1X, THE METHOD FAILS AFTER,1X,I2,1X, *ITERATIONS ) END Abu Hasan Abdullah

34 References 1. Singiresu S. Rao (2002): Applied Numerical Methods for Engineers and Scientists, ISBN X, Prentice Hall 2. Steven C. Chapra, Raymond P. Canale (2006): Applied Numerical Methods for Engineers and Scientists, 10ed, ISBN , McGraw-Hill 3. David Kincaid and Ward Cheney (1991): Numerical Analysis: Mathematics of Scientific Computing, ISBN , Brooks/Cole Publishing Co. 4. Steven C. Chapra (2005): Applied Numerical Methods with MATLAB for Engineers and Scientists, ISBN , McGraw-Hill 5. William J. Palm III (2005): Introduction to Matlab 7 for Engineers, ISBN , McGraw-Hill Abu Hasan Abdullah