Numerical Solutions to Civil Engineering Problems PDF

INTRODUCTION Numerical Solutions to Civil Engineering Problems 1st Semester AY 2023-2024 OUTLINE Ø Mathematical Model -- Conservation Laws Ø Numerical Methods Ø Programming Ø Errors -- Significant Figures -- Accuracy and Precision -- Error Definitions -- Error Types A...

INTRODUCTION Numerical Solutions to Civil Engineering Problems 1st Semester AY 2023-2024 OUTLINE Ø Mathematical Model -- Conservation Laws Ø Numerical Methods Ø Programming Ø Errors -- Significant Figures -- Accuracy and Precision -- Error Definitions -- Error Types A large egg (M ≈ 67 g) at room temperature (Tegg = 21 °C) in boiling water (Twater = 100 °C) will take 3.6 minutes to reach a temperature Tyolk = 63 °C when soft-boiled, and 8.2 minutes to reach a temperature Tyolk = 80 °C when hard-boiled. Source: https://newton.ex.ac.uk/teaching/CDHW/egg/ A large egg (M ≈ 67 g) at room temperature (Tegg = 21 °C) in boiling water (Twater = 100 °C) will take 3.6 minutes to reach a temperature Tyolk = 63 °C when soft-boiled, and 8.1 minutes to reach a temperature Tyolk = 80 °C when hard-boiled. Source: https://newton.ex.ac.uk/teaching/CDHW/egg/ egg image: Horst Frank (CC BY-SA 3.0) MATHEMATICAL MODEL Dependent æ independent forcing ö = fç , parameters, ÷ variable è variables functions ø Dependent variable – Reflects the behavior or state of the system Independent variables – Dimensions along which the system’s behavior is being determined MATHEMATICAL MODEL Dependent æ independent forcing ö = fç , parameters, ÷ variable è variables functions ø Parameters – Reflects the system’s properties and composition Forcing functions – External influences acting upon the system MATHEMATICAL MODEL Ø The actual mathematical expression can range from a single algebraic relationship to large complicated sets of differential equations. EXAMPLE: Newton’s 2nd Law of Motion F F = ma Þ a= -- Describes a natural process in m mathematical terms What if we wanted to -- Represents an idealization and predict the velocity of a simplification of reality free-falling body near -- Yields reproducible results and the Earth’s surface as a can be used for predictive function of time? purposes MATHEMATICAL MODEL *The mathematical model becomes: F dv F a= Þ = m dt m Ø The net force is expressed in terms of measurable variables and parameters. FD = mg ( gravity ) F = FD + FU FU = -cd v 2 ( air resistance ) MATHEMATICAL MODEL *The mathematical model becomes: dv cd 2 =g- v dt m Ø Solution (if jumper is initially at rest): gm æ gcd ö v (t ) = tanhç t÷ Analytical or cd ç m ÷ closed-form solution è ø *However, most mathematical models cannot be solved exactly… MATHEMATICAL MODEL CONSERVATION LAWS Change = Increases - Decreases MATHEMATICAL MODEL CONSERVATION LAWS NUMERICAL METHODS Why study Numerical Methods? Ø These are techniques by which Math problems are formulated in order to be solved using arithmetic and logical operations -- Reduces higher Math to basic arithmetic operations; reinforces understanding of Math Ø Greatly expands the types of problems to be solved -- Large system of equations, nonlinear systems’ complicated geometries, etc. NUMERICAL METHODS Why study Numerical Methods? Ø Efficient way to learn how to use computers -- Numerical methods are designed for computer implementation -- Allows the designing of programs/scripts to solve problems involving Math models NUMERICAL METHODS PREVIEW OF TOPICS Ø Root-finding (Nonlinear systems) -- Solve for x so that f(x) = 0. NUMERICAL METHODS PREVIEW OF TOPICS Ø Linear Algebraic Equations -- Given the a’s and the b’s, solve for the x’s. a11 x 1 + a12 x2 = b1 a21 x 1 + a22 x2 = b2 NUMERICAL METHODS PREVIEW OF TOPICS Ø Curve fitting Ø Integration Ø Differentiation NUMERICAL METHODS PREVIEW OF TOPICS Ø Differential Equations -- Given dy Dy = = f (t , y ) dt Dt Solve for y as a function of t. y i + 1 = y i + f ( ti , y i ) Dt PROGRAMMING Ø Process of creating a set of instructions that tell a computer how to perform a task Ø Done for automation purposes Ø “Garbage in, garbage out” / GIGO PROGRAMMING LANGUAGES/SOFTWARES Ø C/C++, Java, SQL, VBA, Python Ø Mathcad, MATLAB, Octave, Maple, Mathematica Source: https://www.boredpanda.co m/14-year-old-draws-comics- on-whiteboard-jakes-door/ PROGRAMMING TERMINOLOGIES Ø Algorithm -- Sequence of logical steps required to perform a specific task Ø Pseudocode -- English description of a program Ø Flowchart -- Visual/graphical representation of a program PROGRAMMING EXAMPLE: Write a computer program to calculate sin(x) using the following Taylor series expansion: ( ) 2n+1 n ¥ -1 x 3 x 5 x 7 x sin( x ) = å = x - + - + n=0 (2n + 1 )! 3! 5! 7! PROGRAMMING Algorithm Ø Enter x and the number of terms to use, N. Ø Calculate sin(x) using the Taylor series. Ø Print the result. PROGRAMMING Pseudocode Flowchart PROGRAMMING PROGRAMMING ERRORS (BUGS) Ø Syntax error -- Will not compile. Compiler will help you to find it. -- (e.g. Writing primtf instead of printf) Ø Run-time error -- Will compile but may stop running at some point. -- (e.g. Division by zero) PROGRAMMING PROGRAMMING ERRORS (BUGS) Ø Logical error -- Will compile and run, but the result is wrong. -- (e.g. From the previous example, all terms of sin(x) are positive) *Run-time and logical error need careful debugging of the program. PROGRAMMING NOTES: Ø For clarity: -- Put comments to explain variables, purpose of code segments, etc. Ø For testing: -- Run with typical data. -- Run with unusual but valid data. -- Run with invalid data to check error handling. PROGRAMMING NOTES: Ø Refresh your programming knowledge. Ø Find an introductory level programming book in the library/internet; check for tutorials. Ø Approach the problem through the algorithm-pseudocode-computer code link. Ø Practice at home (or during your free time). Ø Do not be afraid of programming; try to feel the power of it. =) ERRORS SIGNIFICANT FIGURES Ø Digits of the number which are known to be correct Ø Designate the reliability of a number *The number of significant figures to be used in a number depends on the origin of the number. EXAMPLE: Calculation of the area of a circle A = p D2 / 4 ERRORS ACCURACY and PRECISION Ø Computers used to obtain numerical solution are imperfect tools; limited to represent the magnitudes and precision of numbers. Ø Errors from calculations (and measurements) can be characterized with regard to their accuracy AND precision. ERRORS Ø Precision -- How closely a computed (individual) or measured values agree with each other Ø Accuracy -- How closely a computed or measured value agrees with the true value ERRORS Imprecision Ø Also called uncertainty Ø Magnitude of the scatter Inaccuracy Ø Also called bias Ø Systematic deviation from the truth *ERROR = Imprecision + Inaccuracy ERRORS NOTES: Ø Precision is governed by the number of digits being carried in the numerical calculations. Ø Accuracy is governed by the errors in the numerical approximation. ERROR DEFINITIONS TRUE ERROR (Et) Ø Exact/absolute value of the error Ø Doesn’t consider order of magnitude of the value under examination Et = True Value - Approximation ERROR DEFINITIONS TRUE ERROR (Et) Ø The error can be normalized to the true value -- Fractional Relative True Error True Value - Approximation et = True Value -- Percentage Relative True Error True Value - Approximation et = ´ 100% True Value ERROR DEFINITIONS APPROXIMATE ERROR (Ea) Ø In actual situations, true value is rarely available Ø Error estimates (or approximations) are determined in absence of knowledge regarding the true value Approximate Error ( E A ) ea = Approximation ERROR DEFINITIONS APPROXIMATE ERROR (Ea) Ø Utilize the best available estimate of the true value, then normalize -- Fractional Relative Approximate Error Present Approx. - Previous Approx. ea = Present Approx. -- Percentage Relative Approximate Error Present Approx. - Previous Approx. ea = ´ 100% Present Approx. ERROR DEFINITIONS NOTES: Ø Many numerical methods work in an iterative* fashion, thus there should be a stopping criterion for these methods. Ø The process stops when the error level drops below a certain specified tolerance value (εs). ea < es *A procedure in which repetition of a sequence of operations yields results successively closer to a desired result. ERROR DEFINITIONS SCARBOROUGH CRITERION (1966) Ø If the tolerance is selected to be e s = 0.5 ´ 10 % 2-n then the approximation is guaranteed to be correct to at least n significant figures. ERROR TYPES ROUND-OFF ERROR Ø Computers cannot -- Represent some quantities exactly * Conversion from base-10 to base-2 number system may create problems * Numbers like π or 1/3 -- Use infinitely many digits to store numbers * Single precision (7-8 digits) vs. Double precision (15-16 digits) ERROR TYPES ROUND-OFF ERROR Ø Sources of Round-off Errors -- Subtractive cancellation * Subtracting two nearly equal numbers -- Large (arithmetic) computations * Computations are often interdependent; consider cumulative effect -- Adding a large and a small number * Usually occur in infinite series calculation * Sum the series in reverse order ERROR TYPES ROUND-OFF ERROR Ø Sources of Round-off Errors -- Smearing * Individual terms in a summation are larger than the summation itself * Example: Series of mixed signs -- Calculation of inner products n åx y i =1 i i =x 1 y 1 + x 2 y 2 +  + x n y n ERROR TYPES ROUND-OFF ERROR Ø These errors cannot be totally eliminated, but clever algorithms may help to minimize them. TRY AT HOME: Add 0.1 thousand times. Use both single and double precision. Compare the results. =) ERROR TYPES TRUNCATION ERROR Ø Result from using an approximation in place of an exact mathematical procedure EXAMPLE: Calculating the sine of a number using finite number of terms from the infinite series will result in truncation error. ERROR TYPES TRUNCATION ERROR ( ) 2n+1 n ¥ -1 x 3 5 x x7 x sin( x ) = å = x - + - + n=0 (2n + 1 )! 3! 5! 7! At x = π/2: ERROR TYPES OVERVIEW OF TAYLOR SERIES Ø Taylor Series is used to evaluate a function at one point, using the value of the function and its derivatives at another point. Known: f ( x i ) , f ' ( x i ) , f '' ( x i ) , etc. Unknown: f ( x i +1 ) 0 Order Approx.: f ( x i + 1 ) » f ( x i ) th ERROR TYPES OVERVIEW OF TAYLOR SERIES Known: f ( x i ) , f ' ( x i ) , f '' ( x i ) , etc. Unknown: f ( x i +1 ) st 1 Order Approx.: df Df f ( x i + 1 ) - f ( x i ) h = x i +1 - x i f '( x i ) = » = dx Dx h \ f ( x i + 1 ) » f ( x i ) + hf ' ( x i )  ERROR TYPES OVERVIEW OF TAYLOR SERIES Ø More derivatives can be added to capture more of the function’s curvature. f '' ( x i ) f ( n) ( x i ) hn + R f ( x i +1 ) = f ( x i ) + f '( x i ) h + h + + 2 n 2! n! -- nth order Taylor series approximation of f(xi+1) -- Being an infinite series, an equal sign replaces the approximate sign -- The approximation improves as h decreases  ERROR TYPES OVERVIEW OF TAYLOR SERIES Ø More derivatives can be added to capture more of the function’s curvature. f '' ( x i ) f ( n) ( x i ) hn + R f ( x i +1 ) = f ( x i ) + f '( x i ) h + h + + 2 n 2! n! -- Rn is the remainder (truncation error) term n+1 h Rn = f ( n+1 ) (x ) where x i < x < x i + 1 ( n + 1 )! ERROR TYPES OVERVIEW OF TAYLOR SERIES Ø nth order Taylor series expansion will be exact if f(x) is an nth order polynomial Ø For practical purposes, inclusion of only a few terms will result in an approximation that is close enough to the true value -- Truncation error can be controlled (h) -- Choice on how far away from x to evaluate f(x) ERROR TYPES OVERVIEW OF TAYLOR SERIES n+1 h Rn = f ( n+1 ) (x ) ( n + 1 )! \ Rn = O ( h n+1 ) Ø Truncation error is of the order of hn+1 Ø Useful in judging the comparative error of numerical methods based on Taylor series expansions ERROR TYPES TOTAL NUMERICAL ERROR Ø Summation of the truncation and round-off errors Choose a large step size to decrease the amount of calculations and round off errors without incurring the penalty of a large truncation error. ERROR TYPES OTHER SOURCES OF ERRORS Ø Gross Errors/Blunders Ø Model Errors Ø Data Uncertainty