Intro to Engineering Modeling
Intro to Engineering Modeling EGR 102
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This 11 page Class Notes was uploaded by Leo Johnson Sr. on Saturday September 19, 2015. The Class Notes belongs to EGR 102 at Michigan State University taught by Timothy Hinds in Fall. Since its upload, it has received 10 views. For similar materials see /class/207513/egr-102-michigan-state-university in Engineering and Tech at Michigan State University.
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Date Created: 09/19/15
g EGR 102 Introduction to Engineering Modeling Ordinary Differential Equations The Midpoint Method Figures from Applied Numerical Methods with MATLABquot Steven Chapra McGraw Hill EGR 102 Lecture 15 1 Objectives Learn how to implement the following RungeKutta RK methods for a single ODE Euler s Method Heun s Method Midpoint Method EGR 102 Lecture 15 2 Ordinary Differential Equations ODEs Equations which are composed of an unknown function and its derivatives are called differential equations Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change I c h v dependent variable 4 g TV d m t independent var1ab1e EGR 102 Lecture 15 3 Ordinary Differential Equations ODEs Methods described here are for solving differential equations of the form dy t dt f y o The methods in this chapter are all onestep methods and have the general format n1 y h where is called an increment function and is used to extrapolate from an old value yto a new value y1 EGR 102 Lecture 15 4 g RungaKutta Methods FirstOrder Equations Euler s Method Heun s Method Midpoint Improved Polygon Method SecondOrder R K Methods Heun Method with Single Corrector Midpoint Method Ralston s Method EGR 102 Lecture 15 g Euler s Method The rst derivative provides a direct estimate of the slope at X f ya where f is the differential equation evaluated at and y This estimate can be substituted into the equation i 1 139 A new value of yis predicted using the slope to extrapolate linearly over the step size h EGR 102 Lecture 15 EGR 102 4 Step size I Euler s Method amp Solutions True solullon Mu Onestep method with step size comparisons Lecture 15 One method to improve the estimate of the slope involves the determination of two derivatives for the interval At the initial point At the end point EGR 102 The two derivatives are then averaged to obtain an improved estimate of the slope for the entire interval 3 amen Heun s Method 393m 8 3 4 Lecture 15 Heun s Method vu v I 39 I I fr39L S ope am P V921quot Slope g a Predictor amp b Corrector EGR 102 Lecture 15 9 l Heun s Method y Euler39s method 5 Heun s method True solution 3 10 Lecture 15 EGR 102 g Midpoint Method The Midpoint Method Also called Improved Polygon or Modi ed Euler Method Uses Euler s method to predict a value of yat the midpoint of the interval JilinZ at f 112 EGR 102 Lecture 15 11 39 Mid oint Method Another improvement to Euler s method is similar to Heun s method but predicts the slope at the midpoint of an interval rather than at the end slope 4hr ml ml EGR 102 Lecture 15 12 Midpoint Method Steps Predict a value of y at the midpoint of interval yi12 y ftiyi Use the predicted value to calculate slope at the mid oint dy p fti12 yi12 dt i12 Use this new slope to extrapolate the solution linearly from the beginning of the interval t to the end rm yi1 y fti129yil2h EGR 102 Lecture 15 13 g Midpoint Method Example Use the Midpoint Method to numerically integrate 2 2x3 12x2 20x85 dx from x0 to x4 with a step size of 05 The initial condition at x0 is y1 EGR 102 Lecture 15 14 Midpoint Method Example x g 2x3 12x2 20x85 y01 at x00 h05 yi1 yi fxi12yi12h K 1 100000 05 310938 1 281250 15 198438 2 175000 EGR 102 m gh2 ydyldx h2 85 025 312500 125 075 342188 15 125 243750 125 175 167188 05 225 187500 Lecture 15 15 g RungeKutta Methods EGR 102 H H H Analytical Euler Heun Midpuint H Halston Lecture 15 g Lecture Homework Homework 14 due today Solutions will be posted Homework 15 posted today Will not be collected or graded Solutions will be posted EGR 102 Lecture 15 17 39 Final Exam Thurs 505 300pm to 500pm C102 Wilson Hall 5 6 problems Format similar to lecture exams Cumulative of all lecture material but emphasis will be placed on lecture material not covered on Exams 1 or 2 EGR 102 Lecture 15 18 I Final Exam All material up to amp including today Cost Engineering Singleannual amp interest payment calculations Curve Fitting Linear amp polynomial least squares amp correlations Iterative Root Calculations Bisection method NewtonRaphson method EGR 102 Lecture 15 19 I Final Exam All material up to amp including today Optimization Newton s method Numerical Integration Trapezoidal rule Simpson s 13 amp 38 rules Composites Ordinary Differential Equations Euler s method Midpoint method EGR 102 Lecture 15 20 10 I Final Exam I Format 56 problems Bring a calculator amp extra batteries Cheat sheet Single 85 x 11 piece of paper Both sides Handwritten only Turn in with exam name PID amp lab on paper Rework Lecture HW Quizzes amp Examples EGR 102 Lecture 15 21 g Grades Check accuracy of grades posted on Angel Only mechanism for determining course grades Grades determined per syllabus All but Lecture HW 14 amp Quiz Lab Project amp Final Exam scores should be in Angel EGR 102 Lecture 15 22 11
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