1. ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 2 Mathematical Modeling and Engineering Problem Solving

2. Objectives Introduce Mathematical Modeling Analytic vs. Numerical Solution

3. Problem Solving Process Understanding of Physical Problem Observation and Experiment Repetition of empirical studies Fundamental Laws

4. Problem Solving Process Physical Problem Mathematical Model DataTheory Numeric or Graphic Results Implementation

5. Mathematical Model A formulation or equation that expresses the essential features of a physical system or process in mathematical terms Dependent Variable =f Independent Variables Forcing Functions Parameters

6. Mathematical Model Dependent Variable Reflects System Behavior Independent Variable Dimensions Space & Time Parameters System Properties & Composition Forcing Function External Influences acting on system

7. Mathematical Model Change = Increase - Decrease Change  0 : Transient Computation Change = 0 : Steady State Computation Expressed in terms of

8. Mathematical Model Fundamental Laws Conservation of Mass Conservation of Momentum Conservation of Energy

9. A Simple Model Dependent Variable Velocity (v) Independent Variable Time (t) Parameters Mass (m), Shape (s) Forcing Function Gravity, Air resistance FuFu FDFD

10. A Simple Model Fundamental Law Conservation of Momentum Force Balance (+) FDFD FiFi FuFu c=Drag Coefficient

11. A Simple Model FuFu FDFD FiFi

12. Describes system in Mathematical Terms Represents an Idealization and Simplification ignores negligible details focuses on essential features Yields Reproducible Results use for predictive purposes

13. Analytic vs Numerical Solution Analytic Solution

14. Analytic vs Numerical Solution m=68.1 kg c=12.5 kg/s g=9.8 m/s 2 t  0 t (s)v (m/s) 0.0 216.40 427.77 635.64 841.10 1044.87 1247.49  53.39 Analytic Solution

15. Analytic vs Numerical Solution Transient Steady State Practical purposes

16. Analytic vs Numerical Solution Numerical Solutions Techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations

17. Analytic vs Numerical Solution Start from Governing Equation Derivative = Slope

18. Analytic vs Numerical Solution vivi titi True Slope

19. Analytic vs Numerical Solution Use Finite Difference to Approximate Derivative vivi titi t i+1 v i+1 True Slope Approximate Slope

20. Analytic vs Numerical Solution

21. Numerical Solution Slope New Value Old ValueStep Size Euler Method

22. Analytic vs Numerical Solution Procedure 1. Select a sequence of time nodes 2. Define initial conditions (e.g. v(t=0) ) 3. For each time node evaluate

23. Analytic vs Numerical Solution t (s)v (m/s) 0.00 219.6 432 639.85 844.82 1047.97 1249.96

24. Homework Problems 1.6, 1.8 Also Resolve parachutist problem using the numerical solution developed in class with: (a) Time intervals 1 (s), (b) Time intervals 0.5 (s), for the first ten sec. of free fall. Plot the solutions and discuss the error as compared to the analytic solution DUE DATE: Wednesday September 3.