Download presentation

Presentation is loading. Please wait.

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.

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google