ECIV 301

Programming & Graphics

Numerical Methods for Engineers

Lecture 2

Mathematical Modeling and Engineering Problem Solving

Objectives

• Introduce Mathematical Modeling

• Analytic vs. Numerical Solution

Problem Solving Process

Understanding of Physical Problem

Observation and Experiment

Repetition of empirical studies

Fundamental Laws

Problem Solving Process

Physical Problem

Mathematical Model

Data Theory

Numeric or Graphic Results

Implementation

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 FunctionsParameters

Mathematical ModelDependent Variable

Reflects System Behavior

Independent VariableDimensions Space & Time

ParametersSystem Properties & Composition

Forcing FunctionExternal Influences acting on system

Mathematical Model

Change = Increase - Decrease

Change 0 : Transient Computation

Change = 0 : Steady State Computation

Expressed in terms of

Mathematical Model

Fundamental Laws

• Conservation of Mass

• Conservation of Momentum

• Conservation of Energy

A Simple ModelDependent Variable

Velocity (v)

Independent VariableTime (t)

ParametersMass (m), Shape (s)

Forcing FunctionGravity, Air resistance

Fu

FD

A Simple ModelFundamental Law

Conservation of MomentumForce Balance

(+)

FD

mgFD Fi

dt

dvmFi

Fu

cvFu c=Drag Coefficient

A Simple Model

0 Dui FFF

0 mgcvdt

dvm

vm

cg

dt

dv

Fu

FD

Fi

A Simple Model

Describes system in Mathematical Terms

Represents an Idealization and Simplificationignores negligible detailsfocuses on essential features

Yields Reproducible Resultsuse for predictive purposes

Analytic vs Numerical Solution

vm

cg

dt

dv

tm

c

ec

gmv 1

Analytic Solution

Analytic vs Numerical Solutionm=68.1 kg

c=12.5 kg/s

g=9.8 m/s2

t 0

t (s) v (m/s)

0.0 0.0

2 16.40

4 27.77

6 35.64

8 41.10

10 44.87

12 47.49

53.39

tm

c

ec

gmv 1

Analytic Solution

Analytic vs Numerical Solution

0

15

30

45

60

0 10 20 30 40

Time (s)

Ve

loc

ity

(m

/s)

Transient Steady State

Practical purposes

Analytic vs Numerical Solution

Numerical Solutions

Techniques by which mathematical problems are formulated so that they can

be solved with arithmetic operations

Analytic vs Numerical Solution

Start from Governing Equation

vm

cg

dt

dv

Derivative = Slope

Analytic vs Numerical Solution

vi

ti

True Slope

Analytic vs Numerical SolutionUse Finite Difference to Approximate Derivative

vi

ti ti+1

vi+1

True Slope

Approximate Slope

ii

ii

tt

tvtv

dt

dv

1

1

Analytic vs Numerical Solution

vm

cg

dt

dv

ii

ii

tt

tvtv

dt

dv

1

1

iii

ii tvm

cg

tt

tvtv

1

1

Analytic vs Numerical Solution

iiiii tttvm

cgtvtv

11

Numerical Solution

SlopeNew Value

Old Value Step Size

Euler Method

Analytic vs Numerical SolutionProcedure

1. Select a sequence of time nodes

2. Define initial conditions(e.g. v(t=0) )

3. For each time node evaluate

iiiii tttvm

cgtvtv

11

Analytic vs Numerical Solution

0

15

30

45

60

0 5 10 15 20 25

Time (s)

Ve

loc

ity

(m

/s)

Analytic Solution

Numerica Solutionl

t (s) v (m/s)

0.0 0

2 19.6

4 32

6 39.85

8 44.82

10 47.97

12 49.96

HomeworkProblems 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.