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Applications ofNumerical Methods inEngineering

CNS 3320

James T. Allison

University ofMichiganDepartment of Mechanical Engineering

January 10, 2005

University of Michigan Department of Mechanical Engineering

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Applications ofNumerical Methods in Engineering

Objectives:

B Motivate the study of numerical methods through discussion of

engineering applications.

B Illustrate the use ofMatlab using simple numerical examples.

University of Michigan Department of Mechanical Engineering January 10, 2005

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Lecture Overview

Quantitative Engineering Activities: Analysis and Design

Selected Categories of Numerical Methods and Applications

Linearization

Finding Roots ofFunctions

Solving Systems ofEquationsOptimization

Numerical Integration and Differentiation

Selected Additional Applications

Matlab Example: Fixed Point Iteration

Matlab Example: Numerical Integration


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QuantitativeEngineering Activities: Analysis and Design

Engineering: Solving practical technical problems using scientific and

mathematical tools when available, and using experience and intuition

otherwise.

B Mathematical models provide a priori estimates of performance very

desirable when prototypes orexperiments are costly.

B Engineering problems frequently arise in which exact analytical solutions

are not available.

B Approximate solutions are normally sufficient for engineering applications,

allowing the use of approximate numerical methods.


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QuantitativeEngineering Activities: Analysis and Design

BAnalysis Predicting the response of a system given a fixed system design and operating

conditions.

060 mph acceleration time of a vehicle (Mechanical Engineering)

Power output of an electric motor (Electrical/Mechanical Engineering)

Gain of an electromagnetic antenna (Electrical Engineering) Maximum load a bridge can support (Civil Engineering)

Reaction time of a chemical process (Chemical Engineering)

Drag force of an airplane (Aerospace Engineering)

Expected return of a product portfolio (Industrial and Operations Engineering)

BDesign Determining an ideal system design such that a desired response is achieved.

Maximizing a vehicles fuel economy while maintaining adequate performancelevels by varying vehicle design parameters.

Minimizing the weight of a mountain bike while ensuring it will not fail

structurally

by varying frame shape and thickness.


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Categories ofNumerical Methods and Applications

Linearization

Finding Roots ofFunctions

Solving Systems ofEquations

Optimization



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2

Linearization

Nonlinear equations can be much more difficult to solve than linearequations.

Taylors series expansion provides a convenient way to approximate a nonlinear

equation or function with a linearequation.

Accurate only near the expansion point a.

f(x) = f(a) + f0(a)(x a)

+

f00(a)

2!(x a)+ . . .

Linear approximation uses first two terms of the expansion.


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Linearization Example: Swinging Pendulum

Sum forces in tangential direction:X

Ft = Wt = mgsin= mat

d2

= mdt2

L = mL

g

sin = 0L

Linearize sin:

sin sin(0) + cos(0)( 0) = T

Equation of motion valid for small

angles: m mg

= 0L

Wr

W=mg

Wt


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Finding Roots of Functions

Find the value ofx such that f(x) = 0

Frequently cannot be solved analytically in engineering applications.

Transcendental equations

Black-box functions

May have multiple or infinitesolutions

Example: static equilibrium problems must satisfy

X

F = 0

X

M = 0


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Root Finding Example- Statically Indeterminate Structural Analysis

L

dnb

beam nb

.

.

.

d3

dr(nb-1)

.rod n

b-1

.

.

lr(nb-1)

beam 3 d

r2

rod 2 lr2

d2

beam2 d

r1

rod 1 lr1 d

1

beam1

F1


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F1

Root Finding Example- Statically Indeterminate Structural Analysis

Force applied to lower beam known

All other forces and displacements unknown

Solution process:

1. Make a guess forthe force on the topbeam

2. Calculate the required applied force to generate this top beam force

3. Compare to actual applied force, iterate until theymatch

Solve: F1(F3) F1 = 0

F3

3

2F

2 r1 1

a dr2

f b e g c


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b

Solving Systems of Equations

Solve for the value of the vector x that satisfies the given system ofequations.

Linear Systems:

a11x1 + a12x2 = b1a21x1 + a22x2 = b2

a11 a12a21 a22

x1

x2

b1=

2 Ax = b

Non-Linear Systems:

f1(x)= b1f2(x)= b2

f(x) = b


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Linear SystemsExample: CircuitAnalysis

Kirchhoffs Laws:

1. The sum of all voltage changes around any closed loop is zero:

neX

i=1

Vi = 0

2. The sum of all currents at any node is zero.

nbX

i=1

Ii = 0

Application of these two laws to an electrical circuit facilitates the formulation of a system

ofn linear equations when n unknown quantities exist.


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I


Given that R1 = 2, R2 = 4, R3 = 1, E1 = 6V , E2 = 9V , and usingequations from Loop 1, Loop 2, andNode A we find:

6 2I1I3 =

0

2 0 1

1

6

9 + 4I2I3 = 0

I1 + I2 + I3 = 0

0 4 1

1 1 1

I2 =

I3

9 Ax = b

0

1 2

Node A

I1

I3

I2

R1

R3

R2

Loop 1 Loop 2

Node B


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Matlab Implementation


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Commonly constructed of

monocrystalline

Inconel.

alloys such as

Nonlinear Systems Example: Turbine Blade Analysis

Turbine blades

are components of

gas-turbine engines (used for aircraft

and electricitygeneration)

Subject to high temperatures,

high inertial forces, and high dragforces.

Structural and thermal

analyses must beperformed simultaneously

(coupled non-linearequations).


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Nonlinear Systems Example: Turbine Blade Analysis

Methods apply to arbitrary non-linear

equations (black-box functions)

T(x) = f1(L)

L = f2(T(x))

vg, T

g

x

L0

T(x)(temperature

profile)

t

fac

w

Thermal

AnalysisStructural

Analysis

L (dilated length)


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Optimization

Find the values of the input variables to a function such that the function is minimized

(or maximized), possibly subject to constraints.

Negative Null Form: minx

f(x)

subject to g(x) 0

h(x) = 0

Applications:

Engineering Design

Regression

Equilibrium in Nature


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Optimization- Engineering Design

Maximizeperformance criteria subject to failure constraints:

Minimize bicycle frame weight subject to structural failure constraints by varying

frame shape and thickness.

Minimize cost subject to performance and failure constraints.

Minimize vehicle cost subject to acceleration, top speed, handling, comfort, and

safety constraints by varying vehicle design variables.


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Optimization- Regression

Regression: technique for approximating an unknown response surface (function).

Sample several points experimentally

Fit an approximating function to the data points, minimizing the errorbetween

the approximating function and the actual data points.

Criteria forbest fit:

SSE =

nX(fif i(p))

2

i=1

minp

SSE(p)


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Optimization- Equilibrium in Nature

Gravitational Potential Energy

Objects seekposition of minimum gravitational potential energy: V = mgh

Bubbles

Energy associated with surface area.

Bubbles seek to minimize surface area spherical shape.

Many small bubble coalesce to form fewer large bubbles.

Atomic Spacing

Atoms seekpositionsthat minimize elastic potential energy.

At large separation distances attractive forces pull atoms together (depends on

bonding type).

At small separation distances repulsive forces due to positively charged nuclei push

atoms apart.

The net force results in an energy well. The steepness of this well determines

material properties, such as thermal expansion.


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Solve:

Z bf(x)dx

a

df(x)

dx

where f(x) is an arbitrary continuous function.

Numerical approaches maybe required when:

f(x) is an analytical function that yields the integration unsolvable

f(x) is known only through discretely sampled data points


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dt

dtdt

2

v

2

Numerical Integration Example: Falling Climber

Falling rock climber possesses kinetic energyT (energy of motion) that must be absorbed

by the belay system.

Z

T = Fdr

T at time of impact can be determinedanalytically. Since F v = mdvv,

and

using the product rule d (v v) = 2dv v

dv v = 1(v

v).

Fdr= mdv v=

Z r2

1 1md(v v) = md

2 2

Z v2

v2

T = F dr=

r1

2 1md v

2

=2 21

1m v

2

2

2

v1

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Numerical Integration Example: Falling Climber

T can be determined analytically, how

the rope deflects requires

numerical

methods.F

Z f

T = V =

0

Fdr

The rope behaves as a nonlinear spring,

and the force the rope exerts F is an

unknown function of its deflection .

F() determined experimentally

with discrete samples.

Approximation ofF()necessitates numerical integration.

Solving forf requires a root

finding technique.

V=T

f


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Numerical Integration Example: Position Calculation

Accelerometer: measures second time derivative ofposition.

Application: determining position from discrete set of acceleration values (robotics).

a = x=d

2x

=dt2

dx

dt

x = x0+

Z txdt

0Z t

x = x0 + xdt

0


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du

Numerical DifferentiationExample: Solid Mechanics

Objective: Determine stress within a loaded

object to predict failure.

ConstitutiveLaw:

du = E = Edx

PhotoelasticityExample:

Displacement u determined experimentally at

discrete points, facilitating the calculation of

dxand .


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Selected Additional Applications

Numerical solutions todifferential equations

Finite Difference Method

Computational Fluid Dynamics

(NavierStokes Equations)

Dynamics (Newton-Euler &

Lagranges equations)

Finite Element Method

Solid Mechanics (Elasticity

equations)

Heat Transfer (Heat equation)

Kinematics Simulation

Complex System

Optimization


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