Application of Numerical Method Eng'Ng

8/12/2019 Application of Numerical Method Eng'Ng

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Applications of Numerical Methods in EngineeringCNS 3320

James T. AllisonUniversity of Michigan

Department of Mechanical Engineering

January 10, 2005

University of Michigan Department of Mechanical Engineering


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Applications of Numerical Methods in Engineering

Objectives:

Motivate the study of numerical methods through discussion of

engineering applications.

Illustrate the use of Matlab 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 of Functions Solving Systems of Equations Optimization Numerical Integration and Differentiation

Selected Additional Applications

Matlab Example: Fixed Point Iteration

Matlab Example: Numerical Integration



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Engineering: Solving practical technical problems using scientic andmathematical tools when available, and using experience and intuitionotherwise.

Mathematical models provide a priori estimates of performance verydesirable when prototypes or experiments are costly.

Engineering problems frequently arise in which exact analytical solutionsare not available.

Approximate solutions are normally sufficient for engineering applications,allowing the use of approximate numerical methods.



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Analysis Predicting the response of a system given a xed system design and operatingconditions. 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)

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

Maximizing a vehicles fuel economy while maintaining adequate performance levelsby 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 of Numerical Methods and Applications

Linearization

Finding Roots of Functions

Solving Systems of Equations

Optimization

Numerical Integration and Differentiation



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Linearization

Nonlinear equations can be much more difficult to solve than linear equations. Taylors series expansion provides a convenient way to approximate a nonlinear equation

or function with a linear equation.

Accurate only near the expansion point a .

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

2! (x a ) 2 + . . .

Linear approximation uses rst two terms of the expansion.



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Linearization Example: Swinging Pendulum

Sum forces in tangential direction:

F t = W t = mgsin = ma t

= md 2dt 2

L = m L

g

L sin = 0Linearize sin :

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

Equation of motion valid for small

angles: gL

= 0m

T

W=mg

m

W t

W r



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Finding Roots of Functions

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

Frequently cannot be solved analytically in engineering applications. Transcendental equations Black-box functions

May have multiple or innite solutions

Example: static equilibrium problems must satisfy

F = 0

M = 0



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Root Finding Example- Statically Indeterminate Structural Analysis

beam 1

beam 2

beam 3

beam nb

.

.

.

F 1

rod 1

rod 2

rod nb-1

d nb

d 3

d 2

d 1

d r(nb-1)

d r2

d r1

L

lr1

lr2

lr(nb-1)

.

.

.



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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 for the force on the top beam2. Calculate the required applied force to generate this top beam force3. Compare to actual applied force, iterate until they match

Solve: F 1(F 3) F 1 = 0

F 33

r2a d

2

f

F 2b

r1

e

1 F 1g c



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Solving Systems of Equations

Solve for the value of the vector x that satises the given system of equations.

Linear Systems:

a 11 x 1 + a 12 x 2 = b1a 21 x 1 + a 22 x 2 = b2 a 11 a 12a 21 a 22 x 1x 2 = b1b2 Ax = b

Non-Linear Systems:

f 1(x ) = b1f 2(x ) = b2 f (x ) = b



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Linear Systems Example: Circuit Analysis

Kirchhoffs Laws:

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

n e

i =1

V i = 0

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

n b

i =1

I i = 0

Application of these two laws to an electrical circuit facilitates the formulation of a systemof n linear equations when n unknown quantities exist.



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Given that R 1 = 2 , R 2 = 4 , R 3 = 1 , E 1 = 6 V , E 2 = 9 V , and usingequations from Loop 1, Loop 2, and Node A we nd:

6 2I 1 I 3 = 09 + 4 I 2 I 3 = 0

I 1 + I 2 + I 3 = 0

2 0 10 4 1

1 1 1

I 1I 2I 3

= 6 90

Ax = b

Loop 1 Loop 2

Node A

Node B

R 1 R 3 R 2

I1 I3 I2

1 2



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Matlab Implementation



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Nonlinear Systems Example: Turbine Blade Analysis

Turbine blades are components of gas-turbine engines (used for aircraftand electricity generation)

Subject to high temperatures, highinertial forces, and high drag forces.

Commonly constructed of monocrystalline alloys such asInconel.

Structural and thermal analysesmust be performed simultaneously(coupled non-linear equations).



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Nonlinear Systems Example: Turbine Blade Analysis

Methods apply to arbitrary non-linearequations (black-box functions)

T (x ) = f 1(L )

L = f 2(T (x ))

w

t

L0

x

vg , T g

f ac

ThermalAnalysis

StructuralAnalysis

T(x) (temperatureprofile)

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

Maximize performance criteria subject to failure constraints:

Minimize bicycle frame weight subject to structural failure constraints by varying frameshape and thickness.

Minimize cost subject to performance and failure constraints.

Minimize vehicle cost subject to acceleration, top speed, handling, comfort, and safetyconstraints 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 error between the

approximating function and the actual data points.

Criteria for best t:

SSE =n

i =1

( f i f i (p )) 2

minp

SSE (p )



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Optimization- Equilibrium in Nature

Gravitational Potential Energy Objects seek position 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 seek positions that 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 pushatoms 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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Numerical Integration and Differentiation

Solve:

b

af (x )dx

df (x )dx

where f (x ) is an arbitrary continuous function.

Numerical approaches may be 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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Numerical Integration Example: Falling Climber

Falling rock climber possesses kinetic energyT (energy of motion) that must be absorbedby the belay system.

T = F d rT at time of impact can be determinedanalytically. Since F v = m d vdt v , andusing the product rule ddt (v v ) = 2

d vdt v

d v v = 12 (v v ) .

F d r = md v v = 1

2

md (v v ) = 1

2

md v 2

T = r 2

r 1F dr =

v22

v 21

12

md v 2 = 12

m v 22 v21



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Numerical Integration Example: Falling Climber

T can be determined analytically, howthe rope deects requires numericalmethods.

T = V = f

0F d r

The rope behaves as a nonlinear spring,and the force the rope exerts F is anunknown function of its deection .

F ( ) determined experimentally withdiscrete samples.

Approximation of F ( ) necessitatesnumerical integration.

Solving for f requires a root ndingtechnique.

V=T

F

f



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Numerical Integration Example: Position Calculation

Accelerometer: measures second time derivative of position.Application: determining position from discrete set of acceleration values (robotics).

a = x = d2xdt 2

= dxdt

x = x 0 + t

0xdt

x = x 0 +

t

0

xdt



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Numerical Differentiation Example: Solid Mechanics

Objective: Determine stress within a loadedobject to predict failure.

Constitutive Law:

= E = E du

dx

Photoelasticity Example:

Displacement u determined experimentally atdiscrete points, facilitating the calculation of dudx and .



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Selected Additional Applications

Numerical solutions to differentialequations 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


Application of Numerical Method Eng'Ng

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