CVEN302-501CVEN302-501

Computer Applications in Computer Applications in

Engineering and ConstructionEngineering and Construction

Dr. Jun Zhang

TeachingTeaching AssistantsAssistantsMr. TBA

(Section 501 and computer lab)

Office: Room

Phone:

Email:

Office Hour:

Lab Hours: Mon & Thur 7:00-9:00pm

Course ObjectivesCourse Objectives To develop the ability to solve engineering problems

numerically

To evaluate numerical solution methods (knowing the

advantages and limitations of numerical methods)

To design, implement, and test computer programs

To improve the skills of writing MATLAB codes

Objective: “Numerical methods”Objective: “Numerical methods”

Numerical methods give solutions to math problems written as algebraic statements that computers can execute

We will learn to formulate the solutions and evaluate their applicability and performance.

Objective: “Design”Objective: “Design”

Writing the solutions as a series of steps a computer can execute flow chart and pseudo-code

Objective: “Implementation”Objective: “Implementation”

Converting the pseudo-code solution into a computer program.

Objective: “Testing”Objective: “Testing”

Checking that the computer program actually solves the equations you mean to solve

Evaluating the success of the numerical solution chosen

Numerical accuracy, stability, and efficiency

Objective: “Evaluation”Objective: “Evaluation”

Critical evaluation of the solution the program gives for the actual engineering problem.

This requires all your engineering and computer skills

Objective: “Presentation”Objective: “Presentation”

Communication of the results of a computer program to the people who need to know the answer

Clients Bosses Regulators Contractors

Chapter 1Chapter 1

Mathematical Modeling and Engineering Problem Solving

Engineering ProblemsEngineering ProblemsEmpiricalobservation and experimentcertain aspects of empirical studies occur repeatedly such general behavior can be expressed as

fundamental laws that essentially embody the cumulative wisdom of past experience

Theoretical / Numerical formulation of fundamental lawsAlgebraic

ODE

PDE qy

T

x

T

dt

xdm

dt

dvmF

E ;maF

2

2

2

2

2

2

Mathematical ModelsMathematical Models

Modeling is the development of a mathematical representation of a physical/biological/chemical/ economic/etc. system

Putting our understanding of a system into math

Problem Solving Tools: Analytic solutions, statistics, numerical methods,

graphics, etc.Numerical methods are one means by which

mathematical models are solved

Mathematical ModelingThe process of solving an engineering or physical problem.

C o m m o n fe a tu reso p era tion

A p p lica tio ns

S o lu tio ns

A n a lytica l & Num erical Methods

F o rm u la tion o r G o vern ingE q ua tio ns

M a th em a tica l M od e ling A p pro x im a tio n & A ssu m p tion

E n g ine e ring o r P h ys ica l p rob le m s(D e sc rip tio n )

Bungee JumperBungee Jumper

You are asked to predict the velocity of a bungee jumper as a function of time during the free-fall part of the jump

Use the information to determine the length and required strength of the bungee cord for jumpers of different mass

The same analysis can be applied to a falling parachutist or a rain drop

Newton’s Second Law

F = ma = Fdown - Fup

= mg - cdv2 (gravity minus air resistance)

Observations / Experiments

Where does mg come from?

Where does -cdv2 come from?

Bungee Jumper / Falling ParachutistBungee Jumper / Falling Parachutist

Now we have fundamental physical laws, so we combine those with observations to model the system

A lot of what you will do is “canned” but need to know how to make use of observations

How have computers changed problem solving in engineering?

Allow us to focus more on the correct description of the problem at hand, rather than worrying about how to solve it.

Newton’s Second Law

Exact (Analytic) SolutionExact (Analytic) Solution

2d

2d

vm

cg

dt

dv

vcmgdt

dvm

Exact Solution

t

m

gc

c

mgtv d

d

tanh)(

Numerical MethodNumerical Method

i1i

i1i

0t

tt

tvtv

t

v

t

v

dt

dv

)()(

lim

What if cd = cd (v) const? Solve the ODE numerically!Solve the ODE numerically!

Assume constant slope (i.e, constant drag force)

over t

Finite difference (Euler’s) method

Numerical (Approximate) SolutionNumerical (Approximate) Solution

2i

d

i1i

i1i

i1i

i1i

tvm

cg

tt

tvtv

tt

tvtv

t

v

dt

dv

)()()(

)()(

Numerical Solution

)()()()( i1i2

id

i1i tttvm

cgtvtv

Example 1.2 Hand CalculationsExample 1.2 Hand CalculationsA stationary bungee jumper with m = 68.1 kg leaps from a stationary hot air balloon. Use the Euler’s method with a time increment of 2 s to compute the velocity for the first 12 s of free fall. Assume a drag coefficient of 0.25 kg/m.

0tv0t

tttvm

cgtvtv

00

i1i2

id

i1i

)( ;

)()()()(

Explicit time-marching scheme

m = 68.1 kg, g = 9.81 m/s2, cd = 0.25 kg/m

sm6008511012312351168

250819312351vs12t

sm312351810180250168

250819180250vs10t

sm18025068298346168

250819298346vs8t

sm29834646413736168

250819413736vs6t

sm43173624620019168

250819620019vs4t

sm620019020168

2508190vs2t

2

2

2

2

2

2

/.)().(.

... ;

/.)().(.

... ;

/.)().(.

... ;

/.)().(.

... ;

/.)().(.

... ;

/.)()(.

.. ;

Euler’s MethodEuler’s Method

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

The solution accuracy depends on time increment

Use a constant time increment t = 2 s

Example: Bungee JumperExample: Bungee Jumper

Olympic 10-m Platform DivingOlympic 10-m Platform Diving

mgvvcmgdt

dvmWater

mgvcmgdt

dvmAir

wdw

a2da

:

:Buoyant Force

cda cdw

Example: Finite Elements and Example: Finite Elements and Structural AnalysisStructural Analysis

Simple truss - force balance Complex truss

Instead of limiting our analysis to simple cases, numerical method allows us to work on realistic cases.