Transcript Introduction - Texas A&M University

CVEN302-501
Computer Applications in
Engineering and Construction
Dr. Jun Zhang
Teaching Assistants
Mr. TBA
(Section 501 and computer lab)
Office: Room
Phone:
Email:
Office Hour:
Lab Hours: Mon & Thur 7:00-9:00pm
Course 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”

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”

Writing the solutions as a series of steps a
computer can execute
 flow chart and pseudo-code
Objective: “Implementation”
 Converting the pseudo-code solution into a
computer program.
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”
 Critical evaluation of the solution
the program gives for the actual
engineering problem.
 This requires all your engineering
and computer skills
Objective: “Presentation”

Communication of the results of a computer
program to the people who need to know the
answer




Clients
Bosses
Regulators
Contractors
Chapter 1
Mathematical Modeling and
Engineering Problem Solving
Engineering Problems
Empirical
 observation and experiment
 certain 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 laws
F  ma ;
  E
 Algebraic
 ODE
 PDE
dv
d2x
F m
m 2
dt
dt
 2T  2T
 2 q
2
 x  y
Mathematical 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 Modeling
The process of solving an engineering or physical problem.
Engineering or Physical problems
(Description)
Mathematical Modeling
Approximation & Assumption
Formulation or Governing
Equations
Analytical & Numerical Methods
Solutions
Applications
Common features
operation
Bungee 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
Bungee Jumper / Falling Parachutist
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?
 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.
Exact (Analytic) Solution
 Newton’s Second Law
dv
m
 mg  c d v 2
dt
cd 2
dv
 g v
dt
m
 Exact Solution
v( t ) 
 gc d 
mg
tanh
t 
cd
 m 
Numerical Method
 What if cd = cd (v)  const?
 Solve the ODE numerically!
dv
v
 lim
dt t 0 t
v v ( t i  1 )  v ( t i )

t
ti1  ti
Assume constant slope
(i.e, constant drag force)
over t
Numerical (Approximate) Solution
 Finite difference (Euler’s) method
dv v v( t i  1 )  v( t i )


dt t
ti1  ti
v( t i 1 )  v( t i )
cd
 g  v( t i ) 2
ti1  ti
m
 Numerical Solution
cd

2
v( t i  1 )  v( t i )   g  v( t i ) ( t i  1  t i )
m


Example 1.2 Hand Calculations
A 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.
cd

2
v( t i  1 )  v( t i )   g  v( t i ) ( t i  1  t i )
m


t0  0; v( t0 )  0
Explicit time-marching scheme
m = 68.1 kg, g = 9.81 m/s2, cd = 0.25 kg/m
Euler’s Method
 Use a constant time increment t = 2 s
Step 1
Step 2
0.25


v  0  9.81 
(0 ) 2  ( 2  0 )  19.6200m / s
68.1


0.25


t  4 s; v  19.6200  9.81 
( 19.6200) 2  ( 4  2 )  36.4317m / s
68.1


t  2 s;
Step 3
t  6 s;
Step 4
t  8 s;
Step 5
Step 6
0.25


v  36.4137  9.81 
( 36.4137) 2  (6  4 )  46.2983m / s
68.1


0.25


v  46.2983 9.81 
( 46.2983) 2  ( 8  6 )  50.1802m / s
68.1


0.25


t  10s; v  50.1802  9.81 
( 50.1802) 2  ( 10  8 )  51.3123m / s
68.1


0.25


t  12s; v  51.3123 9.81 
( 51.3123) 2  ( 12  10)  51.6008m / s
68.1


The solution accuracy depends on time increment
Example: Bungee Jumper
Olympic 10-m Platform Diving
Air :
a
dv
2
m
 mg  c da v 
mg
dt

w
dv
Water : m
 mg  cdw v v 
mg
dt

Buoyant Force
cda  cdw
Example: Finite Elements and
Structural Analysis
Simple truss - force balance
Complex truss
Instead of limiting our analysis to simple cases,
numerical method allows us to work on realistic cases.