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Example: structural resistance of a bridge
Measurements:
describe the geometry,
material properties
Finite element
model:
discrete representation of the
bridge
Numerical
algorithm
Stresses
prediction Numerical solution
Numerical
methods
Physical
validation
Physical description
and modeling
Engineering problems
Data Model
Experimental
expertise
Statistics &
data analysis
Physics &
Mechanics
Mathematics
Numerical
analysis
Computer
programming
Other examples
America’s Cup race boats design Schooling of fishes and collective locomotion
(from National Geographic)
Flying with flexible wings
(from Thomas et al., J. Exp. Biol., 2004)
(from Combes & Daniel, J. Exp. Biol., 2003)
Numerical simulation is an important tool to create models of engineering or
biological systems to understand their properties or optimize a design.
Lecture Outline
I. Solving an engineering problem
1. Examples of engineering problems and general form
2. Four important steps to model an engineering problem
3. Four possible sources of errors
4. Constraints on scientific computing
II. Numerical methods - importance and overview
1. What are numerical methods?
2. Why should we study numerical methods?
3. Why Matlab?
4. Overview of the numerical methods studied in this class
Four steps to model an engineering problem
A. Physical description
B. Mathematical formulation
C. Solution of the mathematical model
D. Interpretation and validation
A simple example: golf ball trajectory
How far does the golf ball travel before hitting the ground?
Given an initial swing velocity U, what is the optimal angle α to
maximize this distance?
U
!
Four steps to model an engineering problem
A. Physical description
- identify the important physical phenomena
- give an accurate description of the system
- make relevant assumptions
B. Mathematical formulation
C. Solution of the mathematical model
D. Interpretation and validation
Golf ball trajectory: physical description
!
U
A. Physical description
B. Mathematical formulation
C. Solution of the mathematical model
D. Interpretation and validation
✓ Physical description of the problem
• Measurements of the ball characteristics: radius, mass M, surface structure...
• Approximation: represent the ball as a perfect sphere (only an approximation!)
• Swing characteristics: initial velocity U, initial angle α.
✓ What are the forces applied on the ball after it is hit? How significant are they?
• Weight
• Air resistance
• Lift force
• Coriolis force
• Wind effect
Mg (Δz<<Rearth: assume uniform g)
quadratic drag -c|v|v (neglect surface structure effect)
neglect ball spin
neglect Earth rotation on this time-scale
assume no wind
Four steps to model an engineering problem
A. Physical description
B. Mathematical formulation
- translate the physical ideas into mathematical
equations (e.g. write conservation laws)
- use the physical assumptions to simplify the
model as necessary
C. Solution of the mathematical model
D. Interpretation and validation
Golf ball trajectory: mathematical formulation
!
U
A. Physical description
B. Mathematical formulation
C. Solution of the mathematical model
D. Interpretation and validation
✓ Parameters:
✓ Variables:
✓ Physical principle: conservation of momentum (Newton second law)
✓ Initial conditions:
M
d2x
dt2 =!Fext =! M
d2x
dt2 = M
dv
dt
= Mg − c|v|v
Mass M, radius a, drag coefficient c = !"a2
2 CD
Position x = (x, y), velocity v = (x˙ , y˙)
x(t = 0) = (0, 0), v(t = 0) = (U cos !, U sin !)
Can we neglect the quadratic drag?
?
Simplification: how much is too much?
While modeling a physical system, some assumptions are introduced to
simplify its formulation. How much should the problem be simplified?
➡ If the model is too simplified, some of the physics will be lost and the
solution might not be accurate or relevant for the application
considered.
➡ If the model is not simplified enough, its solution can be difficult (or
impossible) to obtain, even on a modern computer.
Depending on the application and the computational time we can dedicate
to it, the level of simplification will be varied.
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