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problem and the reason why the different assumptions were made.
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 possible sources of errors
Measurements: Finite element
model
Stresses Numerical algorithm
prediction Numerical solution
Numerical
methods
Physical
validation
Physical description
and modeling
➡ Measurement error
➡ Modeling error
➡ Truncation error
➡ Round-off error
Four possible sources of errors
A. Measurement error
➡ Any instrument has a limit on its precision, and an experimental result is
always obtained with a tolerance: e.g. 20.2 ± 0.1 cm
B. Modeling error
➡ Difference between the real system and the simplified description used.
➡ Example of the bridge: representing the elements as homogeneous or
with a simplified geometry.
➡ Example of the golf ball: neglecting the aerodynamic drag
Four possible sources of errors
C. Truncation error (see lecture on Wednesday)
➡ This error arises due to the discrete or iterative nature of the numerical
methods used.
➡ For example, an iterative scheme can be developed to obtain the physical
quantity G.
- If the scheme is well-designed then G(n) approaches the true
value of G0 when n→∞.
- However, we always have to stop at a finite value of n=N.
The truncation error is the difference between G(N) and G(∞).
➡ A truncation error also arises when approximating a continuous quantity
by a discrete form or a derivative by a discrete limit:
D. Round-off error (see lecture on Wednesday)
➡ This error is intrinsic to the use of a computer.
➡ A computer does not use the real number but finite-precision numbers
(e.g. some decimals are discarded.)
df
dx !
f(x + !x) − f(x)
!x
Constraints on scientific computing
✓ In a modern computer, the numerical computations are done in binary
form and the actual calculation takes place in the processor.
✓ Before the operation can be performed, the data on which the
operation is performed must be transmitted to the processor from its
storage location.
✓ The cost (in time) of a computation is the result of two limitations:
- the power of the processor measured by its clock time in
operations per second
- the busing of data from the memory to the processor
Constraints on scientific computing
✓ There are several types of memory: they differ by their size and their
access time (time it takes to load it to the processor)
✓ The further from the processor, the longer the access time and the
larger the memory space.
✓ Large numerical codes must be optimized to minimize the time lost in
data transfer from the memory to the processor.
Processor
Cache
RAM
Secondary Memory (hard-drive)
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
What are numerical methods?
✓ A numerical method provides an approximation for a mathematical
problem using a finite number of simple arithmetic or logical
operations.
✓ For engineers and scientists, numerical methods are a set of “recipes”
to be used and combined to solve a larger problem.
✓ Numerical methods are very old (more than 2000 years!)
✓ Computers have just made them much more popular and easier to use
as computers are well designed to execute a large number of repetitive
and simple operations.
Why should we study numerical methods?
✓ They provide an approximation to the solutions of problems that can
not be solved analytically (i.e. most mathematical problems).
✓ Most (~ all) engineering problems have to be solved numerically.
✓ Numerical methods are the elementary pieces of any simulation
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