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PROGRAM ALIH TAHUN
TEKNIK MESIN-PPS-UB
MALANG-2014
ACHMAD. A. SONIEF
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Requires understanding of engineering
systems
By observation and experiment
Theoretical analysis and generalization
Computers are great tools, however,
without fundamental understanding of
engineering problems, they will be useless.
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Mathematical Problem
It has a solution.
This solution is unique
How do we find the solution ?
How do we find approximate solution?
Error estimates
Algorithm
Programming
Computation with guaranteed accuracy
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Fig. 1.1
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A mathematical model is represented as a functional
relationship of the form
Dependent independent forcing
Variable =f variables, parameters, functions
Dependent variable: Characteristic that usually reflects the stateof the system
Independent variables: Dimensions such as time and spacealong which the systems behavior is being determined
Parameters: reflect the systems properties or composition
Forcing functions: external influences acting upon the system
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Newtons 2nd
law of Motion States that the time rate change of momentum of a body
is equal to the resulting force acting on it.
The model is formulated asF = m a (1.2)
F=net force acting on the body (N)
m=mass of the object (kg)
a=its acceleration (m/s2)
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Formulation of Newtons 2nd law has several
characteristics that are typical of mathematical models of
the physical world:
It describes a natural process or system in mathematical
terms
It represents an idealization and simplification of reality
Finally, it yields reproducible results, consequently, can be
used for predictive purposes.
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Some mathematical models of physical phenomena
may be much more complex.
Complex models may not be solved exactly or require
more sophisticated mathematical techniques than
simple algebra for their solution
Example, modeling of a falling parachutist:
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m
cvmg
dt
dv
cvF
mgF
FFF
m
F
dt
dv
U
D
UD
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This is a differential equation and is written interms of the differential rate of change dv/dt ofthe variable that we are interested in predicting.
If the parachutist is initially at rest (v=0 at t=0),using calculus
tmcec
gmtv
)/(1)(
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vm
cg
dt
dv
Independent variable
Dependent variable ParametersForcing function
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Conservation Laws and Engineering Conservation laws are the most important and
fundamental laws that are used in engineering.
Change = increasesdecreases (1.13) Change implies changes with time (transient). If the
change is nonexistent (steady-state), Eq. 1.13becomes
Increases =Decreases
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For steady-state incompressible fluid flow in pipes:
Flow in = Flow outor
100 + 80 = 120 + Flow4
Flow4
= 60PPS-UB-PAT-TM-2010 12
Fig 1.6
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Numerical Computations
Model of natural or
Social phenomena
Mathematical model
Modeling
algorithm
programming
Computing
Analysis of Results
Development of stable algorithm
Flow chart, Language ( C,Visual C)
Software ( Matlab, Mathematica )
Error Estimates. Correctness.
Exact solution.
Approximation solution.
Execution on a computer
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Numerical MethodsNavier-Stokes equations analytically solvable only in
special cases
approximate the solution numerically
use a discretization method to approximate thedifferential equations by a system of algebraicequations which can be solved on a computer
Finite Differences (FD) Finite Volume Method (FVM)
Finite Element Method (FEM)
Boundary Element Method (BEM)PPS-UB-PAT-TM-2010
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Numerical methods, gridsGrids
Structured grid
all nodes have the same number
of elements around it only for simple domains
Unstructured grid
for all geometries
irregular data structure
Block-structured grid
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Numerical methods, properties
ConsistencyTruncation error : difference between discrete eq and the exact one
Truncation error becomes zero when the mesh is
refined. Method order n if the truncation error is proportional
to or
Stability
Errors are not magnified Bounded numerical solution
nx)(
nt)(
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Numerical methods, properties (2)
Convergence
Discrete solution tends to the exact one as the gridspacing tends to zero.
Lax equivalence theorem (for linear problems):
Consistency + Stability = Convergence
For non-linear problems: repeat the calculations insuccessively refined grids to check if the solutionconverges to agrid-independent solution.
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Hierarchical model of errors in Scientific computing
Model of natural or
Social phenomena
Mathematical model
Numerical computation
model
Computed numerical
model
Fluid flow
Partial differentialequations
Linear system
of equations
Approximate
solution
Errors in
modelization
Truncation
errors
Round-off
errors
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Language
C, C++, C-XSC
PASCAL, PASCAL-SC
FORTRAN, FORTRAN-SC
BASIC, VISUAL BASIC
PROFIL - Programmers Runtime
Optimized Fast Interval LibraryMATHEMATICA
MATLAB, INTLAB
MAPLE -V
software
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