Ch 1 ComputationBase

7/25/2019 Ch 1 ComputationBase

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PROGRAM ALIH TAHUN

TEKNIK MESIN-PPS-UB

MALANG-2014

ACHMAD. A. SONIEF

PPS-UB-PAT-TM-2010 1


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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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PPS-UB-PAT-TM-2010

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)(


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

PPS-UB-PAT-TM-2010


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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)(

PPS-UB-PAT-TM-2010


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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.

PPS-UB-PAT-TM-2010


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PPS-UB-PAT-TM-2010

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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PPS-UB-PAT-TM-2010

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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Documents

Ch 1 ComputationBase