optimization of welding process parameter
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CHAPTER 1
INTRODUCTION
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INTRODUCTION
Joints of dissimilar metal combinations are employed in different
applications requiring certain special combination of properties as well as to
save cost incurred towards costly and scarce materials. Conventional fusion
welding of many such dissimilar metal combinations is not feasible owing to
the formation of brittle and low melting intermetallics due to metallurgical
incompatibility, wide difference in melting point, thermal mismatch, etc.
Solid-state welding processes that limit extent of intermixing are generally
employed in such situations. Friction welding is one such solid-state welding
process widely employed in such situations. The joining of materials by
conventional welding techniques becomes difficult if the physical properties
such as melting temperature and thermal expansion coefficient of the two
materials differ a lot, as it is necessary to have controlled melting on both
sides of weld joints simultaneously. Even if this criterion is met, it may notbe possible to have an appropriate joint when the two materials are
metallurgically incompatible. At the same time, the welding process,
generally involves many input and output parameters. To produce the joints
with highest quality, the proper process parameters have to be selected. The
suitable process parameters, to obtain the required output, need many
experiments and thus make the process to consume more time and money.
Here the interest is to screen the experiments using the computational
methods to analyze and optimize friction welding and other welding process
parameters respectively.
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1.1.
FRICTION WELDING PROCESS
Friction welding is a class of solid-state welding processes that generates heat
through mechanical friction between a moving work piece and a stationary
component, with the addition of a lateral force called "upset" to plastically
displace and fuse the materials. Technically, because no melt occurs, friction
welding is not actually a welding process in the traditional sense, but a
forging technique. However, due to the similarities between these techniques
and traditional welding, the term has become common. Friction welding is
used with metals and thermoplastic in a wide variety of aviation and
automotive applications.
1.1.1 PROCESS MECHANISM
The mechanism of friction welding is schematically illustrated in
Fig. 1.2 for better understanding of the process. As the pieces are initially
contacted, micro asperities come into contact area.
i. One component is rotated while the other is advanced into pressurecontact with it.
ii. Heat is produced at the faying surfaces. Overheating of metals cannotoccur as the weld zone temperature is always stabilized below melting
point.
iii.Softened material begins to extrude in response to the applied pressure,creating an annular upset.
iv.Heat is conducted away from the interfacial area for forging to takeplace.
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v. Rotation is stopped and a forge force is applied to complete the weld.vi.The joint undergoes hot working to form a homogenous, full surface,full diameter, high-integrityweld.
Fig 1.1. Process mechanism of the friction welding process.
1.2. INTRODUCTION TO OPTIMIZATION:
Optimization is getting the best under the given circumstances.
Optimization is the process of getting the maximum or minimum value of thefunction. (S.S. Roa, 2009). Best in the sense may be maximum or minimum
under the given circumstances. One may need the maximum salary or the
minimum expense. Optimizationcan be defined as the process of finding the
conditions that give the maximum or minimum value of a function.
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1.2.1.STATEMENT OF AN OPTIMIZATION PROBLEM:
Optimization problem may be stated as follows (S.S.Roa, 2009):Find
X={ x1, x2, x3, x4 xn } which minimize or maximize the
function f (X) ,
Subjected to constraints,
gi(X) 0, i= 1, 2, 3, 4, n
li (X) = 0, j = 1, 2, 3,4., n
Where,
X is an ndimensional design vector,
f (X) is the objective function,
gi (X) is the inequality constraints
lj (X) is the equality constraints
The above is said to be unconstraint optimization problem.
The constrained optimization technique is stated as follows:
Find
X={ x1, x2, x3, x4 xn } which minimize or maximize the function
f(X)
1.2.2.TYPES OF OPTIMIZATION PROBLEM:
The various methods of optimization are defined as the
follows:
i. Calculus methodsii. Calculus of variations
iii. Nonlinear programmingiv. Geometric programming
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v. Quadratic programmingvi. Linear programming
vii.
Dynamic programmingviii. Integer programming
ix. Stochastic programmingx. Separable programming
xi. Multi objective programmingThe above is said to be traditional optimization technique.
Following are some of the modern optimization techniques:
i. Genetic algorithmsii. Simulated annealing
iii. Ant colony optimizationiv. Particle swarm optimizationv. Neural networks
vi. Fuzzy optimization.
1.2.3.APPLICATION OF OPTIMIZATION TECHNIQUE:
Optimization techniques are used in the application such as
design of aircraft and aerospace structures for minimum weight design of
civil engineering structures such as frames, foundations, bridges, towers,
chimneys, and dams for minimum cost, optimum design of linkages, cams,
gears, machine tools, and other mechanical components, Selection of
machining conditions in metal-cutting processes for minimum production
cost, design of material handling equipment, such as conveyors, trucks,
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and cranes, for minimum cost, design of pumps, turbines, and heat transfer
equipment for maximum efficiency, shortest route taken by a salesperson
visiting various cities during one tour, optimal production planning,controlling, and scheduling, analysis of statistical data and building
empirical models from experimental results to obtain the most accurate,
optimum design of control systems.(S.S. Roa, 2009).
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CHAPTER 2
GENETIC ALGORITHM
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GENETIC ALGORITHM
An algorithm is a series of steps for solving a problem. A genetic
algorithm is a problem solving method that uses genetics as its model of
problem solving. Its a search technique to find approximate solutions to
optimization and search problems. Basically, an optimization problem looks
really simple. One knows the form of all possible solutions corresponding to a
specific question. The set of all the solutions that meet this form constitute the
search space. The problem consists in finding out the solution that fits the
best, i.e. the one with the most payoffs, from all the possible solutions. Each
solution is represented through a chromosome, which is just an abstract
representation. Coding all the possible solutions into a chromosome is the
first part, but certainly not the most straightforward one of a Genetic
Algorithm. A set of reproduction operators has to be determined, too.
Reproduction operators are applied directly on the chromosomes, and are
used to perform mutations and recombination over solutions of the problem.
Appropriate representation and reproduction operators are really something
determinant, as the behavior of the GA is extremely dependant on it.
Frequently, it can be extremely difficult to find a representation, which
respects the structure of the search space and reproduction operators, which
are coherent and relevant according to the properties of the problems.
Selection is supposed to be able to compare each individual in the
population. Selection is done by using a fitness function. Each chromosome
has an associated value corresponding to the fitness of the solution it
represents. The fitness should correspond to an evaluation of how good the
candidate solution is. The optimal solution is the one, which maximizes the
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fitness function. Genetic Algorithms deal with the problems that maximize
the fitness function. But, if the problem consists in minimizing a cost
function, the adaptation is quite easy. Either the cost function can betransformed into a fitness function, for example by inverting it; or the
selection can be adapted in such way that they consider individuals with low
evaluation functions as better. Once the reproduction and the fitness function
have been properly defined, a Genetic Algorithm is evolved according to the
same basic structure. It starts by generating an initial population of
chromosomes. This first population must offer a wide diversity of genetic
materials. The gene pool should be as large as possible so that any solution of
the search space can be engendered. Generally, the initial population is
generated randomly. Then, the genetic algorithm loops over an iteration
process to make the population evolve. Each iteration consists of the
following steps:
i. SELECTION: The first step consists in selecting individuals forreproduction. This selection is done randomly with a probability
depending on the relative fitness of the individuals so that best ones are
often chosen for reproduction than poor ones.
ii. REPRODUCTION: In the second step, offspring are bred by theselected individuals. For generating new chromosomes, the algorithm
can use both recombination and mutation.
iii. EVALUATION: Then the fitness of the new chromosomes isevaluated.
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iv. REPLACEMENT: During the last step, individuals from the oldpopulation are killed and replaced by the new ones.
The general optimization procedure using a genetic algorithm is
Step 1: Choose a coding to represent problem parameters, a selectionoperator, a crossover operator and a mutation operator. Choose
population size n, crossover probability pc, and mutation probability
pm. Initialize a random population of strings of size.. Choose a
maximum allowable generationnumber tmax. Set t=0.
Step 2: Evaluate each string in the population.
Step 3: If t > tmax or other termination criteria is satisfied, terminate.
Step 4: Perform reproductions on the population.
Step 5: Perform crossovers on pair of strings with probabilitypc.
Step 6: Perform mutations on strings with probabilitypm.
Step 7: Evaluate strings in the new population. Set t = t+ 1 and go toStep 3.
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The flowchart showing the process of GA is as shown in Fig. 2.1
YES
NO
FIG 2.1. FLOW CHART FOR GENETIC ALGORITHM
START
CREATE INITIAL RANDOM
POPULATION
EVALUATE FITNESS FOR
EACH
POPULATION
STORE BEST INDIVIDUAL
CREATING MATING POOL
CREATE NEXT GENERATIONBY APPLYING
CROSSOVER
OPTIMALOR GOOD
SOLUTION
REPRODUCE AND IGNORE
FEW
POPULATIONS
PERFORM MUTATION
STOP
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In short, the basic four steps used in simple Genetic Algorithm to solve a
problem are,
The representation of the problem
The fitness calculation Various variables and parameters involved in controlling the algorithm The representation of result and the way of terminating the algorithm
2.1. COMPARISON OF GENETIC ALGORITHM WITH OTHER
OPTIMIZATION TECHNIQUES:
The principle of GAs is simple: imitate genetics and natural selection
by a computer program: The parameters of the problem are coded most
naturally as a DNA-like linear data structure, a vector or a string. Sometimes,
when the problem is naturally two or three-dimensional also corresponding
array structures are used. A set, called population, of these problem dependent
parameter value vectors is processed by GA. To start there is usually a totally
random population, the values of different parameters generated by a random
number generator. Typical population size is from few dozens to thousands.
To do optimization we need a cost function or fitness function as it is usually
called when genetic algorithms are used. By a fitness function we can select
the best solution candidates from the population and delete the not so good
specimens. The nice thing when comparing GAs to other optimization
methods is that the fitness function can be nearly anything that can be
evaluated by a computer or even something that cannot! In the latter case it
might be a human judgment that cannot be stated as a crisp program, like in
the case of eyewitness, where a human being selects among the alternatives
generated by GA. So, there are not any definite mathematical restrictions on
the properties of the fitness function. It may be discrete, multimodal etc. The
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main criteria used to classify optimization algorithms are as follows:
continuous / discrete, constrained / unconstrained and sequential / parallel.
There is a clear difference between discrete and continuous problems.Therefore it is instructive to notice that continuous methods are sometimes
used to solve inherently discrete problems and vice versa. Parallel algorithms
are usually used to speed up processing. There are, however, some cases in
which it is more efficient to run several processors in parallel rather than
sequentially. These cases include among others such, in which there is high
probability of each individual search run to get stuck into a local extreme.
Irrespective of the above classification, optimization methods can be further
classified into deterministic and non-deterministic methods. In addition
optimization algorithms can be classified as local or global. In terms of
energy and entropy local search corresponds to entropy while global
optimization depends essentially on the fitness i.e. energy landscape.
Genetic algorithm differs from conventional optimization techniques in
following ways:
i. GAs operate with coded versions of the problem parameters rather thanparameters themselves i.e., GA works with the coding of solution set
and not with the solution itself.
ii. Almost all conventional optimization techniques search from a singlepoint but GAs always operate on a whole population of points(strings)
i.e., GA uses population of solutions rather than a single solution from
searching. This plays a major role to the robustness of genetic
algorithms. It improves the chance of reaching the global optimum and
also helps in avoiding local stationary point.
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iii. GA uses fitness function for evaluation rather than derivatives. As aresult, they can be applied to any kind of continuous or discrete
optimization problem. The key point to be performed here is to identifyand specify a meaningful decoding function.
iv. GAs use probabilistic transition operates while conventional methodsfor continuous optimization apply deterministic transition operates i.e.,
GAs does not use deterministic rules.
2.2. ADVANTAGES OF GENETIC ALGORITHM
The advantages of genetic algorithm includes,
Parallelism Liability Solution space is wider The fitness landscape is complex Easy to discover global optimum The problem has multi objective function Only uses function evaluations. Easily modified for different problems. Handles noisy functions well. Handles large, poorly understood search spaces easily Good for multi-modal problems Returns a suite of solutions. Very robust to difficulties in the evaluation of the objective function. They require no knowledge or gradient information about the response
surface
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Discontinuities present on the response surface have little effect onoverall optimization performance
They are resistant to becoming trapped in local optima.
They perform very well for large-scale optimization problems Can be employed for a wide variety of optimization problems
2.3. LIMITATION OF GENETIC ALGORITHM
The limitation of genetic algorithm includes,
The problem of identifying fitness function Definition of representation for the problem Premature convergence occurs The problem of choosing the various parameters like the size of the
population,
mutation rate, cross over rate, the selection method and its strength. Cannot use gradients. Cannot easily incorporate problem specific information Not good at identifying local optima No effective terminator. Not effective for smooth unimodel functions Needs to be coupled with a local search technique. Have trouble finding the exact global optimum Require large number of response (fitness) function evaluations Configuration is not straightforward
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2.4. APPLICATIONS OF GENETIC ALGORITHM
Genetic algorithms have been used for difficult problems (such as NP-hardproblems), for machine learning and also for evolving simple programs. They
have been also used for some art, for evolving pictures and music. A few
applications of GA are as follows:
Nonlinear dynamical systemspredicting, data analysis Robot trajectory planning Evolving LISP programs (genetic programming) Strategy planning Finding shape of protein molecules TSP and sequence scheduling Functions for creating images Controlgas pipeline, pole balancing, missile evasion, pursuit Designsemiconductor layout, aircraft design, keyboard configuration,
communication networks
Schedulingmanufacturing, facility scheduling, resource allocation Machine LearningDesigning neural networks, both architecture and
weights, improving classification algorithms, classifier systems
Signal Processingfilter design Combinatorial Optimizationset covering, traveling salesman (TSP),
Sequence scheduling, routing, bin packing, graph coloring and
partitioning.
2.5. TERMINOLOGIES AND OPERATORS OF GA
Genetic Algorithm uses a metaphor where an optimization problem
takes the place of an environment and feasible solutions are considered as
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individuals living in that environment. In genetic algorithms, individuals are
binary digits or of some other set of symbols drawn from a finite set. As
computer memory is made up of array of bits, anything can be stored in acomputer and can also be encoded by a bit string of sufficient length. Each of
the encoded individual in the population can be viewed as a representation,
according to an appropriate encoding of a particular solution to the problem.
For Genetic Algorithms to find a best optimum solution, it is necessary to
perform certain operations over these individuals. This part of the chapter
discusses the basic terminologies and operators used in Genetic Algorithms to
achieve a good enough solution for possible terminating conditions.
2.6. INDIVIDUALS
An individual is a single solution. Individual groups together two forms of
solutionsas given below:
1. The chromosome, which is the raw genetic information (genotype)that the GAdeals.
2. The phenotype, which is the expressive of the chromosome in the termsof themodel.
A chromosome is subdivided into genes. A gene is the GAs
representation of a single factor for a control factor. Each factor in the
solution set corresponds to gene in the chromosome\
2.7. GENES
Genes are the basic instructions forbuilding a Generic Algorithms. A
chromosomeis a sequence of genes. Genes may describe a possible solution
to a problem, without actually being the solution. A gene is a bit string of
arbitrary lengths. The bit string is a binary representation of number of
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intervals from a lower bound. Agene is the GAs representation of a single
factor value for a control factor, where control factor must have an upper
bound and lower bound.
2.8. FITNESS
The fitness of an individual in a genetic algorithm is the value of an
objective function for its phenotype. For calculating fitness, the chromosome
has to be first decoded and the objective function has to be evaluated. The
fitness not only indicates how good the solution is, but also corresponds to
how close the chromosome is to the optimal one.
2.9. POPULATIONS
A population is a collection of individuals. A population consists of a
number of individuals being tested, the phenotype parameters defining the
individuals and some information about search space.
2.10. DATA STRUCTURES
The main data structures in GA are chromosomes, phenotypes,
objective function values and fitness values. This is particularly easy
implemented when using MATLAB package as a numerical tool. An entire
chromosome population can be stored in a single array given the number of
individuals and the length of their genotype representation. Similarly, the
design variables, or phenotypes that are obtained by applying some mapping
from the chromosome representation into the design space can be stored in a
single array. The actual mapping depends upon the decoding scheme used.
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The objective function values can be scalar or vectorial and are necessarily
the same as the fitness values. Fitness values are derived from the object
function using scaling or ranking function and can be stored as vectors.
2.11. SEARCH STRATEGIES
The search process consists of initializing the population and then
breeding new individuals until the termination condition is met. There can be
several goals for the search process, one of which is to find the global optima.
This can never be assured with the types of models that GAs work with.
There is always a possibility that the next iteration in the search would
produce a better solution. In some cases, the search process could run for
years and does not produce any better solution than it did in the first little
iteration. Another goal is faster convergence. When the objective function is
expensive to run, faster convergence is desirable, however, the chance of
converging on local, and possibly quite substandard optima is increased.
Apart from these, yet another goal is to produce a range of diverse, but still
good solutions. When the solution space contains several distinct optima,
which are similar in fitness, it is useful to be able to select between them,
since some combinations of factor values in the model may be more feasible
than others. Also, some solutions may be more robust than others.
2.12. ENCODING
Encoding is a process of representing individual genes. The process can
be performed using bits, numbers, trees, arrays, lists or any other objects. The
encoding depends mainly on solving the problem. For example, one can
encode directly real or integer numbers
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2.12.1. BINARY ENCODING
Each chromosome encodes a binary (bit) string. Each bit in the stringcan represent some characteristics of the solution. Every bit string therefore is
a solution but not necessarily the best solution. Another possibility is that the
whole string can represent a number. The way bit strings can code differs
from problem to problem Binary encoding gives many possible chromosomes
with a smaller number of alleles. On the other hand this encoding is not
natural for many problems and sometimes corrections must be made after
genetic operation is completed. Binary coded strings with 1s and 0s are
mostly used. The length of the string depends on the accuracy.
In this,
Integers are represented exactly Finite number of real numbers can be represented Number of real numbers represented increases with string length
The other encoding methods are
i. Octal Encodingii. Hexadecimal Encoding
iii. Permutation Encodingiv. Value Encodingv. Tree Encoding
2.13. BREEDING
The breeding process is the heart of the genetic algorithm. It is in this
process, the search process creates new and hopefully fitter individuals. The
breeding cycle consists of three steps:
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a. Selecting parents.b. Crossing the parents to create new individuals (offspring or children).c.
Replacing old individuals in the population with the new ones.
2.13.1. SELECTION
Selection is the process of choosing two parents from the population
for crossing.After deciding on an encoding, the next step is to decide how to
perform selection i.e., how to choose individuals in the population that will
create offspring for the next generation and how many offspring each will
create. The purpose of selection is to emphasize fitter individuals in the
population in hopes that their off springs havehigher fitness. Chromosomes
are selected from the initial population to be parents for reproduction. The
problem is how to select these chromosomes.
Selection is a method that randomly picks chromosomes out of the
population according to their evaluation function. The higher the fitness
function, the more chance an individual has to be selected. The selection
pressure is defined as the degree to which the better individuals are favored.
The higher the selection pressured, the more the better individuals are
favored. This selection pressure drives the GA to improve the population
fitness over the successive generations. Selection has to be balanced with
variation form crossover and mutation. Too strong selection means sub
optimal highly fit individuals will take over the population, reducing the
diversity needed for change and progress; too weak selection will result in too
slow evolution. The various selection methods are discussed as follows:
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ROULETTE WHEEL SELECTION
Roulette selection is one of the traditional GA selection techniques.The commonlyused reproduction operator is the proportionate reproductive
operator where a string is selected from the mating pool with a probability
proportional to the fitness. Theprinciple of roulette selection is a linear search
through a roulette wheel with theslots in the wheel weighted in proportion to
the individuals fitness values. A target value is set, which is a random
proportion of the sum of the fit nesses in the population.The population is
stepped through until the target value is reached. This is onlya moderately
strong selection technique, since fit individuals are not guaranteed to be
selected for, but somewhat have a greater chance. A fit individual will
contribute more to the target value, but if it does not exceed it, the next
chromosome in linehas a chance, and it may be weak. It is essential that the
population not be sorted by fitness, since this would dramatically bias the
selection. The above described Roulette process can also be explained as
follows: The expected value of an individual is that fitness divided by the
actual fitness of the population. Each individual is assigned a slice of the
roulette wheel, the size of the slicebeing proportional to the individuals
fitness. The wheel is spun N times, where N is the number of individuals in
the population. On each spin, the individual under the wheels marker is
selected to be in the pool of parents for the next generation Roulette wheel
selection is easier to implement but is noisy. The rate of evolutiondepends on
the variance of fitnesss in the population.
Roulette wheel selection is easier to implement but is noisy. The rate of
evolution depends on the variance of fitnesss in the population.
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RANDOM SELECTION
This technique randomly selects a parent from the population. In terms
of disruptionof genetic codes, random selection is a little more disruptive, onaverage, thanroulette wheel selection.
RANK SELECTION
The Roulette wheel will have a problem when the fitness values differ
very much. If the best chromosome fitness is 90%, its circumference occupies
90% of Roulette wheel, and then other chromosomes have too few chances to
be selected. Rank Selection ranks the population and every chromosome
receives fitness from the ranking. The worst has fitness 1 and the best has
fitness N. It results in slow convergence but prevents too quick convergence.
It also keeps up selection pressure when the fitness variance is low. It
preserves diversity and hence leads to a successful search. In effect, potential
parents are selected and a tournament is held to decide which of the
individuals will be the parent. There are many ways this can be achieved and
two suggestions are,
1. Select a pair of individuals at random. Generate a random number, R,between 0 and 1. IfR < r use the first individual as a parent. If the
R>=r then use the second individual as the parent. This is repeated to
select the second parent. The value ofris a parameter to this method.
2. Select two individuals at random. The individual with the highestevaluation becomes the parent. Repeat to find a second parent.
the other selection method are:
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Tournament Selection Boltzmann Selection
Stochastic Universal Sampling
2.14. CROSSOVER (RECOMBINATION)
Crossover is the process of taking two parent solutions and producing
from them a child. After the selection (reproduction) process, the population
is enriched with better individuals. Reproduction makes clones of good
strings but does not create new ones. Crossover operator is applied to the
mating pool with the hope that it creates a better offspring.
1. Crossover is a recombination operator that proceeds in three steps:2. The reproduction operator selects at random a pair of two individual
strings for the mating.
3. A cross site is selected at random along the string length.Finally, the position values are swapped between the two strings following
the cross site. That is, the simplest way how to do that is to choose randomly
some crossover point and copy everything before this point from the first
parent and then copy everything after the crossover point from the other
parent. The various crossover techniques are discussed as follows:
2.14.1. SINGLE POINT CROSSOVER
The traditional genetic algorithm uses single point crossover, where the
two mating chromosomes are cut once at corresponding points and the
sections after the cuts exchanged. Here, a cross-site or crossover point is
selected randomly along the lengthof the mated strings and bits next to the
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cross-sites are exchanged. If appropriate siteis chosen, better children can be
obtained by combining good parents else it severelyhampers string quality.\
2.14.2. TWO POINT CROSSOVER
Apart from single point crossover, many different crossover algorithms
have been devised, often involving more than one cut point. It should be
noted that adding furthercrossover points reduces the performance of the GA.
The problem with addingadditional crossover points is that building blocks
are more likely to be disrupted. However, an advantage of having more
crossover points is that the problem spacemay be searched more thoroughly.
2.14.3. MULTI-POINT CROSSOVER
There are two ways in this crossover. One is even number of cross-sites
and the otherodd number of cross-sites. In the case of even number of cross-
sites, cross-sites are selected randomly around a circle and information is
exchanged. In the case ofodd number of cross-sites, a different cross-point is
always assumed at the stringbeginning.
2.14.4. UNIFORM CROSSOVER
Uniform crossover is quite different from the N-point crossover. Each
gene in the offspring is created by copying the corresponding gene from one
or the other parent chosen according to a random generated binary crossover
mask of the same length as the chromosomes. Where there is a 1 in the
crossover mask, the gene is copied from the first parent, and where there is a
0 in the mask the gene is copied from the second parent. A new crossover
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mask is randomly generated for each pair of parents. Offspring, therefore
contain a mixture of genes from each parent. The number of effective
crossing point is not fixed, but will average L/2 (where L is the chromosomelength). The other type of cross over are:
Three Parent Crossover Crossover with Reduced Surrogate Shuffle Crossover Precedence Preservative Crossover Ordered Crossover Partially Matched Crossover
2.14.5. CROSSOVER PROBABILITY
The basic parameter in crossover technique is the crossover probability
(Pc). Crossover probability is a parameter to describe how often crossover
will be performed. If there is no crossover, offspring are exact copies of
parents. If there is crossover, offspring are made from parts of both parents
chromosome. If crossover probability is 100%, then all offspring are made by
crossover. If it is 0%, whole new generation is made from exact copies of
chromosomes from old population (but this does not mean that the new
generation is the same!). Crossover is made in hope that new chromosomes
will contain good parts of old chromosomes and therefore the new
chromosomes will be better. However, it is good to leave some part of old
population survive to next generation.
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2.15. MUTATION
After crossover, the strings are subjected to mutation. Mutationprevents the algorithmto be trapped in a local minimum. Mutation plays the
role of recovering thelost genetic materials as well as for randomly disturbing
genetic information. It is an insurance policy against the irreversible loss of
genetic material. Mutation has traditionally considered as a simple search
operator. If crossover is supposed toexploit the current solution to find better
ones, mutation is supposed to help for the exploration of the whole search
space. Mutation is viewed as a background operator to maintain genetic
diversity in the population. It introduces new genetic structures in the
population by randomly modifying some of its building blocks. Mutation
helps escape from local minimas trap and maintains diversity in the
population. It also keeps the gene pool well stocked, and thus ensuring
periodicity. A search space is said to be ergodic if there is a non-zero
probability of generating any solution from any population state. There are
many different forms of mutation for the different kinds of representation. For
binary representation, a simple mutation can consist in inverting the value of
each gene with a small probability. The probability is usually taken about 1/L,
where L is the length of the chromosome. It is also possible to implement
kind of hill-climbing mutation operators that do mutation only if it improves
the quality of the solution. Such an operator can accelerate the search. But
care should be taken, because it might also reduce the diversity in the
population and makes the algorithm converge toward some local optima.
Mutation of a bit involves flipping a bit, changing 0 to 1 and vice-versa.
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2.15.1. FLIPPING
Flipping of a bit involves changing 0 to 1 and 1 to 0 based on amutation chromosome generated
2.15.2. INTERCHANGING
Two random positions of the string are chosen and the bits
corresponding to those positions are interchanged.
2.15.3. REVERSING
A random position is chosen and the bits next to that position are
reversed and child chromosome is produced.
2.15.4. MUTATION PROBABILITY
The important parameter in the mutation technique is the mutation
probability (Pm). The mutation probability decides how often parts of
chromosome will be mutated.If there is no mutation, offspring are generated
immediately after crossover (or directly copied) without any change. If
mutation is performed, one or more parts ofa chromosome are changed. If
mutation probability is 100%, whole chromosome is changed, if it is 0%,
nothing is changed. Mutation generally prevents the GA from falling into
local extremes. Mutation should not occur very often, because then GAwill
in fact change to random search.
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2.16. REPLACEMENT
Replacement is the last stage of any breeding cycle. Two parents aredrawn froma fixed size population, they breed two children, but not all four
can return to the population, so two must be replaced i.e., once off springs are
produced, a method must determine which of the current members of the
population, if any, should be replaced by the new solutions. The technique
used to decide which individual stayin a population and which are replaced in
on a par with the selection in influencing convergence. Basically, there are
two kinds of methods for maintaining the population; generational updates
and steady state updates.
2.16.1. RANDOM REPLACEMENT
The children replace two randomly chosen individuals in the
population. The parentsare also candidates for selection. This can be useful
for continuing the searchin small populations, since weak individuals can be
introduced into the population.
2.16.2. WEAK PARENT REPLACEMENT
In weak parent replacement, a weaker parent is replaced by a strong
child. With thefour individuals only the fittest two, parent or child, return to
population. This processimproves the overall fitness of the population when
paired with a selection technique that selects both fit and weak parents for
crossing, but if weak individuals and discriminated against in selection the
opportunity will never raise to replace them.
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2.16.3. BOTH PARENTS
Both parents replacement is simple. The child replaces the parent. Inthis case, eachindividual only gets to breed once. As a result, the population
and genetic material moves around but leads to a problem when combined
with a selection technique that strongly favors fit parents: the fit breed and
then are disposed of.
2.17. SEARCH TERMINATION (CONVERGENCE CRITERIA)
In short, the various stopping condition are listed as follows:
Maximum generationsThe genetic algorithm stops when thespecified numbers of generations have evolved.
Elapsed timeThe genetic process will end when a specified time haselapsed. Note: If the maximum number of generation has been reached
before the specified time has elapsed, the process will end.
No change in fitnessThe genetic process will end if there is nochange to the populations best fitness for a specified number of
generations. Note: If the maximum number of generation has been
reached before the specified number of generation with no changes has
been reached, the process will end.
Stall generationsThe algorithm stops if there is no improvement inthe objective function for a sequence of consecutive generations of
length Stall generations.
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Stall time limitThe algorithm stops if there is no improvement in theobjective function during an interval of time in seconds equal to Stall
time limit. The termination or convergence criterion finally brings the
search to a halt. The following are the few methods of termination
techniques
2.17.1. BEST INDIVIDUAL
A best individual convergence criterion stops the search once the
minimum fitnessin the population drops below the convergence value. This
brings the search to a faster conclusion guaranteeing at least one good
solution.
2.17.2. WORST INDIVIDUAL
Worst individual terminates the search when the least fit individuals in
the population have fitness less than the convergence criteria. This guarantees
the entire population to be of minimum standard, although the best individual
may not be significantly better than the worst. In this case, a stringent
convergence value may never be met, in which case the search will terminate
after the maximum has been exceeded.
2.17.3. SUM OF FITNESS
In this termination scheme, the search is considered to have satisfaction
converged when the sum of the fitness in the entire population is less than or
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equal to the convergence value in the population record. This guarantees that
virtually all individuals in the population will be within a particular fitness
range, although it is better to pair this convergence criteria with weakest genereplacement, otherwise a few unfit individuals in the population will blow out
the fitness sum. The population size has to be considered while setting the
convergence value.
2.17.4. MEDIAN FITNESS
Here at least half of the individuals will be better than or equal to the
convergence value, which should give a good range of solutions to choose
from.
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CHAPTER 3
LITERATURE REVIEW
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LITERATURE REVIEW
Sathiya.A et.,al., (2008) applied successfully to process similarjoints of stainless steel (AISI 304).The friction processed joints exhibited
comparable strength with the base material and joint strength decreased with
an increase in the friction time. The material shear flow and dimples at the
fractured surface confirms the ductile mode of failure of joints during tensile
testing. Hardness at the joint zone increases with the increase in friction time.
The increase in hardness at the joint zone is due to the thermal history of the
joining process. Friction welding method can be applied successfully to
process similar joints of stainless steel (AISI 430).The processed joints
exhibited better mechanical and metallurgical characteristics, than the fusion
joints. Because of the solid state bonding technique, the problems associated
with fusion joining are minimized in the case of friction welding. The joints
exhibited 95.52% of parent materials tensile strength. The tensile specimen
failures were associated primarily with the weld interface region. The fracture
is predominantly associated with material (shear like) flow. The toughness of
the friction welded ferritic stainless steel is comparatively higher than fusion
processed joints due to the refinement of grain size at the weld zone.
The ultimate capacity of the arc spot welding was modeled based
on the experimental data using Artificial neural network in the literature(Abdulkadir Cevik et,al., 2008). The main objective in the study is to obtain
the explicit formulation of nominal shear stress as a function of the geometric,
and the mechanical properties of the spot welding. The proposed model is
obtained using the Neural network (NN) tool box of MATLAB. The
statistical analysis of the proposed model and the experimental is being
carried out for the train sets and test sets for which the correlation coefficient
is found as 0.984 and 0.969 respectively. The parametric study is made to
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investigate the effect of changing geometric parameters and the yield strength
on the nominal shear strength of arc spot welds.
An intelligent system was being developed by I.S. Kim
et.al(2005).,using MATLAB/SIMULINK software. The mathematical model
incorporating the different welding parameters and complex geometrical
features is developed based on the regression and the neural networks. In the
method of modeling using the multiple regression models, the Analysis Of
Variance technique (ANOVA) is performed on the factorial design to
quantify the effect of welding parameters. Then the multiple correlation
coefficients and the Fratio were to measure the goodness of fit. In the
method of modeling using the ANN method, Back-Propagation (BP) method
is being used and at the same time the generalized delta rule is used as the
learning algorithm. With the learning rate of 0.6 and the momentum term of
0.9, the network is trained for 2,00,000 iteration. The error between the
desired and the actual output is less than 0.001 at the end of the trainingprocess. By comparing the above two methods, the neural network model is
found to be comparatively best, which is capable of making the prediction of
the experimental result with the reasonable accuracy.
According to Cemal Meran, ( 2006) welding was the major joining
process in the industry. Three main parameters such as welding current,
welding velocity, and arc length have a big influence on the quality welding.
The work by Cemal Meran(2006), deal with the use of stochastic search
process, which is the basis for the genetic algorithms in developing estimation
of the welding parameters for the joined brass plates. The obtained result is
being compared with the experimental value for the verification, ehich
showed a good agreement.
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In order to evaluate the effect of process parameters, such as tool
rotational speed, transverse speed and axial force, on the tensile strength of
friction welded RDE-40 aluminium alloy, the Taguchis parametric designand optimization approach was used( A. K. Lakshminarayanan., et.al., 2008).
Through the Taguchis parametric design approach, the optimum level of the
process parameter was determined. The optimal value is found to be 303MPa.
Rather than the wellknown effect of the process parameter on the
quality of the welding, the study made by Serder et.al.,(2008) focuses on the
sensitivity analysis of the process parameters and fine tuning requirements of
the parameters for optimum weld bead geometry. Changeable process
parameters such as welding current, welding voltage and welding speed are
used as design variables. Experimental work is based on the three level
factorial design and the mathematical modeling is based on the multiple
curvilinear regression analysis. Effect of all the three design parameters on
the bead width and bead height show that small changes in theses parametershave very important role in the quality of welding operation.
Conventional regression analysis were carried out by Parikshit
Dutta et.al.,(2007), on some of the experimental data of a tungsten inert gas
(TIG) welding process parameters to find its effect. At the same time,
Artificial Neural Network (ANN) concept is used to find the effect of various
process parameters, in which thousand training data were employed. Back
Prorogation Neural Network (BPNN) and genetic neural system (GA-NN) are
the type of ANN which is used for the modeling purpose. The performance of
all those techniques were being compared and it is concluded that Neural
Network (NN) based approaches were seen to be more adoptive, which may
be due to the reasons that it is based on the principle of steepest descent
method.
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An empirical model was developed by G. Padmanaban et.al., to
predict the maximum tensile strength of AZ31B magnesium alloy. The
process parameter includes laser power, welding speed and focal position.The response surface methodology is used as an optimization tool to predict
the maximum tensile strength. The experiments were conducted based on a
three factor, three level, central composite face centered matrix with full
replication technique. A maximum tensile strength of 212MPa is obtained
under the welding conditions in which the laser power is 2.5Kw, welding
speed of 5.0m/min, and focal position of -1.5mm. Comparatively welding
speed is the most factor which has the greater influence compared to others.
The mathematical model was developed for the analysis and
simulation of the correlation between the friction stir welding of aluminium
plates and mechanical properties using Artificial Neural Network (ANN). The
model can be used to calculate mechanical properties of welded Al plates as
the function of weld speed and tool rotation speed. The combined effect ofweld speed and tool rotation speed on the mechanical properties of welded Al
plates was simulated and then the comparison between the measured and
calculated data. The calculated results were in good agreement with the
measured value.
The works carried out by G. Mahendran et al..,(2008) developed
diffusion bonding windows for joining AZ3113 Magnesium and AA 2024
Aluminium alloys. By the experimental work, the optimal process parameters
were found. The constructed bonding windows may act as the reference map
for selecting appropriate process parameters to obtain high strength bonds. It
is concluded by the author that the highest shear strength is obtained at the
bonding temperature of 4250C, the bonding pressure of 20MPa, and the
holding time of 45min due to the formation of optimum thick diffusion layer
at the interface of MgAl alloys.
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Further experimental studies were carried out by G.Mahendran et
al., (2010) on Magnesium and Copper using diffusion bonding methods.
Three factor five-level, central composite rotatable design matrix method isused for the purpose of designing the experiments. Empirical relationship
were developed in order to predict the diffusion layer thickness, hardness and
strength of the joint. The response surface methodology is used for
optimization and which is found as follows: joints fabricated at the bonding
pressure of 12MPa, the bonding time of 30mins, the bonding temperature of
4500C , which yields a maximum shear strength and bond strength of 66 and
81MPa respectively.
The diffusion bonding process studied in Nickel alloy, Su236, by
Ravishankar B et.al.,(2009) under the temperature ranges from 1123-1323k
and the compressive strength of 90% of yield strength, determined the
importance of the process parameters, the mechanism responsible for bonding
and the joint characteristic. The experimental results were compared with themodel developed by John Pilling. The mechanism of bonding was evaluated
by grain growth equation. The quality of bond was assessed using optical
metallographic and lap shear testing. The bonding mechanism and
composition of the interface were determined using quantified EPMA line
analysis.
The non conventional technique is being used by S. Suresh Kumar
at.al.,(2009). The ultrasonic A-scan method is used to evaluate the quality of
the welded joints and to study the interface properties of the joint of diffusion
bonded of Ti-6Al-4V, in that work the experiment is carried out for the given
material for the pressure ranges from 1.6 4MPa, and the bonding time of
4hours. The shear strength of the bond is to be correlated with the fractional
area bonded which is measured ng metallographic techniques and ultrasonic
technique, which is the non-destructive method.
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The optimal setting of the welding process parameters was
identified by Sathiya. P., et.al.,(2009) using non conventional technique such
as Genitic Algorithm, Simulated Annealing, Particle Swarm Optimization.Thje mathematical model relating the process parameters is developed using
Artificial Neural Network (ANN). The welding is carried out in Stainless
Steel (AISI 304). The optimized value obtained through genetic algorithm
closely resembles the experimental value.
TABLE 3.1. OPTIMIZED PROCESS PARAMETRE AND
EXPERIMENTAL VALUE (Sathiya. P., et.al., 2009)
Process parameter Optimized value Experimental value
Upsetting Pressure(UP) 17.7028 bar 17bar
Upsetting Time (UT) 4.2663 sec 4sec
Heating Time (HT) 35.1078 bar 35 bar
Upsetting Time 4.025 sec 4 sec
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CHAPTER 4
OPTIMIZING THE
PROCESS PARAMETERS
OF FRICTION WELDING
PROCESS
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OPTIMIZING THE PROCESS PARAMETERS OF FRICTION
WELDING PROCESS
4.1. PROCEDURE FOR OPTIMIZING THE PROCESS
PARAMETERS
The optimumprocess parameter of the friction welding process, is to be
found for maximizing the tensile strength of the Al/SS joint
(Purushothaman.L et.al, 2009). From the quoted literature, the experimentaldatas were taken since those work concentrate only on developing the
experimental matrix for the various process parameters and the tensile
strength of the Al/SS joint. From the experimental data the following work
are being carried out using MATLAB.
4.2. DEVELOPING THE MODEL BASED ON THE MULTIPLE
LINEAR REGRESSION METHOD:
4.2.1. FEASIBLE LIMITS AND THE EXPERIMENTAL DESIGN
MATRIX:
The feasible working limits of friction welding process parameters
from the literature is identified and presented in Tables 4.1. The experimental
design matrix is presented in table 4.
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Table 4.1. Working Limits of friction welding
# Parameter Notation Unit
Levels
(-2) (-1) 0 (+1) (+2)
1
Friction
Pressure Frp Ton 1 1.25 1.5 1.75 2
2Friction
Time Frt sec 3 3.75 4.5 5.25 6
3
Forging
Pressure Fop Ton 1 1.25 1.5 1.75 2
4
Forging
Time Fot sec 3 3.75 4.5 5.25 6
Due to wide range of factors, it was decided to use four factors, five
levels, central composite face centered design matrix to optimise the
experimental conditions, which fits the second order response surfaces very
accurately. Central composite rotational design matrix with the star points are
at the center of each face of factorial space was used, so = 2. This variety
requires 5 levels of each factor. The upper limit and lower limit of a factor
were coded as +2 and2 respectively
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Table 4.2. Experimental Design Matrix
Expt.
Coded values Original values
Tensile
strength(MPa)
A B C D TS
1 1 1 1 1 1.75 5.25 1.75 5.25 139
2 0 2 0 0 1.5 6 1.5 4.5 108
3 1 1 1 -1 1.75 5.25 1.75 3.75 106
4 -1 1 1 -1 1.25 5.25 1.5 3.75 88
5 0 0 0 2 1.5 4.5 1.5 6 143
6 1 -1 1 1 1.75 3.75 1.75 5.25 114
7 1 1 -1 -1 1.75 5.25 1.25 3.75 728 -1 1 -1 1 1.25 5.25 1.25 5.25 104
9 0 0 -2 0 1.5 4.5 1 4.5 64
10 0 0 0 0 1.5 4.5 1.5 4.5 186
11 0 0 0 0 1.5 4.5 1.5 4.5 190
12 0 0 2 0 1.5 4.5 2 4.5 112
13 -1 -1 -1 -1 1.25 3.75 1.25 3.75 36
14 -2 0 0 0 1 4.5 1.5 4.5 67
15 0 -2 0 0 1.5 3 1.5 4.5 68
16 -1 -1 1 -1 1.25 3.75 1.75 3.75 67
17 0 0 0 0 1.5 4.5 1.5 4.5 184
18 2 0 0 0 2 4.5 1.5 4.5 104
19 -1 1 1 1 1.25 5.25 1.75 5.25 123
20 1 1 -1 1 1.75 5.25 1.25 5.25 123
21 -1 -1 -1 1 1.25 3.75 1.25 5.25 84
22 0 0 0 0 1.5 4.5 1.5 4.5 185
23 -1 1 -1 -1 1.25 5.25 1.25 3.75 51
24 1 -1 1 -1 1.75 3.75 1.5 3.75 8625 1 -1 -1 -1 1.75 3.75 1.25 3.75 58
26 0 0 0 -2 1.5 4.5 1.5 3 62
27 1 -1 -1 1 1.75 3.75 1.25 1.75 102
28 0 0 0 0 1.5 4.5 1.5 4.5 184
29 -1 -1 1 1 1.25 3.75 1.75 5.25 98
30 0 0 0 0 1.5 4.5 1.5 4.5 180
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4.2.2. DEVELOPING RESPONSE SURFACE MODELS:
Response Surface Models are multivariate polynomial models. They
typically arise in the design of experiments where they are used to determine a set
of design variables that optimize a response.. Squared terms produce the simplest
models in which the response surface has a maximum or minimum, and so an
optimal response. Response surface models are multivariate polynomial models.
They typically arise in the design of, where they are used to determine a set of
design variables that optimize a response. The second order polynomial
(regression) equation used to represent the response surface Y is given by
Y = b0 + bi xi + bii xi2
+ bij xi xj +er
In order to estimate the regression coefficients, a number of experimental
design techniques are available.All the coefficients were obtained applying central
composite rotatable centered design using the MATLAB software package.
Tensile Strength = 184.83+9.29 * X1 + 10.04 * X2 + 11.96 * X3 + 20.21 *X4-
0.062 * X1 * X2 -0.69 * X1 * X3 - 0.69 * X1 * X4 + 1.31 * X2 * X3 + 1.31 * X2 *
X4- 4.31 * X3 * X4 -24.89 * X12- 24.26 * X2
2- 24.26 * X3
2- 20.64 * X4
2
4.2.3. OPTIMIZATION USING GENETIC ALGORITHM:
Genetic algorithms are computerized search and optimization algorithms
based on the mechanics of natural genetics and natural selection. GA is very
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different from traditional search and optimization methods used in engineering
problems. Because of its simplicity, ease of operation, minimum requirements and
global perspective, GA has been successfully used in a wide variety of problems.
GENETIC ALGORITHM TOOLBOX IN MATLAB
In order to estimate the maximum tensile strength, the following steps are to
be followed:
i. Firstly the model which is developed using response surface method whichrelates the tensile stength and the process parameter is typed in editor
window.
ii. Then the editor window is saved using any name.iii. After opening the optimization toolbox, the fitness function is entered with
the filename saved.
iv. Then the number of the variables is entered and the lower and the upperbound is entered.
v. If needed the parameters of the GA toolbox is changed, otherwise it is set asdefault and optimization is started to run
vi. After executing the problem, the results will be displayed.
GENETIC ALGORITHM PARAMETERS
i. The parameters used for GA is given below.ii. Population size = 100
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iii. Length of chromosome = 40iv. Selection operator: Roulette methodv.
Crossover operator : Single point operator
vi. Crossover probability = 0.9vii. Mutation probability = 0.01
viii. Fitness parameter : Tensile strength .
Fig 4.1. Editor window of the MATLAB
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Fig 4.2. GA TOOLBOX IN MATLAB
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4.3. OPTIMIZED PROCESS PARAMETERS OF FRICTION WELDING:
The optimized result and the maximized tensile strength of the
friction welding process parameters and the experimental results are shown as
follows:
Table 4.3. Optimized Process Parameters of Friction Welding
S.N
O
Process
paramete
r
Values of process parameter Tensile strength (MPa)
Predicted Experimental Predicted Experimental
1 Friction
pressure(ton)
1.525 1.5
192.674 1902 Friction
time (sec)
4.65 4.5
3 Forging
pressure
(ton)
1.55 1.5
4 Forging
time (sec)
4.8 4.5
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CHAPTER 5
OPTIMIZING THE
PROCESS PARAMETERS
OF OTHER WELDING
PROCESS
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OPTIMIZING THE PROCESS PARAMETERS OF OTHER
WELDING PROCESS
The optimization process is being carried out in some of the
welding process such as diffusion bonding and laser welding process. The
empirical model relating the output variable and the input process parameter
are taken from the available standard literature already published.
5.1. OPTIMIZING THE LASER WELDING PROCESS PARAMETERS
An empirical model was developed by G. Padmanaban et.al, to
predict the maximum tensile strength of laser welded AZ31B magnesium
alloy. The response surface methodology is used as an optimization tool to
predict the maximum tensile strength. The experiments were conducted based
on a three factor, three level, central composite face centered matrix with full
replication technique.The process parameter includes, laser power, welding
speed, focal position
The empirical model developed by the authors relating the tensile
strength and the process parameter that includes laser power, welding speed
and focal position is shown as below. The process parameters and their
working limits is shown in Table 5.1
Tensile Strength = 206.39-3.60 * X1 + 5.40 * X2 -0.80 * X3 +0.63 * X1 *
X2+0.88 * X1 * X3-2.37 * X2 * X3 +0.77 * X122.23 * X2
2-19.38 * X3
2.(
G. Padmanaban et.al.,)
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Table 5.1. Process Parameters and their Working Limits of Laser Welding
Process
S.No Parameter UnitLevels
(-1) 0 (+1)
1
Laser
power Watts 2500 3000 3500
2
Welding
Speed m/min 4.5 5 5.5
3
Focal
position Mm 0 -1.5 -3
5.1.1. OPTIMIZATION USING GA TOOLBOX IN MATLAB
Genetic algorithms are computerized search and optimization algorithms
based on the mechanics of natural genetics and natural selection. GA is very
different from traditional search and optimization methods used in
engineering problems. Because of its simplicity, ease of operation, minimum
requirements and global perspective, GA has been successfully used in a wide
variety of problems.In order to estimate the maximum tensile strength, the
following steps are to be followed:
i. Firstly the model which is developed using response surface methodwhich relates the tensile stength and the process parameter is typed in
editor window.
ii. Then the editor window is saved using any name.iii. After opening the optimization toolbox, the fitness function is entered
with the filename saved.
iv. Then the number of the variables is entered and the lower and the upperbound is entered.
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v. If needed the parameters of the GA toolbox is changed, otherwise it isset as default and optimization is started to run. After executing the
problem, the results will be displayed.
GENETIC ALGORITHM PARAMETERS
The parameters used for GA is given below.
Population size = 100 Length of chromosome = 40 Selection operator: Roulette method Crossover operator : Single point operator Crossover probability = 0.9 Mutation probability = 0.01 Fitness parameter: Tensile strength.
5.1.2. OPTIMIZED PROCESS PARAMETERS OF LASER WELDING:
The optimized result and the maximized tensile strength of the laser
welding process parameters and the experimental results are shown as
follows:
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Table 5.2. Optimized Process Parameters of Laser Welding
S.NO Process
parameter
Values of process parameter Tensile strength (MPa)
Predicted
(RSM)
Predicted
(GA)
Experimental Predicted
(RSM)
Predicted
(GA)
Experimental
1 Laserpower
(Watts)
2520 2500 2500
212.25 213.712 212
2 WeldingSpeed
(m/min)
5.24 5 5
3 Focal
position
(mm)
-1.19 -1.65 -1.5
5.2. OPTIMIZING THE DIFFUSION BONDING PROCESS
PARAMETERS
Experimental studies were carried out by G.Mahendran et al.,
(2010) on Magnesium and Copper using diffusion bonding methods. Three
factor five-level, central composite rotatable design matrix method is used for
the purpose of designing the experiments. Empirical relationship was
developed in order to predict the diffusion layer thickness, hardness and
strength of the joint using the response surface methodology.
In this part of the work, the shear strength and the bond strength of
the diffusion bonded Magnesium and Copper is to be maximized for the
optimum process parameters.
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The mathematical model relating the shear strength, tensile
strength and the process parameters is shown as below:
Shear Strength = 58.97-4.33 * X1 -3.26 * X2 -3.35 * X3 +1.59 * X12
1.59 * X22+3.42 * X3
2
Bond Strength = 74.46-4.58 * X1 -3.38 * X2 -3.48 * X3 -1.12 * X1 * X2 -
1.69 * X12+1.87 * X2
2-1.51* X3
2
The process parameter and their working limits are shown in table
5.3.
Table 5.3. Process Parameters and their Working Limits of Diffusion Bonding
5.2.1. OPTIMIZATION USING GA TOOLBOX IN MATLAB
In this problem the objective functions are Shear Strength and
Bond Strength of the Diffusion Bonded Magnesium and Copper joints. So the
problem falls under the multhiobjective optimization problem.
S.No Parameter Unit
Levels
-
1.682-1 0 1 1.682
1 BondingTemperature
0C 425 450 475 500 525
2Bonding
PressureMPa 4 8 12 16 20
3Holding
TimeMin 10 20 30 40 50
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GENETIC ALGORITHM PARAMETERS
The parameters used for GA is given below.
GA Solver : GAMULTIOBJ (Multi objective Optimization using GA)
Population size = 60 Population type : double vector Length of chromosome = 40 Selection operator: Tournament Crossover operator : Single point operator Crossover probability = 0.8 Mutation probability = 0.01 Mutation function : Adaptive Feasible Fitness parameter : Shear strength and Bond Strength .
5.2.2. OPTIMIZED PROCESS PARAMETERS OF LASER WELDING:
The optimized result and the maximized tensile strength of the laserwelding process parameters and the experimental results are shown as
follows:
Table 5.4. Optimized Process Parameters of Friction Welding
S.NOProcess
parameter /
Unit
Values of process
parameterShear strength (MPa) Bond strength (MPa)
Predicted
(GA) Experimental
Predicted
(GA) Experimental
Predicted
(GA) Experimental
1
Bonding
Temperature
/0C
449.37 450
69.742 66 80.083 812
Bonding
Pressure /
Mpa
10.62 8
3Holding
Time / min
19.77 20
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CHAPTER 6
CONCLUSION
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CONCLUSION
The following important conclusions are made from the above investigation:
i. From the literature survey friction welding process, the research worksof the welding are studied well. The possibilities of the research in
welding process especially in the friction welding and the diffusion
bonding process were identified.ii. Modeling techniques for the solid state welding processes were
surveyed. Most of the studies showed that the conventional regression
method is used for modeling solid state welding process. Very few
literatures were published on the modeling of the welding process
parameters using Artificial Neural Network (ANN). Finite element
method is also used to model the mechanism of these welding
processes and to simulate it.
iii. Conventional optimization techniques such as Design OfExperiments(DOE), response surface methodology, statistical methods
such as Analysis of variance (ANOVA), correlation analysis were used
as the optimization tool in many literatures, whereas very few
literatures were interested to investigate on the non-traditional
optimization techniques such as Genetic Algorithm, particle swarm
optimization, simulated annealing, etc.,
iv. The optimization of friction welding, laser welding and diffusionbonding process parameters were carried for the specific objective
functions. As a special case the multi objective optimization of
diffusion bonding process parameters in order to maximize the shear
strength and the bond strength of the welded joints.
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v. The optimized results of the process parameters and the objectivefunction of the friction welding, laser welding and diffusion bonding
process are being compared with the experimental results and in some
case it is being compared with the other Conventional optimization
technique results which are all in good agreement.
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REFERENCES
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