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Optimization
MEL 806
Thermal System Simulation (2-0-2)
Dr. Prabal Talukdar Associate Professor
Department of Mechanical Engineering
e
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Introduction• In the preceding lectures, we focused our attention on obtaining a
workable, feasible, or acceptable design of a system. Such a design
satisfies the requirements for the given application, without violating
any imposed constraints. A system fabricated or assembled
ecause o s es gn s expec e o per orm e appropr a e as s
for which the effort was undertaken.
• However, the design would generally not be the best design, where
e e n on o es s ase on cos , per ormance, e c ency or
some other such measure.
• In actual practice, we are usually interested in obtaining the best
qua y or per ormance per un cos , w accep a e env ronmen aeffects. This brings in the concept of optimization, which minimizes
or maximizes quantities and characteristics of particular interest to a
.
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UNCONSTRAINED SEARCH WITH MULTIPLE
VARIABLES
Let us now consider the search for an optimal
design when the system is governed by two ormore independent variables.
However, the complexity of the problem rises
,
therefore, attention is generally directed at the most
important variables, usually restricting these to two
or three.
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• ,
be well characterized in terms of two or threepredominant variables.
• Examples of this include the length and diameter
of a heat exchanger, fluid flow rate andevaporator temperature in a refrigeration
system, and so on.
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•
approach to the optimum design, a
or lines of constant values of the objective
.
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Lattice Search Method
Lattice search method in a two-variable space.
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Univariate Search•
objective function with respect to one variable ata time. Therefore, the multivariable problem is
reduced to a series of single-variable
optimization problems, with the processconverg ng o e op mum as e var a es are
alternated
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Graphical presentation
Various steps in the univariate search method.
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The method A starting point is chosen based on available information on the system
.
First, one of the variables, say x, is held constant and the function is
optimized with respect to the other variable y.
Point A represents the optimum thus obtained. Then y is held constant
at the value at point A and the function is optimized with respect to x to
obtain the optimum given by point B.
Again, x is held constant at the value at point B and y is varied to obtain
the optimum, given by point C.
This process is continued, alternating the variable, which is changedwhile keeping the others constant, until the optimum is attained.
This is indicated b the chan e in the ob ective function from one ste
to the next, becoming less than a chosen convergence criterion or
tolerance
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If y is kept constant, the value of x at the optimum is given by
Similarly, if x is held constant, the value of y at the optimum is given by
let us choose x = y = 0.5 as the starting point.
First x is held constant and y is varied to obtain an optimum value
of U. Then y is held constant and x is varied to obtain an optimum
value of U. In both cases, the recedin e uations are used.
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Calculations
x y u
0.5 1.632993 9.839626
1.944161 0.828139 5.598794
2.437957 0.739531 5.427791
. . .
2.547644 0.723436 5.422363
2.550314 0.723057 5.422359
2.55076 0.722994 5.422359
2.550834 0.722983 5.422359
2.550847 0.722982 5.422359
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Steepest Ascent/Descent Method• The stee est ascent/descent method is a ver efficient
search method for multivariable optimization and iswidely used for a variety of applications, including
.
• It is a hill-climbing technique in that it attempts to move
toward the peak, for maximizing the objective function, ortoward the valley, for minimizing the objective function,
over the shortest possible path.
case and steepest descent in the latter.
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• At each step, starting with the initial trial point, the
rec on n w c e o ec ve unc on c anges a e
greatest rate is chosen for moving the location of theoint, which re resents the desi n on the multivariable
space.
Steepest ascent method, shown in terms of (a) the climb toward the peakof a hill and (b) in terms of constant U contours.
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• It was shown that the radient vector is normal toU∇
the constant U contour line in a two-variable space, tothe constant U surface in a three-variable space, and so
.
• Since the normal direction represents the shortest
distance between two contour lines, the direction of thegradient vector is the direction in which U changes at
the greatest rate.U∇
,written as
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• At each trial oint the radient vector is determined and
the search is moved along this vector, the direction beingchosen so that U increases if a maximum is sought, or U
.
• The direction represented by the gradient vector is given
by the relationship between the changes in theindependent variables. Denoting these by Δx1, Δx2 , ---
Δxn , we have
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First approach• Choose a startin oint. Select Δx. Calculate the
derivatives.• Decide the direction of movement, i.e., whether Δx is
pos ve or nega ve. a cu a e y. a n e new va ues
of x, y, and U.
• Calculate the derivatives a ain at this oint. Re eat
previous steps to attain new point.
• This procedure is continued until the change in the
var a es e ween wo consecu ve era ons s w n adesired convergence criterion.
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Example Problem•
before and apply the two approaches just
method to obtain the minimum cost U.
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• The startin oint is taken as x = = 0.5. The results
obtained for different values ofΔ
x are
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Multivariable Constrained
•
optimization, which is much more involved thanthe various unconstrained optimization cases
considered thus far.
• The number of independent variables must belarger than the number of equality constraints;
otherwise, these constraints may simply be used
possible
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Penalt Function Method• The basic a roach of this method is to convert the
constrained problem into an unconstrained one byconstructing a composite function using the objective
• Let us consider the optimization problem given by the
equations
• The composite function, also known as the penaltyfunction, may be formulated in many different ways.
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• If a maximum in U is being sought, a new objective
•
function is defined as
• Here the r’s are scalar quantities that vary the
importance given to the various constraints and are
.
• They may all be taken as equal or different.
• Higher values may be taken for the constraints that are
critical and smaller values for those that are not as
important.
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• If the penalty parameters are all taken as zero, the constraints
have no effect on the solution and therefore the constraints
are not satisfied.
• On the other hand, if these parameters are taken as large, the
constraints are satisfied but the conver ence to the o timum is
slow.
• Therefore, by varying the penalty parameters we can vary the
rate of conver ence and the effect of the different constraints
on the solution.
• The general approach is to start with small values of the
enalt arameters and raduall increase these as the G’s
which represent the constraints, become small.• This implies going gradually and systematically from an
unconstrained roblem to a constrained one.
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different values of the penalty parameter r.
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Example Problem
In a two-component system, the cost is the objective function given by the
,
where x and y represent the specifications of the two components. Thesevariables are also linked by mass conservation to yield the constraint
G(x, y) = xy -12 = 0
Solve this problem by the penalty function method to obtain minimum cost.
The new objective function V(x, y), consisting of the objective function ande cons ra n , s e ne as
the optimum. An exhaustive search can be used because of the simplicityof the method and the given functions.
, ,
large, the constraints are satisfied, but the convergence is slow.
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We may also derive x and y in terms of the penalty parameter r, by
,
expressions to zero, as
See the spreadsheet