Presentation Slides

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BMS College of Engineering
Mechanical
Engineering
Techniques used for shape, size and topology
optimization of a cantilever beam
Guide: H.K. Rangavitthal
Associate Professor
Name USN
Akshay Varik 1BM10ME005
Amit Kumar Singh 1BM10ME006
Anirudh N Katti 1BM10ME010
Rahul R Kamath 1BM10ME083
S Darshan 1BM10ME093
DEPARTMENT OF MECHANICAL
ENGINEERING

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INTRODUCTION
 Optimization is a process of selecting the best possible result among
many possible results under given circumstances. Mathematically it
is finding out maximum or minimum of function. Design
optimization is applying optimization to conventional design
process to get best value for design variables based on certain
criteria.

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1.1 Applications of optimization
 Designing aircraft parts for minimum weight
 Designing optimum production schedules
 Minimum cost design of various devices
 Inventory control planning
 Optimum design of structural members
 Optimum design of control systems and electrical circuits
 Optimal production planning and controlling
 Finding Optimal nose cone angle for minimum drag
 Designing material handling equipments such as conveyors, cranes,
trucks etc for material handling

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1.2 History of optimization
 The existence of optimization problems and methods trace back to
300 BC, when Euclid solved problems of elementary geometry
which were optimization problems.
 Contributions of Newton and Leibnitz to calculus lead to the
development of differential calculus methods of optimization.
 Numerical methods of unconstrained optimization were developed
in the United Kingdom only in the 1960s.
 Development of the methods of constrained optimization- simplex
method by Dantzig in 1947 for linear programming problems ,
principle of optimality in 1957 by Bellman for dynamic
programming problems.
 Modern methods include genetic algorithms, simulated annealing,
particle swarm optimization, neural network-based optimization,
and fuzzy optimization etc.

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Methods of optimization
.
Traditional Optimization Techniques Modern Optimization Techniques
Calculus Methods Genetic Algorithm
Gradient Based Simulated Annealing
Geometric Programming Ant Colony Optimization
Linear Programming Particle Swarm Optimization
Quadratic Programming Neural Networks
Dynamic Programming Fuzzy Logic
Stochastic Programming
Multi-Objective Programming

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Introduction to beams
Beam
 A beam is a long structural member with a relatively small cross-sectional
dimensions and is generally subjected to bending due to transverse
forces acting on it
Supports for beams
 Beams are provided with various kinds of supports generally depending
on the direction and magnitude of the resultant of the force system
carried by them. Each type of support may exert a particular mode of
reaction that could be a reactive force, reactive moment, or a combination
of both.

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Simple support:
 A beam just placed over two supports is said to be a simply
supported beam. A simple support can exert reaction only in a
direction perpendicular to the contact surface. Rotation of a beam
is not arrested by a simple support and hence it does not exert any
reactive moment.

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Hinged support:
 A beam may be hinged to a support with a pin such that its
translational motion is arrested in all the directions. However a
hinged support too cannot arrest the rotation of a beam and hence
it does not exert a reactive moment. A hinge can exert reaction in
any direction in the plane of loading, depending on the directions of
the loads. Therefore the known reaction R may be found by
considering its any two mutually perpendicular components such as
RAV and RAH.

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Fixed support:
 Both translational and rotational motion of a beam may be arrested
by providing the fixed support that exerts both reactive force and
reactive moment. A beam having only one of its ends provided with
fixed support is called cantilever.

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Moment of Intertia
The moment of inertia of an object about a given axis describes how
difficult it is to change its angular motion about that axis. Therefore, it
encompasses not just how much mass the object has overall, but how far
each bit of mass is from the axis. The farther out the object's mass is, the
more rotational inertia the object has, and the more force is required to
change its rotation rate.
Stress in Beams
Internally, beams experience compressive, tensile and shear stress as a
result of the loads applied on them. Typically, under gravity loads, the
original length of the beam is slightly reduced to enclose a smaller radius
arc at the top of the beam, resulting in compression, while the same
original beam length at the bottom of the beam is slightly stretched to
enclose a larger radius arc, and so is under tension.

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The same original length of the middle of the beam, generally halfway
between the top and bottom, is the same as the radial arc of bending, and
so it is under neither compression nor tension, and defines the neutral
axis (dotted line in the beam figure). Above the supports, the beam is
exposed to shear stress.
There are some reinforced concrete beams in which the concrete is
entirely in compression with tensile forces taken by steel tendons. These
beams are known as pre-stressed beams, and are fabricated to produce a
compression more than the expected tension under loading conditions.
High strength steel tendons are stretched while the beam is cast over
them. Then, when the concrete has cured, the tendons are slowly released
and the beam is immediately under eccentric axial loads. This eccentric
loading creates an internal moment, and, in turn, increases the moment
carrying capacity of the beam. They are commonly used on highway
bridges.

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The primary tool for structural analysis of beams is the Euler-Bernoulli
Equation. Other mathematical methods for determining the deflection of
beams include "method of virtual work " and the "slope deflection
method". Engineers are interested in determining deflections because the
beam may be in direct contact with a brittle material such as glass. Beam
deflections are also minimized for aesthetic reasons. A visibly sagging
beam, even if structurally safe, is unsightly and to be avoided.
Mathematical methods for determining the beam forces (internal forces of
the beam and the forces that are imposed on the beam support) include
the "moment distribution method", the force or flexibility method and
the direct stiffness method.

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General Cross-Sectional Shapes
Most beams in buildings have rectangular cross sections, but a more
efficient cross section for a beam is an I or H section which is typically
seen in steel construction. Because of the parallel axis theorem and the
fact that most of the material is away from the neutral axis, the second
moment of area of the beam increases, which in turn increases the
stiffness.
An I-beam is only the most efficient shape in one direction of bending: up
and down looking at the profile as an I. If the beam is bent side to side, it
functions as an H where it is less efficient. The most efficient shape for
both directions in 2D is a box (a square shell) however the most efficient
shape for bending in any direction is a cylindrical shell or tube. But, for
unidirectional bending, the I or wide flange beam is superior.
Efficiency means that for the same cross sectional area (volume of beam
per length) subjected to the same loading conditions, the beam deflects
less. Other shapes, like L (angles), C (channels) or tubes, are also used in
construction when there are special requirements.

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TRADITIONAL OPTIMIZATION
TECHNIQUES
1.Graphical Method
 This method can be employed for single-objective constrained problems
involving one or two variables.
 It is the simplest type of optimization technique and gives results which
can be used as reasonably accurate initial-guess for more powerful
algorithms.
 The primary design equation, subsidiary design equation and limit
equations are plotted.
 The intersection of these plots gives the optimal results using this
method.

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Methodology
 The design of cantilever beam of rectangular cross section,
subjected to point end load, falls into this category. The
width(x2) and height(x1) of the beam are the design variables
 Input variables are force applied (F), length of the beam (L),
Young’s Modulus of the material (E), Yield Stress (σy), Factor
of Safety (FOS), max deflection (δmax) and limits on depth (‘a’,
‘b’) and width of the beam (‘c’, ‘e’)

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DESIGN EQUATIONS

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INPUT PARAMETERS
Parameter Value Units
L 1000 Mm
F 2000 N
E 200e3 MPa
σy 56 Mpa
Minimum b 40 Mm
Maximum b 200 Mm

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RESULTS
As per this method, the optimum dimensions of the beam are width 38mm and
height 76mm.

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Traditional Method of Optimization
Johnson’s Method

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Introduction
 An optimization technique which extensively considers the
subsidiary design equations and limit equations apart from the
primary design equation
 The objective is to write cost as a function of geometrical
parameters
 The cantilever under consideration is designed to be used as flat
spring

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Formulations
 The primary design equation is C=cW=cwV=cwLbh
 Materials primarily used are berrylium copper, phosphor bronze
and spring steel. Also cost of labor, tooling's are considered
constant for any feasible design
 The subsidiary design equation is K1=(Ebh3/4L3) which arises from
the spring stiffness and maximum deflection equation

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 The maximum shear stress theory is adopted.
Hence τxy = (F/2I)[(h/2)2-y2] .
 Here, on due substitution we get τmax = (Mh)/(4I) =(3FL/bh3) at y=h/2,
since it is the governing equation because of its value being greater
than that at y=0
 From fatigue failure criteria, we obtain a limit equation as
(τmax - (1-p) 0.5 τmax ) <= (Se/2N) that is τmax <= (Se/(1+p)N)
 Space restrictions result in another limit equation, being
bmax >= b >= bmin

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 Eliminating the independent geometric parameters, we get
[K1/ (τmax)2] = (Ebh3/4L3)/(9F2L2/b2h4) , which on rearranging and
substitution in primary design equations yield us
Cmin = (9/4)(F2N2/K1)[(cwE(1+p)2/Se
2)]
 Finally, on incorporating the geometrical limit equations
C=[2(2/3)][K1
(1/3)L2][cw/E(1/3)]b(2/3) and C=[4][K1L4][cw/E]h-2

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Inference
 Plotting above on log plot gives us straight lines having slopes 2/3
and -2 respectively. The optimum region is seen from the graphs

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Other methods of Optimisation
Stochastic Hill Climbing
Simulated Annealing
Neural Network

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Stochastic Hill Climbing
 Stochastic hill climbing usually starts from a random select point.
 Randomly chooses a solution in the neighborhood of current
solution of current solution and retains the new solution if it
improves the fitness function.
 On functions with many peaks (multimodal functions), the
algorithm is likely to stop on the first peak it finds even if it is not
the highest one. Once a peak is reached, hill climbing cannot
progress anymore, and that is problematic when this point is a local
optimum.
 It works well if there is not too many local optima in the search
space. But if the fitness function is very “noisy” with many small
peaks, stochastic hill climbing is definitely not a good method to use.

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Simulated Annealing
 Simulated Annealing was originally inspired by formation of crystal
in solids during cooling.
 The iteration of the simulated annealing consists of randomly
choosing a new solution in the neighbourhood of the actual solution.
If the fitness function of the new solution is better than the fitness
function of the current one, the new solution is accepted as the new
current solution. If the fitness function is not improved, the new
solution is retained with a probability: p=e^[−(f (y )− f (x ))]/(kT)
where [ f (y) − f (x) ] is the difference of the fitness function between
the new and the old solution.

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 The simulated annealing behaves like a hill climbing method but
with the possibility of going downhill to avoid being trapped at local
optima.
 When the temperature is high, the probability of deteriorate the
solution is quite important, and then a lot of large moves are
possible to explore the search space. The more the temperature
decreases, the more difficult it is to go downhill, the algorithm tries
to climb up from the current solution to reach a maximum.
 Usually, the simulated annealing starts from a high temperature,
which decreases exponentially. The slower the cooling, the better it
is for finding good solutions.

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Neural Network
 They process information in the way biological nervous
systems, such as the brain, for optimal decision making.
 Pattern classification is a domain in which neural
networks has shown to perform better than
conventional linear programming methods.
 They exhibit the following key behaviours: interacting
with noisy data, massive parallelism, fault tolerance,
adapting to circumstances that make them perform
better than conventional computations in these class of
problems, where one can't formulate an algorithmic
solution, where one can get lots of examples of the
behaviour we require.

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Other methods include:
 Gradient based method of finding local optima.
 Particle swarm method- Here the leader is the one closest to food
and others orient themselves closest to the leader. Leader might
change as the process progresses.
 Random Search method
 Symbolic Artificial Intelligence

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MODERN OPTIMIZATION
METHODS

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GENETIC ALGORITHM
 GA’s are search algorithms based on the mechanics of natural
selection and natural genetics. They combine survival of the fittest
among string structures with a structured yet randomized information
exchange to form a search algorithm with some innovative flair of
human search.
 In every generation, a new set of artificial strings is created using bits
and pieces of the fittest of the old; an occasional mutation is tried for
good measure.
 GA has been developed by John Holland and his team of students.
 The central theme of research on GA has been robustness, the balance
between efficiency and efficacy necessary for survival in different
environments.

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GA vs. Traditional methods
 GA’s work with a coding of the parameter set, not the
parameters themselves.
 GA’s search from a population of points, not a single
point.
 GA’s use pay-off (objective function) information, not
derivations or other auxiliary knowledge
 GA’s use probabilistic rules, not deterministic rules.

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Terminology
Natural Genetic Algorithm
Chromosome String
Genre feature, character or detector
Allele feature value
Locus string position
Genotype Structure
Phenotype parameter set, alternate solution, a decoded structure
Epistasis non-linearity

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Operators used in GA
 Reproduction is a process in which individual strings are copied
according to their objectivefunction values, F (fitness function).
Copying strings according to their fitness values means that strings
with a higher value have a higher probability of contributing to one or
more offspring in the next generation. This is the philosophy of
Darwin’s Survival of the fittest concept.
 Crossover may proceed in 2 steps
Members of mating pool are mated at random
Each pair of strings undergoes crossover
 An integer position K along the string is selected uniformly
at random between 1 and the string length less one [1, l-1]
 Two new strings are created by swapping all characteristics
between position K+1 and l inclusively.
A1=01101 A1
1=01100
A2=11000 A2
1=11001

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•GA exploits a wealth of information in the strings by reproducing
high quality notions to their performance crossing these notions
with many other high-performance notions from other strings.
Mutations are the occasional (with some probability) random
alteration of the value of a string position.
•By itself, mutation is a random walk through the string place.
When used sparingly with reproduction and crossover, it is an
insurance policy against premature loss of important motions.

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Parameters to be specified in GA
 Probability of Crossover – The probability that crossover will be
performed between a pair of permit chromosome. If crossover is not
performed, the parents (unmodified) are treated as potential children
for the next generation. (Pcrossover)
 Probability of Mutation (Pmotation) – The probability that any given allele
on any given chromosome will mutate.
 Fitness Scaling Coefficient (Cmolt) – A measure of the magnitude of
fitness value attenuation. This should be selected so that a proper
number of the fittest members are chosen to continive into the
subsequent generation, without allowing them to dominate the
generation. This typically represents the desired ratio of the fitness in
a population to the average fitness of the population.
 Population Size – Number of chromosomes in each GA generation

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Optimization by Genetic Algorithm
 Objective: To minimize the dimensions of thickness of web
when a rectangular beam is converted to an I-beam

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Formulation of Objective Function

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Matlab Code
clear all;
close all;
clc;
t = input('Enter the value of thickness of web= ');
h = 78;
b = 36;
F = 2000;
E = 200000;
c=3.076;
w=7.85*(10^(-9));
k=100;
ss= (((((h*(b-t))/(2*t))+((h+b-t)/4))*((6*F*(h+b-t))/(h*((h+b-t)^3)-(h^4)+(t*(h^3)))))^(2));
cost= ((c*w)/(k))*(9*E*F*F)/(4*ss);

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Manual Solution using GA

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Iteration 1
STRING INITIAL
POPULATION
DECIMAL
VALUE
SHEAR STRESS
FUNCTION
VALUE(Mpa)
COST
FUNCTION
VALUE($)
RELATIVE
FITNESS
ACTUAL
COUNT
1 01110 14 4.7261 91.9655 0.14 1
2 11010 26 1.8824 230.8977 0.37 0
3 01101 13 5.3217 81.6732 0.13 2
4 11001 25 1.9901 218.4056 0.35 1
Sum= 622.942
Average=155.7355
Minimum Value=81.6732
MATING POOL MATE
(random)
CROSSOVER
SITE (random)
NEW
POPULATION
DECIMAL
VALUE
SHEAR STRESS
FUNCTION
VALUE
COST
FUNCTION
VALUE
01110 2 4 0111 1 15 4.2389 102.5356
01101 1 4 0110 0 12 6.0619 71.7001
01101 4 2 01 001 9 9.8363 44.1874
11001 3 2 11 101 29 1.6101 269.9383
Sum=488.3614
Average=122.09035
Minimum Value= 44.1874

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Iteration 2
STRING POPULATION DECIMAL
VALUE
COST FUNCTION
VALUE
RELATIVE
FITNESS
ACTUAL
COUNT
1 01111 15 102.5356 0.2099 1
2 01100 12 71.7001 0.1468 1
3 01001 9 44.1874 0.0904 2
4 11101 29 269.9383 0.5527 0
MATING POOL MATE
(random)
CROSSOVER
SITE (random)
NEW
POPULATION
DECIMAL
VALUE
SHEAR STRESS
FUNCTION
VALUE
COST
FUNCTION
VALUE
01111 4 3 011 01 13 5.3216 81.6732
01100 3 4 0110 1 13 5.3217 81.6732
01001 2 4 0100 0 8 12.0693 36.0119
01001 1 3 010 11 11 6.9999 62.0925
Sum=261.45
Average=65.3627
Minimum Value=36.0119
After the second iteration, we see that the average has dropped by 46.46% of 1st
iteration and minimum value as come down to 36.011

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Optimization using packages
1.Ansys Design Optimization
Structural optimization of cantilever beam using
ANSYS design optimization

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Introduction
 ANSYS, Inc. is an engineering simulation software (computer-
aided engineering, or CAE) developer which offers engineering
simulation solution sets in engineering simulation that a design
process requires. Companies in a wide variety of industries use
ANSYS software. The tools put a virtual product through a
rigorous testing procedure (such as crashing a car into a brick wall,
or running for several years on a tarmac road) before it becomes a
physical object.
 Virtually any aspect of design can be optimized: dimensions (such
as thickness), shape (such as fillet radii), placement of supports, cost
of fabrication, natural frequency, material property, and so on.
Here we are concentrating on dimensional optimization.

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 The ANSYS program offers two optimization methods to
accommodate a wide range of optimization problems. The sub problem
approximation method is an advanced zero-order method that can be
efficiently applied to most engineering problems.
 First Order optimization method as employed in ANSYS uses penalty
function approach to minimize the objective function. This is done by
converting a constrained minimization problem into an unconstrained
problem by adding penalty functions. The penalties are added to the
objective function to account for the imposed constraints.

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Statement of problem
 Ansys uses several methods of optimization.
 The motive of this section is to understand one of these methods by
applying it to optimize a rectangular cantilever beam with uniform
cross section area.
 We solve this problem using design optimization by first order
method.
 We express the results through graphs
 In the end we interpret these final results in terms of optimum
dimensions and maximum stresses in the beam.

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Design procedure
Before describing the procedure for design optimization, we will define
some of the terminology: Design variable, State variable, Objective
function, feasible and infeasible designs, the analysis file, iterations, and
loops, design sets, etc.
 Design Variables (DVs): Independent variables that directly affect the
design objective. In this example, the width and height of the beam are the
DVs , changing either variable has a direct effect on the solution of the
problem.
 State Variables (SVs): Dependent variables that change as a result of
changing the DVs. These variables are necessary to constrain the design. In
this example, the SV is the maximum stress in the beam. Without this SV,
our optimization will continue until both the width and height are zero.
This would minimize the weight to zero which is not a useful result.
 Objective Variable (OV): The objective variable is the one variable in the
optimization that needs to be minimized. In our problem, we will be
minimizing the volume of the beam.

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 The step by step procedure is shown below:-
 Model the problem on ANSYS package:
Select a beam element and apply material properties and real
constants.
Model the beam using elements.
Apply the loads and solve for maximum stresses.
The completed beam model is shown in figure 5.1
 Build the Model Parametrically:
The result we obtained in previous step are stored parametrically along
with the basic structural dimensions as scalar parameters.
The parameters are: Width, Height, SmaxI, SmaxJ and Volume
 Preparing the Analysis File:
It is necessary to write the outline of our problem to an ANSYS
command file. This is so that ANSYS can iteratively run solutions to
our problem based on different values for the variables that we will
define.

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6 RESULTS:
Figure 5.1: Original Beam of rectangular Cross section.
Depth= 75 mm
Width=38 mm Length=1000mm Load=1000N

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For this purpose we bring about changes in DB log file of the ANSYS
and replace the graphical codes with optimization ones and save it as a
text file.
 Establish Parameters for Optimization and Declare Optimization
Variables:
At this point, having completed the analysis file, you are ready to
begin optimization. When performing optimization interactively, it is
advantageous (but optional) to first establish the parameters from the
analysis file in the ANSYS database.
The initial parameter values may be required as a starting point for a
first order optimization. Also, for any type of optimization run, having
the parameters in the database makes them easy to access through the
GUI when defining optimization variables. To establish the parameters
in the database, reassign the database file associated with the analysis
file.
ANSYS needs to know which variables are critical to the optimization.
To define variables, we need to know which variables have an effect on
the variable to be minimized.
Here objective is to minimize the volume of a beam which is directly
related to the weight of the beam.

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 Choose Optimization Tool or Method and Initiate Optimization
Analysis:
Choose the optimization tool as the first order method as results
obtained are more accurate. After choosing the method of
Optimization, the analysis is run by deciding the number of iterations
or loops that must be conducted by the system.
 Viewing The Final Results and Obtaining Important Plots:
The graph is shown in figure 3. The conclusions that can be drawn by
the graph are:
1. The width and height of the beam are reducing with increase in
Smax
2. Here Smax is the maximum stress that the material can withstand.
3. As the difference in maximum stresses developed in the beam and the
maximum stress that can be withstood by the material increases the
cross sectional are decreases for a given length. In our analysis the
maximum stress developed in the beam was found to be 56Mpa.
4. The maximum stress of material was set to 200 Mpa. As the material
required can withstand more stress, it is underutilized for the beam.
So the volume is reduced to optimize the weight of the beam.

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Figure 5.2 Final Dimensions Obtained
Depth=51.86 mm
Width=22.36 mm

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Optimization using Packages
2.COMSOL
Topological optimization of cantilever beam using
COMSOL

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Topological Optimization Using COMSOL
 “Topology optimization is a mathematical approach that optimizes
material layout within a given design space, for a given set of loads
and boundary conditions such that the resulting layout meets a
prescribed set of performance targets.”
 An optimization problem can be stated as: Minimize the Compliance
(deflection) of the structure such that the total mass of the part is
within some limit.
 Topology optimization treats the material distribution, as defined
by the finite element mesh, as the design variables in an attempt to
improve the cost function, while satisfying the constraints.

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Methodology
 Model Used : SIMPS (solid isotropic material with penalization)
E = μ(x)pE0, where 0 < μ(x) ≤1.
In the model, p is set to 5.
 Objective Functional : strain energy
 Control Variable : ρdesign which is constrained, using a pointwise
constraint, to a value between 10−4 and 1
 An integral constraints is used to set the volume fraction to 0.5
 An integral inequality constraint which provides regularization is
set up to reduce the dependency on the mesh size and prevent
check-board pattern solution.
 Material Used: Structural Steel

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Results
2D plot showing deflection and von-mises stress

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2D Density plot showing topology optimization

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Optimization using Packages
2.HyperWorks
Structural optimization of cantilever beam using
HyperWorks

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Introduction to HyperWorks
 HyperWorks is a very efficient tool for design optimization,
performance data management, and process automation.
 HyperMesh is an extremely powerful preprocessor for CAE
applications like mesh generation
 HyperMorph is a part of HyperMesh is used for shape generation
 OptiStruct is a part of HyperWorks which is used for Optimization
of Structures.

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Objective
 Minimizing the volume of a cantilever beam subjected to end load
whose deflection at the lower right corner should be limited to 3mm.

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Procedure of Optimization
 Beam is generated using nodes and an appropriate meshing is
provided as per the accuracy required.
 Handles are created in the system. There are two types of handles-
Global and Local Handles.

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Morphing Technique
 Morphing is a technique used
to change one image into
another using seamless
transition.
 HyperMorph is used to morph
a FE model in useful, logical,
and intuitive ways which result
in minimal element distortion.
 It is used here as it allows us to
modify original mesh to meet
new mesh design.

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 Two shape design variables are created using the shapes that were
generated using Morphing.
 Two responses are defined in this problem; a volume response for
the objective function and a displacement response for the
constraint.
 A constraint with upper bound is to defined for the displacement
response ie.. Deflection at the lower right corner should be limited
to 3mm.
 A shape card is defined so as to look at displacement/stress results
at the end.
 Finally for the Optimization step, OptiStruct is run from the
Analysis page.

65
Mechanical
Engineering
Results Obtained
 Optimum beam
shape for the given
condition
 Contour Plot of the
Displacement on
Top of the Shape
Optimized Model.

66
Mechanical
Engineering
Conclusion
 This project gave us a very good opportunity to understand the
various techniques of structural optimization. These methods were
applied to a general problem, pertinent to a cantilever beam of
rectangular cross-section and the results were analyzed
 Graphical method gave the breadth and height of a rectangular
cantilever beam based on basic strength of material equations
 Johnsons method furthered the graphical method and took into
consideration Fatigue strength of material and spring stiffness
 ANSYS method gave us the variation of breadth and height of
rectangular cantilever beam relative to the maximum stresses in the
beam. This method used mathematical iterations to give us the
result

67
Mechanical
Engineering
 Genetic algorithm gave us the thickness of the web of the I beam
which minimizes the cost while preventing failure of the component
due to shear stresses.
 Comsol software gave us a cantilever beam which is topologically
optimized having a truss-like structure which has maximum stiffness
 HyperWorks software gave a 2d shape-optimized cantilever beam
which transformed the straight rectangular beam to a tapered one.

68
Mechanical
Engineering
Future Work
 While the structural optimization is one of the most extensively
researched areas in optimization, its use in product development is
relatively new and it is yet to gain a mainstream popularity in
industry.
 The future research in this area will continue to address:
1.Technologies for structural optimization in conceptual design
2. Technologies for large-scale structural optimization.
 Mutations and further improvements to the basic Genetic
algorithm like particle swarm optimization can also be
incorporated to create a robust code to solve complex and multi-
objective design problems.

69
Mechanical
Engineering
Thank
you

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