Multiobjective Design Optimization of Rolling Element Bearing using NSGA II

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Multi-objective Design Optimization of Rolling
Element Bearing using NSGA II

• Presented by,
• Shantanu Gupta
• Under guidance of
• Dr. S. B. Nair
• Dr. R. Tewari

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OUTLINE

• Rolling Element Bearings
• Problem statement
• Optimization (single and multi-objective)
• Previous work
• NSGA II Multi-objective optimization using
  evolutionary concepts
• Application and Results
• Conclusion

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OUTLINE

• Rolling Element Bearings
• Problem statement
• Optimization (single and multi-objective)
• Previous work
• NSGA II Multi-objective optimization using
  evolutionary concepts
• Application and Results
• Conclusion

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Rolling Element Bearings

• Wide use in mechanical engineering
• Ever growing application
• Multiple performance criteria
• Time for designers to get assistance from
  Computer scientists

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Nomenclature
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Cross section
dm Db ro ri Z
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Important performance measures

• Longest fatigue life
• Dynamic Capacity (Cd)
• Longest wear life
• Minimum film thickness (hmin)
• Static capacity (Cs)

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OUTLINE

• Rolling Element Bearings
• Problem statement
• Optimization (single and multi-objective)
• Previous work
• NSGA II Multi-objective optimization using
  evolutionary concepts
• Application and Results
• Conclusion

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Problem

• It is a multi-objective optimization problem
• 5 parameters
• 8 inequality constraints
• 3 objectives

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Parameters

• Db is the ball diameter
• Dm is the mean diameter
• Z is number of balls
• fi is inner curvature coefficient
• fo is outer curvature coefficient

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Constraints (1) Bounds
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Constraints (2) More

• Phi is the assembly angle
• epsilon is a constant
• Kdmin and Kdmax are by geometric constraints

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Objectives (1) Dynamic Capacity
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Objectives (2) Minimum film thickness

• For increased wearing life
• Few more constants are shown here

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Objectives (3) Static capacity max of inner
and outer static capacity
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OUTLINE

• Rolling Element Bearings
• Problem statement
• Optimization (single and multi-objective)
• Previous work
• NSGA II Multi-objective optimization using
  evolutionary concepts
• Application and Results
• Conclusion

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Optimization (1)

• Common day engineering problem
• Problem could be stated as mathematical functions
  that have to be maximized or minimized
• df/dx 0, implies local maxima or minima
• Single objective is easy to handle !
• What about multiple objectives ?

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Optimization (2)

• f1(p), f2(p)...fn(p) are objectives
• c(p) gt 0 represents constraints on parameter
  space
• The result of this optimization is not a unique
  parameter set p, instead it is satisfied by an n
  dimensional front, called as Pareto front.

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Optimization (3)

• The concept of optimizing one performance on the
  cost of other is termed as Pareto optimality.
• The trade-off curve is also said to be Pareto
  optimal front and the points over it are termed
  as Pareto optimal points.

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Optimization (4)

• Domination
• One solution is said to dominate another if it is
  better in both objectives
• Non-Domination Pareto points
• A solution is said to be non-dominated if it is
  better than other solutions in at least one
  objective

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Example
Non-dominated
f2
Dominated
f1
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An Example Pareto Front

• P feasible parameter space
• f() nonlinear mapping
• F feasible objective space
• dF Pareto front

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Deterministic or Stochastic
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OUTLINE

• Rolling Element Bearings
• Problem statement
• Optimization (single and multi-objective)
• Previous work
• NSGA II Multi-objective optimization using
  evolutionary concepts
• Application and Results
• Conclusion

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Using weighted sum approach

• Have upper and lower bounds for each objective
• Normalize them
• Add them together
• Now it is as good as single objective

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Shortcomings

• No consideration of multi-objective nature of
  problem
• Each run will lead to only one final solution
  point (one point on Pareto front)
• Can not handle non-convexities of the Pareto front

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OUTLINE

• Rolling Element Bearings
• Problem statement
• Optimization (single and multi-objective)
• Previous work
• NSGA II Multi-objective optimization using
  evolutionary concepts
• Application and Results
• Conclusion

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Non Dominated Sorting based Genetic Algorithm II

• Developed at KanGAL (Prof K. Deb)
• Superior to most of the MOEA in the research
  arena today
• Uses Elitism
• Famous for Fast non-dominated search

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Outline of Algorithm

• Take Parent (t) and Child (t) populations of tth
  generation
• Do the fast non-dominated sorting
• Crowding distance assignment
• Sort on the basis of crowding operator
• Make Parent (t1) population of this generation
• Selection Tournament or Roulette
• Crossover Real numbers using distribution index
• Mutation Real numbers using distribution index
• Make Child (t1) population of this generation

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Fast non-dominated sort

• Each layer is a Pareto front
• Rank of a solution is the layer number

1
2
f2
3
4
f1
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Crowding distance assignment
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Crowding operator based sorting
OR

• After this sorting, Parent (t1) is made taking
  top N candidates
• We now do selection, crossover, and mutation to
  obtain Child (t1)

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Graphical representation of Algorithm
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Outline of Algorithm

• Take Parent (t) and Child (t) populations of tth
  generation
• Do the fast non-dominated sorting
• Crowding distance assignment
• Sort on the basis of crowding operator
• Make Parent (t1) population of this generation
• Selection Tournament or Roulette
• Crossover Real numbers using distribution index
• Mutation Real numbers using distribution index
• Make Child (t1) population of this generation

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OUTLINE

• Rolling Element Bearings
• Problem statement
• Optimization (single and multi-objective)
• Previous work
• NSGA II Multi-objective optimization using
  evolutionary concepts
• Application and Results
• Conclusion

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For single optimization

• The results obtained were better than that given
  by previous approaches

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For dual optimization Cd, Cs
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For dual optimization Cd, hmin
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For dual optimization Cs, hmin
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All three objectives together OLD

• Previous approach with weighted sums computed the
  values for two weight assignments

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All three objectives together NEW

• We have a complete gamut of values lying on the
  Pareto front.
• To give an estimate 670 values
• All obtained in a single run
• Gives designer a variety to choose from

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OUTLINE

• Rolling Element Bearings
• Problem statement
• Optimization (single and multi-objective)
• Previous work
• NSGA II Multi-objective optimization using
  evolutionary concepts
• Application and Results
• Conclusion

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Conclusion

• Multi-objective optimization problem in rolling
  element bearings is identified
• Comparison of evolutionary and deterministic
• methods
• NSGA II was found to be an ideal answer
• Very encouraging results are obtained
• Future work would be to analyze these plots
  (mechanical engineering task) and give best
  fitting Pareto points as design specification

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Thank you

• Any Questions ?