1. ME 521 Computer Aided Design 15-Optimization
Assoc.Prof.Dr. Ahmet Zafer Şenalp
Mechanical Engineering Department
Gebze Technical University

2. Mechanical Engineering Department, GTU
15-Optimization
Introduction
What is an Optimization Problem?
Optimization is a process of selecting or converging onto a final solution amongst a number of possible options, such that a certain requirement or a set of requirements is best satisfied.”
i.e., you want a design in which some quantifiable property is minimized or maximized (e.g., strength, weight, strength-to-weight ratio)
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

3. Mechanical Engineering Department, GTU
15-Optimization
Real Parameter(s)
Continuous real parameter(s):
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

4. Mechanical Engineering Department, GTU
15-Optimization
Some Types of
Optimization Problems
Combinatorial:
Examples of
• minimization: cost, distance, length of a traversal, weight, processing time, material, energy
consumption, number of objects
• maximization: profit, value, output, return, yield, utility, efficiency, capacity, number of objects.
The solutions may be combinatorial structures like arrangements, sequences, combinations, choices
of objects, sequences, subsets, subgraphs, chains, routes in a network, assignments, schedules of
jobs, packing schemes, etc.
Discrete optimization or combinatorial optimization means searching for an optimal solution in a finite or countably infinite set of potential solutions. Optimality is defined with respect to some criterion function, which is to be minimized or maximized.
In mathematics, and more specifically in graph theory, a graph is a representation of a set of objects where some pairs of objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

5. Mechanical Engineering Department, GTU
15-Optimization
Some Types of
Optimization Problems
Geometric:
E.g., Minimize waste ratio
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

6. Mechanical Engineering Department, GTU
15-Optimization
Some Types of
Optimization Problems
Structural:
E.g., Minimize weight, maximize strength
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

7. Mechanical Engineering Department, GTU
15-Optimization
Formulating the Problem
To have a set of possible solutions, the design must be parameterized. The objective function must be created in terms of those parameters. Formulation steps: a) Identify decision parameters (e.g., engine_displacement, piston_diameter) b) Define Objective Function (e.g., maximize power_to_weight_ratio) c) Identify constraints (e.g., 1 inch ≤ piston_diameter ≤ 12 inch)
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

8. Mechanical Engineering Department, GTU
15-Optimization
Formulating the Problem
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

9. Mechanical Engineering Department, GTU
15-Optimization
Formulating the Problem
Mathematically, you need to select: Decision Parameters (vector X, solution X*∈Rn ) Eg: b) Objective Function (F(X)) X* ∈Rn so that F(X* ) = min F(X) In other words, the solution is the vector of real numbers X* for which F(X) is minimum. c) Constraints - Bounds: X ≤ X* ≤ Xu - Inequality: Gi (X* ) ≥ 0 i=1,2,…,m eg: - Equality: Hj (X* ) = 0 j=1,2,…,q eg:
This means X* is a vector of n real numbers.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

10. Mechanical Engineering Department, GTU
15-Optimization
Formulating the Problem
Constraints act as a guide for the optimization problem.
Bounds: Are direct limits on the values a parameter can take (e.g., 5 ≤ x1 ≤ 10.)
Inequality: Are expressions that limit the values parameters can take
(e.g., x1 - x2 – 5 ≥ 0.)
Equality: These reduce one design variable for each equality constraint.
(e.g., x1 – x3 – 5 = 0.)
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

11. Mechanical Engineering Department, GTU
15-Optimization
Structural Optimization
Structural optimization is a specific class of optimization problems that uses an FE analysis as part of the objective function or constraints. Structural optimization involves three elements: 1. Automatic FE mesh generation. 2. FE analysis 3. Optimization algorithm
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

12. Mechanical Engineering Department, GTU
15-Optimization
Structural Optimization
Three ways of optimizing structures:
– Parameter optimization
• Feature sizing.
• Modify solid model construction parameters.
– Shape Optimization
• Model is shaped freely.
• Move nodes in FE model.
– Topology Optimization
• Model shape and topology changed freely.
• Change element densities in FE model.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

13. Mechanical Engineering Department, GTU
15-Optimization
Shape Optimization
Optimal Truss Design – Location of nodes and properties of cross-sections are the decision parameters.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

14. Mechanical Engineering Department, GTU
15-Optimization
Shape Optimization
The locations of the FE nodes or the B-Spline control points are the decision parameters.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

15. Mechanical Engineering Department, GTU
15-Optimization
Topological Optimization
In topological optimization, the decision parameters are the amounts of material in each cell. Material is only added where it is needed to carry loads.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

16. Mechanical Engineering Department, GTU
15-Optimization
Topological Optimization
Besides allowing for size and shape changes, topological optimization allows voids to appear or disappear.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

17. Mechanical Engineering Department, GTU
15-Optimization
Choosing a Solution Method
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

18. Mechanical Engineering Department, GTU
15-Optimization
Gradient-Based Solving
Gradient-Based methods are iterative. They choose a better solution by following the downward slope of the curve/surface given by:
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

19. Mechanical Engineering Department, GTU
15-Optimization
Local Minima
Gradient-based methods do not work well when there are several local minima: The “Simulated Annealing” and “Genetic Algorithm” methods were introduced to solve this problem.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

20. Mechanical Engineering Department, GTU
15-Optimization
Genetic Algorithm
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

21. Mechanical Engineering Department, GTU
15-Optimization
Genetic Algorithm
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

22. Mechanical Engineering Department, GTU
15-Optimization
Genetic Algorithm
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU

23. Mechanical Engineering Department, GTU
15-Optimization
Constraint Handling
A common way of handling constraints is to introduce penalty parameters (e.g. ρ k) As multipliers that modify the objective function.
Dr. Ahmet Zafer Şenalp ME 521
Mechanical Engineering Department, GTU