1. 13-Optimization e-mail: Assoc.Prof.Dr. Ahmet Zafer Şenalp e-mail: azsenalp@gmail.comazsenalp@gmail.com Mechanical Engineering Department Gebze Technical University ME 521 Computer Aided Design

2. 13-Optimization 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 2 Mechanical Engineering Department, GTU Introduction

3. 13-Optimization Continuous real parameter(s): Dr. Ahmet Zafer Şenalp ME 521 3 Mechanical Engineering Department, GTU Real Parameter(s)

4. 13-Optimization 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. Dr. Ahmet Zafer Şenalp ME 521 4 Mechanical Engineering Department, GTU Some Types of Optimization Problems 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.

5. 13-Optimization Geometric: Dr. Ahmet Zafer Şenalp ME 521 5 Mechanical Engineering Department, GTU Some Types of Optimization Problems E.g., Minimize waste ratio

6. 13-Optimization Dr. Ahmet Zafer Şenalp ME 521 6 Mechanical Engineering Department, GTU Some Types of Optimization Problems Structural: E.g., Minimize weight, maximize strength

7. 13-Optimization 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 7 Mechanical Engineering Department, GTU Formulating the Problem

8. 13-Optimization Dr. Ahmet Zafer Şenalp ME 521 8 Mechanical Engineering Department, GTU Formulating the Problem

9. 13-Optimization Mathematically, you need to select: Decision Parameters (vector X, solution X* ∈ R n ) Eg: b) Objective Function (F(X)) X* ∈ R n 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* ≤ X u - Inequality: G i (X* ) ≥ 0 i=1,2,…,m eg: - Equality: H j (X* ) = 0 j=1,2,…,q eg: Dr. Ahmet Zafer Şenalp ME 521 9 Mechanical Engineering Department, GTU Formulating the Problem This means X* is a vector of n real numbers.

10. 13-Optimization 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 10 Mechanical Engineering Department, GTU Formulating the Problem

11. 13-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 11 Mechanical Engineering Department, GTU Structural Optimization

12. 13-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 12 Mechanical Engineering Department, GTU Structural Optimization

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

14. 13-Optimization The locations of the FE nodes or the B-Spline control points are the decision parameters. Dr. Ahmet Zafer Şenalp ME 521 14 Mechanical Engineering Department, GTU Shape Optimization

15. 13-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 15 Mechanical Engineering Department, GTU Topological Optimization

16. Besides allowing for size and shape changes, topological optimization allows voids to appear or disappear. 13-Optimization Dr. Ahmet Zafer Şenalp ME 521 16 Mechanical Engineering Department, GTU Topological Optimization

17. 13-Optimization Dr. Ahmet Zafer Şenalp ME 521 17 Mechanical Engineering Department, GTU Choosing a Solution Method

18. 13-Optimization 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 18 Mechanical Engineering Department, GTU Gradient-Based Solving

19. 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. 13-Optimization Dr. Ahmet Zafer Şenalp ME 521 19 Mechanical Engineering Department, GTU Local Minima

20. 13-Optimization Dr. Ahmet Zafer Şenalp ME 521 20 Mechanical Engineering Department, GTU Genetic Algorithm

21. 13-Optimization Dr. Ahmet Zafer Şenalp ME 521 21 Mechanical Engineering Department, GTU Genetic Algorithm

22. 13-Optimization Dr. Ahmet Zafer Şenalp ME 521 22 Mechanical Engineering Department, GTU Genetic Algorithm

23. A common way of handling constraints is to introduce penalty parameters (e.g. ρ k ) As multipliers that modify the objective function. 13-Optimization Dr. Ahmet Zafer Şenalp ME 521 23 Mechanical Engineering Department, GTU Constraint Handling