ME 521 Computer Aided Design 15 Optimization Assoc

ME 521 Computer Aided Design 15 -Optimization Assoc. Prof. Dr. Ahmet Zafer Şenalp e-mail: azsenalp@gmail. com Mechanical Engineering Department Gebze Technical University

Introduction 15 -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-toweight ratio) Dr. Ahmet Zafer Şenalp ME 521 Mechanical Engineering Department, GTU 2

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

Some Types of Optimization Problems 15 -Optimization Combinatorial: 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. 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 Mechanical Engineering Department, GTU 4

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

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

Formulating the Problem 15 -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 Mechanical Engineering Department, GTU 7

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

Formulating the Problem 15 -Optimization Mathematically, you need to select: Decision Parameters (vector X, solution X*∈Rn ) This means X* is a vector of n real numbers. 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 - Equality: Hj (X* ) = 0 Dr. Ahmet Zafer Şenalp ME 521 i=1, 2, …, m eg: j=1, 2, …, q eg: Mechanical Engineering Department, GTU 9

Formulating the Problem 15 -Optimization Constraints act as a guide for the optimization problem. • • • Bounds: Are direct limits on the values a parameter can take (e. g. , 5 ≤ x 1 ≤ 10. ) Inequality: Are expressions that limit the values parameters can take (e. g. , x 1 - x 2 – 5 ≥ 0. ) Equality: These reduce one design variable for each equality constraint. (e. g. , x 1 – x 3 – 5 = 0. ) Dr. Ahmet Zafer Şenalp ME 521 Mechanical Engineering Department, GTU 10

Structural Optimization 15 -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 11

Structural Optimization 15 -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 12

Shape Optimization 15 -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 13

Shape Optimization 15 -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 14

Topological Optimization 15 -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 15

Topological Optimization 15 -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 16

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

Gradient-Based Solving 15 -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 Mechanical Engineering Department, GTU 18

Local Minima 15 -Optimization 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 19

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

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

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

Constraint Handling 15 -Optimization 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 23