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13-Optimization
Assoc.Prof.Dr. Ahmet Zafer Şenalpe-mail: [email protected]
Mechanical Engineering DepartmentGebze Technical University
ME 521Computer Aided Design
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 2Mechanical Engineering Department,
GTU
Introduction
13-Optimization
Continuous real parameter(s):
Dr. Ahmet Zafer Şenalp ME 521 3Mechanical Engineering Department,
GTU
Real Parameter(s)
13-Optimization
Combinatorial:
Examples of • minimization: cost, distance, length of a traversal, weight, processing time, material, energyconsumption, 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, choicesof objects, sequences, subsets, subgraphs, chains, routes in a network, assignments, schedules ofjobs, packing schemes, etc.
Dr. Ahmet Zafer Şenalp ME 521 4Mechanical 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.
13-Optimization
Geometric:
Dr. Ahmet Zafer Şenalp ME 521 5Mechanical Engineering Department,
GTU
Some Types of Optimization Problems
E.g., Minimize waste ratio
13-Optimization
Dr. Ahmet Zafer Şenalp ME 521 6Mechanical Engineering Department,
GTU
Some Types of Optimization Problems
Structural:
E.g., Minimize weight, maximize strength
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 7Mechanical Engineering Department,
GTU
Formulating the Problem
13-Optimization
Dr. Ahmet Zafer Şenalp ME 521 8Mechanical Engineering Department,
GTU
Formulating the Problem
13-Optimization
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:Dr. Ahmet Zafer Şenalp ME 521 9Mechanical Engineering Department,
GTU
Formulating the Problem
This means X* is a vector of n real numbers.
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 10Mechanical Engineering Department,
GTU
Formulating the Problem
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 analysis3. Optimization algorithm
Dr. Ahmet Zafer Şenalp ME 521 11Mechanical Engineering Department,
GTU
Structural Optimization
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 12Mechanical Engineering Department,
GTU
Structural Optimization
13-Optimization
Optimal Truss Design – Location of nodes and properties of cross-sections are the decision parameters.
Dr. Ahmet Zafer Şenalp ME 521 13Mechanical Engineering Department,
GTU
Shape Optimization
13-Optimization
The locations of the FE nodes or the B-Spline control points are the decision parameters.
Dr. Ahmet Zafer Şenalp ME 521 14Mechanical Engineering Department,
GTU
Shape Optimization
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 15Mechanical Engineering Department,
GTU
Topological Optimization
Besides allowing for size and shape changes, topological optimization allows voids to appear or disappear.
13-Optimization
Dr. Ahmet Zafer Şenalp ME 521 16Mechanical Engineering Department,
GTU
Topological Optimization
13-Optimization
Dr. Ahmet Zafer Şenalp ME 521 17Mechanical Engineering Department,
GTU
Choosing a Solution Method
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 18Mechanical Engineering Department,
GTU
Gradient-Based Solving
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 19Mechanical Engineering Department,
GTU
Local Minima
13-Optimization
Dr. Ahmet Zafer Şenalp ME 521 20Mechanical Engineering Department,
GTU
Genetic Algorithm
13-Optimization
Dr. Ahmet Zafer Şenalp ME 521 21Mechanical Engineering Department,
GTU
Genetic Algorithm
13-Optimization
Dr. Ahmet Zafer Şenalp ME 521 22Mechanical Engineering Department,
GTU
Genetic Algorithm
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 23Mechanical Engineering Department,
GTU
Constraint Handling
