Optimization in Engineering Design: Minimizing Stress in Structures, Study Guides, Projects, Research of Design

Download Optimization in Engineering Design: Minimizing Stress in Structures and more Study Guides, Projects, Research Design in PDF only on Docsity!

Introduction to Design

Optimization

Minimum Weight

(under Allowable

Stress)

A PEM Fuel Cell Stack

with Even Compression

over Active Area

(Minimum Stress

Difference)

Various Design

Objectives

Engineering Applications of Optimization

• Design - determining design parameters that lead to the best

performance of a mechanical structure, device, or system.

Core of engineering design, or the systematic approach to

design (Arora, 89)

• Planning
  • production planning - minimizing manufacturing costs
  • management of financial resources - obtaining maximum profits
  • task planning (robot, traffic flow) - achieving best performances
• Control and Manufacturing - identifying the optimal control

parameters for the best performance (machining, trajectory, etc.)

• Mathematical Modeling - curve and surface fitting of given data

with minimum error

Commonly used tool: OPT function in FEA; MATLAB Optimization Toolbox

What are common aspects in

optimization problems?

• There are multiple solutions to the problem; and the

optimal solution is to be identified.

• There exist one or more objectives to accomplish and a

measure of how well these objectives are accomplished

(measurable performance).

• Constraints of different forms (hard, soft) are imposed.
• There are several key influencing variables. The change

of their values will influence (either improve or worsen)

the measurable performance and the degree of

violation of the constraints.

Definition of Design

Optimization

An optimization problem is a problem in

which certain parameters ( design variables )

needed to be determined to achieve the

best measurable performance ( objective

function ) under given constraints.

• Type of design variables
  • optimization of continuous variables
  • integer programming (discrete variables)
  • mixed variables
• Relations among design variables
  • nonlinear programming
  • linear programming
• Type of optimization problems
  • unconstrained optimization
  • constrained optimization
• Capability of the search algorithm
  • search for a local minimum
  • global optimization; multiple objectives; etc.

Classification of the Optimization Problems

1

2

x e g f X Ae Bx

− = +

1 1 2 2

n n

e g f X = c x + c x + K + c x

0 2 4 6 8 10

0 2 4 6 8 10

0

5

10

x

Alpine Function

x

f(x)

An Example Optimization

Problem

Design of a thin wall tray with minimal material :

The tray has a specific volume, V , and a given height, H.

The design problem is to select the length, l, and width,

w, of the tray.

Given

A “workable design”:

Pick either l or w and solve for others

lwh = V h = H

lw

V

H

l

w

h

An Optimal Design

• The design is to minimize material volume, (or weight),

where T is an acceptable small value for wall

thickness.

Minimize

subject to

Design variables: w, l, and h.

V w l h T wl lh wh

m bottom sides

lwh V

h H

l

w

constraints (functions)

l

w

h

Objective

function

Analytical (Closed Form) Solution

• Eliminate the equality constrains, convert the original problem into a

single variable problem, then solve it.

from h = H & l w H = V; solve for l :

thus

from

• Discard the negative value, since the inequality constraint is violated.
• The optimal value for l :

l

V

Hw

w

min

T( )

V

Hw

w

V

Hw

• 2 H + 2 wH

w

min

T( ) ( )

V

H

V

w

• 2 + 2 wH = f w

2 *

( )

0, ,

df w V V

we have w then the design optimum w

dw H H

= = =

• a stationary point

l

V

Hw

V

H

w

= = =

V T

V

H

hw

V

w

M

= ( + 2 + )

2

= T +

V

H

( 4 VH )

l

w

h

Graphic

al

Solution

l

w

h

Procedures for Solving an Eng. Optimization

Problem

• Formulation of the Optimization Problem
  • Simplifying the physical problem
  • identifying the major factor(s) that determine the performance or outcome

of the physical system, such as costs, weight, power output, etc. – objective

• Finding the primary parameters that determine the above major factors
  • the design variables
• Modeling the relations between design variables and the identified major

factor - objective function

• Identifying any constraints imposed on the design variables and modeling

their relationship – constraint functions

• Selecting the most suitable optimization technique or algorithm to

solve the formulated optimization problem.

• requiring an in-depth know-how of various optimization techniques.
• Determining search control parameters
• determining the initial points, step size, and stopping criteria of the

numerical optimization

• Analyzing, interpreting, and validating the calculated results

An optimization program does not guarantee a correct answer, one needs to

• prove the result mathematically.
• verify the result using check points.

Standard Form for Using Software Tools for

Optimization (e.g. MatLab Optimization Tool Box)

Where m are the number of inequality constraints and q the number of

equality constraints

Denoting the optimization variables X , as a n-dimensional vector, where the n

variables are its componets, and the objective function F(X) we search for :

Regional constraints

Behavior constraints