Reflecation paper

- Module 10 - Prototype Optimization

 Overview

• Optimization is an act, process, or methodology of making

something (as a design, system or decision) as fully perfect,

functional or effective as possible.

• Different people may have different perspectives about

optimization.

• For some, it is the process of doing iterations on a product until it

achieves the set conditions of the product definition.

• For others, it is to find values of the variables that minimize or

maximize the objective function while satisfying the constraints.

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- Module 10 – Cont.

• In product or prototype design, optimization is the design of a

product that is the most economical and efficient design possible while

also fulfilling all of the customer requirements.

• The prototype testing, evaluations, and consumer input should be used to fix what is wrong.

• By using product models either by hand or through several different

software programs, optimization on a product can be done.

• There will be trade-offs when all product quality variables are

considered, and this process is called engineering optimization.

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• The design variables are shown in the Table:

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• Optimization is being used in various fields like engineering,

medicine, transportation, etc.

• The following examples are the situations in which one wants to find

the best way to do something:

(i) How should the transistors and other devices be laid out in a

computer chip so that it occupies the minimum area ?

(ii) How should the telephone line be routed between two cities so that

the maximum number of simultaneous calls are possible ?

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A typical engineering design process.

- Module 10 – Cont.

• Modern optimization methods perform shape optimizations on

components generated within a choice of CAD packages.

• Ideally, there is seamless data exchange via direct memory transfer

between the CAD and FEA (finite element analysis) applications

without the need for file translation.

• In the approach, the design optimization process begins before the

FEA model is generated.

• The user simply selects which dimension in the CAD model needs

to be optimized and the design criterion, which may include:

maximum stresses, temperatures, or frequencies.

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• The new FEA model, including a new high-quality solid mesh, to be

analyzed, and the results are compared with the design criterion.

• The FEA model is also updated using the principle of associability,

which implies that constraints and loads are preserved from the

prior analysis.

• This process is repeated until the design criterion is satisfied.

• The optimization method also allows for global constraints to enforce

weight and volume criteria.

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 FORMULATION OF ENGINEERING PROBLEMS FOR

OPTIMIZATION

 DEFINITIONS

• Finding optimal solutions in product prototyping requires one to:

1st. Define the problem,

2nd. Define the required results.

• In order to meet customer requirement, must identify all the

variables of the problem and determine which of the variables must

be fixed and at what value to fix them.

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• Needs to define the constraints, limits, and scope of the problem.

• The third step is to find the solution to the problem.

• Assessing the solution by answering the following questions:

. Is the design optimal ?

. Is the design feasible ?

. Is the design reasonable ?

. How can further modification be done for the optimization model ?

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• There is only a global optimal (maximum or minimum) solution for a

function, and there may be several local optima as shown:

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Various optimal solution points.

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 PROBLEM FORMULATION

• Problem formulation is very critical to the success of the design.

• Formulate the problem into the following standard:

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• The steps to come up with the above formulation include:

1. Specify the design variables:

2. Specify PDP:

3. Specify objective function f ( x ):

4. Specify constraints:

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- Module 10 – Cont.

• EXAMPLE 10.1

Design a 1 liter (1-L) container shaped like a right circular cylinder,

as shown in the Figure, to hold a fixed volume.

Design the container to use the least amount of material. Give the

dimensions that use

the least amount of material. Formulate this problem in standard

format.

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 OPTIMIZATION USING DIFFERENTIAL CALCULUS

• Once the problem is in the form of a mathematical equation, the

next step is to find the solution.

• One solution method is to set up a spreadsheet and iteratively

search the design space defined by the range of design variables.

• The optimization process reduces the trial and error spreadsheet

operation.

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• An extreme point on the function, either maximum or minimum,

exists if the following conditions are met, for both first-order and

second-order equations:

First-order condition: The function may contain an extreme point.

Second-order condition: The value attains a maximum or minimum

depending upon whether the value of the second derivative is

negative or positive, respectively.

for objective function = U(x) , d2U/dx2 = 0

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EXAMPLE 10.2

Using the same conditions as Example 10.1, apply differential calculus

to solve the problem.

Solution:

-First, the volume of the can must be calculated.

-Knowing the height, h

-Knowing the radius, r

the volume is computed using the equation

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- Module 10 – Cont.

 LAGRANGE’S MULTIPLIER METHOD

• One common problem in calculus is that of finding the maximum or

minimum of a function.

• The method of Lagrange multipliers is a powerful tool for solving

this class of problems without the need to explicitly solve the

conditions and use them to eliminate extra variables .

• This method converts a constrained problem to an unconstrained

problem.

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• First, the Lagrangian form is obtained by combining Lagrange

multiplier l, the objective function, the inequality constraint g, and

the equality constraint h.

• The second step involves the derivation of the first-order conditions.

• The steps to solve the problem using the Lagrange multiplier

method are:

1. Format the equations into standard format.

2. Form the Lagrangian function.

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3. Derive the first-order conditions.

4. Use the three equations to solve for x*, y*, and λ*.

5. λ1 to λm need to be nonnegative. If λj is negative, one needs to

ignore gj, as it is a redundant constraint, and resolve the problem

again.

6. There is no need to check λm+1 … λm+k for equality constraints.

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- Module 10 – Cont.

EXAMPLE 10.3

A company is designing an advertisement poster for a new product.

The logo must have a printed area of 200 cm2 , a 0.5 cm margin at the

two sides of the text and 3 cm margins each along the top and the

bottom.

If it is desired to use the minimum material due to large quantity, find

the best choice for the poster dimensions width (W) and height (H).

Formulate this problem in standard format.

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- Module 10 – Cont.

 OPTIMIZATION USING MICROSOFT EXCEL

• Microsoft Excel Solver is a Microsoft Excel Add-in, one of the

features that allows the creation of engineering and financial models in

a spreadsheet

• The optimization model has three parts:

1st. the target cell

2nd. changing cells

3rd. constraints

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 CASE STUDY: APPLICATION OF OPTIMIZATION IN FIXTURE

DESIGN

• Optimization techniques can be used to solve complex problems.

• Capability to summarizes a research case study using linear

programming optimization for fixture planning .

• Linear programming is an optimization problem in which the objective

function and the constraints are all linear.

• The algorithm can be implemented by writing a computer code or

using tools such as Excel

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- Module 10 – Cont.

• A modular fixturing system mainly consists of locators and clamps.

• The function of locators is to properly locate a work part to provide

repeatable manufacturing operations.

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• Friction between the workpart and the locators and clamps is not

considered in this example.

• The sequential steps in the creation of a fixture are:

- creation of a mesh over the surface of the work part as shown in

the Figure.

- determination of the positions of the clamps and locators.

- checking deterministic positioning as shown in the Figure

- checking for clamping stability, a positive clamping sequence.

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