DESIGN OPTIMIZATION


In dealing with DFSS, a frequent euphemism is "design optimization." What is design optimization? Design optimization is a technique that seeks to determine an optimum design. By "optimum design," we mean one that meets all specified requirements but with a minimum expense of certain factors such as weight, surface area, volume, stress, cost, and so on. In other words, the optimum design is one that is as efficient and as effective as possible.

To calculate an optimum design, many methods can be followed. Here, however, we focus on the ANSYS program, as defined by Moaveni (1999), which performs a series of analysis-evaluation-modification cycles. That is, an analysis of the initial design is performed, the results are evaluated against specified design criteria, and the design is modified as necessary. This process is repeated until all specified criteria are met.

Design optimization can be used to optimize virtually any aspect of the design: dimensions (such as thickness ), shape (such as fillet radii), placement of supports, cost of fabrication, natural frequency, material property, and so on. Actually, any ANSYS item that can be expressed in terms of a parameter can be subjected to design optimization. One example of optimization is the design of an aluminum pipe with cooling fins where the objective is to find the optimum diameter, shape, and spacing of the fins for maximum heat flow.

Before describing the procedure for design optimization, we will define some of the terminology: design variables, state variables , objective function, feasible and unfeasible designs, loops , and design sets. We will start with a typical optimization problem statement:

Find the minimum-weight design of a beam of rectangular cross section subject to the following constraints:

Total stress ƒ should not exceed ƒ max

[ ƒ ƒ max ]

Beam deflection should not exceed max

[ max ]

Beam height h is limited to h max

[h h max ]

Design Variables (DVs) are independent quantities that can be varied in order to achieve the optimum design. Upper and lower limits are specified on the design variables to serve as "constraints." In the above beam example, width and height are obvious candidates for DVs, since they both cannot be zero or negative, so their lower limit would be some value greater than zero.

State Variables (SVs) are quantities that constrain the design. They are also known as "behavioral constraints" and are typically response quantities that are functions of the design variables. Our beam example has two SVs: ƒ (the total stress) and (the beam deflection). You may define up to 100 SVs in an ANSYS design optimization problem.

The Objective Function is the quantity that you are attempting to minimize or maximize. It should be a function of the DVs, i.e., changing the values of the DVs should change the value of the objective function. In our beam example, the total weight of the beam could be the objective function (to be minimized). Only one objective function may be defined in a design optimization problem.

A design is simply a set of design variable values. A feasible design is one that satisfies all specified constraints, including constraints on the SVs as well as constraints on the DVs. If even one of the constraints is not satisfied, the design is considered infeasible.

An optimization loop (or simply loop) is one pass through the analysis-evaluation-modification cycle. Each loop consists of the following steps:

  1. Build the model with current values of DVs and analyze.

  2. Evaluate the analysis results in terms of the SVs and objective function.

  3. Modify the design by calculating new values of DVs. These new values are calculated by ANSYS and are used to define the new version of the model.

At the end of each loop, new values of DVs, SVs, and the objective function are available and are collectively referred to as a design set (or simply set).

HOW TO DO DESIGN OPTIMIZATION

Design optimization requires a thorough understanding of the concept of ANSYS parameters, which are simply user -named variables to which you can assign numeric values. The model must be defined in terms of parameters (which are usually the DVs), and results data must be retrieved in terms of parameters (for SVs and the objective function). The usual procedure for design optimization consists of six main steps:

  1. Initialize the design variable parameters.

  2. Build the model parametrically.

  3. Obtain the solution.

  4. Retrieve the results data parametrically and initialize the state variable and objective function parameters.

  5. Declare optimization variables and begin optimization.

  6. Review and verify optimum results.

Details of these steps are beyond the scope of this volume. However, the reader may find the information in Moaveni (1999).

UNDERSTANDING THE OPTIMIZATION ALGORITHM

Understanding the algorithm used by a computer program is always helpful, and this is particularly true in the case of design optimization. Perhaps one of the most important issues is the notion of approximation .

For simple mathematical functions that are continuously differentiable, minima can be found by analytical techniques such as solving for points of zero slope. The mathematical relationship between an arbitrary objective function and the DVs, however, is generally not known, so the program has to establish the relationship by curve fitting. This is done by calculating the objective function for several sets of DV values (i.e., for several designs) and performing a least squares fit among the data points. The resulting curve (or surface) is called an approximation. Each optimization loop generates a new data point, and the objective function is updated. It is this approximation that is minimized, not the actual objective function.

State variables are handled in the same manner. An approximation is generated for each state variable and updated at the end of each loop. (Because approximations are used for the objective function and SVs, the optimum design will be only as good as the approximations.)

CONVERSION TO AN UNCONSTRAINED PROBLEM

State variables and limits on design variables are used to constrain the design and make the optimization problem a constrained one. The ANSYS program converts this problem to an unconstrained optimization problem because minimization techniques for the latter are more efficient. The conversion is done by adding penalties to the objective function approximation to account for the imposed constraints. You can think of penalties as causing an upturn of the objective function approximation at the constraints. The ANSYS program uses extended interior penalty functions. (For more information on penalty functions see sources in the selected bibliography for this chapter.)

The search for a minimum is then performed on the unconstrained objective function approximation using the Sequential Unconstrained Minimization Technique (SUMT), which is explained in most texts on engineering design and optimization.




Six Sigma and Beyond. Design for Six Sigma (Vol. 6)
Six Sigma and Beyond: Design for Six Sigma, Volume VI
ISBN: 1574443151
EAN: 2147483647
Year: 2003
Pages: 235

flylib.com © 2008-2017.
If you may any questions please contact us: flylib@qtcs.net