Modern finite element analysis and design-of-experiments (DOE) techniques can be combined into a powerful quality engineering tool.

FOR MANY YEARS American industry has had access to some very powerful tools, in the form of fast computers and the numerous finite element analysis (FEA) programs that address problems in elasticity, plasticity, heat transfer, fluid dynamics, and many other fields. But much of the use of FEA has been for failure mode analysis, a type of analysis that usually means economic value lost, not added. Instead of being used to design products the right way on the first try, FEA has been used to fix failed designs.

Now we are beginning to see commercially available FEA software that supports parametric modeling, i.e., the definition of finite element models in terms of variables instead of fixed numerical values. The Ansys program from Swanson Analysis Systems Inc. in Houston, Pa., for example, which has supported parametric modeling since the early 1980s and which recently underwent extensive improvements, now supports what can be described as a FORTRAN-like programming language. The Mechanica program from Rasna Corp. in San Jose, Calif., also has extensive parametric capabilities that users can exploit with nearly trivial effort. But even with these and the more powerful FEA tools that undoubtedly will follow, most engineers are unlikely to create quality products. At best, they will create products that work but are a far cry from market-winning quality. The parametric modeling features of the current crop of FEA software, in fact, are likely to be underused because of the inefficient one parameter-at-a-time techniques that most engineers use.

Quality engineering is the practice of engineering and manufacturing products in such a way as to minimize variability in the key performance characteristics of the products. This principle is the driving force behind such well-known quality efforts as the Six-Sigma program at Motorola Inc. in Austin, Tex. It is also the principle behind Taguchi methods, the quality engineering system made popular by Genichi Taguchi during the last decade.

By minimizing variability in product performance, quality engineers ensure that every copy of a product is equally capable of meeting the expectations of the customer that buys it. In automotive terms, it means making no lemons. It means, in fact, making nothing that even remotely resembles a lemon. For customers, the benefit is obvious; every customer gets a product that performs exactly as expected. For the products, manufacturers, the benefit is success in the marketplace. Quality products lead to increased market share and unshakable customer loyalty, as the American auto industry learned during the 1 980s.

The primary tool set of quality engineers is design of experiments (DOE). DOE is a body of knowledge that has been developed by mathematical statisticians over the past 70 years, with the goal of extracting as much useful information as possible from a limited number of experiments. The techniques of DOE include factorial experiments (also known as orthogonal arrays) and response surface methods. These make it possible for quality engineers to solve problems that involve many variables, not just one or two. With such problems, the one-factor-at-a-time method of experimentation is not practical, economical, or efficient.

Consider a design problem that involves just three variables, say, A, B, and C. For this case, the design space, i.e., the set of all possible combinations of the three variables, is a cube. Our task is to determine the best combination of the three variables. Naturally we want to determine the effect of each variable on the performance characteristic that is of interest to us; we'll call this the response function (sec Figure 1).

In using the one-factor-at-a-time approach we might begin by performing one experiment with all three variables at their low settings. Thus w e would learn the magnitude of our response function at one corner of the cube. Next we might change A to its high setting and hold B and (. constant at their low settings. Thus with our second experiment we would learn the magnitude of our response function at a second corner of the cube. If we were to continue this process for B and C, i.e., change them individually while holding the remaining variables constant then we would learn the magnitude of our response function at four corners of the cube.

With the one-factor-at-a-time method of experimentation would explore four combinations out of the eight that are possible and we would learn how our response function changes along 3 of the 12 edges of the cube. In addition each variable would be used at its high setting for only one of the four runs and at its low setting for three runs. In other words the one-factor-at-a-time method of experimentation leads to wasteful repetition and too little information.

This could be devastating. The solution to an important quality problem might be at one of the corners of the design space (cube) that the one-factor-at-a-time method never explores. Indeed, the key to making a product work might escape us entirely, since that key might be one of the combinations we do not test.

With a full factorial experiment, for three variables at two settings, we would perform an eight-run experiment for our design problem, not a four-run experiment, and we would gain more than twice the information. With a full factorial we would learn not only the effect of each variable but also how those variables interact. If one of our variables were a parameter that we could not control, either in the product's end-use environment or at the factory (what Taguchi calls a noise parameter), we would learn how to use the remaining two variables to minimize its effect 011 our response function. In other words, we would gain the information needed to minimize the product,.s performance variability.

The methods of DOE bring great efficiency to physical experiments. Those techniques bring the same efficiency to the numerical experiments that engineers perform with finite element models. By combining FEA with DOE, engineers can convert their analysis tools to incredibly powerful tools for design. The result of the FEA/DOE combination is a bonanza of timely valuable design information, making FEA a powerful tool for quality engineering.

To illustrate the FEA/DOE combination, consider the following case study, during Which the response surface method and a parametric finite element model, developed with the Ansys program, were used to reveal the nonlinear buckling behavior of a short segment of the power conductor of an electronic system.

Figure 2 shows the segment of the power conductor in question. A stiff braided cable is attached to the top end of the conductor via a movable metal post. The cable and other components beyond it apply an axial force, F, to the top end of the conductor. The lower end of the polyethylene-covered conductor is solidly attached to a stiff metal support. The aim is to determine the buckling load of the short segment of copper wire, as a function of length, initial eccentricity of the loaded end, and extent of polyethylene coverage. The ranges of interest are 0.400 to 0.700 inch for length, 0.010 to 0.100 inch for eccentricity, and 10 to 90 percent for polyethylene coverage.

Figure 3 shows the parametric finite element model used in the design effort. The large lower section represents the stiff support to which the copper conductor is attached. The axial load, F, was applied at the top end of the model, which was free to translate but not to rotate. The single contact element was used to bend the conductor sideways prior to the application of the axial load. This step ensured that the proper initial eccentricity was achieved. The nonlinear model took into account large detections, stress stiffening, and plasticity. Plasticity could occur at both the bottom and the top of the conductor. In fact, for some combinations of the design variables, the base of the copper actually developed a plastic hinge. The buckling load was defined as the highest load that the model could sustain prior to the loss of stability.

The parametric finite element model was used to generate 27 data points in the three-parameter design space. The experiment design (the 27 combinations of the variables) was generated with a commercially available DOE software package called Echip from Echip Inc. in Hockessin, Del.

The following equation shows the empirical model used to approximate the highly nolinear buckling behavior of the conductor:

buckling load = [a.sub.0] + [a.sub.1]L + [a.sub.2]E + [a.sub.3]P + [a.sub.4]LP

+ [a.sub.5]EP + [a.sub.6][E.sup.2] + [a.sub.7][P.sup.2] + [a.sub.8][LE.sup.2] + [a.sub.9][E.sup.3 where L = length (0.400 to 0.700 inch); E = eccentricity (0.010 to 0.100 inch); P= polyethylene coverage (10 to 90 percent); and [a.sub.i] = linear regression constants. The 10-term model, fitted With multiple linear regression to experimental data generated with the parametric finite element model, allows us to see the response surface. i.e., the unknown function. This is one of the advantages of the response surface method; it gives us the big picture. A second, equally significant, advantage is that it provides the ability to predict the response. Thus, not only can we see the function but we can also use it to make performance predictions.

Another advantage of the response surface method-and of DOE in general--is that with it we can identify which variables have a significant effect on the response and which do not. For design engineers this is invaluable, particularly if the information is available early in the design stage of a project. Figure 4, for example, shows the response surface for our buckling problem. At a glance, we can tell that the extent of polyethylene coverage has, at best, only a second-order effect. However, before this work began the designers and the author were convinced that the polyethylene provided the design with a generous margin of safety. That engineering judgment was proven wrong by the response surface. In fact, the amount of eccentricity is the only significant variable, since length played a trivial role in the buckling response due to the limited range of interest. The response surface method is just one of the tools in the DOE tool box. There are many others, all of which offer great synergism when combined with a properly verified parametric finite element model. They offer, in fact, the kind of design information that can make the difference between failure and success in the fiercely competitive world market.

Anthony Rizzo is a member of the technical staff at AT&T Bell Laboratories in Whippany, N.J.