This article is one of five papers on modeling and simulation (part two) to be presented exclusively on the web as part of the September 1999 JOM-e—the electronic supplement to JOM. The first part of this topic supplemented the August issue. The coverage was developed by Steven LeClair of the Materials Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base.
	The following article appears as part of JOM-e, 51 (9) (1999), http://www.tms.org/pubs/journals/JOM/9909/Frazier/Frazier-9909.html JOM is a publication of The Minerals, Metals & Materials Society

Holistic Strategies for Designing Multistage Materials Processes

TABLE OF CONTENTS
INTRODUCTION OPTIMAL PROCESS DESIGN Thermomechanical Processing Systems Dynamical Models Trajectory Optimization TRAJECTORY OPTIMIZATION APPLIED TO A MULTISTAGE PROCESS Continuous Discrete Event CONCLUSIONS References

INTRODUCTION

Process design is a challenging endeavor involving difficult decisions, such as choosing the overall sequence of manufacturing operations, selecting the number of forming steps, and specifying the processing conditions. These decisions should be driven by the real need to achieve several possibly competing objectives associated with product specification, processing cost, and productivity. Software tools need to be developed that more directly assist in making design decisions, since designers generally have limited time for performing and interpreting detailed analyses.

Perhaps the most effective and direct means for dealing with modeling, optimization, and robustness issues is through a framework based upon dynamical systems principles¹ and trajectory optimization.² In this framework, a system is viewed as an interconnection of various interacting components, each being described by a set of continuous- or discrete-state equations (relations describing the time and spatial evolution of internal variables) and input-output relationships that when viewed as a whole provides a complete system description. The system is designed via the application of formal methods of constrained optimization, which require the designer to translate the design problem into mathematical expressions corresponding to constraints (needs) and objectives (wants). The choice optimization algorithm to obtain a solution is determined in part by the type and fidelity of the models as well as the objective of the particular design task.

A second issue is the choice of model order. Experience shows that the appropriate model order for a particular process design is highly dependent on the particular task at hand, whether it is developing a cost estimate for bidding, preliminary technical design, or final detailed design; time available for the design effort; or available computing resources. In general, model order should increase as the design progresses. Although this is a well-accepted axiom, what is less understood is that using too high an order at early design stages can be counterproductive as the increased complexity of high-order models, more often than not, introduces more design variables and more inferior local optima in the design space. Some of these optima are often associated with the numerical nature of the model and not the true problem. These local optima create havoc with the fast local search optimization algorithms that are suitable for use with high-order models, forcing the designer to use global optimization methods that require many simulations of the computationally expensive high-order models. If sufficient time and computing resources are available, this may be acceptable, but in most cases they are not. Of course, the disadvantage of using low-order models is that the resulting optimum, while global, is not likely to be the global optimum of the high-order model case; but it is hoped that it is closer to the high-order global optimum than would have been obtained using the high-order models from the outset given finite computing resources using local- or global-search methods. Fortunately, several researchers have become aware of these problems and are developing systematic methods for simultaneously managing model order and the search for the optimum in a single framework.^3,4

OPTIMAL PROCESS DESIGN

Thermomechanical Processing Systems

Figure 1. A schematic of a hot-forging process.

Figure 2. A sequence of dynamical processes.

A typical systems representation for a metal-forging system is illustrated in Figure 1. Typical models for forging processes consist primarily of relations among workpiece material, tooling, and processing equipment. Two levels of coupling are present—models within models at the same time (instant) and models coupled sequentially through the workpiece (material) itself. In most cases, processes are not coupled directly, but are coupled only through the workpiece and its influence on the processes.

Dynamical Models

(t) = f_c(x(t),u(t))

in the continuous-time case or as

in discrete-time, where x and u are vectors of the system's state and control variables, respectively. The functions f_c and f_d define the relationships among the current state of the system, the current control variables, and the change of the state. The important point concerning dynamical models is that they are valid for a broad range of control signals, unlike algebraic models, which are only valid for a particular class of process controls, such as constant temperature and strain rate. The use of dynamical models provides greater predictive capability over algebraic models and provides more degrees of freedom in the time domain for optimal process design. Figure 2 illustrates a sequence of dynamical processes; each process is coupled to the next via the workpiece states.

a	b
Figure 3. A (a) strain-rate change test and (b) constant strain-rate test showing that different dynamical processing conditions can yield similar material microstructures.

Dynamical models of material behavior are especially important because of the time-varying behavior of quantities such as microstructure, flow stress, and defect formation. Figure 3 illustrates how the microstructure and flow stress of a -TiAl alloy responds to different transient processing conditions. In this example, the material responses under constant strain-rate conditions versus increasing step changes in strain rate are compared. Because several different processing histories (trajectories) can lead to nearly the same end result, it is important to be able to find the best trajectory to realize the desired objectives. Consequently, models and design techniques for controlling microstructure during thermomechanical processing have been developed to address critical issues, such as stability, transient and steady-state response, and robustness of material trajectories, which are defined here as the time evolution of material attributes (phase, grain size, dislocation density, etc.).

Dynamical models of equipment systems are also useful, if not essential, for determining the desired adjustable parameter settings for coincident tracking of the equipment response with optimized commands. In general, furnaces, vacuum chambers, and forming machinery possess a range of time-varying performance capabilities that can be tuned to the needs of a given process. An example using a high-fidelity dynamical model of a 907 tonne forge press is described in Reference 5.

Trajectory Optimization

subject to the constraints

ci(x(ti),u(ti))  0, i = 1 to Nc di(x(t),u(t))  0, t [tl,tu]i, i = 1 to Nd

and the material and process models

	w	= f	w j	(x	p j	,xw,uj), j = 1 to Nf
	p j	= f	p j	(x	p j	,xw,uj), j = 1 to Nf

where

	xwf
	xp1		u1
x=		and =
	xpNf		uNf

Figure 4. An illustration of a cogging operation.

A block diagram illustrating the model for the case of three sequential processes is shown in Figure 2, in which J is the overall design objective (a targeted goal); h_j are objective terms defined at a point in time and often used for final workpiece objectives, such as microstructure; g_i is the integrand of objective terms defined on a time interval (e.g., energy required, processing cost, and processing time); c_i are the constraints defined at a point in time (often final value constraints, such as strain, cross head travel limit, etc.); d_i are constraints defined over a time interval, such as temperature limits, strain-rate limits, load limits of equipment, and maximum temperature and pressure; x^w is the vector of workpiece states (e.g., dislocation density, average grain size, phase fractions, temperature distribution, etc.); x^p_j are vectors of process states (e.g., ram speed, pressure, and ambient temperature.); u_j are vectors of process controls, desired ram speed, desired pressure, and desired temperature; and N_f, N_c, N_d, N_g, and N_h are the number of discrete processes, pointwise constraints, interval constraints, pointwise objective terms, and interval objective terms, respectively. The subscript j in the quantities x^p_j and u_j refers to a particular thermomechanical process, such as forging, extrusion, or heat treatment.

Figure 5. A plot of the maximum and minimum billet temperatures throughout the process. The initial heat-up is not shown.

It is important to recognize that optimization is essentially just one method for performing a design, and whether the resulting design is the best one depends entirely on the criteria specified by the designer. As illustrated, optimization techniques require the specification of two types of criteria: objectives (wants) and constraints (needs). To achieve the desired goals, the designer must specify all relevant criteria and must carefully determine the criteria that are objectives and the criteria that are constraints. As an example, it may be desired to minimize the production costs (objective), while maintaining specified product-quality standards (constraint). On the other hand, the opposite scenario may be desired (i.e., maximize the product quality [objective] while not exceeding a specified cost [constraint]). Effective optimization strategies consider the entire processing-design problem, not just some parts, thereby avoiding the over-optimization of parts of the process at the expense of the whole manufacturing enterprise. Although not described in detail, the same formulation and comments apply to discrete-type models and optimization methods.

TRAJECTORY OPTIMIZATION APPLIED TO A MULTISTAGE PROCESS

Continuous

Figure 6. A video clip (~400 kb) of the temperature distribution at the end of the initial heat up.

Application of the design algorithm to a nonlinear, finite-difference, heat-transfer model of the process yielded furnace times that maintained the temperature within the desired constraints throughout the entire workpiece and yielded a total processing time of approximately 7.5 hours—a significant reduction. The characteristics of the solution include a significant amount of differential heating with a rather steep gradient from the surface to the interior. It is also interesting to note that, except for the initial heat-up, the furnace times are monotonically increasing. Figure 5 shows the maximum and minimum temperatures within the billet as a function of time. The blue vertical bands represent the periods when the workpiece is out of the furnace. The middle third of each band corresponds to the period when the billet is being deformed. The figure illustrates the transient nature of the process in that the maximum temperature continues to fall even after the billet reenters the furnace because of the internal thermal gradients. A similar phenomenon occurs with the minimum temperature. Figure 6 comprises a movie (~400 kb) illustrating the temperature distribution during the first five steps (heat-cool-heat-cool-heat).

Discrete Event

A diagram showing the processes and the possible physical design variables is given in Figure 8. Application of the optimization algorithm yielded an initial cost of $5,305 and a final cost of $1,191 for the first run and initial and final costs of $11,614 and $2,245, respectively, for the second run. The resulting process design is intended to determine the manufacturing sequence and process parameters to the first order. For example, in the forging processes, the volume of material required and the nominal shape of the forging dies are determined, but the exact shape of the dies is not. Detailed design, which should only impact cost at the second order, is still required to complete the process design.

CONCLUSIONS

Figure 7. The organizational structure of the design system.	Figure 8. The process routes and design parameters.

The importance of considering dynamical effects in the design of thermomechanical materials processes cannot be overstated since these processes are inherently transient (time-varying) in nature. This design philosophy has been used very successfully by aerospace engineers for more than two decades to design flight trajectories and control parameters of multistage missiles, launch vehicles, and satellites. The design has also shown that the mathematical structure of the problem is precisely the same as multistage materials processes. Certainly the complexities of modeling the behavior of materials during processing, combined with an emphasis on the equilibrium behavior of material systems, have been a bane to the widespread use of dynamical systems engineering principles in materials and manufacturing processes design. However, the evidence of more materials and manufacturing researchers addressing these issues indicates that things are changing.

References
1. R.E. Skelton, Dynamic Systems Control (New York: John Wiley and Sons, 1988).
2. R.E. Stengel, Optimal Control and Estimation (New York: Dover, 1994).
3. N.M. Alexandrov et al., "A Trust Region Framework for Managing Use of Approximation Models in Optimization" (1998), pp. 16–23.
4. N.M. Alexandrov, "On Managing the Use of Surrogates in General Nonlinear Optimization and MDO," Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, AIAA paper AIAA-98-4798 (1998).
5. W.G. Frazier et. al., "Model and Simulation of Metal Forming Equipment." J. of Mat. Engineering and Performance, 6, pp. 153–160.
6. W.G. Frazier et al., "Application of Control Theory Principles to Optimisation of Grain Size during Hot Extrusion," Materials Science and Technology, 14 (1) (1998), pp. 25–31.
7. S. Jacobson (private communication).

W.G. Frazier is with the Air Force Research Laboratory, Wright-Patterson Air Force Base. E.A. Medina is with Austral Engineering and Software.

For more information, contact W.G. Frazier, Air Force Research Laboratory, WPAFB, Ohio; e-mail william.frazier@afrl.af.mil.