Solidification modeling, or computer simulation of the casting process, is being used by more and more foundries around the world to design the process for production of castings before castings are made or before equipment is built or altered. The primary objectives of using casting modeling are generally as follows:

- To improve the quality of the casting produced, both in the first castings made and over the production life of the part.
- To reduce the amount of time for new parts to enter production.
- To reduce costs such as melting and handling by improving yield and reducing cleaning operations.
- To quickly resolve design-related quality problems which may occur during production.

The traditional method of designing casting processes (and which, frankly, is still in use in most foundries today) is to use either experience or a set of rules (or some combination thereof) to arrive at an initial design for gating and feeding a casting. Pattern or mold equipment is then produced according to this initial design, and test castings are made. If the test castings exhibit some unacceptable defect, then the pattern equipment is modified, and new test castings are poured. This sequence may be repeated several times until suitable castings are made, or until time pressures from the customer force the foundry to begin shipping castings even if the design is sub-optimal. Often, in practice, this trial-and-error method of improving the process design continues even while castings are being shipped to the customer, which may have adverse effects on the relationship between the customer and the foundry.

Computer modeling offers the potential of evaluating alternate potential process designs in much less time, and at much less cost, than building equipment and producing sample castings.

The casting modeling systems which are available for foundry users today are, for the most part, software programs which accept a user’s design for a casting production system and then analyze the design to make a prediction of the likelihood of defects. The user provides both the geometric data (the shape of the casting, feeders and gating system) and the material property data for the model to be analyzed. Once an analysis has been completed, the user views the results of the analysis, typically by viewing various graphic images and interpreting these images. If an area of potential defect is found within the casting (such as internal shrinkage porosity), then the user needs to decide as to whether the defect is acceptable or not within the area shown.

If the defect is deemed to be unacceptable, then the user must make some determination as to why the defect formed in the area shown. This may involve viewing a variety of output data in order to develop an understanding of the solidification dynamics involved. Once this is accomplished, the user must then decide as to what change in the process or in the design might improve the situation, and when this decision is made the model must be altered and a new analysis run in order to verify whether the change had the intended effect. If not, then the process must be repeated until the desired result is obtained.

The effect of this process is that the design must be modified through a trial-and-error sequence until the analysis shows that the desired result has been achieved. Thus, in effect, we have replaced trial-and-error on the foundry floor (the traditional method) with trial-and-error on the computer. The advantage of doing this is that the time and cost have been reduced. However, we are still dependent upon the foundry engineer to interpret the results of a given analysis and to decide as to what changes are required for the next design iteration. And, once an acceptable result has been achieved, we still do not know if the result is an optimum. For example, is this the smallest riser size that would produce a sound casting, or could we have gone smaller?

To advance beyond the trial-and-error stage, we have developed the OPTICast™ software module to apply the methodology of optimization to the solidification modeling process. Optimization is a technology wherein a problem is structured such that input variables and constraints are identified, and an objective is specified, and we then search for an optimum solution to the problem using an optimization algorithm. There are numerous techniques for performing this search for the optimum condition, some of which are more efficient than others. The specific technique which we have settled on for performing optimization with casting solidification modeling is known as multi-variable response surface optimization, which basically considers the amount of change which results from modifying one or more design variables and determines whether this brings us closer to, or farther away from, the desired objective.

The effort involved applying Response Surface Methodology to the metal casting rigging process. This is the first time, to our knowledge, that this methodology has been successfully applied to the casting process.

APPLYING OPTIMIZATION TECHNOLOGY

The question to be considered was how to apply this optimization technology to casting modeling, so that the design of a given casting with its rigging could be automatically modified to produce an optimum condition. Optimization requires the identification of three basic parameters:

- DESIGN VARIABLES

These are features of a design that can vary while the system searches for an optimum condition. In the case of a computer model of a casting process, design variables may be geometric features such as the diameter and height of a riser or a riser/sleeve combination. They may also be process specifications such as the pouring temperature of the casting alloy. It is necessary to specify a minimum, a maximum and a nominal value so that the optimization system knows within what “envelope” it can operate.

The aspect of design that is most often varied when optimizing a casting process design is probably geometry. Therefore, it is necessary to develop a methodology by which geometry could be automatically varied. This was complicated by the fact that in model building, shape geometry can be represented through various types of parametric shapes as well as non-parametric geometry imported from CAD systems. It was decided that a universal “scaling” parameter could be applied to any shape, independently in the horizontal and vertical directions, in order to vary any type of shape. This would mean that a system user could select some geometric feature and specify the limits of scaling of this feature horizontally and vertically, and thus create a design envelope for the feature within which the optimization module could vary the size and shape of the feature. There are other, more sophisticated ways of modifying geometry that could be implemented later, but as a first iteration it was felt that the scaling technique would suffice for most foundry applications.

A second type of design variable required is process data, such as initial temperatures or pouring times. The optimization routine was therefore set up to allow the user to select specific process data and specify the allowable range for that item.

- CONSTRAINTS

Constraints are values of some aspect of process data above or below which a result is not allowed. Constraints may be specified as a minimum condition, in which case the result value must be at or above the given constraint value, or as a maximum condition where the result value must be at or below the given constraint value. One or more constraints may be specified for each optimization run. An example constraint would be a maximum allowable porosity level.

- OBJECTIVE FUNCTION

The objective function specifies what, ultimately, is trying to be achieved with a given process design. Selection of an objective function tells the optimization system what process result is to be used to judge whether you have achieved an optimum condition. The system user selects an objective function and specifies whether the value of that function is to be minimized or maximized. For example, you might select predicted shrinkage porosity as an objective function, in which case you would want to minimize its value. On the other hand, you might select material yield (the ratio of casting weight to poured weight) as an objective function and try to maximize its value.

Only one objective function can be specified for each optimization run.

## OPTIMIZATION SEQUENCE

The overall sequence of events which occurs when optimizing a design is shown in the following process flow diagram:

As shown in the diagram, the sequence of operation of an optimization run is, first, for the user to create an initial process design, i.e., a three-dimensional model of the casting with gating and feeding, and all relevant material data. This is the same data required for any casting simulation. The user then selects the design variables, constraints and objective function and launches an optimization run. Optimization basically consists of running a series of simulations automatically, varying the values of the design variables, checking to make sure that constraints are not violated, and searching for a maximum or minimum value for the selected objective function.

In order to allow this process to occur within a reasonable amount of time, the time for each successive simulation must be as short as possible. This means that the simulation algorithm must perform the minimum number of computations to arrive at an accurate result.

There are two basic methods of computing successive heat loss and solidification of castings. The first is the Finite Difference Method (FDM), in which small brick-shaped elements approximate the shape of the casting, and progressive temperatures and material conditions are solved by wave after wave of calculation increments through the mesh which solve for temperatures in each element by considering only the neighboring elements.

The second method is the Finite Element Method (FEM), in which elements of different sizes and shapes more closely approximate the casting geometry. In FEM, a simultaneous solution of temperatures at all nodes is required at each time step; in effect, FEM requires solution of a very large system of simultaneous equations. Although the time steps in FEM can be larger

than in FDM, the FEM method is very calculation-intensive and, in general, the FDM method is faster to produce results. Thus, it is our feeling that FDM is a preferred calculation route for the optimization process.

Another difference between FDM and FEM involves the creation of the mesh, which is the set of elements that represents the casting geometry. An FEM mesh, as stated, involves elements of varying size and shape. Creation of an FEM mesh is a relatively complex process and often requires user interaction. To modify geometry of a casting model in an automatic way and avoid user interaction, as is required by optimization, would generally mean modifying the mesh by compressing or stretching the elements. This could potentially result in invalid elements that would void the simulation calculations. This intractable problem can be avoided by focusing on the FDM method. In this method, the basic geometry of the casting model can be adjusted programmatically, and a new mesh can be created for each simulation, because the meshing process is entirely automated and does not require any user intervention.

Other features which can considerably speed up the calculation process, and which have been used in simulations for optimization, include the use of uniform-shaped (cubical) elements and the use of average material properties. Using a variety of such techniques, we have been able to produce simulations of casting processes which used a million or more elements (thus representing the casting geometry quite accurately) and which have run in less than 30 minutes on up-to-date personal computers. Using these calculations, an optimization requiring, say, 16 successive simulation calculations could run in one day. The same series of simulations in some other types of software systems could conceivably take weeks to perform.

## REFINEMENT OF PARAMETER SELECTIONS

In order to develop a procedure for optimization, it is necessary first to understand what goals you are trying to achieve. Today, many casting simulations are run for the purpose of predicting shrinkage in castings (either in the form of macroporosity or microporosity), with the goal being to produce a sound casting. Therefore, our initial efforts in this area have been to optimize castings with respect to shrinkage formation.

Macroporosity and microporosity form under specific, and sometimes different, circumstances. In our work with simulation, we have found that macroporosity can be predicted by simulating the volumetric changes occurring in solidifying metals, and then simulating the flow of liquid feed metal in response to these volumetric changes. Microporosity, on the other hand, often seems to form in areas of poor directional solidification, which can be measured by consideration of criteria such as temperature gradients, local solidification times and the velocity of the solidification wavefront through various parts of the casting. In order to optimize castings for soundness, it is necessary to select the proper data for constraints and the objective function to achieve a sound casting in the most efficient way possible.

Foundry engineers are also concerned with the cost of producing a casting. One of the major costs is the energy involved in melting the metal to make a casting; the less metal required to be poured, the lower the melt cost. Therefore, when dealing with the concept of optimizing a casting design, you should consider the total amount of metal which is required vs. the net metal in the casting, i.e., the process yield. It may be that several alternate riser designs could result in a sound casting, and the optimization system should be capable of identifying which of these designs is preferable from a cost standpoint.

In order to provide a direction for refinement of the optimization process, a simple casting with a single riser was considered, as shown in the figure below:

Multiple simulation runs were performed with this casting, varying riser height and diameter, and several response surfaces were mapped out (a response surface is merely a plot of a process result over several combinations of process inputs). One of the features of the response surfaces that became obvious from this exercise was that, when considering measures of the soundness of the casting (either macroporosity or microporosity), there are regions where the casting is sound over several combinations of riser height and diameter, as can be seen in the response surface pictured on the next page:

This means that, if we were to use a soundness criterion as our objective function, once the optimization system entered this region it would be unable to differentiate between alternate riser designs; in other words, there are several combinations of riser height and diameter that would result in a sound casting, and the optimization system has no reason to pick one over the other. On the other hand, a response surface which is generated showing process yield mapped over the design variables seems to have a continuous slope as the process inputs are varied, as can be seen in the following image:

This means that it is easier to locate a maximum or minimum point on such a surface.

The preceding results suggest that a rational method for optimizing a given casting for soundness would be to specify one or more porosity predictors as constraints, and then maximize the process yield. This would allow the system to identify those designs which result in a sound casting, and then to select the least-cost design as being that which maximizes the material yield of the process. This scheme would resolve a further problem, in that only one objective function may be specified for an optimization run; it is not possible, for example, to specify criteria for both macro- and microporosity as being objective functions within one optimization run. Considering both criteria as constraints allows both to be considered when searching for an overall optimization. Of course, it is possible that for a given initial design a sound casting may not be possible, in which case such constraints could not be satisfied; the optimization system would report this, and some consideration would have to be given to revising the process design so that a sound, optimized result could be obtained. It may be possible, in the future, to structure a sort of “hyper-optimization” such that the system is able to search for constraints that would be able to produce a valid, sound result.