of having such short calculation times can only be utilized with a computer that can automatically analyze calculat- ed variants with respect to the predefined objectives (e.g. maximum feeding, low porosity, low air entrapment etc.) and subsequently create new variants and analyze them in the same manner to achieve the optimal solution. By inte- grating such software for casting process simulation with an optimization algorithm (in this case the Multi-Objec- tive Genetic Algorithm, [MOGA]10
a computer-based optimization tool
peratures, riser dimensions, alloy composition, process parameters, etc.) are modified by the op- timization algorithm in order to meet the given objectives.
which should not be
confused with the non-elitist MOGA developed by Fon- seca and Fleming),11
is established which is able to determine optimal values of user-defined design variables, thereby optimizing a given casting process with respect to defined objectives. Subse- quently, such a system can readily provide optimal solu- tions for any kind of casting process.12-14
This paper details multi-objective optimization of filling and solidification patterns, together with the riser volume of a steel forging ram cast into a furan sand mould, and presents the results obtained from the study. Numerical multi-objec- tive optimization is carried out by coupling the general-pur- pose casting simulation software package MAGMASOFT® with the add-on optimization module MAGMAfrontier.
This study includes five different layouts. The initial layout is obtained from a foundry that manufactures the forging ram. The second layout with manually rearranged gating system and chills is also provided by the foundry. Three layouts are generated by numerical optimization. The first two designs are analyzed both in terms of filling and solidification, and the results are compared with the experimental casting trials. No computerized optimization is involved. The three opti- mized designs are assessed only in terms of solidification since the filling pattern remains unchanged; however the temperature fields at the beginning of the solidification are inherited from the filling stage. Conclusions and proposals are made from the various investigations and findings in the study and presented in the last section of the paper.
Coupling of Simulation and Optimization Tools
Before the optimization process can be started, a standard proj- ect must be defined in the simulation software environment. This includes a definition of geometry in the pre-processor. Furthermore, a suitable mesh must be generated and all relevant process parameters adequately defined. The optimization itself is based on performing a large sequence of “standard” calcula- tions, each with different design variants. Therefore all design variables must be defined in a parametric way.
In essence, any constrained optimization method is based on the following scheme: a user formulates the problem in a mathematical way by means of several parameters:15
• Input (design) variables together with allowed ranges of variation. Design variables (initial tem-
62
• Output variables usually represent values calcu- lated in a standard casting simulation. The optimi- zation objectives are usually formulated by using the output variables (e.g. when the porosities and the volumes of the feeders in one design need to be assessed for a feeder optimization).
• Constraints are conditions for designs. They limit the design space or the range of solutions. If a de- sign does not satisfy these constraints, it is consid- ered unacceptable.
• Objective functions – These are the aims and tar- gets that are the goal to achieve by means of op- timization. They maximize or minimize certain combinations of output variables, e.g. min (volume of a riser) or min (max. porosity).
When all this information is available an optimization loop can be started.
Optimization Methodology
In the next few sections, a multi-objective optimization problem in the gravity sand casting process of a forging ram is presented. The objectives for this case study are the fol- lowing: minimize the top riser volume, minimize shrinkage porosity, and limit centerline porosity, by means of an opti- mized arrangement of the chills.
The actual optimization cycle is initiated by establishing a first generation, i.e. a set of solutions, containing a user- defined amount of individual variables referred to as the initial population or Design of Experiments (DOE) in the applied software package. Each individual represents one design for the considered design variables, e.g. dimensions of the riser and the chills. For each of these designs a ther- mal (solidification) analysis is started, and the values for the requested output variables are calculated. The filling analysis is not performed during optimization since the gating system is kept the same as in the previous side-filled case. The output data is used to compare and evaluate the different designs. After the first generation has been cal- culated, the optimization algorithm evaluates the designs relative to the objective function(s) and constraint(s). Af- terwards, it generates a new set of solutions using math- ematical mechanisms that follow the concept of genetics, i.e. selection, cross-over and mutation. This strategy is consequently referred to as a genetic algorithm.16
From the
algorithmic point of view, the procedure can be described as follows:
1. Create the initial set of designs (initial population) 2. Calculate all designs and grade their performance in terms of the objectives
International Journal of Metalcasting/Fall 10
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