In many engineering cases, more than two optimization tar- gets are addressed. However, it is more complex or unfea- sible to graphically visualize and reasonably evaluate more than two objectives. The only way for the optimization al- gorithms to take the other optimization targets into consid- eration without prescribing them as objectives is to express them by means of optimization constraints. Of course, one should then expect the presence of unfeasible designs that do not comply with the prescribed constraints. Our optimiza- tion problem is constrained only by the predefined ranges of variation of the design variables (Table 2).
Based on the number of design variables and their ranges of variation, the optimizer generates the total number of fea- sible combinations (the initial DOE sequence) by a DOE sequence technique referred to as Full Factorial Design. Ob- viously, it would be very time-demanding to calculate all possible designs, thus the initial population is provided by the Sobol DOE sequence generating technique,21
which is a
quasi-random sequence. The points in this type of sequence are maximally avoiding each other, so the initial population fills the design space in a uniform manner. The optimization run has been executed using the following parameters:
Initial Population: Sobol Sequence Population Size: 100 Number of Generations: 10
Probability of Directional Cross-Over: 60% Probability of Selection: 30% Probability of Mutation: 10% Elitism: Enabled
Treat Constraints: Penalizing Objectives Algorithm Type: MOGA Steady
The last task is to assess how the optimized riser influences the formation of the centerline porosity, represented by the Niyama criterion22- 24
in the simulation software environ-
ment. The minimization of the centerline porosity is neither prescribed as an objective nor a constraint in the optimizer. The optimization results are simply analyzed with respect to the Niyama criterion in order to select the “most” optimal solution on the Pareto line.
The Niyama criterion is a local thermal parameter defined as the relationship between the gradient (G) in K/mm and the cooling rate (R) in K/s, both of which are assessed at a specified temperature near the end of solidification, when the solidification shrinkage forms (Equation 1). In the pres- ent study, the Niyama criterion is evaluated at a temperature 10% of the solidification range above the solidus tempera- ture. This is important to state, since the choice of Niyama evaluation temperature can remarkably affect the resulting Niyama values.25
Equation 1 66
Figure 5. Manually optimized casting design (the result of the case study 2).
International Journal of Metalcasting/Fall 10
With the help of the Niyama criterion, it is feasible to predict the presence of centerline shrinkage porosity, i.e. micro- and macro-shrinkage in steels caused by shallow temperature gradients.26, 27
It indicates that in regions that solidify quick- ly, there must be hot metal nearby to establish a high gradi- ent to feed the shrinkage during solidification.
It has been proven by numerous trials that for sufficiently large Niyama values, no shrinkage porosity forms. When the Niyama value decreases below a critical value, small amounts of micro-shrinkage begin to form. As the Niya- ma value decreases further, the amount of micro-shrink- age increases until it becomes detectable on a standard ra- diograph. This transition occurs at a second critical value. Both of these threshold values depend on the composition of the alloy and, in some cases, on the casting process conditions.
assumed to be micro-shrinkage that will not be detected via common radiography. The second threshold value for mi-
cro-porosity, above which the material is completely sound, is set to be Nymicro
= 1 (K·s)1/2 /mm. It should be emphasized
that the Niyama criterion only predicts feeding-distance re- lated shrinkage; it does not explicitly predict hot spots in a casting, and it does not predict gas porosity.22
Based on a literature search and results obtained from the manufacturing foundry, it is determined that the critical Ni- yama value for macro-porosity for this particular steel alloy is Nymacro
= 0.45 (K·s) 1/2
/mm. Everything above this value is threshold value for mi-
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