INTRODUCTION
Solidification is an important phase in the casting process influencing quality
and yield of products. Generally the solidification process begins with the
initiation of crystallization at the mould walls and gradually proceeds inwards.
The last freezing sections and locations i.e., the hotspots are the most likely
locations of defects such as shrinkage cavities or porosity. These hotspots
in complex geometries of commercial castings are visualized and identified by
simulating the solidification process using finite difference, finite element
and boundary elements methods (Minkoff, 1986; Codina
et al., 1994; Mechighel and Kadja, 2007).
Such simulation helps in predicting hotspots and understanding the influence
of feeder location, chills or insulations in ensuring a desired directional
solidification pattern. Such a controlled solidification increases the yield
by producing higher quality casting and minimizes casting rejections.
Although the finite element based numerical simulation approach is accurate
and reliable, it is also computationally expensive. Other approaches to solidification
analysis are primarily driven by the geometry of the casting. Such geometrybased
approaches have been developed to reduce the cost of computations in terms of
computer power and time requirements. Since geometric methods are fast and easier
to use, they are also used for complex shapes for preliminary analysis before
carrying out more detailed numerical simulation for further investigation. These
methods are based on the premise that the sequence of solidification and thus
the location of hotspots are driven by the shape of geometry. These approaches
simulate the solidification process using geometric parameters and link these
parameters with the thermal properties of the metal, mould and heat transfer
systems. The relative value of temperature and solidification time assumes greater
significance than the actual values associated with any point inside the casting
for predicting shrinkage defects and for designing appropriate feeders (Lewis
et al., 1997). Most of these geometric methods are based on the Chvorinov’s
classic rule that related solidification time t_{s} of a casting to
its modulus. The term modulus pertains to the ratio of heat content volume V
to its heat transfer area A of the casting:
where, k is a material constant depending on the cast metal and mould material.
Derivatives of Chvorinov rule were further investigated and expanded by many
researchers (Wlodawer, 1966; Berry
et al., 1959; Heine and Uicker, 1983; Neises
et al., 1987; Ravi and Srinivasan, 1996,
1989; Sirilertworakul et al.,
1993).
The objective of the present study is to investigate the use of medial axisbased
geometric technique that could be used to optimize the insulation padding around
the castmould interface.
PROPOSED METHOD
Heuvers (Ravi and Srinivasan, 1996) was the first
to suggest a practical method consisting of inscribing a series of circles (spheres
when viewed threedimensionally), the diameter of which increases in the direction
of the feeder head. Experiments showed that modulus must be larger by a factor
of about 1.1 to ensure the directional solidification with adequate feeding
from thicker sections to thinner sections or it must increase by 10% from casting
across the ingate to the feeder. A model of the casting is sketched and then
the circles are inscribed at points of intersections (Fig. 1)
so that, ΔR/Δl = 0.1 where, ΔR is the radius increase and Δl
is the length of medial axis between two points. In this example, the padding
additions have been designed on the basis of the Heuvers circles as they increase
in diameter towards the feeder head.
This study has utilized this method in combination with Medial Axis Transformation
which automatically calculates the Heuvers’ circles for a given geometry.
The combined method proposes a geometric optimisation technique for ensuring
directional solidification and relocating hotspots in the feeder. The proposed
method outputs optimal values of interfacial heat transfer coefficients which
then can be used in the finite element simulation for further detailed and accurate
simulation. The Medial Axis Transform (MAT) of a twodimensional region is a
locus of the center of an inscribed disc of maximal diameter as it rolls within
the domain by maintaining the contact with the domain boundary (Campell,
1997), as shown in Fig. 2. Medial axis becomes a medial
surface for a three dimensional object.
As we deduce from discussion above, a casting with Heuvers’
circle drawn is in fact hypothetically a new imaginary casting (Fig.
1) whose modulus is increasing in the direction of the feeder. The aim is
to obtain effective Interface Boundary Conditions (IBCs) using Heuvers’
inscribed circles in such a manner that these IBCs can help achieve the same
solidification pattern as would have been provided by the imaginary casting.

Fig. 1(ac): 
Schematics of the Heuvers’ circle method, (a) Original
casting section, (b) Casting with Heuvers’ circles and (c) Additional
padding as per Heuvers’ circles 

Fig. 2: 
Threedimensional object showing its medial surface and inscribed
spheres 
One such option was to relate the geometric parameters (radius information)
with the boundary conditions e.g. modified Heuvers’
radii to interfacial heat transfer coefficient to achieve the desired solidification
pattern. Using Eq. 1, it can be assumed that if the heat transfer
coefficient h is constant:
And conversely, if R is constant:
Therefore, we infer that:
Thus, for an original casting, at any arbitrary point A on medial axis:
and for an equivalent imaginary casting, at point A on medial axis, we will
have:
Heat transfer coefficient needs to be modified if t_{H} and R_{O}
are to be kept constant:
From Eq. 2 and 3:
where, R_{O} and h_{o} are original radius and interfacial
heat transfer coefficient, R_{H} and h_{H} are Heuvers’
radius and the modified interfacial heat transfer coefficient, respectively
and t_{O} and t_{H }are original solidification time and one
achieved after applying Heuvers’
Circle method.
Numerical examples: The method was first tested for a simple Lshaped
casting geometry before a more rigorous test was carried out on a more complex
gear blank casting which we described in the following (Berry
et al., 1959). The gear blank geometry was first created and then
generated its medial axes using MAT, a tool available in CADfix, a Computer
Aided Design based package from FEGS Ltd., UK (Ransing
et al., 2004; Pao et al., 2004; Lewis
et al., 2003). The length of the medial axis, the relative distance
between points and radii (R_{O}) at these selected points were then
obtained using CADfix (Fig. 3). Selected points on the medial
axis together with the calculated medial radius were tabulated in column 1 and
3, respectively in Table 1. The originally assigned h_{o}
is shown in column 2 and the computed Heuvers’ radii is shown in column
4. Using this radius information, direct application of Eq. 4
provided the modified values of interfacial heat transfer coefficient (h_{H})
as shown in column 5 in Table 1.
In order to proceed with the analysis, material properties and boundary conditions
were prescribed to get the temperature profile and solidification pattern based
on finite element analysis (uncoupled solution) using a FE based packagePROFETS
(PROFEssional Thermal Solidification).

Fig. 3: 
Gear blank casting showing medial axis and radius points 
Table 1: 
Original and modified radii and interfacial heat transfer
coefficients 

^{a}Point as given in Fig. 3 
Table 2: 
Material properties for the gear blank simulation 

The initial temperature for Cast (LM24) and Mould (H13) was prescribed at 650
and 150°C, respectively, with thermal conductivity of 186.28 W m^{1}
K^{1} and convection to ambient at 75 W m^{2} K^{1}
at an ambient temperature of 20°C. The original interfacial heat transfer
coefficient (h_{o}) of 3000 W m^{2} K^{1} was prescribed
throughout the interfacial boundary. The required material properties for the
simulation are tabulated in Table 2. The result without any
optimization obtained from this FE simulation is shown in Fig.
4.
The solidification contours in Fig. 4 illustrated that point
Q174 on the medial axis on left hand side of the casting (Fig.
3) indicates the hotspot, as this is the centre of the largest inscribed
circle in the casting and this hotspot needs to be relocated into the feeder
on the extreme right. There is very little doubt about the validity of the solution
shown in Fig. 4 as the inhouse finite element package has
been extensively tested before (Ransing et al.,
2004; Pao et al., 2004; Lewis
et al., 2003; Wong and Pao, 2010). Consequently,
the solution was used as the reference solution for the subsequent optimisation
purpose. The cast boundary was then divided into segments and the modified boundary
conditions (modified values of interfacial heat transfer coefficients from Table
1) were then applied on respective segments around those points.

Fig. 4: 
Finite element solution with uniform interfacial heat transfer
coefficient on all boundaries 

Fig. 5: 
Finite element solution with Heuvers’ radii based modified
interfacial heat transfer coefficient on selected boundaries of casting 
The same finite element analysis was then carried out to obtain the optimized
solution (Fig. 5) based on the model presented in this study
with a combination of Genetic Algorithm (Wong and Pao,
2010; Shirazi et al., 2008).
It is obvious from Fig. 5 that the hot spot has been relocated
successfully and the last point to solidify (Q174) has migrated into the feeder
section. In fact, the proposed method successfully migrated all the hot spot
into the feeder thus ensuring proper thermal control during the solidification
stage. In Table 1, columns 5 and 6 showed the modified factors
and the resulting heat transfer coefficients after the optimization.
At this point, it is instructive to note that in practicality, it is impossible
to find the insulation padding transfer coefficient. This is not the intention
of this study. The primary focus here is the modification factor obtained in
Table 1. Its values, when scaled to unity, gives a good indication
of the relative insulation requirement. This information is thus deem sufficient
for foundries to proceed with the mould design.
CONCLUSION
Thus it is seen that using geometric information we can predict hotspots and
then use it further to obtain effective interface boundary conditions to influence
and achieve desired directional solidification. Since geometric methods are
fast and easier to use, they can be effectively used for complex casting shapes
for preliminary analysis before carrying out more detailed numerical simulation
for further investigation.