ARCHIVES
of
ISSN (1897-3310)
Volume 10
Issue 4/2010
FOUNDRY ENGINEERING
Published quarterly as the organ of the Foundry Commission of the Polish Academy of Sciences
89 – 92
16/4
Application of evolutionary algorithms to
optimize the model parameters of casting
cooling process
S. Kluska-Nawarecka a, Z. Górny a, A. Smolarek-Grzyb b*
a
b
Foundry Research Institute, ul. Zakopiańska 73, 30-418 Kraków, Poland
Department of Industrial Computer Science, AGH University of Science and Technology,
Al. Adama Mickiewicza 30, 30 - 059 Kraków, Poland
*Corresponding author. E-mail address: [email protected]
Received 13.07.2010; accepted in revised form 15.07.2010
Abstract
One of the most commonly used methods of numerical simulation is the finite element method (FEM). Its popularity is reflected in the
number of tools supporting the preparation of simulation models. However, despite its usefulness, FEM is often very troublesome in use;
the problem is the selection of the finite element mesh or shape function. In addition, MES assumes a complete knowledge of the
simulated process and of the parameters describing the investigated phenomena, including model geometry, boundary conditions, physical
parameters, and mathematical model describing these phenomena. A comparison of the data obtained from physical experiments and
simulations indicates an inaccuracy, which may result from the incorrectly chosen shape of element or geometry of the grid. The
application of computational intelligence methods, combined with knowledge of the manufacturing technology of metal products, should
allow an efficient selection of parameters of the mathematical models and, as a consequence, more precise control of the process of the
casting solidification and cooling to ensure the required quality. The designed system has been integrated with the existing simulation
environment, which will significantly facilitate the preparation and implementation of calculations of this type. Moreover, the use of a
distributed model will significantly reduce the time complexity of calculations, requiring multiple repetition of complex simulations to
estimate the quality of the different sets of parameters.
Keywords: Solidification casting process; Casting; Parameters of casting solidification; Evolutionary algorithm; Numerical simulation
1. Introduction
Growing market demands the competitiveness of products
including products of the foundry industry, i.e. castings, which
involves actions taken to promote modernisation for an
improvement of product quality and/or cutting down the
manufacturing costs and, above all, the consumption of energy, as
well as materials and work input, while improving the working
conditions and reducing the level of harmful emissions.
Designing of the casting manufacturing processes can be
significantly upgraded and made more complex through the use
of the opportunities that are created by the application
of numerical methods. An example can be modelling
of calculations of the casting solidification and cooling process
in foundry mould. In this case, the fundamental difficulty
in implementation of the existing computational methods is the
lack of fully reliable values of the coefficients important in the
calculation of the heat exchange process that takes place between
the casting and mould.
ARCHIVES of FOUNDRY ENGINEERING Volume 10, Issue 4/2010, 89-92
89
Complex character of physical phenomena occurring at the
molten/solidifying metal - foundry mould interface can make
various coefficients differ significantly from actual values, and
therefore an important problem is faced how to ensure compliance
between the results of simulation and physical experiment [1]
One of the most commonly used methods of numerical
simulation is the finite element method (FEM). The popularity
of this method is reflected in the number of tools supporting the
preparation of simulation models. However, despite its usefulness,
FEM is often very troublesome in use; the problem is the
selection of the finite element mesh or shape function. In addition,
MES assumes a complete knowledge of the simulated process,
and of the parameters describing the investigated phenomena,
including model geometry, boundary conditions, physical
parameters, and mathematical model describing these phenomena.
A comparison of the data obtained from physical experiments and
simulations indicates an inaccuracy, which may result from the
incorrectly chosen shape of element or geometry of the grid. In
many cases, the knowledge of the simulated process is not
sufficient. The unknown parameters can be calculated only
directly from experimental data, but to do so numerous physical
experiments should be carried out. In the absence of efficient
algorithms for computation of many important issues related with
calculations and decisions, increasing popularity enjoy systems
modelled on the processes of biological evolution, called
evolutionary algorithms. These algorithms can be used for
complex optimisation tasks, which creates opportunities for their
application in parametric optimisation of a simulation model
of the cooling and solidification process, based on data obtained
from physical experiments.
2. Adaptation of parameters of the model
of solidification and cooling process using
evolutionary algorithms
The evolutionary algorithm is a kind of algorithm searching
the space of alternative solutions to find the best ones. These
algorithms are used to solve the difficult tasks of global
parametric and combinatorial optimisation related with areas such
as engineering design, planning and scheduling, forecasting,
modelling of economic and social phenomena. The essence of the
operation of an evolutionary algorithm consists in an iterative
transformation of the population of individuals representing a set
of potential solutions to the task. Evolution is reduced here to
producing successive generations, using so called genetic
operators and selection process. The task of the genetic operators
is random modification (mutation) and exchange (recombination)
of genetic material of the individuals, i.e. search for new
solutions. The selection of best individuals is made on the basis
of the provided by the environment, objective function, which is
a measure of the adaptation of individuals, equivalent to the
quality of solutions they represent. Owing to this, the process
of evolution should aim towards generation of better and better
individuals and finding the searched for (and usually
approximate) solution of the problem [1, 4, 5, 9, 10, 11]. The way
in which the algorithm operates is shown in Fig. 1.
90
Fig. 1. Schematic representation of the operation of classic
evolutionary algorithm
As the use of the finite element method for simulation
of cooling and solidification of castings cannot reflect with
sufficient precision the real physical experiments, evolutionary
algorithms were used to search for the values of simulation
parameters by "fitting" the simulation results until measurements
of true values are obtained. A comparison of the results of the
experiment and simulation allows determination of the "quality"
of the chosen set of parameters of the simulation of casting
solidification.
Evolution is by nature a parallel process: many individuals,
species and populations grow simultaneously and undergo
generational changes in common environment. Applying natural
parallelism of the evolutionary process, parallel calculations were
used in evolutionary algorithms. The standard approach,
sometimes called global parallelisation, consists in a parallel
implementation of selected steps of an algorithm, performed for
different individuals on multiple computing units. In this case, the
population remains non-structuralised, and both selection as well
as the individual matching are made globally (in the whole
population). Thus, these operations because of a relatively low
complexity of the calculations in respect of the cost
of communication, or synchronisation of access to particular
individuals in a population are rarely subjected to parallelisation
(synchronisation of generations). Most often, to parallelisation are
subjected calculations of the value of the individual adaptations,
sometimes also operators of variations.
In master-slave
architecture, sometimes also called a farming model, the entire
population is managed by a primary computing unit (master). It
performs the sequential steps of the algorithm and is responsible
for the distribution of parallelised tasks (sending of individuals or
groups of individuals) among other computing units (slaves), for
coordination of calculations and receiving the results. There is no
communication between the slave computing units. The
advantages of global parallelisation include compatibility with the
classic pattern of an evolutionary algorithm and ease of
implementation. Considerable speeding up of calculations is
obtained in the case of expensive fitness functions, which is often
the case in practical applications, since each time it involves a
conversion of a complex model of the phenomenon (system), as
in the considered case of optimising the FEM simulation
parameters. The master process implements a classic genetic
algorithm [4]. To calculate the value of fitness function for each
ARCHIVES of FOUNDRY ENGINEERING Volume 10, Issue 4/2010, 89-92
individual, it transfers the decoded individuals to slave processes
[6]. The schematic diagram of a genetic algorithm combined with
evolutionary optimisation of model parameters is shown in Fig. 2.
function of temperature. The source of heat has been neglected. A
simple case of the spherically shaped body was taken into
consideration (Fig. 3).
The quantity determining the run of the casting cooling
process is temperature (Fig. 4) - the value and the rate
of temperature changes affect the casting microstructure and the
level of thermal stress. The consistency obtained between the
simulation results and actual data depends on proper selection
of the values of individual parameters and simplifications adopted
in the model.
Fig. 2. The structure of algorithm with FEM simulation and
optimisation of model parameters [2]
3. Mathematical model
The mathematical model of the casting solidification and
cooling process presented below is fundamental in the research
based on computer simulation. The majority of the available
simulation packages enable modelling of phenomena which occur
during cooling of casting using the Fourier-Kirchhoff equation
[3]:
∂T ( X , t )
X ∈ Ω : c (T ) ρ (T )
=
(1)
∂t
= div[λ (T ) gradT ( X , t )] + Q ( X , t )
where: X – the point in a casting, T – the temperature [K],
t – the time [s], c(T) – the specific heat [J/kgK],
ρ - the density [kg/m3], λ(T) – the thermal conductivity [W/mK],
Q(X,t) – the heat source – the amount of heat generated in a unit
volume in a time unit; [W/m3]
Gradient and divergence in the spherical coordinate system
At constant value of λ:
1
∂ 2T
1 ∂ 2 ∂T
(r
)+ 2 2
+ ...
2
r ∂r
∂r
r sin ϑ ∂ϕ 2
1
∂
∂T
... + 2
(sin ϑ
)]
r sin ϑ ∂ϑ
∂ϑ
λdiv ( gradT ) = λ[
At changing value of λ:
1 ∂
∂T
1
∂ 2T
div(λgradT ) = [ 2
(λ r 2
)+ 2
(
) + ...
λ
∂r
r ∂r
r sin 2 ϑ
∂ϕ 2
... +
Fig. 4. Temperature curve plotted for selected point of mould and
casting
The correctness of the obtained simulation model will depend
on the parameters to be optimised and on the available
experimental results. The more degrees of freedom and criterion
components, the better results can be achieved, but with increased
calculation effort [1].
(2)
4. The evolutionary model of
solidification
(3)
1
∂
∂T
(λ sin ϑ
)]
∂ϑ
r 2 sin ϑ ∂ϑ
where: r – the coordinate in the direction of the radius, ϑ - the
ϕ
Fig. 3. Physical model of sphere with gating system and riser
angle in the meridional direction,
- the azimuthal angle
Therefore it has been decided to determine the thermophysical coefficients in accordance with the above model. The
optimised values are density, the heat capacity coefficient and
thermal conductivity; the last two quantities are determined as a
The possibility of using evolutionary algorithms was
illustrated on the example of a problem of searching for a value of
thermal conductivity on a model of the sphere. For the remaining
data (density, specific heat), the values given in respective tables
were adopted. A classic genetic algorithm was used, in which
different individuals represent different relationships between the
values of the thermal conductivity and temperature:
x = [x1 = λ (T1 ),..., x5 = λ (T5 )]
(4)
where: Ti – the temperature determining the point of interpolation
for the
thermal conductivity - temperature relationship,
ARCHIVES of FOUNDRY ENGINEERING Volume 10, Issue 4/2010, 69-76
91
λ(T1)- the discrete values of thermal conductivity in the interval
(0,1] with step 0.05.
The values of objective function are determined by
comparison of actual values with the simulated ones. The
objective function can be consistent with: a point in given time, an
area, and a graph of preset function in time. In the case under
consideration, this function has been defined in the following
form [1]:
fitness ( x) =
100
∑ (T (t ) − T (t ))
i =1
i
n
2
i
(5)
where: ti – the consecutive time points, T(ti) – the values of
temperature obtained at the measuring point during simulation,
Tn(ti) – the values of temperature obtained in physical
experiment.
The standard temperature run during casting cooling was
obtained by simulation of a preset thermal conductivitytemperature relationship. Practical operation of the application
used is illustrated in Fig. 5.
unsolvable problem remains the choice of some coefficients of
mathematical models of these processes. When modelling the
cooling of castings in mould, special difficulties arise in the
determination of several parameters (coefficient of metal-mould
heat transfer, specific heat, thermal conductivity of metal and
mould, density of metal and mould). These coefficients depend
not only on the material properties but also on temperature. The
paper proposes a method for optimisation of the model parameters
based on an evolutionary fitting of the curve representing
simulation results to the curve obtained in a physical experiment.
The application of computational intelligence methods,
combined with the knowledge of the manufacturing technology of
metal products, should enable efficient selection of parameters of
mathematical models and thus allow a more precise control of the
process of casting solidification and cooling to ensure the required
quality. The designed system has been integrated with the existing
simulation environment, which will significantly facilitate the
preparation and implementation of this type of calculations.
Moreover, the use of a distributed model significantly reduces the
time complexity of the calculations, requiring multiple repetition
of complex simulations to estimate the quality of the different sets
of parameters.
References
[1]
Fig. 5. Application using evolutionary algorithm for optimisation
of the casting solidification and cooling parameters
An application to optimise the parameters of casting
solidification and cooling uses data from both the experiment and
simulation. The use of an evolutionary algorithm improves the
value of thermal conductivity (Fig. 6). According to the scheme
shown in Fig. 2, the "corrected" parameter will be used in resimulation of the model. Owing to the operation of the
application, the parameters used in FEM simulations are similar to
the physical data and the model becomes more adequate.
Fig. 6. Plotted temperature curves for physical data obtained by
simulation and „fitting”
5. Summary
Górny Z., Kluska – Nawarecka S., Kisiel – Dorohinicki M.,
Mathematical and simulation models in studies of copper
and aluminium alloys, The Foundry Research Institute,
Krakow 2003
[2] Byrski A., Kisiel-Dorohinicki M., Kluska-Nawarecka S.,
Evolutionary optimisation of FEM simulation parameters of
cooling and solidification process, KomPlasTech 2004,
Krakow 2004
[3] Suchy J., Mochnacki B., Modelling and simulation
solidification process, PWN, Warsaw 1993
[4] Goldberg D.E, Genetic Algorithms in Search, Optimization
and Machine Learning, Kluwer Academic Publishers,
Boston, 1989
[5] J. Arabas: Lectures in evolutionary algorithms, Warszawa:
WNT, 2001.
[6] Kluska – Nawarecka S., Computer-aided methods od the
diagnosis of casting defects, The Foundry Research Institute,
Krakow, 1999
[7] Handzlik P., Trębacz L., Byrski A., Kisiel-Dorohinicki M.,
Application of distributed genetic algorithm to optimisation
of FEM simulation parameters of solidification process,
KomPlasTech 2005, Krakow, 2005
[8] Górny Z., Casting non-ferrous alloys, WNT, 1992
[9] Duda J.: An evolutionary algorithm for scheduling in
a foundry, Archives of Foundry vol. 8, Katowice 2003
[10] Macioł A., Stawowy A.: Discrete event simulation for
foundry system design, Archives of Foundry, 2005, R. 5,
nr 17, str. 155-162.
[11] Stawowy A., Wrona R., Macioł A.: Evolutionary algorithm
for castings cost evaluation, Archives of Foundry, 2006,
R. 6, nr 18, t. 1, str. 21-26.
Finite element method (FEM) finds currently many
applications in simulation of thermal processes. Still, the
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ARCHIVES of FOUNDRY ENGINEERING Volume 10, Issue 4/2010, 89-92