casting design through multi-objective optimization

Casting Design Through Multi-Objective Optimization

Jean Kor*, Xiang Chen*, Zhizhong Sun**, Henry Hu**

*Department of Electrical & Computer Engineering, University of Windsor

Windsor, Ontario, �9B 3P4, Canada

e-mail: [email protected]

** Department of Mechanical, Automotive & Materials Engineering, University of Windsor

Windsor, Ontario, �9B 3P4, Canada

Abstract: The gating and riser system design plays an important role in the quality and cost of a metal casting. Due

to the lack of fixed theoretical procedures to follow, the design process is carried out on a trial-and-error basis. The

casting design optimization problem is characterized by multiple design variables, conflicting objectives and a

complex search space, making it unsuitable for sensitivity-based optimization. In this study, a formal optimization

method using Multi-Objective Evolutionary Algorithm (MOEA) was developed to overcome such complexities. A

framework for integrating the optimization procedure with numerical simulation for the design evaluation is presented.

The proposed optimization framework applied to the gating and riser system of a sand casting gave good results and

provided more flexibility in decision making.

1. INTRODUCTION

Casting design, in particular the gating and riser system

design has a direct influence on the quality of cast

components. The design of gating and riser systems is

largely based on past experience and empirical rules

(Campbell,1991) and simulation has become an

important tool for the design, analysis and optimization

of casting processes (Kor, 2006).

Coupling numerical simulation with formal optimization

methods is one way of adopting a more systematic

approach towards casting design. The casting design

optimization problem is characterized by multiple

control points and multiple conflicting objectives that

involve many parameters (Wang, 2007).

In this study, the casting design process is formalized

and a proposed optimization framework using multi-

objective evolutionary algorithm (MOEA) is presented.

Evolutionary algorithms are guided stochastic search

methods that are well known for their ability to address

such difficulties. They are global optimizers that work

with a population of solutions in parallel and have been

shown to be robust in finding optimal solutions in

complex search spaces regardless of the starting points

(Jones, 2002).

The key aspect of using MOEA for casting design

optimization is that it does not require any analytical

information for the optimization formulation. The main

goal is to improve the efficiency and quality of the

casting design process by combining simulation and

formal optimization techniques. The focus is primarily

on the gating and feeding system where numerical

simulation is employed to predict the performance of a

design.

2. MULTI-OBJECTIVE OPTIMIZATION

In a multi-objective problem, the aim is to find a set of

values for the design variables which optimizes a set of

objective functions simultaneously. When the objectives

are conflicting, finding a solution that minimizes all of

the objectives at the same time becomes impossible.

The operation of the genetic algorithm is shown in

Figure 1. The elitist Non-Dominated Sorting Genetic

Algorithm II (NSGA II) is currently one of the most

popular MOEA method used in real-world multi-

objective optimization problems (Deb, 2003). Some of

the distinguishing features of NSGA II are its fast elitist

sorting strategy that involves a combined pool of both

the parent and child populations and the elimination of

sharing parameters using an autonomous crowding

distance strategy.

NSGA II maintains a Pareto archive and introduces

elitism by comparing the current population with the

previously found best non-dominated solutions. The

selection procedure is guided towards a diverse set of

points on the Pareto front based on the concept of non-

dominated ranking and crowding distance. NSGA II is

used for this study and is interchangeably referred to as

MOEA.

3. CASTING DESIGN OPTIMIZATION USING

MOEA

3.1. Optimization frame work

In virtual product development, the computer aided

design (CAD) and analysis sequence is integrated into an

optimization strategy, whereby the product's geometry

are described by parameters for an efficient process

(Merkel, 2003).

Preprints of the 18th IFAC World CongressMilano (Italy) August 28 - September 2, 2011

Copyright by theInternational Federation of Automatic Control (IFAC)

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In casting design, the two main objectives are: (1)

eliminating casting defects and (2) maximizing casting

yield. To achieve these goals, different designs can be

explored by changing the shapes and sizes of the gating

and riser components. The application of genetic

algorithms have been shown to be effective for cross-

sectional shape generators that involve a marriage of

continuous variables such as shape dimensions, and

discrete variables such as shape type and quantity

(Yoshimura, 2005). For this study, geometrical

descriptors such as length, radius, and height of the

casting components were selected as the design variables

for the optimization process as shown in Figure 2.

Fig. 1 General flowchart of the Genetic Algorithm.

Fig. 2. Binary encoding of the gating and riser design

variables.

A commercial simulation software, MAGMASOFT®

was employed for the gating and riser system design

analysis. Since the details on the complex calculations

involved were not available, the performance of a design

had to be measured from the accessible simulation

outputs.

The optimization strategy had to be connected to the

simulation environment. Here, the design analysis

software was treated like a ‘black box’ with an input

interface to accept new design suggestions and an output

interface to communicate their performance measures to

the optimization engine. These output measures were

used to evaluate the fitness of a design in MOEA.

The optimization framework and its process flow are

illustrated in Figure 3. The structure of the casting

design optimization implementation is divided into four

main parts: pre-processing, simulation, post-processing

and optimization which are described next.

Fig. 3. Optimization process flow chart of the gating

and riser system design

PRE-PROCESSING

Simulation of the cavity filling and solidification process

required the geometrical information for the casting, the

gating system and the mould in advance. To explore

different designs in an autonomous fashion, parametric

geometry functions were employed to create the gating

and riser systems with the aid of command files. By

modifying the parameter values in the geometric

functions, the design could be varied accordingly.

SIMULATION

The actual filling and solidification simulation would

take place once the meshed geometries and the necessary

casting process parameters have been established. The

type of numerical calculations employed was based on

the algorithm (Solver) type chosen. Once the simulation

is completed, the results were processed for analysis in

the next step.

POST-PROCESSING

With the 3-D post processor module in

MAGMASOFT®, the visualization of the fluid flow and

temperature field patterns in the cavity during the casting

process could be graphically analyzed. In order to

Initialize a population

Select parents for mating

Create new offspring by crossover & mutation

Delete undesirable members of population

Evaluate & insert new offspring into population

Termination criterion satisfied?

Return best chromosome

Increment population

Yes

No

Casting

Design

(STL)

Objectives, Weights,

Constraints &

Design Variables,

Initial Gating &

Riser Design

(Parametric Geometry)

Modeling

Filling &

Solidification

Simulation

MeshingExport

ResultsMAGMA API

GA

Evaluation &

Optimization

Termination

Criteria Met?

Return

Optimal

Result(s)

Yes

No

MAGMASOFT

Graphical

Results

Text

Results

Modify Design Parameters

Casting

Parameters


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formalize the casting design optimization process, these

results had to be converted to a proper format that can be

routinely handled. The Application Programming

Interface (API) of MAGMASOFT® was employed to

export the graphical results into text results.

Using the API, customized subroutines were developed

to access the MAGMASOFT® files and data structures,

extract the desired results from each control volume and

convert them into the appropriate format for further

processing. The processed results were further organized

to form the appropriate performance measures required

for the optimization performance evaluation process as

shown in Figure 4.

Fig. 4. The post-processing of simulation results for design

evaluation

OPTIMIZATION

Based on the output measures acquired from the

MAGMASOFT® simulation results, the performance

evaluation of a design was carried out against the

objective functions and constraints specified in the

problem definition. Based on the performance

evaluation process, the next direction of the MOEA

optimization process would then be determined.

In this study, the quality of a cast product was

characterized by its shrinkage porosity and liquid metal

velocity. Using the output results available from the

MAGMASOFT® simulation, the shrinkage porosity

measure was taken as the maximum porosity value

contained in the control volumes of the casting. Since a

shrinkage porosity-free casting is always desired, the

porosity requirement was P = 0%.

Turbulence of the liquid metal in the casting was

assessed in terms of flow velocities obtained also from

the simulation results. Based on Campbell’s rules, the

liquid metal flow should not exceed a velocity of 0.5 m/s

to maintain the stability of the meniscus front. Since the

critical velocity Vcrit was not properly defined in either

the X, Y or Z directions(Campbell, 1998), the constraint

on the entry velocity of the liquid metal was set such that

it must not exceed 0.5 m/s in any direction of the

velocity vector, Vx, Vy and Vz for a design to be

considered feasible.

For the purpose of scoring, the magnitude of the velocity

vector 222zyxxyz VVVV ++= was calculated based on the

Vx, Vy, and Vz components in each control volume of the

casting. The maximum velocity magnitude represented

the velocity objective measure.

For maximizing casting yield, the metal yield was

calculated based on the volume ratio of the actual casting

and the total gating and riser system:

%100% ×+

=+risergatingcast

cast

VolumeVolume

VolumeYield

Since minimization of the objective function was

chosen, the yield loss, Yl = 1-Y was used for the

objective measure.

3.2 Gating and Riser Design Case Study

TEST CASTING

A cylindrical housing model was used to demonstrate

the MOEA optimization strategy. It has an outer radius

of 260 mm and 160 mm at the largest and narrowest

part, an inner radius of 120 mm and 180 mm at the upper

half and bottom part respectively and a height of 245

mm. This casting is relatively large with a total weight of

30 kg. Bottom filling of the mold was employed. The

three-dimensional CAD model of the test casting and its

gating and riser system is shown in Figure 5.

For this experiment, cylindrical shaped gating and riser

components were used for simplicity. Due to the

symmetry of the test casting, the number of design

variables was reduced to four independent parameters.

The design variables were the radii of the top risers ar,

middle risers br and their heights rh as well as the radii of

the ingates and runner gr as shown in Figure 6.

Dimensional constraints were also imposed to the design

variables to maintain realistic shapes as given in the

Table 1.

The optimization problem for this study is defined as

follows:

[ ] where

% 30)(

% 0)(

cm/s 50)(,,

)( subject to

)(

)(

)(

)( Minimize

T

l

l

hrrr

zyx

xyz

rgba

Y

P

VVV

Y

P

V

=

<

=

<

=

=

x

x

x

x

xg

x

x

x

xf


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Fig. 5. The gating and riser components of the test

casting.

Fig. 6. The gating and riser components and design

variables of the test housing casting.

COMPUTER SIMULATION

The initial gating and riser system as well as the housing

model were created using the preprocessor module of

MAGMASOFT®. The enmeshment of the geometry was

made up of approximately 110,000 elements.

Magnesium alloy AZ91 was used as the casting material

and dry silica was used for the sand mould. The

thermophysical properties of both the cast materials were

available in the database module of MAGMASOFT®.

The pouring temperature is 650°C, die initial

temperature is 20°C and the heat transfer coefficient is

800 W/m2K. The filling time was 12 seconds.

For the MOEA implementation, a string size of 20 was

used for the binary encoding. The crossover rate was 0.8

and the mutation rate was set at 0.05. Single point binary

crossover and bit-wise flip mutation operators were

used. The binary tournament size used was 2 for

selection. The initial population was randomly generated

and the number of generations, G was used as the

terminating criterion where it is set according to the

maximum number of designs allowed by the total

acceptable run time.

Table 1 Design variables and the parameter range

Parameters

(mm) ar br gr rh

Upper Bound 40 10 12 20

Lower Bound 70 40 35 100

String Size 4 4 4 4

4. RESULTS AND DISCUSSIONS

In this study, only the velocity and yield objectives were

plotted in the objective space for easier visualization

whereas the porosity objective was treated as a

constraint. Using NSGA II with N=20, a set of non-

dominated solutions was found in each generation, G.

Since constraints were explicitly handled, the solutions

found were ensured to satisfy all the design

requirements.

50.0

55.0

60.0

65.0

70.0

75.0

80.0

85.0

70.0 75.0 80.0 85.0 90.0 95.0

Yield (%)

Vxyz (cm/s)

G1

G4

G8

G12

G16

Fig.7. Shift of the Pareto front in different

generations (7SGA II, 7=20).

50.0

55.0

60.0

65.0

70.0

75.0

80.0

85.0

75.0 80.0 85.0 90.0 95.0

Yield (%)

Vxyz (cm/s)

N12 G20

N20 G16

N40 G8

Fig.8. Shift of the Pareto front in different generations and

using different population sizes.

The shift of the Pareto front found in G = 1, 4, 8, 12 and

16 is illustrated in Figure 7 and it can be seen that the

solutions improved as G progressed while better

solutions were being discovered. At the completion of

one run, 6 Pareto optimal solutions were identified and

are listed in Table 2. The solutions found had a good

spread that gave trade-off information between the

competing velocity and yield objectives as shown in

Figure 7. Based on the proposed optimization

framework, optimal gating and riser system designs

could be successfully obtained even with the absence of

auxiliary knowledge or analytical information in the

problem formulation. Similarly, an initial population

made up of all bad designs did not impede the ability of

the optimization algorithm in finding better feasible

solutions.

Table 2 Pareto optimal set obtained using 7SGAII with

7=20 and G=16


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VARIABLES OBJECTIVES

ar br gr rh Vx (cm/s)

Vy (cm/s)

Vz (cm/s)

Velocity (cm/s)

Porosity (%)

Yield (%)

68.7

10.0

23.1

32.9

46.6 41.9 42.3 75.6 0.0 87.1

58.4

10.0

23.1

63.9

35.2 42.6 43.1 70.1 0.0 86.1

62.3

13.9

26.8

63.9

33.6 39.8 39.5 65.6 0.0 84.4

59.7

12.9

28.3

61.3

39.0 41.1 30.9 64.3 0.0 82.8

54.5

13.9

32.8

63.9

39.0 37.7 20.0 57.8 0.0 79.1

70.0

12.9

34.3

61.3

39.2 35.2 22.4 57.2 0.0 76.8

70.0

12.9

34.3

61.3

39.2 35.2 22.4 57.2 0.0 76.8

4.1 Effects of Population Sizing

Various suggestions for population sizing can be found

in literature. According to the De Jong’s standard

settings (De Jong, 1975), the suggested population size

is 50. Latter research by Schaffer (Schaffer, 1989) and

Grefenstette (Grefenstette, 1986) showed that using a

moderate size of about 20-30 could solve the problem

faster without excessive computational costs. A smaller

population size of 4 to 10 was also recommended in

some recent research on micro-GA to accelerate fitness

convergence (Zhang, 2004; Ye, 2000). For this study,

three different population sizes of 12, 20 and 40 were

employed to compare their ability to find the Pareto front

as shown in Figure 8.

Based on the results, the population size of 20 was found

to be most appropriate for finding the optimal solutions

efficiently. The moderately-sized population allowed a

sufficient pool of genes to exist in the population for

effective crossover operations compared to an

inadequate size of 12. It also had a lower computational

cost compared to the population size of 40.

4.2 Effects of Requirement Change

The MOEA technique gave flexibility in decisions

making since trade-off information was given by the

Pareto optimal set. With a new tightened yield

requirement, Y > 85% imposed, another corresponding

solution that satisfies this constraint could be easily

picked from the existing set in Figure 8. This is

advantageous from the practical standpoint because

objective priorities often change according to current

conditions in real world problems.

4.3. Optimized Design

Based on the results obtained using the proposed

optimization framework, comparisons of some gating

and riser designs are shown in Figures 9 and 10. The

influence of the runner and ingates sizes on the liquid

metal velocity in the casting is shown in Figure 9. It can

be observed that using a large gr for the gating system

increased the gating ratio and allowed the liquid metal to

be slowed down before entering the casting cavity.

The effect of riser size on the shrinkage porosity formed

in the casting is shown in Figure 10. It can be seen that

using top risers with a low height rh and wide radius ar

could not eliminate the porosity in the casting. Similar

infeasible results were also obtained when a tall riser

with a small radius was employed. Porosity can only be

eliminated from the casting with an optimally sized riser.

The identification of the proper gating and riser sizes

was autonomously examined by the optimization

algorithm and did not require any human intervention.

Fig. 9. Filling velocity results in the casting with

gating size of (a) gr = 20 mm, (b) gr = 25 mm, (c) gr =

30 mm.

Fig. 10. Shrinkage porosity result of the casting with

(a) short, wide riser, (b) tall, thin riser, (c) medium

sized riser, and (d) an optimal riser.

(a)

(b)

(c)

(d)

(a) (b)

(c) (d)


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5. CONCLUSIONS

In this study, the proposed optimization framework

managed to autonomously obtain 6 optimal design

choices after 320 design evaluations in a relatively short

period of time. Using the trial-and-error approach to

manually design, iterate and redesign could possibly take

up much more man hours and yet may not necessarily

obtain optimal designs. A population size of 20 was

found to be most efficient for finding the optimal

solutions. Regardless of the infeasible initial population,

optimal gating and riser system designs were

successfully obtained even with the absence of analytical

information. The main advantage was the ability to

obtain trade-off information between the competing

yield and quality objectives. This provided an insight

into the system characteristics and allowed flexibility in

decision making.

6. ACKNOWLEDGEMENTS

The authors would like to acknowledge the support of

NSERC and the University of Windsor for this work.

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casting design through multi-objective optimization

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