casting shape optimisation via process modelling

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Ž . Materials and Design 21 2000 381]386 Casting shape optimisation via process modelling R.W. Lewis U , M.T. Manzari, R.S. Ransing, D.T. Gethin Department of Mechanical Engineering, Uni ¤ ersity of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK Abstract The paper describes the application of numerical optimisation case studies to two practical cast components. The first study highlights the potential benefit of thermal controls to achieve an optimised cast part, whereas, the second focuses on shape optimisation. In comparison with practice, it was noted that the unconstrained optimisation for thermal control identified the improvement path correctly, but the solution could not be applied practically. The shape optimisation study also identified the correct improvement path and the result was supported by trials that had been completed previously in the foundry. Q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Finite element analysis; Optimisation; Thermal; Shape 1. Background and relevant literature review Casting is one of the oldest manufacturing processes. It is used widely in many industry sectors and employs Ž processes that are gravity fed e.g. die, sand and invest- . ment and pressure fed methods that mainly use per- manent dies. In all cases, the purpose of the process is Ž . to produce the part or a cluster of parts . The design of the process must address the supply system for the molten metal, the feeding system as the part solidifies and shrinks and the thermal control to ensure the integrity of the cast component by the elimination of all forms of shrinkage porosity from within the casting. The volume of such a filling and feeding system needs to be minimised to ensure process yield and the sectional area of the interface between the system and the cast component needs to be minimised to reduce any fettling work. Simulation of the casting process is now becoming mature and a number of systems are available specifi- w x cally for this purpose 1 ] 3 . In each case, following the U Corresponding author. analysis, there is a requirement to examine the results and to make a judgement on whether the system design is satisfactory, or whether it needs to be improved in any way to ensure part integrity. Thus, the design of the process is iterative and where the time frame available to do this is finite, there is no certainty that the design will be optimised. To date, optimisation of manufacturing processes by numerical schemes has re- ceived limited attention and has been applied princi- pally in structural optimisation where objectives may w x focus on weight or stiffness 4,5 . The procedure that has been established in these studies is to identify a design function that captures the variables and to com- pute function response with respect to perturbations on the design variables. By performing this sensitivity analysis, it is possible to home in on an optimised Ž . solution that minimises or maximises the design func- tion. One of the first investigations in the field of wx casting is reported in 6 . This work demonstrated the effectiveness of shape optimisation techniques to size the feeding system. It was found to require a feasible feeding system as a starting point, it was not capable of optimising from a non-feasible design. The method was also limited since the calculation used a fixed meshing 0261-3069r00r$ - see front matter Q 2000 Elsevier Science Ltd. All rights reserved. Ž . PII: S 0 2 6 1 - 3 0 6 9 99 00079-5

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Page 1: Casting shape optimisation via process modelling

Ž .Materials and Design 21 2000 381]386

Casting shape optimisation via process modelling

R.W. LewisU, M.T. Manzari, R.S. Ransing, D.T. Gethin

Department of Mechanical Engineering, Uni ersity of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK

Abstract

The paper describes the application of numerical optimisation case studies to two practical cast components. The first studyhighlights the potential benefit of thermal controls to achieve an optimised cast part, whereas, the second focuses on shapeoptimisation. In comparison with practice, it was noted that the unconstrained optimisation for thermal control identified theimprovement path correctly, but the solution could not be applied practically. The shape optimisation study also identified thecorrect improvement path and the result was supported by trials that had been completed previously in the foundry. Q 2000Elsevier Science Ltd. All rights reserved.

Keywords: Finite element analysis; Optimisation; Thermal; Shape

1. Background and relevant literature review

Casting is one of the oldest manufacturing processes.It is used widely in many industry sectors and employs

Žprocesses that are gravity fed e.g. die, sand and invest-.ment and pressure fed methods that mainly use per-

manent dies. In all cases, the purpose of the process isŽ .to produce the part or a cluster of parts . The design

of the process must address the supply system for themolten metal, the feeding system as the part solidifiesand shrinks and the thermal control to ensure theintegrity of the cast component by the elimination ofall forms of shrinkage porosity from within the casting.The volume of such a filling and feeding system needsto be minimised to ensure process yield and thesectional area of the interface between the system andthe cast component needs to be minimised to reduceany fettling work.

Simulation of the casting process is now becomingmature and a number of systems are available specifi-

w xcally for this purpose 1]3 . In each case, following the

U Corresponding author.

analysis, there is a requirement to examine the resultsand to make a judgement on whether the system designis satisfactory, or whether it needs to be improved inany way to ensure part integrity. Thus, the design ofthe process is iterative and where the time frameavailable to do this is finite, there is no certainty thatthe design will be optimised. To date, optimisation ofmanufacturing processes by numerical schemes has re-ceived limited attention and has been applied princi-pally in structural optimisation where objectives may

w xfocus on weight or stiffness 4,5 . The procedure thathas been established in these studies is to identify adesign function that captures the variables and to com-pute function response with respect to perturbations onthe design variables. By performing this sensitivityanalysis, it is possible to home in on an optimised

Ž .solution that minimises or maximises the design func-tion. One of the first investigations in the field of

w xcasting is reported in 6 . This work demonstrated theeffectiveness of shape optimisation techniques to sizethe feeding system. It was found to require a feasiblefeeding system as a starting point, it was not capable ofoptimising from a non-feasible design. The method wasalso limited since the calculation used a fixed meshing

0261-3069r00r$ - see front matter Q 2000 Elsevier Science Ltd. All rights reserved.Ž .PII: S 0 2 6 1 - 3 0 6 9 9 9 0 0 0 7 9 - 5

Page 2: Casting shape optimisation via process modelling

( )R.W. Lewis et al. r Materials and Design 21 2000 381]386382

system that was distorted in response to design changes.This also limited the range of geometric change beforethe finite element meshes became distorted excessively.

As pointed out above, the design variables that areavailable in the thermal analysis of the casting processinclude thermal control and the position and size of thefilling and feeding systems. Thermal control techniquesdepend on the casting method, for example in pressuredie casting it is achieved by positioning the coolantchannels in the die, in gravity die casting, die coats areused extensively to control heat transfer in the early

w xstage of the solidification process 7 . This suggests thatdie coat thickness may be an appropriate design vari-able that may be used to optimise the design of thecasting process. However, die coats are not effectiveafter the formation of any air gap between the die andcast component. The gravity die casting and sand cast-ing processes make use of filling and feeding systems tocontrol heat flow, and, hence, the solidification patternin the casting. This variable leads to the requirementfor shape optimisation with regard to position and sizeof the filling and feeding system. Work that has beencompleted in this area will also be reported within thisstudy.

2. Thermal models and optimisation methodology

The numerical analysis of thermal behaviour in thew xcasting process is well-documented 8,9 . Heat transfer

within the casting and diermould system is governedby conduction, whereas, heat is removed from thesystem, principally by convection from exposed exteriorsurfaces. The transient heat conduction equation iswritten as

d H dT Ž .s=?k =T 1dT d t

For alloy systems, in simple models, phase change isusually represented using enthalpy. More rigorousmodels are available and these make use of the phase

w xdiagram for the alloy system 10 . For the enthalpymodel

d H Ž .Cs 2dT

In the case of gravity processes, the application of a diecoat, the evolution of an air gap or the development ofhigh contact stresses has an impact on the heat transfer

w xoccurring at the interface 7 . The most convenient wayof accounting for this is by means of an interface

w xelement 11 in which the interfacial heat transfer isdetermined by means of a coefficient.

Ž . Ž .qsh T yT 3i 1 2

Numerical optimisation requires the definition of adesign function. In the case of optimisation based onheat transfer, this only needs to include variables thatare affected by heat transfer, such as temperature orfreezing time. In the case of shape optimisation the

Ž .function needs to include dimensional or mass infor-mation as well as thermal contributions. It is usual toexpress this as an ‘objective function’ that needs to beminimised or maximised. Also the process design vari-ables need to be maintained within practical limits andthis imposes constraints on the optimisation process.Thus, the optimisation process can be expressed math-

Ž .ematically as minimising or maximising the objectiveŽ .function F X , subject to the equality constraint

Ž .H X s0 is1, Li

and the inequality constraint

Ž .G X F0 js1,Mj

In these equations, X represents the design variables,such as the heat transfer coefficient at the die wall orthe dimensions of the feeding and filling system. Thevalues L and M represent the number of constrainedand unconstrained design variables. The optimisationprocess was carried out by means of the commercial

w xcode DOT 12,13 . Following a sensitivity analysis, thisevaluates the objective function and automatically per-

Žturbs the design variables so as to minimise or max-

Fig. 1. Flow diagram for the thermal and shape optimisation proce-dure.

Page 3: Casting shape optimisation via process modelling

( )R.W. Lewis et al. r Materials and Design 21 2000 381]386 383

Fig. 2. A sectional view through the wheel casting.

.imise the design function. In this work, the BFGSw xalgorithm 13 has been used due to its efficient conver-

gence rate. The solution flow diagram is shownin Fig. 1.

The form of the objective function depends on thephysical basis on which optimisation is being carriedout. Both models impose a directional solidificationconstraint by imposing a positive gradient between thecast component and the fillingrfeeding system. Thefirst case study will focus on optimisation by the choiceof thermal control as provided by die coats in thegravity die casting process. In this case the position andsize of filling and feeding system remains fixed. In thesecond case study, the size of the fillingrfeeding systemwill be investigated as well, thus, representing multi-objective optimisation.

2.1. Case study 1

The first case study shown in Fig. 2 represents analuminium wheel casting that is produced using a grav-ity die process.

The figure shows one main feeding system. Since thecasting is filled from the bottom, this system plays nopart in feeding the casting during its solidification. The

figure also shows the path along which solidificationwill be enforced to ensure that the part will always besupplied by the feeding system. In this case, the objec-tive function is given by

Sny1

Ž . Ž .Fsn pmax t y t ,0 4Ý f iq1 f iis1

where nsnumber of defined paths, S snumber ofn

points in the nth user defined path, pspenalty termand T s freezing time at the ith design point.fi

The freezing time is interpolated via the shape func-tion:

v t sFreezing time at ith node;fi -nodev tsRun time;v DsTime step;v T t snodal temperature at time t; andiy nodev T ssolidus temperature.sol

Further details concerning this function are given inw x14 . Thermophysical and initial conditions need to besupplied to perform the finite element analysis on thecasting. The thermophysical properties are summarised

Table 1Summary of thermophysical properties

Ž . Ž .Aluminium LM25 Die Steel H13

Ž . Ž . Ž . Ž .Temperature 8C Enthalpy Jrkg 8C Temperature 8C Enthalpy Jrkg 8C

0 0 0 05 5550 6.01=10 550 1.42=105 5615 10.7=10 615 2.49=105 5800 12.7=10 800 5.70=10

Conductivity 186 Wrm 8C Conductivity 34 Wrm 8C3 3Density 2790 kgrm Density 7721 kgrm

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( )R.W. Lewis et al. r Materials and Design 21 2000 381]386384

Fig. 3. Initial variation in interface heat transfer coefficient.

in Table 1. The thermal analysis assumed an uniforminitial aluminium temperature of 6258C and a uniformdie temperature of 1508C. Heat loss from the exposedsurface of the die was assumed to take place by convec-tion and a coefficient of 75 Wrm2 8C together with areference temperature of 258C was used. These valuesrepresent typical foundry practice.

The final requirement for the casting shown in Fig. 2is the identification of interfaces between the castingand the die. These need to be identified as differentsegments to allow the optimisation process to assigndifferent interface heat transfer values to each seg-ment. In this case study, this is the only method ofachieving directional solidification. To start the calcula-tion the values defined by the characteristic curve inFig. 3 was used at all interfaces.

An initial set of calculations was carried out with theheat transfer coefficient at the interface defined asshown in Fig. 3. The result from this set of calculationsis displayed in the form of solidification contours inFig. 4. This figure shows clearly that the last point tosolidify is in the centre of the wheel itself. This clearlyleads to a defect in the centre section of the casting.An examination of the section shown in Fig. 2 alsoconfirms the presence of a porous region in the centresection, thus, confirming the accuracy of the model andanalysis.

Calculation was then carried out using the optimisa-tion procedure described above. The solidification con-

Fig. 4. Baseline analysis with fixed interface heat transfer coeffi-cients.

Fig. 5. Solidification contours through the casting with optimisedinterface heat transfer coefficients.

tours through the casting are now shown in Fig. 5.Clearly, as required, the last location to solidify residesin the feeding system confirming the effectiveness ofthe optimisation procedure.

The variation in heat transfer coefficients over thefour regions defined in Fig. 2 is shown in Fig. 6. In thiscase, the values can not be achieved practically sincethey represent large values that would be associatedwith very high contact pressures, i.e. perfect contactand no air gap.

2.2. Case study 2

The second case study is shown in Fig. 7. Thiscomprises a simple aluminium hub that is cast bymeans of the gravity die process and uses LM25 as thealloy material. In this case the aim is to optimise thesize of the feeding system that is positioned on the topof the casting as shown in Fig. 7.

Since this part is also bottom filled through a smallsection, again it will be assumed that the filling systemdoes not contribute to this phase of the process, eventhough the filling cup represents a large volume ofmetal in the die. Thus, Fig. 7 shows only the cast partand top feeder system. The potential shape changesthat will be available to optimise the shape of the

Fig. 6. Variation in heat transfer coefficient under optimised condi-tions.

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( )R.W. Lewis et al. r Materials and Design 21 2000 381]386 385

Fig. 7. Schematic view of the hub casting and feeding system.

feeding system are also shown in the figure. This allowschanges in feeder size as well as the sectional area thatlinks with the cast part itself. In this case, the objectivefunction must include both thermal and volume com-

w xponents. It is now given by 15 .

Ž . � 4F X s T yT qC qWA B

Volume of Feeder Ž .= 5½ 5Max. Volume of Feeder

where, T , T s temperature at points A and B at 500A Bs, Cs forces the thermal gradient, here Cs20, Wsweighing of the shape optimisation, here Ws1.

This choice of function ensures the directional solidi-fication of the part by means of the thermal contribu-

Fig. 8. Initial casting design scheme } cooling times3 s.

Fig. 9. Final optimised casting design scheme } cooling times3 s.

tion and the effect of the feeder size and shape bymeans of the mass term.

The calculations were carried out for an ambienttemperature of 258C and a convective heat transfercoefficient of 75 Wrm2 8C applied over all externalsurfaces. As shown in Fig. 7, several geometric variantswere investigated, however, only a selected result willbe represented as shown in Figs. 8 and 9 that depict theinitial and final optimised design schemes as finiteelement meshes and the zones of liquid, mushy andsolid materials. Clearly the initial result is a feasiblecasting since the last zone to solidify lies in the feedersystem. However, the feeder is large in comparisonwith the casting and represents a significant energywaste since metal is recycled in its molten form.

The final design shows a much reduced size in thefeeder system which still gives an acceptable part withonly a mushy zone left in the feeder at 3 s highlightingthe reduction in cycle time for the optimum design. Itwas also interesting to note that when other con-strained geometric disturbances were investigated, theoptimised scheme always gave a result that was similarto that shown in Fig. 9 where the constraints allowedsuitable freedom, thus, pointing to the identification ofa true optimum. From a practical viewpoint, trials inthe foundry confirmed the benefit of the feeder sizereduction as highlighted in Figs. 8 and 9.

3. Conclusions

The application of numerical optimisation tech-niques to the casting process has been presented.Optimisation based on thermal control and shape hasbeen presented highlighting the benefit that may bederived. The unconstrained optimisation based on ther-

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( )R.W. Lewis et al. r Materials and Design 21 2000 381]386386

mal control identified the requirements to achieve asound casting in the case considered. However, thesolution was not practical in the foundry since it im-plied the application of an excessive number of diecoats. The shape-based optimisation identified cor-rectly the benefits to be derived by feeder size reduc-tion and the result reflects successful application offoundry practice.

References

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w x2 http:rrphysica.gre.ac.uk.w x3 http:rrwww.swan.ac.ukrmatengrmavis.htm.w x4 Bletzinger K-U, Reitinger R, Kimmich S, Ramm E. Shape

optimisation with program carat; software systems for struc-tural optimisation. Int Ser Numer Math 1992;110:97]124.

w x5 Hinton E, Rao NVR, Seinz J. Finite element structural shapeand thickness optimisation of axisymmetric shells. Eng Comput1992;9:499]527.

w x6 Morthland TE, Byrne BE, Tortorelli DA, Danzig JA. Optimalriser design for metal castings. Metall Mater Trans B

Ž .1995;26B 4 :871]885.

w x7 Bell F, Gethin DT, Anderson JT. A foundry based experimen-tal investigation into the effect of die coats in the gravitycasting process. Cast Met 1995;8:51]56.

w x8 Lewis RW, Roberts PM. Finite element simulation of solidifi-cation problems. Appl Sci Res 1987;44:61]92.

w x9 Shyy W, Udaykumar HS, Ouyang H. Progress in computationalsolidification modelling. Proceedings Solidification Processing97. Sheffield, July 1997:126]129.

w x10 McAdie RL, Cross JT, Lewis RW, Gethin DT. A finite elemententhalpy technique for solving coupled non-linear heat conduc-tionrmass diffusion problems with phase change. Int J NumerMethods Heat Fluid Flow 1995;5:907]921.

w x11 Samonds M, Lewis RW, Morgan K, Symberlist R. Finite ele-ment modelling of the mould]metal interface in casting simu-lation with coincident nodes or thin elements. Computat TechHeat Transfer 1985:1.

w x12 DOT User Manual, 4.20, Vanderplaats R&D inc, 1985.w x13 Vanderplaats GN. Numerical optimisation techniques for engi-

neering design } with applications. McGraw Hill, 1984.w x14 Ransing RS, Lewis RW, Gethin DT. Lewis-ransing correlation

to optimally design the metal]mould heat transfer. Acceptedfor publication in Int J Numer Methods Heat Fluid Flow.

w x15 Manzari MT, Lewis RW, Gethin DT, Cross JT. Towards opti-mum shape in casting design. Proc Opticon ’98. Altair Comput-ing, Newport, CA, 1998.