A Multiphysics Computational Framework and its

PHYSICA : A Multiphysics Computational Framework and its Application to Casting Simulations C. Bailey, G. Taylor, S. M. Bounds, G Moran, and M. Cross Centre for Numerical Modelling and Process Analysis, University of Greenwich, London SE18 6PF, UK e-mail : C. [email protected]

ABSTRACT

Metals casting is a process governed by the interaction of a range of physical phenomena. Most Computational models of this process address only what are conventionally regarded as the primary phenomena - heat conduction and solidi cation. However, to predict phenomena such as porosity requires modelling the interaction of the

uid ow, heat transfer, solidi cation and the development of stress-deformation in the solidi ed part of the casting. This paper will describe a modelling framework called PHYSICA which has the capability to simulate such multiphysics based phenomena.

1 INTRODUCTION When a new casting design (i.e part for a jet engine) is to be manufactured, the general practice of the foundry is to undertake numerous trials and tests to nd the optimum design procedure. This practice is obviously very expensive in man hours, materials and energy costs; in addition, it causes long lead times before production may commence. Until recently, the use of computational mechanics based models to aid the foundry engineer has been negligible (Cross, 1995). This is primarily due to the complex nature of the process and the interacting physics present. During the casting process (Campbell, 1991) a cold mould is lled with molten metal, which cools and

solidi es. As the cooling progresses thermal gradients build up in the metal. In the liquid regions this promotes thermal convection that will redistribute heat around the casting. Before the onset of solidi cation the casting is in full contact with its surrounding mould, but as the casting solidi es it begins to pull away from the mould, developing a gap. This gap will restrict the ow of heat from the metal to the mould, hence in uence the manner in which the cast component solidi es and developes. To simulate such a process the computational modeller must address the following;

 Fluid ow analysis of Mould lling.  Thermal convection once the mould has

lled.  Thermal analysis of solidifcation and evolution of latent heat.  Deformation and residual stress predictions of solidi ed component.  Porosity formation

Computational mechanics based phenomena and resulting discretisation methods have historically followed two distint routes. Finite Element methods have generally been adopted for discretising the solid mechanics equations and Finite Volume methods have been used for the conservation equations governing uid ow. For multiphysics based phenomena, like the casting process,

a uni ed computational framework is required which can handle the interactions between numerous physics within the same code. Lately nite volume methods have been used to analyse solid mechanics problems on unstructured meshes (Onate 1994, Bailey 1995, Taylor 1995). Such methods have proved to be as accurate as classical nite elements in analysing both linear and non-linear materials.

2 A MULTIPHYSICS ENVIRONMENT A common element in all computational mechanics codes is the mesh representing the domain. Various discretisation methods can be used to discretise the equations governing the physics of interest across this mesh. For Finite Volume procedures the evaluation of uxes across cell/element faces, volume sources and coecients of the linear solvers are generic, being essentially based on mesh geometry and materials properties within a cell. PHYSICA is a software framework which has generic tools that concentrate on objects (i.e mesh faces, etc) and their relations in evaluating all the relevant terms (diusion, convection, source, etc) at a generic level. Figure 1 shows the design principles inherent within the PHYSICA framework and Figure 2 shows the var-

PHYSICA: Levels of abstraction. Model.

Flow Module.

Stress Module.

Heat Module.

Control. General Equation.

Algorithm.

Solution Control.

System Matrix.

Matrix Solvers.

Initialisation.

Properties.

Sources.

Convections.

Diffusions.

Utility.

File I/O Unit. Memory Manager.

User Access.

Interpolation.

Geometry Tools.

Matrix Tools.

Database Manager.

Vector Tools.

Figure 2: PHYSICA - Level of code abstraction. new higher order convection scheme may be inserted by the user at the 'Algorithm' level and the modular structure of the code will allow compatibility with all other routines. Such a computational framework will aid in reducing model development times by many orders of magnitude. At present PHYSICA has the following capabilities;

 Data structures for tetrahedral, wedge, hexahedral and full polyhedral elements.

 SIMPLE based solution for incompressible Navier Stokes Equations.

 Free surface ow algorithm.  Range of turbulence models

Solution Space.

Model space.

Element.

PHYSICA: Algorithms are built on methods for "Objects". Unstructured Mesh. Node.

Face.

Traditional approach is to use routines that may do many tasks on many "Objects".

Figure 1: PHYSICA - Design principles. ious levels of abstraction. It is expected that the majority of modellers would only use the top three levels. So, for example a

 Enthalpy based solidi cation/melting procedures.

 Elastoviscoplastic solid mechanics algorithm

 Solvers which include PCCG, SOR and Jacobi

 Hooks to pre and post-processing packages

PHYSICA is now being usd to simulate numerous multiphysics based phenomena for which the casting process is one.

3 GOVERNING EQUATIONS

deformation of metals the elastic strains are given by

The equations that represent the governing physics of heat transfer, uid and solid mechanics in three-dimensional space are:-

3.1 Computational Fluid Dynamics (CFD)

Momentum @ (Ui) + r(V U ) = r(rU ) ? @p + S i i @t @ni i (1) where  and (T ) are the viscosity and density. The source term contains the Darcy and bouyancy terms due to temperature changes and solidi cation. Continuity

@ + r(V ) = 0:0 @t

(2)

Energy @ (h) + r(V h) = r(krT ) + S

@t

h

(3)

where h, k and T are the enthalpy, thermal conductivity and temperature. The evolution of latent heat is included in the source term where; S = @ (Lfl) ? r:(LV f ) (4) h

l

@t

where L and fl are the latent heat and liquid fraction respectively.

3.2 Computational Solid Mechanics (CSM)

The equilibrium equations in incremental form are
ij;j = 0

i; j = x; y; z

(5)


e =
 ?
t ?
vp

(7)

where
,
t and
vp are the total, thermal and viscoplastic strain increments respectively. The viscoplastic strains are given by the Perzyna (Perzyna 1966) constitutive relationship which is given by; !1 dvp = eq ? 1 N 3 s (8) dt  2 y

eq

where eq , y and s are the von-mises stress, yield stress, uidity and deviatoric stress respectively. The total strain increment for nite strains is;
 = [L]
d

(9)

where [L] is the dierential matrix and
d are the displacement increments.

4 DISCRETISATION All the above equations are discretised on unstructured meshes using nite volume procedures. For the uid ow equations and energy equation the control volumes are cell centred (Chow, 1994) while for the stress equations the control volumes are set up using a vertex centred approach (Taylor, 1997). It should be noted that the same mesh is used to represent the domain in question and set up the relevant control volumes. The dependent variables resulting from the discretisation are uid velocities, enthalpy and solid displacement increments. The resulting system of discretised equations are of the form ;

where the stress increments are related to the strain increments via Hooks law;

[A] = b


 = [D]


where [A] is the system matrix and b contains source terms for the dependent variables  which are velocities, enthalpy and displacement increments.

(6)

where [D] is the materials matrix and
 are the elastic strains increments. For the

(10)

5.1 Boundary Conditions

It is important when predicting how a cast shape develops to be able to model the thermal changes correctly. To do this account must be taken of the gap formation at the mould/cast interface. For this analysis coincident nodes have been used to model the interface. This enables the cast component to move freely away from the mould. The boundary condition at this interface is given by

@T = h (T ? T ) (11) @n eff c m where Tc , Tm and heff are the cast temper-

ature, mould temperature and eective heat transfer coecient which is given by

heff = dkgap gap

(12)

where kgap and dgap are thermal conductivity of the gap medium and the gap distance.

The following results will show two test cases that illustrate the complexity involved in modelling the multiphysics phenomena present in the the casting process. The simulations will both assume an initial ll and start with a uniform temperature in the molten metal and mould. The rst test case compares numerical predictions with experimental results (Schmidt and Svensson, 1994) for the die casting of a hollow aluminium cylinder. Figure 3 shows the experimental design where the top and bottom of the component are insulated to ensure a one dimensional heat loss to the mould material. Thermocouples A 25 mm

INSULATOR 50 mm

F

MOULD

Note, in step 7, the geometrical quantities are recalculated due to the changes in displacement from the computational solid mechanics analysis, and the equations in steps 1 and 2 rediscretised over the updated mesh.

6 RESULTS

CASTING

1. Solve CFD equations 2. Solve Energy equation 3. Evaluate other variable quantities, eg. physical properties, etc. 4. Repeat steps 1-3 until convergence 5. Solve CSM equations 6. Update total stress variables. 7. Recalculate geometrical quantities 8. Repeat steps 1-7 for time-step advancement

CORE

The complete solution procedure for solving the coupled uid and solid mechanics equations in metal casting is:-

During this analysis the mould is assumed to be rigid and hence the mechanical contact analysis although time consuming is reasonably straight forward. A node is labelled as being in contact when it penetrates the mould and the displacement calculations are undertaken with this constraint in place. If the stress condition at a contacting node becomes tensile then the constraints at this node are released in further calculations.

30mm

5 SOLUTION PROCEDURE

INSULATOR

100 mm

25 mm

A Thermocouples F Displacement transducers

Figure 3: Experimental design of cylinder casting (side View). where inserted into the mould and a displacement transducer was attached to the cast component to monitor the gap formation as the cast pulls away from the mould. An axisymmetric analysis was undertaken

in PHYSICA to simulate the solidi cation and resulting solid deformation of this component. The cylindrical mould and core are made of steel and the casting used an aluminium alloy. The material properties used in this analysis are given in tables 1 and 2 (see Appendix). The heat transfer heff was assumed to vary linearly between 400W/m2K to 20W/m2K corresponding to gap distances of 0.0mm to 0.5mm. Figure 4 shows the nature in which the cast component deforms over time.

800

Experimental Numerical (Thermo-mechanical) Numerical (Thermal) 600 Temp. (deg. C)

400

200 0

200

400

600

800

Time (s)

Figure 5: Temperature pro les in casting and mould.

Displacement of cylindrical die casting over time. 400 secs.

500 secs.

Y X

Y Z

X

Z

1 600 secs.

2 900 secs.

0.5 Y X

Y Z

X

3

Z

4

Figure 4: Gap formation over time (x10).

0.4 Experimental Numerical Gap 0.3 (mm)

0.2

Figure 5 show comparisons between between experimental data for temperatures in both mould and casting with numerical predictions. It can be seen that by neglecting the eects of gap formation at the cast/mould interface results in cooling of the casting occurring far too quickly. Figure 6 shows the resulting gap formation at the vertical cast/mould interface over time where the insulation is allowing uniform contraction across the casting. It can be seen that the model shows good comparison with experimental data. The next test case shows the resulting stress and porosity distributions in a three dimensional casting. This simulation used the PHYSICA code where uid ow, solidi cation and deformation are solved. Figure 7 illustrates the amount of thermal convection in the initial stages of cooling. It is interesting to note the large ow rates from the feeder (i.e the cylinder) to other parts of the cast-

0.1

0.0 0

200

400

600

800

Time (s)

Figure 6: Gap formation at cast/mould interface.

Figure 7: Thermal convection in 3D casting.

ing. This is what is expected as insulating material has been placed around the feeder to ensure it remains in a liquid state, so that it can feed areas of the casting which are prone to porosity due to solidi cation shrinkage. Figure 8 shows the resulting vonmises stress distribution across this casting where the large magnitudes of stress are located at the centre of the plate between the step and the feeder. This is expected as the

served, where thermal convection in the cast metal allows redistribution of heat from hotter areas to cooler areas near the interface, thus resulting in faster cooling. Figure 11 800

With convection Without convection 600 Temp. (deg. C) 400

200

.25E9 .212E9 .175E9 .137E9 .1E9 .625E8 .25E8

Figure 8: Final residual stress pro le in casting. current design restricts contraction in this area due to the location of the feeder and the step. Figure 9 shows the nal magnitude of the gap between the casting and its surrounding mould along the mid section. These gaps have aected the manner in which the cast component cooled. The

0 0

500

1000

1500

Time (s)

Figure 10: Cooling curves in mould and cast component. shows the likelihood of porosity in speci c regions for this casting using a Niyama criteria (Niyama 1982). These trends seem feasible as the simulations predicted that the feed path from the feeder to the step region of the casting is cut o before the step region completely solidi es.

.345E9 .276E9 .207E9 .138E9 .691E8

Figure 9: Gaps at cast/mould interface. eect of convection in the cast component on the nal cooling pro le was investigated. Figure 10 shows the dierences in cooling curves for analysis with and without convection (note top curves represent cast and lower ones the mould) . It can be seen that dierences of up to 60C can be ob-

Figure 11: Niyama.

Porosity predictions using

7 CONCLUSIONS Increasingly, there is a requirement in engineering processes to model multiphysics phenomena. The rational behind the PHYSICA framework is to supply modellers

with a tool which allows ease in developing computational mechanics base models where interactions between the participating physics can be accomplished in a robust and ecient manner. The Casting process is a classical example of such multiphysics phenomena which involves coupling between uid ow, heat transfer and solid mechanics at the macroscopic level. The paper has shown PHYSICA predictions for two casting simulations, where comparisons between experiment and model data are encouraging. Development of PHYSICA is continuing, where future versions will include the physics of magnetic elds, radiation, combustion etc. Also better models for porosity formation are being developed which will help identify both macro and micro porosity.

ACKNOWLEDGEMENTS

[6]

[7]

[8]

[9]

The nancial assistance of the EPSRC, British Aerospace and Normalair-Garrett Ltd is acknowledged in this work.

[10]

References

[11]

[1] C. Bailey, M. Cross, A nite volume format for structural mechanics. Int. Jnl Num Methods Engg. 10, 17571776, (1995). [2] C. Bailey, P. Chow, M. Cross, Y. Fryer and K. Pericleous, \ Multiphysics modelling of the metals casting process.", Proc. R. Soc. Lond. A, 452, pp 459486, (1996). [3] J. Campbell, Castings, (ButterworthHeinemann, Oxford, (1991). [4] P. Chow, M. Cross, An enthalpy control volume unstructured mesh (CVUM) algorithm for solidi cation by conduction only. Int. Jnl Num Methods Engg. 35 (1992), 1849-1870. [5] P. Chow, Control Volume Unstructured Mesh Procedure for Convection-

[12]

[13]

Diusion Solidi cation Processes. PhD thesis, University of Greenwich, (1993). M. Cross and J. Campbell (eds),\Modeling of Casting, Welding and Advanced Solidi cation Processes VII", published by the Materials, Metals and Minerals Society, Warrendale, PA, (1995) E. Onate, M. Cervera and O. C. Zienkiewicz, \A nite volume format for stuctural mechanics", Int. Journal for Num. Methods in Engg, 37, 181-201 (1994). E. Niyama, T. Uchida, M. Morikawa, S. Saito, \Method of shrinkage prediction and its application to steel castings practice". AFS casting metals research journal. (1982), 52{63. PHYSICA, University of Greenwich, London, UK. Email [email protected] P. Perzyna, \Fundamental problems in viscoplasticity", Advan. Appl Mech, 9, 243-377, (1966) G. Taylor, C. Bailey and M.Cross, \Solution of elastic/viscoplastic constitutive equations : A nite volume approach" Appl. Math Modelling, 19, 746-760, (1995) G. Taylor, A Vertex Based Discretisation Scheme Applied to Material NonLinearity within a Multiphysics Finite Volume Framework, Ph.D Thesis, University of Greenwich (1997). P. Schmidt, I Svensson, Heat transfer and air gap formation in permanent mould casting of aluminium alloys, Technical report TRITA-MAC0541, Materials research centre, Royal Institute of Technology, Stockholm, 1994.

APPENDIX TL TS h k  c   E

Liquidus temperature Solidus temperature Latent heat of fusion Thermal conductivity Density Speci c heat capacity Coecient of thermal expansion Poisson's ratio Young's modulus

Y Yield stress

618:8C 566:4C 440 kJ/kg 150 W/(mK) 2,710 kg/m3 1160 J/(kgK) 5  10?5 /K 0.33 60; 000MPa 20C 34; 000MPa 450C 1  10?2 MPa 566:4C 500MPa 20C ? 4 1  10 MPa 566:4C

Table 1: Material properties of the aluminium casting alloy.

k Thermal conductivity 150 W/(mK)  Density 2,710 kg/m3 c Speci c heat capacity 1160 J/(kgK) Table 2: Material properties for the steel mould material.