MODELLING OF EQUIAXED GRAIN GROWTH IN SOLIDIFICATION PROCESS

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1 76/14 Archives of Foundry, Year 004, Volume 4, 14 Archiwum Odlewnictwa, Rok 004, Rocznik 4, Nr 14 PAN Katowice PL ISSN MODELLING OF EQUIAXED GRAIN GROWTH IN SOLIDIFICATION PROCESS O. WODO 1, N. SCZYGIOL. Institute of Computer and Information Sciences Czestochowa University of Technology Dabrowskiego 73, 4-00 Czestochowa SUMMARY In this paper we discuss the object-oriented software designed to implement simulation of the equiaxed grain growth in solidifying casting using cellular automata (CA). We present example of the numerical simulation for aluminium 7% silicon alloy. Finally, we point out possible extensions of the system. Key words: solidification, cellular automata, grain growth 1. INTRODUCTION In the last decade problem of dendritic grain structure prediction has been intensively analysed. Several approaches have been proposed including the phase field method, methods related to the Monte Carlo model, cellular automata and finite element methods as well as hybrids of above [13,6,]. At the same time, to understand a physical background of the process, several experimental studies have been carried out [1]. Recently several approaches based on the cellular automata have been proposed. In [3] Coxe and Reiter created simple fuzzy automata that allow simulating the growth of snowflakes but without physical basis. In [1] Wang et al. proposed model, also based on cellular automata, of microstructural development and primary spacing by both branching and overgrowth mechanism. The most advanced solution has been developed by M. Rappaz and Gandin in [4,8]. In this method the processes of nucleation and growth are simultaneously considered resulting in very accurate model. In this paper we present object-oriented implementation of the abovementioned Rappaz and Gandin approach. This software has been created to extend a functionality 1 mgr inż., wodo@icis.pcz.pl dr hab. inż., prof. P. Cz., sczygiol@icis.pcz.pl

2 559 of the NuscaS system [9,10]. It allows using different algorithms of automaton generation, as well as nucleation and growth descriptions. The rest of this paper is organised as follows. In Section we briefly present cellular automata based approach to the solidification process while in Section 3 we show its object-oriented implementation, next in Section 4 we provide numerical results for aluminium-7% silicon alloy. Finally we close the paper with some concluding remarks.. GRAIN GROWTH SIMULATION USING CELLULAR AUTOMATA The system is based on the approach proposed by Rappaz and Gandin. In this method two phenomena nucleation and growth of columnar and equiaxed grains are considered. However, in the system only growth of equiaxed grain is analysed. The considered domain is divided into regular square cells; each cell has the attribute of state (solid-1, liquid-0) and value of undercooling. At the beginning of simulation each cell has the liquid state and has initial undercooling that is changed within time processing, depending on solidifying condition. Each cell has neighbourhood defined in unique way. However, the configuration of neighbourhood can be various e.g. cubic von Neumann or cubic Moore, but remains the same for the considered domain. For the network of cells defined in this way the simulation proceeds. For each time step two processes are taken into account: the nucleation and the growth phenomena. The homogeneous nucleation process is described by a Gaussian function with 3 three parameters: the density of sites, n [ m ], the standard deviation, T [ K ] and the mean undercooling, Tmax [ K ], respectively. These values can be determined via experimental observation [8]. The total number of grains which nucleate during a time step is given by the equation (1): T T dn N S dt (1) d T T where dn n T n T exp T Tmax Tmax is nucleation law, S is an area of the casting [ m ] and T. is undercooling [ K ]. The cells, which nucleated in a given time step are randomly chosen among the cells with liquid state. Next, the grain is created and the crystallographic direction is randomly chosen among 48 classes of the orientation corresponding to a class width of less than ( 90 /48). For every grain which has nucleated the growth algorithm is applied in the next time steps.

3 560 In the system the rectangles algorithm has been used. Its detailed description can be found in Ref. [4]. Here, we put only a brief description. The data for this algorithm are: undercooling of the cell, the neighbourhood of the cell and the crystallographic direction. Fig. 1 shows a schematic drawing of the algorithm. Here, for every grain the transformed coordinate is introduced, which is rotated by angle, connected with crystallographic orientation. The algorithm for every grain b egins with calculating the diagonal of the square for central cell c, using KGT model [5]. This model introduces relationship between temperature and velocity of dendritic tip, 3 T, that can be approximated by third order polynomial, T a T a. 3 T When the growing square with time engulf the center of four neighbours n 1, n, n 3 and n 4 four rectangles start to grow in the center of their neighbour and their state changes into solid. Four new coordinates are introduced to simplify calculation. The growth of rectangles is updated with local undercooling,. For these cells theirs neighbours are considered and the algorithm proceeds in analogous way. T ni 3. IMPLEMENTATION Rys. 1. Schemat algorytmu wzrostu (dokładny opis można znaleźć w [4]). Fig. 1. Schematic diagram of growth algorithm (for details, see Ref. [4]). The method presented in the previous section has been implemented using the C++ language and object-oriented methodology. Figure shows the UML [7] diagram of the main components included in the software. The main class is SolidificationProcess, which is responsible for the main iteration of the simulation. This class utilises next three components MeshGenerator, GrowthProcess, NucleationProcess. These represent abstract interfaces to implement various cellular automata generators, algorithms of nucleation and grain growth processes, respectively. Two classes, CA_Cell and CA_CellMesh, represent cellular automata. To represent single grain and grain network the Grain and GrainMesh classes are used.

4 561 Such an approach has a very important feature; thanks to the mechanism of polymorphism we can change components of the system easily introducing new functionality. Rys.. Schemat UML pakietu do symulacji procesu krzepnięcia z wykorzystaniem CA Fig.. UML scheme of the software simulating the solidification process using CA. 4. NUMERICAL RESULTS To verify our system we have performed numerical simulation of solidification process. An aluminium-7% silicon alloy was used for simulation. The nucleation 10 3 parameters of this alloy are: n [ m ], T 0. 1[ K ], T max [ K ], respectively. The growth kinetics was described by KGT model with parameters: a.9 10 [ m sk ] and a [ m sk ]. The casting has the shape of square, [ m ] and was divided into cells with cell spacing dl 10 5 [ m ]. In this simulation dt [ s ] time step was used. The temperature filed was uniform for the considered casting and was changing with the cooling rate T.3 [ K s ]. Figure 3 shows obtained results of the numerical simulation. Development of grain structure is shown in three time steps, which correspond to the undercoolings: T 10. [ K ], T [ K ], T [ K ], respectively.

5 56 As we can see the evolution of grain shape corresponds to what can be expected when considering the uniform temperature field. In Figure 3(a) only few grains had nucleated and begun to grow (the location of grain is random). It is worth to notice, that the grains crystallographic orientation is random, as well. Figure 3(b) shows the consecutive state, more grains had nucleated and grown. The last figure presents the final structure of solidified casting. (a) (b) (c) Rys. 3. Struktura w odlewie Fig. 3. Grain structure in casting Obtained results highlight important property of the method. Application of the cellular automata allows observing the interaction between growing grains with good quality. What is more, the method preserves grains adhesion. This feature is particularly interesting since it allows generation of the hybrid finite elements based on the grains shape. 5. CONCLUS IONS AND FUTURE WORK In this paper we have presented the object-oriented software designed to implement simulation of the equiaxed grain growth in solidifying casting. The system is based on the cellular automata paradigm. The presented package has been created to extended a functionality of the NuscaS system. One of the main aims of this project is to allow generation of the unstructured meshes of the hybrid finite elements [11]. The current work focuses on the improvement of the system efficiency, which is crucial for real life applications. Another important issue is coupling with finite element method, which should give a possibility to introduce the non-isothermal temperature field to the simulation. Because of the high complexity of the problem a parallel implementation is considered.

6 563 REFERENCES [1] J. Alkemper, P.W. Voorhees: Three-dimensional characterization of dendritic microstructures, Acta mater., vol.49, pp , (001). [] W.J. Boettinger et al.: Solidification microstructures: recent developments, future direction, Acta mater., vol. 48, pp.43-70, (1999). [3] A. Coxe, C. Reiter: Fuzzy hexagonal automata and snowflakes, Comput. And Graphics, vol.7,pp (003) [4] Ch. A. Gandin, R.J. Schaefer, M. Rappaz: Analytical and numerical prediction of dendritic grain envelopes, Acta mater., vol. 44, pp ,(1995). [5] W. Kurz, B. Giovanola, R. Trivedi: Theory of microstructural development during rapid solidification, Acta metal., vol.34, pp , (1986). [6] M. Plapp, A. Karma: Multiscale finite-difference-diffusion-monte-carlo method for simulating dendritic solidification, Journal of Comp. Physics, vol.165,pp ,(000). [7] OMG (Object Management Group) http.// [8] M. Rapazz, Ch.-a. Gandin: Probabilistic modelling of microstructure formation in solidification processes, Acta metal. mater., vol.411, pp ,(1993). [9] N. Sczygiol: Object-oriented analysis of the numerical modelling of casting solidification, Comp. Ass. Mech. And Eng. Scien., vol.8, pp.79-98,(001). [10] N. Sczygiol: Modelowanie numeryczne zjawisk termodynamicznych w krzepnącym odlewie i formie odlewniczej, Wyd. P. Cz., Częstochowa 000 [11] G. Szwarz: Modelowanie numeryczne pękania stopów o strukturze równoosiowej w stanie stało ciekłym, praca doktorska P. Cz, 00. [1] W. Wang, P.Lee, M.McLean:A model of solidification microstructure in nickelbased superalloy: prediction primary dendrite spacing selection, Acta mater., vol 51,pp ,(003). [13] J. A. Warren, W.J. Boettinger: Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method, Acta metal. mater., vol. 43, pp , (1994). MODELOWANIE WZROSTU STRUKTURY RÓWNOOSIOWEJ W PROCESIE KRZEPNIĘCIA STRESZCZENIE W pracy przedstawiono autorską obiektową realizację systemu do symulacji wzrostu struktury równoosiowej w krzepnącym odlewie z wykorzystaniem automatu komórkowego. Pokazano wyniki przykładowej symulacji dla stopu dwuskładnikowego (Al-7% Si). Omówiono również możliwe sposoby rozbudowy systemu. Recenzowała Prof. Ewa Majchrzak