COMPUTATIONAL MATERIALS ENGINEERING Episode 11 ppsx

13 + (dphi[1][i][j] - dphi[1][i][j-1])/delta_y ; 14 15 potentialEnergy = A*w(argPhi[i][j]); 16 drivingForce = B*f(argPhi[i][j]); 17 18 // Summation of terms for the update of the phi matrix 19 newPhasefield = argPhi[i][j] + delta_t * ( 2*gamma* 20 divergence - 1/(epsilon*epsilon)*potentialEnergy 21 - 1/epsilon*drivingForce); 22 23 return newPhasefield; 24 } 7.3.7 Parameter File In the following Listing 7-3, we give a detailed description of the parameter file in ASCII format for the configuration of a simulation and explain the meaning of the variables’ names. Lines starting with the symbol sharp are signed as comments and will not be parsed to the main program. LISTING 7-3 Description of Input Data for a Phase-Field Simulation 1 2 ######### 3 # Parameter file - ’params.cfg’ 4 ######### 5# 6 # Boundary conditions: 7 # boundary_type_* defines the type of the boundary condition. 8 # For each boundary a special type can be defined. 9 # Possible values: 10 # 0 = Dirichlet 11 # 1 = Neumann 12 # 2 = Periodic 13 # If the Dirichlet boundary condition is chosen, values 14 # at the boundaries need to be set by rand_wert_*. 15 16 boundary_type_N=0 17 boundary_type_E=0 18 boundary_type_S=0 19 boundary_type_W=0 20 boundary_value_N=1 21 boundary_value_E=1 22 boundary_value_S=1 23 boundary_value_W=1 24 25 # Initial filling: 26 # filling_* is an option to fill a square region. 27 28 filling_x1=40 29 filling_y1=40 30 filling_x2=60 31 filling_y2=60 32 33 # value_interiorbody initializes the filled region 34 # with a defined value. The remaining region is set 35 # to value_exteriorbody. 36 37 value_interiorbody=1 236 COMPUTATIONAL MATERIALS ENGINEERING 38 value_exteriorbody=0 39 40 # Parameters of the phase-field equation: 41 # epsilon is the thickness of the diffuse interface. 42 43 epsilon=1.0 44 45 # gamma is the surface entropy density of the interface. 46 47 gamma=1.0 48 59 # eta is the driving force. 50 51 eta=1.0 52 53 # delta_t is the time step 54 55 delta_t=0.001 56 57 # delta_x, delta_y are the cell spacings in x- and y-directions. 58 59 delta_x=0.1 60 delta_y=0.1 61 62 # A, B are coefficients to switch on/off terms of the rhs. 63 64 A=1 65 B=1 66 67 # Computational parameters - 68 # max_n determines the number of computed frames. 69 70 max_n=1000 71 72 # Nx, Ny are the numbers of cells in x- and y-direction. 73 74 Nx=100 75 Ny=100 76 77 # frames defines the distance between frames to be stored. 78 79 frames=50 7.3.8 MatLab Visualization To illustrate the evolution of the phase-field in time and space, the C++ program may produce a MatLab file “data file.m” as output file. This file contains a number of successive φ matrices at preselected time steps. Applying further MatLab script files “show ij.m,” “show xy.m,” or “show 3d.m” allows the graphical illustration of the phase field either as 2D colored images or as a 3D surface plot. The Listings 7-4 and 7-5 now give examples of the MatLab codes. LISTING 7-4 Structure of a MatLab Output File 1 %%%%% 2 % data_file.m - MatLab output file with the matrices 3 % of the stored frames 4 %%%%% Phase-Field Modeling 237 5 % Deleting of the old data in the matrix A 6 A = []; 7 8 % Setting the x- and y-dimension of the computational domain 9 max_j = 6; max_i = 4; 10 11 % Pause in seconds between the frames used for the movie 12 p = 0.500000; 13 14 % Total Number of frames 15 n=5; 16 17 % Frames 1-5 are stored in the matrix A 18 A(:,:,1) = [0.000000,0.000000,0.000000,0.000000,0.000000,0.000000; 19 1.000000,1.000000,1.000000,0.000000,0.000000,0.000000; 20 1.000000,1.000000,1.000000,0.000000,0.000000,0.000000; 21 1.000000,1.000000,1.000000,0.000000,0.000000,0.000000]; 22 23 A(:,:,5) = [0.211965,0.266338,0.211965,0.114580,0.033851,0.012800; 24 0.397253,0.475835,0.397253,0.211965,0.077528,0.027870; 25 0.475835,0.583743,0.475835,0.266338,0.092771,0.040670; 26 0.397253,0.475835,0.397253,0.211965,0.077528,0.027870]; LISTING 7-5 Matlab Script File for Visualization of the Simulation Data as 2D Images 1 %%%%% 2 % show_xy.m - MatLab Visualization of a colored 2D image 3 %%%%% 4 % Calling of the data file 5 data_file; 6 7 % Loop over n frames illustrating the temporal evolution 8 % of the phase field 9 for k = 1:1:n 10 % imagesc() illustrates the 2D matrices 11 imagesc(A(:,:,k),[0 1]); 12 % axis image keeps the proportions of the graphical axis 13 axis image; 14 axis xy; 15 % Labelling of the axes 16 xlabel(’x’); 17 ylabel(’y’); 18 % colormap() defines the colour table 19 colormap (hot); 20 % pause() sets a pause between the frames to slow down the movie 21 pause (p); 22 end; 7.3.9 Examples The aim of this section is to study the behavior of equation (7.19) numerically for some special cases and to experience the influence of the boundary conditions, of the parameters, and of individual terms in the phase-field equation. For the subsequent case study, the following parameters occurring in equation (7.19) are taken: τ =1.0,  =1.0, γ =1.0 and Nx = 100, Ny =10, N t = 1000 238 COMPUTATIONAL MATERIALS ENGINEERING By choosing appropriate values for A and B given subsequently, the certain terms in the phase- field equation are switched on and off, respectively. The time step δt can be set according to the stability condition in equation (7.20). Case 1: Energy Equation—Dirichlet and Neumann Boundary Conditions In this first application, we consider the case in which the phase-field equation reduces to a heat transport equation of the type ∂ t u(x, t)=D  u(x, t) by setting the control coefficients A =0 and B =0. The influence of different boundary conditions is displayed in Figure 7-12. The diagrams show the profile of the phase-field variable φ(x, t) starting from φ(x, t)=0at time t =0everywhere in the domain. Case 2: Phase-Field Equation To investigate the phase-field equation, we switch on the potential entropy contribution w ,φ (φ) and the bulk driving force f ,φ (φ) by setting the coefficients A =1and B =1. The gray scales indicate φ =1in white, φ =0in black, and the diffuse interface region in varying colors. 1. Diffuse Interface Thickness: A planar solid–liquid front is placed in the center of the domain at Nx/2 with a sharp interface profile, with zero driving force m =0and with Neumann boundary conditions on each side. The effect of different values of the small length scale parameter:  =1and  =10responsible for the thickness of the diffuse interface is shown in Figure 7-13. 1 0.8 0.6 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10 0.2 0.4 0.6 0.8 1 0 X φ(x, t) X φ(x, t) FIGURE 7-12 Left diagram (pure Dirichlet condition): The left (west) boundary is set to φ W =1 and the right (east) boundary to φ E =0. A linear profile is est ablished; Right diagram (combination of the Dirichlet and the Neumann boundary condition): The left (west) boundary is set constant to φ W =1and the right (east) boundary is isolated (Neumann). The complete domain “warms up” approaching the constant value φ =1. Bild-Nr.: 10/10 10 20 30 40 50 60 70 80 90 X Y 8 6 4 2 Bild-Nr.: 10/10 10 20 30 40 50 60 70 80 90 X Y 8 6 4 2 (a) (b) FIGURE 7-13 Diffuse interface profile for (a)  =1and (b)  =10. Phase-Field Modeling 239 2. Driving Force: As a next configuration, the three simulations shown in Figure 7-14(b)–(d) were performed with  =1with different values of the driving force and with the initial configuration of Figure 7-14(a). For m =0, the initial planar front remains stable, for m = −1 the solid phase (light color) grows, whereas for m =1the solid phase shrinks. Bild-Nr.: 10/10 10 20 30 40 50 60 70 80 90 8 6 4 2 Bild-Nr.: 10/10 10 20 30 40 50 60 70 80 90 X Y 8 6 4 2 Bild-Nr.: 10/10 10 20 30 40 50 60 70 80 90 X Y 8 6 4 2 Bild-Nr.: 10/10 10 20 30 40 50 60 70 80 90 X Y 8 6 4 2 (a) (b) (c) (d) FIGURE 7-14 (a) Initial solid–liquid front with a sharp interface and  =1, (b) diffuse interface profile for zero driving force ( m =0), (c), and (d) moving solid–liquid interface for m = −1 and m =1, respectively. 5 10 20 30 40 10 15 20 25 30 35 40 45 Bild-Nr.:15/15 X 5 10 20 30 40 10 15 20 25 30 35 40 45 Bild-Nr.:15/15 X 5 10 20 30 40 10 15 20 25 30 35 40 45 Bild-Nr.:15/15 X 5 10 20 30 40 10 15 20 25 30 35 40 45 Bild-Nr.:15/15 X 5 10 20 30 40 10 15 20 25 30 35 40 45 Bild-Nr.:15/15 X 5 10 20 30 40 10 15 20 25 30 35 40 45 Bild-Nr.:15/15 X (a) (b) FIGURE 7-15 Two-dimensional phase-field simulation with (a) a solid nucleus in the center of the domain and Neumann boundary conditions and with (b) a solid nucleus at the bottom of the domain and periodic boundary conditions. 240 COMPUTATIONAL MATERIALS ENGINEERING Case 3: 2D Phase-Field Simulations of a Growing Nucleus For the simulation in Figure 7-15, a solid nucleus is set in a 2D domain of Nx×Ny =50× 50 grid points, with A =1, B =1and with a driving force m = −2.5. Due to periodic boundary conditions in Figure 7-15(b ) the particle grows across the lower boundary and appears at the top boundary. 7.4 Model for Multiple Components and Phases 7.4.1 Model Formulation Based on the phase-field model for pure substances, a more general approach can be derived in a thermodynamically consistent way allowing for an arbitrary number of phases (or grains) and components [GNS04]. The model that will be described in the following sections uses N-order parameters to describe either different phases in alloy systems or different orientational variants in polycrystals. The formulation can be defined solely via the bulk free energies of the individual phases, the surface energy densities (surface entropy densities, respectively) of the interfaces, the diffusion and mobility coefficients. Thus, the full set of phase-field evolution equations is defined by quantities which can be measured. Since the bulk free energies determine the phase diagrams (see, e.g., Chalmers [Cha77], Haasen [Haa94]), the phase-field model can be used to describe phase transitions, in principal, for arbitrary phase diagrams. The phase- field model for a general class of multicomponent and multiphase (or polycrystalline) alloy systems is formulated consisting of K components and N different phases (or grains) in a domain Ω ⊂ IR 3 . The domain Ω is separated in phase regions Ω 1 , ,Ω N occupied by the N phases as schematically illustrated in the left image of Figure 7-16. The middle and right images show examples of an Al–Si grain structure with grains of different crystallographic orientations and of a real multiphase structure with primary dendrites and an interdendritic eutectic substructure. The concentrations of the components are represented by a vector c(x, t)=(c 1 (x, t), , c K (x, t)). Similarly, the phase fractions are described by a vector-valued-order parameter (x, t)=(φ 1 (x, t), ,φ N (x, t)). The variable φ α (x, t) denotes the local fraction of phase α. The phase-field model is based on an entropy functional of the form S(e, c, )=  Ω  s(e, c, ) −  a(, ∇)+ 1  w()   dx (7.21) (a) (b) (c) FIGURE 7-16 (a) Schematic drawing of a domain separation by four different phase regions, (b) polycrystalline grain structure, and (c) multiphase solidification microstructure with dendrites and an interdendritic eutectic structure. Phase-Field Modeling 241 We assume that the bulk entropy density s depends on the internal energy density e, the concen- trations c, and the phase-field variable . We require that the concentrations of the components and of the phase-field variables fulfill the constraints K  i=1 c i =1 and N  α=1 φ α =1 (7.22) 7.4.2 Entropy Density Contributions It will be convenient to use the free energy as a thermodynamical potential. We therefore pos- tulate the Gibbs relation df = −sdT +  i µ i dc i +  α r α dφ α Here, T is the temperature, µ i = f ,c i are the chemical potentials, and r α = f ,φ α are potentials due to the appearance of different phases. We set e = f + sT (7.23) and hence de = Tds+  i µ i dc i +  α r α dφ α ds = 1 T de −  i µ i T dc i −  α r α T dφ α If we interpret s as a function of (e, c, ), then we have s ,e = 1 T ,s ,c i = −µ i T ,s ,φ α = −r α T (7.24) We note that given the free energy densities of the pure phases, we obtain the total free energy f as a suitable interpolation of the free energies f α of the individual phases in the system. By inserting the free energy f into the phase-field method enables to model systems with a very general class of phase diagrams. In the way it is formulated, the model can describe systems with concave entropies s α (e, c) in the pure phases. This corresponds to free energies f α (T,c) which are convex in c and concave in T . In the case where f(T,c) is not convex in the variable c, the free energy needs to contain gradients of the concentrations (as in the Cahn–Hilliard model). Choosing the liquid phase to be the last component φ N of the phase-field vector , an ideal solution formulation of the bulk free energy density reads f id (T,c, ):= N  α=1 K  i=1  c i L α i T − T α i T α i h(φ α )  + K  i=1  R g v m Tc i ln(c i )  − c v T ln( T T M ) with L N i =0and L α i , i =1, ,K, α =1, ,N − 1, being the latent heat per unit volume of the phase transition from phase α to the liquid phase and of pure component i. Furthermore, T α i , i =1, ,K, α =1, ,N−1 is the melting temperature of the ith component in phase α, T M is a reference temperature. c v , the specific heat and v m , the molar volume are assumed to be constant, R g is the gas constant. With a suitable choice of the function h() as introduced in equations ( 7.4) – (7.6) satisfying h(0)=0and h(1)=1, for example, h(φ α )=φ α or h(φ α )= φ 2 α (3 −2φ α ), the free energy density f is an interpolation of the individual free energy densities f α . We can calculate 242 COMPUTATIONAL MATERIALS ENGINEERING s = −f ,T = − N  α=1 K  i=1  c i L α i T α i h(φ α )  − K  i=1  R v m c i ln(c i )  + c v ln(T ) so that e = f + Ts = − N  α=1 K  i=1  c i L α i h(φ α )  + c v T We note that if L α i = L α for all components i, then e does not depend on c. The chemical poten- tials µ i (T,c, ) are given as the derivative of the free energy density f(T,c, ) with respect to c i leading to µ i (T,c, )= N  α=1  L α i T − T α i T α i h(φ α )  + R v m T (ln(c i )+1) (7.25) A more general expression for alloys is the Redlich–Kister–Muggianu model of subregular solution f sr = f id + K  i=1 K  j=1 c i c j M  ν=0 M (ν) ij (c i − c j ) ν with binary interaction coefficients M (ν) ij depending on the parameter ν.ForM =0, the Redlich–Kister–Muggianu ansatz takes the form of a regular solution model. In most appli- cations, in particular to metallic systems, M takes a maximum value of two. A ternary term ∼ c i c j c k can be added to describe the excess free enthalpy. The thermodynamics of the interfaces gives additional contributions to the entropy given by a Ginzburg–Landau functional of the form −  Ω  a(, ∇)+ 1  w()  dx Here, a(, ∇) is the gradient entropy density which is assumed to be homogeneous of degree two in the second variable; namely, a(,η∇)=η 2 a(, ∇), ∀η ∈ IR + . The simplest form of the gradient entropy density is a(, ∇)= N  α=1 |∇φ α | 2 However, it has been shown [SPN + 96, GNS98, GNS99b] that gradient entropies of the form a(, ∇)=  α<β A αβ (φ α ∇φ β − φ β ∇φ α ) (7.26) where A αβ are convex functions that are homogeneous of degree two, are more convenient with respect to the calibration of parameters in the phase-field model to the surface terms in the sharp interface model. A choice that leads to anisotropic surface terms is a(, ∇)=  α<β γ αβ (a αβ (q αβ )) 2 |q αβ | 2 (7.27) Phase-Field Modeling 243 where γ αβ represents the surface entropy density of the α–β interface and q αβ =(φ α ∇φ β − φ β ∇φ α ) is a generalized gradient vector oriented in the direction of the normal to an α–β interface. The formulation using the generalized gradient vectors q αβ allows to distinguish the physics of each phase (or grain) boundary by providing enough degrees of freedom. Anisotropy of the surface entropy density is modeled by the factor (a αβ (q αβ )) 2 depending on the ori- entation of the interface. Isotropic phase boundaries are realized by a αβ (q αβ )=1. Weakly anisotropic crystals with an underlying cubic symmetry can be modeled in 3D by a straightfor- ward extension of the expression in equation (7.10): a αβ (q αβ )=1− δ αβ  3 − 4 |q αβ | 4 4 |q αβ | 4  (7.28) with δ αβ being the strength of the anisotropy of the α–β interface. The norms are given by |q αβ | 4 4 =  3 i=1 (q 4 i ) and |q αβ | 4 =   3 i=1 (q 2 i )  2 with q i =(φ α ∂ ∂x i φ β − φ β ∂ ∂x i φ α ). For a strongly anisotropic crystal of faceted type, we define a αβ (q αβ ) = max 1≤k≤n αβ  q αβ |q αβ | · η k αβ  (7.29) where η k αβ ,k =1, ,n αβ are the n αβ corners of the Wulff shape of the α–β transition leading to flat crystal faces with sharp edges. These evolve in the direction of the cusps. In principle, equation (7.29) allows to model arbitrary crystal shapes with n αβ corners. For a comparison, we display in Figure 7-17(a) and (b) the smooth and faceted formulation of the function a αβ (q αβ ) for a cubic crystal symmetry. In Figure 7-17(c) the simulation results of two crystals with 45 ◦ rotated orientation growing from adjacent nuclei are shown. Each grain develops its minimum energy surfaces in contact with the melt and at their interface. The interfacial entropy density contribution w() is a nonconvex function with N global minima corresponding to the N phases in the system. As an extension of the standard double well potential in equation (7.9) and Figure 7-4, one may take the standard multiwell potential w st ()=9  α<β γ αβ φ 2 α φ 2 β (7.30) 1 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 0.5 0 -0.5 -1 1 1 0.5 0 -0.5 -1 -1 -0.5 0 0.5 1 0.5 0 -0.5 -1 (a) (b) (c) FIGURE 7-17 Three-dimensional surface plot of, (a) a smooth and, (b) a faceted cubic anisotropy, (c) contour plots of two adjacent growing, 45 ◦ misoriented cubic crystals applying the smooth anisotropy formulation in equation (7.28) with δ = 0.2. 244 COMPUTATIONAL MATERIALS ENGINEERING or a higher-order variant ˜w st ()=w st ()+  α<β<δ γ αβδ φ 2 α φ 2 β φ 2 δ For practical computations the multiobstacle potential yields good calibration properties. The expression is w ob ()= 16 π 2  α<β γ αβ φ α φ β (7.31) with a higher-order variant ˜w ob ()=w ob ()+  α<β<δ γ αβδ φ α φ β φ δ where w ob and ˜w ob are defined to be infinity whenever  is not on the Gibbs simplex. In Figures 7-18 and 7-19, we show a plot of both the multiwell and the multiobstacle potential for the case of three phases ( N =3). We refer to refs. [GNS99a] and [GNS99b] for a further discussion of the properties of the surface terms w st , ˜w st ,w ob , and ˜w ob . We assume for simplicity that a and w and, hence, the interfacial contributions to the entropy, do not depend on (T,c). 7.4.3 Evolution Equations The energy and mass balance equations can be derived from the energy flux J 0 and from the fluxes of the components J 1 , ,J K by ∂e ∂t = −∇·J 0 energy balance (7.32) ∂c i ∂t = −∇·J i mass balances (7.33) 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 -0.2 -0.4 -0.6 (a) -0.6 -0.4 -0.2 0.2 0.4 0.6 0.2 0.4 0.6 0.8 0.8 0.6 0.4 0.2 0 x y 0 0 (b) FIGURE 7-18 Plot of the multiwell potential w st () for N =3 and equal surface entropy den- sities γ αβ . Phase-Field Modeling 245 [...]... the code are necessary, for example, the introduction of indexed loop variables 252 COMPUTATIONAL MATERIALS ENGINEERING An important feature for parallel phase-field simulations is the implementation of an appropriate load balancing mechanism Due to the advancing fronts, which require more calculations, the demand for computational power is locally nonuniform This can be achieved by performing statistics... dimensionless latent heats Lα and a i ˜ rescaled gas constant Rv , the chemical potential µj [equation (7.25)] follows as µj J m3 ˜ = (cv ϑ) · µj with Lα i JK ˜ = (cv ϑ) · Lα , i m3 K 248 COMPUTATIONAL MATERIALS ENGINEERING R J ˜ = vm cv Rv molK ˜ ˜ ˜ ˜ With the dimensionless mobility coefficients [equations (7.35)–(7.37)] L00 , L0j , Li0 , Lij of the form L00 Li0 JK sm m2 K s 2 = = cv ϑ2 ˜ L00 , τ∗... boundary of two adjacent grid cells [see Figure 7-20(b)] For example, the x component of the energy flux reads: L00 ∂ 1 ∂x T L00 i,j,k i+ 1 ,j,k 2 + Dx 1 T i,j,k with L00 i+ 1 ,j,k 2 = 1 L00 2 250 COMPUTATIONAL MATERIALS ENGINEERING i+1,j,k + L00 i,j,k (7.47) and + Dx 1 T i,j,k = 1 ∆x 1 T i+1,j,k − 1 T i,j,k + Dx where indicates the forward difference operator in x-direction As a next step, the divergence operation... T N Lij (T, c, φ)∇ j=1 N Lij (T, c, φ)∇ j=1 δS δcj δS δcj −µj T with mobility coefficients (Lij )i,j=0, ,K To fulfill the constraint (7.22) during the evolution, we assume K Lij = 0, i=1 246 COMPUTATIONAL MATERIALS ENGINEERING j = 0, , K K i=1 ci = 1 in equation which implies K Ji = 0, and, hence, ∂t ( K ci ) = ∇ · ( K Ji ) = 0 We further assume i=1 i=1 i=1 that L is symmetric (Onsager relations) In... on top of the parent α solid phase A small β nucleus was introduced on top of the melting α solid phase Immediately, the β phase grows along the α–L phase boundary Since the supply of 254 COMPUTATIONAL MATERIALS ENGINEERING 1.6 1.4 1.2 T 1 0.8 0.6 0 0.2 0.4 C 0.6 0.8 1 FIGURE 7-23 A phase diagram of a binary peritectic alloy constructed by the method of common tangents FIGURE 7-24 Phase-field simulations... diffusion in the liquid, the excessive B atoms are transported in lateral direction, driving the propagation of the S2 solid phase The A atoms behave in an analogous manner A characteristical 256 COMPUTATIONAL MATERIALS ENGINEERING 1.00 0.95 Temperature L S1 + L (a) (c) 0.90 (d) TE T2 0.85 0.80 S2 + L T1 (b) S1 S2 S1 + S2 0.75 cS 0.70 c1 1 0 cS cE 0.8 0.6 0.4 Concentration of B 0.2 2 1 FIGURE 7-25 Symmetric... are formed If the surface entropies are anisotropic, further effects due to shear forces at the boundaries influence the angle condition as expressed by the Herring torque terms [Woo73] 258 COMPUTATIONAL MATERIALS ENGINEERING (a) 0.3615 0.3610 0 0.2 0.4 0.6 0.8 x-Coordinate (c) 0.4 0.2 0 0.2 0.4 0.6 0.8 x-Coordinate 1 0.6 0.5 0.4 1 0 0.1 0.2 0.3 0.4 x-Coordinate 0.5 (d) Concentration of B Concentration... hollows which deepen Subsequently, the two solid phases compete and alternately overgrow one another During the evolution, a liquid droplet is engulfed by the solid at the solid–solid 260 COMPUTATIONAL MATERIALS ENGINEERING ... be calculated exclusively with right-sided differences of φα within a 7-point stencil, in this case sufficient to guarantee second-order accuracy 7.4.6 Optimization of the Computational Algorithm Several adaptive strategies reduce the computational effort: • The equations for the phase fields are exclusively solved in their respective diffuse inter- face area by the use of an activation flag field This field... the evolutional timescale If equations (7.32)–(7.33) are solved conjointly, the stability criterion demands the use of the small time step width, which is usually the heat diffusion scale To reduce the computational effort, different step widths, integer multiples of the smallest (heat diffusion) scale, can be used to solve the three kinds of equations (7.32), (7.33), and (7.34) By taking integer multiples, . 1.000000,1.000000,1.000000,0.000000,0.000000,0.000000]; 22 23 A(:,:,5) = [0. 2119 65,0.266338,0. 2119 65,0 .114 580,0.033851,0.012800; 24 0.397253,0.475835,0.397253,0. 2119 65,0.077528,0.027870; 25 0.475835,0.583743,0.475835,0.266338,0.092771,0.040670; 26. remaining region is set 35 # to value_exteriorbody. 36 37 value_interiorbody=1 236 COMPUTATIONAL MATERIALS ENGINEERING 38 value_exteriorbody=0 39 40 # Parameters of the phase-field equation: 41. equation (7.19) are taken: τ =1.0,  =1.0, γ =1.0 and Nx = 100, Ny =10, N t = 1000 238 COMPUTATIONAL MATERIALS ENGINEERING By choosing appropriate values for A and B given subsequently, the certain