submitted to Journal of Computational Physics
Fixed-point convergence of modular, steady-state heat transfer models coupling multiple scales and phenomena for melt crystal growth
Arun Pandy and Jeffrey J. Derby∗ Department of Chemical Engineering and Materials Science and Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, MN 55455-0132, USA
December 3, 2004
Abstract A novel, two-parameter coupling scheme is developed to exchange temperature and heat flux information between two, steady-state heat transfer models representing different scales and phenomena important for describing melt crystal growth. The framework of fixed-point iterations is employed to assess the convergence behavior of a modular, block Gauss–Seidel iteration of these independent models as a function of the coupling parameters. Analytic forms describe the convergence behavior of one-dimensional model problems and provide insight to the behavior of more complicated, two-dimensional models. Certain choices of coupling parameters that are physically reasonable are shown to result in algorithms that will never converge, demonstrating that notions based on physical intuition may not be useful for predicting algorithm performance. ∗
Corresponding author. Fax: +1-612-626-7246; e-mail:
[email protected]
1
Introduction
As computational modeling becomes more sophisticated, there is an increasing need for effective ways to couple existing software tools that have been developed for the solution of specific problems. This is especially true for multi-physics and multi-scale problems; see, e.g., [1]. One field where this has long been an issue is that of fluid-structure interaction problems [2–7], including applications in aeroelasticity [8,9] and biomechanics [10]. Models for coupled problems have also been developed for a wide variety of other applications, including thermomechanical response [11], soil-pore fluid interaction [12], turbomachinery problems [13], conjugate heat transfer [15–19], and many others. The present study is motivated by a desire to employ a quantitatively predictive model for melt crystal growth processes by the coupling of two independent process models. One of these models computes high-temperature heat transfer by radiation and conduction in a large-scale furnace surrounding an ampoule in which crystal growth is occurring. The other model focuses on physical phenomena within the ampoule, i.e., the details of melt convection, heat transfer, and crystal-melt interface shape. Such models have rarely been attempted in the past because of the great challenges in realistically representing all of these features in one model. There have been several notable efforts to develop furnace-scale models for melt crystal growth processes. Dupret and co-workers developed the numerical code FEMAG [20] for predicting heat transfer in Czochralski (CZ) crystal growth, and M¨ uller et al. advanced the code CrysVUN [21, 22], originally for predicting heat transfer in vertical gradient freeze processes. Although both have evolved to also represent phase-change phenomena and include convection effects, their fort´e is the depiction of radiation heat transfer in high-temperature systems at the global level. Local crystal growth models have typically focussed on the challenges posed by solving coupled transport (especially fluid dynamics) and interfacial phenomena, while representing heat exchange with the furnace in idealized manners. Excellent overviews of such models are the still-relevant review by Brown [23] and more recent reviews by M¨ uller [24], Lan [25], and Yeckel and Derby [26]. There have been attempts to couple a code for furnace heat transfer to a local crystal growth
2
model that computes melt convection in crystal growth processes. Baumgartl et al. [27] applied temperature conditions from a global heat transport simulation to compute melt convection in a Czochralski crystal growth process. Ouyang and Shyy [28, 29], developed a two-level approach to incorporate heat transfer in a Bridgman system into a crystal growth model. A similar two-level approach was used by Tom´as et al. [30] to model cadmium telluride growth in vertical Bridgman systems. Virbulis et al. [31] employed FEMAG [20] and the commercial code CFD-ACE in a coupled manner to analyze silicon CZ flow. Recently, Lukanin et al. [32] describe the coupling of a two-dimensional global heat transfer code with three-dimensional turbulent flow computations for silicon Czochralski growth analysis. Yeckel et al. [33, 34] have applied two-dimensional furnace heat transfer analyses to supply boundary conditions to three-dimensional local computations. For the most part, the coupling used for these models has been one-way. Namely after computing furnace heat transfer with a coarse-scale representation of the melt and crystal, temperature boundary conditions are supplied to the local model [27–30,33,34]. This approach ignores the change in the furnace heat load caused by such local factors as melt convection and latent heat generation at the melt-crystal interface. Others [31, 32] report some sort of iterations between furnace and local models, but do not supply enough details about the information exchanged nor the convergence tolerances employed to ascertain whether the final solutions truly satisfied the mathematical governing equations of each model. Recently, Pandy et al. [35] presented a multi-scale model to study the growth of single crystals of cadmium zinc telluride in an industrial electrodynamic gradient freeze furnace. This model combined a furnace-scale model based on the commercial code CrsyVUn [21, 22] with a local model, CATS2D [43], that solves for heat transfer, melt flow, and melt-crystal interface shape. This study examined the errors introduced by two different one-way coupling of the models, i.e., providing either flux or temperature boundary conditions from the furnace model to the local model without iteration, compared to a fully self-consistent solution of the multi-scale model. This study also found that the details of how the two models were coupled had a strong bearing on the convergence of the iterative computations. The objective of the current study is to understand the nature of these convergence difficulties and suggest strategies to alleviate them. The art of computational modeling has always involved a choice of which physical phenom-
3
ena are to be included and at what level of detail. One approach is to simultaneously represent all chosen phenomena in all domains in one large mathematical model; such an approach has been referred to as a direct coupling, monolithic, or analytic approach . This approach may be preferred, since all interactions can be accounted for simultaneously and self-consistently. However, such an approach may be undesirable or infeasible, e.g., when the computational or software development costs for solving all of the assembled governing equations is too high or when subprocess models exist but employ commercial software, preventing access to the source code. In these situations a loose coupling, modular , or synthetic approach, in which different models and solvers are used together, may be preferred. This is the situation to be considered here. In a general sense, we will consider two problems, which are typically nonlinear, that are coupled by a common boundary. We’ll denote these models as R1 (x1 ; x ˜2 ) = 0,
(1)
R2 (x2 ; x ˜1 ) = 0,
(2)
where Ri denotes a vector-valued residual function associated with problem i and xi are all of the unknowns associated with model i. The tilde denotes those degrees of freedom of one model that are direct inputs to, or parameters for, the other model. We will denote iterative solvers that are used for each problem as follows, (k+1)
= f1 (x1 ; x ˜2 ),
(k+1)
= f2 (x2 ; x ˜1 ),
x1
x2
(k)
(3)
(k)
(4)
meaning that each solver, fi , iterates only on the unknowns associated with that problem, xi , with the input from the other model, x ˜j , fixed and the superscript (k) denotes an iteration counter. These solvers can be of any form, although we employ Newton–Raphson methods for all of the examples here. For the coupled problem, we choose to employ each solver independently, while exchanging information between the models for each iteration. Thus, we are employing the loose coupling, modular, or synthetic approach described above. Along these lines, we define a block Gauss–Seidel iteration method [36], whereby iterations proceed in the following manner. Given (k)
iterates x1
(k)
and x2
that represent converged solutions to each of the independent models, 4
we first perform (k+1)
x1
(k)
(k)
= f1 (x1 ; x ˜2 ),
(k+1)
until we converge to a new solution for x1
(5)
. Then we carry out the next step of the overall
iteration with this updated information (hence the Gauss–Seidel classification of the scheme), (k+1)
x2 (k+1)
until the next iterate x2
(k)
(k+1)
= f2 (x2 ; x ˜1
),
(6)
is found. Successful iterations eventually converge to a self-
consistent solution to the coupled problem. The actions of above taken together can be thought of as a general fixed-point iteration, where a vector of total unknowns xT ≡ (x1 , x2 )T is updated in an iterative manner as follows, x(k+1) = F x(k)
(7)
where F (x) is a vector map and x(k) and x(k+1) are successive iterates of the map. These iterates will converge to the attracting fixed point x∗ provided we start with an initial guess belonging to its basin of attraction. However, while we know the individual solvers, f1 and f2 , we do not know the explicit form of the global map, F, because of the shared unknowns between the two models. A condition for convergence is defined by the sensitivity matrix of F with respect to x, namely that ||
∂F |x∗ || < 1 ∂x
(8)
where || · || is a suitable matrix norm and x∗ denotes a solution to the coupled problem. The the sensitivity matrix of the mapping function F with respect to the unknowns x, denoted as ∂F/∂x ≡ F0 , is a matrix with coefficients, Fij0 =
∂Fi , ∂xj
(9)
where Fi denotes the ith component of of the vector function F and xj denotes the jth component of vector x. It is readily shown from the fixed-point iteration that ||xk − x∗ || ≤ a||xk−1 − x∗ ||,
5
(10)
where the positive constant a ≤ ||∂F/∂x |x∗ ||. When a < 1, we expect the sequence to converge and the rate of convergence to be linear, i.e., the error in the the k th iterate, ||xk −x∗ ||, will be linearly proportional to the error of the prior iteration at k − 1, ||xk−1 − x∗ || . The rate of convergence of the iterations could be significantly improved with a more involved iteration method. For example, quadratic convergence could be achieved using a full Newton–Raphson method; however, one would need a complete Jacobian matrix for the coupled problem. Knowing all of the components of F0 would allow one to construct this matrix, but, again, we do not know the explicit form of F0 because of the loose coupling of the individual models. Interestingly, Chan [37] proposed an approximate Newton method for solving coupled nonlinear systems similar to the ones considered here. This approach relies on accessing the known individual Jacobian matrices for the subproblems, ∂f1 /∂x1 and ∂f2 /∂x2 , while approximating the unknown cross-block matrices, namely ∂f1 /∂x2 and ∂f2 /∂x1 . Recent work has focussed on implementing such approaches but using only the solvers themselves [38–40], e.g., as would be the case when commercial packages are used for the submodels and the source code is inaccessible. While these approximate-Newton approaches may be quite attractive, they may not represent the best strategy to solve the overall problem. For example, Menck [41] points out that inaccuracies inherent in the approximations made by these algorithms may result in a prohibitive number of iterations per step. The goal of the algorithm presented in [41] was efficiently controlled linear convergence. In addition, the block Gauss–Seidel approach employed here is trivial in its implementation compared to the much more complicated procedures needed for approximate-Newton methods. The choice of modeling approach is dependent upon trade-offs among implementation effort, computational efficiency, and robustness. From a physical perspective, the coupled models considered here must exchange information, x ˜1 and x ˜2 , across their shared boundary to satisfy energy conservation. This is accomplished via continuity of the temperature field and normal heat fluxes along their shared boundary. We introduce a novel, two-parameter coupling scheme that provides a means to mix temperature and fluxes during this exchange between the models. The details of how these quantities are exchanged and how these details affect the convergence of the loosely coupled model will be the primary focus of this paper.
6
Our results first address a set of one-dimensional, model problems meant to represent the dominant physics of the coupled heat transfer problem underlying melt crystal growth. Because of their relative simplicity, the convergence characteristics of these coupled problems are investigated analytically. Next, we study several two-dimensional, numerical models that include the nonlinearities associated with melt crystal growth simulations. These models are assessed using numerical experiments. The outcomes of this work clarify the behavior of the loosely coupled, modular model previously employed by us in [35]. In addition, we believe that the results presented here will provide a useful framework to address issues of convergence for other heat transfer problems whose solution strategies involve the coupling of existing models.
2
One-dimensional model problems
To consider the general outcomes of different iteration strategies, we first consider three onedimensional heat transfer problems that are representative of melt crystal growth. Case 1 is a linear problem that includes conduction and convection and is posed over two domains. Case 2a is a nonlinear problem that includes conduction, convection, and radiation over two domains. Case 2b is a similar mixed-mode problem but posed over three domains.
2.1 2.1.1
Problem definition Case 1
Here, we define a steady-state heat transfer problem over a one-dimensional domain spanning from x = 0 to x = 2, where x is a non-dimensional length, as shown in Figure 1(a). This domain is divided into subdomains Ω1 (0 ≤ x ≤ 1) and Ω2 (1 ≤ x ≤ 2). Subdomain Ω1 represents a simplified, one-dimensional analogue of the crystal growth melt, where we define an conduction-convection equation. Subdomain Ω2 is an analogue for the furnace in the crystal growth system, where we define a conduction equation. The governing equations for the combined problem are, d2 T dT −c =0 2 dx dx d2 T κ 2 =0 dx
7
in
Ω1 ,
(11)
in
Ω2 ,
(12)
where a non-dimensional temperature is denoted by T , κ ≡ k2 /k1 is the ratio of thermal conductivities between the two domains, and c ≡ ρCp vex e/k1 is the Peclet number of the system. The melt velocity and characteristic length are given by ve and x e, respectively; ρ is the melt density; and Cp is the heat capacity of the material in the melt of subdomain Ω1 . Boundary conditions are specified at the ends of the system, x = 0 and x = 2, using Dirichlet conditions: T = T0
at
x = 0,
(13)
T = T2
at
x = 2.
(14)
To connect the heat transfer problem at x = 1, we desire the continuity of temperature and flux: T |x=1−
= T |x=1+ ,
(15)
q |x=1−
= q |x=1+ ,
(16)
where x = 1− and x = 1+ correspond to the location x = 1 belonging to subdomains Ω1 and Ω2 , respectively, and q are non-dimensional heat fluxes given by, q |x=1−
= − dT dx |x=1− ,
(17)
q |x=1+
= −κ dT dx |x=1+ .
(18)
The problem defined above is linear, and the analytical solution is given by, T =
(T1 − T0 ec ) + (T0 − T1 ) ecx in (1 − ec ) T = (T2 − T1 ) (x − 2) + T2 in
Ω1
(19)
Ω2
(20)
where T1 is the temperature at x = 1 and is given by T1 =
κ (1 − ec ) T2 − T0 cec . κ (1 − ec ) − cec
(21)
For this and the following one-dimensional models, we employ a set of system parameter values representative of a crystal growth system, namely c = −9, κ = 0.1, T0 = 0.98, and T2 = 1. For these parameters, the steady-state temperature profile though both domains is plotted in Figure 2(a). The relatively large value of the Peclet number, c, results in a flattened thermal profile in domain Ω1 , which connects to the linear temperature profile in domain Ω2 . 8
2.1.2
Case 2a
This case is motivated by the high-temperature nature of heat transfer in a melt crystal growth furnace, where radiation effects predominate. Here, we consider a non-linear problem by treating the subdomain Ω2 to be transparent to radiation with the point x = 1 exchanging radiation with the point x = 2. All the governing equations, i.e., Equations (11) to (16), defined for the previous problem hold. The only difference comes in the definition of the nondimensional fluxes, which account for radiative terms as well. The non-dimensional fluxes at x = 1− and at x = 1+ are now given by, q |x=1− =
− dT dx |x=1− ,
(22)
4 q |x=1+ = −κ dT dx |x=1+ +R T |x=1+ −T2 ,
4
where R ≡
σ Te3 x e k1
(23)
is the non-dimensional radiation number with Te as the characteristic temper-
ature and σ as the Stefan-Boltzmann constant. Note we assume that both surfaces at x = 1 and x = 2 are black, so that their emissivities are unity. The solution over the entire domain remains the same as that given by Equations (19) and (20); however, now the expression for the temperature at x = 1, namely T1 , is non-linear and given by, (T0 − T1 ) cec − κ (T2 − T1 ) = 0. R T14 − T24 + 1 − ec
(24)
We solve the nonlinear Equation (24) numerically via Newton’s method. The solution for R = 5.67 is plotted along with the linear Case 1 solution in Figure 2(a). Due to the added radiation heat flux at x = 1, this solution differs considerably from the solution to the linear problem without radiation (Case 1). 2.1.3
Case 2b
For reasons that will become clearer in the following section, we also consider a special form for the case including radiation heat transfer, which will be referred to as Case 2b. Here, we introduce a third domain Ωd of small thickness d spanning from x = 1 to x = 1 + d between the subdomains Ω1 and Ω2 , as shown in Figure 1(b). The subdomain Ωd is treated as opaque to radiation, and heat transfer in this domain occurs only by conduction. Continuity of temperatures and heat fluxes apply at its boundaries
9
with subdomains Ω1 and Ω2 . In this modified problem, radiative heat transfer comes into the picture as a flux term in the flux continuity condition at x = 1 + d, which is the boundary between Ωd and Ω2 . The solution to this problem over the entire domain is given below: T =
(T1 − T0 ec ) + (T0 − T1 ) ecx in (1 − ec ) (Td − T1 ) T = (x − 1) + T1 in d (T2 − Td ) T = (x − 2) + T2 in (1 − d)
Ω1 ,
(25)
Ωd ,
(26)
Ω2 .
(27)
The temperatures T1 and Td , at x = 1 and x = 1 + d, respectively, are given by, T0 cec d − Td κd (1 − ec ) , cec d − κd (1 − ec ) κd (T0 − Td ) cec κ (T2 − Td ) R Td4 − T24 + − = 0, c c (κd (1 − e ) − ce d) 1−d T1 =
(28) (29)
where κd is the ratio of conductivity kd in subdomain Ωd to conductivity k1 in Ω1 . For evaluation of this case, we solve Equation (29) using Newton’s method. We consider the behavior of this system as we make Ωd vanishingly thin; Equation (28) yields the relation, lim Td = T1 .
d→0
(30)
Next we substitute Td = T1 in Equation (29), again take the limit d → 0, and find that this equation becomes, (T0 − T1 ) cec R T14 − T24 + − κ (T2 − T1 ) = 0, 1 − ec
(31)
which is exactly Equation (24). Therefore, in the limit of d → 0, the solution to this problem will tend to the solution to the non-linear heat transfer problem, Case 2a, of the previous section. This assertion is demonstrated numerically in Figure 2(b), where Equations (25) to (28) are solved for various values of the thickness, d, of domain Ωd . As d becomes smaller, the solution of Case 2b approaches that of Case 2a (where d = 0). With a value of d = 0.0001, the two solutions are practically indistinguishable. For the following discussions, we employ this value of d = 0.0001 for Case 2b.
2.2
Iterative procedure
10
Let us say we have two solvers, S1 and S2 . S1 will be used to solve Equation (11) in Ω1 with one boundary condition specified by Equation (13) at x = 0. S2 will be used to solve Equation (12) in Ω2 with one boundary condition specified by Equation (14) at x = 2. At the interface between the two domains, at x = 1, we apply mixed boundary conditions at x = 1− and x = 1+ for solvers S1 and S2 , respectively, (1 − α) (T |x=1− −T |x=1+ ) at α (1 − β) + (T |x=1+ −T |x=1− ) at β
q |x=1− = q |x=1+ +
x = 1− ,
(32)
q |x=1+ = q |x=1−
x = 1+ ,
(33)
where α and β are chosen parameters. These boundary conditions act to couple the two models and domains at x = 1 . They are constructed so that, when α = 0 or β = 0 the corresponding boundary condition reduces to the temperature continuity condition, Equation (15), and when α = 1 or β = 1 a pure flux continuity condition, Equation (16), is enforced. Applying these conditions guarantees that both temperature and flux continuity conditions are satisfied and that the entire heat transfer problem is well posed (as long as the cases of α = β = 0 and α = β = 1 are avoided). The iteration begins by the choice of a suitable initial guess for the interface temperature, (0)
T1 . With this value for T1 applied as a Dirichlet condition, solver S1 is used to compute a temperature field. A complete solution of the temperature field in domain Ω1 for iteration k defines the start of a two-step procedure employed to compute the next iterate at k + 1. (k)
Step 1: The temperature, T |x=1− = T1 , and the corresponding flux q |x=1− computed from the solution in Ω1 calculated by solver S1 are input into Equation (33). This defines the boundary condition at x = 1+ for solver S2 , and the temperature field in Ω2 is calculated. Step 2: From the prior step, values for T |x=1+ and q |x=1+ now are available from the temperature field in Ω2 and are substituted into Equation (32). This is imposed as a boundary condition at x = 1− , and solver S1 solves for an updated temperature field in Ω1 , defining the k + 1st solution iterate. The iteration counter is updated and and the two-step procedure is continued until the sequence converges or is deemed to be non-convergent. Overall, this procedure defines a fixed-point iteration, (k+1)
T1
(k) = F T1 ,
11
(34)
where F (T1 ) is the mapping function, which depends on the governing equations and the boundary conditions. As noted in the introduction, a necessary condition for this fixed-point iteration to converge is for |F 0 (T1∗ ) | < 1,
(35)
where F 0 ≡ ∂F/∂T1 and T1∗ in the above formula denotes the converged solution to the heat transfer problem. Of specific interest are the behaviors exhibited by this fixed-point iteration for different cases as a function of the chosen coupling parameters, α and β.
2.3
Results and discussion
Analytic forms for the iteration steps and the mapping functions for all of the one-dimensional cases considered here are presented in [42], along with additional cases. For brevity, these detailed formulae will not be presented, rather their most salient outcomes are now discussed. Figure 3 shows isocontours of |F 0 | plotted over the parameter space of α and β for Case 1. The shaded regions bounded by the isocontours of |F 0 | = 1 are those where the fixed point is repelling. For values of the coupling parameters corresponding to these regions, the iteration sequence will never converge, regardless of the value of the initial guess. These results defy physical intuition; for example, choosing to set α = 0, which corresponds to using a Dirichlet condition for domain Ω1 , will result in an iteration sequence that will not converge under any conditions. Unshaded regions correspond to conditions where |F 0 | < 1 and the iteration will be convergent for suitable initial guesses. From this diagram, we can also compare regions of the parameter space of α and β with respect to the speed of convergence. The smaller the magnitude of |F 0 |, the faster the convegence. For this problem, a special case occurs along the axis defined by α = 1/ (1 + κ), where κ is the ratio of thermal conductivities of the domains. Along this line, |F 0 | = 0, and the first iterate produces the exact solution of T1 , irrespective of the initial guess. The behaviors of the iteration for Case 2a are indicated in Figure 4 for various values of the radiation number, R. The case of R = 0.0 corresponds to Case 1 discussed above. Convergence of the iterations cannot occur in the shaded regions, where |F 0 | ≥ 1. The unshaded regions represent combinations of α and β where convergence of the iteration is possible. Notice that this region shrinks progressively as the radiation number, R, is increased in value. This 12
corresponds to a greater role of radiation heat transfer, and its associated nonlinearity, in the problem. For R = 5.67, corresponding to the temperature field shown in Figure 1(a), there is only a very narrow range of values of α and β that will allow a successful solution strategy for this problem. Figure 5 shows the convergence properties of the iteration method applied to Case 2b for the domain Ωd size of d = 0.0001. Note again that the regions of convergence and non-convergence for R = 0.0 correspond exactly to those of Case 1. As is evident by comparing Figures 4 and 5, the behavior of Case 2b is dramatically different from that of Case 2a as the radiation number is increased. While the specific values of α and β that allow convergence change markedly with increasing R, the area of the unshaded region changes very little. Unlike the prior Case 2a, there many choices of coupling parameters for Case 2b that will lead to converged solutions, even for large values of R. Defining a small domain Ωd to move the nonlinear radiation flux term away from the coupling condition significantly improves the convergence behavior of the algorithm. Notably, this effect holds even when the size of this region Ωd is small enough so as to negligibly affect the original heat transfer problem, as shown previously in Figure 1(b).
3
Two-dimensional problems
We now wish to test the behavior of iterations for more realistic, two-dimensional models of melt crystal growth systems. We specifically choose to model vertical gradient-freeze (VGF) systems, where the unidirectional solidification of a melt is induced by imposing a timedependent thermal gradient vertically across the containing ampoule. This is the type of system considered previously in [35]; a more general description of this process are available in [24]. The approaches considered in this section loosely couple separate models for furnace heat transfer and crystal growth.
3.1 3.1.1
Problem definition Case 3
Here, we construct a coupled model for a VGF system that consists of two submodels, as indicated schematically in Figure 6. The furnace model, denoted as Sf , includes a portion of the ampoule (split evenly between the furnace model and the crystal growth model) surrounded 13
by four heaters that, in turn, are surrounded by insulation; see Figure 6(a). Heat transfer within the domains occurs via conduction, and the steady-state temperature field is represented by, −κi ∇2 T = Qi (r, z) ,
(36)
where κi denotes a thermal conductivity ratio for subdomain i and the non-dimensional heatsource terms Qi (r, z) are nonzero in the heater elements but zero elsewhere. The crystal growth model, shown schematically in Figures 6(b) and (c), consists of axisymmetric domains representing the inner part of the ampoule, the melt , and the crystal. In the melt, heat transfer is by conduction and convection. Convection in the melt is driven by buoyancy effects due to thermal gradients in the system. Conduction occurs in the ampoule and crystal. Solidification is accounted for by solving for a melt-crystal interface, over which latent heat is released according to the local growth rate and the melting-point temperature of the system is maintained. Accounting for all of these effects results in a coupled, nonlinear problem for the temperature field in all domains, the velocity and pressure fields in the melt, and the shape and location of the melt-crystal interface. For the sake of brevity, details of the crystal growth model are not presented here; the interested reader is referred to Refs. [26,35,42] for a more complete discussion. The two models Sf and Sc are coupled along the boundary ∂cf that cuts through the middle of the ampoule; see Figure 6. The inner part of the ampoule belongs to the crystal growth model and the outer part belongs to the furnace model. This choice for ∂cf couples two domains where only conduction occurs and was motivated by the prior results of Case 2b. Similar to the one-dimensional cases discussed above, to make the overall model well-posed, we must enforce continuity of temperature and normal heat flux along the boundary between the crystal growth domain and the furnace domain: Tc = Tf ,
(37)
ncf · qc = ncf · qf ,
(38)
where Tc and Tf are the non-dimensional temperatures along the boundary ∂cf in the crystal growth domain and the furnace domain, respectively, ncf is the unit normal to the boundary ∂cf pointing towards the furnace domain, qi denotes heat flux vectors. For this problem, the
14
heat fluxes are expressed as qc = κa ∇Tc ,
(39)
= κa ∇Tf ,
(40)
qf
where κa denotes a thermal conductivity ratio for the ampoule. Both models are solved using a Galerkin finite element method employing biquadratic basis functions for the temperature field and a mixed basis of biquadratic/piecewise-discontinuous linear elements for the incompressible velocity field in the melt. Details are available in [43]. The geometry, dimensions, and parameters for this system were chosen to represent a model VGF system for the growth of cadmium zinc telluride [42]. The converged, self-consistent solution for this problem is depicted by isotherms in Figure 6(a) and (b) and streamlines in the melt in Figure 6(c). Strong buoyant flows, dominated by a cellular flow with an upward component along the hotter ampoule wall and a downward flow along the cooler centerline, act to effectively mix the melt and flatten the isotherms in it. Elsewhere, curved isotherms indicate more two-dimensional heat transfer. 3.1.2
Case 4
The final case considered is a realistic, multi-scale model developed in [35] to study the growth of single crystals of cadmium zinc telluride in an industrial furnace. The general approach is the same as that described above; however, a commercial code, CrysVUN [21, 22], which employs the finite volume method, serves as the furnace heat transfer model, Sf . With this code comes the ability for a quantitative prediction of high-temperature heat transfer in a crystal growth furnace [35]; however, only Dirichlet boundary conditions can be applied within CrysVUN. We discuss this limitation in the context of the coupled model in the ensuing discussion. For the entire simulation, CrysVUn is coupled with the same finite element, crystal growth model employed in Case 3 that solves for heat transfer, melt flow, and melt-crystal interface shape [43]. A depiction of these coupled models is shown in Figure 7(a), along with a converged solution for the crystal and melt in Figures 7(b) and (c). The isotherms show a vertical gradient across the melt and crystal. Significant convective effects again flatten the isotherms in the melt, as in Case 3. However, latent heat effects are much more pronounced in this system, leading to 15
a significantly curved melt-crystal interface and two-dimensional heat transfer in the crystal. The flow in the melt, depicted by streamlines in Figure 7(c), is dominated by two, nested flow cells rotating in a counter-clockwise direction. More details are available in [35, 42].
3.2
Iterative procedure
Analogous to the coupling relations used in the multi-domain one-dimensional problems, Equations (32) and (33), we construct the following boundary conditions along the boundary, ∂cf , between the crystal growth domain and the furnace domains, 1−α (Tc − Tf ) , α 1−β · qc − (Tf − Tc ) , β
ncf · (−∇Tc ) = ncf · qf +
(41)
ncf · (−∇Tf ) = ncf
(42)
where the subscripts c and f denote quantities computed by the crystal growth model, Sc , and the furnace model, Sf , respectively, and α and β are chosen parameters. As before, α = 0 and β = 0 cause pure temperature (Dirichlet) boundary conditions to be imposed, while α = 1 and β = 1 result in pure flux (Neumann) boundary conditions. Other values of α and β result in mixed (Robin type) conditions at the boundary. The iterative procedure is carried out using the same two-step scheme as described before; however, now the iterations involve the temperature field along the coupling boundary, which we represent as a vector of coupled temperature unknowns, Tc . Briefly, the two-step iteration proceeds beginning with a converged solution from the furnace model, Sf , at iteration k: Step 1: The temperature and flux distributions, Tf and qf , computed from the furnace model, Sf are input into Equation (41). This defines the boundary condition along boundary ∂cf , and the temperature field in the crystal growth model, Sc is calculated. Step 2: From the prior step, values for Tc and qc , now are available from the crystal growth model, Sc , and are substituted into Equation (42). This is imposed as a boundary condition along boundary ∂cf , and the temperature field in the furnace model, Sf is calculated. The resulting temperature fields for both models define the the k + 1st solution iterate, and iterations proceed until convergence or divergence. Overall, the procedure again defines a fixed-point iteration, (k) T(k+1) = F T , c c 16
(43)
(k+1)
where Tc
(k)
and Tc
denote successive iterates of the procedure and F is a non-linear vector
map. This fixed-point iteration relation for Tc will converge only if ||
∂F |T∗ || < 1, ∂Tc c
(44)
where || · || is a suitable matrix norm and T∗c denotes the values of the coupling temperatures evaluated at the fixed point (corresponding to the converged solution to the overall problem). The the sensitivity matrix of the mapping function F with respect to the temperature unknowns Tc , denoted as ∂F/∂Tc ≡ F0 , is a matrix with coefficients, Fij0 =
∂Fi , ∂Tc,j
(45)
where Fi denotes the ith component of of the vector function F and Tc,j denotes the jth component of vector Tc .
3.3
Results and discussion
Unlike the cases of the one-dimensional problems investigated in the prior sections, we do not know the analytical form of the fixed-point iterations for Cases 3 and 4. Therefore, we cannot compute convergence conditions for these models a priori. Instead, we investigate convergence by numerical experiments using various choices of α and β. We generate a convergence diagram for Case 3 over a range of α and β, similar to the ones presented for the one-dimensional problems, in Figure 8, where the outcomes of 31 different computations are indicated. In this diagram, filled boxes represent combinations of α, β for which the iterations diverged. Slow convergence, i.e., requiring over 10 iterations, are marked with filled circles. The boxes marked with X correspond to combinations of α, β that resulted in fast convergence, i.e., fewer than 10 iterations. This diagram depicts trends similar to those of the convergence plots for Case 2b with larger radiation numbers, as shown in Figure 5. Consistent with prior results, there are regions for this two-dimensional model where the iterations do not converge. We present just one result for Case 4, since the computational effort involved with this high-fidelity model is large. Based on the outcome of the prior cases, we expect that only some values of α and β will allow convergence. Furthermore, because one can only specify Dirichlet 17
boundary conditions in CrysVUn, we are restricted to setting β = 0. From the convergence plots, especially those in Figures 5 and 8, we speculate that only small values of α will lead to successful convergence. Indeed, as discussed more completely in [35,42], the iterations diverged for every case with α > 0.1. Figure 9 shows the convergence of the iterative scheme for β = 0 and α = 0.1 by plotting the L2 -norm of the temperature update along the coupling boundary as a function of the iteration number. Clearly, the iteration sequence is convergent.
4
Concluding remarks
We have presented the convergence behavior of several coupled models that represent the relevant physics of crystal growth from the melt under steady-state conditions. Specifically, the problems considered here are based on the of exchange heat transfer information, a mixture of temperatures and normal fluxes, across a physical boundary separating two different model domains. These models are built using a modular, or synthetic, approach by loosely coupling independent models for different subdomains with different physics via a block Gauss–Seidel iteration procedure. Such approaches are attractive because of their simplicity for solving problems that may be represented by linking existing codes, especially when the software of one or more of those codes is inaccessible. Analytical solutions and convergence criteria obtained for the one-dimensional problems considered here showed that convergence was possible only for certain combinations of the coupling parameters α and β. Even for the simplest, linear, one-dimensional problem, certain iteration schemes that seemed physically reasonable, such as providing a pure temperature condition to one model and a pure flux condition to the other model, were shown to be always divergent, irrespective of initial guesses. Thus, notions based on physical intuition may not be useful for predicting algorithm performance. Increasing the role of nonlinearities associated with the coupled models, here by increasing the amount of radiation heat transfer, generally decreased the possibilities for convergence. However, redefining the boundary coupling adjacent domains to involve only conduction heat transfer greatly improved the prospect for a convergent algorithm. This beneficial behavior persisted even with the introduction of an artificial domain at the coupling boundary small enough so that the original problem solution
18
was practically unaffected. Such an approach may be beneficial for other coupled heat transfer problems involving strong nonlinear effects. Two-dimensional, numerical models constructed from coupled, nonlinear furnace and crystal growth submodels were considered. Analytical results defining convergence behavior were not available for these two-dimensional models, but numerical experiments confirmed that they displayed very similar convergence characteristics compared to those exhibited by the nonlinear, one-dimensional problems. This understanding was useful for constructing a convergent algorithm for a more realistic model that coupled a commercial, finite-volume furnace heat transfer code with a sophisticated, finite-element crystal growth code. Finally, we believe that these results provide a useful framework to address issues of convergence for other heat transfer problems whose solution strategies involve the coupling of existing models.
Acknowledgments This material is based upon work supported by the National Science Foundation, under Grant No. 0201486, and the Minnesota Supercomputing Institute.
References [1] Kassab, A.J. and Aliabadi, M.H. (eds.), Advances in Boundary Elements: Coupled Field Problems, , Computational Mechanics, Boston 2001. [2] H. Morand, R. Ohayon: Fluid-Structure Interaction. John Wiley & Sons, Chichester, 1995. [3] Dowell, E., and Hall, K.C., Modeling of Fluid Structure Interaction, Annual Review of Fluid Mechanics, Vol. 33, pp. 445-490, 2001. [4] T. Belytschko, R. Mullen: Two-dimensional fluid-structure impact computations with regularization. Comp. Meth. Appl. Mech. Eng. 27 (1981) 139154 [5] J. Donea, S. Giuliani, J. P. Halleux: An arbitrary Lagrangian-Eulerian finite element method for transient fluid-structure interaction. Comp. Meth. Appl. Mech. Eng. 33 (1982) 689723 19
[6] K.-J. Bathe, H. Zhang, M. H. Wang: Finite element analysis of incompressible and compressible fluid flows with free surfaces and structural interactions. Computers & Structures 56 (1995) 193213 [7] J. Mouro, P. Le Tallec: Fluid structure interaction with large structural displacements. Comp. Meth. Appl. Mech. Eng. 190 (2001) 30393067 [8] Morton SA, Melville RB, Visbal MR. Accuracy and coupling issues of aeroelastic NavierStokes solutions on deforming meshes. J Aircraft 1998;35:798805. [9] S. Piperno, C. Farhat, B. Larrouturou: Partitioned procedures for the transient solution of coupled aeroelastic problems. Comp. Meth. Appl. Mech. Eng. 124 (1995) 79112 [10] Dubini, G.; Pietrabissi, R.; Montevecchi, F.M.: Fluid-structure interaction in bio-fluid mechanics; Med. Eng. Phys. 17(2) (1995), 609617. [11] J. Argyris, I. S. Doltsinis, P. Pimenta, H. W¨ ustenberg: Thermomechanical response of solids at high strains natural approach. Comp. Meth. Appl. Mech. Eng. 32 (1982) 357 [12] O. C. Zienkiewicz, D. K. Paul, A. H. C. Chan: Unconditionally stable staggered solution procedure for soil-pore fluid interaction problems. Int. J. Num. Meth. Eng. 26 (1988) 10391055 [13] E. Divo, E. Steinthorsson, F. Rodriguez, A.J. Kassab and J.S. Kapat, Glenn-HT/BEM Conjugate Heat Transfer Solver for Large-Scale Turbomachinery Models, NASA/CR2003212195 (April 2003). [14] E. Divo, E. Steinthorsson, A. J. Kassab and R. Bialecki, An iterative BEM/FVM protocol for steady-state multi-dimensional conjugate heat transfer in compressible flows, Engineering Analysis with Boundary Elements, Volume 26, Issue 5, May 2002, Pages 447-454. [15] Patankar, S.V., A Numerical Method for Conduction in Composite Materials, Flow in Irregular Geometries and Conjugate Heat Transfer, , Proc. 6th. Int. Heat Transfer Conf. NRC Canada, and Hemisphere Pub. Co., New York, Vol. 3, pp. 297-302, 1978.
20
[16] Comini, G., Saro, O. and Manzan, M., A Physical Approach to Finite Element Modeling of Coupled Conduction and Convection, , Vol. 24, Numerical Heat Transfer, Part. B. pp. 243-261, 1993. [17] Shyy, W. and Burke, J., Study of Iterative Characteristics of Convective Diffusive and Conjugate Heat Transfer Problems, , Vol. 26, Numerical Heat Transfer, Part. B. pp. 21-37, 1994. [18] Hassan, B., Kuntz, D. and Potter, D.L., Coupled Fluid/Thermal Prediction of Ablating Hypersonic Vehicles, . AIAA Paper 98-0168, 1998. [19] Wei Shyy. Multi-scale computation of heat transfer with moving boundaries. Int. J. Heat Fluid Flow, 23:278–287, 2002. [20] F. Dupret, P. Nicod`eme, Y. Ryckmans, P. Wouters, and M.J. Crochet. Global modelling of heat transfer in crystal growth furnaces. Int. J. Heat Mass Transfer, 33:1849, 1990. [21] M. Kurz, A. Pusztai, and G. M¨ uller. Development of a new powerfull computer code CrysVUN++ especially designed for fast simulation of bulk crystal growth processes. J. Crystal Growth, 198:101, 1999. [22] R. Backofen, M. Kurz, and G. M¨ uller. Process modelling of the industrial VGF crystal growth process using the software package CrysVUN++. J. Crystal Growth, 199:210, 2000. [23] R.A. Brown. Theory of transport processes in single crystal growth from the melt. AIChE Journal, 34(6):881, 1988. [24] G. M¨ uller. Modeling of crystal growth from the melt. In P. Vincenzini and A. Lami, editors, Computational Modelling and Simulations of Materials. Techna Srl., 2003. [25] C.W. Lan. Recent progress of crystal growth modeling and growth control. Chem. Eng. Sci., 59:1437–1457, 2004. [26] A. Yeckel and J. J. Derby. Computer modelling of bulk crystal growth. In P. Capper, A. Willoughby, and S. Kasap, editors, Bulk Crystal Growth of Electronic, Optical, and Optoelectronic Materials. John Wiley and Sons, 2004. 21
[27] J. Baumgartl, A. Bune, K. Koai, G. M¨ uller, and A. Seidl. Global simulation of heat transport, including melt convection in a Czochralski crystal growth process - combined finite element / finite volume approach. Mat. Sci. Eng., A173:913, 1993. [28] H. Ouyang and Wei Shyy. Multi-zone simulation of the Bridgman growth process of β-NiAl crystal. Int. J. Heat Mass Transfer, 39(10):2039–2051, 1996. [29] H. Ouyang and Wei Shyy. Numerical simulation of CdTe vertical Bridgman growth. J. Crystal Growth, 173:352–366, 1997. [30] C.M. Tom´as and V. Mu´ noz. CdTe crystal growth process by the Bridgman method: numerical simulation. J. Crystal Growth, 222:435–451, 2001. [31] J. Virbulis, Th. Wetzel, A. Muiznieks, B. Hanna, E. Dornberger, E. Tomzig, A. M¨ uhlbauer, and W. V. Ammon. Numerical investigation of silicon melt flow in large diameter CZ-crystal growth under the influence of steady and dynamic magnetic fields. J. Crystal Growth, 230:92–99, 2001. [32] D. P. Lukanin, V. V. Kalaev, Yu. N. Makarov, T. Wetzel, J. Virbulis, and W. von Ammon. Advances in the simulation of heat transfer and prediction of the melt-crystal interface shape in silicon CZ growth. J. Crystal Growth, 266:20–27, 2004. [33] A. Yeckel, A. Pandy, and J.J. Derby. Representing realistic complexity in numerical models of crystal growth: Coupling of global furnace modeling to three-dimensional flows. In G. De Vahl Davis and E. Leonardi, editors, Advances in Computational Heat Transfer II, volume 2, pages 1193–1200. Begell House Inc., New York, 2001. [34] A. Yeckel, G. Compere, A. Pandy, and J.J. Derby. Three-dimensional imperfections in a model vertical Bridgman growth system cadmium zinc telluride. J. Crystal Growth, 263:629–644, 2004. [35] A. Pandy, A. Yeckel, M. Reed, C. Szeles, M. Hainke, G. M¨ uller, and J.J. Derby, “Analysis of the growth of cadmium zinc telluride in an electrodynamic gradient freeze furnace via a self-consistent, multi-scale numerical model ,” J. Crystal Growth in press (2005).
22
[36] Codina R., Cervera M. Block-iterative algorithms for nonlinear coupled problems. In: Papadrakakis M, Bugeda G, editors. Advanced Computational Methods in Structural mechanics. Barcelona: CIMNE; 1996. [37] Chan T.F. An approximate newton method for coupled nonlinear systems. SIAM J Numer Anal 1985;22:90413. [38] Artlich S, Mackens W. Newton-coupling of fixed point iterations. In: Hackbusch W, Wittum G, editors. Numerical Treatment of Coupled Systems. Braunschweig: Vieweg; 1995. [39] Arnold, M. Constraint partitioning in dynamic iteration methods. Z. Angew. Math. Mech., 81(Supplement 3) S735–S738 (2001). [40] H. G. Matthies, J. Steindorf, Partitioned but strongly coupled iteration schemes for nonlinear fluidstructure interaction Computers and Structures 80 (2002) 19911999 [41] Menck, J. An Approximate Newton-Like Coupling of Subsystems. Z. Angew. Math. Mech. 82 (2002) 2, 101–114. [42] A. Pandy. Global modeling of bulk crystal growth of cadmium zinc telluride in industrial VB/VGF systems. PhD thesis, University of Minnesota, 2004. [43] A. Yeckel and R.T. Goodwin. CATS2D (Crystallization and Transport Simulation), User Manual. Unpublished (available at http://www.msi.umn.edu/yeckel/cats2d.html), 2003.
23
Figure Captions Figure 1 Schematic representation of the computational domains for the one-dimensional model problems. (a) Domains for Cases 1 and 2a. (b) Domains for Case 2b.
Figure 2 Temperature profiles for the one-dimensional model problems. (a) Solutions for Cases 1 and 2a. (b) Solutions for Case 2b for different values of Ωd domain size, d.
Figure 3 Convergence behavior for Case 1 as a function of coupling parameters α and β. Shaded regions denote iteration algorithms that diverge. Contours of |F 0 | mark regions according to speed of convergence.
Figure 4 Convergence behavior for Case 2a as a function of coupling parameters α and β. Shaded regions denote iteration algorithms that diverge. The region of convergence shrinks with increasing radiation number, R.
Figure 5 Convergence behavior for Case 2b as a function of coupling parameters α and β. Shaded regions denote iteration algorithms that diverge. The region of convergence shifts, but does not appreciably shrink, with increasing radiation number, R.
Figure 6 Schematic representation of the computational domains for the two-dimensional model problem, Case 3. (a) Domain for furnace model, showing isotherms of temperature field. (b) Domain for crystal growth model, showing isotherms of temperature field. (c) Domain for crystal growth model, showing streamlines of melt flow field.
Figure 7 (a) Schematic representation of the computational domains for the two-dimensional model problem, Case 4, showing finite-volume furnace model on left and finite-element crystal growth model on right. (b) Domain for crystal growth model, showing isotherms of temperature field. (c) Domain for crystal growth model, showing streamlines of melt flow field.
24
Figure 8 Diagram of convergence results for Case 3, tabulated as a function of coupling parameters α and β. Filled squares denote iteration algorithms that diverged. Filled circles indicate iteration algorithms that converged but required more than 10 iterations. Algorithms that converged in fewer than 10 iterations are marked with an X.
Figure 9 The L2 -norm of the update to dimensionless interface temperatures is plotted as a function of iteration number for Case 4 with α = 0.1 and β = 0. The algorithm is convergent.
25
Figure 1: Pandy et al.
26
Figure 2: Pandy et al. 27
Figure 3: Pandy et al.
28
Figure 4: Pandy et al.
29
Figure 5: Pandy et al.
30
Figure 6: Pandy et al.
31
32
Figure 7: Pandy et al.
Figure 8: Pandy et al.
33
Figure 9: Pandy et al.
34
