SOLUTION ALGORITHM
Following from: Discrete ordinates and finite volume methods
Discrete Form of the Governing Equations
The mathematical formulation of the discrete ordinates method (DOM) and finite volume method (FVM) has been presented in the article “Mathematical formulation.” In that article, it was shown that, in the case of Cartesian coordinates, and when the step scheme is employed, the discretized equations for the DOM may be written as follows for a control volume centered at grid node P:
(1) |
for m = 1, 2, ... M. If the FVM is used, the corresponding discretized equations take the following form:
(2) |
for m = 1, 2, ... M. In both cases, the equations may be rewritten as follows:
(3) |
where the summation extends over all the neighbors of grid node P, which will be denoted as E (east), W (west), N (north), S (south), T (top), and B (bottom). Coefficient a_{P} and the source term b are given by
(4a) |
(4b) |
for the DOM and by
(5a) |
(5b) |
for the FVM. The expressions for coefficients a_{i} depend on the direction/control angle under consideration. For example, if all the direction cosines of the mth direction are positive in the DOM, then I^{m}_{x,in} = I_{W}, I^{m}_{y,in} = I_{S}, I^{m}_{z,in} = I_{B}, and
(6) |
Similarly, if D^{m}_{cx} > 0, D^{m}_{cy} > 0, and D^{m}_{cz} > 0 for the mth control angle in the FVM, then I^{m}_{x,in} = I_{W}, I^{m}_{y,in} = I_{S}, I^{m}_{z,in} = I_{B}, and
(7) |
In these equations, subscripts W, E, S, N, B, and T identify the west, east, south, north, bottom and top neighbors of grid node P, respectively.
Standard Solution Algorithm
These equations show that the different directions/control angles are only coupled via the in-scattering term (hereafter, we will refer only to directions, even though the appropriate terminology for the FVM is control angle) and, in the case of nonblack boundaries, via the boundary conditions. This suggests an iterative solution of the full set of discretized governing equations in such a way that the radiation intensities at the N_{x} × N_{y} × N_{z} control volumes are successively calculated for directions 1, 2, ..., m, ..., M. When the mth direction is considered, the radiation intensities associated to the remaining directions in the in-scattering term, which appear in source term b in Eqs. (4b) and (5b), are treated explicitly. This means that the radiation at grid node P along a direction l ≠ m is taken from the previous iteration if direction l has not yet been treated in the current iteration, or from the current iteration in the opposite case.
Examination of the discretized governing equations further shows that the coefficients a_{i} associated to the cell faces located downstream of the direction under consideration (e.g., E, N, and T for a direction with positive direction cosines in the case of the DOM, and a control angle with D^{m}_{cx} > 0, D^{m}_{cy} > 0, and D^{m}_{cz} > 0 for the FVM) are equal to zero, and only the coefficients a_{i} associated to the upstream cell faces are different from zero. Accordingly, the solution of the discretized set of governing equations may be efficiently carried out using a Gauss-Seidel method that solves the equations following an optimal order.
The optimal marching procedure starts from a control volume located at a corner of the computational domain. That corner is selected from the sign of the direction cosines of the direction under consideration, in such a way that the cell faces that merge at that corner are upstream cell faces coincident with the boundaries of the enclosure, i.e., s · n_{f} > 0 for the three boundary cell faces. For example, the calculations for all directions with positive direction cosines start from the control volume located at the corner where the west, south, and bottom boundaries meet. In each iteration, and for every direction leaving from the selected control volume, the radiation intensities at the boundary cell faces are either known or guessed, based on the values computed in the previous iteration. The surface irradiation is neglected in the first iteration. Therefore, the radiation intensity at that control volume and direction are readily determined from Eqs. (1) or (2).
The solution for all the remaining control volumes proceeds in the direction of orientation of the direction cosines, in such a way that the upstream cell face intensities of the visited control volume are available either from the boundary conditions or from the calculations performed for the previously visited control volumes. In the case of positive directions’ cosines, the control volumes are successively visited in three nested loops from west to east, from south to north, and from bottom to top, the order of the loops being irrelevant as far as the computed solution is concerned. Then, similar calculations are performed for all the other directions. After all the directions have been treated, the radiation intensities leaving the boundary surfaces are updated using the boundary conditions. The iteration process continues until the convergence criterion has been satisfied.
The solution algorithm may be summarized as follows (e.g., Fiveland, 1984):
- Set equal to zero the radiation intensity for all control volumes and all directions.
- Set equal to zero the irradiation at all boundary cell faces and the incident radiation at all control volumes.
- Loop over all directions from m = 1 to m = M, and proceed as follows for every direction:
- Select the order of sweeping the control volumes. The calculations start from the control volume located at the corner of the enclosure such that s · n_{f} > 0 for the three boundary cell faces. The sweeping order is such that all the control volumes are visited following the direction of orientation of the direction cosines, as exemplified above.
- Loop over all control volumes and proceed as follows for every control volume:
- Calculate I^{m}_{x,in}, I^{m}_{y,in}, and I^{m}_{z,in} either from the boundary conditions, in the case of cell faces lying on the boundary of the computational domain, or from the radiation intensities leaving the upstream neighboring control volumes, (I^{m}_{x,out})_{W}, (I^{m}_{y,out})_{S}, and (I^{m}_{z,out})_{B}, in the case of cell faces in the interior of the domain. In the former case, the article “Boundary conditions” explains the calculation procedure. In the last case, the radiation intensities at the upstream cell faces are readily obtained from (I^{m}_{x,in})_{P} = (I^{m}_{x,out})_{W}, (I^{m}_{y,in})_{P} = (I^{m}_{y,out})_{S}, and (I^{m}_{z,in})_{P} = (I^{m}_{z,out})_{B}.
- Calculate the in-scattering term taking the radiation intensity at grid node P along a direction l ≠ m from the previous iteration if direction l has not yet been treated in the current iteration, or from the current iteration in the opposite case.
- Calculate I^{m}_{P} from Eq. (1) for the DOM or from Eq. (2) for the FVM.
- Calculate I^{m}_{x,out}, I^{m}_{y,out}, and I^{m}_{z,out}. In the case of the step scheme, I^{m}_{x,out} = I^{m}_{y,out} = I^{m}_{z,out} = I^{m}_{P}.
- Calculate the contribution of the mth direction to the incident radiation at the control volume under consideration using Eq. (14) of the article “Mathematical formulation” for the DOM or Eq. (29) of the same article for the FVM.
- Calculate the contribution of the mth direction to the heat flux incident at all cell faces on boundary walls of the computational domain using Eq. (13) of the article “Mathematical formulation” for the DOM or Eq. (28) of the same article for the FVM.
- Check if the convergence criteria have been satisfied. If not, increase the iteration counter and return to step 3.
In general, the solution procedure is iterative, not only due to the in-scattering term, but also because the radiation intensity leaving a boundary is generally unknown. However, if there is no scattering and the boundaries are black, with prescribed temperature, then the radiation intensity leaving the boundary is given by σT^{4}_{W} / π. In such a particular case, there is no need for iterations, and the numerical solution is obtained after completion of step 3 of the algorithm given above.
In the case of radiative equilibrium, the temperature field is unknown, and the radiative transfer equation needs to be solved simultaneously with the equation of radiative equilibrium, which is given by (Modest, 2003) as
(8) |
where T is the temperature of the medium and σ is the Stefan-Boltzmann constant. In radiative equilibrium, ∇ · q = 0. The algorithm outlined above may be easily modified to address radiative equilibrium problems. The temperature of the medium is guessed in the first iteration to allow the calculation of radiation intensities in step 3(b)(iii). Then, after step 3(c) of the algorithm, the temperature field is recalculated from Eq. (8). The new temperature field is used in the second iteration, and successively updated, in every iteration, by again applying Eq. (8) after step 3(c).
Convergence Criteria
Different convergence criteria may be used, for example,
(9) |
(10) |
(11) |
In these equations, the superscript k denotes the iteration number and δ_{1}, δ_{2}, and δ_{3} are prescribed tolerances. Equation (9) demands the maximum value, over all control volumes and all directions, of the modulus of the difference between the radiation intensity in two successive iterations, normalized by the radiation intensity at the current iteration, to be smaller than a prescribed tolerance. Equation (10) requires that the maximum value, over all control volumes, of the difference between the incident radiation in two successive iterations, normalized by the incident radiation at the current iteration, is smaller than a prescribed tolerance. The convergence criterion expressed by Eq. (11) is similar, but is based on the incident heat flux on the boundary rather than the incident radiation in the domain. Other quantities may be used as a convergence criterion, such as the net heat flux at the boundary or the radiative heat source of the energy conservation equation, given by the divergence of the radiative heat flux. It is also common to combine two or more of the above criteria.
Instead of requiring the maximum value (L_{∞} norm, also referred to as maximum norm) of the above quantities to be lower than a prescribed tolerance, it is common to require that the sum of a quantity extended over all control volumes (and directions, in the case of the radiation intensity) is lower than a given tolerance (L_{1} norm). Sometimes, L_{2} norm, defined as the square root of the sum of the squares of a quantity extended over all control volumes, is used. It is also possible to use absolute errors rather than relative errors in the above criteria. In such a case, the denominators in Eqs. (9)–(11) are omitted. The absolute errors may be normalized by a fixed quantity, characteristic of the problem under investigation. In any case, the tolerance should be small enough to guarantee that the solution is fully converged and independent of the selected convergence criteria.
Modifications for High-Order Resolution Schemes
Equations (1) and (2) hold for the step discretization scheme. It was stated in the article “Spatial discretization schemes” that the step scheme, despite its simplicity, has major shortcomings, and many other spatial discretization schemes were presented. All of them may be easily implemented using the deferred correction procedure, expressed by Eq. (22) of that article. In this way, Eqs. (1) and (2) are written as follows (see details in Coelho, 2002), respectively,
(12) |
(13) |
for m = 1, 2, ... M. In these equations, I^{m}_{x,out}, I^{m}_{y,out}, and I^{m}_{z,out} are the cell face radiation intensities leaving the control volume centered at grid node P, and are evaluated using the high-order discretization scheme. Equations (3), (4a), and (5a) remain unchanged for high-order resolution schemes, but Eq. (4b) for the DOM and Eq. (5b) for the FVM are modified as follows, respectively:
(14) |
(15) |
The new terms in Eqs. (12)–(15) associated to a high-order spatial discretization scheme are treated explicitly, i.e., they are evaluated using the values of the radiation intensities from the previous iteration. The explicit treatment of the new terms slows down the convergence process, but pays off in terms of increased accuracy and computational efficiency in comparison to the step scheme.
Modifications for Body-Fitted and Unstructured Meshes
In the case of body-fitted structured meshes or unstructured meshes, the DOM and FVM discretized equations for the step scheme were presented in the article “Mathematical formulation” [Eqs. (10) and (25), respectively]. If a high-order resolution scheme is used, these equations are written as follows:
(16) |
(17) |
for m = 1, 2, ... M. Here, I^{m}_{U,f} and I^{m}_{HO,f} are the radiation intensities for direction m and at cell face f, which are evaluated from the step and the high-order resolution scheme, respectively. Therefore, I^{m}_{U,f} = I^{m}_{P} in the last term of these equations. This term is evaluated explicitly, as in the case of Cartesian coordinates.
The standard solution algorithm described above may be extended to unstructured grids with minor changes. First, the sweeping order of the control volumes in step 3(a) of the solution algorithm is determined for every direction, as described below, and stored. While this procedure is straightforward for Cartesian grids, it is a little more involved for unstructured meshes, and is carried out in a preprocessing stage. Hence, step 3(a) of the solution algorithm consists of retrieving the sweeping order for the direction under consideration from the stored data. Second, in step 3(b)(i), the values of I^{m}_{HO,f} in the first term of the right-hand side of Eqs. (16) and (17) are determined from the boundary conditions, in the case of a cell face lying on the boundary, or set equal to the radiation intensity at that cell face, which was formerly calculated for the upstream control volume, in the case of an internal cell face. Third, in step 3(b)(iv), the values of I^{m}_{HO,f} are determined using the high-order resolution scheme.
The sweeping order may be found as described, e.g., in Moder et al. (2000), Joseph et al. (2005), and Asllanaj et al. (2007). Here, the method described in Moder et al. (2000) is summarized. First, the direction or control angle is defined. All interior cell faces of all control volumes are flagged as unknown, while all boundary cell faces are flagged as known. Then, all the control volumes are visited, in an arbitrary order, and the control volume with more flagged known faces such that s^{m} · n_{f} < 0 (DOM) or D^{m}_{cf} < 0 (FVM) is selected and stored (if there is more than one control volume that satisfies this condition, any one of them may be selected, the choice being arbitrary), and all its cell faces are flagged as known. This last process is repeated, excluding the control volumes formerly selected for the direction under consideration, until all the control volumes have been selected, i.e., the sweeping order defined. The full process is then repeated for the remaining directions.
An alternative solution has been employed by Chai et al. (1995), which avoids the definition of a sweeping order for every direction. They sweep all control volumes from all corners of the domain (four, in their 2D calculations), as in the original algorithm. This implies that the radiation intensity at some upstream cells may be unavailable, and needs to be taken from the previous iteration, which is likely to slow down the convergence process, even though the authors claim rapid convergence for most situations.
Acceleration Methods
The Gauss-Seidel method employed in the solution algorithm described above often converges fast because, due to the nature of propagation of radiation, the coefficients associated to the downstream cell faces (relative to the direction of propagation) of a control volume are zero whenever the step scheme is employed. This allows the choice of an optimal sequence for the solution of the discrete equations such that only the in-scattering source term and the deferred correction terms are treated explicitly. However, the convergence rate degrades if these terms are large, as well as when the temperature of the medium is unknown.
The radiation intensities in the in-scattering and in the deferred correction terms are calculated using the values available from the previous iteration. This explicit treatment causes a significant increase in the number of iterations required to achieve convergence if the scattering coefficient is large, or the contribution from the deferred correction terms is important. Similarly, the convergence rate decreases with the increase of the absorption coefficient of the medium when the temperature field is unknown, e.g., in radiative equilibrium problems. In such a case, the RTE and the conservation equation for energy are solved simultaneously. Acceleration methods to overcome or mitigate the decrease of the convergence rate have been proposed by Chui and Raithby (1992), Fiveland and Jessee (1996), Koo et al. (2001), and Raithby and Chui (2004), as described below. Other solution algorithms based on the Krylov subspace or multigrid methods are addressed in the following subsections.
Chui and Raithby (1992) proposed an implicit scheme based on the solution of a new set of implicit equations that provides an improved estimation of the radiation intensities in the in-scattering term. In their method, the discrete form of the RTE for every control volume is summed over all the solid angles, yielding a set of equations that expresses the conservation of radiative energy, which is equivalent to integration of Eq. (8) over every control volume. Then, a directionally dependent phase weight is defined for a grid node as the ratio of the radiation intensity to the average radiation intensity. The radiation intensity along a given direction is written as the product of the phase weight for that direction by the average radiation intensity, and inserted into the formerly derived equations. This results in a new set of implicit equations for the average radiation intensity in every control volume, where all the phase weights are included in the coefficients, and there are no directionally dependent terms to be treated explicitly in the source term. These equations may be solved together with the equations for conservation of energy to obtain the temperature and the average radiation intensity fields. In the case of radiative equilibrium, the solution of the new set of implicit equations directly yields the average radiation intensity field.
In the solution algorithm, after the radiation intensity at all control volumes and for all directions has been determined for the current iteration, an intermediate average radiation intensity field and the phase weights are computed. Then, the coefficients of the implicit equations for the average radiation intensity field are calculated using the new phase weights, and the set of implicit equations is solved, yielding an improved estimate of the average radiation intensity field. The new average radiation intensity field and the phase weights are used to calculate the in-scattering term. The full process is repeated until the convergence criteria are satisfied. This acceleration method has been applied in the framework of the FVM to simple 2D radiative equilibrium problems, and it was found that the convergence rate is greatly accelerated in comparison with the standard algorithm when the optical thickness of the medium is large.
Fiveland and Jessee (1996) compared three different acceleration schemes formerly applied to neutron transport problems, namely, successive overrelaxation of the in-scattering term, mesh rebalance, and synthetic acceleration. Although successive overrelaxation and synthetic acceleration improve convergence, mesh rebalance method performs significantly better. The mesh rebalance method is similar to that presented by Chui and Raithby (1992), the concept of phase weights being replaced by correction factors. However, it was found that the mesh rebalance method fails if the spatial discretization becomes finer and the scattering coefficient is large. In order to overcome this failure, a coarse the mesh rebalance method was proposed. This is identical to the original version, except that a coarser mesh is used for the rebalance equations (corresponding to the new implicit equations in the method of Chui and Raithby) rather than for the discrete equations for the radiation intensity. This means that the control volumes are grouped to form a coarser mesh, and the correction factors are defined for the coarser mesh rather than the finer one. An adaptive rebalance scheme was proposed to select the coarse mesh such that the coarse control volumes have optical thicknesses between 1 and 2.
More recently, Koo et al. (2001) presented a fully implicit method aimed at the improvement of convergence when the temperature field is unknown and the absorption coefficient of the medium is large. The RTE and the equation for energy conservation are solved simultaneously. The overall energy balance procedure can be expressed by describing the radiation intensity in terms of the blackbody radiation intensity, yielding a set of equations for the blackbody radiation intensity field. A successive iteration method is devised to iteratively solve this set of equations using an explicit, a semiexplicit, or a fully implicit method. It is shown that the fully implicit method reduces the number of iterations, but the total computational time may be increased. In addition, the number of explicit iterations increases asymptotically with mesh refinement.
Raithby and Chui (2004) reported an improved version of the implicit scheme of Chui and Raithby (1992), which was found to fail for pure scattering in the case of optically intermediate media and fine grids. The reason for this failure was attributed to the lack of diagonal dominance of the set of equations for the average radiation intensity, and the strong coupling between the equations for the radiation intensities and the average radiation intensities. A simple remedy was proposed to overcome this problem, based on a modification of that system of equations, and on the underrelaxation of the phase weights.
Krylov Subspace Iterative Methods
In Krylov subspace methods, the solution of the system of governing equations, Ax = b, is obtained by minimizing the residual over the so-called p-Krylov subspace, which is spanned by vectors r_{o}, Ar_{o}, A^{2}r_{o}, ..., A^{p}r_{o}, where r_{o} is the initial residual. In general, these methods are optimally suited for large sparse systems of equations, and converge faster than stationary iterative methods, such as the Gauss-Seidel method, and the convergence rate may be improved using preconditioning.
The conditioned conjugate gradient squared (CCGS) method of Sonneveld (1989) was used by Ben Salah et al. (2005a,b) and Grissa et al. (2007, 2010). Ben Salah (2005b) found no significant variation of iterations and CPU time with the increasing of the scattering coefficient when the CCGS solver was used. However, Axelsson (1996) claims that Lanczos-type methods, such as this one, are not based on any minimization property, and may exhibit rather erratic convergence behavior.
Krishnaprakas et al. (2001) compared three different generalized conjugate gradient methods of the Krylov subspace family for the solution of 2D radiation problems using the FVM. They used the generalized conjugate gradient (GCG) method of Young and Jea (1980), the generalized minimum residual (GMRES) method of Saad and Schultz (1986), and the generalized conjugate gradient least-squares (CGS-LS) method of Axelsson (1996). All of these methods were employed using successive underrelaxation preconditioning, and restarted after p iterations, where p denotes the dimension of the Krylov subspace. The equations for all directions and all control volumes were solved simultaneously, which is only feasible due to the small number of equations (grid sizes and directions) employed. It was found that preconditioning greatly improves the convergence rate, but the underrelaxation parameter plays a major role, and there is no rule to obtain the optimum value.
Godoy and Desjardin (2010) used a Newton-Krylov iterative method (GMRES) to solve 3D radiation problems using the DOM. The equations for all directions and all control volumes were solved simultaneously using a domain decomposition strategy and parallel computing (see article “Parallel implementation”). Otherwise, it would not have been possible to store all the data in a single processor, as pointed out by Murthy and Mathur (1999). Once an initial guess of the radiation intensities is obtained, the iterative procedure consists of a step of Newton’s method, followed by p iterations of GMRES, which are repeated until convergence is achieved.
Multigrid Methods
Multigrid methods (Wesseling, 1992) are based on the idea that iterative methods generally effectively reduce high-frequency error components, but not low-frequency error components. However, these may be efficiently removed on a coarse mesh, which suggests the acceleration of convergence of an iterative method by global correction based on the solution carried out using coarser grids. Two different kinds of multigrid methods may be distinguished, namely, full approximation storage (FAS) (Brandt, 1977), also referred to as geometric multigrid, and algebraic multigrid methods (Hutchinson and Raithby, 1986).
Multigrid methods have been employed by Murthy and Mathur (1998, 1999) and Balsara (2001) to improve the convergence rate of radiative transfer problems. Murthy and Mathur (1998) used an algebraic multigrid procedure that constructs coarse-level equations by grouping a number of fine-level discrete equations. The residuals from the fine-level equations were restricted to form the source terms of the coarse-level equations, while the solution of the coarse-level equations was prolonged to yield corrections at the fine level. A V cycle was used along with a Gauss-Seidel relaxation procedure at each multigrid level. Coupling of the RTE and the energy equation was also addressed. A convergence acceleration procedure, referred to as the coupled ordinates method, based on an algebraic multigrid method, was reported in Murthy and Mathur (1999) and applied to solve the coupled RTE and energy equations. The solution was carried out on a sequence of nested meshes formed by grouping the control volumes of the finest mesh. A V cycle was used to prolongate and restrict residuals between mesh levels. The discrete equations on the coarser meshes were derived by a mixed geometric/additive correction strategy. At each control volume, the energy and the RTE equations were solved in a point-coupled fashion by the inversion of a local matrix, the values of the spatial neighbours being assumed known at prevailing values. This point-coupled procedure was used as a relaxation sweep in the multigrid method. Great accelerations were obtained for a range of optical thicknesses.
Balsara (2001) used the FAS multigrid method in conjunction with a Newton-Krylov method (GMRES). The RTE was solved for 2D problems with isotropic scattering, but the coupling with the energy equation was not addressed. A good convergence rate was found for both transparent and participating media, including strongly absorbing and/or scattering media. The convergence rate was even improved by increasing the scattering coefficient.
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Использованная литература
- Axelsson, O., Iterative Solution Methods, New York: Cambridge University Press, 1996.
- Asllanaj, F., Feldheim, V., and Lybaert, P., Solution of Radiative Heat Transfer in 2-D Geometries by a Modified Finite-Volume Method Based on a Cell Vertex Scheme Using Unstructured Triangular Meshes, Numer. Heat Transfer, Part B, vol. 51, pp. 97−119, 2007.
- Balsara, D., Fast and Accurate Discrete Ordinates Methods for Multidimensional Radiative Transfer, Part I, Basic Methods, J. Quant. Spectrosc. Radiat. Transfer, vol. 69, pp. 671−707, 2001.
- Ben Salah, M., Askri, F., and Ben Nasrallah, S., Unstructured Control-Volume Finite-Element Method for Radiative Heat Transfer in a Complex 2-D Geometry, Numer. Heat Transfer, Part B, vol. 48, no. 5, pp. 477−497, 2005a.
- Ben Salah, M., Askri, F., Rousse, D., and Ben Nasrallah, S., Control Volume Finite Element Method for Radiation, J. Quant. Spectrosc. Radiat. Transfer, vol. 92, no. 1, pp. 9−30, 2005b.
- Brandt, A., Multi-Level Adaptive Solutions to Boundary Value Problems, Math. Comput., vol. 31, no. 138, pp. 333−390, 1977.
- Chai, J. C., Parthasarathy, G., Lee, H. S., and Patankar, S. V., Finite Volume Radiative Heat Transfer Procedure for Irregular Geometries, J. Thermophys. Heat Transfer, vol. 9, no. 3, pp. 410−415, 1995.
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