International Conference on Mathematics and Computational Methods Applied to Nuclear Science & Engineering (M&C 2013) Sun Valley, Idaho, USA, May 5-9, 2013, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2013)
MONTE CARLO SOLUTION METHODS IN A MOMENT-BASED SCALE-BRIDGING ALGORITHM FOR THERMAL RADIATIVE TRANSFER PROBLEMS: COMPARISON WITH FLECK AND CUMMINGS H. Park, J. D. Densmore*, A. B. Wollaber, D. A. Knoll, R. M Rauenzahn, Los Alamos National Laboratory Los Alamos, NM 87545
[email protected];
[email protected];
[email protected];
[email protected] *Bettis Atomic Power Laboratory West Mifflin, PA 15122
[email protected]
ABSTRACT We have developed a moment-based scale-bridging algorithm for thermal radiative transfer problems. The algorithm takes the form of well-known nonlinear-diffusion acceleration which utilizes a low-order (LO) continuum problem to accelerate the solution of a high-order (HO) kinetic problem. The coupled nonlinear equations that form the LO problem are efficiently solved using a preconditioned Jacobian-free Newton-Krylov method. This work demonstrates the applicability of the scale-bridging algorithm with a Monte Carlo HO solver and reports the computational efficiency of the algorithm in comparison to the well-known Fleck-Cummings algorithm. Key Words: scale-bridging algorithm, thermal radiative transfer, Monte Carlo method, Jacobian-free Newton-Krylov method
1. INTRODUCTION Thermal radiative transfer (TRT) problems appear in many different physical applications such as combustion systems, inertial confinement fusion, and astrophysics. Many realistic simulations require the solution of a kinetic (transport) problem in some regions, while the continuum description (e.g., diffusion approximation) may be adequate in the remaining regions. The transport equation has seven independent variables, which makes its solution computationally intensive. Furthermore, scattering and/or absorption-emission physics strongly couple energy and angular variables in thick diffusive regions such that standard fixed-point iteration converges very slowly. To seamlessly couple the high-order (HO) kinetic and low-order (LO) continuum scale physics, we have developed a moment-based, scale-bridging algorithm [1] [2]. The algorithm utilizes well-known nonlinear diffusion acceleration [3] [4] in order to efficiently converge absorption-emission physics at every time step. Moreover, recent work has demonstrated that, through a predictor-corrector approach, the algorithm can achieve a temporal second-order convergence rate. The other advantage of this algorithm is that upon nonlinear convergence between the HO and LO systems, the solution of the discrete LO system is guaranteed to be same as the solution of the HO system, which enables a tightly-coupled multiphysics simulation through the LO system.
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Recently, we have extended the moment-based acceleration concept to a Monte Carlo (MC) HO solver. The combination of continuous energy-deposition particle tracking and an asymptotic shortcut have enabled using the MC method as a HO solver. In this work, we compare the algorithmic performance of a MC moment-based, scale-bridging algorithm with the well-known Fleck-Cummings linearized Monte Carlo (FC-MC) method (often referred to as the Implicit Monte Carlo (IMC) method elsewhere)[5].
2. THERMAL RADIATIVE TRANSFER EQUATION We are interested in solving the following set of (grey) TRT equations, 1 ∂I ˆ · ∇I + σI = σac T 4 , +Ω c ∂t 4π ∂T ρcv + σacT 4 − ∂t
Z
dΩσI = 0,
(1)
(2)
4π
where I and T are the angular intensity and material temperature, σ, ρ, cv , a and c are the opacity, material density, specific heat, radiation constant, and speed of light, respectively. With proper initial and boundary conditions, Eqs. (1) and (2) are well defined. We refer to Eq. (1) as the HO equation. Eqs. (1) and (2) are nonlinearly coupled through the emission term (σacT 4 ). One might try to solve the HO system with the following fixed-point iteration for I n+1 and T n+1 (using backward Euler discretization for simplicity of presentation),
k In+1 − In ˆ σac 4 k−1 k k + Ω · ∇In+1 + σIn+1 = (T )n+1 , c∆t 4π Z k Tn+1 − Tn k 4 k ρcv dΩσIn+1 = 0. + σac(T )n+1 − ∆t 4π
(3)
(4)
Here, the superscript k denotes the iteration level of the fixed-point iteration, while the subscript n denotes the time level. Because the opacity and the emission term are strong functions of the material temperature, a fixed-point iteration can be slow to converge in highly absorbing media with a large material temperature. Rather than solving the nonlinear system directly via a fixed-point iteration, many approaches utilize linearization, such as the one of Fleck and Cummings [5], to avoid nonlinear iterations.
2.1. Linearized TRT Equation of Fleck and Cummings We briefly review the linearization method introduced by Fleck and Cummings [5]. The first step is to rewrite Eq. (2) in terms of Θ = aT 4 as follows, 1 ∂Θ + σcΘ − β ∂t
Z
dΩσI = 0,
(5)
4π
where β=
∂Θ ∂T
ρcv
=
4aT 3 . ρcv
(6) 2/12
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Integrating over the time interval [tn , tn+1 ] yields Z tn+1 Z dΩσI − βσcΘ , dt β Θn+1 − Θn = 4π tn Z ¯ ¯ dΩI − c [αΘn+1 + (1 − α)Θn ] , ≈ βσ ¯ ∆t
(7)
4π
R ¯ = dtβ, and integration of the where variables with an overbar denote average quantities, e.g., β∆t emission source term is approximated with a linear combination of Θn and Θn+1 , together with the “implicitness” parameter α. Solving Eq. (7) for Θn+1 yields Z 1 ¯ ¯ ¯ Θn+1 = dΩI + 1 − β σ ¯ ∆tc(1 − α)Θn βσ ¯ ∆t . 1 + β¯σ ¯ ∆tcα 4π
With substitution of Eq. (8), the time-integrated HO TRT equation becomes Z σ ¯c αβ¯σ ¯ ∆t In+1 − In ˆ 1 ¯+ + Ω · ∇I¯ + σ ¯ I¯ = dΩ I Θ n , c∆t 4π 1 + αβ¯σ ¯ c∆t 4π 1 + αβ¯σ ¯ c∆t
(8)
(9)
Eq. (9) is typically solved by MC simulation but without approximating the time derivative. However, this can cause difficulties if the corresponding term in the LO system is time discretized. Although there is a remedy to overcome this issue [6], we use a time-implicit discretization of Eq. (9) in this work (I¯ = In+1 ). Finally, we have the following time-implicit transport equation, Z σ ¯ (1 − f ) In 1 ˆ + dΩIn+1 + σcf Θn , (10) In+1 = Ω · ∇In+1 + σ ¯+ c∆t c∆t 4π 4π where f = 1+αβ¯1σ¯ c∆t is the Fleck factor. The overbar quantities are often evaluated explicitly using the ¯ σ previous time-step solutions. The factor (1 − f ) can be seen as the effective scattering ratio. When β, ¯, and/or ∆t are large, the Fleck factor approaches zero. In this limit, the system becomes scattering dominated, and MC particle tracking is computationally expensive. The FC method can thus become inefficient in thick diffusive regions. Recently, several geometric hybrid methods have been developed to overcome this difficulty [7–9]. We propose a different approach based on a moment-based, scale-bridging algorithm that does not introduce effective scattering at all and thus avoids this issue entirely. Our approach shares some similarities with the Symbolic Implicit Monte Carlo (SIMC) method [10,11] in which the emission term and angular intensity are evaluated truly implicitly through a response matrix reconstruction at the end of each time step. Because it is truly implicit, SIMC does not employ effective scattering, either. However, our approach is different from SIMC because instead of using a response matrix, it computes the emission term directly from discretely consistent LO solution.
3. MOMENT-BASED SCALE-BRIDGING ALGORITHM In order to create the LO system, we take the first two angular moments of Eq. (1) to get ∂E + ∇ · F~ + σcE = σacT 4 , ∂t 1 ∂ F~ + c∇ · EE + σ F~ = 0, c ∂t
(11) (12) 3/12
Park et al. 1 c ˆ ˆ RdΩΩΩI dΩI
where E = R
R
dΩI and F~ =
R
ˆ are the radiation energy density and the radiation flux, respectively. dΩΩI
is the Eddington tensor. In continuum, Eqs. (11) and (12) are equivalent to Eq. (1). However, E= once discretized, they may not be consistent due to a mismatch in the truncation errors [1]. In order to have discrete consistency between the LO and HO systems, we modify and discretize Eq. (11) and (12) as Aij + σi cEi = σi acTi4 , Vi
(13)
c2 + − (∆x E)ij + cσij Fij = γij cEi − γij cEj , 3
(14)
∆t Ei +
X j
∆t Fij +
Fij
where ∆t and ∆x denote appropriate differencing formula for the time and spatial derivatives, and Ei and Fij denote the average radiation energy density in cell i and the radiation flux on the face between cells i and j, respectively. Vi is the volume of cell i, and Aij is the area of the face between cells i and j. In Eq. (14), we have made the following two changes from the orginal first moment equation (12), 1. the Eddington tensor is replaced by 1/3 (i.e., the P1 approximation), and ± are added to match the truncation errors. 2. consistency terms γij
± We can evaluate γij by substituting the HO solution moments (i.e., E HO , F HO ) into Eq. (14). In order to do so, we rewrite R Eq. (14) as the difference of the following two equations written in terms of the partial ˆ where ~nij is the outward normal vector of the face between cells i and dΩ|~nij · Ω|I, fluxes Fij∓ = ~nij ·Ω≶0 ˆ j,
c2 + cEi , (∆x E)ij + cσij Fij+ = γij 6 c2 − cEj . ∆t Fij− − (∆x E)ij + cσij Fij− = γij 6
∆t Fij+ +
(15) (16)
We emphasize here that the consistency term is derived with an assumption that both the HO and LO systems satisfy the same discrete energy balance, Eq. (13). The final LO system consists of Eqs. (13), (14), and (after discretizing) (2). Upon (nonlinear) convergence, the LO solution is guaranteed to be consistent with the HO solution as a result of the consistency term. We utilize the Jacobian-free Newton-Krylov method [12] [13] to solve the coupled LO system. In order to be truly consistent between HO and LO solutions, we must also iterate between the HO and LO systems within a time step. However, we will not perform this iteration. Recent work [2] has shown that a predictor-corrector time stepping algorithm with one HO transport sweep per time step can achieve second-order asymptotic convergence and achieve adequate consistency between the HO and LO solutions. To simplify algorithm comparison, we only consider the first-order, backward-Euler (BE1) time integration in this work. 3.1. Monte Carlo HO Solver For our moment-based, scale-bridging algorithm, the HO-TRT equation is simplified to the following, In 1 n+1 ˆ . (17) I n+1 = S + Ω · ∇I + σ+ c∆t c∆t 4/12
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We specifically denote the emission term as S, indicating it is a fixed source problem. Eq. (17) is seen as a purely absorbing problem, and therefore, it can be readily solved with a transport sweep. However, in an optically-thick, diffusive region, the MC statistics can be very poor, and a large amount of noise may be introduced in the system. To effectively utilize the MC method as a HO solver of the moment-based, scale-bridging algorithm, we utilize two techniques: continuous energy-deposition tracking, and an asymptotic shortcut. 3.1.1. Continuous energy-deposition particle-tracking Continuous energy-deposition (CED) particle tracking [5][6] can be considered as a variance reduction technique that uses the expected value of the absorption removal along the flight path. Because the HO-TRT equation is purely absorbing, we can follow the MC particle deterministically along the path and the MC particle can continuously deposit its energy. One of the advantages of using the CED particle tracking is that a combination of the CED collision and surface flux tallies satisfies the discrete energy balance, Eq. (13), to a machine round-off regardless of the number of particles used. On the other hand, the combination of the usual tracklength tallies and surface flux tallies satisfies the discrete energy balance only at a limit of an infinite number of particles. Thus, in a practical situation, discrete energy balance ~ may have more than 10% error [6]. evaluated from combination of tracklength E and surface crossing F This energy balance is crucial because our scale-bridging algorithm is based on the assumption of the discrete energy balance. The weight of a particle traveled a distance l can be computed as w(l) = w0 e−
Rl 0
dl′ σ(l′ )l′
.
(18)
The deposited energy within distance l is simply w(l) − w0 . Due to the deterministic nature of CED, there is no need to sample distance to collision. However, in this work, we allow each particle to travel 5 mean-free-paths before switching to an analog MC in order to avoid tracking particles with extremely small weights. 3.1.2. Asymptotic shortcut for thick diffusive regions The second technique is to use an asymptotic solution in optically thick regions. To briefly describe this method, we use the following scaled TRT equation (in 1D) that represents the equilibrium-diffusion limit of the TRT equation [14], ǫ2 ǫ2 ∂I + σ+ In , (19) I = σB + ǫµ ∂x c∆t c∆t
where B = 21 acT 4 and ǫ is a small scaling factor. We denote Bi (x) as the emission source within cell i, but the shape of Bi (x) is not yet specified. We expand the specific angular intensity in terms of ǫ as X Ii = ǫk Ik,i . (20) k
Substituting Eq. (20) into Eq. (19) yields X k
ǫk+1 µ
∂Ik,i X k ǫ2 ǫk+2 + In,i . Ik,i = σBi + ǫ σ+ ∂x c∆t c∆t
(21)
k
Equating the term of the same order of ǫ in Eq. (21) gives 5/12
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• O(1) (k = 0) I0,i = Bi .
(22)
• O(ǫ) (k = 1) ∂I0,i + ǫσI1,i = 0, ∂x
(23)
1 ∂Bi I1,i = − µ . σ ∂x
(24)
ǫµ which leads to
Thus the O(ǫ) asymptotic solution can be written as Ii ≈ Bi −
ǫ ∂Bi µ . σ ∂x
(25)
To O(ǫ), the radiation energy density and the radiation flux can be expressed as 2 Bi (x), c 2 ∂Bi Fi (x) = − . 3σ ∂x
Ei (x) =
(26) (27)
We consider two different representations of the spatial dependence of the emission source. The first example approximates the emission source with a linear interpolation of T 4 . In this case, the emission source within the cell i can be expressed as Bi (x) =
1 ac [1 + mi (x − xi )] Ti4 , 2
(28)
where mi is the slope for cell i. Then the spatial derivative of Bi (x) becomes ac ∂Bi = mi Ti4 . ∂x 2
(29)
With this expression, the outgoing partial flux at x = xi−1/2 from cell i, Fiout (xi−1/2 ) can be written to O(ǫ) as Z 0 out dµ|µ|Ii (xi−1/2 ), Fi (xi−1/2 ) = −1 " # Z 0 1 ∂Bi (xi−1/2 ) dµ|µ| Bi (xi−1/2 ) − µ = , σ ∂x −1 i−1/2 i ac h mi ac = 1− ∆xi Ti4 + mi Ti4 , (30) 4 2 6σ while the outgoing partial flux from cell i − 1 becomes Z 1 out dµ|µ|Ii−1 (xi−1/2 ), Fi−1 (xi−1/2 ) = 0 h i mi−1 ac ac 4 4 1+ − ∆xi−1 Ti−1 mi−1 Ti−1 . = 4 2 6σ Note that if B is a continuous function in x, then i h i h mi−1 mi 4 . ∆xi Ti4 = 1 + ∆xi−1 Ti−1 1− 2 2
(31)
(32) 6/12
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In this case, the total flux becomes out Fi−1/2 = Fi−1 − Fiout = −ac
m
i
6σ
Ti4 +
mi−1 4 . T 6σ i−1
(33)
Eq. (33) indicates that we must include at least the O(ǫ) term in the asymptotic solution in order to have a non-zero total flux across cell interfaces when T is continuous across the cell interface. The second representation uses the following form of Bi (x), ac 4 T + g(x), with 2 i ac 4 4 Ti + Ti±1 , Bi (xi±1/2 ) = 4 ∂B Bi±1 − Bi ± = . ∂x xi±1/2 ∆x Bi (x) =
Z
xi+1/2
dxg(x) = 0, xi−i/2
(34)
Note that an explicit form of g(x) is not necessary. Then, the outgoing partial fluxes can be written as ac 4 ac 4 4 Ti + Ti−1 Ti4 − Ti−1 + , 8 6σ∆x ac ac 4 4 4 out Ti + Ti−1 Ti4 − Ti−1 − , Fi−1 (xi−1/2 ) = 8 6σ∆x Fiout (xi−1/2 ) =
(35) (36)
The corresponding total flux is
out Fi−1/2 = Fi−1 − Fiout = −
ac 4 Ti4 − Ti−1 , 3σ∆x
(37)
which is consistent with a finite difference approximation of the equilibrium-diffusion limit of the TRT equations themselves. In this work, all the numerical results with the asymptotic shortcut use this form. To implement the asymptotic shortcut, we first identify ”optically thick” regions and ”optically thin” regions based on cell optical depth. No particles are tracked within optically thick regions; instead, particles are emitted from the interfaces between optically thin and optically thick regions according to Eqs. (30) and (31) or Eqs. (35) and (36), while particles entering optically thick regions through these interfaces are immediately killed. The consistency terms for cells in optically thick regions are then updated using the aforementioned equations along with tallies of particles entering these cells. Note that the asymptotic shortcut only affects the Monte Carlo solution of the HO equation; no modifications are made to the LO system.
4. RESULTS 4.1. 1D Two Material Problem The first example problem is a one-dimensional, two-material problem. The system has a length of 1.0 cm and is initially in equilibrium at 50 eV. A 500 eV isotropic incident intensity is applied at x=0 cm. Table I summarizes the material properties for this problem. With this example problem, we first compare the spatial accuracy of the asymptotic shortcut HOLO algorithm and the FC-MC algorithm. Fig. 1 shows the comparison of the radiation temperature between 7/12
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Table I. Summary of material properties for the 1D two material problem. material 1
material 2
x-range [cm]
< 0.5
> 0.5
σ [cm−1 ]
0.2
2000
g ρ [ cc ]
0.01
10.0
cv [ eVerg −g ]
1012
FC-MC and HOLO-MC with O(ǫ) asymptotic shortcut for different mesh sizes at t = 5 sh (1 sh = 10−8 s). The maximum time-step size was set to 10−2 sh. The time-step size was increased 5% each time step starting from the initial time-step size of 10−3 sh. 104 particles per time step were used in each simulation. The solutions of HOLO-MC and FC-MC match as the mesh is refined (e.g., ∆x = 0.0015625 cm). Furthermore, it shows that the HOLO-MC solution changes little between the coarse and fine mesh cases. On the other hand, the coarse mesh solution (∆x = 0.05 cm) for FC-MC shows a significant error in the location of the wave front. Fig. 2(a) depicts the CPU time versus simulation time obtained by both FC-MC and HOLO-MC using the above time-stepping strategy. The number of particle histories (n) was set to 104 or 105 per MC solve. We used a uniform mesh size of 0.0125 cm. As can be seen from Fig. 2(b), The HOLO-MC solution is less noisy with the same n compared to the FC-MC solution. Fig. 2(c) compares CPU time per 10 time steps (i.e., grind time) versus time-step number. The CPU cost per time step is almost constant in HOLO-MC, while FC-MC increases the CPU cost per time step significantly over the course of the simulation because of the increasing time-step size and material temperature, which introduces more (expensive) effective scattering events. From Table II, we see that the total CPU time required by HOLO-MC with n = 105 is less than that of FC-MC with n = 104 , and HOLO-MC has reduced the CPU time by a factor of 7∼11. Also note that the total CPU time for HOLO-MC does not scale with n. This is because of the fixed cost for the LO solve, which becomes more prominant in the case with smaller n. Furthermore, considering that HOLO-MC solutions are less noisier than FC-MC solutions, the overall gain becomes even greater. Table II. Total CPU time required for the 1D two material problem. A total of 520 time steps is taken at each run. HOLO-MC FC-MC n = 104
11.6 s
78.4 s
n = 105
67.9 s
771.9 s
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Figure 1. Comparison of radiation temperature solution at t=5 sh for the 1D two material problem.
(a) CPU vs simulation time
(b) radiation temperature at t = 5sh.
(c) CPU per 10 time steps (grind time) vs time-step number
Figure 2. Comparison of CPU time between FC-MC and HOLO-MC with two different particle histories per time step for the 1D two material problem.
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4.2. 2D Crooked Pipe Problem The second example is a 2D crooked pipe problem [7]. Although the original problem in [7] is an axisymmetric cylindrical RZ problem, we use a Cartesian xy geometry in this work. The problem consists of optically thin and thick regions, and each has temperature-independent (constant) opacities. Material properties for this problem is shown in Table I. The problem domain spans 0 < x < 2 cm and 0 < y < 7 cm. The system is initially in equilibrium at 50 eV. We used a uniform mesh size of 0.025 cm in both x and y directions. The initial time-step size was set to 10−3 sh and it was increased 5% each time step until it reached 10−1 sh. Each simulation is run with 106 particles per time step. Fig. 3 depicts the radiation temperature solutions at 50 sh. The left-most solution is obtained by the FC-MC method, while the middle- and right-most solutions are obtained by the HOLO-MC method with the O(1) and O(ǫ) asymptotic shortcut, respectively. As can be seen from Fig. 3, the O(1) asymptotic solution (i.e., a constant emission source within cell) is highly diffusive. On the other hand, the O(ǫ) asymptotic solution is less diffusive than the FC-MC solution, which is consistent with the previous example (Fig. 1). To reach a simulation time of 50 sh, the FC-MC method required about 138 CPU-hours, while the HOLO-MC with O(ǫ) asymptotic shortcut required about 3 CPU-hours∗ . Fig. 4(a) shows the CPU time vs simulation time in both FC-MC and HOLO-MC. At the beginning of the simulation, FC-MC is faster than HOLO-MC. This is due to the whole system being relatively cold and the Fleck factor being close to one. Thus, there are very few effective-scattering events. At this point, the CPU time corresponding to the LO solve is still significant. However, once the system has heated up and there exists a large number of effective-scattering events, FC-MC requires a large CPU time for each MC solve. On the other hand, Fig. 4(b) shows that HOLO-MC uses a relatively constant CPU time per time step (grind time) regardless of the problem evolution.
Figure 3. Comparison of radiation temperature solution at t=50 sh for the 2D XY crooked pipe problem. left: the solution from FC-MC, middle: the solution from HOLO-MC with O(1) asymptotic shortcut, right: the solution from HOLO-MC with O(ǫ) asymptotic shortcut. ∗ The code we used is not optimized either for FC-MC or HOLO-MC. However, we believe that the CPU time comparison between FC-MC and HOLO-MC using similar coding on the same machine with the same compiler is relevant.
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(a) CPU vs simulation time
(b) CPU per 10 time steps vs time-step number
Figure 4. Comparison of CPU time vs simulation time for the 2D crooked pipe problem.
5. CONCLUSION We have demonstrated an applicability of a moment-based, scale-bridging algorithm with a MC HO solver for TRT problems. Combination of continuous energy-deposition particle tracking and an O(ǫ) asymptotic shortcut in thick, diffusive regions reduces the statistical noise of the MC solution so that it can be used in this hybrid deterministic/stochastic method. We have shown a comparison of the proposed algorithm with the popular Fleck-Cummings linearization approach. For examples provided in this work, we have seen that the O(ǫ) asymptotic shortcut solution has a smaller spatial error than the FC-MC solution in a simple two material problem. As a system heats up, the rising number of effective-scattering events in FC-MC can lead to a large increase in CPU time per time step. On the other hand, the computational cost of the HOLO-MC algorithm is almost independent of the time-step size. As a result, the HOLO-MC algorithm reduced the CPU solution time by more than an order of magnitude compared to FC-MC. Necessary extensions to our HOLO-MC algorithm in order for it to be employed in realistic calculations include accounting for radiation-frequency dependence [2] and developing coupling techniques for multiphysics applications.
ACKNOWLEDGEMENTS This work was performed under US government contract DE-AC52-06NA25396 for Los Alamos National Laboratory, which is operated by Los Alamos National Security, LLC, for the US Department of Energy.
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