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Radiation transport modeling using extended quadrature method of moments
Abstract
The radiative transfer equation describes the propagation of radiation through a material medium. While it provides a highly accurate description of the radiation field, the large phase space on which the equation is defined makes it numerically challenging. As a consequence, significant effort has gone into the development of accurate approximation methods. Recently, an extended quadrature method of moments (EQMOM) has been developed to solve univariate population balance equations, which also have a large phase space and thus face similar computational challenges. The distinct advantage of the EQMOM approach over other moment methods is that it generates moment equations that are consistent with a positive phase space density and has a moment inversion algorithm that is fast and efficient. The goal of the current paper is to present the EQMOM method in the context of radiation transport, to discuss advantages and disadvantages, and to demonstrate its performance on a set of standard one-dimensional benchmark problems that encompass optically thin, thick, and transition regimes. Special attention is given in the implementation to the issue of realizability—that is, consistency with a positive phase space density. Numerical results in one dimension are promising and lay the foundation for extending the same framework to multiple dimensions.
... The three most popular alternatives are sectional methods, moment methods, and Monte Carlo methods. 10,[13][14][15] In the sectional method (SM), the size domain is divided into sections or "bins" such that the particle size distribution is treated like a histogram. Moment methods conserve computational resources relative to the SM by only tracking the evolution of the lower-order statistical moments of a particle size distribution instead of tracking the entire distribution. ...
... Monte Carlo methods explicitly model the behavior of a finite sub-population of particles 10,15 and thus may require high runtimes relative to moment methods to achieve a given accuracy. 13 MC simulation is generally considered too computationally expensive for incorporation with computational fluid dynamics code, 11,17 but it is a reasonable alternative for batch reactor-type systems (e.g., laboratory studies). 14 Importantly, there are an impressive number of variants upon, and alternatives to, these three major approaches. ...
... The Beta EQMOM defines the particle size distribution on the domain [0, 1], rather than upon the finite interval [m min , m max ]. 13,43 At each time step, the first step in the BEQMOM is thus to perform the following coordinate transformation on the moment set calculated during the previous time step 43 ...
Particle size affects the toxicity and environmental fate of engineered nanoparticles (NPs). Although size effects have been widely studied in the experimental NP fate and toxicity literature, no numerical models
published to date have attempted to identify and compare state-of-the-art particle modeling methods
commonly applied in closely related fields. We compare four numerical frameworks for modeling changes
in the size distribution of a NP suspension undergoing dissolution and aggregation: the Sectional Method
(SM), Direct Simulation Monte Carlo (DSMC), the Direct Quadrature Method of Moments (DQMOM), and
the Extended Quadrature Method of Moments (EQMOM). The SM and the DQMOM were faster or more
accurate than the EQMOM and DSMC in nearly every trial. For cases simulating aggregation, the DQMOM
took seconds to achieve solutions with ≤2% error, while the SM (a rigorous implementation of the most
popular population balance method to date for NPs) took up to 1.5 hours. The SM was far more accurate
than the DQMOM for dissolution test cases; however, up to 50 size bins were required to achieve ≤10%
error. This raises questions about the validity of the current practice of using five or fewer bins in models of
NP fate in rivers. Because runtimes become prohibitive as environmental complexity and particle properties
are added, the DQMOM is promising for field-scale models and models that describe NPs with complex
morphologies or compositions, such as non-spherical NPs, coated NPs, and nanohybrids. MATLAB code
for all models is provided in the ESI.
... Another moment closure, the extended quadrature method of moments (EQ-MOM) was recently investigated in the context of radiative transfer in slab geometry [16]. The EQMOM uses an efficient procedure to a reconstruct convex combination of beta distributions to approximate the energy density from the moment vector, and good results were observed when the method was applied to standard test problems. ...
... The extended quadrature method of moments (EQMOM) [16] reconstructs an ansatzÎ B using a conical combination of beta distributions. The beta distribution on 1, 1¥ is given by ...
... We will call this method the B n closure. An algorithm to determine the parameters w i , γ i , and δ from the moment vector E is given in [16], but it remains unclear whether this algorithm succeeds for all realizable moment vectors E. It is also unclear whether the resulting moment equations (3) with ...
We consider the simplest member of the hierarchy of the extended quadrature method of moments (EQMOM), which gives equations for the zeroth-, first-, and second-order moments of the energy density of photons in the radiative transfer equations in slab geometry. First we show that the equations are well-defined for all moment vectors consistent with a nonnegative underlying distribution, and that the reconstruction is explicit and therefore computationally inexpensive. Second, we show that the resulting moment equations are hyperbolic. These two properties make this moment method quite similar to the attractive but far more expensive M2 method. We confirm through numerical solutions to several benchmark problems that the methods give qualitatively similar results.
... Similar ideas are also known in hydrodynamics and other fields as the Quadrature Method of Moments (QMOM) [36,13,19,18,7]. These ideas have been recently applied to radiative transfer [46]. The general idea is to use an N -atomic discrete measure ...
... Typically one uses n = 2N −1 to obtain 2N equations and 2N unknowns (N weights and abscissas). Plugging ψ QMOMN into the moment problem (2.7) yields a nonlinear system which can be solved using the so-called Wheeler-algorithm [37,46] which diagonalizes a tridiagonal matrix to find the weights and abscissas. This results in a robust and efficient algorithm for the inversion of the moment problem. ...
... This results in a robust and efficient algorithm for the inversion of the moment problem. Additionally a major advantage of the QMOM approach is the fact that it can correctly reproduce moments which lie on the realizability boundary [46] since there the distribution is (uniquely) atomic [29]. ...
We study mixed-moment models (full zeroth moment, half higher moments) for a
Fokker-Planck equation in one space dimension. Mixed-moment minimum-entropy
models are known to overcome the zero net-flux problem of full-moment minimum
entropy Mn models. Realizability theory for these mixed moments of arbitrary
order is derived, as well as a new closure, which we refer to as Kershaw
closures. They provide non-negative distribution functions combined with an
analytical closure. Numerical tests are performed with standard first-order
finite volume schemes and compared with a finite-difference Fokker-Planck
scheme.
... Generally, the numerical methods for radiative transfer equations can be categorized into the deterministic method and the stochastic method. The deterministic methods include the macroscopic moment methods [5,6,7,8,9,10] and microscopic discrete ordinate SN method [11,12,13,14]. The moment methods propose a closure to the radiant intensity by expanding it in a specific functional space [15]. ...
... With these notations in (6), (7) and (8), equation (1) turns to an equivalent multi-group radiative transfer system. ...
In this paper, we extend the unified kinetic particle (UGKP) method to the frequency-dependent radiative transfer equation with both absorption-emission and scattering processes. The extended UGKP method could not only capture the diffusion and free transport limit, but also provide a smooth transition in the physical and frequency space in the regime between the above two limits. The proposed scheme has the properties of asymptotic-preserving, regime-adaptive, and entropy-preserving, which make it an accurate and efficient scheme in the simulation of multiscale photon transport problems. The methodology of scheme construction is a coupled evolution of macroscopic energy equation and the microscopic radiant intensity equation, where the numerical flux in macroscopic energy equation and the closure in microscopic radiant intensity equation are constructed based on the integral solution. Both numerical dissipation and computational complexity are well controlled especially in the optical thick regime. 2D multi-thread code on a general unstructured mesh has been developed. Several numerical tests have been simulated to verify the numerical scheme and code, covering a wide range of flow regimes. The numerical scheme and code that we developed are highly demanded and widely applicable in the high energy density engineering applications.
... In this section, a diffusive Marshak-wave problem is investigated; this problem is a standard test case in the literature [7,25,28,41]. This problem consists of a semi-infinite medium of material with the opacity σ = 300/θ 3 . ...
... 41) where α ∈ [0, 2], i = 1, . . . , N z , l = 1, . . . ...
The thermal radiative transfer (TRT) equations form an integro-differential system that describes the propagation and collisional interactions of photons. Computing numerical solutions of the TRT equations accurately and efficiently is challenging for several reasons, the first of which is that TRT is defined on a high-dimensional phase space that includes the independent variables of time, space, and velocity. In order to reduce the dimensionality of the phase space, classical approaches such as the PN (spherical harmonics) or the SN (discrete ordinates) ansatz are often used in the literature. In this work, we introduce a novel approach: the hybrid discrete (HNT) approximation to the radiative thermal transfer equations. This approach acquires desirable properties of both PN and SN, and indeed reduces to each of these approximations in various limits: HN1≡ PN and H0T≡ ST. We prove that HNT results in a system of hyperbolic partial differential equations for all T≥1 and N≥0. Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions, especially in the diffusive (i.e., highly collisional) regime. This stiffness challenge can be partially overcome via implicit time integration, although fully implicit methods may become computationally expensive due to the strong nonlinearity and system size. On the other hand, explicit time-stepping schemes that are not also asymptotic-preserving in the highly collisional limit require resolving the mean-free path between collisions, making such schemes prohibitively expensive. In this work we develop an asymptotic-preserving numerical method that is based on a nodal discontinuous Galerkin discretization in space, coupled with a semi-implicit discretization in time. In particular, we make use of a second order explicit Runge–Kutta scheme for the streaming term and an implicit Euler scheme for the material coupling term. Furthermore, in order to solve the material energy equation implicitly after each predictor and corrector step, we linearize the temperature term using a Taylor expansion; this avoids the need for an iterative procedure, and therefore improves efficiency. In order to reduce unphysical oscillation, we apply a slope limiter after each time step. Finally, we conduct several numerical experiments to verify the accuracy, efficiency, and robustness of the HNT ansatz and the numerical discretizations.
... The method of moments is commonly used to derive reduced models for kinetic transport [22,24,44,49,54]. To solve such a reduced model is often less expensive, regarding computational time, than to solve the complete model. ...
... It can be shown that a moment vector on the realizability boundary can be uniquely represented by an atomic distribution function. Thus, QM OM is able to exactly reproduce this behavior in such a case [27,49]. ...
In this paper we investigate the application of a maximum entropy moment approach for the population balance equations. A numerical investigation is presented comparing the maximum entropy approach with other moment methods for the population balance equations. We consider different types of moment closures, namely polynomial (P_N) closures, maximum entropy (M_N) closures and the quadrature method of moments QMOM_N. Moreover, a finite volume scheme is discussed for comparison. Realizability issues and implementation details are discussed. The numerical examples range from spatially homogeneous cases to population balance equations coupled to fluid dynamic equations for a lid-driven cavity. A detailed numerical discussion of accuracy, order of the moment method and computational time is given.
... In a previous study [3], we analyzed the second order extended quadrature method of moments (EQMOM) introduced in [16] which we call the B 2 model. In this work, we propose an approximation of the M 2 model in 3D space by extending the B 2 model studied in [3] to 3D. ...
... In a previous work [3], we examined the properties of second order extended quadrature method of moments (EQMOM) proposed in [16] in slab geometry, and the model was referred as the B 2 model. In EQMOM, the ansatzÎ is reconstructed by a combination of beta distributions. ...
We extend to three-dimensional space the approximate M_2 model for the slab geometry studied in our previous paper. The B_2 model therein, as a special case of the second order extended quadrature method of moments (EQMOM), is proved to be globally hyperbolic. The model we proposed here extends EQMOM to multiple dimensions following the idea to approximate the maximum entropy closure for the slab geometry case. Like the M_2 closure, the ansatz of the new model has the capacity to capture both isotropic and beam-like solutions, while the new model has fluxes in closed-form, thus is applicable to practical numerical simulations. The rotational invariance, realizability, and hyperbolicity of the model are studied.
... In order to overcome these limitations, Yuan et al. [1] proposed an Extended Quadrature Method of Moments (EQMOM), which enables the shape of the particle size distribution to be reconstructed from a moment set using kernel density functions instead of Dirac delta functions. EQMOM was evaluated for 13 benchmark test cases and further applied to model radiation transport [49], ...
... It should be mentioned that a similar moment transformation strategy has been used by Vikas et al. [49], who applied EQMOM to model radiation transport. Eq. (26) can then be simplified tõ ...
The scope of this study is the application of the recently developed univariate moment method Extended Quadrature Method of Moments (EQMOM) (Yuan et al., 2012) to model soot formation in flames. Furthermore, it is combined with another advanced moment approach, called the Conditional Quadrature Method of Moments (CQMOM) (Yuan and Fox, 2011), and this extension leads to a bivariate model.Retaining the efficiency of a moment method, EQMOM enables the reconstruction of the number density function. CQMOM is a numerically robust multivariate moment method which allows a bivariate soot particle description in terms of particle volume and surface to take into account aggregation. The joint Extended Conditional Quadrature Method of Moments (ECQMOM) model combines the advantages of the two methods to arrive at a numerically efficient bivariate moment method which captures both the particle size distribution and the formation of aggregates.Both the EQMOM and the ECQMOM model are validated against experimental results for premixed burner-stabilized ethylene flames. Thereby, the gas phase is modeled using a modified version of a very detailed, well-established kinetic scheme, which is adapted to be consistent with the moment methods introduced. The results demonstrate the suitability of the applied models to describe both soot precursors and soot evolution in flames. Furthermore, the ability of the moment approaches to represent the statistical soot model accurately is evaluated comparing EQMOM and ECQMOM to other numerical approaches, which are based on the Monte Carlo method, the standard Gaussian Quadrature Method of Moments and the Gaussian-Radau Quadrature Method of Moments, respectively.
... 19,20 Numerical methods for solving the TRT equations based on these models can generally be categorized as either deterministic or stochastic. Deterministic methods, such as the macroscopic moment method, 16,[20][21][22][23][24] the microscopic discrete ordinate (or SN) method, [25][26][27][28][29] and the lattice Boltzmann method, [30][31][32] are formulated explicitly and produce analytical results. As the most commonly used stochastic method, the Monte Carlo (MC) method [33][34][35][36] uses random numbers to simulate the interactions of individual radiation particles with the background material. ...
For a long time, efficient algorithms for high-dimensional equations, represented by photon radiation transport, have been one important topic in the development of computational methods for particle transport processes. In this paper, we present an implicit unified gas-kinetic particle (IUGKP) method for multiscale gray radiative transfer. Based on the integral solution of the radiative transfer equation, the photon transport processes are categorized into non-equilibrium transport processes with a large photon free path and equilibrium transport processes with a small photon free path. The long-path processes are solved by an implicit Monte Carlo (IMC) method, and the short-path processes are solved by an implicit diffusion system. The closure formulation of photon distribution is derived from the local integral solution of the radiative transfer equation to couple the IMC and diffusion system. The improvement of the proposed IUGKP method over UGKP method is that particles can be tracked continuously instead of just until the first collision, making simulation with large time steps possible. The IUGKP method has the properties of asymptotic-preserving (AP) and regime-adaptive (RA). The AP property states that the IUGKP method converges to the consistent numerical methods for the asymptotic limiting equations of RTE in the limiting regimes. The RA property states that the computational accuracy of the IUGKP method adapts to the regimes. In this paper, the mathematical proof of the AP and RA properties is presented, and the multiscale numerical tests are performed to demonstrate the accuracy and efficiency of the IUGKP method.
... Generally, the numerical methods for radiative transfer equations can be categorized into the deterministic method and the stochastic method. The deterministic methods include the macroscopic moment methods [5][6][7][8][9][10] and microscopic discrete ordinate SN method [11][12][13][14][15]. The moment methods propose a closure to the radiant intensity by expanding it in a specific functional space [16]. ...
... The Poisson kernel was also involved in a discretized spectral approximation method in the one-dimensional neutron transport theory [27], which is different from our attempt. On the other hand, to our best knowledge, the reported combinations of EQMOM with KE-VC are only for the one-dimensional radiative transfer equations [1,29]. As such, our method, denoted Poisson-EQMOM, is a novel way to formulate hydrodynamic theories of planar flows for overdamped active matter. ...
This work is concerned with kinetic equations with velocity of constant magnitude. We propose a quadrature method of moments based on the Poisson kernel, called Poisson-EQMOM. The derived moment closure systems are well defined for all physically relevant moments and the resultant approximations of the distribution function converge as the number of moments goes to infinity. The convergence makes our method stand out from most existing moment methods. Moreover, we devise a delicate moment inversion algorithm. As an application, the Vicsek model is studied for overdamped active particles. Then the Poisson-EQMOM is validated with a series of numerical tests including spatially homogeneous, one-dimensional and two-dimensional problems.
... A possible w ay forw ard would be to resort to higher order methods, e.g. Vikas et al. ( 2013 ) and Le vermore ( 1996 ). Ho we ver, a stable and efficient high-order method has not been discussed in the astrophysics literature, as far as we are a ware. ...
We present a new radiative transfer method (sph-m1rt) that is coupled dynamically with smoothed particle hydrodynamics (sph). We implement it in the (task-based parallel) swift galaxy simulation code but it can be straightforwardly implemented in other sph codes. Our moment-based method simultaneously solves the radiation energy and flux equations in sph, making it adaptive in space and time. We modify the m1 closure relation to stabilize radiation fronts in the optically thin limit. We also introduce anisotropic artificial viscosity and high-order artificial diffusion schemes, which allow the code to handle radiation transport accurately in both the optically thin and optically thick regimes. Non-equilibrium thermo-chemistry is solved using a semi-implicit sub-cycling technique. The computational cost of our method is independent of the number of sources and can be lowered further by using the reduced speed of light approximation. We demonstrate the robustness of our method by applying it to a set of standard tests from the cosmological radiative transfer comparison project of Iliev et al. The sph-m1rt scheme is well-suited for modelling situations in which numerous sources emit ionising radiation, such as cosmological simulations of galaxy formation or simulations of the interstellar medium.
... If this is the case, then the HyQMOM closure presented in this work can be used to investigate convergence with increasing n. Besides the kinetic equation and population balances, other potential applications of the HyQMOM closure include radiation transport [35,9,10] and multiphase-flow models derived from a kinetic equation [27]. ...
A solution is proposed to a longstanding open problem in kinetic theory, namely, given any set of realizable velocity moments up to order 2n, a closure for the moment of order 2n+1 is constructed for which the moment system found from the free-transport term in the one-dimensional (1-D) kinetic equation is globally hyperbolic and in conservative form. In prior work, the hyperbolic quadrature method of moments (HyQMOM) was introduced to close this moment system up to fourth order (n 2). Here, HyQMOM is reformulated and extended to arbitrary even-order moments. The HyQMOM closure is defined based on the properties of the monic orthogonal polynomials Qn that are uniquely defined by the velocity moments up to order 2n -- 1. Thus, HyQMOM is strictly a moment closure and does not rely on the reconstruction of a velocity distribution function with the same moments. On the boundary of moment space, n double roots of the characteristic polynomial P2n+1 are the roots of Qn, while in the interior, P 2n+1 and Qn share n roots. The remaining n + 1 roots of P2n+1 bound and separate the roots of Qn. An efficient algorithm, based on the Chebyshev algorithm, for computing the moment of order 2n + 1 from the moments up to order 2n is developed. The analytical solution to a 1-D Riemann problem is used to demonstrate convergence of the HyQMOM closure with increasing n.
... In this section, a diffusive Marshak-wave problem is investigated; this problem is a standard test case in the literature [6,19,21,34]. This problem consists of a semiinfinite medium of material with the opacity = 300/ 3 . ...
The thermal radiative transfer (TRT) equations form a system that describes the propagation and collisional interactions of photons. Computing accurate and efficient numerical solutions to TRT is challenging for several reasons, the first of which is that TRT is defined on a high-dimensional phase space. In order to reduce the dimensionality, classical approaches such as the P (spherical harmonics) or the S (discrete ordinates) ansatz are often used in the literature. In this work, we introduce a novel approach: the hybrid discrete (H) approximation. This approach acquires desirable properties of both P and S, and indeed reduces to each of these approximations in various limits. We prove that H results in a system of hyperbolic equations. Another challenge in solving the TRT system is the inherent stiffness due to the large timescale separation between propagation and collisions. This can be partially overcome via implicit time integration, although fully implicit methods may become expensive due to the strong nonlinearity and system size. On the other hand, explicit time-stepping schemes that are not also asymptotic-preserving in the highly collisional limit require resolving the mean-free path between collisions. We develop a method that is based on a discontinuous Galerkin scheme in space, coupled with a semi-implicit scheme in time. In particular, we make use of an explicit Runge-Kutta scheme for the streaming term and an implicit Euler scheme for the material coupling term. Furthermore, in order to solve the material energy equation implicitly after each step, we linearize the temperature term; this avoids the need for an iterative procedure. In order to reduce unphysical oscillation, we apply a slope limiter after each time step. Finally, we conduct several numerical experiments to verify the accuracy, efficiency, and robustness of the method.
... A possible way forward would be to resort to higher order methods, e.g. Vikas et al. (2013) and Levermore (1996). However, a stable and efficient high-order method has not been discussed in the astrophysics literature, as far as we are aware. ...
We present a new radiative transfer method (SPH-M1RT) that is coupled dynamically with smoothed particle hydrodynamics (SPH). We implement it in the (tasked-based parallel) SWIFT galaxy simulation code but it can be straightforwardly implemented to other SPH codes. Our moment-based method simultaneously solves the radiation energy and flux equations in SPH, making it adaptive in space and time. We modify the M1 closure relation to stabilize radiation fronts in the optically thin limit which performs well even in the case of head-on beam collisions. We also introduce anisotropic artificial viscosity and high-order artificial diffusion schemes, which allow the code to handle radiation transport accurately in both the optically thin and optically thick regimes. Non-equilibrium thermochemistry is solved using a semi-implicit subcycling technique. The computational cost of our method is independent of the number of sources and can be lowered using the reduced speed of light approximation. We demonstrate the robustness of our method by applying it to a set of standard tests from the cosmological radiative transfer comparison project of Iliev et al. The SPH-M1RT scheme is well-suited for modelling situations in which numerous sources emit ionising radiation, such as cosmological simulations of galaxy formation or simulations of the interstellar medium.
... This work has been followed by an extension to radiation hydrodynamics, with the possibility of taking into account relativistic effects [33,34]. Two others approach can also be cited, namely, one based on a modified system of moments proposed in [35], and another one based on a discontinuous Galerkin approach credited to [36]. In both cases, applications remain restricted to one-dimensional situations. ...
a mixed FE formulation is proposed for a direction averaged radiation model. To circumvent issues related to inf-sup condition, the formulation is stabilized using the VMS framework. The formulation is tested on illustrative examples, where an immerse volume method is used to represent domains with heterogenous opacities.
... The two-beam instability problem is designed to test a closure's ability to handle multi-modal distributions [50]. The maximum entropy model (M N ) yields unphysical shocks [5,25] in this problem. ...
Linear models for the radiative transfer equation have been well developed, while nonlinear models are seldom investigated even for slab geometry due to some essential difficulties. We have proposed a moment model in MPN for slab geometry which combines the ideas of the classical PN and MN model. Though the model is far from perfect, it was demonstrated to be quite efficient in numerically approximating the solution of the radiative transfer equation, that we are motivated to further improve this model. Consequently we propose in this paper a new model following the chartmap in MPN with some significant theoretic progresses. The new model is derived with global hyperbolicity, and meanwhile some necessary physical properties are preserved. We give a complete analysis to the characteristic structure and propose a numerical scheme for the new model. Numerical examples are presented to demonstrate the numerical performance of the new model.
... The PDEs obtained in this way are then discretized on the domain of interest and solved numerically, typically using the finite-volume method [12,13]. The literature reports many example applications where moment methods were used [9,10,, including applications to aerosols [11,17,18], gas-particle flows [37,44,45], gas-liquid flows [22,22,39,46], combustion [27,43,47], sprays [24,32,35,48] and radiation transport [14], to mention a few. In this work we focus on the transport of moments of a univariate distribution, where contains only one scalar positive quantity, as it happens in the solution of population balance equations describing the evolution of the particle size in a particle population. ...
... The PDEs obtained in this way are then discretized on the domain of interest and solved numerically, typically using the finite-volume method [12,13]. The literature reports many example applications where moment methods were used [9,10,, including applications to aerosols [11,17,18], gas-particle flows [37,44,45], gas-liquid flows [22,22,39,46], combustion [27,43,47], sprays [24,32,35,48] and radiation transport [14], to mention a few. In this work we focus on the transport of moments of a univariate distribution, where contains only one scalar positive quantity, as it happens in the solution of population balance equations describing the evolution of the particle size in a particle population. ...
The second-order realizable ζ moment advection scheme developed in Laurent and Nguyen, (2017) is extended to the case of unstructured grids with cells of arbitrary shape. The necessary modifications to the scheme and the conditions under which the scheme ensures the realizability of the advected moment set are presented. The implementation of the scheme in the OpenFOAM ® CFD toolbox is verified comparing the results obtained in one-dimensional test cases involving moment sets well inside the moment space, and at the boundary of the moment space. Results obtained with the proposed scheme are compared to the corresponding analytical solution. The scheme is then tested considering two-dimensional cases of pure moment advection with an imposed irrotational velocity field. First, a quadrilateral grid is considered to determine the order of the scheme and compare it to the results reported in Laurent and Nguyen (2017) with the same grid resolution. Then, the accuracy of the scheme on two-dimensional triangular grids is determined.
... (1)) describes the conservation of radiative intensity in a direction of space. Solving the RTE involves solving all directions, which can be achieved by a previous integration through simplifications (Rosseland (Habibi et al., 2007), model P-n (Ge et al., 2015;Radice et al., 2013), view factors (Drake, 1990), method of moments (Avila et al., 2011;Vikas et al., 2013) Fig. 2). Next, the integral is calculated as the sum of all of the discrete ordinates. ...
This work describes adaptive quadrature, a new feature designed to improve the Discrete Ordinate Method (DOM) in parallel and cone-shaped radiation sources. OpenFOAM was chosen as the simulation framework to implement the new feature. It already included a Discrete Ordinate Method (fvDOM), but it focuses on thermal radiation and is limited to isotropic emission, diffuse phenomena, and non-scattering media. The model was completed to cover volumetric and superficial absorption, isotropic and anisotropic scattering (with user-defined phase functions), diffuse and specular reflection, diffuse and parallel transmission, and three types of superficial emission sources, i.e., isotropic, cone-shaped, and parallel. Additionally, the model is prepared to work in grey mode or with wavelength bands. Multiple regions with different optical properties are also allowed. Most of the model has been validated by individual simulations of every feature in simple geometries that permit an analytical solution, with errors between 0% and 6.08%. These simulations were also verified in a comparison with an established version of the standard DOM, with differences between model implementations below 2.5%. Some advantages of the developed adaptive quadrature are also analysed using simulation results for different radiative sources and angular discretization. The main conclusion is that adaptive quadrature better defines view angle and light direction of emission sources compared to the established DOM, improving significantly the accuracy of the simulation of non-isotropic sources.
... There are typically two categories of numerical methods for solving the radiative transfer equations. The first category consists of the deterministic methods with different ways of discretizing and modeling, such as the discrete ordinate method (DOM) [4,5,6,7] and the moment methods [8,9,10,11]. The second category is the stochastic methods, for example, the Monte Carlo method [12,13,14]. ...
In this paper, we extend the unified gas-kinetic wave-particle (UGKWP) method to the multiscale photon transport. In this method, the photon free streaming and scattering processes are treated in an un-splitting way. The duality descriptions, namely the simulation particle and distribution function, are utilized to describe the photon. By accurately recovering the governing equations of the unified gas-kinetic scheme (UGKS), the UGKWP preserves the multiscale dynamics of photon transport from optically thin to optically thick regime. In the optically thin regime, the UGKWP becomes a Monte Carlo type particle tracking method, while in the optically thick regime, the UGKWP becomes a diffusion equation solver. The local photon dynamics of the UGKWP, as well as the proportion of wave-described and particle-described photons are automatically adapted according to the numerical resolution and transport regime. Compared to the -type UGKS, the UGKWP requires less memory cost and does not suffer ray effect. Compared to the implicit Monte Carlo (IMC) method, the statistical noise of UGKWP is greatly reduced and computational efficiency is significantly improved in the optically thick regime. Several numerical examples covering all transport regimes from the optically thin to optically thick are computed to validate the accuracy and efficiency of the UGKWP method. In comparison to the -type UGKS and IMC method, the UGKWP method may have several-order-of-magnitude reduction in computational cost and memory requirement in solving some multsicale transport problems.
... There are typically two categories of numerical methods for solving the radiative transfer equations. The first category consists of the deterministic methods with different ways of discretizing and modeling, such as the discrete ordinate method [14,4,3,26] and the moment methods [11,2,32,1]. The second category consists of the stochastic approach, for example, the Monte Carlo method [10,22,13]. ...
In this work, we present a unified gas-kinetic particle (UGKP) method for the simulation of multiscale photon transport. The multiscale nature of the particle method mainly comes from the recovery of the time evolution flux function in the unified gas-kinetic scheme (UGKS) through a coupled dynamic process of particle transport and collision. This practice improves the original operator splitting approach in the Monte Carlo method, such as the separated treatment of particle transport and collision. As a result, with the variation of the ratio between numerical time step and local photon's collision time, different transport physics can be fully captured in a single computation. In the diffusive limit, the UGKP method could recover the solution of the diffusion equation with the cell size and time step being much larger than the photon's mean free path and the mean collision time. In the free transport limit, it presents an exact particle tracking process as the original Monte Carlo method. In the transition regime, the weights of particle free transport and collision are determined by the ratio of local numerical time step to the photon's collision time. Several one-dimensional numerical examples covering all transport regimes from the optically thin to optically thick are computed to validate the accuracy and efficiency of the current scheme. In comparison with the discrete ordinate method, the UGKP method is based on particles and avoids the discretization of particle velocity space, which does not suffer from the ray effect.
... The work of Buet is also notable [248,249] and [250], with an extension to radiation hydrodynamics, with the possibility of taking into account relativistic effects. Two other approaches can also be quoted, namely, one based on a discontinuous Galerkin approach credited to [251], and another one based on a modified system of moments proposed in [252]. In both cases, applications remain restricted to one-dimensional situations. ...
For heating and quenching operations occurring during material forming processes, thermal radiation is the the predominant physical phenomenon. Hence, when one tries to simulate such processes, it is important to have at disposal powerful tools for the numerical modelling of thermal radiation.The numerical simulation of these processes often rises numerous problems and questions, as the representation of a complex environment, involving several components ( ingots, burners, nozzles, walls), to deal with different coupled physical phenomena ( flow, heat transfer, boiling, thermal radiation). In this regard, some “immersed” numerical methods, allows a generalist treatment of these different problems, have gained popularity and drag interest of the scientific community in the recent years.The Thost project, aiming to produce a software for heat transfer during material forming processes, fits in the framework, and this PhD is part of this project. The goal is therefore to design tools for numerical modelling of thermal radiation within the immersed volume method of the Thost software. Two approaches are presented: one consisting in the adaptation of an existing method to the context of the immersed volume method, another concerning the development of a formulation for a specific model of radiation. These methods are then tested on industrial applications provided by our partners.
... This work has been followed by an extension to radiation hydrodynamics, with the possibility of taking into account relativistic effects [33,34]. Two other approaches can also be cited, namely, one based on a modified system of moments proposed in [35], and another one based on a discontinuous Galerkin approach credited to [36]. In both cases, applications remain restricted to one-dimensional situations. ...
In this work, we present a computational approach for the numerical simulation of thermal radiation. Radiation is modeled by solving a set of two coupled partial differential equations, the so-called M1 model. A Variational Multiscale method is developed for this system, and tested on some illustrative benchmarks. The question of dealing with heterogeneous physical properties is also considered, and treated by means of an immersed method. It combines a level-set approach for representation of interfaces, mixing laws to build effective physical properties and an anisotropic mesh adaptation process, all these ingredients leading to an accurate description of the interface.
... 100 Page 2 of 24 Z. Banach and W. Larecki ZAMP bosonic radiation [12][13][14][15][16][17]. The natural question then arises as to whether the Kershaw-type closure procedure can also be applied to the moment equations of fermionic radiation. ...
Besides the maximum entropy closure procedure, other procedures can be used to close the systems of spectral moment equations. In the case of classical and bosonic radiation, the closed-form analytic Kershaw-type and B-distribution closure procedures have been used. It is shown that the Kershaw-type closure procedure can also be applied to the spectral moment equations of fermionic radiation. First, a description of the Kershaw-type closure for the system consisting of an arbitrary number of one-dimensional moment equations is presented. Next, the Kershaw-type two-field and three-field transport equations for fermionic radiation are analyzed. In the first case, the independent variables are the energy density and the heat flux. The second case includes additionally the flux of the heat flux as an independent variable. The generalization of the former two-field case to three space dimensions is also presented. The fermionic Kershaw-type closures differ from those previously derived for classical and bosonic radiation. It is proved that the obtained one-dimensional systems of transport equations are strictly hyperbolic and causal. The fermionic Kershaw-type closure functions behave qualitatively in the same way as the fermionic maximum entropy closure functions, but attain different numerical values.
... The closure of the system of moment equations obtained by employing the ideas of Kershaw [22] is not the only closure procedure that yields the results alternative and comparable to those resulting from the maximum-entropy closure approach. In the context of bosonic radiative transfer in slab geometry, another closure method, the extended quadrature method of moments (EQMOM), was investigated in [52] to effectively construct the approximate moment-dependent distribution functions by using a convex combination of beta distributions. The simplest unique member of the EQMOM hierarchy, where a single beta distribution is used, leads to the system of transport equations for the first three moments of the distribution function. ...
The mixed three-moment hydrodynamic description of fermionic radiation transport based on the Boltzmann entropy optimization procedure is considered for the case of one-dimensional flows. The conditions for realizability of the mixed three moments chosen as the energy density and two partial heat fluxes are established. The domain of admissible values of those moments is determined and the existence of the solution to the optimization problem is proved. Here, the standard approaches related to either the truncated Hausdorff or Markov moment problems do not apply because the non-negative fermionic distribution function, denoted f, must satisfy the inequality f < 1 and, at the same time, there are three different intervals of integration in the integral formulae defining the mixed moments. The hydrodynamic equations are obtained in the form of the symmetric hyperbolic system for the Lagrange multipliers of the optimization problem with constraints. The potentials generating this system are explicitly determined as dilogarithm and trilogarithm functions of the Lagrange multipliers. The invertibility of the relation between moments and Lagrange multipliers is proved. However, the inverse relation cannot be determined in a closed analytic form. Using the H-theorem for the radiative transfer equation, it is shown that the derived system of hydrodynamic radiation equations has as a consequence an additional balance law with a non-negative source term.
... Similarly, the results are not restricted to the minimum-entropy approach but can be transferred to other realizable closures like Kershaw [22,38,39] or the quadrature method of moments [12,13,45,46]. ...
We derive an implicit-explicit (IMEX), realizability-preserving first-order scheme for moment models with Lipschitz-continuous source terms. In contrast to fully-explicit schemes the time step does not depend on the physical parameters, removing the stiffness from the system. Furthermore, a wider class of collision operators (e.g. the Laplace-Beltrami operator) can be used. The derived scheme is applied to minimum-entropy models.
... Especially, several of them are only derived in one space dimension (see e.g. [50,53,3]), and it is not entirely clear how to generalize the ideas to multiple dimensions. In this paper we deal with the case of multiple dimensions. ...
Particle transport in radiation therapy can be modelled by a kinetic equation which must be solved numerically. Unfortunately, the numerical solution of such equations is generally too expensive for applications in medical centers. Moment methods provide a hierarchy of models used to reduce the numerical cost of these simulations while preserving basic properties of the solutions. Moment models require a closure because they have more unknowns than equations. The entropy-based closure is based on the physical description of the particle interactions and provides desirable properties. However, computing this closure is expensive. We propose an approximation of the closure for the first two models in the hierarchy, the and models valid in one, two or three dimensions of space. Compared to other approximate closures, our method works in multiple dimensions. We obtain the approximation by a careful study of the domain of realizability and by invariance properties of the entropy minimizer. The model is shown to provide significantly better accuracy than the model for the numerical simulation of a dose computation in radiotherapy. We propose a numerical solver using those approximated closures. Numerical experiments in dose computation test cases show that the new method is more efficient compared to numerical solution of the minimum entropy problem using standard software tools.
... Especially, several of them are only derived in one space dimension (see e.g. [50,53,3]), and it is not entirely clear how to generalize the ideas to multiple dimensions. In this paper we deal with the case of multiple dimensions. ...
The transport of photons and electrons is studied in the field of radiotherapy to compute the dose, that is, the quantity of energy transferred to the medium by a beam of particles at each position. A kinetic model is proposed, and to decrease the computation times, it is reduced through a moment extraction. Entropy-based angular moment models of order up to two (M1 and M2 models) are shown to provide accurate results compared to a reference code with much lower computational costs.
... We note that the combinations of B-distribution was used as an approximation for the specific intensity in [15], though for a different purpose. ...
We propose an approximate second order maximum entropy () model for
radiative transfer in slab geometry. The model is based on the ansatz of the
specific intensity in the form of a -distribution. This gives us an
explicit form in its closure. The closure is very close to that of the maximum
entropy, thus an approximation of the model. We prove that the new model
is globally hyperbolic, sharing most of the advantages of the maximum entropy
closure. Numerical examples illustrate that it provides solutions with
satisfactory agreement with the model.
... The LnEQMOM algorithm will succeed if the transformed moments are realizable. To this purpose, it is necessary to have an algorithm to check the moment realizability (Dette, 1997;Shohat and Tamarkin, 1943;Vikas et al., 2013;Yuan et al., 2012). A set of real values is a set of moments of a distribution function if the Hankel determinants are non-negative. ...
An extended quadrature method of moments (EQMOM) with log-normal kernel density functions is developed in this work, and applied to the solution of a population balance equation (PBE) for aggregation and breakup, coalescence, and condensation problems. The cases with one and two kernel density functions are studied analytically, and the existence of an analytical solution is shown. A numerical procedure based on the work of Yuan et al. (2012) is adopted to address cases with a larger number of kernel density functions. Results for the reconstructed number density function (NDF), the time evolution of the zero-order moment and of the mean particle size are compared with those obtained from the rigorous solution of the PBE reported by Vanni (2000) for the cases of aggregation and breakup. A problem concerning coalescence and one regarding condensation, both with analytical solution, are also examined. The results obtained with the proposed approach are compared to those provided by EQMOM with gamma kernel densities. Satisfactory results were obtained for the reconstructed distribution. Excellent agreement was observed between the rigorous solution and the approximated one for the time evolution of the total number density and the mean particle size.
Highlights
§ New quadrature scheme, UCWQ, for the DOM, which effectively overcomes the limitations of standard SN and TN quadratures.
§ A novel pyramidal mesh design enabling unlimited angular refinement, preventing distortions in temperature and radiative flux profiles.
☆Matlab code is provided as supplementary material for this article, with the listings included in the appendix.
§ Mitigating shadowing effects and minimizing errors caused by ray effects.
§ Industrial applications, such as studying combustion, atmospheric pollution layers and radiation contribution of greenhouse gases.
§ Astrophysical systems, such as stellar atmospheres and structures within the interstellar medium.
https://www.sciencedirect.com/science/article/pii/S2590123024009836
https://doi.org/10.1016/j.rineng.2024.102728
This paper describes the reduction in memory and computational time for the simulation of complex radiation transport problems with the discrete ordinate method (DOM) model in the open-source computational fluid dynamics platform OpenFOAM. Finite volume models require storage of vector variables in each spatial cell; DOM introduces two additional discretizations, in direction and wavelength, making memory a limiting factor. Using specific classes for radiation sources data, changing the store of fluxes and other minor changes allowed a reduction of 75% in memory requirements. Besides, a hierarchical parallelization was developed, where each node of the standard parallelization uses several computing threads, allowing higher speed and scalability of the problem. This architecture, combined with optimization of some parts of the code, allowed a global speedup of x15. This relevant reduction in time and memory of radiation transport opens a new horizon of applications previously unaffordable.
In this paper, two modifications are introduced for improving the accuracy, versatility, and robustness of a class of hybrid methods for radiation transport. In general, such methods are constructed by splitting the radiative flux into collided and uncollided components to which low- and high-resolution angular approximations are applied, respectively. In this work we focus on discrete ordinates discretizations of high and low order. The first modification we introduce changes the way in which the collided component is mapped into the uncollided component at the end of each time step in a simulation. The new mapping is a Nyström-type reconstruction that is applicable to arbitrary discrete ordinates quadratures, is guaranteed to preserve positivity of the solution provided that all ordinate weights are positive, is significantly more accurate than previous methods, and can be readily extended to other discretizations such as moment methods, finite element methods, and diffusion approximations. The second modification leverages integral deferred correction (IDC) to iteratively correct for the splitting error introduced by the inconsistency in angular discretization between the collided and uncollided components, in addition to improving the accuracy of the low-order temporal error that is treated by traditional IDC methods. Numerical tests in one- and two-dimensional geometries are used to demonstrate the increased accuracy and efficiency of the proposed modifications. It is found that the two techniques combined yield methods with solution accuracy and memory requirements comparable to that of monolithic discrete ordinates methods while reducing runtime by as much as a factor of between two and ten, depending on the problem.
The outset of realistic rendering is a desire to reproduce the appearance of the real world. Rendering techniques therefore operate at a scale corresponding to the size of objects that we observe with our naked eyes. At the same time, rendering techniques must be able to deal with objects of nearly arbitrary shapes and materials. These requirements lead to techniques that oftentimes leave the task of setting the optical properties of the materials to the user. Matching the appearance of real objects by manual adjustment of optical properties is however nearly impossible. We can render objects with a plausible appearance in this way but cannot compare the appearance of a manufactured item to that of its digital twin. This is especially true in the case of translucent objects, where we need more than a goniometric measurement of the optical properties. In this survey, we provide an overview of forward and inverse models for acquiring the optical properties of translucent materials. We map out the efforts in graphics research in this area and describe techniques available in related fields. Our objective is to provide a better understanding of the tools currently available for appearance specification when it comes to digital representations of real translucent objects.
In this paper, we extend the unified gas-kinetic wave-particle (UGKWP) method to multiscale photon transport. In this method, the photon free streaming and scattering processes are treated in an un-splitting way. The photon distribution is described by both discrete simulation particle and analytic distribution function. By accurately recovering the multiscale modeling of the unified gas-kinetic scheme (UGKS), the UGKWP method presents a smooth transition for photon transport from optically thin to optically thick regimes according to the cell's Knudsen number. In the optically thin regime, the UGKWP method performs as a Monte Carlo type particle tracking method, while in the optically thick regime it recovers a diffusion process without particles. The proportion of wave-described and particle-described photons is automatically adapted according to the numerical resolution and local photon scattering physics, i.e., the so-called cell Knudsen number. Compared to the discrete ordinates-based UGKS, the UGKWP method requires less memory and does not suffer from ray effect. Compared to the implicit Monte Carlo (IMC) method, the statistical noise of the UGKWP method is greatly reduced, and the computational efficiency is significantly improved in the optically thick regime. Several numerical examples covering all transport regimes from the optically thin to optically thick ones are computed to validate the accuracy and efficiency of the UGKWP method. In comparison with the UGKS and IMC method, the UGKWP method may have a several-order-of-magnitude reduction in computational cost and memory requirement in solving some multiscale transport problems.
In numerical weather prediction models, previous approaches have employed bulk parameterization schemes based on presumed-number-density-functions or quadrature methods of moments (QMOM). In the present work, a new parameterization based on the extended quadrature method of moments (EQMOM) introduced in Yuan et al. (2012) is applied to the case of pure sedimentation of rain drops (one-dimensional “rain-shaft” test case). In EQMOM, the drop size distribution is represented by a weighted sum of kernel density functions, combining elements of quadrature and presumed functional form methods. In cloud microphysics, moment parameterization is frequently based on Gamma distributions, which guided the choice of the kernel shape employed here. EQMOM allows inclusion of a number of prognostic moments in the method (e.g. M⁽⁰⁾-M⁽⁶⁾), which improves flexibility in the representation of a continuous size distribution. QMOM and EQMOM up to order 3 were applied in two drop sedimentation test cases previously presented in the literature, in which initial states consist of different cloud heights and drop size distribution shapes. Results were compared to a spectral reference model using a number of transported bin sizes showing good agreement. The analysis was focused in the sedimentation induced errors obtained by the different approaches. In QMOM, size sorting due to different fall velocities generates step patterns in the moment profiles. With EQMOM, on the other hand, these artifacts are significantly suppressed. Furthermore, predictions of the number concentration, total liquid content, radar reflectivity, mass mean diameter and rain rates are shown to be greatly improved when EQMOM is employed. Quantitatively, EQMOM is capable of reducing global error measures by nearly one order of magnitude, when compared to results obtained by previous methods in a common benchmark, showing the great potential of the method in the field of meteorology.
One of the important attractions of employing the transported probability density function (PDF) methods for turbulent reacting flows is the fact that the chemical source terms are treated exactly. The composition probability density function (PDF) evolves with convective transport in physical space due to the mean velocity (macromixing), the turbulent diffusivity which transports the composition PDF in physical space (mesomixing), the transport in composition space due to molecular mixing (micromixing) and chemical reactions. The key model term of our interest in this study in the transport PDF equation is the molecular mixing term, which describes how molecular diffusion affects both the shape of the PDF and the rate of scalar variance decay. Molecular mixing is expressed using the Fokker-Planck model (Fox, 2003, 1994), which is an extension of the interaction-by-exchange-with-the-mean (IEM) model. The IEM model, widely used in chemical-reaction engineering and computational combustion due to its simple form, assumes the linear relaxation of the scalar towards its mean, while the Fokker-Plank model considers, in addition, the effect of the differential diffusion process to mimic the mixing. Differential diffusion occurs when the molecular diffusivities of the scalar fields are not the same. An extended quadrature-based moment method (EQMOM) (Yuan et al., 2012; Chalons C. Fox R.-O. Massot M., 2010) is used to close the transport equation for the composition PDF. The β kernel density function is used for EQMOM since the scalar composition is represented by a scalar bounded between -1 and 1. The PDF predicted by the EQMOM model in the same conditions studied in the direct numerical simulation of Eswaran and Pope (1988) are reported and compared to the DNS predictions.
References:
Chalons, C., Fox, R. O., Massot, M., 2010. A multi-Gaussian quadrature method of moments for gas-particle flows in a LES framework. Studying Turbulence Using Numerical Simulation Databases, Center for Turbulence Research, Summer Program 2010, Stanford University 347–358.
Eswaran, V., Pope, S.B., 1988. Direct numerical simulations of the turbulent mixing of a passive scalar. Physics of Fluids 31, 506–520.
Fox, R.O., 2003. Computational Models for Turbulent Reacting Flows. Cambridge University Press.
Fox, R.O., 1994. Improved Fokker–Planck model for the joint scalar, scalar gradient PDF. Physics of Fluids 6, 334–348.
Yuan, C., Laurent, F., Fox, R.O., 2012. An extended quadrature method of moments for population balance equations. Journal of Aerosol Science 51, 1–23.
The evolution of the particle size distribution in polydisperse systems is regulated by the population balance equation (PBE) [1]. A successful approach to solve the PBE is the quadrature method of moments (QMOM), in which the particle size distribution is approximated by a discrete number of Dirac delta functions uniquely determined from a truncated set of the moments of the distribution [2]. The accuracy of the QMOM approach depends on the number of quadrature nodes, and it is limited by the potentially ill-conditioned moment inversion problem [3]. Recently, an extended quadrature method of moment (EQMOM) was proposed, which uses a base of continuous non-negative kernel functions to reconstruct the particle size distribution from its first 2N + 1 moments [3]. Beta, gamma [3], and Gaussian [4,5] distributions were used as kernel density function for EQMOM. However, the particle size distribution in many systems and multiphase flows can be represented by a log-normal distribution function. Based on this consideration, we propose a variant of the extended quadrature method of moments [3] based on log-normal kernel density functions. We investigate the property of the method when one or two kernel density functions are used, since in these cases an analytical solution can be found to the reconstruction problem. We then validate the method against the rigorous solution of a PBE for aggregation and breakage problems obtained in [6], the solution for coalescence and breakup problems presented in [7], and for the condensation problem examined in [3]. Results for aggregation and breakage problems are compared also to the QMOM results reported in [8].
References
[1] D. Ramkrishna, Population balances: theory and applications to particulate systems in engineering, Academic Press, San Diego, CA, 2000.�
[2] R. McGraw, Description of aerosol dynamics by the quadrature method of moments, Aerosol Science and Technology. 27 (1997) 255–265.
[3] C. Yuan, F. Laurent, R.O. Fox, An extended quadrature method of moments for population balance equations, Journal of Aerosol Science. 51 (2012) 1–23.
[4] C. Chalons, R.O. Fox, M. Massot, A multi-Gaussian quadrature method of moments for gas-particle flows in a LES framework, in: Proceedings of the Summer Program 2010, Center for Turbulence Research, Stanford University, 2010: pp. 347 – 358.
[5] A. Vie, C. Chalons, R.O. Fox, F. Laurent, M. Massot, A multi-Gaussian quadrature method of moments for simulating high Stokes number turbulent two-phase flows, in: Annual Research Briefs 2011, Center for Turbulence Research, Stanford University, 2011: pp. 309 – 320.
[6] M. Vanni, Approximate Population Balance Equations for Aggregation–Breakage Processes, Journal of Colloid and Interface Science. 221 (2000) 143–160.
[7] P.L.C. Lage, On the representation of QMOM as a weighted-residual method—The dual-quadrature method of generalized moments, Computers & Chemical Engineering. 35 (2011) 2186–2203.
[8] D. L. Marchisio, R. Dennis Vigil, R. O. Fox, Implementation of the quadrature method of moments in CFD codes for aggregation-breakage problems, Chemical Engineering Science. 58 (2003) 3337–3351.
We provide a brief introduction to quadrature-based moment methods that can be used to model polydisperse multiphase flows. A more detailed description can be found in Marchisio and Fox (2013). Our focus here is to introduce the reader to the principal topics and to provide insight into the numerial algorithms. An example application of gas-particle flow is used to illustrate the methods.
Full transport solutions of time-dependent problems can be computationally very expensive. Therefore, considerable e!ort has been devoted to developing approximate solution techniques that are much faster computationally and yet are accurate enough for a particular application. Many of these approximate solutions have been used in isolated problems and have not been compared to each other. This paper presents two test problems that test and compare several approximate transport techniques. In addition to the di!usion and P approximations, we will test several di!erent #ux-limited di!usion theories and variable Eddington factor closures. For completeness, we will show some variations that have not yet appeared in the literature that have some interesting consequences. For example, we have found a trivial way to modify the P equations to get the correct propagation velocity of a radiation front in the optically thin limit without modifying the accuracy of the solution in the optically thick limit. Also, we will demonstrate nonphysical behavior in some published techniques.
We present an analysis of three means of treating the material energy equation coupled with the nonlin-ear radiation transport equation in slab geometry. The radiation transport equation is discretized using a linear discontinuous Galerkin method. The material temperature is then treated with either 1) two tem-perature unknowns per cell, 2) one temperature unknown per cell, or 3) one temperature unknown per cell with a linear reconstruction of the emission source. Only the two temperatures per cell discretization is robust in the equilibrium diffusion limit. Though it is not robust in the diffusion limit, the reconstructed emission treatment is much more accurate than the one temperature per cell treatment. We use all three methods to solve a diffusive Marshak wave problem. These numerical solutions agree with the analysis. On a problem representing an ablating hohlraum where the solution is strongly influenced by a transport boundary layer between an optically thick and thin region, both the two temperature and reconstructed emission source treatments required 10 µm resolution to resolve the solution. The one temperature per cell solution was not yet converged at this fine resolution.
A suite of analytical benchmarks, called AZURV1, has been developed for verification of time-dependent neutral particle transport algorithms in the one group approximation. Only 1-D sources in infinite media are considered. The 1-D sources include infinite plane, point, infinite line, spherical shell and solid spherical pulsed emissions. The analytical solution for each source is based on the solution to the pulsed plane source. Transformations between geometries then give scalar flux expressions to be numerically evaluated to analytical benchmark accuracy for the fundamental pulsed sources considered.
Acetonitril [75-05-8]Acrolein [107-02-8]Acrylsäure [79-10-7]Aldrin [309-00-2]Allylalkohol [107-18-6]Ametryn [834-12-8]Anilin [62-53-3]Arsen [7440-38-2] und Arsenverbindungen Asbest [1332-21-4]Atrazin [1912-24-9]Benzol [71-43-2]Benzotrichlorid [98-07-7]Benzoylchlorid [98-88-4]Benzylchlorid [100-44-7]Beryllium [7440-41-7] und ein BerylliumverbindungenBlausäure [74-90-8]Blei [7439-92-1] und BleiverbindungenBromoform [75-25-2]Butylbenzylphthalat [85-68-7]
A well-known asymptotic analysis describes the transition of transport theory of diffusion theory in the limit of optically thick systems with small absorption and sources. Recently, this analysis has been applied to discretized transport algorithms. The results of this analysis, which provide information on accuracy and iteration efficiency, cannot be obtained from standard truncation error analyses because in the asymptotic limit, the optical thickness of a spatial cell generally tends to infinity. The ideas underlying this analysis are described, the main results are reviewed, and some open questions are discussed.
A diffusion theory for radiative transfer is derived which is naturally flux limited. i.e., the magnitude of the flux can be no greater than the density times the maximum transport speed. Numerical comparisons with exact solutions of the equation of transfer indicate that this approximate theory is significantly more accurate than classical isotopic diffusion theory (the Eddington approximation) and asymptotic diffusion theory.
A generic Monte Carlo model of a photon therapy machine is described. The model, known as McRad, is based on EGS4 and has been in use since 1991. Its primary function has been the characterization of the incident photon fluence for use by dose calculation algorithms. The accuracy of McRad is examined by comparing the dose distributions in a water phantom generated using only the Monte Carlo data with measured dose distributions for two machines in our clinic; a 6 MV Varian Clinac 600C and the 15 MV beam from a Clinac 2100C. The Monte Carlo generated dose distributions are computed using a dose calculation algorithm based on the use of differential pencil beam kernels. It was found that the match to measured data could be improved if the model is tuned by adjusting the energy of the electron beam incident on the target. The beam profiles were found to be more sensitive indicators of the electron beam energy than the depth dose curves. Beyond the depths reached by contaminant electrons, the computed and measured depth dose curves agree to better than 1%. The comparison of beam profiles indicate that in regions up to within 1 cm of the field edge, the measured and computed doses generally agree to within 2%–3%.
The use of the Monte Carlo method in radiative heat transfer is reviewed. The review covers surface-surface, enclosure, and participating media problems. Discussion is included of research on the fundamentals of the method and on applications to surface-surface interchange in enclosures, exchange between surfaces with roughness characteristics, determination of configuration factors, inverse design, transfer through packed beds and fiber layers, participating media, scattering, hybrid methods, spectrally dependent problems including media with line structure, effects of rising parallel algorithms, practical applications, and extensions of the method. Conclusions are presented on needed future work and the place of Monte Carlo techniques in radiative heat transfer computations.
A class of second-order numerical schemes for the compressible Euler equations is described, and their L 1 stability (i.e., ρ≥0, T≥0) is proved. Following Van Leer’s approach [B. van Leer, J. Comput. Phys. 32, 101-136 (1979)] the solution (ρ,u,T here) is represented as piecewise linear functions. The necessity of a slope limitation appears naturally in the derivation of the schemes, but it can be less strict than the slope reconstructions usually used. These schemes are written in terms of explicit flux splitting formula and are naturally multidimensional in space; the upwinding is obtained through a very generalized notion of characteristics: the kinetic one.
Gas-particle and other dispersed-phase flows can be described by a kinetic equation containing terms for spatial transport, acceleration, and particle processes (such as evaporation or collisions). In principle, the kinetic description is valid from the dilute (non-collisional) to the dense limit. However, its numerical solution in multi-dimensional systems is intractable due to the large number of independent variables. As an alternative, Lagrangian methods “discretize” the density function into “parcels” that are simulated using Monte-Carlo methods. While quite accurate, as in any statistical approach, Lagrangian methods require a relatively large number of parcels to control statistical noise, and thus are computationally expensive. A less costly alternative is to solve Eulerian transport equations for selected moments of the kinetic equation. However, it is well known that in the dilute limit Eulerian methods have great difficulty to describe correctly the moments as predicted by a Lagrangian method. Here a two-node quadrature-based Eulerian moment closure is developed and tested for the kinetic equation. It is shown that the method can successfully handle highly non-equilibrium flows (e.g. impinging particle jets, jet crossing, particle rebound off walls, finite Stokes number flows) that heretofore could not be treated accurately with the Eulerian approach.
Simulations of radiative transfer within an air-lift photobioreactor (PBR) are demonstrated by coupling it to the fluid hydrodynamics and employing wavelength dependant properties for the participating media. The radiative properties of the algal media are determined by matching the numerical predictions against measurements of radiative intensity distributions. To assist towards the design, scale-up and optimization of such reactors, a parametric investigation of the relative importance of the angular resolution of the radiation calculations, scattering phase functions of the bubbles, air mass flow rate and the bubble size are investigated by examining the 3D distributions of radiation within the PBR. Two hundred and eighty eight solid angles in the finite volume (FV) formulation of the radiative transfer equation (RTE) provide an optimum combination of speed and accuracy in the resulting calculations. While scattering results in a more effective redistribution of the energy, the results from employing isotropic and forward-scattering phase functions for the bubbles are found to be similar. The importance of bubble scattering diminishes at algal concentrations (>0.5 g/L) where the radiation attenuation is dominated by the absorption coefficient of the algal media. While 1 μm sized bubbles more effectively redistribute the radiation downstream of the radiators compared to larger sized bubbles, the differences in the radiation profiles obtained from 10 μm and 100 μm-sized bubbles were small within this reactor. The radiation distributions are also influenced by the mass flow rate of air. The calculations demonstrate the need to rigorously account for the air flow rate, bubble size and the scattering effects of bubbles through fully coupled numerical solutions to the fluid flow and radiative transfer equations and provide some best practice guidelines for increasing the fidelity of radiation simulations in PBR’s.
Population balance equations (PBE) for a number density function (NDF) arise in many applications of aerosol technology. Thus, there has been considerable interest in the development of numerical methods to find solutions to PBE, especially in the context of spatially inhomogeneous systems where moment realizability becomes a significant issue. Quadrature-based moment methods (QBMM) are an important class of methods for which the accuracy of the solution can be improved in a controlled manner by increasing the number of quadrature nodes. However, when a large number of nodes is required to achieve the desired accuracy, the moment-inversion problem can become ill-conditioned. Moreover, oftentimes pointwise values of the NDF are required, but are unavailable with existing QBMM. In this work, a new generation of QBMM is introduced that provides an explicit form for the NDF. This extended quadrature method of moments (EQMOM) approximates the NDF by a sum of non-negative weight functions, which allows unclosed source terms to be computed with great accuracy by increasing the number of quadrature nodes independent of the number of transported moments. Here, we use EQMOM to solve a spatially homogeneous PBE with aggregation, breakage, condensation, and evaporation terms, and compare the results with analytical solutions whenever possible. However, by employing realizable finite-volume methods, the extension of EQMOM to spatially inhomogeneous systems is straightforward.
Three methods are reviewed for computing optimal weights and abscissas which can be used in the quadrature method of moments (QMOM): the product-difference algorithm (PDA), the long quotient-modified difference algorithm (LQMDA, variants are also called Wheeler algorithm or Chebyshev algorithm), and the Golub–Welsch algorithm (GWA). The PDA is traditionally used in applications. It is discussed that the PDA fails in certain situations whereas the LQMDA and the GWA are successful. Numerical studies reveal that the LQMDA is also more efficient than the PDA.
Exact and approximate solutions for the penetration of radiation with ;
constant driving temperature are presented. (auth)
Photon radiation transport is described by the Boltzmann equation. Because this equation is difficult to solve, many different approximate forms have been implemented in computer codes. Several of the most common approximations are reviewed, and test problems illustrate the characteristics of each of the approximations. This document is designed as a tutorial so that code users can make an educated choice about which form of approximate radiation transport to use for their particular simulation.
The numerical stability, equilibrium diffusive limit, and accuracy of the variable Eddington factor (VEF) methods and flux-limited diffusion methods for radiation transport calculations are considered. The diffusive limit analysis proves that three VEF closures and their associated flux-limiters retain full first-order accuracy in the equilibrium diffusion limit while achieving the correct propagation speed in the optically thin streaming limit. The stability analysis reveals that the flux-limited diffusion methods are unconditionally stable, but the VEF equations with an arbitrary nonlinear closure can be numerically unstable for certain commonly used differencing schemes. However, regular solutions to the VEF equations are obtainable by Godunov-type schemes. Numerical comparisons among various solutions for a test problem show that flux-limited diffusion methods are only slightly less accurate than their corresponding VEF methods, and the Minerbo VEF method and the Minerbo flux-limited diffusion method are in general more accurate than other approximations.
The purpose of this book is to present the basic elements of numerical methods for compressible flows. It is appropriate for advanced undergraduate and graduate students and specialists working in high speed flows. The focus is on the unsteady one-dimensional Euler equations which form the basis for numerical algorithms in compressible fluid mechanics. The book is restricted to the basic concepts of finite volume methods, and even in this regard is not intended to be exhaustive in its treatment. Although the practical applications of the one-dimensional Euler equations are limited, virtually all numerical algorithms for inviscid compressible flow in two and three dimensions owe their origin to techniques developed in the context of the one-dimensional Euler equations. The author believes it is therefore essential to understand the development and implementation of these algorithms in their original one-dimensional context. The text is supplemented by numerous end-of-chapter exercises.
The quadrature method of moments (QMOM) has been widely used for the simulation of the evolution of moments of the aerosol general dynamic equations. However, there are several shortcomings in a crucial component of the method, the product-difference (P-D) algorithm. The P-D algorithm is used to compute the quadrature points and weights from the moments of an unknown distribution. The algorithm does not work for all types of distributions or for even reasonably high-order quadrature. In this work, we investigate the use of the Chebyshev algorithm and show that it is more robust than the P-D algorithm and can be used for a wider class of problems. The algorithm can also be used in a number of applications, where accurate computations of weighted integrals are required. We also illustrate the use of QMOM with the Chebyshev algorithm to solve several problems in aerosol science that could not be solved using the P-D algorithm.
Precise modeling of radiative heat exchange between the furnace and the glass preform is a very important part of the modeling of the fiber drawing process in a high temperature furnace. Most earlier studies on this process have used the optically thick approximation, i.e., the radiative heat exchange is assumed to depend only on the preform surface temperature while the transmission, emission, and absorption within the preform are approximated as a diffusion process. The validity of this approximation in the modeling of the fiber drawing process is dubious since the diameter of the preform undergoes a drastic reduction during the drawing process. The objectives of this research are to use a more accurate approach - the zonal method - to replace the optically thick approximation for computing the radiative heat exchange between the furnace and the preform, and to determine if the optically thick approximation is valid for this process. In applying the zonal method, the preform surface is assumed to be diffuse to both transmission and reflection. An enclosure analysis is performed for the radiative exchange between the furnace and the outer surface of the preform and the zonal method is employed to consider the radiative exchange within the glass preform. The emissivity for the glass preform has been calculated based on the diffuse surface assumption and applied to the computation of radiative heat flux with the optically thick approximation for the purpose of comparison with the present work. The results obtained by the zonal method show that the radiative heat flux is strongly influenced by the radial temperature variation within the preform, while those obtained by the optically thick approximation do not show this effect, as expected. Comparisons of the results obtained by these two approaches reveal that the optically thick approximation predicts the radiative heat flux satisfactorily for a range of axial temperature variations, but only when the radial temperature variation within the preform is small. The diameter change in the neck-down region has almost no effect on the validity of the optically thick approximation.
Radiation codes have been developed based on the S2 and S4 discrete ordinates approximations which involve solving the radiation transport equation in 4 and 12 directions, respectively. Evaluation of the codes against exact numerical solutions and experimental data show that both these approximations predict radiative fluxes with acceptable accuracy. The models are also evaluated treating them as part of an overall predictive scheme for pulverized coal combustion. The comparative importance of the various input parameters on the model predictions is evaluated via a detailed sensitivity study based on Fourier analysis technique. This analysis shows that the predictions arc most sensitive to the particle number densities and temperatures, while little sensitivity to the absorption and scattering efficiencies is detected.
Benchmark solutions to nontrivial radiation transport problems are crucial to the validation of transport codes. This paper gives an analytical transport solution for non-equilibrium radiative transfer in an infinite and isotropically scattering medium. The radiation source in the medium is isotropic in angle and constant in time (but only exists in a finite period of time), and is allowed to be uniformly distributed in a finite space or to be located at a point. The solution is constructed by applying the Fourier transform with respect to spatial variable and the Laplace transform with respect to temporal variable. The integration over angular variable is treated exactly. The resulting solution, as a function of space and time and in the form of a double integral, is evaluated numerically without much difficulty. Tables and figures are given for the resulting benchmark solution.
The fundamental problem of applying the method of discrete ordinates to radiative transfer predictions is the selection of the discrete directions and their associated weights. Both the accuracy of the solution and the computational effort depend on the angular discretization. This paper provides a sound mathematical methodology for the derivation of angular quadratures. By applying the collocation principle, the errors introduced by a quadrature are analysed and the constituting equations of angular quadratures are identified. Special emphasis is placed on the rotational invariance of the quadrature schemes. Multidimensional radiative transfer in participating media with isotropic and anisotropic scattering is accounted for throughout the analysis. A major goal of the present study is the construction of a new principle for multidimensional angular quadratures which is essentially a generalization of the principles employed for the well-known Sn quadratures. The new construction principle has two major advantages. First, it enables a very flexible tailoring of quadratures according to the actual requirements. Second, compared to the Sn quadratures, the new types of quadratures provide a higher accuracy while using the same number of nodal points.
The limitations of asymptotic methods for numerically solving highly forward peaked scattering (HFPS) problems are reviewed before resorting to a discrete ordinates solution for such problems based on biased angular quadrature formulas to increase the precision of the angular representation and on source evaluation from cell-averaged angular fluxes to reduce memory requirements. Also, a twice-collided source is introduced to avoid numerical representation of singularities in the solution. As an example the propagation and spreading of a collimated particle beam in an HFPS medium has been calculated with a discrete ordinates diamond-differenced numerical solution of the transport equation in two-dimensional curvilinear cylindrical coordinates. The calculation was carried out for a strongly forward peaked Henyey-Greenstein scattering law for which Fokker-Planck asymptotic models are not valid. The results show promise for numerically calculated reference solutions based on accurate spatial representations for checking the accuracy of standard asymptotic models for these types of problems.
The equations of radiative transfer are systematically analyzed by asymptotic methods. To lowest order, the classical equilibrium diffusion approximation is recovered. The next order analysis leads to the equilibrium diffusion differential equations and initial condition, but with a boundary condition containing a linear extrapolation distance alpha. This quantity is related to the solution of a canonical halfspace problem and is computed by deriving an appropriate variational principle. For the case of no scattering, an exact Wiener-Hopf solution is available. The F(N) solution technique is also applied to the problem of obtaining alpha with good results. Higher order asymptotic radiative transfer descriptions are discussed and, while not immediately constituting practical calculational techniques, do have implications for computing the parameters in the multiband treatment of the frequency variable.
A new nonlinear solution method is developed and applied to a non-equilibrium radiation diffusion problem. With this new method, Newton-like super-linear convergence is achieved in the nonlinear iteration, without the complexity of forming or inverting the Jacobian from a standard Newton method. The method is a unique combination of an outer Newton-based iteration and and inner conjugate gradient-like (Krylov) iteration. The effects of the Jacobian are probed only through approximate matrix–vector products required in the conjugate gradient-like iteration. The methodology behind the Jacobian-free Newton–Krylov method is given in detail. It is demonstrated that a simple, successive substitution, linearization produces an effective preconditioning matrix for the Krylov method. The efficiencies of different methods are compared and the benefits of converging the nonlinearities within a time step are demonstrated.
The nature of anomalous computational effects due to the discretization of the angular variable in transport theory discrete ordinates approximations is described and analyzed. The origin of these effects within the derivation of the Sn discrete ordinates equations is shown, and the effects are related to the non-equivalence of the general geometry discrete ordinates equations and the corresponding spherical harmonics equations. Procedures are given for the definition of two-dimensional discrete ordinates equations that are equivalent to the spherical harmonics equations. Elimination of ray effects from the two-dimensional S2 equations by reduction to the diffusion theory equations is verified in a numerical example. Recipes for the elimination of ray effects are analyzed in the analytic solution of the infinite medium, isotropic line-source problem in the rectangular geometry, S2 approximation. Optimum magnitudes for corrective source terms are indicated by the analysis. It is concluded that ray effects may be eliminated by modification of the discrete ordinates formulation, but that the extra computational effort may be more expensive than the alternative of increasing the order of angular quadrature and that the presence of discretization effects may serve as an indicator of the adequacy of the angular quadrature used.
The nature of anomalous computational effects due to the discretization of the angular variable in transport theory discrete ordinates approximations is described and analyzed. The origin of these effects within the derivation of the Sn discrete ordinates equations is shown, and the effects are related to the non-equivalence of the general geometry discrete ordinates equations and the corresponding spherical harmonics equations. Procedures are given for the definition of two-dimensional discrete ordinates equations that are equivalent to the spherical harmonics equations. Elimination of ray effects from the two-dimensional S2 equations by reduction to the diffusion theory equations is verified in a numerical example. Recipes for the elimination of ray effects are analyzed in the analytic solution of the infinite medium, isotropic line-source problem in the rectangular geometry, S2 approximation. Optimum magnitudes for corrective source terms are indicated by the analysis. It is concluded that ray effects may be eliminated by modification of the discrete ordinates formulation, but that the extra computational effort may be more expensive than the alternative of increasing the order of angular quadrature and that the presence of discretization effects may serve as an indicator of the adequacy of the angular quadrature used.
The S/sub n/ method is a powerful tool for tne analysis and numerical
solution of a large variety of reactor problems. Basically, S/sub n/ is a finite
difference technique which proceeds in the following general way: The coatinuous
angular variable in the transport equation is replaced by a discrete variable,
whereupon the transport equation becomes a system of coupled partial differential
equations, and, if certain specified conditions are met, a system of equations
called the S/sub n/ equations. These have a number of desirable properties, in
particular, the S/sub n/ equations may be readily solved successively. The
solution of the S/sub n/ system approaches the solution of the transport equation
as the number of discrete angles is increased, and the solution of the S/sub n/
difference equations converges to the solution of the S/sub n/ equations as the
space mesh is refined. The transport equation is basically a conservation law
and this feature can be carried to the S/sub n/ difference equations. The S/sub
n/ method is then generalized to the cases of multiple velocity groups,
inhomogeneous source terms, and geometrics of higher order. To solve the X
difference equations for the many problem types annd physical situations which
arise, a number of numerical techniques are required. Many of these are
explained, in particular, the basic iteration methods annd the procedures used to
establish and speed connvergennce. The accuracy of the S/sub n/ method is stated
as a function of n, the number of space points, and other parameters. The
accuracy of the S/sub n/ methods for reactor cell calculations and disadvantage
factor calculations is discussed in some detail. Finally, a number of practical
topics are reviewed, such as the selection of velocity groups, the problem of
choosing cross section, and the comparison of calculation with experiment.
(auth)
This text provides a foundation in both the theoretical and practical aspects of radiative transfer, for advanced students of atmospheric, oceanic and environmental sciences. The transfer of solar and infrared radiation through optically-thick clouds, aerosol layer, and the oceanic mixed layer is presented through the use of heuristic models of scattering and absorption, and a systematic approach to formulation and solution of the radiative transfer equation. Problems such as the the transmission of ultraviolet radiation through the atmosphere and ocean, remote sensing, solar heating and infrared cooling processes, UV biological dose rates, and Greenhouse warming are solved using a variety of methods. This self-contained, systematic treatment will prepare students from a range of disciplines in problems concerning the effects of solar and infrared radiation on natural systems. The hardback edition received excellent reviews.
Presented in two volumes, The Physics of Astrophysics is ideally suited
for a year-long astrophysics course for university seniors and
first-year graduate students. The first volume deals with the emission,
absorption, and scattering of radiation by matter, as well as covering
related topics such as radiative transfer, statistical physics,
classical electrodynamics, and atomic and molecular structure.