Maximum Entropy Closure Research Articles

A maximum-entropy length-orientation closure for short-fiber reinforced composites

We describe an algorithm for generating fiber-filled volume elements for use in computational homogenization schemes which accounts for a coupling of the fiber-length and the fiber-orientation. For prescribed fiber-length distribution and fiber-orientation tensor of second order, a maximum-entropy estimate is used to produce a fiber-length-orientation distribution which mimics real injection molded specimens, where longer fibers show a stronger alignment than shorter fibers. We derive the length-orientation closure from scratch, discuss its integration into the sequential addition and migration algorithm for generating fiber-filled microstructures for industrial volume fractions and investigate the resulting effective elastic properties. We demonstrate that accounting for the length-orientation coupling permits to match the measured Young’s moduli in principal fiber direction and transverse to it more accurately than for closure approximations ignoring the length-orientation coupling.

Application of Approximate Maximum-Entropy Moment Closure to the Wigner Equation

This paper investigates the potential of moment closures as a possible path to future quantum hydrodynamics models. Moment closures are known to produce accurate predictions of continuum and non-equilibrium classical gas flows while offering modeling and numerical advantages over other commonly used methods. The maximum-entropy hierarchy of moment closures holds the promise of robustly hyperbolic stable moment equations, however, the predicted distribution function in phase space is positive by design, which is in disagreement with the quasi-distribution function described by the Wigner equation. In this paper, predictions made by an interpolative one-dimensional five-moment system are compared to direct numerical solutions of the Wigner equation for a simple low-energy quantum state in unbounded double-well potentials. Numerical solutions for a particle in various potentials described by quartic polynomials are presented. The capacity of moment methods to provide accurate and efficient models for quantum systems is explored.

Maximum Entropy Closure Relation for Higher Order Alignment and Orientation Tensors Compared to Quadratic and Hybrid Closure

A closure relation expresses the fourth order orientation tensor as a function of the second order one. Two well-known closure relations, the hybrid closure and the maximum entropy closure, are compared in the case of a rotation symmetric orientation distribution function. The maximum entropy closure predicts a positive fourth order parameter in the whole range of the second order parameter, whereas the hybrid closure results in negative fourth order parameters for small values of the second order one. For the maximum entropy closure quadratic fit polynomials are presented. For a general distribution without rotation symmetry, the expression for the entropy is exploited to derive an explicit form for the maximum entropy distribution. Lowest order approximation of this distribution function leads to simple closure forms for the fourth order alignment tensor and also for higher order alignment tensors.

Principal-axis Analysis of the Eddington Tensor for the Early Post-bounce Phase of Rotational Core-collapse Supernovae

Using full Boltzmann neutrino transport, we performed 2D core-collapse supernova simulations in axisymmetry for two progenitor models with 11.2 and 15.0 M ⊙, both rotational and nonrotational. We employed the results obtained in the early post-bounce phase (t ≲ 20 ms) to assess performance under rapid rotation of some closure relations commonly employed in the truncated moment method. We first made a comparison in 1D under spherical symmetry, though, of the Eddington factor p defined in the fluid rest frame (FR). We confirmed that the maximum entropy closure for the Fermionic distribution (MEFD) performs better than others near the proto–neutron star surface, where p < 1/3 occurs, but does not work well even in 1D when the phase-space occupancy satisfies e < 0.5 together with p < 1/3, the condition known to be not represented by MEFD. For the 2D models with the rapid rotation, we employed the principal-axis analysis of the Eddington tensor. We paid particular attention to the direction of the longest principal axis. We observed in FR that it is aligned neither with the radial direction nor with the neutrino flux in 2D, particularly so in convective and/or rapidly rotating regions, the fact not accommodated in the moment method. We repeated the same analysis in the laboratory frame and found again that the direction of the longest principal axis is not well reproduced by MEFD because the interpolation between the optically thick and thin limits is not very accurate in this frame.

A two species micro–macro model of wormlike micellar solutions and its maximum entropy closure approximations: An energetic variational approach

Wormlike micelles are self-assemblies of polymer chains that can break and recombine reversibly. In this paper, we derive a thermodynamically consistent two-species micro–macro model of wormlike micellar solutions by employing an energetic variational approach. The model incorporates a breakage and combination process of polymer chains into the classical micro–macro dumbbell model of polymeric fluids in a unified variational framework. We also study different maximum entropy closure approximations to the micro-macro model by “variation-then-closure” and “closure-then-variation” approaches. By imposing a proper dissipation in the coarse-grained level, the closure model, obtained by “closure-then-variation”, preserves the thermodynamical structure of both mechanical and chemical parts of the original system. Several numerical examples show that the closure model can capture the key rheological features of wormlike micellar solutions in shear flows.

Gaussian Process Regression for Maximum Entropy Distribution

Maximum-Entropy Distributions offer an attractive family of probability densities suitable for moment closure problems. Yet finding the Lagrange multipliers which parametrize these distributions, turns out to be a computational bottleneck for practical closure settings. Motivated by recent success of Gaussian processes, we investigate the suitability of Gaussian priors to approximate the Lagrange multipliers as a map of a given set of moments. Examining various kernel functions, the hyperparameters are optimized by maximizing the log-likelihood. The performance of the devised data-driven Maximum-Entropy closure is studied for couple of test cases including relaxation of non-equilibrium distributions governed by Bhatnagar-Gross-Krook and Boltzmann kinetic equations.

Stationary response probability density of nonlinear random vibrating systems: a data-driven method

A data-driven method is established to derive the (approximately) analytical expression of the stationary response probability density of nonlinear random vibrating system, which explicitly includes system features and intensity of excitation. The stationary response probability density is first assumed as an exponential form by using the principle of maximum entropy. Through the rule of dimensional consistency, the power of exponential function is expressed as a linear combination of a set of nondimensional parameter clusters which are constituted by system features, intensity of excitation, and state variables. By comparing the power of exponential function with the approximate logarithm probability density evaluated from simulated data statistically, the determination of unknown coefficients comes down to the solution of (overdetermined) simultaneous linear algebraic equations. The data-driven method rediscovers the exact stationary response probability density of random-excited Duffing oscillator and derives an approximately analytical expression of stationary response probability density of van der Pol system from the simulated data of six cases with different values of system features and intensity of excitation. This data-driven method is a unique method which can explicitly include the information of system and excitation in the analytical expression of stationary response probability density. It avoids the solution of simultaneous nonlinear algebraic equations encountered in the maximum entropy method and closure methods and, in the meanwhile, avoids the sophisticated selection of weighting functions in closure methods.

A Maximum Entropy-Inspired Interpolative Closure for the Prediction of Radiative Heat Transfer in Laminar Co-Flow Diffusion Flames

ABSTRACT An interpolative-based first-order maximum entropy (M1) moment closure for providing approximate solutions to the equation of radiative transfer in non-gray participating media is proposed and described. This newly developed non-gray moment closure technique results in significant computational savings compared to an approach that makes use of the direct numerical solution of the optimization problem for entropy maximization. Its predictive capabilities are also assessed by comparing its solutions to those of the more commonly adopted first-order spherical harmonics (P1) moment closure technique, as well as the popular discrete ordinates method (DOM), which is used as a benchmark for the model comparisons. The evaluations are performed for sooting co-flow laminar diffusion flames for blends of ethanol and methane fuels at atmospheric as well as at elevated pressures and include comparisons to available experimental data for soot volume fraction and flame temperature. The strong spectral dependence exhibited by the absorption coefficient of real gases is treated herein using the statistical narrow-band correlated-k method, and the chemical kinetics of the underlying species are modeled using a reduced mechanism for methane and ethanol fuels. Theoretical details of the proposed interpolative M1, along with a description of the proposed Godunov-type finite-volume scheme developed for the numerical solution of the resulting system of hyperbolic moment equations are briefly discussed. The finite-volume method makes use of limited second-order solution reconstruction, multi-block body-fitted quadrilateral meshes with anisotropic adaptive mesh refinement (AMR), and an efficient Newton–Krylov–Schwarz (NKS) iterative method for solution of the resulting nonlinear algebraic equations arising from the finite-volume discretization procedure. The numerical results for laminar co-flow flames show that the flame solutions and predictions of soot formation of the non-gray M1 maximum-entropy moment closure are very promising and in very good agreement with those of both the DOM and P1 spherical harmonics model, while offering a substantial reduction in the number of dependent solution variables, compared to the DOM.

Entropic quadrature for moment approximations of the Boltzmann-BGK equation

Using the method of moments to approximate the solution of kinetic equations has become a standard technique especially in the context of solving the Boltzmann equation to model rarefied gases. One attractive example is the maximum-entropy closure, which, however, is computationally barely affordable. In this paper, we will combine the maximum-entropy approach with the quadrature method of moments (QMOM) – two methods which have in some sense diametric properties, thereby introducing the “Entropic Quadrature” (EQ) closure. EQ calculates a sparse quadrature-based reconstruction of the unknown velocity distribution like the QMOM where the physical meaningful maximum-entropy principle is employed as a criterion for selecting a quadrature among different quadratures fitting the given moments. We will discuss the construction of the closure and give several numerical examples of its performance in one and two dimensions.

Kershaw-type transport equations for 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.

Analytic closures for M1 neutrino transport

Carefully accounting for neutrino transport is an essential component of many astrophysical studies. Solving the full transport equation is too expensive for most realistic applications, especially those involving multiple spatial dimensions. For such cases, resorting to approximations is often the only viable option for obtaining solutions. One such approximation, which recently became popular, is the M1 method. It utilizes the system of the lowest two moments of the transport equation and closes the system with an ad hoc closure relation. The accuracy of the M1 solution depends on the quality of the closure. Several closures have been proposed in the literature and have been used in various studies. We carry out an extensive study of these closures by comparing the results of M1 calculations with precise Monte Carlo calculations of the radiation field around spherically symmetric protoneutron star models. We find that no closure performs consistently better or worse than others in all cases. The level of accuracy that a given closure yields depends on the matter configuration, neutrino type and neutrino energy. Given this limitation, the maximum entropy closure by Minerbo on average yields relatively accurate results in the broadest set of cases considered in this work.

Efficient algorithms and implementations of entropy-based moment closures for rarefied gases

We present efficient algorithms and implementations of the 35-moment system equipped with the maximum-entropy closure in the context of rarefied gases. While closures based on the principle of entropy maximization have been shown to yield very promising results for moderately rarefied gas flows, the computational cost of these closures is in general much higher than for closure theories with explicit closed-form expressions of the closing fluxes, such as Grad's classical closure. Following a similar approach as Garrett et al. (2015) [13], we investigate efficient implementations of the computationally expensive numerical quadrature method used for the moment evaluations of the maximum-entropy distribution by exploiting its inherent fine-grained parallelism with the parallelism offered by multi-core processors and graphics cards. We show that using a single graphics card as an accelerator allows speed-ups of two orders of magnitude when compared to a serial CPU implementation. To accelerate the time-to-solution for steady-state problems, we propose a new semi-implicit time discretization scheme. The resulting nonlinear system of equations is solved with a Newton type method in the Lagrange multipliers of the dual optimization problem in order to reduce the computational cost. Additionally, fully explicit time-stepping schemes of first and second order accuracy are presented. We investigate the accuracy and efficiency of the numerical schemes for several numerical test cases, including a steady-state shock-structure problem.

The 35-Moment System with the Maximum-Entropy Closure for Rarefied Gas Flows

This paper presents a robust implementation of the maximum-entropy closure in the context of rarefied gas dynamics. Moment systems supplied with the maximum-entropy closure have attractive mathematical properties: They are hyperbolic in the interior of the domain of definition of the dual minimization problem and endowed with an entropy law. In contrast to Grad’s classical closure theory, the maximum-entropy closure allows for applications to strongly non-equilibrium gas flows. The 35-moment system studied in this paper includes as basis functions all monomials up to order four, so that evolution equations for important non-equilibrium quantities, such as the stress tensor and heat flux vector, are contained in the system. To remove the singularity in the maximum-entropy closure, we consider a bounded underlying velocity domain and approximate moments of the reconstructed maximum-entropy distribution with a fixed, block-wise Gauss–Legendre quadrature rule. The convex dual minimization problem is solved with a Newton type algorithm. We show that the Hessian matrix used in the Newton iteration can become ill-conditioned even for equilibrium states if monomial basis functions are used. To improve the robustness of the Newton iteration, we consider partially and fully adaptive basis algorithms and demonstrate that the 35-moment system allows for accurate and robust simulations of non-equilibrium rarefied gas flows in the transition regime by applying the model to one-dimensional gas processes, including a continuous shock-structure problem.

Maximum Entropy Closure of Balance Equations for Miniband Semiconductor Superlattices

Charge transport in nanosized electronic systems is described by semiclassical or quantum kinetic equations that are often costly to solve numerically and difficult to reduce systematically to macroscopic balance equations for densities, currents, temperatures and other moments of macroscopic variables. The maximum entropy principle can be used to close the system of equations for the moments but its accuracy or range of validity are not always clear. In this paper, we compare numerical solutions of balance equations for nonlinear electron transport in semiconductor superlattices. The equations have been obtained from Boltzmann–Poisson kinetic equations very far from equilibrium for strong fields, either by the maximum entropy principle or by a systematic Chapman–Enskog perturbation procedure. Both approaches produce the same current-voltage characteristic curve for uniform fields. When the superlattices are DC voltage biased in a region where there are stable time periodic solutions corresponding to recycling and motion of electric field pulses, the differences between the numerical solutions produced by numerically solving both types of balance equations are smaller than the expansion parameter used in the perturbation procedure. These results and possible new research venues are discussed.

Hydrodynamic equations for an electron gas in graphene

In this paper we review, and extend to the non-isothermal case, some results concerning the application of the maximum entropy closure technique to the derivation of hydrodynamic equations for particles with spin-orbit interaction and Fermi-Dirac statistics. In the second part of the paper we treat in more details the case of electrons on a graphene sheet and investigate various asymptotic regimes.

Spectral maximum entropy hydrodynamics of fermionic radiation: a three-moment system for one-dimensional flows

The spectral formulation of the nine-moment radiation hydrodynamics resulting from using the Boltzmann entropy maximization procedure is considered. The analysis is restricted to the one-dimensional flows of a gas of massless fermions. The objective of the paper is to demonstrate that, for such flows, the spectral nine-moment maximum entropy hydrodynamics of fermionic radiation is not a purely formal theory. We first determine the domains of admissible values of the spectral moments and of the Lagrange multipliers corresponding to them. We then prove the existence of a solution to the constrained entropy optimization problem. Due to the strict concavity of the entropy functional defined on the space of distribution functions, there exists a one-to-one correspondence between the Lagrange multipliers and the moments. The maximum entropy closure of moment equations results in the symmetric conservative system of first-order partial differential equations for the Lagrange multipliers. However, this system can be transformed into the equivalent system of conservation equations for the moments. These two systems are consistent with the additional conservation equation interpreted as the balance of entropy. Exploiting the above facts, we arrive at the differential relations satisfied by the entropy function and the additional function required to close the system of moment equations. We refer to this additional function as the moment closure function. In general, the moment closure and entropy–entropy flux functions cannot be explicitly calculated in terms of the moments determining the state of a gas. Therefore, we develop a perturbation method of calculating these functions. Some additional analytical (and also numerical) results are obtained, assuming that the maximum entropy distribution function tends to the Maxwell–Boltzmann limit.

A hierarchy of maximum entropy closures for Galerkin systems of incompressible flows

We propose a maximum-entropy closure strategy for dissipative dynamical systems building on and generalizing earlier examples (Noack & Niven (2012) [11]). Focus is placed on Galerkin systems arising from a projection of the incompressible Navier–Stokes equation onto orthonormal expansion modes. The maximum-entropy closure is motivated by a simple analytical example and elaborated to a hierarchical framework with sufficient conditions for the existence of solutions.

Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution

The ability to predict continuum and transition-regime flows by hyperbolic moment methods offers the promise of several advantages over traditional techniques. These methods offer an extended range of physical validity as compared with the Navier–Stokes equations and can be used for the prediction of many non-equilibrium flows with a lower expense than particle-based methods. Also, the hyperbolic first-order nature of the resulting partial differential equations leads to mathematical and numerical advantages. Moment equations generated through an entropy-maximization principle are particularly attractive due to their apparent robustness; however, their application to practical situations involving viscous, heat-conducting gases has been hampered by several issues. Firstly, the lack of closed-form expressions for closing fluxes leads to numerical expense as many integrals of distribution functions must be computed numerically during the course of a flow computation. Secondly, it has been shown that there exist physically realizable moment states for which the entropy-maximizing problem on which the method is based cannot be solved. Following a review of the theory surrounding maximum-entropy moment closures, this paper shows that both of these problems can be addressed in practice, at least for a simplified one-dimensional gas, and that the resulting flow predictions can be surprisingly good. The numerical results described provide significant motivations for the extension of these ideas to the fully three-dimensional case.

Maximum-entropy closure of hydrodynamic moment hierarchies including correlations

Generalized hydrodynamic moment hierarchies are derived which explicitly include nonequilibrium two-particle and higher-order correlations. The approach is adapted to strongly correlated media and nonequilibrium processes on short time scales which necessitate an explicit treatment of time-evolving correlations. Closure conditions for the extended moment hierarchies are formulated by a maximum-entropy approach, generalizing related closure procedures for kinetic equations. A self-consistent set of nonperturbative dynamical equations are thus obtained for a chosen set of single-particle and two-particle (and possibly higher-order) moments. Analytical results are derived for generalized Gaussian closures including the dynamic pair distribution function and a two-particle correction to the current density. The maximum-entropy closure conditions are found to involve the Kirkwood superposition approximation.

Maximum-entropy closure for a Galerkin model of an incompressible periodic wake

AbstractA statistical closure is proposed for a Galerkin model of an incompressible periodic cylinder wake. This closure employs Jaynes’ maximum entropy principle to infer the probability distribution for mode amplitudes using exact statistical balance equations as side constraints. The analysis predicts mean amplitude values and modal energy levels in good agreement with direct Navier–Stokes simulation. In addition, it provides an analytical equation for the modal energy distribution.