Alberto Passalacqua
email: albertop@iastate.edu

Frédérique Laurent

Rodney O Fox

A second-order realizable scheme for moment advection on unstructured grids

Keywords: Moment advection, realizability, unstructured grid, advection, schemes, moment methods, quadrature method of moments

 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.

Introduction

Moment methods are a class of mathematical methods used to determine approximate solutions to 2 problems involving the evolution of a distribution function. Notable examples of equations describing this evolution are the population balance equation [START_REF] Ramkrishna | Population balances: theory and applications to particulate systems in engineering[END_REF][START_REF] Marchisio | Computational Models for Polydisperse Particulate and Multiphase Systems[END_REF], the Boltzmann [START_REF] Cercignani | Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations[END_REF][START_REF] Cercignani | The Boltzmann Equation and Its Applications[END_REF], the Boltzmann-Enskog [START_REF] Ramkrishna | Population balances: theory and applications to particulate systems in engineering[END_REF], the Fokker-Planck and Kolmogorov [START_REF] Fokker | Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld[END_REF][START_REF] Planck | Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie, Sitzungsberichte Preuss[END_REF][START_REF] Kolmogoroff | Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung[END_REF] equations, in addition to several others. These equations can be written in compact form as

𝜕𝑓 𝜕𝑡 + ∇ 𝐱 ⋅ (𝑓𝐯) + 𝒮 = 0 (1.1)
where 𝑓(𝑡, 𝐱, 𝛏) is the distribution, 𝐱 is the position vector, 𝛏 is the vector of internal coordinates specific to the problem (e.g. size, composition, velocity, charge…), 𝐯 is the velocity and 𝒮 contains other terms depending on the evolution equation for 𝑓 under consideration.

Moment methods [START_REF] Chapman | The mathematical theory of non-uniform gases[END_REF][START_REF] Struchtrup | The BGK-model with velocity-dependent collision frequency[END_REF][START_REF] Mcgraw | Description of aerosol dynamics by the quadrature method of moments[END_REF] consist in deriving partial differential equations (PDEs) for the spatiotemporal evolution of a finite vector of moments of the distribution function characteristic of the problem under consideration, in order to reduce the dimensionality of the problem and, consequently, make it computationally more treatable. This is achieved by applying the definition of moment of 𝑓

𝑚 𝑖𝑗𝑘… = ∫ 𝜉 1 𝑖 𝜉 2 𝑗 𝜉 3 𝑘 … 𝑓𝑡, 𝐱, 𝛏)d𝛏 Ω (1.2)
to both sides of Eq. (1.1), which leads to conservation equations for the quantities 𝑚 𝑖𝑗𝑘… , which are the moments about the origin of 𝑓. The PDEs obtained in this way are then discretized on the domain of interest and solved numerically, typically using the finite-volume method [START_REF] Leveque | Finite Volume Methods for Hyperbolic Problems[END_REF][START_REF] Ferziger | Computational Methods for Fluid Dynamics[END_REF]. The literature reports many example applications where moment methods were used [START_REF] Chapman | The mathematical theory of non-uniform gases[END_REF][START_REF] Struchtrup | The BGK-model with velocity-dependent collision frequency[END_REF], including applications to aerosols [START_REF] Mcgraw | Description of aerosol dynamics by the quadrature method of moments[END_REF][START_REF] Wright | Bivariate Extension of the Quadrature Method of Moments for Modeling Simultaneous Coagulation and Sintering of Particle Populations[END_REF][START_REF] Yoon | Representation of generally mixed multivariate aerosols by the quadrature method of moments: I. Statistical foundation[END_REF], gas-particle flows [START_REF] Passalacqua | A fully coupled quadrature-based moment method for dilute to moderately dilute fluid-particle flows[END_REF][START_REF] Kong | A solution algorithm for fluid-particle flows across all flow regimes[END_REF][START_REF] Kong | Euler-euler anisotropic gaussian mesoscale simulation of homogeneous cluster-induced gas-particle turbulence[END_REF], gas-liquid flows [START_REF] Buffo | Multidimensional population balance model for the simulation of turbulent gas-liquid systems in stirred tank reactors[END_REF][START_REF] Buffo | Multidimensional population balance model for the simulation of turbulent gas-liquid systems in stirred tank reactors[END_REF][START_REF] Petitti | Simulation of coalescence, breakup and mass transfer in a gas-liquid stirred tank with CQMOM[END_REF][START_REF] Heylmun | A quadrature-based moment method for polydisperse bubbly flows[END_REF], combustion [START_REF] Donde | A multivariate quadrature based moment method for LES based modeling of supersonic combustion[END_REF][START_REF] Raman | Modeling of Fine-Particle Formation in Turbulent Flames[END_REF][START_REF] Laurent | Accurate treatment of size distribution effects in polydisperse spray diffusion flames: multi-fluid modelling, computations and experiments[END_REF], sprays [START_REF] De Chaisemartin | Eulerian models for turbulent spray combustion with polydispersity and droplet crossing[END_REF][START_REF] Kah | A high order moment method simulating evaporation and advection of a polydisperse liquid spray[END_REF][START_REF] Laurent | Quadrature method of moments for modeling multicomponent spray vaporization[END_REF][START_REF] Doisneau | On Multi-Fluid models for spray-resolved LES of reacting jets[END_REF] and radiation transport [START_REF] Vikas | Radiation transport modeling using extended quadrature method of moments[END_REF], to mention a few. In this work we focus on the transport of moments of a univariate distribution, where 𝛏 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.

A key difficulty in formulating numerical methods for the solution of PDEs describing the evolution of a moment vector is the discretization of the advection term. As illustrated by Wright [START_REF] Wright | Numerical advection of moments of the particle size distribution in Eulerian models[END_REF], this difficulty consists in preserving the moment realizability 2 , which is systematically compromised if classical discretization schemes, different from first-order upwind, are used to discretize the advection term [START_REF] Wright | Numerical advection of moments of the particle size distribution in Eulerian models[END_REF]. The first-order upwind scheme, while used in most of the applications of moment methods found in the literature in order to avoid compromising moment realizability, is too dissipative, and requires extremely refined grid resolutions to achieve satisfactory results, seriously compromising the feasibility of simulations in applications involving large-scale domains or large gradients of the transported moment vector. Wright [START_REF] Wright | Numerical advection of moments of the particle size distribution in Eulerian models[END_REF] examined several solutions to the problem of moment corruption due to advection, including the adoption of augmented schemes for vector transport, the transport of surrogate quantities related to the moment vector that needs to be advected, of quadrature weights and abscissae associated to the moment set in certain quadrature-based moment methods [START_REF] Marchisio | Solution of population balance equations using the direct quadrature method of moments[END_REF],

and moment correction algorithms [START_REF] Wright | Numerical advection of moments of the particle size distribution in Eulerian models[END_REF][START_REF] Mcgraw | Correcting transport errors during advection of aerosol and cloud moment sequences in Eulerian models[END_REF]. This latter approach consists in replacing the compromised moment vector with a valid one, obtained either by removing negative second-order differences [START_REF] Marchisio | Computational Models for Polydisperse Particulate and Multiphase Systems[END_REF] or enacting an optimization procedure to identify a moment vector that maximized ln 𝑚 𝑘 [START_REF] Mcgraw | Correcting transport errors during advection of aerosol and cloud moment sequences in Eulerian models[END_REF].

Kah et al. [START_REF] Kah | A high order moment method simulating evaporation and advection of a polydisperse liquid spray[END_REF] developed a second-order realizable advection scheme for moment vectors of distribution with compact support, suitable for structured grids, in cases with moment vectors in the interior of the moment space. In their approach, instead of performing a direct spatial reconstruction of the moment vector of interest, the corresponding vector of canonical moments is considered. The boundedness of the canonical moments, which are necessarily positive scalars defined over ]0, 1[ is leveraged to formulate a limiter which ensures not only the boundedness of the numerical solution, but also the realizability of the vector of transported moments under a condition on the integration time step. The resulting scheme was applied together with adaptive mesh refinement in [START_REF] Essadki | Adaptive Mesh Refinement and High Order Geometrical Moment Method for the Simulation of Polydisperse Evaporating Sprays[END_REF].

Vikas et al. [START_REF] Vikas | Realizable high-order finite-volume schemes for quadrature-based moment methods[END_REF] proposed a quasi-high-order realizable advection scheme in the context of quadraturebased moment methods. In their scheme, which was formulated on unstructured grids, the advection term is computed as a function of the quadrature approximation of the NDF [START_REF] Marchisio | Computational Models for Polydisperse Particulate and Multiphase Systems[END_REF][START_REF] Mcgraw | Description of aerosol dynamics by the quadrature method of moments[END_REF], using a MUSCLtype limited scheme for the reconstruction of the quadrature weights and a first-order upwind scheme for quadrature abscissae. The scheme ensures moment realizability if a condition on the integration time-step is satisfied. However, the formal order of accuracy of the scheme is limited by the first-order reconstruction of the quadrature abscissae. Vikas et al. [START_REF] Vikas | Realizable high-order finite-volume schemes for quadraturebased moment methods applied to diffusion population balance equations[END_REF] proposed a realizable scheme for diffusion problems, always in the context of quadrature-based moment methods.

Alldredge and Schneider [START_REF] Alldredge | A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension[END_REF] formulated a discontinuous Galerkin scheme in the context of entropybased moment closures, coupling it with a strong stability preserving Runge-Kutta method for time integration. Since this approach relies on entropy-based closures, it is suitable for problems with moment vectors in the interior of the moment space but does not deal with the case of moment vectors are the boundary of the moment space.

Laurent and Nguyen [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF] developed a second-order realizable scheme by considering the reconstruction of positive quantities related to the set of transported moments in what they have called 𝜁 scheme.

This numerical scheme, which requires a CFL-like condition to ensure the realizability of the moment vector, was successfully applied to the transport of moment vectors of a regular NDF and of a bimodal NDF, with moments possibly at the boundary of the moment space. In both cases the scheme has shown the capability of preserving the realizability of the advected moment vector, while maintaining its accuracy.

The nature of the 𝜁 scheme and, in particular, the version of this scheme called 𝜁 simplified scheme by Laurent and Nguyen [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF], makes it an ideal candidate for its extension to unstructured grids with cells of arbitrary shapes because the scheme relies on a traditional MUSCL reconstruction, and only requires a local additional limitation to be applied to the reconstructed quantities on cell faces. The extension of this scheme to unstructured grids and its implementation into the OpenFOAM framework [START_REF] Weller | A tensorial approach to computational continuum mechanics using object-oriented techniques[END_REF] are the topics of this article, the remainder of which is organized as follows: in Sec. 2 the problem of moment transport is introduced. The 𝜁 simplified scheme for hexahedral structured grids of Laurent and Nguyen [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF] is summarized in Sec. 3. Its generalization to unstructured grids with cells of arbitrary shapes is discussed in Sec. 4. Finally, the same one-and two-dimensional test cases used by Laurent and Nguyen [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF] are used in Sec. 5 to verify the implementation of the numerical scheme, using hexahedral uniform grids. Then, the two-dimensional cases are repeated using a triangular grid, and the order of accuracy of the 𝜁 simplified scheme on unstructured grids is assessed.

Moment transport and moment advection

The focus of the present work is on the pure advection problem of a moment vector 𝐦 N = (𝑚 0 , … , 𝑚 N-1 ) associated to a measure with support over ℝ + , with a known velocity field 𝐔, according to the set of equations

𝜕𝑚 𝑘 𝜕𝑡 + ∇ 𝐱 ⋅ (𝑚 𝑘 𝐔) = 0, ∀𝑚 𝑘 ∈ 𝐦 N . (2.1)
The solution of Eq. (2.1) is non-trivial because the numerical schemes used to discretize the time derivative and the divergence term need to guarantee the moment vector 𝐦 N remains realizable. This condition can be expressed through a set of non-linear relationships the components of 𝐦 N need to satisfy. These conditions are represented, for moments associated to a measure with support on ℝ + , through the Hankel determinants [START_REF] Dette | The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis[END_REF][START_REF] Dette | Matrix measures, moment spaces and Favard's theorem for the interval[END_REF]:

𝐻 2𝑝 = | 𝑚 0 … 𝑚 𝑝 ⋮ ⋱ ⋮ 𝑚 𝑝 … 𝑚 2𝑝 | , 𝐻 2𝑝+1 = | 𝑚 1 … 𝑚 𝑝+1 ⋮ ⋱ ⋮ 𝑚 𝑝+1 … 𝑚 2𝑝+1 |, (2.2)
with 𝐻 -2 = 𝐻 -1 = 1.

Necessary and sufficient condition for 𝐦 N to be in the interior of the moment space is that 𝐻 2𝑝 > 0 and 𝐻 2𝑝+1 > 0. Moreover, when 𝐦 N is at the boundary of the moment space, some of these Hankel determinants are null.

A discussion on realizable time integration schemes can be found, for example, in [START_REF] Vikas | Realizable high-order finite-volume schemes for quadrature-based moment methods[END_REF]. Since the focus of this article is on advection schemes, it suffices to remember that any time-integration scheme convex combination of explicit Euler steps is realizable under some restriction on the integration time step.

As anticipated in the introduction, the objective of this article is to extend the second-order realizable advection scheme of Laurent and Nguyen [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF] to unstructured grids. To such a purpose, let us consider a computational grid made of cells of arbitrary shape. The volume of a generic cell c is Ω c , and the number of faces of the same cell is N f,𝑐 . The subscript own indicates the reconstructed values of a variable at the cell face, assuming outgoing flux; the subscript nei indicates the reconstructed values of a variable at the cell face if the flux is going into the cell. For consistency with the convention adopted in OpenFOAM, we assume the flux of a transported property to be positive when outgoing with respect to the computational cell owning the face used to define the flux.

Following Vikas et al. [START_REF] Vikas | Realizable high-order finite-volume schemes for quadrature-based moment methods[END_REF], and using the nomenclature introduced above, Eq. (2.1) can be rewritten in semi-discrete form to obtain the evolution equation of the moment 𝑚 𝑘 defined at the center of each computational cells. Such equation reads

𝜕𝑚 𝑘,𝑐 𝜕𝑡 + 1 Ω 𝑐 ∑ [𝑚 𝑘,own,𝑓 max(𝐔 𝑓 ⋅ 𝐒 𝑓 , 0) + 𝑚 𝑘,nei,𝑓 min(𝐔 𝑓 ⋅ 𝐒 𝑓 , 0)] = 0 N f,𝑐 -1 𝑓=0 , (2.3)
where 𝐒 𝑓 is the vector normal to the surface of the cell face 𝑓, belonging to cell 𝑐, with magnitude equal to the surface area of the cell face |𝐒 𝑓 | and pointing outward of the cell, and 𝐔 𝑓 is the reconstructed value of 𝐔 at the same face.

A procedure needs to be developed to compute the moments 𝑚 𝑘,own,𝑓 and 𝑚 𝑘,nei,𝑓 to ensure the realizability of the advected moment vector 𝐦 N because, as discussed in the introduction, conventional finite-volume advection schemes, relying on reconstructions of order higher than one, do not guarantee the realizability of the transported moment set [START_REF] Wright | Numerical advection of moments of the particle size distribution in Eulerian models[END_REF].

The simplified 𝜻 advection scheme

This section summarizes the realizable 𝜁 simplified advection scheme [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF], which represents the foundation of the scheme for unstructured grids subject of the present work. This scheme was developed for one-dimensional problems with uniform spatial discretization and extended to multiple dimensional problems with Cartesian grids through dimensional splitting. Only the one-dimensional scheme is presented here.

In the 𝜁 simplified scheme, the moment vector 𝐦 N is advected by considering the auxiliary vector

𝛇 N-1 = (𝜁 0 , … , 𝜁 𝑁-2 ), with 𝜁 𝑝 = 𝐻 𝑝+1 𝐻 𝑝-2 𝐻 𝑝 𝐻 𝑝-1 , 𝑝 = 0, 1, … , N -2, (3.1)
where 𝐻 2𝑝 and 𝐻 2𝑝+1 are the Hankel determinants in Eq. (1.2).

These quantities are efficiently calculated using the QD algorithm [START_REF] Dette | The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis[END_REF], the Chebychev algorithm [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF] or the Wheeler algorithm [START_REF] Wheeler | Modified moments and Gaussian quadratures[END_REF]. The adoption of the last two approaches is particularly convenient in the context of quadrature-based moment methods because the coefficients required to calculate the quantities 𝜁 𝑝 correspond to the coefficients of the recurrence relationship of the orthogonal polynomials used to determine the quadrature formula.

Once the vector 𝛇 N-1 has been defined, the 𝜁 simplified advection scheme consists of the following steps:

1. The zero-order moment 𝑚 0 is advected using a conventional MUSCL scheme with the minmod limiter.

2. The values of 𝜁 𝑝 are reconstructed at each side of each cell face using the same MUSCL-type reconstruction as for 𝑚 0 , which ensures the positivity of the quantities 𝜁 𝑝 is preserved.

3. An additional limitation is conditionally applied to the slopes used to perform the reconstruction at the previous point, if needed to ensure the realizability of the advected moment set. This additional limitation is defined based on the realizability of the moment vector [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF][START_REF] Berthon | Stability of the MUSCL Schemes for the Euler Equations[END_REF]]

𝐦 * = 3𝐦 -𝐦 + -𝐦 -, (3.2)
where 𝐦 is the moment vector at the cell center, 𝐦 + and 𝐦 -are the moment vectors computed from the values 𝜁 𝑝 + and 𝜁 𝑝 -reconstructed at cell faces. If the moment vector defined by Eq. (3.2) is not realizable, an iterative procedure is used to reduce the slopes used to reconstruct 𝜁 𝑝 on cell faces, to ensure the realizability of the moment vector at the cost of the accuracy of the reconstruction. Details of this procedure are reported in Laurent and Nguyen (2017), and its generalization to unstructured grids is discussed in this work. The scheme summarized above was proven [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF] to preserve the realizability of the advected moment vector if the time step is adapted to maintain the value of the CFL number below 1/3, while converging to second-order accuracy with a sufficiently refined spatial discretization.

The extension of the 𝜁 simplified scheme to unstructured grids requires the generalization of the definition of 𝐦 * in Eq. (3.2), and of the CFL condition, which is expected to become a function of the number of faces of the grid cell.

Generalization to unstructured grids

The generalization of the 𝜁 simplified scheme to unstructured grids is achieved in this work by considering a co-located grid arrangement, where all the variables are stored at cell centroids, with the exclusion of the values at the boundary of the computational domain, where values are stored at face centroids. This approach is illustrated, for example, in Ferziger and Peric [START_REF] Ferziger | Computational Methods for Fluid Dynamics[END_REF], and was adopted in the computational toolkit [START_REF] Weller | A tensorial approach to computational continuum mechanics using object-oriented techniques[END_REF][START_REF] Jasak | Error analysis and estimation for the finite volume method with applications to fluid flows[END_REF][START_REF] Openfoam | The OpenFOAM Foundation[END_REF] used in this work.

In order to evaluate the advection term considering the grid arrangement described above, values of the velocity and of the advected quantities need to be known at cell faces. The reconstruction of the advected scalars, moments in our case, is illustrated in Sec. 4.1, following the approach implemented into OpenFOAM [START_REF] Weller | A tensorial approach to computational continuum mechanics using object-oriented techniques[END_REF][START_REF] Jasak | Error analysis and estimation for the finite volume method with applications to fluid flows[END_REF], based on the work of several authors [START_REF] Bruner | Parallelization of the Euler Equations on Unstructured Grids[END_REF][START_REF] Chakravarthy | High resolution applications of the Osher upwind scheme for the Euler equations[END_REF][START_REF] Roe | Characteristic-based schemes for the Euler equations[END_REF][START_REF] Roe | Some contributions to the modelling of discontinuous flows[END_REF][START_REF] Sweby | High resolution schemes using flux limiters for hyperbolic conservation laws[END_REF][START_REF] Van Leer | Towards the ultimate conservative difference scheme III. Upstream-centered finitedifference schemes for ideal compressible flow[END_REF][START_REF] Van Leer | Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme[END_REF] and summarized by Darwish and Moukalled [START_REF] Darwish | TVD schemes for unstructured grids[END_REF].

Reconstruction on unstructured grids

Let us consider a generic advected quantity whose value, stored at the centroid C of a generic cell of the computational mesh, is indicated with 𝜓 C . The value of 𝜓 reconstructed on the cell face f is [START_REF] Roe | Some contributions to the modelling of discontinuous flows[END_REF][START_REF] Darwish | TVD schemes for unstructured grids[END_REF]]

𝜓 f = 𝜓 C + 1 2 ℒ(𝑟 f )(𝜓 D -𝜓 C ), (4.1)
where ℒ(𝑟 f ) is the limiter function, which depends on the ratio [71]

𝑟 f = 2∇𝜓 C ⋅ 𝐫 CD 𝜓 D -𝜓 C -1, (4.2)
with 𝑟 𝑓 ≥ 0 for TVD schemes [START_REF] Darwish | TVD schemes for unstructured grids[END_REF].

In these expressions, 𝜓 D represents the downwind value at the center of the neighbor cell sharing face f with the cell whose center is situated in C. The vector 𝐫 CD joins the centers C and D of the same two neighbor cells. The definition of the limiter function for a minmod limiter is [START_REF] Roe | Characteristic-based schemes for the Euler equations[END_REF][START_REF] Darwish | TVD schemes for unstructured grids[END_REF]:

ℒ(𝑟 f ) = max(0, min(1, 𝑟 f )). (4.
3)

The evaluation of the gradient of 𝜓 over the cell C in Eq. (4.2) is detailed in Darwish and Moukalled [START_REF] Darwish | TVD schemes for unstructured grids[END_REF]. A second-order least-squares method is used in this work to ensure accuracy on grids with cells of arbitrary shape and non-uniform spatial distribution.

Additional limitation for moment sets at the boundary of the moment space

The condition in Eq. (3.2), used to identify cases when an additional limitation is needed for the reconstructed values of the quantities 𝜁 𝑝 , needs to be generalized to be applicable to unstructured grids. To this purpose, let N f be the maximum number of faces of a cell of the computational grid.

Since arbitrary shaped cells are considered, this value can be different for any cell, without additional complication of the numerical scheme. Let also the flux of the quantities be positive when outgoing with respect to the computational cell under consideration. The number of faces characterized by outgoing flux with respect to the owner cell will be indicated with N f,o,c . The first step required to extend the 𝜁 simplified scheme [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF] to unstructured grids consists in computing the value of the zeroorder moment 𝑚 0 and of the components 𝜁 𝑝 of the vector 𝛇 N-1 at both sides of each face internal to the computational domain 3 . The symbols 𝑚 0,own and 𝑚 0,nei are used to indicate the values of 𝑚 0 reconstructed, respectively, at the side of the owner and neighbor cell. Similarly, 𝜁 𝑝,own and 𝜁 𝑝,nei indicate the values of 𝜁 𝑝 at the side of the owner cell and of the neighbor cell to the face, as discussed for 𝑚 0 .

Once the reconstruction of 𝑚 0 and 𝛇 N-1 is obtained, the moments at each side of each cell face are recomputed from the vectors (𝑚 0,own , 𝛇 N-1,own ) and (𝑚 0,nei , 𝛇 N-1,nei ) using the algorithm proposed by Skibinsky [START_REF] Skibinsky | Extreme nth moments for distributions on [0, 1] and the inverse of a moment space map[END_REF].

At this point, it is determined if an additional limitation needs to be applied to the reconstructed values of 𝜁 𝑝 , to guarantee the realizability of the advected moment set. This is achieved by defining, in each computational cell 𝑐, the quantity

𝑚 𝑘,𝑐 + = ∑ 𝑚 𝑘,𝑐,𝑓 N f,o,c -1 𝑓=0 , (4.4)
where 𝑚 𝑘,𝑐,𝑓 = 𝑚 𝑘,own,𝑓 is the value of the moment 𝑚 𝑘 reconstructed on face 𝑓 of cell 𝑐, where the flux is outgoing. The quantities 𝑚 𝑘,𝑐 + are then used to define the components of the 𝐦 * vector as

𝑚 𝑘,𝑐 * = (N f,o,c + 1)𝑚 𝑘,𝑐 -𝑚 𝑘,𝑐 + , 𝑘 = 0, … , N -1. (4.5)
If the number of realizable moments in 𝐦 * is smaller than the number of realizable moments in the cell, an additional limitation needs to be applied to the reconstructed values of the 𝜁 𝑝 quantities. The procedure is analogous to the one described in [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF], which is summarized here in a slightly modified fashion to clarify the details of its implementation into OpenFOAM.

While not strictly necessary, the reconstructed values 𝜁 𝑝,f at cell faces are decomposed, for convenience in the implementation, into the value obtained from a first-order reconstruction 𝜁 𝑝,f,up , and into a high-order correction 𝜁 𝑝,f,h.o. This correction is then multiplied by a coefficient 𝜆 𝑝,f ∈ [0, 1], which is the limiter that needs to be determined to ensure the realizability of the advected moment set:

𝜁 𝑝,f = 𝜁 𝑝,f,up + 𝜆 𝑝,f 𝜁 𝑝,f,h.o. . (4.6)
The value of 𝜆 𝑝,f is found by considering the values of the limiters determined based on the owner and the neighbor cell of face f, and taking their minimum:

𝜆 𝑝,f = min(𝜆 𝑝,own , 𝜆 𝑝,nei ). (4.7)

In each cell, the value of the limiter 𝜆 𝑘,𝑐 is determined as follows:

1. Set 𝜆 𝑝,𝑐 = 1.

2. Compute 𝐦 𝑐 + using 𝜁 𝑙,f with 𝑙 ≤ 𝑘, and with 𝜁 𝑝,f,up , for 𝑙 > 𝑝.

3. Recompute 𝐦 𝑐 * using the values of 𝐦 𝑐 + found at the previous step.

7.

Repeat from 1 for all 𝑝 to define the limiter for each of the elements of 𝛇 N-1 .

Once the cell values 𝜆 𝑝,𝑐 are defined for each cell, the face limiters are computed according to Eq.

(4.7), and the reconstructed values of 𝛇 N-1,own and 𝛇 N-1,nei are updated applying Eq. (4.6) to each of their components. The updated moment vectors 𝐦 N-1,own and 𝐦 N-1,nei are then found using the Skibinsky [START_REF] Skibinsky | Extreme nth moments for distributions on [0, 1] and the inverse of a moment space map[END_REF] algorithm to account for the additional limitation determined according to the procedure described above. These updated moment sets are used to define the advection term in the moment conservation equations (Eq. (2.3)).

Courant -Friedrichs -Lewy condition for realizability

The proposed scheme ensures the realizability of the advected moment set if the time step Δ𝑡 used to perform the integration satisfies a CFL condition obtained at the cell faces at which the flux is outgoing. If we consider an arbitrary cell with N f,c faces, N f,o,c of which have an outgoing flux, the maximum value of the CFL number is

CFL max = 1 2 max c ( ∑ 𝐔 𝑓 ⋅ 𝐒 𝑓 𝑓 𝑉 c ) Δ𝑡, (4.8)
where 𝑉 c is the cell volume, 𝐔 𝑓 the flow velocity at the cell face 𝑓 with surface 𝐒 𝑓 , belonging to cell c, and Δ𝑡 is the time step. Using the same notation, the maximum value of the time step to ensure moment realizability is defined by the condition

max 𝑐 [max 𝑓 ( 𝐔 𝑓 ⋅ 𝐒 𝑓 𝑉 𝑐 )] Δ𝑡 ≤ min 1 N f,o,c + 1 , (4.9)
where the minimum function on the right-hand side of Eq. (4.9) is introduced to account for the different number of faces cells may have in unstructured grids.

This is proven by considering the moment vector 𝐦 N,𝑐 𝑛 at time 𝑡 𝑛 , and integrate it according to Eq.

(2.3) with a backward Euler step to obtain the updated moment vector 𝐦 N,𝑐 𝑛+1 at time 𝑡 𝑛+1 . Assuming

𝐔 𝑓 ⋅ 𝐒 𝑓 > 0 for 𝑓 = 1, … , N f,o,c and 𝐔 𝑓 ⋅ 𝐒 𝑓 ≤ 0 for 𝑓 = N f,o,c , … , N f,c
, it is possible to write:

𝐦 N,c 𝑛+1 = 𝐦 N,c 𝑛 - Δ𝑡 𝑉 c ∑ 𝐦 N,own,𝑓 𝑛 𝐔 𝑓 ⋅ 𝐒 𝑓 N f,o,c 𝑓=1 ⏟ 𝒜 - Δ𝑡 𝑉 c ∑ 𝐦 N,c 𝑛 𝐔 𝑓 ⋅ 𝐒 𝑓 N f,c 𝑓=N f,o,c +1 ⏟ ℬ . (4.10)
in which only the term 𝒜 is of interest in the proof because ℬ is necessarily realizable. The moment vector 𝐦 N,c 𝑛 can be expressed in terms of

𝐦 c * 𝐦 N,c 𝑛 = 1 N f,o,c + 1 (𝐦 c * + ∑ 𝐦 N,own,𝑓 𝑛 N f,o,c 𝑓=1 ), (4.11)
consequently:

𝒜 = 1 N f,o,c + 1 𝐦 c * + ∑ ( 1 N f,o,c + 1 - Δ𝑡 𝑉 c 𝐔 𝑓 ⋅ 𝐒 𝑓 ) 𝐦 N,own,𝑓 𝑛 N f,o,c 𝑓=1 . (4.12)
By imposing the term in the summation is positive to ensure the realizability of the moment vector, the condition

1 N f,o,c + 1 ≥ Δ𝑡 𝑉 c 𝐔 𝑓 ⋅ 𝐒 𝑓 (4.13)
is obtained, which, applied to each value of 𝑓 and 𝑐 leads to Eq. (4.9).

It is worth observing that the definition in Eq. (4.9) automatically includes faces internal to the computational domain and faces lying on the boundary. Consequently, no special consideration for boundary conditions where the flux is imposed is needed to correctly define the maximum CFL required to ensure moment realizability.

The condition in Eq. In comparison, the upwind scheme ensures realizability of the advected moment vector if [START_REF] Vikas | Realizable high-order finite-volume schemes for quadraturebased moment methods applied to diffusion population balance equations[END_REF] Δ𝑡 ≤ Δ𝑥 𝑈 𝑓 . (4.18)

Test cases

One-dimensional advection of moments well inside the moment space

The first test case for the implementation of the numerical scheme consists in a one-dimensional advection problem on [0,1] with periodic boundary conditions and with constant velocity equal to 𝐔 = (1, 0, 0). The considered moment set, used as initial condition for the test case, is in the interiod of the moment space and defined through the following auxiliary functions:

𝛼(𝑥) = 3.5 + 1.5 sin 2𝜋𝑥, (5.1)
𝛽(𝑥) = 3.5 -1.5 cos 2𝜋𝑥, (5.2)

𝑚 ̃𝑘(𝑥) = { 1 𝑘 = 0 𝑚 𝑘-1 (𝑥)[𝛼(𝑥) + 𝑘 -1] 𝛼(𝑥) + 𝛽(𝑥) + 𝑘 -1 𝑘 < N , (5.3)
Θ(𝑥) = { 1 2 {1 + tanh [tan (2𝑥 - 1 2 )]} 𝑥 ≤ 1 2 1 2 {1 + tanh [tan ( 1 2 -2𝑥)]} 𝑥 > 1 2 , (5.4)
which allow the values of the initial moments to be computed as 𝑚 𝑘 (𝑥) = 𝑚 ̃𝑘(𝑥)Θ(𝑥).

(5.5)

The solutions obtained considering eight moments with 50, 100, 500 and 1000 uniformly distributed cells, after two flow-through times, are reported in Fig. 1. It is possible to observe from Fig. 1(a) that the grid resolutions of 50 and 100 cells are not sufficient to accurately preserve the moments because of the numerical diffusion caused by coarse grid resolution. However, the realizability of the moment set is preserved. The accuracy of the results significantly improves in the case of a grid resolution of 500 cells (Fig. 1(c)), which allows the analytical solution to be reproduced nearly exactly. The solution obtained on the grid with 1000 cells (Fig. 1(d)) is visually indistinguishable from the one calculated on 500 cells. Results are in agreement with those reported in the literature [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF].

In order to quantify the order of accuracy of the numerical scheme, the error of the numerical prediction with respect to the exact solution is calculated in each computational cell as 𝑒 𝑘 = |𝑚 𝑘,𝑒𝑥𝑎𝑐𝑡 -𝑚 𝑘,𝑛𝑢𝑚 | 𝑚 𝑘,𝑒𝑥𝑎𝑐𝑡 .

(5.6)

The logarithm of the error is reported as a function of the logarithm of the grid size ℎ = Δ𝑥 in Fig. 2, which illustrates that the second order of accuracy of the advection scheme is recovered for all the moments of the transported moment vector. Numerical values of the errors reported in Fig. 2 are shown in Table 1.

The test case presented in this section serves as preliminary verification of the implementation of the 𝜁 simplified scheme into OpenFOAM. However, the additional limitation of the 𝜁 𝑝 quantities was not triggered while performing the simulations required to produce the results in Fig. 1. Therefore, a different test case was considered to also examine this aspect of the implementation of the scheme, as

One-dimensional advection of moments at the boundary of the moment space

The second test case considers moments of a bimodal NDF, which can be at boundary of the moment space. The initial condition is defined with the help of three weight functions, whose value depends on the position in the computational domain 𝑥 as indicated in Eqs. (5.7) -(5.9). With these definitions, the value of the initial moments is found according to the expression:

𝑚 𝑘 (𝑥) = 𝑤 1 (𝑥)𝑥 𝑖,1 𝑘 + 𝑤 2 (𝑥)(2𝑥 𝑖,1 ) 𝑘 + 𝑤 3 (𝑥)𝜆 𝑘 (𝑥)Γ (1 + 𝑘 𝜅 2 ), (5.12)
where 𝑥 𝑖,1 = 0.02. The same cases considered in Sec. 5.1 were repeated with the initial condition defined by Eq. (5.12).

The comparison of the numerical solution and the exact solution is reported in Fig. 3. Similarly, to what is observed in the case of the moments of a regular NDF, also in the case of the moments of the bimodal distribution, the grid resolutions of 50 and 100 cells do not allow to exactly preserve the moments, which are satisfactorily preserved on the grid with 500 and 1000 cells. However, the 𝜁 scheme ensures the realizability of the transported moment vector. The behavior of the error, defined by Eq. (5.7), in the case of the bimodal NDF is reported in Fig. 4 and the corresponding numerical values are shown in Table 2.

Table 2: Numerical values of the errors reported in Fig. 4 and corresponding order of the reconstruction of each moment. The chart shows that nearly second-order accuracy is achieved for low-order moments. However, the intervention of the additional limiter on the 𝜁 quantities impacts the accuracy of high-order moments.

In any case, the order of accuracy of the 𝜁 scheme remains higher than one in all the cases considered in this test, which represents an extreme condition, with moments close to the boundary of the moment space.

-

Log(e4)

Log(e5) Log(e6) Log(e7) Slope 1 Slope 2
Fig. 4: Logarithm of the error in the numerical prediction of the moments of the bimodal NDF case as a function of the logarithm of the grid size -Zeta scheme.

Two-dimensional advection of moments of a regular NDF

The implementation of the second-order moment-preserving 𝜁 scheme was tested in a twodimensional case considering a computational domain defined as [0, 0.5] × [0, 0.5], with a steady Taylor-Green velocity field defined as:

𝐔 = (sin 2𝜋𝑥 cos 2𝜋𝑦 , -cos 2𝜋𝑥 sin 2𝜋𝑦) (5.13)
The velocity on cell faces, needed to evaluate the advection flux, was computed by linearly interpolating the velocity field defined in Eq. (5.13) onto the cell faces. The initial value of the moments was defined as follows:

𝑚 𝑘 (𝑧) = 𝑚 𝑘-1 (𝛼 2𝐷 (𝑧) + 𝑘 -1) 𝛼 2𝐷 (𝑧) + 𝛽 2𝐷 (𝑧) + 𝑘 -1 𝛾 2𝐷 (𝑧), (5.14)
where

𝑧 = 8 √ (𝑥 - 1 8 ) 2 + (𝑦 - 1 8 ) 2 , (5.15)
𝛼 2𝐷 (𝑧) = 3.5 + 1.5 sin 2𝜋(1 -𝑧), (5.16
)

𝛽 2𝐷 (𝑧) = 3.5 -1.5 cos 2𝜋(1 -𝑧), (5.17)
𝛾 2𝐷 (𝑧) = 1 2 {1 + tanh [tan 𝜋 ( 1 2
-𝑧)]}.

(5.18)

Calculations on quadrilateral grids

The case illustrated in this section was simulated considering five grid resolutions with increasing level of refinement, respectively with 64, 128, 256, 512 in each spatial direction, with a maximum CFL number of 0.2. Results showing the predicted zero-order moment at 𝑡 = 0.8 are shown in Fig. 5.

It is apparent from Fig. 5 that the numerical solution obtained on the two finest grid resolutions with 256 2 and 512 2 cells, do not show significant differences. However, to provide a quantitative measure of the accuracy of the zeta scheme, and to validate its implementation into OpenFOAM in comparison to [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF], the order of accuracy of the 𝜁 scheme is quantified numerically by taking the solution obtained on a uniform quadrilateral grid with 1024 2 cells as reference. The graph of the error is reported in Fig. 6 (numerical values in Table 3), which shows that the slope of the error curves tends to 2 for higher grid resolutions, with a reduction of the formal of order of accuracy for moments of higher order. The time evolution of the first-order moment of the NDF and of the value of the corresponding limiter 𝜆 0,c are reported in Fig. 7.

It is observed that 𝜆 0,c = 1 in most of the solution domain. The regions where a complete limitation is applied (𝜆 0,c = 0), reducing the local accuracy to first order, is often where the solution presents very small positive values of the moment. The additional limitation through 𝜆 0,c preserves the realizability of the moment vector, while showing a limited impact on the accuracy of the solution, as previously illustrated by the numerical study of the order of the scheme (Fig. 6).

Calculations on triangular grids

The two-dimensional case presented in this section was also considered to establish the accuracy of the 𝜁 advection scheme on triangular grids. Four grids were considered in these simulations, whose properties are summarized in Table 4. The value of the grid spacing ℎ was calculated according to [START_REF] Asme | Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications[END_REF], as

ℎ = √ 1 𝑁 𝑐𝑒𝑙𝑙𝑠 ∑(Δ𝐴 𝑖 ) 𝑖 , (5.19)
where Δ𝐴 𝑖 is the area of each cell polygon. The contour plots of the zero-order moment obtained with the 𝜁 scheme on the triangular grids at 𝑡 = 0.8s, are reported in Fig. 8, which shows the results are visually identical to those obtained on quadrilateral grids and reported in Fig. 5.

The behavior of the formal order of accuracy of the 𝜁 simplified scheme on triangular grids is shown in Fig. 9 (numerical values in Table 5), which shows comparable trends for all the moments to those observed on quadrilateral grids (Fig. 6), with the second-order accuracy recovered on the finest grids, for the lower-order moments.

Comparison of first-order scheme and 𝜁 simplified scheme

Results obtained with the first-order scheme on a quadrilateral grid are shown in Fig. 10 the significantly higher dissipation introduced by the first-order scheme, typically used for moment transport to avoid compromising the realizability of the advected moment vector. The difference between the first-order scheme and the second-order zeta scheme is further highlighted in Fig. 11, where the zero-order moment obtained at 𝑡 = 0.8s on a grid with 128 2 cells with the two methods is compared. The results obtained with the 𝜁 simplified scheme clearly shows the reduced dissipation compared to the results obtained with the first-order scheme. Similarly to what done for the quadrilateral case, the time evolution of the first-order moment of the NDF is shown in Fig. 12, where the fields of 𝑚 1 at different times appear to be identical to those shown in Fig. 7 for the quadrilateral case. However, a more careful examination of the behavior of the moment limiter 𝜆 0,c shows that the region of the computational domain where the limitation is applied is larger for the case with triangular grid compared to the quadrilateral case. Several cells on the front of the region where 𝑚 1 ≠ 0 are affected by the limitation, while, in the quadrilateral case, a smaller number of cells with 𝜆 0,c ≠ 1 is observed. The last snapshot (Fig. 12(d)) also shows a small number of cells where 𝜆 0,c = 0.5.

Two-dimensional advection of moments of a bimodal NDF

The 𝜁 simplified scheme is applied in this section to the case transport of moments of a bi-modal NDF in two dimensions. Only one triangular grid (Grid B in Table 4) is considered.

The initial conditions for the moments are obtained using Eq. (5.12), where the 𝑥 coordinate is

Conclusions

The second-order realizable 𝜁 simplified scheme proposed in Laurent and Nguyen [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF] was extended to unstructured grids and implemented into the computational framework OpenQBMM [START_REF] Openqbmm | An open-source implementation of Quadrature-Based Moment Methods[END_REF], an extension for OpenFOAM [START_REF] Openfoam | The OpenFOAM Foundation[END_REF] that implements quadrature-based moment methods. The implementation was verified by considering first two one-dimensional cases, the first involving moments of a regular NDF and the second where moments of a bimodal NDF are transported.

The proposed implementation of the scheme into OpenQBMM was able to reproduce the results for the same cases reported in Laurent and Nguyen [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF], showing convergence to second-order accuracy with grid refinement. The scheme was further verified on a two-dimensional case with uniform hexahedral discretization where the first four moments of a regular NDF are transported. Also in this case, results from Laurent and Nguyen [START_REF] Laurent | Realizable second-order finite-volume schemes for the advection of moment sets of the particle size distribution[END_REF] were satisfactorily reproduced. Finally, the same twodimensional case considered for verification on hexahedral grids was used to establish the accuracy of the scheme on triangular grids. The 𝜁 simplified scheme was able to reproduce the results and the behavior of the error affecting the solution observed on hexahedral grids also on triangular grids.

Future work will consider the development of realizable numerical schemes for multivariate problems, where closure is provided by the conditional quadrature method of moments [START_REF] Yuan | Conditional quadrature method of moments for kinetic equations[END_REF].

Source code and test cases

The source code of the OpenQBMM framework [START_REF] Openqbmm | An open-source implementation of Quadrature-Based Moment Methods[END_REF] used to perform the calculations described in ---------------------------------------------------------------------------

4 . 5 .

 45 Once the reconstructed face values of 𝜁 𝑝 which ensure the realizability of the entire advected moment set are found, the moment vectors 𝐦 + and 𝐦 -are recomputed from the values ζ 𝑝 + and ζ 𝑝 -[61]. The integration of the advection term proceeds computing the flux of each moment at each cell face based on the direction of the velocity at the centroid of the same face and updating the values of the moments at the cell center, as prescribed by Eq. (2.3).

  (4.13) reduces, in the case of a uniform one-dimensional discretization with step N f,o,c = 1 , assuming there is no mass and momentum source terms localized in the computational cell. The time-step restriction in multi-dimensional cases requires to consider how many cell faces have outgoing fluxes. Assuming a uniform discretization in both directions of a twodimensional case, the condition in Eq. (4.13) becomes: ,o,c ∈ {1, 2, 3}, the maximum restriction on the time-step size is possible when N f,o,c = 3, reasoning, in a triangular grid, assuming triangles are equilateral with side length ℓ, the most restrictive constraint is achieved when two faces have outgoing flux (N f,o,c = 2),

Fig. 1 :

 1 Fig. 1: Moments of the regular NDF. Analytical solution (dashed lines) and numerical prediction (solid lines) with (a) 50, (b) 100, (c) 500 and (d) 1000 cells -Zeta scheme.

Fig. 2 :

 2 Fig. 2: Logarithm of the error in the numerical prediction of the moments of the regular NDF case as a function of the logarithm of the grid size -Zeta scheme.

Fig. 3 :

 3 Fig. 3: Moments of the bimodal NDF. Analytical solution (dashed lines) and numerical prediction (solid lines) with (a) 50, (b) 100, (c) 500 and (d) 1000 cells -Zeta scheme.

Fig. 5 :

 5 Zero-order moment of the regular NDF used in the two-dimensional test case on quadrilateral grids at t = 0.8. (a) 64 2 cells, (b) 128 2 cells, (c) 256 2 cells, (d) 512 2 cells. Zeta scheme.

Fig. 6 :

 6 Fig. 6: Logarithm of the error in the two-dimensional numerical tests on hexahedral grid with regular NDF as a function of the logarithm of the grid size.

Fig. 7 :

 7 Fig. 7: Time evolution of the first-order moment of the regular NDF considered in the test case on quadrilateral grid (resolution shown: 1024 2 ): (a) t = 0.2 s; (b) t = 0.4 s; (c) t = 0.6 s; (d) t = 0.8 s.

Fig. 8 :Fig. 9 :

 89 Fig. 8: Zero-order moment of the regular NDF used in the two-dimensional test case on triangular grids at t = 0.8. (a) Grid A, (b) Grid B, (c) Grid C, (d) Grid D. Zeta scheme.

Fig. 10 :

 10 Zero-order moment field at t = 0.8 on a quadrilateral grid with (a) 128 2 , (b) 256 2 and (c) 512 2 cells -First-order scheme.

Fig. 11 :

 11 Fig. 11: Comparison of zero-order moment field obtained on a 128 2 quadrilateral grid at t = 0.8, with (a) first-order advection scheme and (b) zeta scheme.

Fig. 12 :Fig. 13 Fig. 13 :

 121313 Fig.13shows the time evolution of 𝑚 0 and 𝑚 1 together with the limiter 𝜆 0,c used in the additional

Fig. 14 :

 14 Fig. 14: (a) Second-order moment of the bi-modal NDF in the test case on tetrahedral grid (Grid B) at t = 0.8, and, (b), the corresponding field of the limiter 𝜆 1,𝑐 used in the additional limitation of the ζ simplified scheme. (a) (b)

  -License This file is derivative work of OpenFOAM. OpenFOAM is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. OpenFOAM is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with OpenFOAM. If not, see <http://www.gnu.org/licenses/>. \*-------------------------------------------------------------------------*/ #include "zetaUnivariateAdvection.H" #include "upwind.H" #include "addToRunTimeSelectionTable.H" // * * * * * * * * * * * * * * Static Data Members * * * * * * * * * * * *

  4. Calculate the number of realizable moments in 𝐦 𝑐 * . If this number is smaller than the number of realizable moments in 𝐦 𝑐 , set 𝜆 𝑝,𝑐 = 1/2.

5. Recompute 𝜁 𝑝,𝑐 from Eq. (4.6).

6.

Repeat the calculation at points 2 -4 with the new value of 𝜁 𝑝,𝑐 . If the number of realizable moments in the updated 𝐦 𝑐 * is smaller than the number of realizable moments in 𝐦 𝑐 , set 𝜆 𝑝,𝑐 = 0.

Table 1 :

 1

	Cells	h	e0	Order m0	e1	Order m1	e2	Order m2	e3	Order m3
	50 0.02 0.0423		0.0555		0.0665		0.0759
	100 0.01 0.0132	1.6851	0.0173	1.6792	0.0210	1.6625	0.0244	1.6359
	500 0.002 0.0007	1.8318	0.0009	1.8512	0.0011	1.8509	0.0012	1.8481
	1000 0.001 0.0002	2.0240	0.0002	1.9997	0.0003	2.0047	0.0003	2.0055
	Cells	h	e4	Order m4	e5	Order m5	e6	Order m6	e7	Order m7
	50 0.02 0.0849		0.0934		0.1013		0.1087
	100 0.01 0.0276	1.6187	0.0307	1.6048	0.0337	1.5862	0.0368	1.5614
	500 0.002 0.0014	1.8426	0.0016	1.8331	0.0018	1.8235	0.0020	1.8160
	1000 0.001 0.0004	2.0023	0.0004	2.0006	0.0004	1.9994	0.0005	1.9851

Numerical values of the errors reported in Fig.

2

and corresponding order of the reconstruction of each moment.

Table 3 :

 3 Numerical values of the errors reported in Fig.6and corresponding order of the reconstruction of each moment.

	Cells	h	e0	Order m0	e1	Order m1	e2	Order m2	e3	Order m3
	64 0.0156 0.2253		0.2451		0.3089		0.3970
	128 0.0078 0.0946	1.2527 0.1506	0.7025	0.2115	0.5466	0.2823	0.4921
	256 0.0039 0.0419	1.1748 0.0746	1.0130	0.1110	0.9299	0.1531	0.8829
	512 0.0020 0.0139	1.5931 0.0229	1.7030	0.0403	1.4637	0.0649	1.2384

Table 4 :

 4 Properties of the unstructured grids considered in the numerical test cases.

	Grid	N. of cells Domain area	h
	A	3938	0.25	7.97 ⋅ 10 -3
	B	15894	0.25	3.97 ⋅ 10 -3
	C	64688	0.25	1.97 ⋅ 10 -3
	D	255596	0.25	9.89 ⋅ 10 -4

Table 5 :

 5 Numerical values of the errors reported in Fig.9and corresponding order of the reconstruction of each moment.

	h	e0	Order m0	e1	Order m1	e2	Order m2	e3	Order m3
	0.0080	0.5123		0.4844		0.4327		0.4483
	0.0040	0.2858	0.8365	0.2720	0.8270	0.2640	0.7084	0.3034	0.5593
	0.0020	0.1383	1.0343	0.1367	0.9805	0.1540	0.7681	0.1983	0.6061
	0.0010	0.0494	1.4991	0.0534	1.3674	0.0702	1.1437	0.0965	1.0480

A moment vector is said realizable on a support if a positive measure exists on the same support whose moments are the same as those in the considered moment vector. It is then in the moment space.

Faces of the computational grid are indicated as internal when they do not belong to a boundary of the computational domain. Values of moments at boundary faces are either known, if imposed at the boundary, or obtained from the only neighbor cell which owns the boundary face.

this article is available in [START_REF] Openqbmm | An open-source implementation of Quadrature-Based Moment Methods[END_REF][START_REF] Passalacqua | OpenQBMM 5.0.0 for OpenFOAM 7[END_REF], where also the test cases presented in the articles are available. The OpenFOAM class implementing the 𝜁 scheme on unstructured grids described in the manuscript is reported in Appedix A.

Acknowledgements

The authors would like to gratefully acknowledge the support of the US National Science Foundation under the SI2-SSE award NSF-ACI 1440443 and the support of the French National Research Agency (ANR) under grant ANR-13-TDMO-02 ASMAPE for the ASMAPE project.

A.P. would like to acknowledge the support of the Jean d'Alembert fellowship program Idex Paris-Saclay, which supported his stay at Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay.

Appedix A

The C++ source code of the main class that implements the 𝜁 scheme in OpenQBMM is reported in this appendix. This code is contained in OpenQBMM 5.0.0 for OpenFOAM 7 [START_REF] Passalacqua | OpenQBMM 5.0.0 for OpenFOAM 7[END_REF]. ------------------------------------------------------------------------* ---------------------------------------------------------------------------License This file is derivative work of OpenFOAM.

Header file: zetaUnivariateAdvection.H /*-

OpenFOAM is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

OpenFOAM is distributed in the hope that it will be useful, but


Sahin Buyukdagli
email: buyukdagli@fen.bilkent.edu.tr

T Ala-Nissila
email: tapio.ala-nissila@aalto.fi

Ralf Blossey
email: ralf.blossey@univ-lille1.fr

Comment on "Nonlocal statistical field theory of dipolar particles in electrolyte solutions" by Y.A. Budkov

Keywords:

The article [START_REF] Budkov | Nonlocal statistical field theory of dipolar particles in electrolyte solutions[END_REF] by Budkov introduces a nonlocal fieldtheoretic model of solvent-explicit electrostatics. Despite giving a detailed introduction to the early literature on the topic, the article misses out on a series of articles that we published several years ago. Consequently, Ref. [START_REF] Budkov | Nonlocal statistical field theory of dipolar particles in electrolyte solutions[END_REF] essentially rederives without mention several results that were derived by us for the first time in Refs. [START_REF] Buyukdagli | Microscopic formulation of nonlocal electrostatics in polar liquids embedding polarizable ions[END_REF][START_REF] Buyukdagli | Alteration of gas phase ion polarizabilities upon hydration in high dielectric liquids[END_REF][START_REF] Buyukdagli | Nonlocal and nonlinear electrostatics of a dipolar Coulomb fluid[END_REF][START_REF] Buyukdagli | Dipolar correlations in structured solvents under nanoconfinement[END_REF]. The work by Budkov also considers variations of our model that are based on a different Yukawa-like solvent structure factor, which enables the exact evaluation of the average electrostatic potential.

In Refs. [START_REF] Buyukdagli | Microscopic formulation of nonlocal electrostatics in polar liquids embedding polarizable ions[END_REF][START_REF] Buyukdagli | Alteration of gas phase ion polarizabilities upon hydration in high dielectric liquids[END_REF][START_REF] Buyukdagli | Nonlocal and nonlinear electrostatics of a dipolar Coulomb fluid[END_REF][START_REF] Buyukdagli | Dipolar correlations in structured solvents under nanoconfinement[END_REF], we developed the first field-theoretic model of nonlocal electrostatics embodying explicitly the extended charge structure of multipolar solvent molecules and ions with intrinsic polarizability. These articles showed that the consideration of the extended solvent charge structure allows to reproduce the non-local dielectric permittivity fluctuations observed in Molecular Dynamics simulations and solves the problem of the UVdivergence of the free energy without the introduction of an arbitrary cutoff.

In Sec. 2.1 of Ref. [START_REF] Budkov | Nonlocal statistical field theory of dipolar particles in electrolyte solutions[END_REF], the Author first derives the nonlocal field theoretic partition function from a formally more general expression by introducing a general solvent structure factor g(r -r ). Then, this generalization is dropped to switch to the dipolar case. The resulting field theoretic Hamiltonian Eq.( 18) becomes exactly identical to the Hamiltonian functionals of our articles [START_REF] Buyukdagli | Nonlocal and nonlinear electrostatics of a dipolar Coulomb fluid[END_REF] (Eq.( 4)) and [START_REF] Buyukdagli | Dipolar correlations in structured solvents under nanoconfinement[END_REF] (Eq.( 1)), and the dipolar limit of the more general Hamiltonian of Ref. [START_REF] Buyukdagli | Microscopic formulation of nonlocal electrostatics in polar liquids embedding polarizable ions[END_REF] (Eq.( 9)). The Author does not cite our previous works in any of these derivations.

The strong overlaps and similarities of Ref. [START_REF] Budkov | Nonlocal statistical field theory of dipolar particles in electrolyte solutions[END_REF] with our published work extend beyond Sec. 2.1. In Sec. 2.2, the dipole limit of the non-local Poisson-Boltzmann (NLPB) equation ( 22) corresponds to the MF-level NLPB equation of our article [START_REF] Buyukdagli | Nonlocal and nonlinear electrostatics of a dipolar Coulomb fluid[END_REF] (Eq.( 5)) and a restricted case of the more general NLPB equation in Ref. [START_REF] Buyukdagli | Microscopic formulation of nonlocal electrostatics in polar liquids embedding polarizable ions[END_REF] (Eq.( 12)). Moreover, in Sec. 2.3, the linearization of the NLPB equation and its solution in Fourier space (Eqs.( 24)-( 30)) bears strong similarities to our articles. Most importantly, the dipolar dielectric function in Eq.( 31) that follows from this solution is exactly identical to the dielectric permittivity function of Ref. [START_REF] Buyukdagli | Nonlocal and nonlinear electrostatics of a dipolar Coulomb fluid[END_REF] (Eq.( 10)). The latter also corresponds to the restricted case of the more general dielectric function previously derived in Ref. [START_REF] Buyukdagli | Microscopic formulation of nonlocal electrostatics in polar liquids embedding polarizable ions[END_REF] (Eq.( 21)). Again, no credit is given by the Author to our earlier articles.

The overlaps of Ref. [START_REF] Budkov | Nonlocal statistical field theory of dipolar particles in electrolyte solutions[END_REF] with our early works go beyond the MF treatment of the model. In Sec. 2.5, the inverse Green's function in Eq.( 56) is similar to Eq.( 17) of our article [START_REF] Buyukdagli | Dipolar correlations in structured solvents under nanoconfinement[END_REF]. More precisely, the dipolar limit of Eq.( 56) corresponds to the one-loop limit of the (variational) inverse kernel Eq.(17) of Ref. [START_REF] Buyukdagli | Dipolar correlations in structured solvents under nanoconfinement[END_REF]. Finally, in the conclusion part of Ref. [START_REF] Budkov | Nonlocal statistical field theory of dipolar particles in electrolyte solutions[END_REF], the Author expresses his intention to generalize the theory by considering an "arbitrary electric structure", via the introduction of "a probability distribution function of distance for each pair of oppositely charged groups", without any citation to our non-local field theoretic models of Refs. [START_REF] Buyukdagli | Microscopic formulation of nonlocal electrostatics in polar liquids embedding polarizable ions[END_REF][START_REF] Buyukdagli | Alteration of gas phase ion polarizabilities upon hydration in high dielectric liquids[END_REF] where the ionic polarizability was taken into account within a Drude oscillator model.

In summary, we find it disconcerting that the most pertinent references to Ref. [START_REF] Budkov | Nonlocal statistical field theory of dipolar particles in electrolyte solutions[END_REF] were not properly acknowledged. This is a severe shortcoming since the Author of Ref. [START_REF] Budkov | Nonlocal statistical field theory of dipolar particles in electrolyte solutions[END_REF] knows about our work; he has already referred to our articles [START_REF] Buyukdagli | Microscopic formulation of nonlocal electrostatics in polar liquids embedding polarizable ions[END_REF][START_REF] Buyukdagli | Dipolar correlations in structured solvents under nanoconfinement[END_REF] in his publications [START_REF] Budkov | A modified Poisson-Boltzmann theory: Effects of co-solvent polarizability[END_REF][START_REF] Budkov | On the theory of electric double layer with explicit account of a polarizable co-solvent[END_REF] on point-like dipoles.


Samuel Nowakowski
email: samuel.nowakowski@univ-lorraine.fr

Guillaume Bernard

From Digital Humanities to a Renewed Approach to Digital Learning and Teaching

Keywords: computer environments, LMS, human values, digital support, teaching and learning. Research area: psychology; pedagogy

In a world in which digital interfaces, dematerialization, automation, so-called tools of artificial intelligence aim to drive away the human or eliminate the relationship with humans! The way other beings see us is important. What would happen if we took the full measure of this idea? How would this affect our understanding of society, culture, and the world we inhabit? How would this affect our understanding of the human, since in this world beyond the human, we sometimes find things that we prefer to attribute only to ourselves? What impacts on education, learning, teaching? After having explored the field opened by these questions, we will bring an answer with a reinvention of the learning platform named KOALA (KnOwledge Aware Learning Assistant). KOALA is a new online learning platform that comes back to internet sources. Symmetrical and acentric, KOALA combines analyzes from the digital humanities and answers to the challenges of education in the 21 st century.

Introduction

this question is central today -not to ask it, it is to take the risk of entrusting to the machines our capacity to decide, to think, … our humanity … and without possible turning back [START_REF] Dick | If this world displeases you … and other tests [Si ce monde vous déplaît… et autres essais[END_REF]! First, a little digression! With a book entitled "How do forests think?" "As we settled down to sleep under the thatch of our hunting camp on the foothills of the Sumaco volcano, Juanicu warned me, 'Sleep on your back!' If a jaguar comes, he will see that you can look back and he will not bother you. If you sleep on your stomach, he'll think you're aicha, a prey or literally meat in Quichua and he'll attack you" [START_REF] Kohn | How Forests Think [Comment pensent les forêts[END_REF]. "He said that if a jaguar sees you as a being able to look at him in return -a self like him, a you -he will leave you alone. But if he came to see you as a prey -one of them -you could end up dead meat. The way other beings see us is important. […] and more especially and more strongly in a world in which digital interfaces, dematerialization, automation, so-called tools of artificial intelligence aim to distance the human or eliminate the relationship with humans!

The way other beings see us is important. What would happen if we took the full measure of this idea? How would this affect our understanding of society, culture, and the world we inhabit? How does this affect our understanding of the human, since in this world beyond the human, we sometimes find things that we prefer to attribute only to ourselves? Why tell you this? Because the question that animates me and must animate us today is what we are going to do with all the tools, more and more powerful, that we are developing. Recall with Plato or Bernard Stiegler, that any tool is a pharmakos, both remedy and poison, emancipator and destroyer, and our powerful tools from the digital do not escape! Destroyers, these tools are, I want for proof their ecological impact. To take a concrete example, each consultation of a web page results in the emission of 2 g of greenhouse gases in the atmosphere and the consumption of 3 centilitres of water.

Globally, the internet is a 6th continent that "weighs" annually 1037 TWh of energy, 608 million tons of greenhouse gases and 8.7 billion cubic meters of fresh water. About 2 times the footprint of France! In addition to being colossal, the environmental impacts of digital are multiple: depletion of non-renewable natural resources, pollution of the air, water and soil inducing health impacts contributing to the destruction of ecosystems and biodiversity, emissions of greenhouse gases that contribute to climate change, etc. And they reinforce each other. It is therefore essential to adopt a multi-criteria approach when studying these impacts and not to limit oneself to a single environmental indicator.

These environmental impacts occur at each stage of the life cycle. But they are concentrated especially in the manufacture of equipment and their end of life. It is therefore essential to extend the active life of equipment by promoting their reuse and by pushing the inevitable stage of recycling as much as possible.

It is therefore essential to put the human in the center! Not to subordinate the intelligence to the tools, and not to forget it in the implementation of our tools.

Humanity and digital

Human beings obviously have genius to design, manufacture and use tools. Our innate talent for technological invention is one of the main qualities that distinguish our species from others and one of the main reasons why we have taken such a hold on the planet and its destiny. But if our ability to see the world as a raw material, something we can alter and manipulate as we please, gives us tremendous power, it also carries great risks. The first risk is that we ourselves become a technical instrument, optimized and programmed, a technology among the others. The anxiety of seeing machines attack our humanity is as old as the machines themselves. Max Weber and Martin Heidegger have described how a narrow and instrumentalist vision of existence influences our understanding of ourselves and shapes the kind of societies we create.

With our smartphones and other digital devices ubiquitous, most of us are permanently connected to the computer network (Internet Live Stats, 2019). The companies that control the network seek to know as much as possible about the users to control their senses until their thoughts through the applications, sites and services that they "propose". At the same time, the proliferation of connected objects, networked machines and devices, from home to workplaces always immerses us in computer environments designed to anticipate our needs (Une Rentree Pour Tout Changer, 2019).

There are of course many advantages to an existence more and more mediatized.

Many activities, once difficult or lengthy, have become easier, requiring less effort and reflection, and the risk of losing, if we are not careful, our ability to act on our own in the world. However, by transferring the initiative to computers and software, we give control of our desires and decisions to programmers and the companies that employ them.

Already, many people rely on computer programs to choose which movie to watch, which meal to cook, which news to follow, even which person to meet!

Why think when you can click?

By giving such choices to strangers, we inevitably open ourselves to manipulation.

Since the design and operation of algorithms is almost always hidden from us, it can be difficult, if not impossible, to know whether the choice made on our behalf reflects our own interests or those of corporations, governments or other third parties. We want to believe that technology strengthens our control over our lives and circumstances, but if it is used without consideration, technology is just as likely to turn us into puppets of those who master and deploy these technologies (Zamiatine, 1971).

Thomas Hughes' "technological impetus" is a powerful force, and opposing this force becomes possible when we have a keen awareness of how technologies are designed and used. If we do not accept this responsibility, we risk going from creator to creature status. What will become of us when we install the externalization of the calculus in "rational" machines in a will to get rid of the uncertainties of the human "reason" which is fallible, unpredictable, submissive, subject to the feelings. This reduction of human contingencies to sequential protocols makes it possible to associate economic rationality with algorithmic rationality. Philip K. Dick expresses it very well: "To become […], for want of a more appropriate term, […] an android,

[which] means, […] to be transformed into an instrument, to let oneself be crushed, manipulate, become an instrument without his knowledge or consent -it's the same thing. But you cannot turn a human into an Android if that human tends to break the law whenever he has the opportunity. Androïdization requires obedience. And, above all, predictability. It is precisely when the reaction of a given person to a given situation can be predicted with scientific precision that the doors to the Trojan horse are opened wide. For what purpose would a flashlight be if, when the button is pressed, the bulb only comes on once in a while? Every machine must walk without firing to be reliable. The Android, like any other machine, must walk on the finger and the eye [START_REF] Dick | If this world displeases you … and other tests [Si ce monde vous déplaît… et autres essais[END_REF]." It would then be "like Ray Bradbury's story in which a Los Angeles man discovers in horror that the police car chasing him has no driver -and that he pursues it of his own accord! What is frankly horrible is not that the car has its own tropism in pursuing the protagonist, but it is the fact that inside the car there is a void. An empty place. The absence of something vital -that is what is horrible! "The person, once gone, cannot be replaced in any way. Whatever our feelings about her, we cannot do without them. And once gone, nothing can make her come back. What's more, if this person is turned into an android, she will never come back to the human state either [START_REF] Dick | If this world displeases you … and other tests [Si ce monde vous déplaît… et autres essais[END_REF]."

What is the human in a world increasingly machinized? What crossroads where to decide alone the path to take at the expense of the dream promise of the other way? Remember that there is never "too much human" in a world of dehumanized multitudes. So how to find a way out?

Find a way out

Let's take these words from Etienne Klein. "It's not because there are frogs after the rain that we have the right to say that it's raining frogs". But why? Because today, big data is erected as a means to access the ultimate knowledge! Indeed, learning analytics in prediction, profiling in artificial intelligence, to profile, to induce from behavioral regularities, to model consumer behaviors, to identify regularities, we infer laws that we consider as general or even universal then even that they are only the digest of what has already been given! Whereas only human thought alone can predict the existence of new dimensions of reality! Let's take a few examples! It is not thanks to big data but thanks to the equations of particle physics that we have been able to predict the existence of the Higgs boson and allow its detection in 2012! It is not thanks to big data but thanks to the equations of gravitation that we have been able to predict the gravitational waves that were detected in 2016, a century after their prediction by the general relativity formulated in 1916 by Einstein.

Thus, THE THOUGHT IT alone makes it possible to go beyond the limits of the observable world, the empirical world, the world as it is given to us. Thought cannot be dissociated from our nature of being situated and experimenting. The most demonstrative example is that of Einstein who proceeded by "thought experiment", that is to say by experiments capable of keeping the empirical world at a distance and prolonging, in a sort of elsewhere, the implications of a theory: what would happen in this or that situation that I can imagine, if this physical law was really accurate? What would the equations say if they could talk? Big Data is obviously a fantastic opportunity but may lead us to look only at correlations between the data that are available, or a correlation is not the same as a cause-and-effect relationship.

The moral of this beautiful story is that a theory allows to emerge new data but the reciprocal is not true. Data does not always make it possible to bring out a theory leading to understanding. So, to all the data gurus, to all who think to read our future in the marc of the algorithms, remember that without the thought in action, nothing that you set up as new gods would exist! That mechanical intelligence, programmed, calculated, is in no way comparable to human intelligence. Collecting and processing masses of data, to act intelligences cannot replace this ability of human thought to go beyond frames, to look beyond the hill blocking the horizon.

A question of education -towards a reinvention?

The understanding of all this necessarily involves learning, listening to nature and others ("we always learn alone but never without others" Philippe Carré). As science is the most effective method we have found to understand the world and the democratic modes that are the best way we have found to organize the collective decision-making process, education must be based on tolerance, debate, rationality, the search for common ideas, learning, listening to the opposite point of view, awareness of the relativity of its place in the world. In order to be aware that we can be wrong, to retain the possibility of changing our mind when we are convinced by an argument, and to recognize that views opposed to ours could prevail.

Every step forward in the scientific understanding of the world is also a transgression from what was happening before. So scientific thought always has something subversive, volutionary (a term that I borrow from Alain Damasio because a revolution always returns to the point of departure). Whenever we redraw the world, we change the very grammar of our thoughts, the frame of our representation of reality.

To be open to knowledge is to be open to the subversive. Unfortunately, in school, on the contrary, science is often taught as a list of "established facts" and "laws" or as a problem-solving training. This way of teaching betrays the very nature of scientific thought [START_REF] Le Guin | The dispossessed [Les dépossédés[END_REF].

We must teach critical thinking, not the respect of textbooks. We must invite students to question preconceptions and teachers, not blindly believe them. Science and thought must lead us to recognize our ignorance, and that in "others" there is more to learn than to dread. That the truth is to be sought in a process of exchange, and not in the certainties or the common conviction that "we are the best". Teaching must therefore be the teaching of doubt and wonder and we must be careful to put our tools at its service.

Reinventing education? Towards openness, the participative, the agile, the connected and the human Martin Luther King said in his sermon pronounced in New York on April 4, 1967:

"When machines, computers and the quest for profit are more important than people, the fatal triptych of materialism, militarism and racism is invincible."

What is more premonitory than such information? Half a century away from a world in which the machine and materialism have never been so overwhelming! So, me too, in this present, in France today, I have a dream! I dream of a society in which industry, economy, democracy, education will be conceived in symbiosis. I dream to see people living in green cities, planted with bamboo oases that will purify the waters and produce biomass for a local industry and adapted to the needs. Participatory workshops or FabLabs will allow everyone to upgrade their skills, learn, learn to learn and accompany, to repair objects rather than replace them, to produce what is needed without passing by massive industrialization.

I dream of a society in which we will have fewer objects. I dream of a measured and intelligent use of the internet, an internet whose resources will be relocated, adapted to the needs, an internet which will allow to share vehicles, various tools and any device with intermittent uses. I dream of a digital resource that will be local, "populated by artificial intelligences" accompanying, adaptable and benevolent. I dream of a sustainable internet, responsible and respectful of humans, life and the Earth.

I dream of a company where to rent phones and computers rather than buy them, will encourage builders not to promote planned obsolescence. A society in which the bulk of the goods would be made from recyclable materials, all within a circular economy.

I dream of a society that can reduce, reuse, recycle, repair, rent, share in a logic of interconnectivity and preservation of diversity, diversity fostered by the education of a conscious citizen, enlightened, responsible and able to reject the clutches of Silicon Valley engineers, advertisers, media directors who seek to make them more vulnerable, impressionable and receptors of standardized thinking. So, I dream of a school where, from their early years, children are taught cooperation, in addition to mathematics, the digital humanities in addition to grammar and history, the art of communicating better with others express their needs and resolve conflicts instead of competition and coercion.

I dream of an education that accompanies throughout life the citizen in his understanding of digital and its issues. I dream of a school that supports the construction of a digital culture accessible to all by facing the difficulty of the uses that have become part of our professional and personal lives. I dream of a school that, beyond familiarity, recall the foundations of the digital sciences and human sciences. So, at a time when we agree on the importance for every citizen to have a critical spirit to escape all forms of obscurantism, I dream of a school that gives the keys to understanding issues related to digital, to avoid falling into a too angelic or too demonizing vision of the contributions of digital. I dream answers, tracks, alternative ways. I dream with Ivan Illich of a convivial society "which gives to man the possibility of exercising the most autonomous and creative action, using tools less controllable by others. Productivity is conjugated in terms of having, usability in terms of being" (Illich, 1973). I dream of a universe-city, imagining a friendly university society open to all and without distinction of sexes, ages and origins. I dream of a world in which one considers with Deleuze, that "[…] to make your questions, [it must be] with elements coming from everywhere, from anywhere". I dream of thinking of the actions that will create the conditions for the inclusion of the university in the city, a connected university, which makes it possible to "think, in things, among things, [that makes it possible] to make rhizome, and not root". A university that "grows between and among other things [START_REF] Deleuze | Dialogues[END_REF]". I dream of a universe-city that grows where you do not expect, interstices. I dream of doing universe-city, of working "from" and not "on the side", of working from society, of these issues, of the needs and expectations of students, and not "beside" stakes of the beginning of the 21 st century. It is careful not to "substitute the awakening of education for the awakening of knowledge" so as not to "stifle in man the poet, [and] freeze his power to give meaning to the world".

I dream of the university as an active part of the city to welcome, give a place, guide, accompany the student.

I dream the universe-city open on the city, which give a role to the student by adapting and accompanying the project of all to the city.

To universe-city is to consider that the learning and the construction of the personal and professional project does not take place only in formal teachings but, in a university involved in the city, in which a true university citizenship apply.

To universe-city is to be free, not to live in a flock by thinking yourself free!

To universe-city is to give the conditions for the emancipation of the individual! So my idea for France is to dare the Universe-city to remember that people more important than "machines, computers and the quest for profit"!

Our answer in a digital learning platform

Our answer is called KOALA, for "Knowledge Aware Learning Assistant". Based on the observations above, in the idea of giving meaning to the action of each, we make ours the statement of Van Jacobson in 1995, "How to kill internet?" It's easy, just invent the web.

The educational platforms are not immune to this. We sought to introduce a new logic of definition of the authority, the states and the exchanges of roles between the actors of the educational act.

Our guiding principle is to work on the notion of commitment to all by proposing a space that facilitates commitment by enabling cognitive engagement, behavioral engagement, social engagement and emotional engagement. The goal is then to lead to emancipation, the re-enchantment of the educational act and the well-being in learning and teaching (Smolyaninova, Ovchinnikov, 2010). Based on these observations and on the experience of previous projects and the challenges of the early 21 st century of a learning society, we have developed a new platform that allows us to return to the sources of the Internet. KOALA is a space for facilitating engagement by installing a new dynamic of the occupation of the learning and teaching space. KOALA will promote accessibility, continuity and porosity where the other platforms enclose.

We thus have a vision that goes from centric to a-centered, from a-symmetrical to symmetrical. KOALA leaves a place on the periphery and puts another approach to the Authority. Access to educational resources, the realization of learning activities, the exercise of the educational relationship is built within the spaces that everyone Fig. 1. KOALA-LMS (Learning Management System) is now at the following address: https://www.koala-lms.org/fr/ wants to invest, and that it will appropriate in its own way. KOALA is thus a living space that guarantees socially and physically situated actions. It promotes an action that fits in a lived and proxemic space and redefines the articulation between private / individual and the collective at the level of each user. Moreover, KOALA authorizes the indeterminacy of the conditions of realization of an action: invention, freedom, autonomy, capacitive environment and potential of situation. In order to provide an answer to our questions, KOALA thus opens up a renewed approach to digital support that allows us to integrate the recommendation of content adapted to the needs, contexts, objectives of learning and training of each according to its needs.

KOALA is then an ecology of the learning experience (Fig. 1).

Conclusion

You know the story Dr. Frankenstein. Dr. Frankenstein "makes" a man, he "manufactures" it. An act so frightening that he himself abandons this creature who has no name. The monster as we call it, is monster because it is both similar to us and so different. The monster is therefore a creature who will make his education alone but will sink into violence when the abandonment of its creator will be combined with the stupidity of men.

But what a foolish enterprise to want to make a human! Yet this is what we strive for each time we want to "build a subject by adding knowledge," or "make a student stacking knowledge." We all want, more or less, to "do something with someone" after "doing something". But like Dr. Frankenstein, we do not always quite understand what "something" and "someone" is, it's not quite the same and we often do not know that this confusion condemns us, despite all the goodwill we can deploy, to failure, to conflict, to suffering, and sometimes even to misfortune.

And in this world of connected machines, so-called "intelligent machines," "the question of what it means to be human in the face of machines is no longer so trivial. We should not be so afraid of robots as we are afraid of becoming robots ourselves. We need to introduce human values into technology rather than technology introducing its values into our humanity. For this, one must be able to measure when a technology is dehumanizing or when humans do not think or behave in humans" [START_REF] Meirieu | Frankenstein pedagogue [Frankenstein pédagogue[END_REF].

In addition, today, some voices are raised, relayed by the higher authorities of power to say that learning can be explained only through measurable and optimizable mechanisms, by a procedural decomposability optimizable and modelable, by an

"

  It's our dreams that turn us from machines into full-fledged human beings." K. Dick Why this quote? It is from one of Philip K. Dick's last lectures, you know K. Dick, you have certainly seen Minority report, Blade Runner (obviously the 1982 version), Total Recall, … In his end-of-life lectures, K. Dick poses the question of the place of man and technology, of what makes us human beings. In this world of technology,

approach to optimal control of learning and especially forgetting the child as a human becoming and evacuating the political dimension of choices in terms of education.

Education is not just a process to be optimized. Education is a social milieu, education is a political project of elaboration of society.

What are we? A brain only, or a complex complex in perpetual construction who's learning. To educate is not to manufacture! To educate is not to equip, to educate it is perhaps there in a few words in this text of the Comité Invisible:

"Do not wait anymore, Do not hope anymore.

Do not be distracted anymore, unseat.

Breaking. Return the lie to the ropes.

Believe in what we feel. Act in consequence.

Force the door of the present.

Try. Miss. To try again. To miss better.

Be proactive. Attack. Build.

Defeat, perhaps.

In any case, overcome. Go his way.

Live, then.

Now."

Adopt KOALA, and join the experience! Smolyaninova, O., Ovchinnikov, V. (2010). University Electronic library for human resources development in Siberia: school content. In 9 th Europen Conference on Elearning (ECEL 2010), 828-835.

Une Rentree Pour Tout Changer [A Return to Change Everything] (2019).

Available at http://multimedia.ademe.fr/infographies/infographie-travail-ademe-logoqqf/ (accessed 25 July 2019).

Zamiatine, E. (1971). We [Nous autres]. Gallimard. 200 p.

От цифровых технологий к обновленному подходу в сфере цифрового обучения Самуэль Новаковски и Гильом Бернард

Университет Лотарингии Франция, Нанси, 54000, Лориа UMR7503 Мы живем в мире, где цифровые интерфейсы, дематериализация, автоматизация и так называемые инструменты искусственного интеллекта стремятся свести к минимуму взаимоотношения между людьми. Но ведь то, как нас видят другие существа, очень важно. Что будет, если мы в полной мере осознаем эту идею? Как она повлияет на наше понимание общества, культуры и мира, в котором мы живем? Как изменится восприятие нами человека, тем более что в этом мире за пределами человеческого мы иногда находим то, что предпочитаем приписывать только себе? Что влияет на образование, обучение, преподавание? Изучив очерченную этими вопросами область, мы дадим ответ: KOALA (KnOwledge Aware Learning Assistant). Это новая образовательная онлайн-платформа, которая отсылает к интернет-источникам. Будучи симметричной и ацентрической, KOALA ведет анализ цифровых материалов в сфере гуманитарных наук и дает ответы на вызовы образованию в XXI веке.

Ключевые слова: компьютерная среда, СДО, человеческие ценности, цифровая поддержка, преподавание и обучение.

Научная специальность: 19.00.00 -психологические науки; 13.00.00 -педагогические науки.


Ploux E, J-M Billard

/ Pfv Ra>o

D-SERINE CONTRIBUTES TO β-AMYLOID-DEPENDENT PATHOPHYSIOLOGY IN ALZHEIMER'S DISEASE

Key regulators of the structural and funcFonal brain plasFcity, the N-methyl-Daspartate subtype of glutamate receptors (NMDARs) requires the binding of the coagonist D-serine to be acFvated. In Alzheimer's disease (AD), soluble oligomers of the beta-amyloid pepFde (Aßo) affect NMDARs possibly through mechanisms involving changes in D-serine levels since Aßo sFmulate in vitro the producFon of the co-agonist. In this study, we asked whether D-serine contributes in vivo to morpho-funcFonal NMDAR-related deregulaFons mediated by Aßo. Behavioral analysis combined to electrophysiological recordings at CA1/CA3 hippocampal synapses have been thus conducted in the 5xFAD transgenic mice model of amyloïdogenesis displaying marked increase in Aßo rates and compared to 5xFAD animals in which the homozygous gene of the serine racemase (SR) that synthesizes D-serine, has been jointly invalidated.

Our results therefore show that deleFon of serine racemase prevents memory-related behavioral deficits observed in mice with prominent features of amyloidogenesis as well as impairment of NMDAR-dependent funcFonal plasFcity, suggesFng a significant contribuFon of D-serine in NMDAR-dependent β-amyloid-related pathophysiology of Alzheimer's disease.

EXPERIMENTAL PROCEDURES 1) Behavioral analysis: 8-min spontaneous alternaFon test was performed in a Y maze apparatus to assess working memory performances in 10-12 months of aged mice. Successive entry of the three arms of the maze was considered as an alternaFon. The percentage of alternaFon was calculated as follows: number of alternaFons / (total number of arms visited -2) x 100.

2) Electrophysiology: Hippocampal slices (400 µm thickness) were cut from two groups of WT, 5xFAD/SR +/+ and 5xFAD/SR -/-mice aged 3-4 or 10-12 months. Field excitatory postsynapFc potenFals (fEPSPs) and presynapFc fiber volley (PFV) were extracellularly recorded in CA1 stratum radiatum aner electrical sFmulaFon of Schaffer collaterals. Input/output curves of the fEPSP/PFV raFo of isolated NMDAr-mediated fEPSPs were constructed in a low magnesium medium supplemented with the non-NMDAr antagonist NBQX (10µM) before and 15 min aner addiFon of D-serine (100 µM). High frequency (HFS)-induced long-term potenFaFon (LTP) was studied in control medium aner tetanic sFmulaFon consisFng in one train at 100 Hz delivered for 1 sec. TesFng sFmulaFon was then resumed for 60 min aner HFS.

3) Semi-quanFtaFve immunoblopng analysis: Hippocampal Fssue was homogenized in protein lysis buffer. The membranes were probed with anFbodiesaginst GluN1 (1:750), GluN2A (1:2500), GluN2B (1:800), GluA2 (1:500), serine racemase (1:400) or ß-acFn (1:7000). Proteins bands of interest were analyzed by scanning densitometry and normalized to ß-acFn density.

Low frequency-induced NMDAR acFvaFon is similar in basal condiFons in all experimental groups (A) but is significantly increased by exogenous D-serine only in WT and 5xFAD/SR -/-mice (B)

Behavioral analysis Hippocampal phenotype of the experimental groups

  When in WT and SR -/-mice displayed only traces of Aßo, significant levels are found in both 5xFAD and 5xFAD/SR -/-animals SR deleFon reverses LTP deficits displayed by both young (A) and aged (B) 5xFAD mice ) 5XFAD (n=6) 5XFAD SR-/-(n=7)GluN2B/actin (AI)Total protein expression for each NMDAR subunit is not significantly affected in 5xFAD mice although it is slightly decreased for GluN2B as compared to WT and 5xFAD/SR -/-mice