Lattice Nomenclature Survey from LGA to Modern LBM

Lattice configuration is a core parameter in Lattice-Boltzmann (LB) methods, both from theoretical and implementation standpoints. As LB methods have progressed over the past decades, a variety of lattice configurations have been proposed and referred to according to a plurality of lattice nomenclature systems that usually include the Euclidean space dimensionality , the lattice velocity count and, in fewer instances, the discretization order in their format. This work surveys lattice nomenclature systems, or lattice naming schemes, along the history of LB methods, starting from their Lattice Gas Automata (LGA) predecessor method, up to the present time. Findings include multiple lattice categories, competing naming standards, ambiguous names particularly in higher-order models, naming systems of varying model parameter scopes, and lack of unambiguous naming schemes even for space-filling, Bravais lattice types.


. Introduction
Historically, the lattice-Boltzmann (LB) method had its origins in the frame of Lattice-Gas Automata [ ], and has been intensely developed since its inception [ , , , ].
One important conceptual and implementation parameter of LB methods is the employed lattice stencil-understood as the lattice geometry, velocity set, weights, and scale parameters [ , , ], although some authors may include in the stencil designation additional modeling elements, such as the relaxation time scheme [ ].
Both LB and LGA methods can be implemented on a variety of lattices, and historically many such lattices (along with their corresponding names or naming systems) have been developed. This work presents a LB literature survey focused on lattice naming schemes, or model nomenclature systems, from its Lattice-Gas Automata (LGA) predecessor method until the present time, in a somewhat chronological timeline.
lattices is thermal ows. On reference [ ] an unnamed D, hexagonal (triangular), -velocity lattice having velocity magnitudes of , , and lattice units [ ] was employed for adiabatic sound propagation and heat transfer Couette ow, whose results were shown to be in agreement with corresponding analytical solutions.
Some 'nDmV' lattices, with n being the Euclidean space dimension and m the lattice velocity count, namely, D V, D V, and D V, were introduced in [ ] for shock wave front structure and shear wave ow application cases. The D V lattice, for instance, was said to be comprised of four sublattices-a term that appeared in subsequent referenceswith each sublattice having discrete velocities of same magnitude and forming adjacent right angles, which led to possibly multiple sublattices per lattice energy level ε ≡ 2e = c 2 , with c being the microscopic (lattice) velocity magnitude, and e the corresponding speci c kinetic energy, as was the case with the ε = 1 2 + 2 2 = 5 energy level of a square lattice, represented by the discrete velocities obtained from permutations of (±{1, 2}, ±{2, 1}) in lattice units, which were grouped in two distinct sublattices. This is in contrast to later works in which energy levels are treated as single groups.

Late 's to mid-'s:
Most likely borrowing from mesh-based continuous mechanics numerical methods, a study [ ] has proposed a LB algorithm for non-uniform mesh grids, by decoupling spatial and momentum space discretizations in the LB scheme. The underlying momentum space discretization was the well-known D Q lattice, referred to in the study as ' -bit BGK model in D space' and other semantically equivalent sentences, in which BGK stands for kinetic theory's Bhatnagar-Gross-Krook collision model for the continuous Boltzmann equation [ , , ].
Nine years after the debut of LB methods, a study [ ] showed that they could be directly derived from the continuous Boltzmann equation with linearized collision operator under the BGK approximation [ ], while lattice stencils from requirements of matching continuous and discrete velocity moments up to a desired order-a decisive publication, not only in making LB methods theory independent from its LGA historical predecessor, but also to pave the way towards later methods for lattice weights determination for the lattice velocity set based on some discrete-to-continuous equivalences [ , ]. The lattices in [ ] were verbosely referred to as 'd-dimensional b-bit g lattice model', with d being the Euclidean space dimension, b the lattice velocity count, and g a geometry term, such as 'triangular,' etc.
A review article by Shen and Doolen [ ] published a decade after McNamara and Zanetti's premiere LB publication [ ] and seven years after Qian's paper introducing the now-prevailing 'DdQb' lattice naming scheme [ ], would still refer to LB lattices either with LGA-style or verbose nomenclatures, and to overall LB schemes based on its collision term treatment, such as 'lattice BGK (LBGK)' models.
Higher-order lattices were proposed in [ ] for two-and three-dimensional Euclidean spaces. They were referred to as 'octagonal grid ( -bit),' and as ' D "octagonal" -bit' models, respectively, and were isotropic up to the sixth-order. Since octagons are not space-lling, plane-tiling geometries, the proposed lattices were not of the Bravais type, meaning they impose a decoupling between the spatial and the momentum space discretizations, as with the non-uniform mesh [ ], and the method has to resort to interpolations, which was later shown to cause spurious numerical di usion [ , p. ].  [ , ].

. The Year of
The year of is seemingly a landmark for multivelocity, higher-order LB schemes-and incidentally for lattice naming schemes-as evidenced by the appearance of three key publications, namely those of Shan, X., Yuan, X.-F., and Chen, H., [ ], of Philippi, P. C., Hegele, L. A., dos Santos, L. O. E. and Surmas, R., [ ], and of Chikatamarla, S. S. and Karlin, I. V., [ ].

Shan and Coauthors:
A systematic discretization framework for the Boltzmann equation was proposed by Shan and coauthors in [ ]. From kinetic theory [ , ], the authors pointed out that successive Chapman-Enskog approximations of the Boltzmann equation obtain the (i) Euler, (ii) Navier-Stokes, (iii) Burnett, and (iv) higher-order macroscopic equations-meaning progressively higher-order moments of the continuous Boltzmann equation express progressively higher-order macroscopic thermohydrodynamic descriptions. Moreover, the authors demonstrated that projecting the Boltzmann equation onto order-N truncated tensorial Hermite polynomial expansion bases [ ], lead to discrete LB models of corres-ponding order-N moments, since resulting Hermite expansion coe cients correspond to the velocity moments up to the chosen order.
In this discretization framework, the lattice is viewed as a Hermite expansion quadrature, and the naming convention was de ned in terms of three parameters, namely, an Euclidean space dimension D, a quadrature velocity count d, and an algebraic degree of precision n encoded in an 'E d D,n ' naming scheme-an order-N Hermite expansion requires a quadrature degree n 2N. Citing Qian and coauthors' lattices [ ], they established the following comparisons, which were o only by a scaling factor: D2Q9 ∝ E 9 2,5 , D3Q15 ∝ E 15 3,5 , and D3Q19 ∝ E 19 3,5 . Additionally, they established that Gauss-Hermite quadratures of the Boltzmann equation yield LB models with minimum velocity count for a given degree of precision and Euclidean spacial dimension, without, however, the ability to prede ne (choose) the discrete velocity abscisae, which apart from special cases fails to produce a space-lling, Bravais lattice-recalling that for LB methods, this means lower memory requirements but decoupled spatial and momentum 'meshes' that require interpolations, thus introducing artifacts such as spurious numerical di usion.
In the Appendix of reference [ ], the authors include a brief overview on deriving quadratures on prede ned Cartesian abscissae, which is the main requirement for spacelling, Bravais lattices for non-interpolating, exact advection LB schemes. The brief overview, however, is of scalar nature, while a tensorial treatment is needed for full clarity. Results for the space-lling E 17 2,7 and E 39 3,7 quadratures were listed among the ones obtained with Gauss-Hermite quadratures.

Philippi and Coauthors:
Tackling the aspects associated in deriving space-lling, Bravais lattices aiming at su ciently high orders as to approach thermal hydrodynamic transport problems, Philippi and coauthors [ ] have proposed a new Method of Prescribed Abscissas, MPA, for obtaining lattice weight values and scaling factor from prede ned lattice arrangements.
Departing from the continuous Boltzmann equation, the derivation of discrete velocity sets, i.e., the lattice vectors, and corresponding weights, was considered as a quadrature problem aiming at (i) matching discrete equilibrium mass distribution function with its continuous counterpart and at (ii) warranting even-ranked velocity tensor isotropy, which, in turn, translates into isotropic uid transport properties.
The Method of Prescribed Abscissas, MPA, yields implicit equations for lattice weights and lattice scale factor in the form of polynomial tensor products, which are gener-ally excessively numerous, especially for higher-order cases. They have to be selected (reduced) and converted either into a non-linear system of equations. The apparent lack of literature guidance in tensor component equation selection criteria and solution approach has motivated works [ , ].
In their prescribed abscissas quadrature discussion, authors state that [ , p. ]: which is homologous to many Shan and coauthor's statements in [ ]. Observations like these, allied to the new and consistent methods of deriving higher-order LB stencils by Gauss-Hermite or Prescribed Abscissas quadratures, allowed for the immediate and subsequent appearances of lattices in D-and in D-Euclidean spaces with increased velocity counts, many of which requiring changes or adaptations in the naming scheme, as the sequence will show. Immediate examples [ ] include (i) two forms of bidimensional, -velocity ones, named D Q and D V for distinction; (ii) a D Q one; as well as (iii) two forms of bidimensional, -velocity ones, named D V (W ) and D V (W ), containing the energy levels ε ∈ {0, 2, 4, 8, 9, 16, 18}, and ε ∈ {0, 1, 2, 4, 8, 9, 16}, respectively-thus without the energy levels (hence, weights) labeled ' ' and ' ' for '(W )' and '(W )', respectively-and (iv) a fourth-order (N = 4) Hermite D V lattice, suitable for thermal ow LB simulations.

Chikatamarla and Karlin:
Seeking to systematically derive stable and Galilean invariant LB models, Chikatamarla and Karlin [ ] set about the problem of LB stencil construction from a discrete form of Boltzmann's H-theorem-in which the (−H) quantity represents a sort of generalized thermodynamic entropy for non-equilibrium states in the Boltzmann Gas Limit (BGL) that increases according to the second law of thermodynamics until equilibrium is reached [ ]. By maximizing the entropy, i.e., by minimizing H in the discrete description, with appropriately chosen weights W i under a set of macroscopic constraints of mass and energy conservation as well as constitutive relations for higher-order macroscopic tensors, authors arrived at explicit expressions for the weights W i and for the stencil reference temperature for several onedimensional lattice models having from to velocities. Methods obtained by this systematic were named 'entropic' LB methods (ELBM).
Remarkably, due to a pattern in Gauss-Hermite quadrature [ ], Chikatamarla and Karlin proposed a straightforward way of obtaining higher-dimension lattice stencils [ ]: As the immediate examples of [ ] evidence, as soon as one allows for including several energy levels (but not necessarily all, nor in monotonic order) in a Bravais lattice, velocity count no longer uniquely identi es lattices, and thus, velocity count-based lattice naming schemes are bound to be ambiguous and require additional information as to uniquely identify the lattice. Usually, the additional information is laying out all velocity vectors, whether by energy level listings, a quiver-like lattice picture, or tabulated lattice velocities (plus weights, and either scaling factor or reference temperature)-all of which seem, to varying degrees, excessively wordy and lengthy. Moreover, the very need for providing additional information as to uniquely identify an already named lattice seems to defeat the purpose of naming it, at least in part.

. Higher-order lattice proliferation
Late 's: The onset of systematic techniques for LB stencils fabrication in [ , , ], allied to the expansion of LB simulation applications and domains, contributed to the proliferation of LB stencils over the following years. This further highlighted the ambiguities in velocity count-based naming schemes, as well as prompted the appearance of further lattice naming variety.
The following 'DdVb' lattices were derived by Ortiz [ ], using prescribed abscissas quadrature [ ]: second-order hexagonally regular (Bravais) D V ; third-order irregular, i.e., not space-lling, D V ; fourth-order irregular D V , D V , D V , and regular D V ; fth-order irregular D V a, D V b, and regular D V a and D V b; sixthorder regular D V . In three-dimensional Euclidean space, the following: second-order irregular D V , third-order irregular D V , and fourth-order irregular D V and D V . A ' -velocity model in three dimensions' is said to be of third order Hermite expansion, with sixth-order tensor isotropy in [ ]; however, no weights, velocity list nor scaling factors of such lattice are given. One-dimensional D Q , D Q , D Q , and D Q , as well as two-dimensional D Q and D Q lattices are presented in [ ], withnite Knudsen number applications in view. Reference [ ] studies various two-and three-dimensional lattices of up to velocities, while referring to a 'DdQq' notation as being 'standard'.
On It is worth noting that pruning operations deal with discrete velocity groups of same magnitude; hence, energy levels-so that pruning remove entire energy levels from a departure lattice con guration.
Several lattice stencils are given in [ ] mostly in the 'DdVb' naming scheme from [ ], but also including an 'n' su x, as in D V n, D V n, and D V n, as to indicate the lattice is not space-lling, and also in the '{0, ±a, ±b}' format, in which b can be an explicit multiple of a, as in {0, ±a, ±3a}, naming schemes. Bravais lattices are given up to D V , D V , and D V in one-, two-, and three-Euclidean spacial dimensions. Appendix tables list full velocity sets as the 'DdVb[n]' naming scheme, in which the 'n' su x is optional, is not uniquely determined.

From to :
The following lattices are referenced in the following works: a D Q in [ , p. ] Mattila and coauthors [ ] have shown that spurious currents emerge along liquid-vapor interface in multiphase simulations. They have shown that higher-order stencils, such as the fourth-order D V , yields more localized and isotropic spurious currents than lower order ones, such as third-order D V and D Q -ZOT ones. Thus, multiphase ows became another application requiring multispeed, higher-order LB methods [ ]. In fact, shortly after, the group published [ ] prescribed abscissas-derived D V and D V lattices, along with an equivalent, however far simpler, form of the prescribed abscissas method in their Section .

Reference [ ] lists lattice velocities, weights and scaling factor for D Q , D Q (reference [ ]'s D V and reference [ ]'s E 17 2,7 ), D Q (reference [ ]'s D V ), and for a D Q , originally unnamed on reference [ ].
Situations like this-in which a given lattice is referred to by di erent names in di erent sources yet without ruling out ambiguities-illustrate the need for improved lattice naming schemes.

(d times)
, then ( ) It is worth noting that the D3Q7 3 and D3Q11 3 lattices have and velocities, respectively. Supersonic and hypersonic ow speeds are comparable to and greater than molecular thermal velocity scales, respectively; moreover, many supersonic ows have a well de ned prevailing ow direction, especially when simple, slender objects move with high speeds through quiescent media. Changing from a rest to the object's reference frame causes molecule velocity populations to be shifted by the object's speed. References [ , ] present the D2Q7 2 lattice, i.e., a D Q one, in the (i) symmetric variety, rest reference frame, and (ii) shifted variety, comoving reference frame of U x = 1 lattice units. Better yet, authors demonstrate that departing from a Galilean-invariant symmetric, rest reference frame lattice, reference-frame shiftings do not change the lattice weights, meaning for arbitrary U, where W i 's are the lattice weights in terms of the lattice reference speed and temperature, and U is the reference frame speed shifting. Better still, the Galilean invariance property allow for the construction of higher-order lattices through tensor products of lower-order Galilean invariant ones, whether they are shifted or not. Therefore, let a symmetrical D1Q7 lattice, with velocity abscissas V 7 = {−3, −2, −1, 0, +1, +2, +3}, produce a unit U x -shifted lattice with velocity abscissas V 7 = {−2, −1, 0, +1, +2, +3, +4}, then the velocity set of the unit U x -shifted D2Q7 2 lattice is given by V 7x ⊗V 7y .
Body-centered cubic, BCC, lattice arrangements arising from emphasizing spatial discretization over the momentum one in the discretization of the Boltzmann equation Finally, a space-lling regular Bravais D V lattice, comprised of velocity vectors, with corresponding weights and scaling factor appears in [ , p. ].

. Discussion
From the lattice naming systems survey of the previous Section, one nds that many lattice naming schemes have appeared over the years, with each new variety either introduced as to accommodate or re ect a new aspect brought in the research, as with [ , , , , ], or to organize and distinguish multiple lattices within a publication, as with [ , , , , ].
The present survey is unaware of any published e ort in the direction of major standardizations across multiple lattice types and features, as well as of the existence of any concise naming system that would allow for uniqueness by ruling out name ambiguity.
Lattice naming variety appears to be due to (i) the inherent decentralized nature of research, (ii) the inherent novelty and discovery associated to the practice of research, making future features, ideas, and demands unforeseeable to previous studies, as well as to (iii) the variety of lattice types.
On this last aspect, the history of the method has seen (a) space-lling, regular, Bravais types in linear, square, triangular (hexagonal), cubic, and projected hypercubic geometries; (b) irregular, non-space-lling ones; (c) those based on spherical coordinate systems; (d) those with shifted reference velocity frame; and (e) those more heavily focused on spatial space discretization rather than on momentum space. This facet alone may at best di cult e orts in creating concise, unambiguous lattice naming schemes of general scope.
With respect to the continuation and adoption of lattice naming systems, the Euclidean dimension, velocity counting based 'DdQb' template of due to Qian [ ], and of of Qian and coauthors [ ] seems to be the closest thing to a present-time de-facto standard for LB stencil naming, being thus acknowledged on research [ ] and on review [ ] papers, as well as textbooks [ , ]. Nonetheless, reaching this current status hasn't been a quick process, as it seemingly took a considerable amount of years until the naming scheme became widely adopted in the LB literature, as evidenced by the lack of its usage in the review paper of Chen and coauthors, which refer to some of them as 'LBM models based on and velocities' [ , p. ], and on the review paper by Succi and coauthors, which refer to Qian and coworkers' model as 'LBGK' [ , p. ] after the collision model.
As opposed to velocity listing. Lattice Nomenclature Survey from LGA to Modern LBM -/ Moreover, other naming conventions such as the 'E d D,n ' one due to Shan and coworkers [ ], the 'DdVb' one due to Philippi and coworkers [ ], and the 'DdQb'-based variations such as '-ZOT' su x and integer power velocity count ones due to Chikatamarla and Karlin [ ] frequently reappear in many subsequent publications, but its adoption seem to be more or less con ned to the proposing author's research groups, and to direct citations-so that one may perceive then to be in competition.
It is worth noting that all 'mainstream' lattice naming schemes-whether 'E d D,n ', 'DdQb' and its variations-are able to describe space-lling, Bravais lattices. Yet, all such lattice naming schemes are velocity count based and therefore su er from ambiguity, as, for instance, the sole 'D Q ' (or 'D V ', or 'D Q5 2 ') information can mean many different lattice con gurations, having completely di erent envelope shapes, conception strategy, set of populated energy levels, and optional shiftings, since it only speci es a set of discrete velocities in two Euclidean space dimensions, and, in the case of the 'E d D,n ' scheme, the resulting order of approximation.
Aiming at space-lling, regular, Bravais types in one-to three-dimensional Euclidean spaces with complete, fullypopulated energy levels, the authors conjecture [ ] that a scheme with (i) energy-level-based primitives, (ii) that allows for operations such as (tensor product) extensions and shiftings; is able to produce relatively concise and unambiguous lattice names, while being su ciently generic within its category.

. Citations by Year
The following are the citations indexed by year in chronological order: :

. Conclusions
A survey of lattice naming systems for lattice-Boltzmann (LB) methods, from its Lattice-Gas Automata (LGA) historic predecessor to the present time has been performed, which correspond to the period of years of our Lord from to . From the survey, key ndings include: (i) the appearance (and discontinuance) of many lattice naming schemes over the years, (ii) an apparent lack of published e orts solely geared towards major lattice name standardizations, (iii) the existence of a great diversity of lattice types, (iv) the prominence of Qian's (and coworkers)' [ , ] velocity-count based 'DdQb' naming scheme-such as D Q -being the closest thing to a de-facto standard in the LB literature; (v) the existence of other, seemingly competing, velocity count naming standards; and (vi) the ambiguity of velocitycount based lattice naming schemes, plainly evident in [ , ].
From the survey and from the diversity of lattice types, it becomes somewhat clear that (a) probably there will be no generic and concise 'one-size-ts-all' naming scheme for all surveyed models, let alone, published ones, and (b) a concise, unambiguous naming scheme, at least for the more regular lattice types is in order, as to enable the necessary distinctions between models of same dimensionality and velocity count. An upcoming work from the authors [ ] is to make a proposition.

Con ict of Interest
The author declares that there is no con ict of interest in this work.