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International Journal of Computational Fluid Dynamics, Vol. 00, No. 00, December 2005, 115
Lattice-Boltzmann simulations in reconstructed parametrized porous media
BENJAMIN AHRENHOLZ, JONAS TÖLKE, MANFRED KRAFCZYK
(Received 00 Month 200x; In nal form 00 Month 200x)
Computations of ows in explicitly resolved porous media reported in the literature so far are based on binarized porous media data
mapped to uniform Cartesian grids. The voxel set is directly being used as the computational grid and thus the geometrical representation
is usually only rst-order accurate due to staircase patterns. In this work we pursue a more elaborate approach: Starting from a highly
resolved tomographic grey value data set we utilize a marching-cube algorithm to reconstruct the surface of the porous medium as a
set of planar triangles. The numerical resolution of the Cartesian grid for the simulation can then be chosen independently from the
voxel set. As we take into account the subgrid distances between the nodes of the Cartesian grid and the planar triangle surfaces, one
can utilize a second-order accurate Lattice-Boltzmann ow solver to eciently compute e.g. permeabilities. As these interpolation-based
no-slip boundary conditions are not mass preserving, we also present a local modication of the no-slip boundary condition restoring
mass conservation. Our numerical results demonstrate, that for saturated ow simulations this coupled approach allows a substantial
acceleration of saturated ow computations in porous media.
Keywords: lattice-Boltzmann; porous media; marching cubes
1 Introduction
Numerical models based on the lattice Boltzmann equation (LBE) have matured as a well established
tool for solving complex problems in uid dynamics. A popular LBE model for simulating ows in porous
media is the well known LBGK (lattice Bhatnagar-Gross-Krook) model (Qian et al. 1992), in combination
with a simple bounce back scheme for the solid-uid boundaries (Ginzburg and d'Humières 2003). This
approach has two disadvantages. First its numerical stability is relatively limited. Second, the simplied
nature of the bounce-back scheme implies that the true position of the wall is depending on the value of
the relaxation time used for the simulation, i.e. the permeability of a porous medium becomes viscosity
dependent (Ginzbourg and Adler 1994). These problems can be drastically reduced by using a multiplerelaxation-time model (d'Humières 1992, Lallemand and Luo 2000, d'Humières et al. 2002), which not only
improves numerical stability but also eliminates the viscosity dependence of the permeability (Ginzburg and
d'Humières 2003). Furthermore, the simple bounce back scheme has to be replaced by a boundary condition
whose convergence behaviour is consistent with the methods bulk behaviour, e.g. (Bouzidi et al. 2001).
Simulations investigating ow in porous media using MRT and higher order boundary conditions have been
carried out for simple analytic geometries such as body centred arrays of spheres (Ginzburg and d'Humières
2003), as well for polydisperse sphere packings (Pan et al. 2005). In this paper we combine a MRT-LB model
with second-order boundary conditions for the simulation of single-phase ow in reconstructed porous
media. The paper is organized as follows: In Section 2 we describe the reconstruction of the geometry of
porous media by using a marching cube algorithm to decouple the numerical grid from its voxel based data.
Section 3 describes the numerical model and its implementation. In section 4.1 numerical experiments for
a body centred cubic array of spheres to validate the approach have been carried out. In section 4.2 ow
simulations in reconstructed parameterized porous media using second order boundary conditions have
been performed and demonstrate the eciency of our approach.
∗ Corresponding
author. Email: {ahrenholz,toelke,kraft}@cab.bau.tu-bs.de
International Journal of Computational Fluid Dynamics
ISSN 1061-8562 print / ISSN 1029-0257 online c 2005 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/10618560xxxxxxxxxxxxx
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2 Decoupling the numerical grid from the tomography based voxel set
2.1
Motivation
Most uid simulations in resolved porous media that are based on natural or real-world geometries are
derived from tomography scans or similar extraction methods. These tomography scans are usually converted to binary voxel sets which are directly utilized as the computational grid and thus have the same
resolution as the voxel matrix. The immediate consequence of this method is an implicit coupling of the resolution of the geometry with the computational (i.e. numerical) grid. This has two distinct disadvantages:
a) an intrinsical stair-case representation of the geometry and b) convergence studies are hard to perform,
because from a numerical point of view, it would be necessary to have scans of dierent resolutions. To
evade this limitations we are introducing an approach, which is based on voxel geometries (Lehmann et al.
2006) represented by grey values and that will reconstruct a plane triangle mesh to obtain a second order
accurate geometric representation of the pore space. A sample of a porous medium (Kaestner et al. 2005)
with a dimension of 8003 voxels and a triangulated section of the same sample are shown in Figure 1 and
2.
Figure 1. X-ray scan of a porous medium measured at HASYLAB (courtesy of Kaestner and Lehmann (Lehmann et al. 2006))
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Figure 2. Section of a triangulated surface of a porous medium
2.2
Isocontour algorithm
For triangulation one can utilize dierent types of isocontour/isosurface algorithms. In this work we adopted
a variant of the marching cube family which for our application is favourable due to its ability to conserve
topological quantities to a large extent (Lewiner et al. 2003) and which is designed to avoid cracks in
the surface. Marching Cubes (Lorensen and Cline 1987) is a computer graphics algorithm for extracting a
polygonal mesh of an isosurface from a 3D discrete scalar eld. The algorithm successively scans the scalar
eld, groups 8 voxels to an imaginary cube and determines the polygons whose cutting plane represents the
isosurface of the predened scalar value in this cube. This computation can be accelerated by creating an
index array of the 28 = 256 possible polygon congurations which serve as a look-up table in all subsequent
cube computations. The look-up table can be further condensed by symmetry arguments to as few as fteen
unique cases. These basic congurations are shown in gure 3.
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Figure 3. Representation of the 15 base congurations of the Marching Cube algorithm
Finally each vertex of the generated polygons is placed on the appropriate position along the cube's
edge by linearly interpolating the two scalar values of its corresponding two nodes. The result is a mesh of
triangles representing the geometric structure of a voxel set.
Especially in the case of a natural porous medium with its arbitrary geometrical complexity it is crucial to
have a consistent mesh of triangles, because cracks will lead to leaks which again will lead to wrong surface
orientations (A pore may be interpreted as solid and vice versa). It turned out, that even our implemented
type of Marching Cubes can lead to wrong triangle congurations under certain conditions. In that case we
are forced to correct the mesh by checking the connectivity of the triangle mesh. If a triangle has less than
three neighbouring faces, it is part of rare triangulation conguration, which can appear in very narrow
channels. Then it is safe to delete the aected triangle to keep the topological quantity. After the geometric
reconstruction the triangulation is used to generate the numerical grid, whose resolution can now be freely
chosen. While the grid is adapted to the mesh, we calculate the distances from the Cartesian grid nodes
to the triangles describing the outer walls. These distances will be used later as so-called q-values for the
boundary conditions (see section 3.2). A recursive ll algorithm then separates uid, boundary and solid
nodes for the upcoming simulation.
2.3
Calculation of subgrid distances
To apply higher order boundary conditions based on linear or quadratic interpolations (Bouzidi et al. 2001)
or the even more complex multi-reection boundary scheme (Ginzburg and d'Humières 2003), the projected
distances between the boundary grid node and the reconstructed surface along the links of the stencil are
required. A representation of boundary nodes (quadratic dots) in pore space with their corresponding links
qi , consistent with the D3Q19 model, is shown in Figure 4.
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Second order accurate Lattice-Boltzmann ow simulations in reconstructed parametrized porous media
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Figure 4. Boundary nodes with subgrid distances qi
The links and their length have only been calculated once as a part of the pre-processing, as the geometry
of the porous medium is assumed not to depend on time. For calculations with voxel scans up to a size of
8003 the triangulation algorithm in the pre-processing step required between 100 and 200 million triangles
for a reconstructed surface for the medium under consideration and generally depends on the mediums
topology as well its porosity. The time consumption is moderate: triangulation and computation of the
subgrid distances requires about 4 hours on a 2 GHz Opteron and 32 GB memory. The algorithm has an
optimum complexity of N (logN ) where N is the number of triangles.
3 The LB Method for single-phase ow with second order accuracy
3.1
The Multi Relaxation Time Model (MRT)
In the following section x represents a 3D vector in space and f a b-dimensional vector, where b is the
number of microscopic velocities. We use the so-called d3q19 model (Qian et al. 1992) with the following
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microscopic velocities,
(
{ei , i = 0, . . . , 18} =
0 c −c 0 0 0 0 c −c c −c c −c c −c 0 0 0 0
0 0 0 c −c 0 0 c −c −c c 0 0 0 0 c −c c −c
0 0 0 0 0 c −c 0 0 0 0 c −c −c c c −c −c c
)
,
where c is a constant velocity related to the speed of sound by c2s = c2 /3. The generalized lattice Boltzmann
(GLB) equation using the multi-relaxation time model introduced by (d'Humières 1992, Lallemand and
Luo 2000) is used in this paper in a slightly modied version (Tölke et al. 2004). The lattice Boltzmann
equation is given by
fi (t + ∆t, x + ei ∆t) = fi (t, x) + Ωi ,
i = 0, . . . , b − 1,
(1)
where ∆t is the time step, the grid spacing is ∆x = c∆t and the collision operator is given by
Ω = M−1 S ((Mf ) − meq ) .
(2)
The matrix M given in appendix A transforms the distributions into moment space. The resulting moments
m = Mf are labelled as
m = (δρ, e, , jx , qx , jy , qy , jz , qz , 3pxx , 3πxx , pww , πww , pxy , pyz , pxz , mx , my , mz ),
where δρ is a density variation related to the pressure variation δp by
δp = c2s δρ,
(3)
and (jx , jy , jz ) = ρ0 (ux , uy , uz ) the momentum, where ρ0 is a constant reference density. The moments
e, pxx , pww , pxy , pyz , pxz are related to the stress tensor by
σxx = −(1 −
σyy = −(1 −
σzz = −(1 −
σxy = −(1 −
σyz = −(1 −
σxz = −(1 −
sν 1
)( e + pxx − ρ0 u2x )
2 3
sν 1
1
1
)( e − pxx + pww − ρ0 u2y )
2 3
2
2
sν 1
1
1
)( e − pxx − pww − ρ0 u2z )
2 3
2
2
sν
)(pxy − ρ0 ux uy )
2
sν
)(pyz − ρ0 uy uz )
2
sν
)(pxz − ρ0 ux uz ).
2
(4a)
(4b)
(4c)
(4d)
(4e)
(4f)
Here sν is a collision rate explained later. The other moments of higher order have no physical meaning
for the incompressible Navier-Stokes equations. The vector meq is composed of the equilibrium moments
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given by
meq
0 = δρ
(5a)
eq
2
2
2
meq
1 = e = ρ0 (ux + uy + uz )
(5b)
meq
3 = ρ0 u x
(5c)
meq
5
meq
7
= ρ0 u y
(5d)
= ρ0 u z
(5e)
2
2
2
= 3peq
xx = ρ0 (2 ux − uy − uz )
(5f)
eq
2
2
meq
11 = pzz = ρ0 (uy − uz )
(5g)
eq
meq
13 = pxy = ρ0 ux uy
(5h)
eq
meq
14 = pyz = ρ0 uy uz
(5i)
meq
9
eq
meq
15 = pxz = ρ0 ux uz
meq
2
=
meq
4
=
meq
6
=
meq
8
=
meq
16
=
meq
17
(5j)
=
meq
18
=0
(5k)
The matrix S is a diagonal collision matrix composed of relaxation rates {si,i , . . . , b − 1}, also called the
eigenvalues of the collision matrix M−1 S M. The rates dierent from zero are
s1,1 = −se
s2,2 = −s
s4,4 = s6,6 = s8,8 = −sq
s10,10 = s12,12 = −sπ
s9,9 = s11,11 = s13,13 = s14,14 = s15,15 = −sν
s16,16 = s17,17 = s18,18 = −sm .
The relaxation rate sν is related to the kinematic viscosity ν by
sν =
1
3 c2ν∆t
+
1
2
.
(6)
The other relaxation rates se , s , sq , sπ and sm can be freely chosen in the range [0, 2] and may be tuned
to improve accuracy and/or stability (Lallemand and Luo 2000). The optimal values depend on the specic system under consideration (geometry, initial and boundary conditions) and can not be computed in
advance for general cases.
These values will be discussed in more detail in the sections dealing with the numerical examples. Using
either a Chapman-Enskog expansion Frisch et al. (1987) or an asymptotic expansion using the diusive
scaling (Junk et al. 2005), it can be shown that the LB method is a scheme of rst order in time and second
order in space for the incompressible Navier-Stokes equations.
3.2
Boundary conditions
The macroscopic ow quantities can only be set implicitly via incoming particle distribution functions on
the boundary nodes. A well known and simple way to introduce no-slip walls is the so-called bounce back
scheme which allows spatial second order accuracy if the boundary is aligned with one of the lattice vectors
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ei and rst order otherwise (Ginzburg and d'Humières 2003). As we have arbitrarily shaped objects, we use
the modied bounce back scheme developed in (Bouzidi et al. 2001, Lallemand and Luo 2003) for velocity
boundary conditions, which is second order accurate for arbitrarily shaped boundaries (gure 5).
q<1/2
q
1
F
q>1/2
D
1
W
A
t
F
t
fi,A
t
fi,F
q
A
W
D
W
t
fi,A
t
fI,A
2q-1
2q
F
D
D
A
W
t+1
F
A
t+1
fI,A
t+1
fI,A
Figure 5. Interpolations for second order bounce back scheme.
Here we identify two cases:
(i) the wall has a distance less then 0.5 ei ∆t from the node and
(ii) the wall has a distance between 0.5 and 1.0 ei ∆t from the node.
ei u w
,
c2
2q − 1 t
1
ei u w
t
=
· fI,A +
· fi,A
−3 2 ,
2q
2q
qc
t+1
t
t
(i) fI,A
= (1 − 2q) · fi,F
+ 2q · fi,A
−6
0.0 < q < 0.5
(7)
(ii)
0.5 ≤ q ≤ 1.0
(8)
t+1
fI,A
Distributions at time t/t + 1 are post/pre-collision values, qei ∆t is the distance to the wall and uw the
velocity at the wall. Therefore we obtain second order accurate results in space even for curved geometries
(Geller et al. 2006). For a detailed discussion of LBE boundary conditions we refer to (Ginzburg and
d'Humières 2003). In contrast to the simple bounce-back scheme the use of this interpolation based no-slip
boundary conditions result in a notable mass loss across the no-slip lines. To circumvent this problem we
transfer the mass dierence to the rest particle distribution. This results in a bounce back scheme which is
conservative in mass (and thus pressure) while introducing a higher-order disturbance of the stress tensor
which does not change the results signicantly. The results obtained with the rst-order bounce-back were
inferior to the second-order scheme which highlights the importance of a proper geometric resolution of
the ow domain.
Pressure boundary conditions are implemented by setting the incoming distributions as (Thürey 2003)
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Second order accurate Lattice-Boltzmann ow simulations in reconstructed parametrized porous media
fI = −fi + fIeq (P0 , u) + fieq (P0 , u)
9
(9)
where P0 is the prescribed pressure, eI = −ei and u is obtained by extrapolation.
3.3
Implementation
A simple lattice Boltzmann algorithm can be implemented easily, however, more advanced approaches in
terms of accuracy, speed and memory consumption require careful programming. The simulation kernel
used for the numerical experiments has been implemented using the Fortran 90 standard. The eciency of
matrix based data structures and its convenient syntax for numerical problems were the determining factor.
The parallelization follows a distributed memory approach using a Message Passing Interface (MPI-Forum
2006). The kernel has been optimized with respect to speed in the rst place, by e.g. using two arrays
for storing the distribution functions. Collision and propagation have been combined into a single nested
loop. The array containing the subgrid distances (q-values) is indexed by a list as they are only needed
for the uid-solid boundary nodes. The simple bounce back scheme does not require the exchange of all
neighbouring distributions due to its local nature. Therefore it would be sucient to exchange only the ve
distributions pointing into communication direction. However, while using the interpolated bounce back
schemes it is necessary to exchange also the distributions necessary for the interpolation, which will slightly,
increase the parallelization overhead.
4 Numerical experiments
4.1
Body centred periodic array of spheres
Before running simulations in natural porous media with the introduced framework we validated the approach with an analytic geometry for which a semi-analytical solution is available: a body centred periodic
array of spheres as shown in Figure 6. For this conguration one can derive a solution following the work
of (Hasimoto 1959) and (Sangani and Acrivos 1982).
The drag force of one sphere can be computed as:
(10)
F = Cd · 6πµud a
where Cd is the dimensionless drag, µ the dynamic viscosity and ud the Darcy velocity. The drag can be
determined as a function of the solid volume fraction c by a series expansion:
Cd =
30
X
n=0
n
αn χ , χ = (
c
cmax
1/3
)
8πa3
, c=
, cmax =
3L3
√
3π
8
(11)
The drag only depends on the characteristics of the geometry, which are dened by L and a (see also
Fig. 7).
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Figure 7. 2D projection of the sphere
array.
Figure 6. A BCC array of spheres in periodic space.
The permeability of the medium can then be derived by considering the region depicted in Fig. 7. Taking
into account that the region contains two spheres (Adler 1992) in sum we can compute the average pressure
gradient in z-direction as
−∂z p =
2F
.
L3
(12)
The Darcy velocity using equation (17) is then
u=
κ 2F
.
µ L3
(13)
Substituting equation 10 in equation (13) we obtain for the intrinsic permeability κ
κ=
L3
.
12πaCd
(14)
The dimensionless permeability κ∗ is
κ∗ =
κ
1
=
·
a2
12πCd
3
L
.
a
(15)
The geometry used in the simulation has been modelled using a common geometric modelling tool, such
as AutoCAD (AutoCAD 2006) or Microstation (Microstation 2006). The triangle mesh was generated
using the grid generation tool as described in section 2. Therefore it was easy to create grids in dierent
resolutions using always the same input-model. The smallest grid used 103 nodes and the largest up to 1203 .
The setup for the simulation is a LB-simulation core using the MRT-model and the 'magic'-relaxation
parameters given below according to (Ginzburg 2005). These parameters eliminate the permeability
dependence on viscosity for simulations using Stokes ow (i.e. neglecting the second order terms of the
equilibrium distributions) for simple bounce-back and reduce them substantially in the case of interpolation
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Second order accurate Lattice-Boltzmann ow simulations in reconstructed parametrized porous media
11
based bounce back by relaxing the even and odd moments dierently:
se = s = sπ = sν , sq = sm = 8
(2 − sν )
(8 − sν )
(16)
Permeability is determined by measuring the volume averaged uid velocity (Darcy velocity) after reaching
a stationary state. From Darcy's law one can calculate the uid permeability,
ud = −
κ
(∂z p + ρg)
ρν
(17)
which is then compared to the theoretical solution given by (Hasimoto 1959, Sangani and Acrivos 1982).
Figures 8 and 9 show the comparison for dierent values of χ. Fig. 8 reveals that the accuracy of the
simulation using the linear interpolated bounce back schemes for a resolution of about 143 grid nodes is
comparable to the one from the truncated series expansion of the theoretical solution (relative error below
1%).
The variations of the numerical results even for low resolutions can be explained by the type of setup.
The numerical grid is intentionally not aligned to the geometry but is created using a small oset, which is
always as large as half the grid spacing. This displacement of the grid results in a non-symmetrical setup
and is considered to t better to a natural porous medium and more importantly is supposed to avoid
the cancellation and therefore the reduction of errors due to a symmetrical setup. At higher resolutions
this out of alignment will have less impact to the obtained results, which explains the relatively high error
at low resolutions. The newly introduced simple mechanism for a mass conservative interpolated bounce
back scheme gives the same order of convergence, but it is not as accurate as the classical boundary
condition. However, it may be useful for setups where a mass conservative scheme is mandatory and
quadratic convergence is required. The simple bounce back scheme cannot compete here; for an accurate
solution one would require resolutions beyond 1003 which would require almost three orders of additional
CPU-power and memory.
rel. error [%]
1,0E+02
K [SBB]
K [LIBB]
K [LIBB FIX]
N -1
1,0E+01
Accuracy of Ref.-Sol.
1,0E+00
1,0E-01
N -2
1,0E-02
10
100
resolution [voxel]
Figure 8. Plot comparing the accuracy of simple and linear interpolated BB with χ = 0.76.
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1,0E+02
rel. error [%]
K [SBB]
K [LIBB Fix]
K [LIBB]
1,0E+01
N-1
Accuracy of Ref.-Sol.
1,0E+00
N-2
1,0E-01
10
100
resolution [voxel]
1000
Figure 9. Plot comparing the accuracy of simple and linear interpolated BB with χ = 0.96.
4.2
Convergence study for a natural porous medium
Present neutron tomography or X-rays from synchrotron scanners are capable of scanning grey value images
reaching voxel resolutions of a few microns. Thus one can perform uid simulations in natural porous media
with voxel resolutions up to 109 voxels. However, the demands of a simulation with a computational grid of
the same size are very high. To demonstrate the gain of decoupling the numerical grid from the geometrical
voxel description we performed a convergence study based on tomography scans of sand cubes. The data
sources were subsets of the sand sample introduced in section 2 with a size of 2503 which corresponds to a
size of 1.25 mm2 which is app. 5-7 times the diameter of a typical pore diameter. Multiple simulations at
dierent resolutions have been carried out and the corresponding results are shown in table 1 and gure
10. The resolution shown in the rst column indicates the factor in grid size from the original voxelbased geometry resolution of 250. The reference permeability of the medium has been calculated using a
Richardson extrapolation.
Table 1. Relative permeability at dierent resolutions.
4.3
Resolution
Voxel
rel. K (SBB)
rel. K (LIBB)
rel. error SBB
[%]
rel. error LIBB
[%]
1.75
1.5
1.0
0.75
0.5
0.3
436
374
250
187
125
76
3.8137092E-05
3.7605696E-05
3.5834207E-05
3.4072388E-05
2.9753718E-05
1.7191929E-05
4.1587915E-05
4.1800425E-05
4.2637626E-05
4.3666093E-05
4.6620740E-05
5.4151301E-05
8.915E+00
1.045E+01
1.591E+01
2.191E+01
3.960E+01
1.416E+02
1.222E-01
6.300E-01
2.581E+00
4.876E+00
1.090E+01
2.329E+01
Computational issues
At the highest resolution used in the simulations, which was 4363 , the memory consumption is quite large.
Even this is a single ow simulation and basically only one set of 19 double precision values are necessary,
the array containing the q-values for each boundary node consumes also 19 oat values. Additional ghost
nodes, which are mandatory for non local interpolation schemes in combination with parallelization cause
additional hardware resources. However, the time used for the collision and propagation steps of the LB
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rel. error [%]
Second order accurate Lattice-Boltzmann ow simulations in reconstructed parametrized porous media
1,0E+03
13
rel. error SBB
rel. error LIBB
1,0E+02
1,0E+01
N -1
1,0E+00
N -2
1,0E-01
10
100
1000
resolution [grid nodes]
Figure 10. Plot comparing absolute permeability depending on the resolution.
scheme is comparably large. Thus a very good parallel eciency is still observed; especially in the case of
asynchronous communication.
Based on the results of the convergence study we nd, that using a lower grid resolution than the original
voxel set, one can obtain results which are of a comparable accuracy as the reference solution, using full
resolution and simple bounce back. Thus from the corresponding runtimes we can see that the computational eciency rises by at least one order of magnitude when using the parameterized reconstruction. For
the porous medium used in our simulations we achieve a relative error below 10% in permeability using a
grid with a resolution factor of 0.5 and interpolated bounce back. With a simple bounce back scheme one
has to use at least a resolution factor of 1.5 to obtain a result of comparable accuracy. That leads to a gain
in eciency - saving CPU time, memory consumption, etc. - by a factor of 16.
5 Conclusions
In this work we demonstrate that second order accuracy and a signicant speedup in convergence can be
achieved not only for ow simulations based on CAD-Type geometries, but also from scanned data such as
porous media ow in sand. In contrary to LBGK models with single-relaxation-time in combination with
the simple bounce back scheme, the MRT model in combination with higher order boundary conditions
are more accurate and due to its faster convergence to a steady state far more ecient. The additional
costs in terms of pre-processing and decreased locality of the boundary stencil are more than balanced by
the increase in numerical eciency. Finally, we would like to point out, that for the simulation of ows in
deformable porous media (uid-structure-interaction) the parameterization of the geometry is mandatory
to consistently compute the structural deformation including contact problems.
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REFERENCES
14
Acknowledgments
Financial support by the Deutsche Forschungsgemeinschaft in the framework of the German LatticeBoltzmann Research Group is gratefully acknowledged. Also many thanks to Peter Lehmann and Anders
Kästner from ETH Zürich for providing the high quality x-ray scans of sand samples.
Appendix A: Transformation matrix M

1·
(1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1)

 2

 c · (−1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1) 


 4

 c · (1 −2 −2 −2 −2 −2 −2 1 1 1 1 1 1 1 1 1 1 1 1) 




 c· (0 1 −1 0 0 0 0 1 −1 1 −1 1 −1 1 −1 0 0 0 0) 


 3

 c · (0 −2 2 0 0 0 0 1 −1 1 −1 1 −1 1 −1 0 0 0 0) 


 c· (0 0 0 1 −1 0 0 1 −1 −1 1 0 0 0 0 1 −1 1 −1) 




 c3 · (0 0 0 −2 2 0 0 1 −1 −1 1 0 0 0 0 1 −1 1 −1) 




 c· (0 0 0 0 0 1 −1 0 0 0 0 1 −1 −1 1 1 −1 −1 1) 




 c3 · (0 0 0 0 0 −2 2 0 0 0 0 1 −1 −1 1 1 −1 −1 1) 


 2

 c · (0 2 2 −1 −1 −1 −1 1 1 1 1 1 1 1 1 −2 −2 −2 −2) 


 4

 c · (0 −2 −2 1 1 1 1 1 1 1 1 1 1 1 1 −2 −2 −2 −2) 


 2

 c · (0 0 0 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 0 0 0 0) 


 4

 c · (0 0 0 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 0 0 0 0) 


 2

 c · (0 0 0 0 0 0 0 1 1 −1 −1 0 0 0 0 0 0 0 0) 


 2

 c · (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 −1 −1) 


 c2 · (0 0 0 0 0 0 0 0 0 0 0 1 1 −1 −1 0 0 0 0) 




 c3 · (0 0 0 0 0 0 0 1 −1 1 −1 −1 1 −1 1 0 0 0 0) 




 c3 · (0 0 0 0 0 0 0 −1 1 1 −1 0 0 0 0 1 −1 1 −1) 


3
c · (0 0 0 0 0 0 0 0 0 0 0 1 −1 −1 1 −1 1 1 −1)
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