Smoothed particle hydrodynamics for fluid dynamics
Xin Bian and George Em Karniadakis
Division of Applied Mathematics, Brown University, USA
class APMA 2580on Multiscale Computational Fluid Dynamics
April 5, 2016
1 / 50
Homework: you should start
Hw # 4: Finite difference method + MPI for Helmholtz equations
If you need a multi-core machine: apply for a CCV accounthttps://www.ccv.brown.edu/
2 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
3 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
4 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
5 / 50
Conservation law of mass: continuity equation
Total mass flows out of the volume V (per unit time) by surface integral∮ρv · df, (1)
where ρ density, v velocity and df is along the outward normal.The decrease of the mass in the volume (per unit time)
− ∂
∂t
∫ρdV . (2)
For a mass conservation, we have an equality
− ∂
∂t
∫ρdV =
∮ρv · df =
∫∇ · (ρv)dV , (3)
which is valid for an arbitrary V . The continuity equation reads
∂ρ
∂t+∇ · (ρv) = 0. (4)
6 / 50
Conservation law of momentum: Euler equations
Total pressure force acting on the volume (surface to volume integral)
−∮
pdf = −∫∇pdV . (5)
For a unit of volume, momentum equations in Lagrangian form read
ρdv
dt= −∇p or
dv
dt= −∇p
ρ. (6)
Considering the particle derivative is related to the partial derivatives as
d
dt=
∂
∂t+ v · ∇, (7)
the Euler equations in Eulerian form read
∂v
∂t+ v · ∇v = −∇p
ρ. (8)
7 / 50
Conservation law of momentum: Navier-Stokes equations
For real fluids, we need to add in viscous stress due to irreversible processand assume that the viscous stress depends only linearly on derivatives ofvelocity.Without derivation, the Navier-Stokes equations read
∂v
∂t+ v · ∇v = −1
ρ[∇p + η4 v + (ζ + η/3)∇∇ · v] (9)
For an incompressible fluid ρ = const. and ∇ · v = 0.Therefore, the momentum equations simplify to
∂v
∂t+ v · ∇v = −1
ρ(∇p + η4 v) . (10)
For a compressible fluid, an equation of state is called for
p = p(ρ,T = T0). (11)
8 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
9 / 50
Mesh-based discretizations
Prof. Karniadakis will cover the mesh-based methods in other lectures
finite difference method
spectral h/p element method
... ...
10 / 50
Mesh-free discretizations: mesh-free = mess-free?
Some particle methods
smoothed particle hydrodynamics (SPH)moving least square methods (MLS)vortex methodVoronoi tesselation... ...
mesh-free ≈ mess-free
no mesh generationLagrangian, no v · ∇vcomplex moving boundaryincorporation of new physics... ...
11 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
12 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
13 / 50
Integral representation of a function
It was invented in 1970s (author?) [12, 8].
Given a scalar function f (r) of spatial coordinate r , its integralrepresentation reads
f (r) =
∫f (r ′)δ(r − r ′)dr ′, (12)
where the Dirac delta function reads
δ(r − r ′)
{∞, r = r ′
0, r 6= r ′(13)
and the constraint of normalization is∫ ∞∞
δ(r)dr = 1. (14)
14 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
15 / 50
SPH 1st step: smoothing or kernel approximation
(author?) [8]; (author?) [12]
Replace δ with another smoothly weighting function w :
f (r) ≈ fk(r) =
∫f (r ′)w(r − r ′, h)dr ′, (15)
where kernel w has properties
1 smoothness
2 compact with h as parameter
3 limh→0 w(r − r ′, h) = δ(r − r ′)
4∫
w(r − r ′, h)dr ′ = 1
5 symmetric
6 ... ...
compact: B-splines, Wendland functions ...(author?) [15, 16, 19]
Gaussian: 1a√π
e−r2/a2
16 / 50
Compact kernel and its normalization
a cubic function as reads (h = 1 for simplicity)
w(r) =
{CD(1− r)3, r < 1;0, r ≥ 1.
(16)
If we require the constraint of normalization in two dimension∫ 2π
0
∫ 1
0C2(1− r)3rdrdθ = 1⇐⇒ C2 =
10
π(17)
a piecewise quintic function reads
w(r) = CD
(3− s)5 − 6(2− s)5 + 15(1− s)5, 0 ≤ s < 1(3− s)5 − 6(2− s)5, 1 ≤ s < 2(3− s)5, 2 ≤ s < 30, s ≥ 3,
(18)
where s = 3r/h and C2 = 7/(478πh2) and C3 = 1/(120πh3).
f (r) = fk(r) + error(h)
17 / 50
SPH 2nd step: summation or particle approximation
Integral ⇐⇒ summation
fk(r) =
∫f (r ′)w(r − r ′, h)dr ′(19)
fs(r) =
Nneigh∑i
fiw(r − ri , h)Vi , (20)
where Vi is a distance, area and volumein 1D, 2D, and 3D, respectively.Therefore,
fk(r) ≈ fs(r) (21)
fk(r) = fs(r) + error(∆r , h)
summation within compact support
18 / 50
An example: evaluation of density and arbitrary function
∀i of particle index, mass mi , density ρi , and Vi = mi/ρi .
fs(r) =
Nneigh∑i
fiw(r − ri , h)Vi =
Nneigh∑i
mi
ρifiw(r − ri , h). (22)
What is the density at an arbitrary position r?
ρs(r) =
Nneigh∑i
mi
ρiρiw(r − ri , h) =
Nneigh∑i
miw(r − ri , h). (23)
What is the density of a particle at rj?
ρs(rj) =
Nneigh∑i
miw(rj − ri , h). (24)
What is the value of a function of a particle at rj?
fs(rj) =
Nneigh∑i
fimi
ρiw(rj − ri , h) =
Nneigh∑i
fiWji = fj (25)
19 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
20 / 50
Gradient of a function
f (r) ≈ fk(r) =
∫f (r ′)w(r − r ′, h)dr ′ (26)
⇓∇r f (r) ≈ ∇r fk(r) = ∇r
∫f (r ′)w(r − r ′, h)dr ′ (27)
⇓
∇r fk(r) =
∫∇r f (r ′)w(r − r ′, h)dr ′ +
∫f (r ′)∇rw(r − r ′, h)dr ′ (28)
⇓∇r fk(r) =
∫f (r ′)∇rw(r − r ′, h)dr ′ (29)
⇓
∇r f (r) ≈ ∇r fk(r) ≈ ∇r fs(r) =
Nneigh∑i
mi
ρif (r ′)∇rw(r − r ′, h) (30)
21 / 50
Other first derivatives
∇r · f (r) ≈ ∇r · fk(r) ≈ ∇r · fs(r) =
Nneigh∑i
mi
ρif (r ′) · ∇rw(r − r ′, h) (31)
∇r×f (r) ≈ ∇r×fk(r) ≈ ∇r×fs(r) =
Nneigh∑i
mi
ρif (r ′)×∇rw(r−r ′, h) (32)
22 / 50
Second derivatives
Note that we have following identity (author?) [4]∫dr′[f (r′)− f (r)
] ∂w(|r′ − r|)∂r ′
1
rijeij
[5
(r′ − r)α(r′ − r)β
(r′ − r)2− δαβ
]= ∇α∇βf (r) +O(∇4fh2). (33)
Therefore,
1
ρi
(∇2v
)= −2
Nneigh∑j
mj
ρiρj
∂wij
∂rvij (34)
1
ρi(∇∇ · v) = −
Nneigh∑j
mj
ρiρj
∂wij
∂r(5eij · vijeij − vij) (35)
(36)
23 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
24 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
25 / 50
Particles for hydrodynamics
continuity equation is accounted for by
ρi =
Nneigh∑j
mjWij , ri = vi (37)
pressure force: −∇p/ρ
FCi =
Nneigh∑j
FCij =
Nneigh∑j
−mj
(pj
ρ2j
)∂w
∂rijeij , (38)
bad: not antisymmetric by swapping i and jrecognize −∇p/ρ = − p
ρ2∇ρ−∇ pρ
FCij = −mj
(pi
ρ2i
+pj
ρ2j
)∂w
∂rijeij , (39)
26 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
27 / 50
viscous force
In general
FDij =
mj
ρiρj rij
∂w
∂rij
[(5η
3− ζ)vij +
(5ζ +
5η
3
)eij · vijeij
](40)
For inompression flows ∇ · v = 0, therefore,
N∑j
5
ρiρj rij
∂wij
rijeij · vijeij ≈
N∑j
1
ρiρj rij
∂wij
∂rijeij . (41)
FDij = 2η
mj
ρiρj rij
∂wij
∂rijvij ≈ 10η
mj
ρiρj rij
∂wij
∂rijeij · vijeij . (42)
Either choice is fine, but they are different.
28 / 50
Summary of SPH for isothermal Navier-Stokes equations
continuity equation:
ρi =
Nneigh∑j
mjWij , ri = vi (43)
momentum equations: (author?) [4, 11]
vi =
Nneigh∑j 6=i
(FCij + FD
ij
), (44)
FCij = −mj
(pi
ρ2i
+pj
ρ2j
)∂w
∂rijeij , (45)
FDij =
mj
ρiρj rij
∂w
∂rij
[(5η
3− ζ)vij +
(5ζ +
5η
3
)eij · vijeij
](46)
weakly compressible: (author?) [2, 13]
p = c2Tρ, or p = p0
[(ρ
ρr
)γ− 1
](47)
p0 relates to an artificial sound speed cT29 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
30 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
31 / 50
Density estimate or sampling
Given a set of point particles with mass mi , what is the density estimatefor a position at r .
ρs(r) =
Nneigh∑i
miw(r − ri , h). (48)
where kernel w has properties1 smoothness2 compact with h as parameter3∫
w(r − r ′, h)dr ′ = 14 symmetric5 ... ...
Eq. (48) is more fundamental than the summation form presented early
fs(r) =
Nneigh∑i
fimi
ρiw(r − ri , h). (49)
32 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
33 / 50
Least action
Define the Lagrangian L as
L = T − U, (50)
where T and U are kinetic and potential energies, respectively. For a setof particles
L =N∑i
mi
(1
2v 2i + ui (ρi , s)
)(51)
Define the action as
S =
∫Ldt. (52)
Minimizing S such that δS =∫δLdt = 0, where δ is a variation with
respect to particle coordinate δr. We have (author?) [17]
δS =
∫ (∂L
∂v· δv +
∂L
∂r· δr)
= 0 (53)
34 / 50
Euler-Lagrangian equations
δS =
∫ (∂L
∂v· δv +
∂L
∂r· δr)
= 0 (54)
consider δv = d(δr)/dt and d/dt = ∂/∂t + v · ∇
⇓
δS =
∫ {[− d
dt
(∂L
∂v
)+∂L
∂r
]· δr}
dt +
[∂L
∂v· δr]tt0
= 0 (55)
assume variation vanishes at start and end times and furthermore, δr isarbitrary. Therefore, we have the Euler-Lagrangian equations
d
dt
(∂L
∂vi
)− ∂L
∂ri= 0. (56)
35 / 50
Equation of motions for particles
From the Lagrangian L =∑N
i mi
(12 v 2
i + ui
)we know
∂L
∂vi= mivi ,
∂L
∂ri= −
Nneigh∑j
mj∂uj
∂ρj
∂ρj∂rj
(57)
Some basic thermodynamics: dU = TdS − PdVSince V = m/ρ, so dV = −mdρ/ρ2. For per unit mass we have
du = Tds − P
ρ2dρ. (58)
For a reversible process ds = 0, therefore ∂ui/∂ρi = p/ρ2. Put everythingknown into the Euler-Lagrangian equations, we get
vi =
Nneigh∑j 6=i
−mj
(pi
ρ2i
+pj
ρ2j
)∂w
∂rijeij . (59)
36 / 50
Conservation laws
Euler hydrodynamics
total mass M =∑N
i mi .
total linear momentum
d
dt
N∑i
mivi =N∑i
midvidt
=N∑i
N∑j
−mimj
(pi
ρ2i
+pj
ρ2j
)∂w
∂rijeij = 0.
(60)
total angular momentum
d
dt
N∑i
ri ×mivi =N∑i
mi
(ri ×
dvidt
)(61)
=N∑i
N∑j
−mimj
(pi
ρ2i
+pj
ρ2j
)∂w
∂rij(ri × eij) = 0. (62)
Similarly for the viscous forces.37 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
38 / 50
Errors in density estimate: kernel error
Recall the kernel approximation
ρk(r) =
∫ρ(r′)w(r − r′, h)dr′, (63)
Expanding ρ(r ′) by Taylor series around r
ρk(r) = ρ(r)
∫w(r − r′, h)dr′ +∇ρ(r) ·
∫(r′ − r)w(r − r′, h)dr′
+∇α∇βρ(r)
∫δr′αδrβw(r − r′, h)dr′ + O(h3). (64)
Recall∫
w(r − r′, h)dr′ = 1 and odd terms vanish due to symmetric w ,
ρ(r) = ρk(r) + O(h2). (65)
39 / 50
Errors for a function f : kernel error and summation error
Similarly as for density estimate:
f (r) = fk(r) + O(h2). (66)
Recall the particle approximation or summation form
fk(ri ) ≈ fs(ri ) =
Nneigh∑j
mj
ρjfjw(ri − rj , h). (67)
Let us do Taylor series on f (rj) around ri
fs(ri ) = fi
Nneigh∑j
mj
ρjw(rij , h) +∇fi ·
Nneigh∑j
rjimj
ρjw(rij , h) + O(h2). (68)
To have error of O(h2), we need
N∑j
mj
ρiw(rij) = 1,
N∑j
rjimj
ρiw(rij) = 0, (69)
which is not guranteed in practice (depends on configurations, ∆x , and h).40 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
41 / 50
Challenges
error analysis due to particle configurations
consistency and conservation at the same time
convergence for a practical purpose
coarse-graining from molecular dynamics
42 / 50
Outline
1 Backgroundhydrodynamic equationsnumerical methods
2 Mathematics of smoothed particle hydrodynamicssome facts and basic mathematicskernel and particle approximations of a functionfirst and second derivatives
3 Particles for hydrodynamicscontinuity and pressure forceviscous force
4 Classical mechanics for particles ⇒ hydrodynamicsdensity estimateequations of motion
5 Numerical errors
6 Research challenges
7 A short excursion to other particle methods
43 / 50
Algorithmic similarity: pairwise forces within short range rc
in a nutshell, ∀ particle i in SPH, SDPD, DPD, or MD, the EoM:
vi =∑j 6=i
(FCij + FD
ij + FRij
)(70)
options for different components
weighting kernel or potential gradient in MDequation of statedensity fieldthermal fluctuationscanonical ensemble / NVT: thermostat... ...
SPH: (author?) [14]
SDPD: (author?) [4]
DPD: (author?) [10]; (author?) [5]; (author?) [9]
MD: (author?) [1]; (author?) [7]; (author?) [6]; (author?) [18]
44 / 50
academic toy: several interesting elements
∼ 12,000 SDPD particles (author?) [3]
hypnotized?
45 / 50
academic toy: several interesting elements
∼ 12,000 SDPD particles (author?) [3]
hypnotized?
46 / 50
References I
[1] M. P. Allen and D. J. Tildesley. Computer simulation of liquids.Clarendon Press, Oxford, Oct. 1989.
[2] G. K. Batchelor. An introduction to fluid dynamics. CambridgeUniversity Press, Cambridge, 1967.
[3] X. Bian, S. Litvinov, R. Qian, M. Ellero, and N. A. Adams.Multiscale modeling of particle in suspension with smootheddissipative particle dynamics. Phys. Fluids, 24(1), 2012.
[4] P. Espanol and M. Revenga. Smoothed dissipative particle dynamics.Phys. Rev. E, 67(2):026705, 2003.
[5] P. Espanol and P. Warren. Statistical mechanics of dissipative particledynamics. Europhys. Lett., 30(4):191–196, May 1995.
[6] D. J. Evans and G. Morriss. Statistical Mechanics of nonequilibriumliquids. Cambridge University Press, second edition, 2008.
47 / 50
References II
[7] D. Frenkel and B. Smit. Understanding molecular simulation: fromalgorithms to Applications. Academic Press, a division of Harcourt,Inc., 2002.
[8] R. A. Gingold and J. J. Monaghan. Smoothed particlehydrodynamics: theory and application to non-spherical stars. Mon.Not. R. Astron. Soc., 181:375–389, 1977.
[9] R. D. Groot and P. B. Warren. Dissipative particle dynamics:Bridging the gap between atomistic and mesoscopic simulation. J.Chem. Phys., 107(11):4423–4435, 1997.
[10] P. J. Hoogerbrugge and J. M. V. A. Koelman. Simulating microsopichydrodynamics phenomena with dissipative particle dynamics.Europhys. Lett., 19(3):155–160, 1992.
[11] X. Y. Hu and N. A. Adams. A multi-phase SPH method formacroscopic and mesoscopic flows. J. Comput. Phys.,213(2):844–861, 2006.
48 / 50
References III
[12] L. B. Lucy. A numerical approach to the testing of the fissionhypothesis. Astron. J., 82:1013–1024, 1977.
[13] J. J. Monaghan. Simulating free surface flows with SPH. J. Comput.Phys., 110:399–406, 1994.
[14] J. J. Monaghan. Smoothed particle hydrodynamics. Rep. Prog.Phys., 68(8):1703 – 1759, 2005.
[15] J. J. Monaghan and J. C. Lattanzio. A refined particle method forastrophysical problems. Astron. Astrophys., 149:135–143, 1985.
[16] J. P. Morris, P. J. Fox, and Y. Zhu. Modeling low Reynolds numberincompressible flows using SPH. J. Comput. Phys., 136(1):214 – 226,1997.
[17] D. J. Price. Smoothed particle hydrodynamics andmagnetohydrodynamics. J. Comput. Phys., 231:759 – 794, FEB 12012.
49 / 50
References IV
[18] Mark E. Tuckerman. statistical mechanics: theory and molecularsimulation. Oxford University Press, 2010.
[19] Holger Wendland. Piecewise polynomial, positive definite andcompactly supported radial functions of minimal degree. Advances inComputational Mathematics, 4(1):389–396, 1995.
50 / 50