christophe berthon, matthieu de leffe, victor...
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A well-balanced SHP schemeChristophe Berthon, Matthieu de Leffe, Victor Michel-Dansac
HydrOcean and NEXTFLOW Software
Created in 2007 by Erwan Jacquin
Main objectives : naval, offshore, sailing and marineenergies
Strong collaboration with LHEEA
About twenty employees, essentially engineers and PhD
Working with DCNS, Total, Michelin (for instance)
SPH-Flow codeBased on the SPH method
Developped in collaboration with LHEEA
Meshless Lagrangian method
Explicit scheme, weakly compressible flows
free surface flows
Navier-Stokes equations
8>>>>>>>><
>>>>>>>>:
@⇢
@t+r · (⇢u) = 0
@⇢u
@t+r ·
⇣⇢u⌦ u+ p ¯̄I
⌘�r · ¯̄⌧ = ⇢g
@⇢E
@t+r · ((⇢E + p)u)�r · (¯̄⌧ · u) = ⇢g · u
withgravity source termviscosity tensor ¯̄⌧
Pressure law in the form: p = ⇢� �B
Particule reformulations (basic idea)
Reformulation of a function
⇧(f)(x) = (f ⇤ �)(x) =Z
Rdf(y)�(x� y) dy = f(x)
The kernel
1 even function2 compact support
3
Z
K
W (x� y, h) dy = 1
4
Z
K
rW (x�y, h) dy = 0
Approximate particule reformulationApproximate reformulation of a fonction
⇧(f)(x) = (f ⇤ �)(x) =Z
Rdf(y)�(x� y) dy = f(x)
⇧h(f)(x) = (f ⇤W )(x) =
Z
Rdf(y)W (x� y, h) dy
Approximate reformulation of a gradient
Z
K
rf(y)W (x� y, h) dy =
Z
K
f(y)rW (x� y, h) dy
r⇧h(f)(x) =
Z
Rdf(y)rW (x� y, h) dy
Particule approximation methodBased on quadrature formulae
Z
Rdf(y) dy '
X
j2P!(x
j
)f(xj
) =X
j2P!j
fj
⇧h(f)(x) =
Z
K
f(y)W (x� y, h) dy �! ⇧h(f)i
=X
j2P!j
fj
Wij
Main difficultiesX
j2P!j
Wij
6= 1
X
j2P!j
rWij
6= 0 andX
j2P!j
xkj
rWij
6= 1k, k 2 J1, dK
ImprovementsDerivation improvement: exact for constant functions
Dh
(f)i
= r⇧h(f)i
� fi
r⇧h(1)i
=X
j2P!j
(fj
� fi
)rWij,i
Variable supports of the kernel
h = h(x) W (xi
� x
j
, h) = W (xi
� x
j
, hi
) = Wij,i
⇧h(f)i
=X
j2P!j
fj
Wij,i
r⇧h(f)i
=X
j2P!j
fj
rWij,i
Associated weak formulationDiscret scalar product: hf, gi
h
=X
j2P!j
fj
gj
Adjoint of Dh
hDh
(f), gih
= �hf,D⇤h
(g)ih
to get
D⇤h
(f)i
=X
j2P!j
(fj
rWij,j
+ fi
rWij,i
)
this formulation is conservative:X
i2P!i
D⇤h
(f)i
= 0
Problèmes de cette formulationGlobal consistance, but lost of the local consistance with thestrong operator
Consequences
PEPS AMIES
SPHINXSPH Improvement for Numerical approXimations
Objectives
To get a consistant weak operateur useful for industrialapplicationsobtention d’un opérateur faible localement consistant
Suggested approach
To modify the weak operator to be consistant and to preservethe duality relation strong / weak operator
A new weak operator
Based on the Lacôme approach: weak formulation with avariable h
⇧h
weak
(f)(x) =Z
Rdf(y)
✓1 +
x� y
h(y).rh(y)
◆W (x� y, h(y)) dy
From weak to strong to get a approximate gradient
r⇧h
strong
(f)(x) =
Z
Rdf(y)⇠strong (x� y, h(x)) dy
where ⇠strong is consistent with rWThe definition of ⇠strong is long and contains rh
Modified formulation
Modification of the strong formulation to preserve the derivationof constant functions
Dh
strong
(rf)(x) = r⇧h
g
(f)(x)� f(x)r⇧h
g
(1)(x)
=
Z
Rd(f(y)� f(x)) ⇠strong (x� y, h(x)) dy
From strong to weak
Dh
weak
(rf)(x) =
Z
Rd
hf(y)⇠weak (x� y, h(y)) + f(x)⇠strong (x� y, h(x))
idy
où ⇠
weak (x� y, h(y)) = �⇠
strong (y � x, h(y))
Collaboration continuation:Well-balanced SPH-scheme for shallow-water
1D model⇢
@t
h+ @x
(hu) = 0,@t
(hu) + @x
(hu2 + g
2h2) = �hg@
x
Z,
SPH-discretization gives8>>>><
>>>>:
hn+1i
� hni
�t+
X
j2P2!
j
(hu)ij
W 0ij
= 0
hn+1i
un+1i
� hni
uni
�t+
X
j2P2!
j
✓hu2 +
1
2gh2
◆
ij
W 0ij
= si
.
si
is a discretization of the source termFij
finite volume hybridation
Reformulation of the modelMain idea
Notations: H = h+ Z and X = h/H
Consider the free surface H instead of h in the numericalflux functionInitial data for the lake at rest: H = cste and u = 0
Main reformulation8<
:
@t
h+ @x
XHu = 0
@t
hu+ @x
⇣X
�Hu2 + gH2/2
�� g
2hZ
⌘+ gh@
x
Z = 0
Set W =
✓HHu
◆to write
@t
w + @x
✓Xf(W )�
✓0
g
2hZ
◆◆+
✓0
gh@x
Z
◆
Remark:
SPH discretization
To rewrite the scheme for the shallow-water equations using thereformulation8>>>>>>>><
>>>>>>>>:
hn+1i
� hni
�t+
X
j2P2!
j
Xij
(Hu)ij
W 0ij
= 0
hn+1i
un+1i
� hni
uni
�t+
X
j2P2!
j
Xij
✓Hu2 +
1
2gH2
◆
ij
W 0ij
=
⇣g2@x
(hZ)� gh@x
Z⌘
i
Source term discretization
To get a well-balanced and conservative scheme, we adopt thefollowing consistent source term discretization:
⇣g2@x
(hZ)� gh@x
Z⌘
i
=g
2
X
j2P2!
j
Xij
H2ij
(1�Xij
)W 0ij
�
gX̄i
H̄i
X
j2P2!
j
(1�Xij
)W 0ij
+
2gH̄2i
X̄i
(1� X̃i
)X
j2P!j
W 0ij
averages Hij
, H̄i
, X̄i
, X̃i
and Xij
still need to be determined
Main result
Assume
Hij
= H̄i
= H, as soon as Hi
= Hj
= H
X̄i
=1
2
Pj2P !
j
X2ij
W 0ijP
j2P !j
(Xij
� 1)W 0ij
+ (X̃i
� 1)P
j2P !j
W 0ij
The scheme defined by the SPH hybridization and sourceterm discretization preserves the lake at rest, for anyexpression of X
ij
and X̃i
consistent with X.The scheme is conservative
X̄i
is consistent with X = h/H
Thank you for your attention
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