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