Université de Strasbourg Strasbourg, 2010...

Pressure laws and fast Legendre transform

Philippe HELLUY, Hélène MATHIS

Université de Strasbourg

Strasbourg, 2010 April

Hélène MATHIS (Université de Strasbourg) Pressure laws and fast Legendre transform April 14, 2010 1 / 34

Motivations

Euler equations for a compressible fluid

∂tρ +∂x(ρu) = 0,

∂t(ρu)+∂x(ρu2 +π) = 0,

∂t(ρQ)+∂x((ρQ+π)u) = 0.

The pressure is related to e = Q−u2/2 and τ = 1/ρ by a pressure lawπ = p(τ,e).It has to satisfy some convexity properties in order that the Eulersystem is stable (hyperbolic).

General framework for constructing pressure laws of fluid mixtures withphase transition?


Outlines

1 Legendre transform

2 Mixtures thermodynamics

3 Fast Legendre Transform

4 Numerical experiments


Outlines






Legendre transform

Let f : Rn → R∪{+∞}.In the following, we always suppose that

1 dom f := {x, f (x) < +∞} 6= /0;2 f is minorized by an affine function.

The conjugate f ∗ of f is defined by

f ∗(s) = supx

(s · x− f (x)) , (1)

and the transformation f 7→ f ∗ is called the Legendre transform.

The conjugate f ∗ also satisfies the conditions 1 and 2.


Graphical interpretation


Example: Conjugate of a regular convex function

If f : Rn → R is C2 strictly convex and

f (x)‖x‖

→‖x‖→∞

∞, (2)

thenf ∗(s) = sx− f (x), with ∇ f (x) = s.

We deduce that∇ f ∗(s) = x ⇔ ∇ f (x) = s.

It is then easy to check that in this case the Legendre transform isinvolutive

f ∗∗ = f .


Example: enthalpyIn thermodynamics, the energy E is a function of the volume V and theentropy S.The temperature and the pressure are defined by

T = ∂SE p =−∂V E

Thermodynamics first principle: T dS = dE + pdV .The partial Legendre transform of E with respect to V is formally

E∗,V = V ′V −E, with V ′ = ∂V E =−p.

ThusE∗,V =−(E + pV ) =−H

is the opposite of the enthalpy (expressed as a function of (−p,S)).And we have the relation

T dS = dH−V d p.


Main properties [HL01]

• f ∗ is convex and lower semi-continuous (lsc).• f ∗∗ = co( f ) is the lsc convex enveloppe (or convex hull) of f

∀x ∈ Rn, co( f )(x) = sup{g(x),g affine function, g ≤ f}.

• The Legendre transform satisfies : max( f ∗,g∗) = (comin( f ,g))∗.• For two convex functions f and g, we define the inf-convolution

operation byf �g(x) = inf

y( f (y)+g(x− y)) , (3)

then( f �g)∗ = f ∗+g∗.


Outlines






Mixtures thermodynamics [Cal85]

We consider two fluids (i), i = 1,2 with volume Vi ≥ 0, mass Mi ≥ 0 andentropy Si ≥ 0. Each fluid is characterized by its energy function

Ei : (Vi,Mi,Si) 7→ Ei(Vi,Mi,Si).

We extend Ei by +∞ outside C = (R+)3. We suppose that Ei is convex,lsc and “extensive”, i.e.

∀λ ≥ 0,∀W ∈ R3,Ei(λW ) = λEi(W ).


Example: perfect gas

The perfect gas energy is

E = V(

MV

)γ

exp(

SCvM

)(4)

where γ > 1 is the polytropic exponent and Cv > 0 is the specific heat atconstant volume.


Mixture equilibrium

The mixture energy is

E(W1,W2) = E1(W1)+E2(W2).

At equilibrium the mixture achieves a minimum of the energy.The mixture volume, mass and entropy are noted respectively V , Mand S. We also note W = (M,V,S).Because of mass conservation,

M = M1 +M2.

The entropy being an additive variable,

S = S1 +S2.

For the volume, two cases are possible depending on the miscibility ofthe mixture


Non-miscible case

We setV = V1 +V2,

thusW = W1 +W2.

The mixture equilibrium energy is then given by an inf-convolutionoperation

E(W ) = minW1∈R3

E1(W1)+E2(W −W1) = E1�E2.


Miscible case

We haveV = V1 = V2,

andZi = (Mi,Si), Z = (M,S) = Z1 +Z2.

The mixture equilibrium energy is also given by an inf-convolutionoperation

E(V,Z) = minZ1∈R2

E1(V,Z1)+E2(V,Z−Z1),

E(V, ·) = E1(V, ·)�E2(V, ·).


Intensive variables

The function α : W 7→ α(W ) is “intensive” if

∀λ ≥ 0,α(λW ) = α(W ).

We introduce the following intensive quantities• specific volume τi = 1

ρi= Vi

Mi, density ρi = Mi

Vi,

• specific entropy si = Si/Mi, volumic entropy σi = Si/Vi = ρisi.


Intensive energies

We also introduce two intensive energy functions defined by• specific energy

ei(τi,si) =1

MiEi(Vi,Mi,Si) = Ei(

Vi

Mi,1,

Si

Mi) = Ei(τi,1,si),

• volumic energy

εi(ρi,σi) =1Vi

Ei(Vi,Mi,Si) = Ei(1,Mi

Vi,

Si

Vi) = Ei(1,ρi,σi).

The intensive energy functions are convex lsc and

εi(ρi,σi) = ρiei(1ρi

,σi

ρi).


Non-miscible case

For a non-miscible mixture, the intensive equilibrium energies aregiven by

e = comin(e1,e2)

andε = comin(ε1,ε2).


Miscible case

For a miscible mixture, the equilibrium volumic energy is given by

ε = ε1�ε2.


Conjugate of the volumic energy

We also havedε = µdρ +T dσ ,

which implies that ∂ρε = µ and ∂σ ε = T . The Legendre transform of ε

is thusε∗ = µρ +T σ − ε (5)

considered as a function of (µ,T ). From the expression of µ above wededuce that the Legendre transform of the volumic energy is thepressure

ε∗(µ,T ) = p(µ,T ).


Physical interpretation

• non-miscible case:

ε∗ = (comin(ε1,ε2))∗ = max(ε∗1 ,ε∗2 ),

p(µ,T ) = max(p1(µ,T ), p2(µ,T )).

The coexistence of the two phases corresponds top(µ,T ) = p1(µ,T ) = p2(µ,T ) (saturation curve).

• miscible case:

ε∗ = (ε1�ε2)∗ = ε

∗1 + ε

∗2 ,

p(µ,T ) = p1(µ,T )+ p2(µ,T ).

The pressure is the sum of the partial pressures (Dalton’s law).

→ (max,+) analysis [Mas87]


Outlines






Discrete conjugate: 1D case

Let f : [a,b]→ R. We extend f by +∞ outside [a,b].We consider a subdivision of [a,b] with N +1 points

a = x0 < x1 < · · ·< xN = b.

We suppose that f is piecewise linear

f (xi) = yi

f (x) = yi + si(x− xi), x ∈ [xi,xi+1], i = 0 · · ·N−1. (6)


Fast algorithm [Luc97]

We know that f ∗ = (co f )∗, thus we first compute the convex hull of f ,which is still piecewise linear on [a,b] (and continuous). Without loss ofgenerality, we can thus suppose that the sequence of slopes(si)i=0···N−1 is strictly increasing. Then

f ∗(s) =

as− f (a) if s < s0xis− f (xi) if si−1 < s < si, i = 1 · · ·N−1bs− f (b) if s > sN−1

The global algorithm to compute N′ values of f ∗ has a O(N +N′)complexity.We observe that f ∗ is linear outside [s0,sN−1].The algorithm to compute the Legendre transform of a functionextended by affine maps is very similar.


2D version

Let f : [a,b]× [c,d]→R. We extend f by +∞ (or by linear maps) outside[a,b]× [c,d].

f ∗(s, t) = supx,y

(sx+ ty− f (x,y)) = supy

(ty+ supx

(sx− f (x,y))

= supy

(ty+ f ∗,x(s,y))

= (− f ∗,x)∗,y . (7)

In practice, we apply the 1D algorithm on all the rows and then all thecolumns of the samples.


Outlines






Maxwell equal area ruleWe consider a van der Waals gas with dimensionless specific energy

e(τ,s) =

{es

(3τ−1)8/3 − 3τ

if τ > 1/3,s > 0,

+∞ elsewhere.

This energy is not convex.Therefore we replace it by

e∗∗ = coe.

⇔ Maxwell equal area ruleconstruction.


Numerical illustration

It is classical to plot the isotherms in the (p,τ) plane.In practice, it is very easy using the FLT.First we compute

f (τ,T ) := e∗,s(τ,T ) = T s− e(τ,s),

with T = ∂se.Then we fix T and plot p = ∂τ f for a varying τ.


Isotherms of e

5e+06

1e+07

1.5e+07

2e+07

2.5e+07

3e+07

3.5e+07

4e+07

4.5e+07

5e+07

5e-05 0.0001 0.00015 0.0002 0.00025 0.0003

Pre

ssur

e

Specific volume

Isotherms


Isotherms of e∗∗


A simple non-miscible mixture

We consider a mixture of two perfect gases (γ1 > γ2)

ei(τi,si) = exp(si)τ1−γii

It is possible to compute analytically e(τ,s) = comin(e1,e2) and p(τ,s)[1].

The saturation curve is simply

p = κT

(κ = κ(γ1,γ2) has a complicatedexpression).


IsothermsWe simply set e = min(e1,e2) and the isotherms.

7.5

8

8.5

9

9.5

10

10.5

11

11.5

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

Pre

ssur

e

Specific volume

Isotherms


A simple miscible mixtureWe take the same perfect gases and now construct the misciblemixture

ε = ε1�ε2

The isotherms are very similar to those of a perfect gas

2

4

6

8

10

12

0.2 0.4 0.6 0.8 1 1.2 1.4

Pre

ssur

e

Specific volume

Isotherms

T = 6.47 T = 6.95 T = 7.43


Mass fractionThe mass fraction ϕ of the gas (1) reads :

ϕ =

∂ p∂ µ

(µ,T )

∂ p1

∂ µ(µ,T )

.

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

2 4 6 8 10 12

Phi

Pressure

Mass fraction

T = 6.47 T = 6.95 T = 7.43


Conclusion

• We have found a natural mathematical framework for studyingEOS of mixtures.

• The FLT is a useful numerical tool for practically computetabularized EOS of mixtures.

Next steps:• real EOS for more than two fluids,• coupling with CFD computations,• use the ideas of idempotent analysis for more general mixtures.


H. B. Callen. Thermodynamics and an introduction tothermostatistics, second edition. Wiley and Sons, 1985.

P. Helluy, H. Mathis. Pressure laws and fast Legendre transform.http://hal.archives-ouvertes.fr/hal-00424061/fr/

J.-B. Hiriart-Urruty and C. Lemaréchal. Fundamentals of convexanalysis. Grundlehren Text Editions. Springer-Verlag, Berlin, 2001.

JAOUEN, S., Etude mathématique et numérique de stabilité pourdes modèles hydrodynamiques avec transition de phase, PhDThesis, 2001.

Y. Lucet. Faster than the fast Legendre transform, the linear-timeLegendre transform. Numer. Algorithms 16 (1997), no. 2, 171–185(1998).

V. Maslov. Méthodes opératorielles. Éditions Mir, Moscou, 1987.


Appendix


Proof

E∗i (V ′,M′,S′) = supV,M,S≥0VV ′+MM′+SS′−Ei(V,M,S) (8)

= supV,ρ,σ≥0V (V ′+ρM′+σS′− εi(ρ,σ)) (9)= supV≥0V (V ′+ ε∗i (M′,S′)) (10)

={

0 if ε∗i (M′,S′)≤−V ′,+∞ else.

(11)

We introduce Ai = {(V ′,M′,S′),ε∗i (M′,S′)≤−V ′}.Then E∗

i is the convex indicator of Ai.Then, E∗ = E∗

1 +E∗2 is the convex indicator of

A1∩A2 = {(V ′,M′,S′),max(ε∗1 (M′,S′),ε∗2 (M′,S′)

)≤−V ′}

We deduce that ε∗ = max(ε∗1 ,ε∗2 ) and then ε = comin(ε1,ε2).�


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