anisotropic hetergeneous n-d heat equation with boundary … · 2019-07-05 · thermodynamic...
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Thermodynamic Foundation of Mathematical Systems Theory - 3rd TFMST 2019
Anisotropic hetergeneous n-D heat equation with boundarycontrol and observation as port-Hamiltonian system:
— Modeling & Discretization —
Denis [email protected]
Universite de Toulouse, ISAE-SUPAERO, FranceJoint work with: Anass SERHANI and Ghislain HAINE
Louvain-la-Neuve, Belgium. July 4th, 2019.D. Matignon Heat as pHs: Modeling & Discretization 1 / 37
Introduction
Introduction
1 Modeling:Mathematical model: the Hamiltonian is a quadratic functional.1st thermodynamical model:the Hamiltonian is the entropy S of the system.2nd thermodynamical model:the Hamiltonian is the internal energy U of the system.
2 Discretization:Use a Structure-Preserving Discretization taking advantage of theFinite Element Method: PFEM (”Partitioned Finite ElementMethod”, see [Cardoso-Ribeiro, Matignon, Lefevre, in proc. IFACLHMNLC’2018]).Perform simulation in an object-oriented environment, such asPython/FEniCS.
D. Matignon Heat as pHs: Modeling & Discretization 2 / 37
Overview
1 Thermodynamical hypotheses
2 Lyapunov functional as Hamiltonian
3 Entropy as Hamiltonian
4 Energy as Hamiltonian
Overview
1 Thermodynamical hypotheses
2 Lyapunov functional as Hamiltonian
3 Entropy as Hamiltonian
4 Energy as Hamiltonian
Thermodynamical hypotheses
Thermodynamical hypotheses 1/2 Notations
Spatial domain and physical parameters:Ω ⊂ Rn≥1, a bounded open connected set.−→n , the outward unit normal on the boundary ∂Ω.ρ(x), the mass density.λ(x), the conductivity tensor.
Notations:T, the local temperature.β := 1
T , the reciprocal temperature.u, the internal energy density.s, the entropy density.−→J Q, the heat flux.−→J S := β
−→J Q, the entropy flux.
CV :=(
dudT
)V, the isochoric heat capacity.
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Thermodynamical hypotheses
Thermodynamical hypotheses 2/2 hypotheses
Assumption: Constant volume and no chemical reaction.1st law of thermodynamics:
ρ(x) ∂tu(t, x) = −div(−→
J Q(t, x)).
Gibbs’ relation:
dU = T dS =⇒ ∂tu(t, x) = T(t, x) ∂ts(t, x).
Entropy evolution:
ρ(x) ∂ts(t, x) = −div(−→
J S(t, x))
+ σ(t, x),
with σ(t, x) :=−−−→grad(β) · −→J Q, the irreversible entropy production.
Fourier’s law:−→J Q(t, x) = −λ(x) ·
−−−→grad (T(t, x)) .
Dulong-Petit’s law: u(t, x) = CV(x) T(t, x) .D. Matignon Heat as pHs: Modeling & Discretization 5 / 37
Overview
1 Thermodynamical hypotheses
2 Lyapunov functional as HamiltonianModelingDiscretizationSimulation
3 Entropy as Hamiltonian
4 Energy as Hamiltonian
Lyapunov functional as Hamiltonian Modeling
Lyapunov functional - Modeling 1/3 quadratic Hamiltonian
Consider the quadratic Hamiltonian:
H(t) := 12
∫Ωρ(x)(u(t, x))2
CV(t, x) dx
Choose u as energy variable, and compute the co-energy variableδuH = u
CV, as variational derivative in L2
ρ.
dtH(t) =∫
Ωρ(x)∂tu(t, x) u(t, x)
CV(t, x) dx− 12
∫Ωρ(x)∂tCV(t, x) (u(t, x))2
(CV(t, x))2 dx,
= −∫
Ωdiv(−→
J Q(t, x)) u(t, x)
CV(t, x) dx− 12
∫Ωρ(x)∂tCV(t, x)
(u(t, x))2
(CV(t, x))2dx,
=∫
Ω
−→J Q(t, x) ·
−−−→grad
(u(t, x)
CV(t, x)
)dx−
∫∂Ω
u(t, γ)CV(t, γ)
−→J Q(t, γ) · −→n (γ) dγ
−12
∫Ωρ(x)∂tCV(t, x) (u(t, x))2
(CV(t, x))2 dx.
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Lyapunov functional as Hamiltonian Modeling
Lyapunov functional - Modeling 2/3 port-Hamiltonian system
Defining as flows and efforts:
fu := ∂tu, eu := uCV−→
f Q := −−−−→grad
(u
CV
), −→e Q := −→J Q
Port-Hamiltonian system:(ρfu−→f Q
)=(
0 −div−−−−→grad 0
)(eu−→e Q
).
Particular choices for boundary control v∂ :either the Dirichlet trace of eu (temperature with Dulong-Petit),or normal trace of −−→e Q (inward heat flux).
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Lyapunov functional as Hamiltonian Modeling
Lyapunov functional - Modeling 3/3 power balance
Constitutive relations:Dulong-Petit’s law =⇒ eu = T and
−→f Q = −
−−−→grad(T)
Fourier’s law =⇒ −→e Q = −λ ·−−−→grad(T) =⇒ −→e Q = λ ·
−→f Q
Power balance:
dtH(t) = −∫
Ω
−→e Q(t, x) ·−→f Q(t, x) dx +
∫∂Ω
v∂(t, γ) y∂(t, γ) dγ
− 12
∫Ωρ(x)∂tCV(t, x)(eu(t, x))2 dx
with the constitutive relations (CV(x) only):
dtH(t) = −∫
Ω
−→f Q(t, x) · λ ·
−→f Q(t, x) dx︸ ︷︷ ︸
dissipation
+∫∂Ω
v∂(t, γ) y∂(t, γ) dγ︸ ︷︷ ︸supplied power
D. Matignon Heat as pHs: Modeling & Discretization 8 / 37
Lyapunov functional as Hamiltonian Discretization
1 Thermodynamical hypotheses
2 Lyapunov functional as HamiltonianModelingDiscretizationSimulation
3 Entropy as HamiltonianModelingDiscretizationSimulation
4 Energy as HamiltonianModelingDiscretizationSimulation
D. Matignon Heat as pHs: Modeling & Discretization 9 / 37
Lyapunov functional as Hamiltonian Discretization
Lyapunov functional - Discretization 1/4 weak form
1 Weak form: Taking arbitrary test functions ϕ and −→ϕ ∫Ω ρfuϕ dx = −
∫Ω div
(−→e Q)ϕ dx,∫
Ω−→f Q · −→ϕ dx = −
∫Ω−−−→grad (eu) · −→ϕ dx.
For instance for the control of the inward heat flux −−→e Q · −→n = v∂2 Apply Green’s formula: ∫
Ω ρfuϕ dx =∫Ω−→e Q ·
−−−→gradϕ dx +
∫∂Ω v∂ ϕ dγ,∫
Ω−→f Q · −→ϕ dx = −
∫Ω−−−→grad (eu) · −→ϕ dx.
D. Matignon Heat as pHs: Modeling & Discretization 10 / 37
Lyapunov functional as Hamiltonian Discretization
Lyapunov functional - Discretization 2/4 approximation bases
Finite-dimensional bases:
X := spanΦ := span(ϕi)1≤i≤N,X := span
−→Φ := span(−→ϕ k)1≤k≤−→N ,X∂ := spanΨ := span(ψm
∂ )1≤m≤N∂,
Approximation:
fu(t, x) ' f du (t, x) := Φ>(x) · fu(t) :=
∑Ni=1 f i
u(t)ϕi(x)−→f Q(t, x) '
−→f d
Q(t, x) := −→Φ>(x) ·−→f Q(t) :=
∑−→Nk=1−→f k
Q(t)−→ϕ k(x)v∂(t, γ) ' vd
∂(t, γ) := ψ>∂ (γ) · v∂(t) :=∑N∂
m=1 vm∂ (t)ψm
∂ (x)
and similarly for the other variables.
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Lyapunov functional as Hamiltonian Discretization
Lyapunov functional - Discretization 3/4 matrix form
The discrete weak formulation on X ×X ×X∂ reads:Mρ fu(t) = D eQ(t) + B v∂(t),−→M fQ(t) = −D> eu(t),M∂ y∂(t) = B> eu(t),
with the following sparse matrices:
Mρ :=∫
Ω Φ · Φ>ρ dx ∈ RN×N ,−→M :=
∫Ω−→Φ · −→Φ> dx ∈ R
−→N×−→N ,
D :=∫Ω−−−→grad (Φ) · −→Φ> dx ∈ RN×−→N , B :=
∫∂Ω Φ ·Ψ> dγ ∈ RN×N∂
M∂ :=∫∂Ω Ψ ·Ψ> dγ ∈ RN∂×N∂ .
Compact form:Mρ 0 00−→M 0
0 0 M∂
︸ ︷︷ ︸
Md
fu(t)−→f Q(t)−y∂(t)
︸ ︷︷ ︸
−→f d
=
0 D B−D> 0 0−B> 0 0
︸ ︷︷ ︸
Jd
eu(t)−→e Q(t)v∂(t)
︸ ︷︷ ︸
−→e d
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Lyapunov functional as Hamiltonian Discretization
Lyapunov functional - Discretization 4/4 structure
Structure-preservation: −→e >d Md−→f d = 0 (thanks to Jd = −J ∗d )
Discrete closure relations:Dulong-Petit’s law:∫
ΩρCV ∂teu ϕ dx =
∫Ωρ fu ϕ dx =⇒ MρCV
ddt
eu = Mρ fu,
Fourier’s law:∫Ω
−→e Q · −→ϕ dx =∫
Ω(λ ·−→f Q) · −→ϕ dx =⇒ −→
MeQ = −→Λ fQ
Discrete Hamiltonian:
Hd(t) = 12
eu>(t) MρCV eu(t)
Discrete power balance:
dtHd(t) = −−→f >Q (t)−→Λ
−→f Q(t)︸ ︷︷ ︸
discrete dissipation
+ v∂>(t) M∂ y∂(t)︸ ︷︷ ︸discrete supplied power
D. Matignon Heat as pHs: Modeling & Discretization 13 / 37
Lyapunov functional as Hamiltonian Simulation
1 Thermodynamical hypotheses
2 Lyapunov functional as HamiltonianModelingDiscretizationSimulation
3 Entropy as HamiltonianModelingDiscretizationSimulation
4 Energy as HamiltonianModelingDiscretizationSimulation
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Lyapunov functional as Hamiltonian Simulation
Lyapunov functional - Simulation 1/2
Domain: Ω = (0,Lx)× (0,Ly) with Lx = 2 and Ly = 1.Finite element: X ×X ×X∂ = P1 × RT0× γ0 (P1)
ρ(x) := x1(2−x1)+1, CV = 3, λ(x) =
5 + x1x2 (x1 − x2)2
(x1 − x2)2 3 + x2
x1 + 1
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Lyapunov functional as Hamiltonian Simulation
Lyapunov functional - Simulation 2/2 time-space simulation
D. Matignon Heat as pHs: Modeling & Discretization 16 / 37
Lyapunov functional as Hamiltonian Simulation
What about... thermodynamics?
1 Read Distributed port-Hamiltonian modelling for irreversibleprocesses, by W. Zhou, B. Hamroun, F. Couenne, Y. Le Gorrec.in Mathematical and Computer Modelling of Dynamical Systems,vol. 23, n. 1, 2017.
2 Have the chance to talk with Francoise Couenne (LAGEP).
D. Matignon Heat as pHs: Modeling & Discretization 17 / 37
Overview
1 Thermodynamical hypotheses
2 Lyapunov functional as Hamiltonian
3 Entropy as HamiltonianModelingDiscretizationSimulation
4 Energy as Hamiltonian
Entropy as Hamiltonian Modeling
Entropy functional - Modeling 1/3 Hamiltonian Entropy
Consider the entropy of the system as Hamiltonian:
S(t) :=∫
Ωρ(x) s(u(t, x)) dx,
Choose u as energy variable, and compute the co-energy variable
δuS = dsdu
= β.
dtS(t) =∫
Ω ρ(x)∂tu(t, x)β(t, x) dx,=∫
Ω ρ(x)∂ts(t, x) dx,= −
∫Ω div
(−→J S(t, x)
)dx +
∫Ω σ(t, x) dx,
= −∫∂Ω−→J S(t, γ) · −→n (γ) dγ +
∫Ω σ(t, x) dx,
= −∫∂Ω β(t, γ)−→J Q(t, γ) · −→n (γ) dγ +
∫Ω σ(t, x) dx .
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Entropy as Hamiltonian Modeling
Entropy functional - Modeling 2/3 port-Hamiltonian system
Defining as flows and efforts:
fu := ∂tu, eu := β−→f Q := −
−−−→grad (β) , −→e Q := −→J Q
Port-Hamiltonian system:(ρfu−→f Q
)=(
0 −div−−−−→grad 0
)(eu−→e Q
).
Particular choices for boundary control v∂ :either the Dirichlet trace of eu (reciprocal temperature),or normal trace of −−→e Q (inward heat flux).
D. Matignon Heat as pHs: Modeling & Discretization 19 / 37
Entropy as Hamiltonian Modeling
Entropy functional - Modeling 3/3 power balance
Constitutive relations:Dulong-Petit’s law: u = CV
β=⇒ eu = CV
u
Fourier’s law: −→e Q = 1β2λ ·
−−−→grad(β) =⇒ −→e Q = − 1
e2uλ ·−→f Q
Power balance:
dtS(t) = −∫
Ω
−→f Q(t, x) · −→e Q(t, x) dx +
∫∂Ω
v∂(t, γ) y∂(t, γ) dγ.
with the constitutive relations:
dtS(t) =∫
Ωσ(t, x) dx︸ ︷︷ ︸
entropy production
≥ 0
+∫∂Ω
v∂(t, γ) y∂(t, γ) dγ︸ ︷︷ ︸supplied part
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Entropy as Hamiltonian Discretization
Entropy functional - Discretization 1/3 weak form
1 Weak form: Taking arbitrary test functions ϕ and −→ϕ ∫Ω ρfuϕ dx = −
∫Ω div
(−→e Q)ϕ dx,∫
Ω−→f Q · −→ϕ dx = −
∫Ω−−−→grad (eu) · −→ϕ dx.
For instance for the control of the inward heat flux −−→e Q · −→n = v∂2 Apply Green’s formula: ∫
Ω ρfuϕ dx =∫Ω−→e Q ·
−−−→gradϕ dx +
∫∂Ω v∂ ϕ dγ,∫
Ω−→f Q · −→ϕ dx = −
∫Ω−−−→grad (eu) · −→ϕ dx.
D. Matignon Heat as pHs: Modeling & Discretization 21 / 37
Entropy as Hamiltonian Discretization
Entropy functional - Discretization 2/3 matrix form
The discrete system:Mρ fu(t) = D eQ(t) + B v∂(t),−→M fQ(t) = −D> eu(t),M∂ y∂(t) = B> eu(t),
Mρ :=∫
Ω Φ · Φ>ρ dx ∈ RN×N ,−→M :=
∫Ω−→Φ · −→Φ> dx ∈ R
−→N×−→N ,
D :=∫Ω−−−→grad (Φ) · −→Φ> dx ∈ RN×−→N , B :=
∫∂Ω Φ ·Ψ> dγ ∈ RN×N∂
M∂ :=∫∂Ω Ψ ·Ψ> dγ ∈ RN∂×N∂ .
Compact form:Mρ 0 00−→M 0
0 0 M∂
︸ ︷︷ ︸
Md
fu(t)fQ(t)−y∂(t)
︸ ︷︷ ︸
−→f d
=
0 D B−D> 0 0−B> 0 0
︸ ︷︷ ︸
Jd
eu(t)eQ(t)v∂(t)
︸ ︷︷ ︸−→e d
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Entropy as Hamiltonian Discretization
Entropy functional - Discretization 3/3 structure
Structure-preservation: −→e >d Md−→f d = 0 (thanks to Jd = −J ∗d )
Discrete closure relations:Dulong-Petit’s law:∫
Ωρ β ϕ dx =
∫Ωρ
CV
uϕ dx =⇒ (1st Non-linear closure equation),
Fourier’s law:∫Ω
−→e Q·−→ϕ dx = −∫
Ω
1e2
u
−→f Q·λ·−→ϕ dx =⇒ (2nd Non-linear closure eqn.),
Discrete power balance:
dtSd(t) = −−→f >Q (t)−→M−→e Q(t)︸ ︷︷ ︸
discrete entropy production
≥ 0
+ v∂>(t) M∂ y∂(t)︸ ︷︷ ︸discrete supplied power
D. Matignon Heat as pHs: Modeling & Discretization 23 / 37
Entropy as Hamiltonian Simulation
Entropy functional - Simulation 1/
D. Matignon Heat as pHs: Modeling & Discretization 24 / 37
Overview
1 Thermodynamical hypotheses
2 Lyapunov functional as Hamiltonian
3 Entropy as Hamiltonian
4 Energy as HamiltonianModelingDiscretizationSimulation
Energy as Hamiltonian Modeling
Energy functional - Modeling
Consider the internal energy of the system as Hamiltonian:
U(t) :=∫
Ωρ(x) u(s(t, x)) dx,
Choosing s as energy variable, compute the co-energy variable
δsU = duds
= T.
dtU(t) =∫
Ω ρ(x)∂ts(t, x)T(t, x) dx,=∫
Ω ρ(x)∂tu(t, x) dx,= −
∫Ω div
(−→J Q(t, x)
)dx,
= −∫∂Ω−→J Q(t, γ) · −→n (γ) dγ,
= −∫∂Ω T(t, γ)−→J S(t, γ) · −→n (γ) dγ.
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Energy as Hamiltonian Modeling
Energy functional - Modeling 2/3 port-Hamiltonian system
Defining as flows and efforts:
fs := ∂ts, eu := T−→f S := −
−−−→grad (T) , −→e S := −→J S
Primary port-Hamiltonian system:(ρfu−→f Q
)=(
0 −div−−−−→grad 0
)(eu−→e Q
)+(σ0
).
Following [Zhou & al., 2017], we introduced new ports:
fσ = T, eσ = −σ
Port-Hamiltonian system ρfu−→f Q
fσ
=
0 −div −1−−−−→grad 0 01 0 0
eu−→e Q
eσ
.D. Matignon Heat as pHs: Modeling & Discretization 26 / 37
Energy as Hamiltonian Modeling
Entropy functional - Modeling 3/3 power balance
Particular choices for boundary control v∂ :either the Dirichlet trace of es (temperature),or normal trace of −−→e S (inward entropy flux).
Constitutive relations:Dulong-Petit’s law and Gibbs’ relation:CVρ∂tT = Tρ∂ts =⇒ CVρ∂tes = es fsFourier’s law: es
−→e S = λ ·−→f S
Entropy closure relation:−→f S · −→e S + fσ eσ = 0
Power balance:
dtU(t) =∫∂Ω
v∂(t, γ) y∂(t, γ) dγ
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Energy as Hamiltonian Discretization
Entropy functional - Discretization 1/3 weak form
1 Weak form: Taking arbitrary test functions ϕ and −→ϕ∫
Ω ρfs ϕ dx = −∫
Ω div(−→e S
)ϕ dx−
∫Ω σ ϕ dx,∫
Ω−→f S · −→ϕ dx = −
∫Ω−−−→grad (es) · −→ϕ dx,∫
Ω fσ ϕ dx =∫
Ω es ϕ dx.
For instance, the control is the temperature es = v∂2 Apply Green’s formula:
∫Ω ρfs ϕ dx = −
∫Ω div
(−→e S)ϕ dx−
∫Ω σ ϕ dx,∫
Ω−→f S · −→ϕ dx =
∫Ω es div(−→ϕ ) dx,+
∫∂Ω v∂ (−→ϕ · −→n ) dγ∫
Ω fσ ϕ dx =∫Ω es ϕ dx.
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Energy as Hamiltonian Discretization
Energy functional - Discretization 2/3 matrix form
The discrete system:Mρ fs(t) = D eS(t)−M eσ,−→M fS(t) = −D> es(t) + B v∂(t),M fσ(t) = M es(t)M∂ y∂(t) = B> eu(t),
D := −∫Ω div
(−→Φ) Φ> dx ∈ R−→N×N , B :=
∫∂Ω(−→Φ · −→n ) ·Ψ> dγ ∈ R
−→N×N∂
Compact form:Mρ 0 0 00−→M 0 0
0 0 M 00 0 0 M∂
︸ ︷︷ ︸
Md
fs(t)fS(t)fσ(t)−y∂(t)
︸ ︷︷ ︸
−→f d
=
0 D −M 0−D> 0 0 B
M 0 0 00 −B> 0 0
︸ ︷︷ ︸
Jd
es(t)eS(t)eσ(t)v∂(t)
︸ ︷︷ ︸−→e d
D. Matignon Heat as pHs: Modeling & Discretization 29 / 37
Energy as Hamiltonian Discretization
Energy functional - Discretization 3/3 structure
Structure-preservation: −→e >d Md−→f d = 0 (thanks to Jd = −J ∗d )
Discrete closure relation:Dulong-Petit’s law and Gibbs’ relation:
CVρ∂tes = es fs =⇒ (Solve ODE)
Fourier’s law:∫Ω
es−→e S · −→ϕ dx =
∫Ω
−→f S · λ ·ϕ dx =⇒ (1st non-linear closure eqn.),
Entropy closure relation:∫Ω
−→f S ·−→e S ϕ dx = −
∫Ω
fσ eσ ϕ dx =⇒ (2nd non-linear closure eqn.)
Discrete power balance:
dtUd(t) = v∂>(t) M∂ y∂(t)
D. Matignon Heat as pHs: Modeling & Discretization 30 / 37
Energy as Hamiltonian Simulation
Energy functional - Simulation 2/2 time-space simulation
D. Matignon Heat as pHs: Modeling & Discretization 31 / 37
Energy as Hamiltonian Simulation
Energy functional - Simulation 2/2 time-space simulation
D. Matignon Heat as pHs: Modeling & Discretization 32 / 37
Conclusion
Conclusion
The integration by parts of one of the weak-form equationsnaturally leads to skew-symmetric representation with theboundary input/output ports;2D (or 3D) problems are straightforward to address;Interconnection structure and Constitutive relations are discretizedseparately;Same method can be used for other port-Hamiltonian systems(higher-order differential operators like Euler-Bernoulli beam, orKirchhoff-Love plate equations, etc.);Space varying coefficients can be easily taken into account;Nonlinear equations: non-quadratic Hamiltonian and non-linearinterconnection structure;Thermodynamically consistent potentials can be dealt with for theheat equation;PFEM can be easily implemented using available Finite Elementsoftware allowing for complex geometries.
D. Matignon Heat as pHs: Modeling & Discretization 33 / 37
Conclusion
Ongoing work and open questions
Other (mixed) choices of input/output are possible;Ongoing convergence analysis;Numerical methods for DAEs;Multiphysics systems modelling: some useful 2D testcases(fluid-structure interaction (FSI), thermal-structure coupling,fluid-thermal coupling);Design and implementation of control laws.
D. Matignon Heat as pHs: Modeling & Discretization 34 / 37
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