Anisotropic hetergeneous n-D heat equation with boundary … · 2019-07-05 · Thermodynamic Foundation of Mathematical Systems Theory - 3rd TFMST 2019 Anisotropic hetergeneous n-D

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





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.


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


2 Lyapunov functional as HamiltonianModelingDiscretizationSimulation



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.



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).



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


Lyapunov functional as Hamiltonian Discretization



3 Entropy as HamiltonianModelingDiscretizationSimulation

4 Energy as HamiltonianModelingDiscretizationSimulation



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 = −




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.



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



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


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



Lyapunov functional - Simulation 2/2 time-space 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).


Overview





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 .



Entropy functional - Modeling 2/3 port-Hamiltonian system


fu := ∂tu, eu := β−→f Q := −

−−−→grad (β) , −→e Q := −→J Q

Port-Hamiltonian system:(ρfu−→f Q

)=(


)(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).



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


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 = −


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 = −




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



Entropy functional - Discretization 3/3 structure


Discrete closure relations:Dulong-Petit’s law:∫

Ωρ β ϕ dx =

∫Ωρ

CV

uϕ dx =⇒ (1st Non-linear closure equation),


−→e Q·−→ϕ dx = −∫

Ω

1e2

u

−→f Q·λ·−→ϕ dx =⇒ (2nd Non-linear closure eqn.),


dtSd(t) = −−→f >Q (t)−→M−→e Q(t)︸︷︷︸

discrete entropy production

≥ 0

+ v∂>(t) M∂ y∂(t)︸︷︷︸discrete supplied power


Entropy as Hamiltonian Simulation

Entropy functional - Simulation 1/


Overview





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.



Energy functional - Modeling 2/3 port-Hamiltonian system


fs := ∂ts, eu := T−→f S := −

−−−→grad (T) , −→e S := −→J S

Primary port-Hamiltonian system:(ρfu−→f Q

)=(


)(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


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


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.



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



Energy functional - Discretization 3/3 structure


Discrete closure relation:Dulong-Petit’s law and Gibbs’ relation:

CVρ∂tes = es fs =⇒ (Solve ODE)


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.)


dtUd(t) = v∂>(t) M∂ y∂(t)


Energy as Hamiltonian Simulation

Energy functional - Simulation 2/2 time-space simulation


Energy as Hamiltonian Simulation

Energy functional - Simulation 2/2 time-space simulation


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.


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.


Bibliography (on PFEM) I

C. BEATTIE, V. MEHRMANN, H. XU AND H. ZWART. Port-Hamiltonian descriptor sytems.Mathematics of Control, Signals, and Systems, 30:17, 2018.

A. BRUGNOLI, D. ALAZARD, V. POMMIER-BUDINGER, AND D. MATIGNON, Port-Hamiltonianformulation and Symplectic discretization of Plate models. Part I : Mindlin model for thickplates, Applied Mathematical Modelling (2019). https://doi.org/10.1016/j.apm.2019.04.035,in Press

, Port-Hamiltonian formulation and Symplectic discretization of plate models. Part II :Kirchhoff model for thin plates, Applied Mathematical Modelling (2019).https://doi.org/10.1016/j.apm.2019.04.036, in Press

A. BRUGNOLI, D. ALAZARD, V. POMMIER-BUDINGER, AND D. MATIGNON, Control byinterconnection of the Kirchhoff plate within the port-Hamiltonian framework, in 58th IEEEConference on Decision and Control (CDC), 2019.(submitted).

A. BRUGNOLI, D. ALAZARD, V. POMMIER-BUDINGER, AND D. MATIGNON, Partitioned FiniteElement Method for the Mindlin Plate as a Port-Hamiltonian system, in 2019 3rd IFACworkshop on Control of Systems Governed by Partial Differential Equations (CPDE), 2019,8 p.

F. L. CARDOSO-RIBEIRO, A. BRUGNOLI, D. MATIGNON, AND L. LEFEVRE, Port-Hamiltonianmodeling, discretization and feedback control of a circular water tank, in 2019 58th IEEEConference on Decision and Control (CDC), 2019. (submitted)

Bibliography (on PFEM) II

F. L. CARDOSO-RIBEIRO, D. MATIGNON, AND L. LEFEVRE, A structure-preservingPartitioned Finite Element Method for the 2D wave equation, in IFAC-PapersOnLine,vol. 51(3), 2018, pp. 119–124.6th IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control LHMNC2018.

, A Partitioned Finite-Element Method (PFEM) for power-preserving discretization ofopen systems of conservation laws, 2019. arXiv:1906.05965

F. L. CARDOSO-RIBEIRO, D. MATIGNON, AND V. POMMIER-BUDINGER, A port-Hamiltonianmodel of liquid sloshing in moving containers and application to a fluid-structure system,Journal of Fluids and Structures, 69 (2017), pp. 402–427.

O. FARLE, D. KLIS, M. JOCHUM, O. FLOCH, AND R. DYCZIJ-EDLINGER, A port-Hamiltonianfinite-element formulation for the Maxwell equations, in 2013 International Conference onElectromagnetics in Advanced Applications (ICEAA), September 2013, pp. 324–327.

E. HAIRER, C. LUBICH, AND G. WANNER, Geometric numerical integration:structure-preserving algorithms for ordinary differential equations, vol. 31 of Springer Seriesin Computational Mathematics, Springer-Verlag Berlin Heidelberg, 2006.

P. KUNKEL AND V. MEHRMANN, Differential-Algebraic Equations: Analysis and NumericalSolution, EMS Textbooks in Mathematics, European Mathematical Society, 2006.

Bibliography (on PFEM) III

M. KURULA AND H. ZWART, Linear wave systems on n-D spatial domains, InternationalJournal of Control, 88 (2015), pp. 1063–1077.

A. MACCHELLI, A. VAN DER SCHAFT, AND C. MELCHIORRI, Port Hamiltonian formulation ofinfinite dimensional systems I. Modeling, in 2004 43rd IEEE Conference on Decision andControl (CDC), vol. 4, 2004, pp. 3762–3767.

G. PAYEN, D. MATIGNON, AND G. HAINE, Simulation of plasma and Maxwell’s equationsusing the port-Hamiltonian approach. Internship report, ISAE-SUPAERO, December 2018.

A. SERHANI, D. MATIGNON, AND G. HAINE, Partitioned Finite Element Method forport-Hamiltonian systems with Boundary Damping: Anisotropic Heterogeneous 2-D waveequations, in 2019 3rd IFAC workshop on Control of Systems Governed by PartialDifferential Equations (CPDE), 2019, 8 p.

, A partitioned finite element method for the structure-preserving discretization ofdamped infinite-dimensional port-Hamiltonian systems with boundary control, in 2019 3rdIEEE workshop on Geometric Science of Information (GSI), 2019, 10 p.

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Anisotropic hetergeneous n-D heat equation with boundary … · 2019-07-05 · Thermodynamic Foundation of Mathematical Systems Theory - 3rd TFMST 2019 Anisotropic hetergeneous n-D