Optimal Control of Two-Phase Flow

(1)

Optimal Control of Two-Phase Flow

Harald Garcke, Michael Hinze, Christian Kahle

RICAM special semester on Optimization WS1: New trends in PDE constrained optimization

14.10. – 18.10.2019

Christian Kahle Optimal Control of Two-Phase Flow 10/2019 1/32

(2)

Optimal control of two-phase flow

Figure:without control

Figure:with control

(3)

Optimal control of two-phase flow

Figure:without control

Figure:with control

(4)

Outline

Setting

The time discrete setting The fully discrete setting Numerical examples

(5)

Outline

Setting

(6)

Diffuse interface approach

Setting: Two subdomainsΩ1andΩ2separated by unknown Γ. Assumption: Γof small thicknessO() >0and components are mixed

inside.

Representation: Continuous order parameterϕforΩ1andΩ2.

Ω1

Ω2 1

−1 ϕ

Γ

ϕ(x) =1⇔x∈Ω1

ϕ(x) = −1⇔x∈Ω2

−1<ϕ(x) <1⇔x∈Γ

−1 0

˜1 ρ2

˜ ρ1

Γ,O()

Ω2 Ω1

ϕ ρ2(x)

ρ1(x)

ϕ(x) =ρ1(x)

˜

ρ1 −ρ2(x)

˜ ρ2

(7)

Diffuse interface approach

inside.

Ω1

Ω2 1

−1 ϕ

Γ

ϕ(x) =1⇔x∈Ω1

ϕ(x) = −1⇔x∈Ω2

−1<ϕ(x) <1⇔x∈Γ

−1 0

˜1 ρ2

˜ ρ1

Γ,O()

Ω2 Ω1

ϕ ρ2(x)

ρ1(x)

ϕ(x) =ρ1(x)

˜

ρ1 −ρ2(x)

˜ ρ2

(8)

Diffuse interface approach

inside.

Ω1

Ω2 1

−1 ϕ

Γ

ϕ(x) =1⇔x∈Ω1

ϕ(x) = −1⇔x∈Ω2

−1<ϕ(x) <1⇔x∈Γ

−1 0

˜1 ρ2

˜ ρ1

Γ,O()

Ω2 Ω1

ϕ ρ2(x)

ρ1(x)

ϕ(x) =ρ1(x)

˜

ρ1 −ρ2(x)

˜ ρ2

(9)

The two-phase flow model

[Abels, Garcke, Grün, 2012]

v velocity, ppressure,

ϕphase field variable,µchemical potential

ρ∂tv+ ((ρv+J) ⋅ ∇)v− div(2ηDv) + ∇p= −ϕ∇µ+ρg, divv=0,

∂tϕ+v⋅ ∇ϕ−div(m∇µ) =0,

−σ∆ϕ+σ⁻¹W^′(ϕ) =µ, where2Dv= ∇v+ (∇v)^t, J= −ρ^′(ϕ)m(ϕ)∇µ.

g gravity,

interfacial width, σ surface tension,

σ=cWσ^{phy s},

ρ(ϕ) density, η(ϕ) viscosity, m(ϕ) mobility.

ϕ 1

−1 1

W^s

(10)

The free energy density W

logarithmic:

W^log(ϕ) = ^θ2((1+ϕ)log(1+ϕ) + (1−ϕ)log(1−ϕ)) +^θ2^ϕ(1−ϕ²), polynomial: W^poly(ϕ) = ¹4(1−ϕ²)²,

double-obstacle: W^∞(ϕ) = ¹2(1−ϕ²)iff∣ϕ∣ ≤1, ∞else, relaxed double-obstacle:

W^s(ϕ) = ¹2(1− (ξϕ)²) +^s2(max(0, ξϕ−1)²+min(0, ξϕ+1)²) +θ.

ϕ 1

−1 1

W^log

ϕ 1

−1 1

W^poly

ϕ 1

−1 1

W^∞

∞

ϕ 1

−1 1

W^s

(11)

Functions depending on ϕ

ϕ ρ(ϕ) =

ρ1+ρ2

2 +^ρ²^−ρ₂ ¹ϕ

η(ϕ) =

η1+η2

2 +^η²^−η₂ ¹ϕ

W(ϕ)

(12)

The formal energy inequality

Theorem

Letv , ϕ, µdenote a sufficiently smooth solution (if exists) and let E(t) = ∫_Ω1

2ρ(t)∣v(t)∣²dx+σ∫_Ω

2∣∇ϕ(t)∣²+1

W(ϕ(t))dx denote the energy of the system. Letv∣^∂Ω=0hold.

Then it holds d

dtE(t) = − ∫_Ω2η(ϕ)∣Dv∣²dx− ∫_Ωm(ϕ)∣∇µ∣²dx+ ∫_Ωgv dx,

E(t2) + ∫_t₁^t²∫_Ωm(ϕ(s))∣∇µ(s)∣²dxds+ ∫_t₁^t²∫_Ω2η(ϕ(s))∣Dv(s)∣²dxds

=E(t1) + ∫_t₁^t²∫_Ωgv(s)dxds

(13)

Applied Controls

BuV =

∑^si=^V1fi(x)uV[i], fi∈L²(Ω)ⁿ

BuB=

∑^si=^B1gi(x)uB[i], gi∈H¹^/²(∂Ω)ⁿ

ϕ0= BuI=uI

uV ∈L²(0, T;R^s^V) =UV, uB∈L²(0, T;R^s^B) =UB,

uI∈ K ∶= {v∈H¹(Ω) ∩L^∞(Ω) ∣ ∣v∣ ≤1,(v ,1) =const} =UI, u= (uV, uB, uI) ∈U=UV ×UB×UI.

(14)

The two-phase flow model with controls

v velocity, ppressure,

ϕphase field variable,µchemical potential

ρ∂tv+ ((ρv+J) ⋅ ∇)v−div(2ηDv) + ∇p= −ϕ∇µ+ρg+BuV, divv=0,

∂tϕ+v⋅ ∇ϕ−div(m∇µ) =0,

−σ∆ϕ+σ⁻¹W^′(ϕ) =µ, where2Dv= ∇v+ (∇v)^t,J= −ρ^′(ϕ)m(ϕ)∇µ, v∣^∂Ω=BuB,ϕ(0) =uI.

g gravity,

interfacial width, σ surface tension,

σ=cWσ^{phy s},

ρ(ϕ) density, η(ϕ) viscosity, m(ϕ) mobility.

ϕ 1

−1 1

W^s

(15)

The optimal control problem

The optimal control problem ϕd: desired distribution, αV +αB+αI=1

minJ(uI, uV, uB, ϕ) ∶=1

2∥ϕ(T) −ϕd∥² +α

2(αI∫_Ω

2∣∇uI∣²+⁻¹Wu(uI)dx αV∥uV∥²L²(0,T;R^sV)+αB∥uB∥²L²(0,T;R^sB)) s.t. two-phase fluid dynamics,

i.e. ϕ≡ϕ(uV, uB, uI)

(P)

(16)

Outline

Setting

(17)

A weak formulation

Abbreviate

a(u, v , w) ∶=1

2((u⋅ ∇)v , w) −1

2((u⋅ ∇)w , v) The model satisfies

∂tρ(ϕ) +div(ρ(ϕ)v+J) = −∇µ⋅ ∇ρ^′(ϕ) Ifρ(ϕ)is linear (mass conservation)

ρ∂tv+ ((ρv+J) ⋅ ∇)v−div(2ηDv) =µ∇ϕ,

∂t(ρv) +div(ρv⊗v) +div(v⊗J) −div(2ηDv) =µ∇ϕ.

Then a weak formulation is 1

2(ρ∂tv+∂t(ρv), w) +a(ρv+J, v , w) +2(ηDv , Dw) = (µ∇ϕ, w) ∀w ∈Hσ

(18)

An energy stable time discretization

[Garcke, Hinze, K. 2016]

u_⋆^k ∶= ¹τ∫^t^tk−1^k u_⋆(t)dt,v^k∣^∂Ω= Bu_B^k, ϕ⁰=uI

1

τ ∫_Ω(ρ^k⁻¹+ρ^k⁻²

2 v^k−ρ^k⁻²v^k⁻¹)w dx +a(ρ^k⁻¹v^k⁻¹+J^k⁻¹, v^k, w) + ∫_Ω2η^k⁻¹Dv^k ∶Dwdx

+ ∫_Ωϕ^k⁻¹∇µ^k⋅w dx− ∫_Ωρ^k⁻¹g⋅w dx− ∫_ΩBu_V^kw dx=0∀w ∈Hσ(Ω), 1

τ ∫_Ω(ϕ^k−ϕ^k⁻¹)Ψ dx− ∫_Ωϕ^k⁻¹v^k⋅ ∇Ψdx

+ ∫_Ωm∇µ^k⋅ ∇Ψdx=0∀Ψ∈H¹(Ω), σ∫_Ω∇ϕ^k⋅ ∇Φdx− ∫_Ωµ^kΦdx

+σ

∫_Ω((W₊)^′(ϕ^k) + (W₋)^′(ϕ^k⁻¹))Φdx=0∀Φ∈H¹(Ω). (CHNSτ)

(19)

Energy inequality

Theorem

Letk≥2,ϕ^k, µ^k, v^k be a solution to (CHNSτ), anduB≡0.

Then the following energy inequality holds

1

2∫_Ωρ^k⁻¹∣v^k∣²dx+σ∫_Ω

2∣∇ϕ^k∣²+1

W(ϕ^k)dx +1

2∫_Ωρ^k⁻²∣v^k−v^k⁻¹∣²dx+σ

2 ∫_Ω∣∇ϕ^k− ∇ϕ^k⁻¹∣²dx +τ∫_Ω2η^k⁻¹∣Dv^k∣²dx+τ∫_Ωm∣∇µ^k∣²dx

≤ 1

2∫_Ωρ^k⁻²∣v^k⁻¹∣²dx+σ∫_Ω

2∣∇ϕ^k⁻¹∣²+1

W(ϕ^k⁻¹)dx + ∫_Ωρ^k⁻¹gv^k dx+ ∫_Ω(Bu_V^k)v^k dx

(20)

Existence of a unique solution

Theorem

LetΩdenote a polygonally / polyhedrally bounded Lipschitz domain.

Letv^k⁻¹∈Hσ(Ω), ϕ^k⁻²∈H¹(Ω) ∩L^∞(Ω), ϕ^k⁻¹∈H¹(Ω) ∩L^∞(Ω), andµ^k⁻¹∈W^1,3(Ω)be given data. Further letBu_V^k ∈L²(Ω)ⁿ, Bu^k_B∈H¹²(∂Ω), BuI∈H¹(Ω) ∩L^∞(Ω)be given data.

Then there exists a weak solutionϕ^k∈H¹(Ω) ∩C(Ω),µ^k ∈W^1,3(Ω), v^k ∈Hσ(Ω)to(CHNSτ).

Furthermore, it can be found by Newton’s method.

(21)

Initialization step

Fork=1we solve: v¹∣^∂Ω= Bu¹_B,ϕ⁰=uI

1

τ∫_Ω(ρ¹+ρ⁰

2 v¹−ρ⁰v⁰)w dx+a(ρ¹v⁰+J¹, v¹, w)

+ ∫_Ω2η¹Dv¹∶Dwdx− ∫_Ωµ¹∇ϕ⁰w dx− ∫_ΩBu¹_Vw dx− ∫_Ωρ⁰g⋅w =0∀w ∈Hσ(Ω), 1

τ ∫_Ω(ϕ¹−ϕ⁰)Ψdx− ∫_Ωϕ⁰v⁰⋅ ∇Ψdx

+ ∫_Ωm∇µ¹⋅ ∇Ψdx=0∀Ψ∈H¹(Ω), σ∫_Ω∇ϕ¹⋅ ∇Φdx− ∫_Ωµ¹Φdx

+σ

∫_Ω((W₊)^′(ϕ¹) + (W₋)^′(ϕ⁰))Φdx=0∀Φ∈H¹(Ω). (CHNS^I_τ)

(22)

Stability

Theorem

LetΩdenote a polygonally / polyhedrally boundedLipschitzdomain.

Letv⁰∈Hσ(Ω),(uI, uV, uB) ∈U be given. Then there exist sequences (v^k)^Kk=1∈Hσ(Ω)^K,(ϕ^k)^Kk=1∈ (H¹(Ω) ∩C(Ω))^K,(µ^k)^Kk=1∈W^1,3(Ω)^K such that(v^k, ϕ^k, µ^k)is the unique solution to (CHNS^I_τ)for k=1and to(CHNSτ)for k=2, . . . , K. Moreover there holds

∥(v^k)^Kk=1∥`^∞(H¹(Ω))≤C(v⁰, uI, uV, uB),

∥(ϕ^k)^Kk=1∥`^∞(H¹(Ω)∩C(Ω))≤C(v⁰, uI, uV, uB),

∥(µ^k)^Kk=1∥^`^∞(W^1,3(Ω))≤C(v⁰, uI, uV, uB).

(23)

Stability in stronger norms

Theorem

LetΩbe polygonally / polyhedrally bounded andconvex or of classC^1,1. Letv⁰∈Hσ(Ω) ∩L^∞(Ω)ⁿ,(uI, uV, uB) ∈Ube given. Then there exist sequences(v^k)^Kk=1∈Hσ(Ω)^K,(ϕ^k)^Kk=1∈H²(Ω)^K,(µ^k)^Kk=1∈H²(Ω)^K such that(v^k, ϕ^k, µ^k)is the unique solution to (CHNS^Iτ)for k=1and to(CHNSτ)for k=2, . . . , K. Moreover there holds

∥(v^k)^Kk=1∥^`^∞(H¹(Ω))≤C(v⁰, uI, uV, uB),

∥(ϕ^k)^Kk=1∥`^∞(H²(Ω))≤C(v⁰, uI, uV, uB),

∥(µ^k)^Kk=1∥`^∞(H²(Ω))≤C(v⁰, uI, uV, uB).

(24)

The optimal control problem

Theorem

LetΩbe polygonally / polyhedrally bounded andconvex or of classC^1,1. The optimization problem

minJ(uI, uV, uB,(ϕ^k)^Kk=1) ∶=1

2∥ϕ^K−ϕd∥² +α

2(αI∫_Ω

2∣∇uI∣²+⁻¹Wu(uI)dx αV∥uV∥²L²(0,T;R^sV)+αB∥uB∥²L²(0,T;R^sB)) s.t. (CHNS^Iτ)and(CHNSτ)

(P^τ) has at least one solution and first order optimality conditions can be derived by Lagrangian calculus.

(25)

Outline

Setting

(26)

Finite element approximation

Th^k triangulation ofΩat time instancetk,

V₁^k∶= {v∈C(Ω) ∣v∣^T∈P¹∀T ∈ Th^k},

V₂^k∶= {v∈C(Ω)ⁿ∣v∣^T ∈ (P²)ⁿ∀T ∈ Th^k,(di v(v), q) =0∀q∈V₁^k},

P^k∶H¹(Ω) →V₁^k prolongation, e.g. H¹-prolongation.

ϕ^k_h, µ^k_h∈V₁^k, v_h^k ∈V₂^k

(27)

The fully discrete setting

u_⋆^k ∶= ⨏^t^tk−1^k u_⋆(t)dt,v_h^k∣^∂Ω=Πh(Bu_B^k), ϕ⁰_h=uI

1

τ ∫_Ω(ρ^k_h⁻¹+ρ^k_h⁻²

2 v_h^k−ρ^k_h⁻²v_h^k⁻¹)w dx +a(ρ^k_h⁻¹v_h^k⁻¹+J_h^k⁻¹, v_h^k, w) + ∫_Ω2η^k_h⁻¹Dv_h^k ∶Dw dx

− ∫_Ωµ^k_h∇ϕ^k_h⁻¹w dx− ∫_Ωρ^k_h⁻¹g⋅w dx− ∫_Ω(Bu_V^k)w dx=0∀w ∈V₂^k, 1

τ ∫_Ω(ϕ^k_h−P^kϕ^k−1_h )Ψ dx+ ∫_Ω(v_h^k⋅ ∇ϕ^k_h⁻¹)Ψ dx

+ ∫_Ωm∇µ^k_h⋅ ∇Ψ dx=0∀Ψ∈V₁^k, σ∫_Ω∇ϕ^k_h⋅ ∇Φdx− ∫_Ωµ^k_hΦdx

+σ

∫_Ω((W₊)^′(ϕ^k_h) + (W₋)^′(P^kϕ^k−_h ¹))Φdx=0∀Φ∈V₁^k. (CHNSh)

(28)

Energy inequality in the fully discrete setting

Theorem

Letk≥2,ϕ^k_h, µ^k_h, v_h^k be a solution to (CHNSh), anduB≡0.

Then the following energy inequality holds

1

2∫_Ωρ^k_h⁻¹∣v_h^k∣²dx+σ∫_Ω

2∣∇ϕ^k_h∣²+1

W(ϕ^k_h)dx +1

2∫_Ωρ^k_h⁻²∣v_h^k−v_h^k⁻¹∣²dx+σ

2 ∫_Ω∣∇ϕ^k_h− ∇P^kϕ^k⁻¹^h∣²dx +τ∫_Ω2η_h^k⁻¹∣Dv_h^k∣²dx+τ∫_Ωm∣∇µ^k_h∣²dx

≤ 1

2∫_Ωρ^k_h⁻²∣v_h^k⁻¹∣²dx+σ∫_Ω

2∣∇P^kϕ^k_h⁻¹∣²+1

W(P^kϕ^k_h⁻¹)dx + ∫_Ωρ^k_h⁻¹gv_h^k dx+ ∫_Ω(Bu_V^k)v_h^k dx

(29)

Stability in the fully discrete setting

Theorem

LetΩbe polygonally / polyhedrally bounded andconvex.

Letv⁰∈Hσ(Ω) ∩L^∞(Ω),u∈U be given. Then there exist sequences (v_h^k)^Kk=1∈ (V₂^k)^Kk=1,(ϕ^k_h)^Kk=1,(µ^k_h)^Kk=1∈ (V₁^k)^Kk=1, such that(v_h^k, ϕ^k_h, µ^k_h)is the unique solution to(CHNSh)fork =1, . . . , K. Moreover it holds

∥(v_h^k)^Kk=1∥^`^∞(H¹(Ω))≤C(v⁰, uI, uV, uB),

∥(ϕ^k_h)^Kk=1∥`^∞(W^1,4(Ω))≤C(v⁰, uI, uV, uB),

∥(µ^k_h)^Kk=1∥^`^∞(W^1,3(Ω))≤C(v⁰, uI, uV, uB).

(30)

The optimal control problem in the fully discrete setting

Theorem

The optimization problem minJ(uI, uV, uB,(ϕ^k_h)^Kk=1) ∶=1

2∥ϕ^K_h −ϕd∥² +α

2(αI∫_Ω

2∣∇uI∣²+⁻¹Wu(uI)dx αV∥uV∥²L²(0,T;R^sV)+αB∥uB∥²L²(0,T;R^sB)) s.t. (CHNSh)

(P^h) has at least one solution and first order optimality conditions can be derived by Lagrangian calculus.

(31)

The limit h → 0

Theorem

Let(u_h^⋆, v_h^⋆, ϕ^⋆_h, µ^⋆_h)denote a stationary point of (P^h). Then there exists a stationary point(u^⋆, v^⋆, ϕ^⋆, µ^⋆)of (P^τ), such that

u^⋆_V,h⇀u_V^⋆ ∈UV, u_B,h^⋆ ⇀u_B^⋆ ∈UB, ϕ^k,_h^⋆⇀ϕ^k,^⋆∈W^1,4(Ω),

u_I,h^⋆ →u_I^⋆∈H¹(Ω), ϕ^k,_h^⋆→ϕ^k,^⋆∈H¹(Ω), µ^k,⋆_h →µ^k,⋆∈W^1,3(Ω), v_h^k,⋆→v^k,⋆∈Hσ(Ω).

(32)

Outline

Setting

(33)

Validity of the energy inequality

10⁻² 10⁻¹

10⁻^0.5

10⁰ E(t) O(t⁻¹)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

−4

−2 0 ⋅10⁻⁶

ζ

(34)

Rising Bubble, setup

Boundary control, setup from first [Hysing et al, 2009] Benchmark, ρ1=1000,ρ2=100,η1=10, η2=1,σ=15.6, T =1.0

Figure:left to right: ϕ⁰,ϕd, four Ansatzfunctions

(35)

Rising Bubble, results

0 0.2 0.4 0.6 0.8 1

0 5 10 15 20

time

∥u(t)∥

Strength of control

0 0.2 0.4 0.6 0.8 1

0.5 0.6 0.7 0.8

time Center of mass

(36)

Initial value identification

Optimization with a phase field as control works best with non-smooth free energy densities.

Wu(ϕ) =W^∞(ϕ) =⎧⎪⎪

⎨⎪⎪⎩

1

2(1−ϕ²) if ∣ϕ∣ ≤1,

∞ else.

ϕ 1

−1 1

W^∞

∞

Results in constraint minimization problem

uI∈H¹(Ω)∩minL^∞(Ω),∣uI∣≤1J(uI) Solved by VMPT [Blank, Rupprecht, SICON 2017].

(37)

Initial value problem, setup

Initial value control, setup from second [Hysing et al] Benchmark, ρ1=1000,ρ2=1, η1=10,η2=0.1,σ=1.96,T =1.0

Figure:left to right: ϕd,ϕ0=u_I⁰= −0.8

(38)

Initial value problem, result

Figure:left to right: u_I^opt,ϕ(u_I^opt)at final time with zero level line ofϕd

(39)

Summary

Energy stable time discretization concept for two-phase flow time discrete

fully discrete

Time discrete optimal control of two-phase flow with three kinds of control actions

time discrete fully discrete

Convergence analysis for h→0.

Thank you for your attention. [email protected]

(40)

Summary

fully discrete

(41)

Summary

fully discrete

(42)

Summary

fully discrete

Thank you for your attention.

[email protected]