Optimal Shape Design of Wave MakerIntroduction Robin Goix 1 Introduction 2 Modelling of wave...

Optimal Shape Design of Wave Maker

Robin Goix

June 23, 2014

Introduction Robin Goix

WavegardenSource: Youtube.(2014, Jan. 30) WAVE GARDEN avec SURF REPORT

Wavegarden’s Technology: an underwater object pulled in pool

Underwater moving Object The entire apparatus

Wavemaker project main goals:

Develop the computational framework to run simulations of wavesgenerated by an underwater moving object

Optimize the shape of that moving object to obtain the best waveaccording to criteria (high, shape, energy...)

1 Introduction

2 Modelling of wave generationBoussinesq system derived by PeregrineWeak formulation and time discretizationAssessment of the code: the solitary wave solutionResults

3 Optimization of the shapeControl of the optimization problemCoupling with a transport equationFirst resultsImprovement possibilities and other solutions

Modelling of wave generation Robin Goix

Boussinesq system derived by Peregrine

Seabed profil:h(x, y, t) = D(x, y) + εζ(x, y, t)

Free surface: η(x, y, t)

Fluid horizontal velocity: u(x, y, z, t)

Scaling: ε =a0

h0, σ =

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

ηt + ζt +∇ · [(h+ εη)u] = 0

Boussinesq system derived by Peregrine

Seabed profil:h(x, y, t) = D(x, y) + εζ(x, y, t)

Free surface: η(x, y, t)

Fluid horizontal velocity: u(x, y, z, t)

Scaling: ε =a0

h0, σ =

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

ηt + ζt +∇ · [(h+ εη)u] = 0

Weak formulation and time discretization

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

ηt + ζt +∇ · [(h+ εη)u] = 0u · n = 0 on ∂Ω

Let (v, ξ) ∈ V×H1(Ω), with V = w ∈ H1(Ω)2,w · n = 0 on ∂Ω,integrating and using divergence theorem, we get:

∫Ωut · v dx−

∫Ωη (∇ · v) dx + ε

∫Ω(u · ∇)u · v dx +

∫Ω(∇ · (hut)) (∇ · (hv)) dx

−σ2

∫Ω(∇ · ut) (∇ · (h2v)) dx +

∫Ωζtt (∇ · (hv)) dx

∫Ωηt ξ dx +

∫Ωζt ξ dx−

∫Ω∇ξ · [(h+ εη)u] dx

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

ηt + ζt +∇ · [(h+ εη)u] = 0u · n = 0 on ∂Ω

∫Ωut · v dx−

∫Ωη (∇ · v) dx + ε

∫Ω(u · ∇)u · v dx +

∫Ω(∇ · (hut)) (∇ · (hv)) dx

−σ2

∫Ω(∇ · ut) (∇ · (h2v)) dx +

∫Ωηt ξ dx +

∫Ωζt ξ dx−

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

ηt + ζt +∇ · [(h+ εη)u] = 0u · n = 0 on ∂Ω

∫Ωut · v dx−

∫Ωη (∇ · v) dx + ε

∫Ω(u · ∇)u · v dx +

∫Ω(∇ · (hut)) (∇ · (hv)) dx

−σ2

∫Ω(∇ · ut) (∇ · (h2v)) dx +

∫Ωηt ξ dx +

∫Ωζt ξ dx−

With the following schemes:

fn+θ = (1− θ)fn + θfn+1

fn+1t =

fn+1 − fn

dt, fn+1

tt =fn+1 − 2fn + fn−1

for each time-step, knowing un and ηn, our problem is therefore:Find (un+1, ηn+1) ∈ V×H1(Ω),∀(v, ξ) ∈ V×H1(Ω):

∫Ωun+1t · v dx−

∫Ωηn+θ (∇ · v) dx + ε

∫Ω(un+θ · ∇)un+θ · v dx

∫Ω(∇ · (hn+θun+1

t )) (∇ · (hn+θv)) dx−σ2

∫Ω(∇ · un+1

t ) (∇ · ((hn+θ)2v)) dx

∫Ωζn+1tt (∇ · (hn+θv)) dx

∫Ωηn+1t ξ dx +

∫Ωζn+1t ξ dx−

∫Ω∇ξ · [(hn+θ + εηn+θ)un+θ] dx

+ stabilization

Assessment of the code: the solitary wave problem

Fixed simulation time and mesh

Different time-stepping methods

With or without added stabilization

Error L2 and L∞

Results: some simulations

Optimization of the shape Robin Goix

Control of the optimization problem

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

ηt + ζt +∇ · [(h+ εη)u] = 0h(x, y, t) = D(x, y) + εζ(x, y, t)

A time-dependent parameter

But Dolfin-Adjoint does not make it easy to optimize such aparameter:

m = InitialConditionParameter(u)

m = ScalarParameter(s)

However it is completely determined by an initial condition and atrajectory:

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

However it is completely determined by an initial condition and atrajectory:ζ(x, y, t) = ζ0(x− Ux(t), y − Uy(t))

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

ζt + εU · ∇ζ = 0

ζ(x, y, 0) = ζ0(x, y)

Coupling with a transport equation

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

ηt + ζt +∇ · [(h+ εη)u] = 0h(x, y, t) = D(x, y) + εζ(x, y, t)ζt + εU · ∇ζ = 0ζ(x, y, 0) = ζ0(x, y)

We solve each system separately, step by step.

However, hyperbolic equations are difficult to be solved using FE.The following weak formulation:

∀τ ∈ H10(Ω),

τζn+1 − ζn

∆t− ε∫

∇τ ·Un+θζn+θ = 0

is not enough to transport the object with a good accuracy.

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

∀τ ∈ H10(Ω),

τζn+1 − ζn

∆t− ε∫

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

∀τ ∈ H10(Ω),

τζn+1 − ζn

∆t− ε∫

ut +∇η + ε(u · ∇)u− σ2 h

2∇(∇ · (hut)) + σ2 h

6∇(∇ · ut)− σ2 h

2∇ζtt = 0

∀τ ∈ H10(Ω),

τζn+1 − ζn

∆t− ε∫

Coupling with a transport equation : a second-order Taylor-Galerkinapproximation

ζn+1 − ζn

(∂ζ

(∂2ζ

+O(∆t2)

∂t= −εU · ∇ζ,

∂2ζ

∂t2= −ε

∂t· ∇ζ − εU · ∇

ζn+1 − ζn

∆t= −εUn · ∇ζn −

ε∆t

∂t· ∇ζn+θ − εUn+θ · ∇(Un+θ · ∇ζn+θ)

)+O(∆t2)

which lead us to the following weak formulation: ∀τ ∈ H10(Ω),∫

Ωτζn+1 − ζn

∆t− ε∫

Ω∇τ ·Unζn −

ε∆t

∫Ω∇τ ·

∂tζn+θ

+ε2∆t

(∇τ ·U)(∇ζn+θ ·Un+θ) = 0

First resultsFunctional to minimize:

infζ0∈Uad

J(ζ0) =

j(η(T )) dx

,withj(η(T )) = −1

2(|η(T )|2+|∇η(T )|2)

Domain ω on which the wave isassessed

Shape of the object - Uad = ζ0 ∈H1, ζmin ≤ ζ0 ≤ ζmax

Two optimization algorithms used:• Coupling between FEniCS and IPOPT • Dolfin-Adjoint and scipy

First results

Convergence of IPOPT

First results

A bigger and steeperwave;

however moreperturbations in thewake ...

... which means a lossof energy.

Improvement possibilities and other solutions

Find a better functional which consider the energy aspect throughthe pressure on the object

Improve the optimization:

More accurate way to solve the transport equation(Discontinuous-Galerkin?)Code our own gradient algorithm to get rid of dolfin-adjointconstraints

Develop an other model which could simulate wave breaking

Optimal Shape Design of Wave MakerIntroduction Robin Goix 1 Introduction 2 Modelling of wave...

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