# Optimization of Injectors Location in a Water Reservoirs Aeration Problem

Authors: Mohamed Abdelwahed
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The aeration process via injectors is used to combat the lack of oxygen in water reservoirs due
to the eutrophication problem. The optimization of the injectors location allows to generate a
better flow dynamic in order to remedy to this phenomena an preserve the water quality. We
propose to adapt the topological gradient method to this problem. The idea consists of studing
the topological sensitivity analysis of the considered model based on the three-dimensional-dependent time Navier-Stokes equation with respect to the insertion of an injector in the flow
domain. Finally the obtained results are used in a numerical algorithm and some numerical
tests are presented to validate the used approach.


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## Body

## 1. Introduction

During the warmer months of the year many lakes and water reservoirs experience some degree of thermal stratification. This phenomena inhibit vertical mixing between the surface layer and the bottom water. This can lead to anoxic conditions in the lower region. The lack of oxygen accelerates the eutrophication of the water which is characterized by a number of damaging effects on its quality. The dynamic aeration is one of used methods to treat the water eutrophication problem. This technique consists of inserting air by the means of injectors located at the bottom of the lake in order to generate a vertical motion mixing up the water of the bottom with that on the top, thus oxygenating the lower part by bringing it in contact with the surface air.We focus in this work on the optimization of the injectors location in order to generate the best dynamic in the fluid with respect to the aeration purpose. The optimal injectors location is characterized as the solution to a topological optimization problem. The topological sensitivity method is used to solve this problem [1–3]. To present this method, we consider the case in which Ω contains a single injector ωz,ε located at z∈ℝ3 and having a diameter ε. The topological sensitivity method provides an asymptotic expansion of an objective function j of the form
(1)j(Ω∖ωz,ε¯)=j(Ω)+εδj(z)+o(ε),∀z∈Ω.
This expression is called the topological asymptotic expansion and δj is the topological sensitivity function. In order to minimize the cost function, the best location to insert the injector in Ω is where δj is the most negative. In fact if δj(z)<0, we have j(Ω∖ωz,ε¯)<j(Ω) for small ε. The function δj can be used as a descent direction in the domain optimization process.This method is used in [4] to optimize the injector position. In this work a stationary model based on generalized Stokes equations is considered. This paper extend this approach to dependent time model based on three dimensional Navier-Stokes equations. In order to obtain a topological sensitivity analysis of the used model with respect to the insertion of an injector, the model is firstly discretized on time using the characteristics method. A generalized Stokes problem is obtained at each time step. The topological sensitivity analysis for this problem presented in [4] is used and then extended to the Navier-Stokes problem.The paper is organized as follows. The used model is presented in Section2. Section 3 is devoted to the optimization problem. A topological sensitivity analysis for the generalized Stokes equations is presented and extended to Navier-Stokes equations. Finally, we illustrate the efficiency of the proposed method by several numerical experiments in Section 4.


## 2. Physical Model

LetΩ be a three-dimensional flow domain representing the eutrophized lake. Different models can be used to describe the resulting two phase water air bubbles flow problem. A general framework is proposed in [5]. In our case, such a problem leads to many difficulties mainly due to the complexity of the model itself and to the inherent high computing costs for such large-scale applications. For this reason, Legendre in [6] studied the case of one air bubble and its interactions with the fluid flow. This work has been extended to several air bubbles by Clement in [7] but in two-dimensional case. Using the fact that the water phase is dominant, a two-dimensional simplified model is proposed in [8]. In this work, the water phase is governed by the three-dimensional dependent time Navier-Stokes equations in which we integrate the effect of momentum released by the injected bubbles by adding a local boundary condition for the velocity on the injector holes. In the presence of an injector ωinj⊂Ω (see Figure 1), the velocity u(x,t) and the pressure p(x,t) solve the following system:
(2)∂u∂t+u·∇u-νΔu+∇p=𝒢inΩi×]0,T],∇·u=0inΩi×]0,T],u(·,0)=u0inΩi,σ(u,p)·n=0onΓn×]0,T],u=uwindonΓs×]0,T],u=0onΓw×]0,T],u=uinjon∂ωinj×]0,T],
where Ωi=Ω∖ωinj¯ is the lake domain in the presence of the injector ωinj, ν is the water viscosity, 𝒢 is the gravitational force, Γn is a free boundary (inlet/outlet), Γs is the surface in contact with the atmosphere, uwind is the wind velocity, Γw is the bottom lake boundary, T is the final time of simulation, u0 is the initial velocity field, and σ(u,p)=ν(∇u+∇uT)-pI is stress tensor with n is unit outward normal vector.Figure 1
DomainΩ.For the sake of simplicity, we denote byΓi=Γs∪Γw∪∂ωinj and by
(3)ud={uwindonΓs,0onΓw,uinjon∂ωinj.


## 3. Optimization of Injectors Location

This section is dedicated to design an efficient method to optimize the injectors location in order to generate the best motion of the fluid. The topological optimization method is used for this purpose.


### 3.1. Topological Optimization Problem

For the sake of simplicity, we will assume that the injectors are well separated and have the geometry formωzk,ε=zk+εωk,1≤k≤m, where ε is the shared diameter and  ωk⊂ℝ3 are bounded and smooth domains containing the origin. The points zk∈Ω,1≤k≤m determine the location of the injectors. The domains ωk describe the injectors geometries.In the presence of injectors, the velocityuε and the pressure pε satisfy the following system:
(4)∂uε∂t+(uε·∇)uε-νΔuε+∇pε=𝒢inΩ∖⋃k=1mωzk,ε¯×]0,T],∇·uε=0inΩ∖⋃k=1mωzk,ε¯×]0,T],uε=udonΓ×]0,T],σ(uε,pε)·n=0onΓn×]0,T].
Note that for ε=0, (u0,p0) is solution to
(5)∂u0∂t+(u0·∇)u0-νΔu0+∇p0=𝒢inΩ×]0,T],∇·u0=0inΩ×]0,T],u0=udonΓ×]0,T],σ(u0,p0)·n=0onΓn×]0,T].
For the optimization criteria, we assume that a “good” lake oxygenation can be described by a target velocity 𝒰g. Then, the cost function 𝒥ε to be minimized is given by
(6)𝒥ε(uε)=∫0TJε(uε)dtwithJε(uε)=∫Ωm|uε-𝒰g|2dx,
where Ωm⊂Ω is the measurement domain (the top layer, see Figure 2).Figure 2
The top layerΩm.We denote byj the design function defined by
(7)j(Ω∖⋃k=1mωzk,ε¯)=𝒥ε(uε),
where uε is the solution of (4).The identification problem can be formulated as a topological optimization problem: find the optimal location of the injectorsωzk,ε=zk+εωk,1≤k≤m, inside the lake domain Ω minimizing the optimal design function j:
(8)(𝒪ε){Findzk*∈Ω,1≤k≤m,suchthatj(Ω∖⋃k=1mωzk*,ε¯∪k=1m)=minωzk,ε⊂Ωj(Ω∖⋃k=1mωzk,ε¯).To solve this optimization problem (𝒪ε) we will use the topological sensitivity analysis method [2, 3, 9–11]. It consists of studying the variation of the design function j with respect to a small topological perturbation of the domain Ω.


### 3.2. Time Discretization

The Lagrangian representation of the flow is based on the functionX:(x,s;t)∈Ω×(0,T)2→X(x,s;t), where X(x,s;t) is the position at time t of the particle of fluid which is at point x at time t=s. Thus t→X(x,s;t) is the parametric representation of the ajectory of the particle. Given the velocity u, the trajectory may be determined from the initial value problem:
(9)ddtX(x,s;t)=u(X(x,s;t),t),X(x,s;s)=x.Lemma 1.
If|s-t| is sufficiently small, then the Jacobian of the homeomorphism x→X(x,s;t) is equal to 1.Proof.
See [12].LetNT be a positive integer, Δt=T/NT and tn=nΔt. For x∈Ω, s=tn+1, tn≤t≤tn+1, 0≤n≤N, (9) implies using Euler scheme that
(10)x-X(x,tn+1;tn)=∫tntn+1u(X(x,tn+1;t),t)dt≈Δtu(X(x,tn+1;tn+1),tn+1)=Δtu(x,tn+1).
Then numerically the position X(x,tn+1;tn) is determined by
(11)X(x,tn+1;tn)=x-Δtu(x,tn+1).
Using (10) it is shown in [12] that material derivative is as follows:
(12)dudt=∂u∂t+(u·∇)u=u(x,tn+1)-u(X(x,tn+1;tn),tn)Δt.
This approximation is called characteristic method [13] and using Lemma 1 we show that (12) is L2 stable [14].Using (12), problem (4) is equivalent to solve the following generalized Stokes equation:
(13)Findun+1andpn+1solutionsofαuεn+1-νΔuεn+1+∇pεn+1=Fεn+1inΩ∖⋃k=1mωzk,ε¯,∇·uεn+1=0inΩ∖⋃k=1mωzk,ε¯,uεn+1=udonΓ,σ(uεn+1,pεn+1)·n=0onΓn,
where α=1/Δt, Fεn+1=(1/Δt)uεn(X(x,tn+1;tn))+𝒢,  uεn+1, and pεn+1 are the approximations of uε and pε on time tn+1. Then, at each time step, we have to solve a steady state generalized Stokes problem having the following generic form:
(14)αuε-νΔuε+∇pε=FεinΩ∖⋃k=1mωzk,ε¯,∇·uε=0inΩ∖⋃k=1mωzk,ε¯,uε=udonΓ,σ(uε,pε)·n=0onΓn.
Note that for ε=0, at each time step (u0,p0) is solution to
(15)αu0-νΔu0+∇p0=F0inΩ,∇·u0=0inΩ,u0=udonΓ,σ(u0,p0)·n=0onΓn.
Then, in order to obtain an asymptotic expansion of the cost function 𝒥ε for the Navier-Stokes problem, we study first the topological sensitivity analysis for the generalized Stokes problem by giving an asymptotic expansion of the cost function Jε.


### 3.3. Topological Sensitivity Analysis for the Generalized Stokes Problem

In this section we present the topological asymptotic expansion of the cost functionJε defined in (6) with respect to the insertion of a small injector ωz,ε=z+εω inside the domain Ω. From the week formulation of (14) one can show that uε is the solution to
(16)uε∈𝒱ε,𝒜ε(uε,w)=bε(w),∀w∈𝒱ε0,uε∣Γ=ud,uε∣∂ωz,ε=uinj,
where the functional space 𝒱ε, the bilinear form 𝒜ε, and the linear form bε are defined by
(17)𝒱ε={θ∈H1(Ω∖ωz,ε¯)3,divθ=0inΩ∖ωz,ε¯},𝒱ε0=𝒱ε∩H01(Ω∖ωz,ε¯)3,𝒜ε(u,w)=α∫Ω∖ωz,ε¯uwdx+ν∫Ω∖ωz,ε¯∇u∇wdx∀u,w∈𝒱ε,bε(w)=∫Ω∖ωz,ε¯Fεwdx∀w∈𝒱ε.
Next, we denote by u^ε the following extension of uε in Ω:
(18)u^ε={uεinΩ,uinjinωz,ε.
We begin first by presenting the topological asymptotic expansion of a cost function Jε defined on 𝒱ε and verifying the following hypothesis.Hypothesis 1.
(i)J0 is differentiable with respect to u; its derivative is denoted by DJ0(u).
(ii) There exists a real numberδJ such that ∀ε≥0(19)Jε(uε)-J0(u0)=DJ0(u0)(u^ε-u0)+εδJ+o(ε).
(iii) The adjoint problem(20)𝒜0(v0,w)=-DJ0(u0)w,∀w∈𝒱00,
has a unique solution v0∈𝒱00.The asymptotic expansion described in the following theorem is valid for arbitrary shaped holes and all cost function verifying the Hypothesis1. The case of spherical holes is given in (24) [11]. Finally we present the sensitivity analysis corresponding to the cost function Jε defined in (6).Theorem 2.
If Hypothesis1 holds, the function Jε has the following asymptotic expansion:
(21)Jε(uε)-J0(u0)=ε[(-∫∂ωη(y)ds(y))·v0(z)+δJ]+Δto(ε),
where η∈H-1/2(∂ω)3 is the solution to the boundary integral equation
(22)∫∂ωE(y-x)η(x)ds(x)=uinj-u0(z),∀y∈∂ω,
with (E, Π) is the fundamental solution of the Stokes equations: E(y)=(1/8πνr)(I+ererT), Π(y)=y/4πr3 with r=||y||, er=y/r and erT is the transposed vector of er.Proof.
This proof is an adaptation of the proof presented in [11].In the particular case whereω=B(0,1), the density η is given explicitly by [10]
(23)η(y)=-3ν2(u0(z)-uinj),∀y∈∂ω.
In this case and under the hypothesis of Theorem 2, the asymptotic expansion becomes
(24)Jε(uε)-J0(u0)=ε[6πν(u0(z)-uinj)·v0(z)+δJ]+Δto(ε).Proposition 3.
The functionJε given by (6) satisfies the Hypothesis 1 with
(25)DJ0(u0)(w)=2∫Ωm(u0-𝒰g)wdx,∀w∈𝒱0,δJ=0.
Then, The function Jε has the following expansion
(26)Jε(uε)-J0(u0)=2∫Ωm(u0-𝒰g)(u^ε-u0)dx+Δto(ε).Proof.
See [15].


### 3.4.  Topological Sensitivity Analysis for the Navier-Stokes Problem

In this section we consider the in stationary Navier-Stokes problem (4). We want to compute the variation j(Ω∖ωz,ε¯)-j(Ω) with respect to the insertion of a small injector in the fluid flow domain.Using (6), the variation of the design function (7) is given by
(27)j(Ω∖ωz,ε¯)-j(Ω)=𝒥ε(uε)-𝒥0(u0)=∫0TJε(uε)dt-∫0TJ0(u0)dt=∑n=1NT∫tn-1tn(Jε(uε)-J0(u0))dt,
where NT∈ℕ* and tn=nΔt, with Δt=T/NT is the time step.By trapezoidal formula, we get(28)∫tn-1tn(Jε(uε(·,t))-J0(u0(·,t)))dt=Δt2[(Jε(uεn)-J0(u0n))+(Jε(uεn-1)-J0(u0n-1))].
Using the sensitivity analysis for the generalized Stokes equations (21) and Proposition 3, we deduce the following asymptotic expansion
(29)j(Ω∖ωz,ε¯)-j(Ω)=12εΔtg(z)+o(ε),
where g(z)=∑n=1NT‍(δjn(z)+δjn-1(z)), with
(30)δjn(z)=(-∫∂𝒪ηn(y)ds(y))·v0n(z).
Here v0n and ηn are, respectively, the approximations at time tn=nΔt of the adjoint solution v0(·t) of problem (20) and η defined in (22). If ω is the unit ball B(0,1), by (24)
(31)δjn(z)=6πν(u0n(z)-uinj)·v0n(z).
In this case, using (25). The design function j defined in (7) has the following expansion:
(32)j(Ω∖ωz,ε¯)-j(Ω)=3πνεΔt×∑n=1NT‍[(u0n(z)-uinj)·v0n(z)+(u0n-1(z)-uinj)·v0n-1(z)]+o(ε),
where u0n and v0n are, respectively, solutions to the generalized Stokes problem (15) and its associated adjoint problem (20).


## 3.1. Topological Optimization Problem

For the sake of simplicity, we will assume that the injectors are well separated and have the geometry formωzk,ε=zk+εωk,1≤k≤m, where ε is the shared diameter and  ωk⊂ℝ3 are bounded and smooth domains containing the origin. The points zk∈Ω,1≤k≤m determine the location of the injectors. The domains ωk describe the injectors geometries.In the presence of injectors, the velocityuε and the pressure pε satisfy the following system:
(4)∂uε∂t+(uε·∇)uε-νΔuε+∇pε=𝒢inΩ∖⋃k=1mωzk,ε¯×]0,T],∇·uε=0inΩ∖⋃k=1mωzk,ε¯×]0,T],uε=udonΓ×]0,T],σ(uε,pε)·n=0onΓn×]0,T].
Note that for ε=0, (u0,p0) is solution to
(5)∂u0∂t+(u0·∇)u0-νΔu0+∇p0=𝒢inΩ×]0,T],∇·u0=0inΩ×]0,T],u0=udonΓ×]0,T],σ(u0,p0)·n=0onΓn×]0,T].
For the optimization criteria, we assume that a “good” lake oxygenation can be described by a target velocity 𝒰g. Then, the cost function 𝒥ε to be minimized is given by
(6)𝒥ε(uε)=∫0TJε(uε)dtwithJε(uε)=∫Ωm|uε-𝒰g|2dx,
where Ωm⊂Ω is the measurement domain (the top layer, see Figure 2).Figure 2
The top layerΩm.We denote byj the design function defined by
(7)j(Ω∖⋃k=1mωzk,ε¯)=𝒥ε(uε),
where uε is the solution of (4).The identification problem can be formulated as a topological optimization problem: find the optimal location of the injectorsωzk,ε=zk+εωk,1≤k≤m, inside the lake domain Ω minimizing the optimal design function j:
(8)(𝒪ε){Findzk*∈Ω,1≤k≤m,suchthatj(Ω∖⋃k=1mωzk*,ε¯∪k=1m)=minωzk,ε⊂Ωj(Ω∖⋃k=1mωzk,ε¯).To solve this optimization problem (𝒪ε) we will use the topological sensitivity analysis method [2, 3, 9–11]. It consists of studying the variation of the design function j with respect to a small topological perturbation of the domain Ω.


## 3.2. Time Discretization

The Lagrangian representation of the flow is based on the functionX:(x,s;t)∈Ω×(0,T)2→X(x,s;t), where X(x,s;t) is the position at time t of the particle of fluid which is at point x at time t=s. Thus t→X(x,s;t) is the parametric representation of the ajectory of the particle. Given the velocity u, the trajectory may be determined from the initial value problem:
(9)ddtX(x,s;t)=u(X(x,s;t),t),X(x,s;s)=x.Lemma 1.
If|s-t| is sufficiently small, then the Jacobian of the homeomorphism x→X(x,s;t) is equal to 1.Proof.
See [12].LetNT be a positive integer, Δt=T/NT and tn=nΔt. For x∈Ω, s=tn+1, tn≤t≤tn+1, 0≤n≤N, (9) implies using Euler scheme that
(10)x-X(x,tn+1;tn)=∫tntn+1u(X(x,tn+1;t),t)dt≈Δtu(X(x,tn+1;tn+1),tn+1)=Δtu(x,tn+1).
Then numerically the position X(x,tn+1;tn) is determined by
(11)X(x,tn+1;tn)=x-Δtu(x,tn+1).
Using (10) it is shown in [12] that material derivative is as follows:
(12)dudt=∂u∂t+(u·∇)u=u(x,tn+1)-u(X(x,tn+1;tn),tn)Δt.
This approximation is called characteristic method [13] and using Lemma 1 we show that (12) is L2 stable [14].Using (12), problem (4) is equivalent to solve the following generalized Stokes equation:
(13)Findun+1andpn+1solutionsofαuεn+1-νΔuεn+1+∇pεn+1=Fεn+1inΩ∖⋃k=1mωzk,ε¯,∇·uεn+1=0inΩ∖⋃k=1mωzk,ε¯,uεn+1=udonΓ,σ(uεn+1,pεn+1)·n=0onΓn,
where α=1/Δt, Fεn+1=(1/Δt)uεn(X(x,tn+1;tn))+𝒢,  uεn+1, and pεn+1 are the approximations of uε and pε on time tn+1. Then, at each time step, we have to solve a steady state generalized Stokes problem having the following generic form:
(14)αuε-νΔuε+∇pε=FεinΩ∖⋃k=1mωzk,ε¯,∇·uε=0inΩ∖⋃k=1mωzk,ε¯,uε=udonΓ,σ(uε,pε)·n=0onΓn.
Note that for ε=0, at each time step (u0,p0) is solution to
(15)αu0-νΔu0+∇p0=F0inΩ,∇·u0=0inΩ,u0=udonΓ,σ(u0,p0)·n=0onΓn.
Then, in order to obtain an asymptotic expansion of the cost function 𝒥ε for the Navier-Stokes problem, we study first the topological sensitivity analysis for the generalized Stokes problem by giving an asymptotic expansion of the cost function Jε.


## 3.3. Topological Sensitivity Analysis for the Generalized Stokes Problem

In this section we present the topological asymptotic expansion of the cost functionJε defined in (6) with respect to the insertion of a small injector ωz,ε=z+εω inside the domain Ω. From the week formulation of (14) one can show that uε is the solution to
(16)uε∈𝒱ε,𝒜ε(uε,w)=bε(w),∀w∈𝒱ε0,uε∣Γ=ud,uε∣∂ωz,ε=uinj,
where the functional space 𝒱ε, the bilinear form 𝒜ε, and the linear form bε are defined by
(17)𝒱ε={θ∈H1(Ω∖ωz,ε¯)3,divθ=0inΩ∖ωz,ε¯},𝒱ε0=𝒱ε∩H01(Ω∖ωz,ε¯)3,𝒜ε(u,w)=α∫Ω∖ωz,ε¯uwdx+ν∫Ω∖ωz,ε¯∇u∇wdx∀u,w∈𝒱ε,bε(w)=∫Ω∖ωz,ε¯Fεwdx∀w∈𝒱ε.
Next, we denote by u^ε the following extension of uε in Ω:
(18)u^ε={uεinΩ,uinjinωz,ε.
We begin first by presenting the topological asymptotic expansion of a cost function Jε defined on 𝒱ε and verifying the following hypothesis.Hypothesis 1.
(i)J0 is differentiable with respect to u; its derivative is denoted by DJ0(u).
(ii) There exists a real numberδJ such that ∀ε≥0(19)Jε(uε)-J0(u0)=DJ0(u0)(u^ε-u0)+εδJ+o(ε).
(iii) The adjoint problem(20)𝒜0(v0,w)=-DJ0(u0)w,∀w∈𝒱00,
has a unique solution v0∈𝒱00.The asymptotic expansion described in the following theorem is valid for arbitrary shaped holes and all cost function verifying the Hypothesis1. The case of spherical holes is given in (24) [11]. Finally we present the sensitivity analysis corresponding to the cost function Jε defined in (6).Theorem 2.
If Hypothesis1 holds, the function Jε has the following asymptotic expansion:
(21)Jε(uε)-J0(u0)=ε[(-∫∂ωη(y)ds(y))·v0(z)+δJ]+Δto(ε),
where η∈H-1/2(∂ω)3 is the solution to the boundary integral equation
(22)∫∂ωE(y-x)η(x)ds(x)=uinj-u0(z),∀y∈∂ω,
with (E, Π) is the fundamental solution of the Stokes equations: E(y)=(1/8πνr)(I+ererT), Π(y)=y/4πr3 with r=||y||, er=y/r and erT is the transposed vector of er.Proof.
This proof is an adaptation of the proof presented in [11].In the particular case whereω=B(0,1), the density η is given explicitly by [10]
(23)η(y)=-3ν2(u0(z)-uinj),∀y∈∂ω.
In this case and under the hypothesis of Theorem 2, the asymptotic expansion becomes
(24)Jε(uε)-J0(u0)=ε[6πν(u0(z)-uinj)·v0(z)+δJ]+Δto(ε).Proposition 3.
The functionJε given by (6) satisfies the Hypothesis 1 with
(25)DJ0(u0)(w)=2∫Ωm(u0-𝒰g)wdx,∀w∈𝒱0,δJ=0.
Then, The function Jε has the following expansion
(26)Jε(uε)-J0(u0)=2∫Ωm(u0-𝒰g)(u^ε-u0)dx+Δto(ε).Proof.
See [15].


## 3.4.  Topological Sensitivity Analysis for the Navier-Stokes Problem

In this section we consider the in stationary Navier-Stokes problem (4). We want to compute the variation j(Ω∖ωz,ε¯)-j(Ω) with respect to the insertion of a small injector in the fluid flow domain.Using (6), the variation of the design function (7) is given by
(27)j(Ω∖ωz,ε¯)-j(Ω)=𝒥ε(uε)-𝒥0(u0)=∫0TJε(uε)dt-∫0TJ0(u0)dt=∑n=1NT∫tn-1tn(Jε(uε)-J0(u0))dt,
where NT∈ℕ* and tn=nΔt, with Δt=T/NT is the time step.By trapezoidal formula, we get(28)∫tn-1tn(Jε(uε(·,t))-J0(u0(·,t)))dt=Δt2[(Jε(uεn)-J0(u0n))+(Jε(uεn-1)-J0(u0n-1))].
Using the sensitivity analysis for the generalized Stokes equations (21) and Proposition 3, we deduce the following asymptotic expansion
(29)j(Ω∖ωz,ε¯)-j(Ω)=12εΔtg(z)+o(ε),
where g(z)=∑n=1NT‍(δjn(z)+δjn-1(z)), with
(30)δjn(z)=(-∫∂𝒪ηn(y)ds(y))·v0n(z).
Here v0n and ηn are, respectively, the approximations at time tn=nΔt of the adjoint solution v0(·t) of problem (20) and η defined in (22). If ω is the unit ball B(0,1), by (24)
(31)δjn(z)=6πν(u0n(z)-uinj)·v0n(z).
In this case, using (25). The design function j defined in (7) has the following expansion:
(32)j(Ω∖ωz,ε¯)-j(Ω)=3πνεΔt×∑n=1NT‍[(u0n(z)-uinj)·v0n(z)+(u0n-1(z)-uinj)·v0n-1(z)]+o(ε),
where u0n and v0n are, respectively, solutions to the generalized Stokes problem (15) and its associated adjoint problem (20).


## 4. Numerical Results

This section is devoted to some numerical examples. Based on the previous theoretical results, we propose fast and accurate algorithms for the detection of optimal injectors location.As described in the previous section, each injectorInjk,1≤k≤m, is modeled as a small hole ωzk,ε=zk+εωk around zk∈Ω, having an injection velocity uinjk. In order to use the explicit topological sensitivity expression we assume that ωk=B(0,1).We recall that the optimal injectors location is characterized as the solution of the topological optimization problem(33)minωzk,ε⊂Ω∫0T(∫Ωm|uε-𝒰g|2dx)dt,
where Ωm is a measurement domain (see Figure 2), 𝒰g is an objective fluid flow, and uε is the solution to the Navier-Stokes equations in Ω∖∪k=1mωzk,ε¯.The detection procedure is based on the following algorithm.Algorithm 4.
(i) Forn=1,NT(a)
computeu0n, a solution to the generalized Stokes equations (15),(b)
computev0n, a solution to the associated adjoint problem (20).
(ii) For1≤k≤3, compute the sensitivity
(34)δjk(z)=∑n=1NT[(u0n(z)-uinjk)·v0n(z)+(u0n-1(z)-uinjk)·v0n-1(z)],∀z∈Ω.We note that the topological sensitivity functionδjk depends on u0n and v0n solutions, respectively, to the direct and adjoint problems computed in the domain Ω (not in the perturbed one Ω∖ωz,ε¯) which is an interesting feature of the used approach. The mixed finite element method “P1+bubble/P1” [16] is used for the space approximation of the problems (15) and (20). The obtained linear system is solved using Uzawa method [13]. One can consult [12] for the existence, uniqueness of the solution and the analysis of the error estimates of the discrete solution (uh,ph).In numerical tests, different values forNT and Δt are studied. The presented results in Figures 3–9 correspond to NT=60 and Δt=1s.Figure 3
Injector radius effect.Figure 4
Mesh step effect.Figure 5
The initial flow.Figure 6
The wanted velocity: a 2D cut on each injector.2D cuts of the topological sensitivityδj showing the exact location (black dots) and the local minima of δj (red zones).
(a)
(b)Figure 8
Obtained injectors location.Figure 9
The obtained velocity: a 2D cut on each injector.Next, some numerical examples are presented. The first and second tests presented in Sections4.1 and 4.2 concern, respectively, the study of the injector radius and the mesh step h effects on the optimization precess. We consider Ω=[0,1]×[0,1]×[0,1]. The domain Ω is discretized by tetrahedral mesh with mesh step h.The third example concerns the identification of separated small injectors in a water reservoir. In the last example we discuss the detection of an injector line containing a sequence of small injectors representing the injection holes. For the two last tests presented in Sections4.3 and 4.4, a 3D reservoir with length 8 m, width 4 m, and maximum height 3 m is considered. The domain is discretized using 16007 nodes and 85112 tetrahedral.


### 4.1. Sensitivity to the Injector’s Radius

The goal is to study the injector (considered as a sphere) radius effectr. We consider the case of a unique injector ω centered at z* and having radius r. We introduce the following error function:
(35)er(h,r)=∥zh-z*∥2r,
where ∥·∥ is the usual norm in ℝ3, z* is the exact injector location, and zh is the computed location using the mesh step h. We presenet in Figure 3 the obtained errors for h=0.02 and different injector radius r varying from 0.02 to 0.1. We remark that error rises when r increases.


### 4.2. Sensitivity to the Mesh Step

The mesh step effect is considered in this section. The error (35) is plotted in Figure 4 for a fixed radius r=0.02 and different mesh step h varying from 0.02 to 0.08. We remark that error rises when h increases.


### 4.3. Detection of Separated Small Injectors

This test concerns the detection of optimal location of separated small injectors. Each injector is characterized by a small hole. In the numerical computation, we have considered the case of three injectorsωk=zk+rB(0,1),1≤k≤3, where r<h and h is the mesh size equal in this case to 0.02.The wanted velocity field𝒰g is chosen as solution to the Navier-Stokes problem in Ω∖∪k=13ωzk*¯, where ωzk*=zk*+hB(0,1),1≤k≤3 are the injectors to be detected. The exact locations zk* and the injection velocities uinjk are described in Table 1.Table 1
The exact locationszk*,1≤k≤3 and their associated injection velocities.
InjectorInjk
Locationzk*
Injection velocityuinjk
Inj
1
x
=
-
1.599, y=-0.765, z=-1.041
u
x
=
-
0.3, uy=0.0, uz=-0.4
Inj
2
x
=
-
0.035, y=-0.058, z=-1.565
u
x
=
0.0, uy=0.0, uz=0.5
Inj
3
x
=
1.0, y=0.800, z=-1.023
u
x
=
-
0.2, uy=0.0, uz=0.4In Figure5, we present the initial flow. The wanted flow is shown in Figure 6.The isovalues of the topological sensitivity are plotted in Figure7. Using Algorithm 4, the injectors locations are given by the local minima of the topological sensitivity δj. We show in Figure 7, respectively, on the left a vertical cut and on the right a horizontal cut of the topological sensitivity function δj at each injector. One can observe that the local minima of the obtained topological sensitivity coincides with the exact location described in Table 1.The obtained optimal locations are shown in Figure8. The obtained velocity is shown in Figure 9. Since we have detected the exact location, the obtained velocity is identical to the wanted one.


### 4.4. Identification of an Injector Line

The aim here is the detection of an injector lineσ (see Figure 10). It is approximated by a sequence of small injectors (holes).Figure 10
Injector.In order to detectσ, we propose here an iterative process based on the following algorithm.Algorithm 5.
(i) Initialization: chooseΩ0=Ω, and set k=0.
(ii) Repeat until the target is reached:The injectorσ is detected iteratively, σk+1=σk∪Hk, with σ0=∅. The constant ck depends on the most negative value of δjk. In the presented test, ck=0.8δk with δk=minz∈Ωkδjk(z).The obtained results are shown in Figures11–14. Figure 11(b) illustrates the obtained fluid flow created by the detected injector. It is nearly identical to the wanted one (see Figure 11(a)).The wanted and obtained fluid flow on the measurement domainΩm.
(a)
𝒰g
(b)
u
ε after 3 iterations.Isovalues ofδjk showing the detected zone Hk (red zone) during the optimization process: a vertical cut (a), a horizontal cut (b).
(a)
(b)Figure 13
Injector position with respect toδj3.Figure 14
Objective function𝒥ε(uε).In Figure12 we plot the isovalues of δjk showing the detected zone Hk (where δjk is most negative) during the optimization process.One can note that the proposed algorithm (Algorithm5) permits to detect the injector line in only three iterations (Figure 13). Finally Figure 14 plots the objective function 𝒥ε(uε).


## 4.1. Sensitivity to the Injector’s Radius

The goal is to study the injector (considered as a sphere) radius effectr. We consider the case of a unique injector ω centered at z* and having radius r. We introduce the following error function:
(35)er(h,r)=∥zh-z*∥2r,
where ∥·∥ is the usual norm in ℝ3, z* is the exact injector location, and zh is the computed location using the mesh step h. We presenet in Figure 3 the obtained errors for h=0.02 and different injector radius r varying from 0.02 to 0.1. We remark that error rises when r increases.


## 4.2. Sensitivity to the Mesh Step

The mesh step effect is considered in this section. The error (35) is plotted in Figure 4 for a fixed radius r=0.02 and different mesh step h varying from 0.02 to 0.08. We remark that error rises when h increases.


## 4.3. Detection of Separated Small Injectors

This test concerns the detection of optimal location of separated small injectors. Each injector is characterized by a small hole. In the numerical computation, we have considered the case of three injectorsωk=zk+rB(0,1),1≤k≤3, where r<h and h is the mesh size equal in this case to 0.02.The wanted velocity field𝒰g is chosen as solution to the Navier-Stokes problem in Ω∖∪k=13ωzk*¯, where ωzk*=zk*+hB(0,1),1≤k≤3 are the injectors to be detected. The exact locations zk* and the injection velocities uinjk are described in Table 1.Table 1
The exact locationszk*,1≤k≤3 and their associated injection velocities.
InjectorInjk
Locationzk*
Injection velocityuinjk
Inj
1
x
=
-
1.599, y=-0.765, z=-1.041
u
x
=
-
0.3, uy=0.0, uz=-0.4
Inj
2
x
=
-
0.035, y=-0.058, z=-1.565
u
x
=
0.0, uy=0.0, uz=0.5
Inj
3
x
=
1.0, y=0.800, z=-1.023
u
x
=
-
0.2, uy=0.0, uz=0.4In Figure5, we present the initial flow. The wanted flow is shown in Figure 6.The isovalues of the topological sensitivity are plotted in Figure7. Using Algorithm 4, the injectors locations are given by the local minima of the topological sensitivity δj. We show in Figure 7, respectively, on the left a vertical cut and on the right a horizontal cut of the topological sensitivity function δj at each injector. One can observe that the local minima of the obtained topological sensitivity coincides with the exact location described in Table 1.The obtained optimal locations are shown in Figure8. The obtained velocity is shown in Figure 9. Since we have detected the exact location, the obtained velocity is identical to the wanted one.


## 4.4. Identification of an Injector Line

The aim here is the detection of an injector lineσ (see Figure 10). It is approximated by a sequence of small injectors (holes).Figure 10
Injector.In order to detectσ, we propose here an iterative process based on the following algorithm.Algorithm 5.
(i) Initialization: chooseΩ0=Ω, and set k=0.
(ii) Repeat until the target is reached:The injectorσ is detected iteratively, σk+1=σk∪Hk, with σ0=∅. The constant ck depends on the most negative value of δjk. In the presented test, ck=0.8δk with δk=minz∈Ωkδjk(z).The obtained results are shown in Figures11–14. Figure 11(b) illustrates the obtained fluid flow created by the detected injector. It is nearly identical to the wanted one (see Figure 11(a)).The wanted and obtained fluid flow on the measurement domainΩm.
(a)
𝒰g
(b)
u
ε after 3 iterations.Isovalues ofδjk showing the detected zone Hk (red zone) during the optimization process: a vertical cut (a), a horizontal cut (b).
(a)
(b)Figure 13
Injector position with respect toδj3.Figure 14
Objective function𝒥ε(uε).In Figure12 we plot the isovalues of δjk showing the detected zone Hk (where δjk is most negative) during the optimization process.One can note that the proposed algorithm (Algorithm5) permits to detect the injector line in only three iterations (Figure 13). Finally Figure 14 plots the objective function 𝒥ε(uε).


## 5. Conclusion

In this paper a numerical study of the optimal injector location problem in lake aeration process has been studied. The topological sensitivity analysis method is used to solve the optimization problem. The proposed numerical algorithm is fast, is easily implemented, and only needs to solve the direct and the adjoint problems on a fixed grid. In practice we have obtained interesting numerical results. Like all numerical methods, the proposed approach has its own drawbacks: on one hand, in the case of the simultaneous creation of multiple holes. The topological sensitivityδj provides information on where to create holes independently of its number. Then the obtained solution can be seen rather as a suggestion then as an optimal solution. On the other hand, the numerical realization via discretization may not take properly into account the correct influence of the shape of the hole assumed to be spherical of radius ε→0. To this aim we need a very fine grid which leads to many difficulties mainly due to the high computing costs necessary for such three-dimensional application. For this reason a parallel implementation of the numerical algorithm can be investigated in a forthcoming work.



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*Source: 875979-2013-02-28.xml*