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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Reconstruction of Tesla micro-valve using topological sensitivity analysis

M. Abdelwahed / N. Chorfi / R. Malek
Published Online: 2019-06-16 | DOI: https://doi.org/10.1515/anona-2020-0014

Abstract

In this paper, we deal with topology optimization attributed to the non stationary Navier-Stokes equations. We propose an approach where we analyze the sensitivity of a shape function relating to a perturbation of the flow domain. A numerical optimization algorithm based on topological gradient method is built and applied to the 2D Tesla micro valve reconstruction. Some numerical results confirm the efficiency of the proposed approach.

Keywords: Non-stationary Navier-Stokes equations; Tesla valve; Shape optimization; Topological sensitivity method

1 Introduction

Tesla valves are no-moving-part valves that utilize fluidic inertial forces to inhibit flow in the reverse direction. It was patented in 1920 by Nikola Tesla as a “Valvular conduit” [1] (see figure 1), and has since made the subject of various applications in micro-satellite [2], drug delivery [3], microbiology [4, 5] and hydrocephalus treatment in medicine [6, 7].

A rotated scanning electron microscope photograph of a Tesla valve by Forster et al. [8]
Fig. 1

A rotated scanning electron microscope photograph of a Tesla valve by Forster et al. [8]

The Tesla micro-valve performance is evaluated by the diodicity parameter (represents the ratio of the pressure drop in backward and forward direction) which evaluates the ability of allowing forward flow while inhibiting the reverse one,

Di=ΔpbackwardΔpforward.

Different works have focused on the optimal shape of the tesla micro-valve. However, the majority of works concerns stationary Partial Differential Equations (PDE). Forster et al. [8] proved the possibility of using Tesla valves in micro-fluidics and determined experimentally the diodicity for Reynolds number (Re) ≃ 180. Truong et al. in [9] derived numerically the optimum geometry of Tesla valve for 100 < Re < 600 with better diodicity than [8]. Bardell et al [10] analyzed the mechanism of the diodicity and proposed a Tesla valve optimal design for low Re. In the case when Re = 100, they finished with Di = 1.4. Gamboa et al. [11] optimized the shape of Tesla valve for application with piezoactuated plenums. The obtained fluid domain related to Re = 100 is characterized by a diodicity number Di = 1.1. In 2008, Pingen et al. [12] used the Lattice Bolzmann Method for the optimization of a micro Tesla valve without any information on diodicity. The used objective function was the pressure drop between inlet and outlet. After that in 2010, Lin et al. [13] used a topology optimization technique based on the power dissipation energy [14] of forward flow as objective function and the diodicity was built into the model as a constraint. For Re = 100, they found a new design of Tesla valve given Di = 1.2. Next in 2015, Lin et al. [15] solved the Tesla valve topology optimization using the approach of material distribution with inverse diodicity as objective function and fluid volume fraction as the constraint.

Until recently, there were no investigations dealing with the non-stationary case. We propose in this paper a new reconstruction method using the sensitivity analysis approach [16, 17, 18, 19] for a non stationary flow.

The principal results of this work concern both theoretical and numerical aspects associated with the Tesla micro-valve problem. The theoretical part is related to the analysis of the topological sensitivity for the non stationary Navier-Stokes equations. The numerical part concerns the 2D optimization of the Tesla micro-valve shape. The optimal shape is constructed by inserting obstacles in the considered initial domain. We build a simple and fast numerical reconstruction algorithm based on the topological gradient technique. The efficiency of the presented approach is confirmed by some numerical tests.

The paper is presented as following: Firstly we formulate the problem in section 2. Section 3 concerns the theoretical aspects. The numerical aspects are given in section 4. Finally section 5 includes Theorems proofs.

2 Problem formulation

Let Ω ⊂ ℝd, d = 2, 3 a bounded domain with regular boundary Γ = ∂Ω. We consider the blood as an incompressible viscous fluid flow described by the non stationary Navier-Stokes equations [20]. The velocity w and the pressure p satisfy the following system:

wt+wwνΔw+p=Gin Ω×]0,T[,divw=0in Ω×]0,T[,w=wdon Γ×]0,T[,w(.,0)=0in Ω,(1)

where ν is the kinematic viscosity coefficient, G is the gravitational force, T is the computational time and wd is a given Dirichlet boundary data. Because of the divergence free condition on w, wd must necessarily satisfy the compatibility condition,

Γwd(x,t).nds(x)=0,a.e. t]0,T[

where n is the unit outward normal vector along Γ.

Remark 2.1

Problem (1) has at least one solution (see [21](Ch.II, eq.(1.89)). If |w|1,Ω < ν/k, with

k=223meas(Ω)1/6ifd=3,12meas(Ω)1/2ifd=2,

then problem (1) has a unique solution (see [17]).

The topological sensitivity method idea is to study the variation of a given shape function j relating to a perturbation in the fluid flow domain geometry.

In structural shape optimization case (respectively electromagnetism and fluid dynamics cases) a geometry perturbation means removing some material (respectively the insertion of an obstacle).

Let 𝓞z,ε = z + ε𝓞, a small obstacle inserted in Ω characterized by its center z, its size ε and its shape 𝓞. 𝓞 is a bounded domain of ℝd containing the origin and 𝓞 (its boundary) is connected and piecewise 𝓒1.

The shape function variation is written

j(ΩOz,ε¯)j(Ω)=ρ(ε)δj(z)+o(ρ(ε)),zΩ

where

  • ερ(ε), a positive scalar function going to zero with ε

  • zδj(z), called the topological gradient, describes the shape function variation when an obstacle is inserted in z. It plays the role of descent direction in the algorithm of optimization.

To our knowledge, the majority of works leading with topological sensitivity method concern the stationary case such as Stokes problem [16, 18], quasi-Stokes [19], stationary Navier Stokes problem [17]. We extend this method to the nonlinear unsteady Navier Stokes flow. To overcome the difficulty due to the non linear operator and its associated adjoint problem we extend the perturbed velocity by zero in the inclusion which permits to use the adjoint method in the whole domain. For the time dependent term we will use the fundamental solution of the non stationary Stokes operator and decompose the velocity variation.

We define the time dependent shape function as:

j(ΩOz,ε¯)=0TJε(wε(.,t))dt,(2)

where Jε in H1(Ω ∖ 𝓞z,ε)d and wε is solution to

wεt+wεwενΔwε+pε=Gin Ωz,ε×]0,T[,divwε=0in Ωz,ε×]0,T[,wε=wdon Γ×]0,T[,wε=0on Oz,ε×]0,T[,wε(.,0)=0in Ωz,ε,(3)

with Ωz,ε = Ω ∖ 𝓞z,ε is the perturbed domain. Note that if ε = 0 (without obstacle), (w0, p0) verify (1) and Ω0 = Ω.

In the following, we will derive a general mathematical analysis for Jε satisfying the following assumption:

Assumption (𝓐)

  1. ε ≥ 0, tJε(wε(., t)) ∈ L1(0, T).

  2. J0 is differentiable in H1(Ω) and we denote DJ0(w) its derivative.

  3. ρ : ℝ+ ⟶ ℝ+ and δ𝓙 ∈ ℝ such that ∀ε ≥ 0

    0T[Jε(wε(.,t))J0(w0(.,t))]dt=0TDJ0(w0(.,t))(wε(.,t)w0(.,t))dt+ρ(ε)δJ+o(ρ(ε)).

3 Main results

We deal in this section with the non stationary Navier-Stokes topological sensitivity relating to the domain perturbation. We consider the shape functions verifying the assumption (𝓐).

3.1 Asymptotic behavior of the velocity variation

We first study the influence on the velocity vε = wεw0 of inserting a small obstacle Oz,ε in Ω. From (1) and (3), it is straightforward to show that (vε, pvε) satisfy the system

vεt+vεvενΔvε+vεw0+w0vε+pvε=0in Ωz,ε×]0,T[,divvε=0in Ωz,ε×]0,T[,vε=0on Γ×]0,T[,vε=w0on Oz,ε×]0,T[,vε(.,0)=0in Ωz,ε.(4)

We will distinguish in the following the 2D and 3D cases.

3.1.1 Three dimensional case

Theorem 3.1

There exists c > 0 independent of ε, such that

vε(x,t)W(x,t)L2(0,T;H1(Ωz,ε))cε,

where W = (W1, W2, W3) ∈ H1(Ωz,ε)3 is defined by

Wj(x,t)=Uj(xzε).w0(z,t),(x,t)R3O¯ε×]0,T[,(5)

with Uj is solution of (exterior Stokes problem)

νΔUj+Pj=0inR3O¯,divUj=0inR3O¯,Uj0at,Uj=ejonO,(6)

with {ej}j=1,2,3 is the3 canonical basis.

We show by using a single layer potential (see [22]) that

Uj(y)=OE(yx)ηj(x)ds(x),Pj(y)=OΠ(yx)ηj(x)ds(x),yR3O¯.

where

E(y)=18πνr(I+ererT),Π(y)=y4πr3yR3.

with r = ∥y∥, er = yr,erT is the transpose of er and ηjH–1/2(𝓞)3 is a solution of the boundary integral equation

OE(yx)ηj(x)ds(x)=ej,yO.(7)

Using Theorem 3.1 we obtain the following corollary.

Corollary 3.2

We have

vε(x,t)=W(x,t)+O(ε),xΩz,ε,t]0,T[.

3.1.2 Two dimensional case

Theorem 3.3

There exists c > 0 independent on ε, verifying

vε(x,t)1log(ε)W(x,t)L2(0,T;H1(Ωz,ε))clog(ε),

where

W(x,t)=4πνj=12[Ej(xz)w0(z,t)]ej,(x,t)Ωz,ε×]0,T[,(8)

with Ej(y) = E(y)ej, 1 ≤ j ≤ 2, {ej}j=1,2 is the2 canonical basis and

E(y)=14πν(log(r)I+ererT),Π(y)=y2πr2,yR2,

represents the fundamental solution of the Stokes System in2 with r = ∥yand er = yr.

Using Theorem 3.3 it follows the velocity estimation in the perturbed fluid flow domain.

Corollary 3.4

We have

vε(x,t)=W(x,t)+O(1log(ε)),xΩz,ε,t>0.

3.2 Asymptotic behavior of the shape function

The topological sensitivity analysis for the non stationary Navier-Stokes operator in three and two dimensional cases is given in this section. The presented results are satisfied by all shape functions j defined by (2) and Jε verifies the Assumption (𝓐).

3.2.1 Three dimensional case

Theorem 3.5

If Jε satisfies the Assumption (𝓐) with ρ(ε) = ε, then j defined by (2) verifies

j(ΩOz,ε¯)=j(Ω)+ε[0Tw0(z,t).MOu0(z,t)dt+δJ]+o(ε),

where

  1. the matrix 𝓜𝓞 is given by

    MOij=Oηji(y)ds(y),1i,j3.

  2. u0 is the solution to the adjoint problem

    u0tu0w0+w0Tu0νΔu0+pu0=DJ0(w0)inΩ×]0,T[,divu0=0inΩ×]0,T[,u0=0onΓ×]0,T[,u0(.,T)=0inΩ.(9)

Corollary 3.6

If 𝓞 = B(0, 1) (the unit ball), ηj(y) = 3ν2ej,yO and

j(ΩOz,ε¯)=j(Ω)+ε[0T6πνw0(z,t).u0(z,t)dt+δJ]+o(ε).

3.2.2 Two dimensional case

Theorem 3.7

If Jε satisfies the Assumption (𝓐) then j defined by (2) verifies

j(ΩOz,ε¯)=j(Ω)+1log(ε)[4πν0Tw0(z,t)u0(z,t)dt+δJ]+o(1log(ε)),

where u0 is the adjoint state solution to the problem (9).

The proofs of Theorems 3.1, 3.3, 3.5 and 3.7 are relegated to section 5. The variation δ𝓙 depends on the shape functions expressions. Some useful examples in numerical applications will be presented in section 3.3.

3.3 Shape function examples

3.3.1 First example

We define the shape function

j(ΩOz,ε¯)=0TΩz,εwεWd(.,t)2dxdt

where 𝓦dL1(0, T; H1(Ω)) is a datum representing a desired fluid flow state.

This example concerns the L2-norm shape function that has been used in geometric control problems like the optimization of location of some obstacle in a tank to approximate an object flow 𝓦d (see [16]).

Proposition 3.8

The function

Jε(w)=Ωz,εwWd(.,t)2dx,wH1(Ωz,ε),

satisfies the assumption (𝓐) with

DJ0(w0(.,t))v=2Ω(w0(.,t)Wd(.,t))vdx,vH1(Ω),δJ(z)=0,zΩ.

3.3.2 Second example

We define the shape function which corresponds to the dissipation energy minimization

j(ΩOz,ε¯)=0TΩz,εwεWd(.,t)2dxdt,

where 𝓦dL1(0, T; H2(Ω)) is a given datum. It was used in several optimization problems such as minimum drag problem [23], pipe bend design [10?], cavity example [24], reconstruction of Tesla valve [13].

Proposition 3.9

The function

Jε(w)=Ωz,ενwWd(.,t)2dx,wH1(Ωz,ε),

satisfies the assumption (𝓐) with

DJ0(w0(.,t))v=2νΩ(w0(.,t)Wd(.,t))vdx,vH1(Ω),δJ(z)=4πν0T|w0(z,t)|2dt,zΩifd=2,0T(Oη(y)ds(y)).w0(z,t)dt,zΩifd=3.

4 Numerical results

In this section, we deal with some numerical applications to validate the obtained theoretical results given in section 3.

4.1 Validation of the asymptotic expansion

To establish the numerical validation of Theorem 3.7, we consider the variation relating to ε of

Δz(ε)=j(ΩOz,ε¯)j(Ω)+1log(ε)δj(z),

where

δj(z)=4πν0Tw0(z,t)u0(z,t)dt+δJ.(10)

We expect to prove numerically that Δz(ε) satisfies the previously derived theoretical estimate Δz(ε) = o(1log(ε)).

To this aim, we consider the following data:

  • Ω = ]0, 1[×]0, 1[ is a square domain.

  • The locations zεi = zi + εB(0, 1) of the considered obstacles are arbitrary chosen (see Table 1).

    Table 1

    Location of obstacles

  • The shape function j is defined by the semi-norm

    j(Ωz,ε)=0TΩz,εwεWd2dxdt,(11)

    where 𝓦d is a given velocity state.

    In this case, the function Δzi(ε) is defined by (see Theorem 3.7 and Proposition 3.9):

    Δzi(ε)=j(ΩOεi¯)j(Ω)+4πνlog(ε)(0Tw0(zi,t)u0(zi,t)dt+0T|w0(zi,t)|2dt).

The validation algorithm uses the following steps:

The validation algorithm:

  • Step 1:

    • compute the solution w0 and the associated adjoint state u0 in the domain Ω.

    • determine j(Ω) defined by (11).

  • Step 2: For each obstacle zεi = zi + εB(0, 1), i = 1, …, 4:

    • determine the variation δj(zi) given in (10),

    • choose ε0i = max {ε > 0, such that zi + ε0i B(0, 1) ⊂ Ω},

    • compute an approximation of the function εj(ΩOεi¯),ε]0,ε0i].

  • Step 3: Deduce numerically the function εlog(|Δzi(ε)|),ε]0,ε0i].

For each considered obstacle Oεi=zi+εB(0,1), we plot in Figure 2 the variation of log(|Δzi(ε)|) relating to log(– log(ε)).

Variation of log(|Δzi(ε)|) relating to log(–log(ε)).
Fig. 2

Variation of log(|Δzi(ε)|) relating to log(–log(ε)).

We define βi to describe the behavior of εΔzi(ε) relating to –log(ε), i.e.

|Δzi(ε)|=Olog(ε)βi.

It corresponds to the slope of the line approximating the variation ε ↦ log(|Δzi(ε)|) relating to log(–log (ε)) for each obstacle zεi, i = 1, .., 4.

From the plotted curves in Figure 2, one deduce the slopes βi, i = 1, …, 4 in table 2.

Table 2

The obtained slopes βi of the lines associated with the obstacles Oεi, i = 1, ..., 4.

We deduce that the numerical results confirm the behavior predicted by the theoretical estimate

Δzi(ε)=o(1log(ε)).

4.2 The Tesla micro-valve application

The hydrocephalus treatment is a very important application in medicine. The problem is to optimize numerically the design of the 2D Tesla micro-valve at Re = 100. To solve this problem we consider the objective function as the forward energy dissipation and the diodicity as a constraint. The optimal domain is constructed through the insertion of some obstacles in the initial one. The problem leads to optimize the location of obstacles.

4.2.1 Shape optimization problem

We define Ω as the pentagon [15] having one inclined inlet Γin and one horizontal outlet Γout (see Figure 3).

Considered pentagon design domain
Fig. 3

Considered pentagon design domain

The aim is to find the fluid flow optimal domain Ω which minimizes the dissipated energy by the forward fluid flow and reproducing the original Tesla valve design given in Figure 1. This can be formulated:

Find Ω solution to minΩDadj(Ω),(12)

where

Dad={DΩ such that ΓinΓD,ΓoutΓD and |D|Vdesired},

with |.| and Vdesired represents respectively the Lebesgue measure and the target volume.

We recall that the performance of the Tesla valve is measured by diodicity Di which is known as the ratio of the pressure drop in backward direction to that in forward direction, which is equivalent to the ratio of dissipation of reverse and forward flows [15]:

Di=Φ(wr)Φ(wf) with Φ(w)=Ω[ν2i,j(wixj+wjxi)2],

with (wf, pf) and (wr, pr) are respectively the solution to the Navier Stokes system in the forward and the reverse flows. Then, diodicity can be maximized by minimizing forward dissipation while maximizing reverse dissipation. That is why our optimization problem is defined with the diodicity Di as a constraint; Di > 1.

Using the above definitions, the optimization problem [15] for reconstructing Tesla valve can be expressed as

  • *

    Objective: power dissipation of forward flow

    j(Ω)=Φ(wf)=ν0TΩ|wf|2dxdt,

  • *

    Constraints

    • -

      Volume fraction |Vdesired| < 0.8 |V0|.

    • -

      Diodicity Dic > 1.

    • -

      Navier Stokes equations for forward and backward directions.

We use the obtained theoretical results in 3.2.2 to solve (12).

4.2.2 The topology optimization process

To obtain the optimal domain, an iterative process is applied to construct a sequence of geometries (Ωk)k≥0 with Ω0 = Ω and Ωk+1 = Ωk ∖ 𝓞k where 𝓞k is an obstacle inserted in Ωk. To define the obstacle location and size, we find the function δjk defined by (see Theorem 3.7)

δjk(z)=4πν[0T(wk(z,t)uk(z,t)+wk(z,t)2)dt],zΩk,(13)

where

  • -

    wk represents the velocity, solution to the Navier-Stokes problem in Ωk

    wkt+wkwkνΔwk+pk=Gin Ωk×]0,T[,divwk=0in Ωk×]0,T[,wk=wdon Γ×]0,T[,wk=0on Σk×]0,T[,wk(.,0)=w0in Ωk.(14)

  • -

    uk is the adjoint state, solution to

    uktukwk1+wk1TukνΔuk+qk=DJ0(wk1)in Ωk×]0,T[,divuk=0in Ωk×]0,T[,uk=0on Γ×]0,T[,uk=0on Σk×]0,T[,uk(.,T)=0in Ωk,(15)

where Σk=(l=0kOl) is the obstacle boundary inserted during the previous iterations. The optimization steps are summarized as:

The Algorithm:

  1. Initialization: Set Ω0 = Ω, and k = 0

  2. Repeat until |Ωk| ≤ Vdesired:

    1. The topological sensitivity function:

      • -

        compute wk, solution to the non stationary Stokes problem (14) in Ωk,

      • -

        compute vk, solution to the associated adjoint problem (15) in Ωk,

      • -

        compute the term δ𝓙k and deduce the function δjk(z), ∀zΩk.

    2. The obstacle to be inserted:

      • -

        determine ρk[0,1] such that j(ΩkOρkk¯)j(ΩkOρk¯),ρ[0,1],

      • -

        set Ok={xΩk;δjk(x)ρkδmink}, where δmink=min(δjk(z)).

    3. The new domain:

      • -

        set Ωk+1 = Ωk ∖ 𝓞k,

    4. kk + 1 and go to (2).

The stopping criteria is defined by the natural optimality condition

δjk(x)0,xΩk.

This algorithm is like a descent method where δjk represents the descent direction and |𝓞k| = |ΩkΩk+1| the step length. The parameter ρk is chosen to allow ρj(ΩkOρk¯) to decrease as much as possible. The computation of ρk in (b) can be viewed as line search step.

The numerical discretization of problems (14) and (15) is done by P1-bubble/P1 finite element method [25]. The computation of the approximated solutions is achieved by the Uzawa’s algorithm. The function δjk is computed piecewise constant over elements.

Next, we will apply the proposed algorithm to reconstruct Tesla micro valve.

4.2.3 Reproducing the Tesla micro valve

We illustrate in this section the strengths of topology optimization method, namely the ability to find optimal design using only information on boundary conditions and constraints without the need of initial design.

The considered design domain is the pentagon domain (see Figure 3). This problem example has already been studied by S. Lin and al. in [15] in the steady state regime using projection method.

For the forward direction, the inlet boundary velocity has a parabolic behavior (Re = 100 relating to the inlet dimension). At the outlet boundary, the pressure is taken constant and no-slip condition is considered on the walls. For the backward flow direction, we reverse these boundary conditions. Besides, we prescribe solid regions close to the inlets/outlets to minimize the boundary effect on the final design solution.

We illustrate the geometries obtained during the optimization process in Figure 4. The optimal domain is obtained after four iterations. It is nearly identical to literature [1, 15] (see Figure 5).

Geometries obtained during the optimization process
Fig. 4

Geometries obtained during the optimization process

mesh of obtained design (left) and reference Tesla valve (right)
Fig. 5

mesh of obtained design (left) and reference Tesla valve (right)

4.2.4 Discussion

In the previous Figure, thanks to the topological gradient, we deduce an easy reconstruction of tesla valve. Now, we normalize the obtained tesla valve behavior by plotting the obtained forward and reverse flows respectively in figures 6(a) and 6(b). It is clear that the velocity field is strongly different for the two cases.

Forward and backward flow velocity field
Fig. 6

Forward and backward flow velocity field

To study the obtained tesla valve performance, we calculate the diodicity. Using the energy view point expression of diodicity, the experimentally derived value is 1.137. In bibliography [26], the diodicity is well predicted using

Di1+4.78105(N0.16Re1.72)

with N is the number of tesla valves and Re is the Reynolds number. Based on this expression, we found Di ≅ 1.1316 which ensures an agreement between the obtained diodicity and the experimental one.

5 Mathematical analysis

This section deals with the proofs of Theorems 3.1, 3.3, 3.5 and 3.7.

5.1 Proof of Theorem 3.1

Let Q be the pressure associated with the velocity W:

Q(x,t)=1εP(xzε).w0(z,t)=1εj=13Pj(xzε)w0j(z,t),(16)

where Pj is the pressure associated with the velocity Uj solution to (6). Setting the variation

zε=vεW and pzε=pvεQ.(17)

From (4) and (6), we can verify that (zε, pzε) is solution to

zεtνΔzε+zε(w0+W)+(w0+W)zε+zεzε+pzε=Wtw0WWw0WWin Ωz,ε×]0,T[,divzε=0in Ωz,ε×]0,T[,zε=Won Γ×]0,T[,zε=w0(x,t)+w0(z,t)on Oε×]0,T[,zε(.,0)=0in Ωz,ε.(18)

The last boundary condition follows due to the fact that Uj = –ej on 𝓞.

Moreover, since |w0|L2(0,T;H1(Ω)) < ν/k, then ε sufficiently small,

|w0+W|L2(0,T;H1(Ωz,ε))α<ν/k.

Let R > 0 such that 𝓞z,εB(z, R) and B(z, R) ⊂ Ω. Using the trace theorem, we obtain

zεL2(0,T;H1(Ωz,ε))c(WtL2(0,T;L2(Ωz,ε))+WL2(0,T;H1(ΩR))+w0(x,t)w0(z,t)L2(0,T;L2(Ωz,ε))+w0W+Ww0+WWL2(0,T;H1(Ωz,ε))),(19)

where ΩR = ΩB(z, R).

Using (5) and the variable change x = z + εy, we obtain

WtL2(0,T;L2(Ωz,ε))=w0t(z,.)L2(0,T)U(xzε)L2(Ωz,ε)=ε3/2w0t(z,.)L2(0,T)UL2((Ωz,ε)/ε).

By the same way, we have

WL2(0,T;H1(ΩR))w0(z,.)L2(0,T)(U(xzε)L2(ΩR)+xU(xzε)L2(ΩR)),w0(z,.)L2(0,T)(ε3/2UL2((ΩR)/ε)+ε1/2yUL2(ΩR)/ε)).

Using [19] (see also [27]), the velocity field Uj, solution to the exterior Stokes problem, satisfies the estimate

UjL2((ΩR)/ε)cε1/2 and yUjL2(ΩR)/ε)cε1/2.

Then, using the smoothness of w0 and the previous estimates, one can deduce

WtL2(0,T;L2(Ωz,ε))cε and WL2(0,T;H1(ΩR))cε.(20)

For the third term in (19). Expanding w0(x, t) = w0(z, t) + εw0(ξy, t)y with ξy ∈ 𝓞z,ε and using the fact that ∇ w0 is uniformly bounded, it follows that

w0(x,t)w0(z,t)L2(0,T;L2(Ωz,ε))cε.(21)

We now examine the last term in (19). Since w0L(Ω),

w0W+Ww0+WWL2(0,T;H1(Ωz,ε))c(WL2(0,T;H1(Ωz,ε))+WL2(0,T;H1(Ωz,ε))+WWL2(0,T;H1(Ωz,ε))),c(WL2(0,T;L2(Ωz,ε))+|W|L2(0,T;H1(Ωz,ε))WL2(0,T;H1(Ωz,ε))),

according to Lemma 4.2 in [17].

In addition, by Lemma 4.5 in [17], the variable change and the continuity of w0, we can deduce

WL2(0,T;L2(Ωz,ε))cε,|W|L2(0,T;H1(Ωz,ε))cε1/2(22)

and then

w0W+Ww0+WWL2(0,T;H1(Ωz,ε))cε.(23)

Finally, combining (20), (21) and (23) we deduce that

zεL2(0,T;H1(Ωz,ε))cε.

5.2 Proof of Theorem 3.3

Let Q be the pressure associated with the velocity W:

Q(x,t)=4πνΠ(xz).w0(z,t)=4πνj=12Πj(xz)w0j(z,t),

where Πj is the pressure associated with the velocity Ej.

Setting

zε=vε1log(ε)W and sε=pvε1log(ε)Q.(24)

From (1) and (3), we obtain that (zε, sε) is solution to

zεtνΔzε+zε(w0+1log(ε)W)+(w0+1log(ε)W)zε+zεzε+sε=1log(ε)[Wtw0WWw01log(ε)WW]in Ωz,ε×]0,T[,divzε=0in Ωz,ε×]0,T[,zε=1log(ε)Won Γ×]0,T[,zε=w0(x,t)4πνlog(ε)E(xz)w0(z,t)on Oz,ε×]0,T[.(25)

Using the relation E((xz)/ε)=E(xz)+log(ε)4πνI, the last boundary condition can be rewritten as

zε=w0(x,t)+w0(z,t)4πνlog(ε)E((xz)/ε)w0(z,t) on Oz,ε×]0,T[.

Then, by an energy inequality [28], it follows

zεL2(0,T;H1(Ωz,ε))clog(ε)[WtL2(0,T;L2(Ωz,ε))+WL2(0,T;H1/2(Γ))+log(ε)w0(z+εy,t)w0(z,t)L2(0,T;H1/2(Oz,ε))+4πνE((xz)/ε)w0(z,t)L2(0,T;H1/2(Oz,ε))+w0W+Ww0+1log(ε)WWL2(0,T;L2(Ωz,ε))].(26)

We estimate in the following each term in (26) separately.

We remark that:

  • Since 𝓞 is an open domain containing the origin, ∃r > 0 such that B(0, r) ⊂ 𝓞.

  • Ω is a bounded domain in such a way that ∃R > 0 such that ΩB(z, R), ∀zΩ.

  • We have Ωz,εz = {xz, xΩz,ε} ⊂ C(0, , R) = {y ∈ ℝ2; < |y| < R}.

From the fact that C(0, , R) ⊂ ℝ2 ∖ {0}, it follows that the function ψ : y ↦ log(|y|) is smooth in C(0, , R) and we have ∥ψ0,C(0,,R)c. Then, using the cylindrical coordinate system, one can prove that ∃c > 0, independent of ε, such that

E(xz)L2(0,T;L2(Ωz,ε))E(y)C(0,rε,R)c,(27)

E(xz)L2(0,T;L2(Ωz,ε)clog(ε).(28)

  • Estimate of the first term in (26): Using that w0H1(0, T; H1(Ω)), we obtain

    WtL2(0,T;L2(Ωz,ε))=4πνw0t(z,t)L2(0,T)E(xz)L2(Ωz,ε)=O(1).

  • Estimate of the last term of (26):

    Since w0 and ∇ w0 belong to L(Ω), we have

    w0W+Ww0+1log(ε)WWL2(0,T;L2(Ωz,ε))c(WL2(0,T;L2(Ωz,ε))+WL2(0,T;L2(Ωz,ε))+1log(ε)WL2(0,T;L2(Ωz,ε))WL2(0,T;L2(Ωz,ε))).

    Using the definition of W, we can deduce the following estimates

    WL2(0,T;L2(Ωz,ε))c,WL2(0,T;L2(Ωz,ε))clog(ε).(29)

    Yet, we have

    w0W+Ww0+1log(ε)WWL2(0,T;L2(Ωz,ε))clog(ε).(30)

  • Estimate of boundary condition imposed on Γ:

    Let > 0 such that 𝓞z,εB(z, ) and B(z, ) ⊂ Ω. Since zΩ = ΩB(z, ), the function xE(xz) belongs to 𝓒1(Ω). By the trace theorem, we have

    WL2(0,T;H1/2(Γ)=4πνw0(z,t)L2(0,T)E(xz)H1/2(Γ)4πνw0(z,t)L2(0,T)[E(xz)L2(ΩR)+E(xz)L2(ΩR)].

    Therefore, ∥WL2(0,T;H1/2(Γ) is uniformly bounded with respect to ε.

  • Estimate of boundary condition imposed on 𝓞z,ε:

    Using the theorem of trace and the smoothness of w0 in 𝓞z,ε×]0, T[, one can obtain

    w0(x,t)w0(z,t)L2(0,T;H1/2(Oz,ε))cε.

    Then, the first boundary term on 𝓞z,ε satisfies

    log(ε)w0(x,t)w0(z,t)L2(0,T;H1/2(Oz,ε))=o1log(ε).

    To estimate the last boundary term, we use that 𝓞 contains the origin.

    Setting 𝓞r = 𝓞∖B(0, r) and 𝓞r, ε = z + ε𝓞r. Using the theorem of trace and the variable change x = z + εy, we obtain

    E((xz)/ε)w0(z,t)L2(0,T;H1/2(Oz,ε))w0(z,t)L2(0,T)(E((xz)/ε)L2(Or,ε)+xE((xz)/ε)L2(Or,ε))w0(z,t)L2(0,T)(εE(y)L2(Or)+ε1/2yE(y)L2(Or)).

    From the fact that yE(y) is sufficiently smooth in 𝓞r ⊂ ℝ2 ∖ {0}, the last quantity is uniformly bounded and then

    4πνlog(ε)E((xz)/ε)w0(z,t)L2(0,T;H1/2(Oz,ε))cε1/2log(ε).

    Finally, combining the above estimates, we obtain, ∃c > 0, independent of ε, such as

    zεL2(0,T;H1(Ωz,ε))clog(ε)

    which ends the proof of Theorem 3.3.

5.3 Asymptotic analysis

This section deals with the proofs of the Theorems presented in paragraphs 3.2 and 3.3. Using the assumption (𝓐),

j(ΩOz,ε¯)j(Ω)=0TJε(wε(.,t))dt0TJ0(w0(.,t))dt=0TDJ0(w0(.,t))(wε(.,t)w0(.,t))dt+ρ(ε)δJ+o(ρ(ε)),(31)

where wε is extended by zero inside the domain 𝓞z,ε.

Using Green formula and that wε = 0 in 𝓞ε, it follows

j(ΩOz,ε¯)j(Ω)=ν0TΩz,εvεu0dxdt0TΩz,εvεtu0dxdt+0TOz,εw0tu0dxdt+ν0TOz,εw0u0dxdt0TΩz,ε(vεw0+w0vε)u0dxdt+20TOz,ε(w0w0)u0dxdt+ρ(ε)δJ+o(ρ(ε)),

where u0 is the solution to the associated adjoint problem.

From (4) and the fact that w0 = 0 on Γ×]0, T[, we obtain

ν0TΩz,εvεu0dxdt0TΩz,εvεtu0dxdt0TΩz,ε(vεw0+w0vε)u0dxdt=0TOz,εσ(vε,pvε)nu0dsdt+0TΩz,ε(vεvε)u0dxdt.(32)

Therefore,

j(ΩOz,ε¯)j(Ω)=0TOz,εw0tu0dxdt+ν0TOz,εw0u0dxdt+20TOz,ε(w0w0)u0dxdt0TOz,εσ(vε,pvε)nu0dsdt+0TΩz,ε(vεvε)u0dxdt+ρ(ε)δJ(z)+o(ρ(ε)).(33)

We begin by giving the estimate of the first three terms in (33).

Lemma 5.1

The integral terms in (33) satisfy the estimate

0TOz,εw0tu0dxdt+ν0TOz,εw0u0dxdt+20TOz,ε(w0w0)u0dxdt=O(εd).

Proof

Using the variable change x = z + εy, the first integral term in (33) can be written

0TOz,εw0tu0dxdt=εd0TO(w0(z+εy,t)tu0(z+εy,t)w0(z,t)tu0(z,t)dydt+εd|O|0Tw0(z,t)tu0(z,t)dt,

where |𝓞| denotes the Lebesgue measure of 𝓞.

Using that w0 and u0 are smooth near z, one can deduce that

0TOz,εw0tu0dxdt+ν0TOz,εw0u0dxdt+20TOz,ε(w0w0)u0dxdt=O(εd).

By the same arguments, we can estimate the two other terms in (33).

The shape function variation can be rewritten

j(ΩOz,ε¯)j(Ω)=0TOz,εσ(vε,pvε)nu0dsdt+0TΩz,ε(vεvε)u0dxdt+ρ(ε)δJ(z)+o(ρ(ε)).

We are now ready to prove the established results in Theorems 3.5 and 3.7 and propositions 3.8 and 3.9.

5.3.1 Proof of Theorem 3.5

Using an integration by parts and the fact that div(vε) = 0 yield

|0TΩz,ε(vεvε)u0dxdt|=|0TΩz,ε((u0)(vε))vεdxdt|u0L(Ωz,ε)vεL2(Ωz,ε)22u0L(Ωz,ε)(zεL2(Ωz,ε)2+WL2(Ωz,ε)2)cε2.(34)

Then, the shape function variation can be written

j(ΩOz,ε¯)j(Ω)=0TOz,εσ(vε,pvε)nu0dsdt+εδJ(z)+o(ε).

From the definition of (zε, sε) and the variable change x = z + εy, we have

0TOz,εσ(vε,pvε)nu0dsdt=0TOz,εσ(zε,sε)nu0dsdt+ε0Tw0(z,t).(Oσ(U,P)(y)n(y)u0(z+εy,t)ds(y))dt,

where σ(U, P)n is the 3 × 3 matrix defined by

(σ(U,P)n)ij=(σ(Uj,Pj)(y)n(y))i,1i,j3.

By the trace theorem, Theorem 3.1 and that u0 is smooth in 𝓞z,ε,

|0TOz,εσ(zε,sε)nu0dsdt|σ(zε,sε)nL2(0,T;H1/2(Oz,ε))u0L2(0,T;H1(Oz,ε))=o(ε).

Making the variable change x = z + εy, expanding u0(z + εy, t) = u0(z, t) + εu0(ξy, t)y with ξy ∈ 𝓞z,ε and using that ∇u0 is uniformly bounded, we obtain

0TOz,εσ(vε,pvε)nu0dsdt=ε0Tw0(z,t).(Oσ(U,P)(y)nds(y))u0(z,t)dt+ε0Tw0(z,t)(Oσ(U,P)(y)n(y)[u0(z+εy,t)u0(z,t)]ds(y))dt+o(ε).

Due to the jump condition of the single layer potential σ(Uj, Pj)n = –ηj + σ(Vj, Sj)n, where (Vj, Sj) is the solution to the interior problem

νΔVj+Sj=0 in O,divVj=0 in O,Vj=Uj on O.

By the fact that div σ(Vj, Sj) = νΔVj – ∇ Sj = 0 in 𝓞, we have Oσ(Vj,Sj)(y)nds=0.

Then, we obtain

0TOz,εσ(vε,pvε)nu0dsdt=ε0Tw0(z,t).(Oη(y)ds(y)u0(z,t))dt+o(ε).

Consequently, the shape function j admits the asymptotic expansion

j(ΩOz,ε¯)=j(Ω)+ε[0Tw0(z,t).MOu0(z,t)dt+δJ]+o(ε),

where 𝓜𝓞 is the matrix given by

MOij=Oηji(y)ds(y),1i,j3.

5.3.2 Proof of Theorem 3.7

The shape function variation is given by

j(ΩOz,ε¯)j(Ω)=0TOz,εσ(vε,pvε)nu0dsdt+0TΩz,ε(vεvε)u0dxdt+1log(ε)δJ(z)+o(1log(ε)).

Recall that the term (W, Q) describing the perturbation due to the presence of a small obstacle 𝓞z,ε is given by: ∀(x, t) ∈ Ωz,ε×]0, T[,

W(x,t)=4πνj=12[Ej(xz)w0(z,t)]ej,Q(x,t)=4πνj=12Πj(xz)w0j(z,t),

where Ej(y) = E(y)ej and Πj(y) = Π(y).ej, 1 ≤ j ≤ 2.

Applying an integration by parts and using the fact that div(vε) = 0 provides

0TΩz,ε(vεvε)u0dxdt=0TΩz,ε(u0vε)vεdxdt.

Then,

|0TΩz,ε(vεvε)u0dxdt|u0L(Ωz,ε)vεL2(Ωz,ε)22u0L(Ωz,ε)[zεL2(Ωε)2+WL2(Ωε)2]c(1log(ε))2=o(1log(ε)).(35)

It follows that

j(ΩOz,ε¯)j(Ω)=0TOz,εσ(vε,pvε)nu0dsdt+1log(ε)δJ(z)+o(1log(ε)).

Then, from the decomposition (24), one can derive

0TOz,εσ(vε,pvε)nu0dsdt=0TOz,εσ(zε,sε)nu0dsdt+4πνlog(ε)0Tw0(z,t)(Oz,εσ(E,Π)(xz)nu0(x,t)ds(x))dt,(36)

where σ(E, Π)n is the 2 × 2 matrix defined by (σ(E, Π)n)i,j = (σ(Ej, Πj)n)i, 1 ≤ i, j ≤ 2.

Using Theorem 3.3 and the smoothness of u0 in 𝓞z,ε, it follows

|0TOz,εσ(zε,sε)nu0dsdt|=o(1log(ε)).

The second term in (36) can be written

Oz,εσ(E,Π)(xz)nu0(x,t)ds(x)=Oz,εσ(E,Π)(xz)n[u0(x,t)u0(z,t)]ds(x)+Oz,εσ(E,Π)(xz)nu0(z,t)ds(x).

Using the trace theorem and the variable change x = z + εy, one can obtain

|0Tw0(z,t)(Oz,εσ(E,Π)(xz)n[u0(x,t)u0(z,t)]ds(x))dt|cw0(z,t)L2(0,T)σ(E,Π)(xz)nH1/2(Oz,ε)u0(x,t)u0(z,t)L2(0,T;H1/2(Oz,ε)).

By the fact that u0 is smooth in 𝓞z,ε, it follows

limε0u0(x,t)u0(z,t)L2(0,T;H1/2(Oz,ε))=0.

Recall that B(0, r) ⊂ 𝓞, 𝓞r = 𝓞 ∖ B(0, r) and 𝓞r,ε = z + ε𝓞r ⊂ 𝓞z,ε. Here, one can check that the function xσ(E, Π)(xz) is smooth in 𝓞r,ε. Using the trace theorem, we prove that the quantity ∥σ(E, Π)(xz)nH–1/2(𝓞z,ε) is bounded with respect to ε, which implies

4πνlog(ε)0Tw0(z,t)(Oz,εσ(E,Π)(xz)n[u0(x,t)u0(z,t)]ds(x))dt=o(1log(ε)).

Combining the above estimates, one can deduce

0TOz,εσ(vε,pvε)nu0dsdt=4πνlog(ε)0Tw0(z,t)(Oz,εσ(E,Π)(xz)nds(x))u0(z,t)dt+o(1log(ε)).

Since div (σ(Ej, Πj)(xz)) = δzej in 𝓞z,ε, it follows

Oεσ(E(xz),Π(xz))nds=I,

where I is the 2 × 2 identity matrix.

Then, the last estimate becomes

0TOz,εσ(vε,pvε)nu0dsdt=4πνlog(ε)0Tw0(z,t)u0(z,t)dt+o(1log(ε)).(37)

Consequently, all shape functions j satisfying the assumption (𝓐) admit the asymptotic expansion

j(ΩOz,ε¯)=j(Ω)+1log(ε)[4πν0Tw0(z,t)u0(z,t)dt+δJ(z)]+o(1log(ε)).

5.3.3 Proof of Proposition 3.8

Since the desired fluid flow state 𝓦dL2(0, T; H1(Ω)), the function J0 is differentiable at w0(., t) and we have

DJ0(w0(.,t))(v)=2Ω(w0(.,t)Wd(.,t))vdx,vH1(Ω).

The variation of the associated shape function j is given by

j(Ωz,ε)j(Ω)=0TΩz,ε|wεWd|2dxdt0TΩ|w0Wd|2dxdt=0TDJ0(w0)(wεw0)dt+0TΩz,ε|wεw0|2dxdt+0TOz,ε|w0|2dxdt0TOz,ε|Wd|2dxdt.

Using the smoothness of w0 and 𝓦d in Ω, one can conclude that

0TOz,ε|w0|2dxdt=o(ε) and 0TOz,ε|Wd|2dxdt=o(ε).

  • -

    For the two-dimensional case: Using the decomposition (24), it follows

    0TΩz,ε|wεw0|220TΩz,ε|zε|2dxdt+1(log(ε))20TΩz,ε|W|2dxdt.

    From Theorem 3.3, one can check

    0TΩz,ε|zε|2dxdt=o(1log(ε)).

    Making use of (27), one can deduce

    WL2(0,T;L2(Ωz,ε))=4πνw0(z,t)L2(0,T)E(xz)L2(Ωz,ε)=O(1).

    Then, it follows

    1(log(ε))20TΩz,ε|W|2dxdt=o(1log(ε)).

  • -

    For the three-dimensional case: Using the decomposition (17), it follows

    0TΩz,ε|wεw0|22(0TΩz,ε|zε|2dxdt+0TΩz,ε|W|2dxdt).

    Using Theorem 3.1 and the change of variable, one can check

    0TΩz,ε|zε|2dxdt=o(ε) and 0TΩz,ε|W|2dxdt=o(ε).

Therefore the function Jε satisfies the assumption (𝓐) with

DJ0(w0(.,t))(v)=2Ω(w0(.,t)Wd(.,t))vdx,vH1(Ω),δJ(x)=0,xΩ.

5.3.4 Proof of Proposition 3.9

The function J0 is differentiable at w0(., t) and we have

DJ0(w0(.,t))(v)=2νΩ(w0(.,t)Wd(.,t))vdx,vH1(Ω).

The variation of the associated shape function j is given by

j(Ωz,ε)j(Ω)=0TDJ0(w0)(wεw0)dtν0TOz,ε|Wd|2dxdt+ν0TOz,ε|w0|2dxdt+ν0TΩz,ε|wεw0|2dxdt.(38)

Thanks to the regularity of w0 and 𝓦d in 𝓞z,ε, one can derive

0TOz,εν|w0|2dxdt=o(ε),0TOz,εν|Wd|2dxdt=o(ε).

  • -

    For the two-dimensional case: By an adaptation of the technique used in the proof of Theorem 3.7, one can derive

    0TOz,εσ(zε,sε)nw0dsdt=4πνlog(ε)0Tw0(z,t)2dt+o(1log(ε)).

    Therefore, the function Jε satisfies the assumption (𝓐) with

    DJ0(w0(.,t))(v)=2νΩ(w0(.,t)Wd(.,t))vdx,vH1(Ω),and δJ(x)=4πν0T|w0(z,t)|2dt,xΩ.

  • -

    For the three-dimensional case: By an adaptation of the technique used in the proof of Theorem 3.5, one can derive

    0TOz,εσ(zε,sε)nw0dsdt=ε[0Tw0(z,t).MOw0(z,t)dt]+o(ε).

    Therefore, the function Jε satisfies the assumption (𝓐) with

    DJ0(w0(.,t))(v)=2νΩ(w0(.,t)Wd(.,t))vdx,vH1(Ω),and δJ(z)=0Tw0(z,t).MOw0(z,t)dt,zΩ.

6 Conclusion

This paper deals with non-stationary Navier-Stokes topological optimization problem. In the theoretical part of this work, we have established a topological asymptotic formula describing the shape function variation related to a small Dirichlet geometric perturbation.

The obtained theoretical results are exploited for building a topological optimization algorithm for solving the Tesla micro-valve optimization problem. We illustrate the strengths of this approach namely the ability to find optimal design based only on boundary conditions and constraints information without the need of an initial design.

Acknowledgement

The authors acknowledge funding from the Research and Development (R&D) Program (Research Pooling Initiative), Ministry of Education, Riyadh, Saudi Arabia, (RPI-KSU). We are very grateful to Professor Maatoug Hassine for the interesting discussions that have improved the quality of this document.

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About the article

Received: 2018-11-18

Accepted: 2019-01-14

Published Online: 2019-06-16

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 567–590, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0014.

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© 2020 M. Abdelwahed et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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