On the problem of detecting source points acting on a fluid

Mohamed Abdelwahed; Nejmeddine Chorfi

doi:10.1515/dema-2023-0108

Article Open Access

On the problem of detecting source points acting on a fluid

Mohamed Abdelwahed and Nejmeddine Chorfi

Published/Copyright: November 24, 2023

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From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

The detection problem of a finite number of source points acting on a steady incompressible fluid flow from overdetermined boundary data was studied. The approach used in this study deals with the topological sensitivity technique. An asymptotic analysis of a prescribed cost function with respect to the domain perturbation was developed. Some numerical results to illustrate the efficiency and robustness of the developed source point detection algorithm were presented.

Keywords: optimization; source point detection; topological sensitivity; Stokes equations

MSC 2010: 35Q30; 49Q10; 74P05

1 Introduction

Consider Ω ⊂ R 2 , an open-bounded domain occupied by a stationary viscous incompressible fluid flow governed by the Stokes equations:

(1) − ν Δ u + ∇ p = F in Ω div u = 0 in Ω u = u * on Γ ,

where u is the fluid velocity, p is the pressure, ν is the kinematic viscosity, u * is a given boundary velocity on the boundary Γ = ∂ Ω , and

F = ∑ i = 1 N ℱ i δ x i

is a source term representing the total action of a finite number of particles, where δ x i represents the Dirac function at x i ∈ Ω and ℱ i is a constant source point force. We suppose that a given other boundary condition σ ( u , p ) n = ϕ on the stress tensor, σ ( u , p ) = − p I + μ ( ∇ u + ∇ u T ) is given on Γ , where I denotes the 2 × 2 identity matrix and n is the unit normal vector along the boundary Γ .

The aim is to identify the unknown source points [1]. Such a problem can be motivated by the study of the flow near colloidal particle phenomena [2,3]. This problem has been studied in [4] using a game strategy approach.

Our purpose is to use the topological sensitivity analysis method [5–11] associated with Kohn-Vogelius formulation [12–15] to solve this problem. We begin in Section 2 by the state of the detection point source inverse problem. Section 3 is devoted to the asymptotic development analysis based on the topological sensitivity method. In Section 4, an iterative algorithm based on the developed approach is presented. Finally, some two- and three-dimensional numerical tests are presented in Section 5 to illustrate the efficiency of the presented algorithm.

2 Detection of source point problem

2.1 Direct problem

The direct problem consists in studying the action of source points on a viscous incompressible fluid described by the Stokes equations (1).

Because of the divergence-free condition, we suppose that u * satisfies the following compatibility condition:

(2) ∫ Γ u * . n d s = 0 .

To discuss the well-posedness of the direct problem, we introduce the fundamental solution ( E , P ) of the Stokes equations in the two-dimensional case

E ( y ) = 1 4 π ν − ( log ( r ) I + e r e r T ) and P ( y ) = y 2 π r 2 ,

with r = ‖ y ‖ , e r = y ∕ r , and e r T is the transposed vector of e r .

The solution ( u , p ) to (1) can be decomposed as:

u = u s + u r and p = p s + p r ,

where ( u s , p s ) is a function with a finite number of singularities coinciding with the locations of the source points. It is the solution to:

(3) − ν Δ u s + ∇ p s = ∑ i = 1 N ℱ i δ x i in R 2 , div u s = 0 in R 2 .

The second part ( u r , p r ) is a regular function solution to:

(4) − ν Δ u r + ∇ p r = 0 in Ω , div u r = 0 in Ω , u r = u * − v s on Γ .

We have the following result ([16,17] for the proof).

Theorem 2.1

Suppose that the boundary Γ is of class C m , m ≥ 1 . If u * ∈ H m − 1 ∕ 2 ( Γ ) verifying (2), then Problem (1)admits a unique solution ( u , p ) satisfying

u = u s + u r and p = p s + p r ,

with ( u s , p s ) solution to (3), given by:

u s = ∑ i = 1 N U ( . − x i ) ℱ i and p s = ∑ i = 1 N P ( . − x i ) . ℱ i ,

and ( u r , p r ) ∈ H m ( Ω ) × H m − 1 ( Ω ) solution to (4).

2.2 Inverse problem

The inverse problem consists in finding the number N and detecting the source point locations x i using measurement of the stress tensor σ ( u , p ) . n on the boundary Γ . The following theorem gives an identifiability result (see [12] for the proof).

Theorem 2.2

Let ( u k , p k ) , k = 1 , 2 , be the solutions to:

− ν Δ u k + ∇ p k = ∑ i = 1 N k ℱ i k δ x i k i n Ω , div u k = 0 i n Ω , u k = u * o n Γ .

Then, if σ ( u 1 , p 1 ) ⋅ n ∣ Γ = σ ( u 2 , p 2 ) ⋅ n ∣ Γ , we have

N 1 = N 2 = N , ℱ i 1 = ℱ i 2 , x i 1 = x i 2 , a n d i = 1 , … , N .

We conclude that using Theorem 3.1, the number and locations of source points are uniquely determined by a single measurement of the stress tensor σ ( u , p ) . n on the boundary Γ .

3 Topological sensitivity approach

3.1 Variation of the cost function with respect to small topological perturbation

3.1.1 General results

For 1 ≤ p < 2 , let

(5) G : L p ( Ω ) 2 → R f ↦ G ( f )

be a differentiable functional on L p ( Ω ) 2 . We have

(6) ∇ G ( f ) δ f = ∫ Ω ℛ ( x ) δ f ( x ) d x , ∀ δ f ∈ L p ( Ω ) 2 ,

where ℛ denotes the Riesz representative of the differential ∇ G ( f ) . We want to study the variation of the functional G with respect to a finite topological perturbation of f on the form:

(7) δ f ε ( x ) = ℱ if x ∈ ϖ ε , 0 if x ∈ Ω \ ϖ ε ¯ ,

where ℱ ∈ R 2 is a constant vector and ϖ ε is a geometry perturbation of size ε > 0 small enough.

We make the following assumptions.

Hypothesis 3.1

The function G is differentiable, and there exists constant γ 1 > 0 such that
(8) for all f ∈ L p ( Ω ) 2 , ∣ G ( f + δ f ) − G ( f ) − ∇ G ( f ) δ f ∣ ≤ γ 1 ∥ δ f ∥ L p ( Ω ) 2 2 , ∀ δ f ∈ L p ( Ω ) 2 .
The function ℛ is Lipschitz continuous, i.e., there exists a constant γ 2 > 0 such that
(9) ∥ ℛ ( x ) − ℛ ( y ) ∥ ≤ γ 2 ∥ x − y ∥ , ∀ x , y ∈ Ω .

The following theorem gives the variation of G .

Theorem 3.1

If Hypothesis 3.1 holds, we have

(10) G ( f + δ f ε ) − G ( f ) = ρ ( ε ) { ℱ ⋅ ℛ ( x 0 ) } + o ( ρ ( ε ) ) ,

where ρ ( ε ) = meas ( ϖ ε ) .

Proof

First, we have

(11) ∣ G ( f + δ f ε ) − G ( f ) − ρ ( ε ) ℱ ⋅ ℛ ( x 0 ) ∣ ≤ ∣ G ( f + δ f ε ) − G ( f ) − ∇ G ( f ) δ f ε ∣ + ∣ ∇ G ( f ) δ f ε − ρ ( ε ) ℱ ⋅ ℛ ( x 0 ) ∣ .

Thanks to (8), we derive

∣ G ( f + δ f ε ) − G ( f ) − ∇ G ( f ) δ f ε ∣ ≤ γ 1 ∥ δ f ε ∥ L p ( Ω ) 2 2 .

Using (7) and the fact that p < 2 , we obtain

∥ δ f ε ∥ L p ( Ω ) 2 2 ≤ c ( ρ ( ε ) 1 ∕ p ) 2 = c ρ ( ε ) 2 ∕ p = o ( ρ ( ε ) ) .

Therefore,

(12) ∣ G ( f + δ f ε ) − G ( f ) − ∇ G ( f ) δ f ε ∣ = o ( ρ ( ε ) ) .

For the second part in (11), we have

∫ Ω ℛ ( x ) ⋅ δ f ε ( x ) d x − ρ ( ε ) ℱ ⋅ ℛ ( x 0 ) = ∫ Ω ( ℛ ( x ) − ℛ ( x 0 ) ) ⋅ δ f ε ( x ) d x .

It follows from (9) that

(13) ∫ Ω ℛ ( x ) δ f ε ( x ) d x − ρ ( ε ) ℱ ⋅ ℛ ( x 0 ) ≤ γ 2 ∫ Ω ∥ x − x 0 ∥ ∥ δ f ε ( x ) ∥ d x = γ 2 ∥ ℱ ∥ ∫ ϖ ε ∥ x − x 0 ∥ d x ≤ γ 2 ∥ ℱ ∥ ε ρ ( ε ) .

Finally, using (12) and (13), we deduce the desired result.□

3.1.2 Perturbation of the Stokes problem

Consider now the Stokes problem case:

(14) − ν Δ u + ∇ p = F in Ω , div u = 0 in Ω , u = u * on Γ .

For all F ∈ L 2 ( Ω ) and u * ∈ H 1 ∕ 2 ( Γ ) , Problem (14) has one solution ( u , p ) ∈ H 1 ( Ω ) × L 0 2 ( Ω ) . For more details and proof, one may consult [18] or [17].

Using a weak formulation of the previous system, one can show that u is the solution to the following variational problem:

u ∈ V , A ( u , w ) = L ( w ) , ∀ w ∈ V 0 , u = u * on Γ ,

with

V = { w ∈ H 1 ( Ω ) , div w = 0 } , V 0 = { w ∈ V , w ∣ Γ = 0 } , A ( v , w ) = ν ∫ Ω ∇ v ∇ w d x , ∀ v , w ∈ V , L ( w ) = ∫ Ω F w d x , ∀ w ∈ V .

Obviously, A is a continuous bilinear form on V × V and L is a continuous linear form on V .

For all f ∈ L p ( Ω ) 2 , we denote by L f the following linear form:

L f : V ⟶ R w ⟼ L f ( w ) = L ( w ) + ∫ Ω f ( x ) w ( x ) d x .

We have the following lemma.

Lemma 3.1

The map

(15) L p ( Ω ) 2 ⟶ ℒ ( V ) f ⟼ L f

is continuous for p > 1 .

Proof

Thanks to [19] (Theorem IX.16), we know that H 1 ( Ω ) ⊂ L q ( Ω ) for 1 ≤ q < + ∞ . It is well known that if v ∈ L p 1 ( Ω ) and w ∈ L p 2 ( Ω ) , the product v ⋅ w ∈ L p ( Ω ) with 1 p = 1 p 1 + 1 p 2 .

Consequently, the map f ↦ L f is continuous as soon as

(16) 1 p + 1 q = 1 .

Then, combining equation (16) and the conditions on q for which we have H 1 ( Ω ) ⊂ L q ( Ω ) , we deduce that the map is continuous for p > 1 .□

In the following, we denote by u f ∈ V the unique solution of the following variational problem:

(17) A ( u f , w ) = L f ( w ) ∀ w ∈ V 0 ,

and we consider the cost function G defined by:

(18) G : L p ( Ω ) 2 ⟶ R f ⟼ G ( f ) = J ( u f ) ,

where J is a given functional defined on Ω .

If J is differentiable, using the Lagrangian method, one can prove that the function G is differentiable and we have

(19) ∇ G ( f ) δ f = − ∫ Ω δ f ( x ) . v f ( x ) d x , ∀ δ f ∈ L p ( Ω ) 2 ,

where v f ∈ V 0 is the solution to the associated adjoint problem:

(20) A ( w , v f ) = − D J ( u f ) w , ∀ w ∈ V 0 .

Relations (6) and (19) involve that

(21) ℛ = − v f .

We note that the regularity of ℛ depends on that of the functional J .

Let us consider now the following particular case:

f ≡ 0 , and δ f = δ f ε .

Posing j ( ε ) = J ( u ε ) , where u ε is the solution u 0 + δ f ε . We deduce the following result using Theorem 3.1.

Proposition 3.1

The cost function j has the following asymptotic expansion:

(22) j ( ε ) − j ( 0 ) = ρ ( ε ) { − ℱ . v 0 } ( x 0 ) + o ( ρ ( ε ) ) .

3.2 Cost function examples

We now discuss Hypothesis 3.1. We consider the standard example of a cost function that we will use in numerical tests:

(23) J ( u ) = ∫ Ω ∣ u − U ∣ 2 d x ,

where U ∈ H 1 ( Ω ) .

Proposition 3.2

The cost function (23) satisfies Hypothesis 3.1 with

∇ G ( f ) ( δ f ) = ∫ Ω v f . δ f d x , ∀ δ f ∈ L p ( Ω ) 2 ,

with v f ∈ V 0 solution to the following adjoint problem:

A ( w , v f ) = − 2 ∫ Ω ( u f − U ) w d x , ∀ w ∈ V 0 .

Proof

It is easy to show that J is differentiable on V and we have

D J ( u ) w = 2 ∫ Ω ( u − U ) . w d x , ∀ w ∈ V .

Then, the adjoint solution v f is the solution to:

(24) v f ∈ V 0 , A ( w , v f ) = − 2 ∫ Ω ( u f − U ) w d x , ∀ w ∈ V 0 .

Using (18), (19), and (23), we obtain

(25) G ( f + δ f ) − G ( f ) − ∇ G ( f ) δ f = ∫ Ω ∣ u f + δ f − U ∣ 2 d x − ∫ Ω ∣ u f − U ∣ 2 d x − ∫ Ω v f ⋅ δ f d x = 2 ∫ Ω ( u f − U ) . ( u f + δ f − u f ) d x + ∥ u f + δ f − u f ∥ 0 , Ω 2 − ∫ Ω v f ⋅ δ f d x .

Recall that u f ∈ V is the solution to:

(26) A ( u f , w ) = L f ( w ) , ∀ w ∈ V , u = u * on Γ .

Then, ( u f + δ f − u f ) ∈ V 0 satisfies

(27) A ( u f + δ f − u f , w ) = ∫ Ω δ f ⋅ w d x , ∀ w ∈ V 0 .

Choosing w = u f + δ f − u f in (24) and w = v f in (27), we deduce

2 ∫ Ω ( u f − U ) . ( u f + δ f − u f ) d x = ∫ Ω δ f ⋅ v f d x .

Hence,

(28) G ( f + δ f ) − G ( f ) − ∇ G ( f ) δ f = ∥ u f + δ f − u f ∥ 0 , Ω 2 .

To estimate the term ∥ u f + δ f − u f ∥ 0 , Ω 2 , we need to suppose that the perturbation δ f has the form:

δ f ( x ) = ℱ , if x ∈ ϖ ε , 0 , if x ∈ Ω \ ϖ ε ¯ ,

where ℱ ∈ R 2 is a constant vector and ε > 0 is small enough.

Posing w f = u f + δ f − u f and s f = p f + δ f − p f , then ( w f , s f ) ∈ H 0 1 ( Ω ) × L 0 2 ( Ω ) is the solution to:

− ν Δ w f + ∇ s f = δ f in Ω , div w f = 0 in Ω , w f = 0 on Γ .

Then, there exists a positive constant c , independent of f , such that

∥ w f ∥ 1 , Ω 2 ≤ c ∫ Ω δ f . w f d x .

We choose p = 13 ∕ 10 and q = 13 ∕ 3 . Using the Holder inequality and the Sobolev embedding theorem, we derive

∫ Ω δ f ⋅ w f d x ≤ ∥ δ f ∥ L 13 ∕ 10 ( ϖ ε ) ∥ w f ∥ L 13 ∕ 3 ( Ω ) = c ρ ( ε ) 10 ∕ 13 ∥ w f ∥ 1 , Ω .

Hence,

(29) ∥ u f + δ f − u f ∥ 0 , Ω 2 = ∥ w f ∥ 1 , Ω 2 = o ( ρ ( ε ) ) .

Using (28) and (29), we deduce

G ( f + δ f ) − G ( f ) − ∇ G ( f ) δ f = o ( ρ ( ε ) ) .

□

4 Numerical algorithm

As an application of the previous theoretical results, we use Formula (22) to detect locations, intensity, and number of source points acting on fluid governed by the Stokes equations.

Let Ω be a two-dimensional domain occupied by the fluid containing some source points S = { s i = ( X i , ℱ i ) , 1 ≤ i ≤ N S } , where X i and ℱ i denote, respectively, the position and intensity of the source point i and N s the total source point number.

Our aim is to identify S using a boundary measurement ϕ of the stress tensor.

4.1 The inverse problem

The inverse problem that we consider can be formulated as follows:

Given u * a velocity on the boundary Γ , satisfying the compatibility Condition (2), find the set of source points S such that the solution ( u s , p s ) to

− ν Δ u s + ∇ p s = ∑ i = 1 N s ℱ i δ X i in Ω , div u s = 0 in Ω , u s = u * on Γ

satisfies σ ( u s , p s ) n ∣ Γ = ϕ .

4.2 The cost function

Since the boundary condition is overspecified, we use the Kohn-Vogelius criterion:

j ( S ) = J ( u D s , u N s ) = ∫ Ω ∣ u D s − u N s ∣ 2 d x ,

where ( u D s , p D s ) is the solution to the Stokes equations with the Dirichlet condition on Γ :

− ν Δ u D s + ∇ p D s = ∑ i = 1 N s ℱ i δ X i in Ω , div u D s = 0 in Ω , u D s = u * on Γ ,

and ( u N s , p N s ) is solution to the Stokes equations with Neumann condition on Γ :

− ν Δ u N s + ∇ p N s = ∑ i = 1 N s ℱ i δ X i in Ω , div u N s = 0 in Ω , u N s = 0 on Γ 1 , σ ( u N s , p N s ) n = ϕ on Γ .

We note that Γ = Γ 1 ∪ Γ 2 with Γ 1 ∩ Γ 2 = ∅ and u * is equal to u 1 on Γ 1 and 0 on Γ 2 .

4.3 Topological gradient

To compute the topological gradient, we need to solve the following problems:

• The two direct problems

(30) − ν Δ u D + ∇ p D = 0 in Ω , div u D = 0 in Ω , u D = u * on Γ , − ν Δ u N + ∇ p N = 0 in Ω , div u N = 0 in Ω , u N = 0 on Γ 1 , σ ( u N , p N ) n = ϕ on Γ .

• The two adjoint problems

(31) − ν Δ v D + ∇ q D = − 2 ( u D − u N ) in Ω , div v D = 0 in Ω , v D = 0 on Γ , − ν Δ v N + ∇ q N = 2 ( u D − u N ) in Ω , div v N = 0 in Ω , σ ( v N , v N ) ⋅ n = 0 on Γ .

Using Proposition 3.1, the topological gradient is given by:

(32) δ j ( x ) = − ℱ ⋅ { v D ( x ) + v N ( x ) } , ∀ x ∈ Ω .

4.4 Algorithm

Our identification problem can be formulated as a topological optimization problem: find the optimal location of the disks s i = B ( X i , r i ) of center X i and radius r i , 1 ≤ i ≤ N , inside the domain Ω minimizing the considered objective function. We propose, in this case, a multi-iteration process in the following sense:

At the first iteration, we keep only one force location corresponding to the “most negative” topological gradient. The next iteration repeats the same procedure in the new domain obtained after inserting the force detected during the previous iteration.

We use the following numerical algorithm:

Initialization: choose Ω 0 = Ω , and set k = 0 .
Repeat until target is reached:
1. solve (30) and (31) in Ω k ,
2. compute the topological sensitivity δ j k given by (32),
3. set Ω k + 1 = { x ∈ Ω k , δ j k ( x ) > c k + 1 } where c k + 1 is the chosen constant,
4. k ← k + 1 .

We propose an adaptation of the previous algorithm to our context. We consider the set:

{ x ∈ Ω k ; δ j k ( x ) < c k + 1 } .

Each connected component of this set is a neighborhood of a point source detected by the algorithm. In numerical computations, we have used c k + 1 = 0.8 κ , where κ is the most negative minimum of δ j k ( x ) .

In the aforementioned algorithm, Systems (30) and (31) are discretized by a finite element method. The numerical simulations are carried out using a developed 2D numerical software.

In practice, certain stopping criteria can be implemented successfully, such as the volume of material to remove [6], the number of the holes to insert [5], or the number of obstacles to detect [7]. In our case, we have imposed that the value of δ j must be positive as a stopping criterion.

5 Numerical tests

The numerical test concerns the detection of the optimal location of the well-separated source points in a two-dimensional case.

Consider Ω = [ 0 , 1 ] × [ 0 , 1 ] a two-dimensional domain discretized by a non-uniform triangular mesh consisting of 441 vertices and 800 triangles. The goal is to detect some source points X i = ( x i , y i ) in Ω having forces ℱ i = ( f i 1 , f i 2 ) , 1 ≤ i ≤ N . At each iteration, a new force is localized on the detected point and its support is represented by a disk of center X i and radius 0.01. We impose the velocity boundary conditions u * = 0 and ϕ = ( 0 ; − 2 ) on the boundary of the domain.

We consider two test cases. The first corresponds to the detection of a certain number of source points having forces of the same intensity. The second corresponds to the case of forces of different intensities. Finally, we studied the effect of the distance between two source points on the detection algorithm.

5.1 Source points with the same force

We consider as an example the case of five source points with the same force ℱ i = ( 1 , 1 ) , 1 ≤ i ≤ 5 . Using Algorithm 4.4, we show in Figure 1 the obtained isovalues of δ j at each iteration. The obtained exact source point location, corresponding to the minimums found, is given in Figure 2. They represent the exact force point location sought.

$Figure 1 Isovalues of δ j \delta j at each iteration showing the position of the detected source point (black dots) corresponding to the minimum found: (a) Iteration 1, (b) Iteration 2, (c) Iteration 3, (d) Iteration 4, (e) Iteration 5, and (f) Iteration 6.$

Figure 1

Isovalues of δ j at each iteration showing the position of the detected source point (black dots) corresponding to the minimum found: (a) Iteration 1, (b) Iteration 2, (c) Iteration 3, (d) Iteration 4, (e) Iteration 5, and (f) Iteration 6.

Figure 2

Exact location of the source points (black dots) corresponding to the minimums found after five iterations.

We remark in Figure 1(f) that the value of δ j is positive, which corresponds to the convergence of the algorithm, and confirm that the algorithm determines the exact number of points, taken equal to 5 in the numerical test. In Figure 2, we present all the five source points detected.

Note in this test case that the algorithm used can determine all the source point locations in a single iteration (Figure 1(a)). This was made possible in this case because the forces are equal.

5.2 Source points with different forces

We consider as an example the case of five random source points with different forces ℱ i , 1 ≤ i ≤ 5 defined in Table 1. Using Algorithm 4.4, we show in Figure 3 the obtained isovalues of δ j at each iteration. The obtained global minimum at each iteration corresponds to one of the exact source point locations.

Table 1

Source points location X i , 1 ≤ i ≤ 5 and their associated forces

Point source	Location ( X i )	Force ( ℱ i )
1	x = 0.8 , y = 0.2	ℱ x = 1.0 , ℱ y = 1.0
2	x = 0.3 , y = 0.25	ℱ x = 1.0 , ℱ y = 0.0
3	x = 0.75 , y = 0.45	ℱ x = ‒ 1.0 , ℱ y = ‒ 2.0
4	x = 0.25 , y = 0.65	ℱ x = 0.0 , ℱ y = ‒ 1.0
5	x = 0.7 , y = 0.75	ℱ x = ‒ 1.0 , ℱ y = 1.0

$Figure 3 Isovalues of δ j \delta j showing the exact location corresponding to each global minimum found (black dots): (a) Iteration 1, (b) iteration 2, (c) iteration 3, (d) iteration 4, (e) iteration 5, and (f) iteration 6.$

Figure 3

Isovalues of δ j showing the exact location corresponding to each global minimum found (black dots): (a) Iteration 1, (b) iteration 2, (c) iteration 3, (d) iteration 4, (e) iteration 5, and (f) iteration 6.

We remark in Figure 3(f) that the value of δ j is positive, which corresponds to the convergence of the algorithm, and confirm that the algorithm determines the exact number of points, taken equal to 5 in the numerical test.

5.3 Effect of the distance between two source points

We study the effect of the distance d , separating two source points, on the identification result.

We consider two source points s 1 = B ( X 1 , r ) and s 2 = B ( X 2 , r ) having the same size r = 0.02 and separated by a variable distance d .

For each value of d , we have used Algorithm 4.4 for detecting the corresponding optimal source points location. We present in Figure 4 the isovalues of the obtained δ j for different values of d . We show that the exact location, corresponding to the most negative value of δ j , can be found when the distance between the two source points decreases until a weak distance between the source points corresponding to the mesh size ( d ≥ 0.05 in this test case). Less than this distance, the algorithm detects only the region containing the two source points, which can be seen as a single “equivalent” one.

$Figure 4 Isovalues of δ j \delta j showing the exact locations (black dots) and the local minimums (obtained locations) for different values of d d .$

Figure 4

Isovalues of δ j showing the exact locations (black dots) and the local minimums (obtained locations) for different values of d .

6 Comments and conclusion

In this work, a new procedure for the detection of source point location from overspecified boundary data based on minimizing a Kohn-Vogelius-type function is presented. This detection procedure has two main characteristics that make it an interesting method.

The first is its simplicity to be extended to practical situations involving other applications using other partial differential equations such as tumor localization and the detection of wild wells in hydrology.

The second interesting characteristic is the speed and accuracy of the presented algorithm.

This work focuses on the topological sensitivity analysis and its numerical application. Another mathematical question of great interest could be discussed. It corresponds to the detection of source points with incomplete boundary conditions, because several causes can lead to such a situation, in particular when parts of the boundary are not accessible to measurements, or when the boundary conditions cannot be precisely given on them.

Acknowledgment

The authors extend their sincere appreciation to deputyship for Research and innovation, “Ministry of Education” in Saudi Arabia for funding this research (IFKSUOR3-592-1).

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state that there is no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2023-05-18

Accepted: 2023-08-31

Published Online: 2023-11-24

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2023-0108

Keywords for this article

optimization; source point detection; topological sensitivity; Stokes equations

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