Abstract
A direct method for the localization of obstacles in a Stokes system is presented and theoretically justified. The method is based on the so-called reciprocity gap functional and is illustrated with several numerical simulations.
1 Introduction
In this work, we consider the following inverse problem: a rigid body (obstacle) occupies a region and is immersed in an incompressible viscous fluid, occupying a bounded domain . Inside this domain, Stokes system holds. At the boundary, Dirichlet conditions are considered: a prescribed velocity at the accessible part of the boundary, , and null velocity at the boundary of the obstacle, . On the other hand, the obstacle produces a stress tensor on (measured data) from where we want to determine the location of the obstacle. This inverse problem is an example of an inverse obstacle problem in nondestructive testing. Such problems have many applications in several engineering areas (see, for instance, some examples in the book [Citation1]). For the problem here considered we refer the work,[Citation2] where theoretical results concerning obstacle identification from boundary measurements and local stability were established. In [Citation3], an iterative method based on the topological derivative and the Kohn–Vogelius functional is proposed for the identification of multiple 3D small obstacles and their location. In [Citation4], a numerical shape reconstruction method based on integral equations was presented and tested. In this case, the location of the obstacle was assumed to be known. In [Citation5], an optimization method for the reconstruction of both obstacle shape and location was proposed and tested (see also the work [Citation6]). In this case, both shape and location were retrieved simultaneously with an iterative method.
Here, we propose a direct method for the location of a single 2D obstacle based on Green’s formula for the Stokes equations. The method requires the computation of the so-called reciprocity functional at appropriate test functions (see [Citation7] for an application to the determination of point forces in a Stokes system) and can be easily adapted for the 3D case. This type of approach was presented in [Citation8] for the reconstruction of cavities or inclusions in a Laplace problem (see also [Citation9] for the localization of several circular obstacles using the reciprocity gap functional in a transmission problem for the Laplace equation). This work was partially presented by the authors at a conference on the Portuguese Naval School (cf. [Citation10]) and is organized as follows: In Section 2, we define the inverse problem here addressed and the associated direct problem. In Section 3, we present some theoretical results concerning the identification of obstacles from a pair of Cauchy boundary data. Section 4 addresses retrieving of the location using the reciprocity gap functional. We show that the proposed formulae for the location of the obstacle are related to the centre of mass coordinates, for some density functions. Moreover, we show that, for some prescribed boundary velocities, the retrieved centre of mass is related to the centre of the whole domain, . We conclude the paper with a section containing several numerical simulations to illustrate the proposed method.
2 Direct and inverse problems
Let be a simply connected, open and bounded domain with regular boundary , which we shall call a regular domain. Let be a regular domain such that , meaning that . Denote by the boundary of . Define the domain of propagation and notice that . The domain will represent the region occupied by the fluid whereas is the region occupied by the obstacle.
Given a boundary velocity of the fluid at , and the no-slip boundary condition at , the system of equations governing the fluid flow here considered is given by1 1 where is the fluid velocity, the pressure and the dynamic viscosity, which we shall assume to be . Recall that is the Laplacian and the gradient of the pressure. The condition means that is solenoidal and represents the incompressibility of the fluid. This condition implies that the prescribed boundary velocity must satisfy the no flux compatibility condition2 2 where is the normal field at , pointing outwards with respect to . The stress tensor associated to the flux iswhereis the stress strain tensor of . In particular, problem (Equation11 1 ) can be written asThe direct problem consists in, given a boundary velocity at (satisfying the no flux condition (Equation22 2 )), determine the generated traction at ,3 3 where satisfies Stokes system (Equation11 1 ). It is well known that, taking , whereproblem (Equation11 1 ) is well posed, with . The space is defined as the subspace of such that in . Recall also that the pressure is unique up to an additive constant and thatwhereIn this functional setting, we have .
The inverse problem consists in: given a pair of Cauchy data , where is defined by (Equation33 3 ), determine the location of . Here, we are assuming only the knowledge of the exterior boundary and that at the boundary of the obstacle, , we have a no-slip boundary condition (the geometry of is, therefore, unknown).
3 Identification and reconstruction results
The following is a well-known identification result of obstacles from a single pair of Cauchy boundary data.
Proposition 3.1
([Citation2, Citation4]) Let be regular domains, and , the corresponding solutions of (Equation11 1 ). Ifthen
Using analytic continuation arguments, the previous identification result is also valid for Cauchy data in an open part of the boundary .
3.1 Reconstruction of circular shaped obstacles
As a consequence of the above result, let us see the following particular case of localization of circular-shaped obstacles. Letand suppose, by simplicity, that . Define the set of admissible circular-shaped obstaclesGiven consider the vector field4 4 It can be easily seen that the pair satisfies Stokes system in the domain of propagation , considering the boundary velocity5 5 On the other hand,6 6 whereThus, we can define the mapwith defined by (Equation55 5 ) and (Equation66 6 ), respectively. By Proposition 3.1, is injective and, in particular, we have the following result.
Lemma 3.2
Let be such that . Then,where is the traction at generated by the obstacle and considering the boundary velocity .
Proof
The map (respectively ) denotes the projection of onto (respectively ). Notice thatwhere is the traction at generated by and considering the velocity . In particular,and the triplet is a solution of the above minimization problem. We now prove that it is unique. Let be such thatThen,and the result follows from the injectivity of .
This lemma shows that the reconstruction of circular-shaped obstacles can be obtained by solving the above minimization problem. However, it requires several boundary measurements which is a drawback when comparing with the iterative method proposed in [Citation5]. Moreover, it relies on an explicit solution of Stokes system for circular-shaped obstacles and it cannot be generalized for other type of obstacles.
4 Recovering the location of an obstacle using the reciprocity functional
Here, we propose a method based on Green’s formula for the location of the obstacle. It requires only one boundary measurement and does not depend on the shape of the obstacle.
Given a regular domain and a regular function with constant sign, the mass centre of with density is the pair defined byIn particular, when the density is constant, we obtain the so-called centre of gravity 7 7 where is the area of . Suppose that has boundary. Since for a density function , there exists an unique satisfyingthen, second Green’s formula (e.g. [Citation11]) yields, for a test function ,where is the normal derivative at . In particular, taking the harmonic test function we haveTherefore, we can also write the centre of mass using only boundary integrals,8 8 In particular, for the centre of gravity, that is assuming , we get9 9 where is given byand is the harmonic function in such that
4.1 Centre of a circle
For the particular case of circular domains , it is well known that (e.g. [Citation12])10 10 for any harmonic function in . Thus, considering the harmonic functions and we have, by (Equation1010 10 ), the following identity for the geometric centre of ,Notice that this formula coincides with (Equation77 7 ). In fact, for , the functionsatisfiesHence,
4.2 Reciprocity functional for Stokes equations
Let and , where and are any regular domains. Assuming that the normal at points outwards with respect to , we have the following Gauss-Green formula11 11 Given , we define the reciprocity functional at 12 12 In the following, we shall assume that satisfies Stokes system (Equation11 1 ). Hence,13 13 where is the pair of Cauchy data at (recall that this data is assumed to be available in the inverse problem). Gauss-Green formula gives14 14
4.3 Formulae for the centre of mass
Suppose that the pair of test functions satisfyThen, using (Equation1313 13 ) we can write the identityfrom where we can obtain the following reconstruction method for the centre of mass.
Take and15 15 where is the standard basis in . We have,On the other hand,hence, assuming , we get the approximation16 16 for the first coordinate. For the second coordinate, we consider17 17 and obtain18 18
Remark 1
The above formulae (Equation1515 15 ) and (Equation1717 17 ) require only a single pair of Cauchy data on . On the other hand, no information regarding the shape of is considered.
The following result shows that in certain cases, the formulae (Equation1515 15 ) and (Equation1717 17 ) provides the coordinates of the obstacle, for some density functions.
Proposition 4.1
Suppose that . There exists such thatMoreover,19 19 where is the centre of gravity of .
Proof
We show the above identities for the first coordinate (the second can be obtained in the same manner). Let . Since the bilaplace problemis well posed in (e.g. [Citation11]) then, for Green’s formula yieldsfor every harmonic test function . Therefore, taking we getSince we also have then,Identity (Equation1818 18 ) follows from the linearity of since
We now obtain a connection between centre of mass of the obstacle and centre of the whole domain .
Corollary 4.2
Suppose that the prescribed velocity at satisfies the orthogonal conditions20 20 where and are the fields defined in (Equation1414 14 ) and (Equation1616 16 ), respectively. Then, there exists such that and21 21
Proof
From (Equation1212 12 ), the identities (Equation1919 19 ) implyLet be the extension of by zero to the whole . Then,
Remark 2
We can take, for instance, boundary velocities such that . In fact, sincethenSince satisfies the no flux condition (Equation22 2 ) it follows that satisfies the hypothesis (Equation1919 19 ).
Remark 3
Suppose that is a circle. From the above result, if the traction data , where and are constants then the obstacle should be centred with , that is, and .
5 Numerical examples
5.1 Numerical solution of the direct problem using Stokeslets
In this section, we will make a brief reference to the application of the Method of Fundamental Solutions (MFS) (e.g. [Citation13]) to solve the direct Stokes problem. Such method was used in this work with the aim of generating stress data, , in a finite number of boundary points .
A fundamental solution for the 2D Stokes system satisfieswhere is the Dirac delta (distribution) centred at the origin. Here, we consider the Stokesletswhere stands for the tensor product. The MFS for the 2D Stokes flow consists in taking the approximations for the velocity and pressure (see [Citation14]) as22 22 and23 23 where are source points placed at an artificial boundary located outside the physical domain and are sources on an interior artificial boundary contained in . The basis functions are the canonical components and .
The coefficients that will determinate the approximation, can be computed by fitting the Dirichlet boundary conditions at some collocations points, that is, () and ().
Here, we considered a least squares fitting which leads to the following system of linear equations,where is given by 2D blocks,andwhereand , with .
5.2 Location reconstruction
In this section, we will present three examples in order to illustrate the feasibility and stability of the proposed method. The location of the obstacle is retrieved as where the coordinates are given by the formulae (Equation1515 15 ) and (Equation1717 17 ). In other words,If the boundary velocity satisfies the orthogonal conditions (Equation1919 19 ) then the previous expressions can be written asWe approximate the above line integrals using a trapezoidal rule. The observation points, that is, the points where we have the measured data will be represented by (blue) dots. The shape of the obstacle will be represented by a full blue line and the retrieved location of the obstacle by a bold red dot.
5.2.1 Example 1
We start by studying the influence of boundary velocity . We test two situations: One considering boundary velocity satisfying the orthogonal condition (Equation1919 19 ) and other not satisfying this condition. The considered obstacle was the kite, defined by the parametrization
We considered and the boundary velocities and . Notice thatand satisfies the orthogonal condition (Equation1919 19 ) (see remark 2). We took the measured data at 60 observation points uniformly distributed over , without adding noise. As we can see in Figure (right plot) we were able to retrieve the location of the obstacle, when the boundary velocity satisfies the orthogonal condition (Equation1919 19 ). For this case, the retrieved location was . When we considered the boundary velocity we obtained the location , which is a bad result (see the left plot of the mentioned figure). The results deteriorate by increasing the size of the obstacle (see Figure ).
5.2.2 Example 2
In this second example, we tested the sensibility of the method with respect to the size and location of the obstacle. We started by testing the effect of the size on the reconstruction results. We considered the circular obstacle bounded by immersed in two regions: first, a region bounded by and then bounded by . The velocity field prescribed at was .
The centre was retrieved using 10, 30 and 60 observations on . The numerical results are summarized in Tables and , respectively. Overall, we obtained good reconstruction results even in the presence of noisy data.
Next, we considered several star-shaped obstacles with different geometries and locations. The boundary velocity was for the first example and for the others. We took 10 noise-free boundary measurements and obtained the results presented in Figure . Notice that the last obstacle (plot (d) of Figure ) is not symmetric. The parametrization of the corresponding boundary is
5.2.3 Example 3
In this example, we considered two different geometries for the enclosing domain . In both cases, the domains are star shaped and nonconvex (see Figure ). The boundary velocity was and the noise-free measured data were obtained at 50 observation points. As we can see in Figure , the location of the obstacle was retrieved accurately.
Last simulations concern a nonconvex shark-shaped obstacle. The boundary is given by the parametrization24 24 and the enclosing domain is the open ball of radius 30 centred at the origin. The velocity considered was and we obtained the corresponding (noise free) measurement and 10 observation points (see Figure for the reconstruction results). Other location and dimension for the shark was also considered (Figure ). Moreover, we tested for noisy data and obtained good reconstruction results.
Last simulations are an attempt to recover the centre of the object from partial data. In one case, we took 30 observation points at the first quadrant (left plot of Figure ). A second situation where the observations’ points were located on the first, second and third quadrants is illustrated by right plot of Figure . In both cases, the results were not good and an a priori data completion method is required.
6 Conclusions
In this paper, we proposed a reconstruction method for the location of a single 2D obstacle in a Stokes flow, using the so-called reciprocity gap functional. The method is sufficiently general and can be easily adapted for 3D problems and other type of inverse obstacle problems. It is very fast and the numerical simulations shows that is accurate and stable. The good performance of the method can be exploited in, for instance, inverse geometric problems, where the geometry of the obstacle is also the goal. These problems are usually tackled as an optimization problem for both shape and location (e.g. [Citation5]). However, a more direct approach can be applied when the location of the obstacle is known (e.g. [Citation15]).
As seen in the last couple simulations, it requires data on the whole , which can be a drawback.
References
- Isakov V. Inverse problems for partial differential equations. Vol. 127, Applied mathematical sciences. New York (NY): Springer; 1998.
- Alvarez C, Conca C, Friz L, Kavian O, Ortega JH. Identification of immersed obstacles via boundary measurements. Inverse Probl. 2005;21:1531–1552.
- Caubet F, Dambrine M. Localization of small obstacles in Stokes flow. Inverse Probl. 2012;28:105007.
- Alves CJS, Kress R, Silvestre AL. Integral equations for an inverse boundary value problem for the two-dimensions Stokes equations. J. Inverse Ill posed Probl. 2007;15:461–481.
- Martins NFM, Silvestre AL. An iterative MFS approach for the detection of immersed obstacles. Eng. Anal. Bound. Elem. 2008;32:517–524.
- Karageorghis A, Lesnic D. The pressure-stream function MFS formulation for the detection of an obstacle immersed in a two-dimensional Stokes flow. Adv. Appl. Math. Mech. 2010;2:183–199.
- Alves CJS, Silvestre AL. On the determination of point-forces on a Stokes system. Math. Comput. Simulation. 2004;66:385–397.
- Alves CJS, Martins NFM. Reconstruction of inclusions or cavities in potential problems using the MFS. In: Chen C, Karageorghis A, Smyrlis Y,editors. The method of fundamental solutions – a Meshless method. Atlanta (GA): Dynamic Publishers; 2008. p. 51–71.
- Talbott S, Spring H. Thermal imaging of circular inclusions within a two dimensional region. Terre Haute (IN): Department of Mathematics, Rose-Hulman Institute of Technology; 2005.
- Martins NFM, Soares D . Localização de obstáculos submersos a partir de dados na fronteira [Localization of immersed obstacles from boundary data]. Conf. Jornadas do Mar; Portugal: Escola Naval; 2012. p. 12–16.
- Chen G, Zhou J. Boundary element methods. London: Academic Press; 1992.
- Evans LC. Partial differential equations. Vol. 19, Graduate studies in mathematics. Providence (RI): American Mathematical Society; 1998.
- Bogomolny A. Fundamental solutions method for elliptic boundary value problems. SIAM J. Numer. Anal. 1985;22:644–669.
- Alves CJS, Silvestre AL. Density results using Stokeslets and a method of fundamental solutions for the Stokes equations. Eng. Anal. Bound. Elem. 2004;28:1245–1252.
- Alves CJS, Martins NFM. The direct method of fundamental solutions and the inverse Kirsch-Kress method for the reconstruction of elastic inclusions or cavities. J. Int. Equat. Appl. 2009;21:153–178.