EP1579353A1

EP1579353A1 - Method for evaluating a physical quantity representing an interaction between a wave and an obstacle

Info

Publication number: EP1579353A1
Application number: EP03782525A
Authority: EP
Inventors: Dominique Placko; Nicolas Liebeaux; Tribikram Kundu
Original assignee: Centre National de la Recherche Scientifique CNRS; Universite Paris Sud Paris 11; University of Arizona
Current assignee: Centre National de la Recherche Scientifique CNRS; Universite Paris Sud Paris 11; Arizona Board of Regents of University of Arizona
Priority date: 2002-11-12
Filing date: 2003-11-06
Publication date: 2005-09-28
Also published as: WO2004044790A1; FR2847051A1; AU2003290161A1; FR2847051B1; US7403879B2; AU2003290161A8; US20060129342A1

Abstract

The invention concerns the modelling of interactions between an incident wave and an obstacle, in particular in the field of non-destructive control. The invention is characterized in that it consists in: meshing the obstacle surface and assigning at least one source (Si) to each surface element (dSi); then calculating conditions at the boundaries in each mesh of the obstacle and deducing therefrom source values; based on an interaction matrix and said source values, estimating a physical quantity representing the interaction between the wave and the obstacle, in any point of space.

Description

Method for evaluating a physical quantity representative of an interaction between a wave and an obstacle

The invention relates to the modeling of the interactions between an incident wave and an obstacle of this wave, in particular in the field of non-destructive testing.

Modeling the interactions between a wave and an obstacle receiving this wave, such as a target placed in the sensitive area of a sensor, finds an advantageous application in non-destructive testing.

A so-called "finite element" modeling method is known which consists in applying a tiling of the three-dimensional space surrounding the obstacle and in evaluating the aforementioned interactions for all the tiles in space.

The finite element calculation methods provide a solution to a problem posed in the form of partial differential equations. They are based on a representation of the study space by an assembly of finite elements, within which approximation functions are determined in terms of nodal values of the physical quantity sought. The continuous physical problem therefore becomes a discrete finite element problem where the nodal quantities are the new unknowns. Such methods therefore seek to approach the global solution, rather than the original partial spatial derivative equations. The discretization of the space taken into account ensures that the latter is entirely covered by finite elements (lines, surfaces <~> π volume-), this operation is called "mesh" in two-dimensional space (2D) or of "paving" in three-dimensional (3D) space. The elements involved are either rectangular or triangular in 2D, or parallelepipedic or tetrahedral in 3D. They can be of different sizes, distributed evenly over the surface or not.

In general, the physical quantity sought, such as an electrostatic potential or a pressure value, is known on the boundary of the domain. This border can be fictitious. Boundary conditions are imposed there. The potential is therefore unknown within the same domain. We then define a node as being a vertex of an element. The unknowns of the problem are therefore the values of the potential at each node of the whole domain.

By way of illustration, FIG. 6 of the prior art represents an example of a surface, constituted by two materials Ml and M2, of different electromagnetic properties, and meshed by triangular elements each comprising three nodes Ai, Bi and Ci. L whole area is bounded by an F border.

Once the mesh is defined, several approaches exist to transform the physical formulation of the problem into a discrete finite element modeling. If the problem is formulated by differential equations and consists in minimizing a functional, we apply usually a variational method. This transformation results in a matrix formulation whose resolution gives the nodal solutions, the solutions at the non-nodal points being obtained by linear interpolation.

However, such calculations, in three dimensions, require significant computer resources and generate very long calculation times, despite the increase in the performance of software enabling these calculations to be implemented.

Admittedly, 2D problems, often simplified by advantageous symmetry conditions to model only part of the geometry, are quickly resolved. But this is not the case for 3D problems, which are the most frequent. Figure 6 shows how much the fineness of the mesh, ie the ratio between the size of an element and that of the smallest detail in the domain, increases the number of nodes.

Consequently, the quantity of equations and unknowns increases proportionally, and, from there, the computation time necessary to solve the problem. It is important to specify that the generation of the mesh, namely the discretization of the workspace, and the generation of the list of the nodes consumes a computation time higher than that necessary for the resolution of the problem.

The present invention improves the situation. To this end, it proposes a method for evaluating a physical quantity associated with an interaction between a wave and an obstacle, in a region of three-dimensional space, in which: a) a plurality of surface samples is determined by mesh at least part of which represents the surface of an obstacle receiving a main wave and emitting, in response, a secondary wave, and each surface sample is assigned at least one source emitting an elementary wave representing a contribution to said wave secondary, b) a matrix system is formed comprising: an interaction matrix, invertible, applied to a given region of space and comprising a number of columns corresponding to a total number of sources, a first column matrix of which each coefficient is associated with a source and characterizes the elementary wave which it emits, and a second column matrix, obtained by a multiplication of the first mat rice column by the interaction matrix and whose coefficients are values of a physical quantity representative of the wave emitted by all the sources in said given region, c) to estimate the coefficients of the first column matrix, we assigns values of physical quantity chosen at predetermined points, each associated with a surface sample, said values chosen being stored in the second column matrix, and one multiplies this second column by 1 matrix inverse of the interaction matrix applied to said predetermined points, d) to assess the ^ ^"one physical magnitude representing the wave you emitted by all sources in a given region of space three-dimensional, the interaction matrix is applied to said given region and this interaction matrix is multiplied by the first column matrix comprising the coefficients estimated in step c).

Thus, according to one of the advantages which the present invention provides, the meshing step a) relates to only one or more surfaces, while the modeling method of the "finite element" type requires paving of the whole space in the vicinity of the obstacle, which makes it possible to reduce, in the implementation of the method according to the invention, the memory resources and the calculation times required.

The method according to the invention applies both to a main wave emitted by a distant source and to a main wave emitted in the near field.

Advantageously, to evaluate a physical quantity representative of an interaction between an element radiating a main wave in the near field and an obstacle receiving this main wave,

- in step a), a plurality of surface samples are also determined, by mesh, representing together an active surface of the element radiating the main wave, and each sample of the active surface is assigned at least one source emitting a wave elementary representing a contribution to said main wave,

- steps b), c) and d) are applied to the samples of the active surface, and - said physical quantity representing the interaction between the radiating element and the obstacle in a given region of the three-dimensional space is evaluated, by taking into account the contribution, in said given region, of the main wave emitted by all the sources of the active surface and the contribution of the secondary wave emitted by all the sources of the surface of

1 obstacle.

The terms "radiating element" are understood to mean both a transmitter of the main wave, such as a wave generator, and a receiver of the main wave, such as a sensor of this wave.

In a first embodiment, the physical quantity to be evaluated is a scalar quantity and, in step a), a unique source is assigned to each surface sample.

In a second embodiment, the physical quantity to be evaluated is a vector quantity expressed by its three coordinates in three-dimensional space, and three sources are assigned, in step a), to each surface sample.

In an advantageous embodiment, to estimate, in step d), the contribution of the secondary wave in the region given space, the values of physical quantity chosen in step c) are a function of a predetermined coefficient of reflection and / or transmission of the main wave by each surface sample of the obstacle.

Thus, it will be understood that the secondary wave may as well correspond to a reflection of the main wave, as to a transmission of the main wave, or even to a diffraction of the main wave. In this advantageous embodiment, step c) finally corresponds to a determination of the boundary conditions at the surface of the obstacle, as an interface between two distinct environments, in particular in a heterostructure.

Furthermore, for a non-destructive control of a target forming an obstacle of a main wave, a reflection or transmission coefficient chosen is assigned to all the predetermined points on the surface of the target, and a simulation obtained by the implementation of the process within the meaning of the invention with an experimental measurement. Thus, the points on the surface of the target which, in the experimental measurement, do not verify the simulation correspond to inhomogeneities or to impurities on the surface of the target.

In another approach, the overall properties of the obstacle are known, in particular in transmission and / or in reflection. By implementing the method of the invention, the position in space of a sensor or even the shape of this sensor is then optimized, for application to a 4/044790

non-destructive testing, this sensor being intended to analyze a target forming an obstacle of the main wave.

To this end, in an advantageous embodiment, a plurality of values of the physical quantity estimated in step d) of the method within the meaning of the invention are compared, obtained for a plurality of regions of space, to select a region candidate for the arrangement of a radiating element intended to interact with the obstacle.

As indicated above, the term "radiating element" means both a sensor and a wave generator. It will thus be understood that the optimization of the position of the radiating element can also be applied to the optimization of the arrangement or of the shape of a wave generator. For example, the present invention also finds an advantageous application for the provision of loudspeakers in a closed volume delimited by obstacles, such as for example the passenger compartment of a motor vehicle.

Other characteristics and advantages of the invention will become apparent on examining the detailed description below, and the appended drawings in which: FIG. 1A schematically represents the respective surfaces of a radiating element ER emitting a wave and of an obstacle OBS receiving this wave, meshed in order to evaluate a scalar quantity representative of the wave at a point M of three-dimensional space; FIG. 1B represents in detail a surface sample dSi corresponding to a mesh of FIG. 1A, as well as a source S _± associated with the surface sample dSi; FIG. 2A schematically represents the respective surfaces of a radiating element ER emitting a wave and of an obstacle OBS receiving this wave, meshed in order to evaluate a vector quantity representative of the wave at a point M of three-dimensional space ; FIG. 2B represents in detail a surface sample dSi corresponding to a mesh of FIG. 2A, as well as three associated sources SAi, SBi and SCi; FIG. 2C represents, in front view, a mesh surface of which each surface sample comprises three sources SAj., SBi and SCi, for the estimation of a vector quantity; FIG. 3A represents, by way of illustration, the armatures of a capacitor, of respective electrical potentials VI and V2, for the estimation of an electrical potential at point M of the three-dimensional space, at each surface sample dSi of the FIG. 3A being associated with a single source Si; FIG. 3B represents, by way of illustration, the armatures of a capacitor, of respective electric potentials VI and V2, for the estimation of an electric field E (M), at point M of the three-dimensional space, at each sample dSi surface of Figure 3B being associated three sources SAi, SBi and SCi; / 044790

10

FIG. 4A represents, like FIGS. 1A and 2A, an interaction between a radiating element ER and an obstacle OBS, to evaluate a physical quantity

(scalar or vector) at a point M in a portion of the space delimited by the surface of the radiating element and the surface of the obstacle, this point in space M receiving both the wave emitted by l 'radiating element and the wave reflected by one obstacle; FIG. 4B, complementary to FIG. 4A, represents a transmission by the obstacle OBS of the wave emitted by the radiating element ER, at a point M of a half-space delimited by the plane formed by the surface of the OBS obstacle; - Figure 5A schematically shows an obstacle OBS, of finite dimensions, with sources associated with the surface samples arranged to estimate a quantity representative of a reflection of the wave on one obstacle; - Figure 5B, in addition to Figure 5A, schematically shows an obstacle OBS, of finite dimensions, with the sources associated with the surface samples arranged to estimate a quantity representative of the transmission of the wave by the obstacle; - Figure 5C shows a simulation of an ultrasonic wave emitted by a radiating element ER and propagating towards an obstacle OBS; FIG. 6 represents a mesh of three-dimensional media, for the application of a calculation method by "finite elements", within the meaning of the state of the art; FIG. 7A shows in detail a surface element and an observation point M, the relative positions of which are identified by an angle θ; and FIG. 7B schematically represents a surface to be meshed with a complex shape, in particular with an observation point M located in a gray area with respect to certain sources of the surface.

First of all, reference is made to FIG. 1A, on which the surface of an obstacle OBS, receiving a wave, is meshed according to a plurality of surface samples dSi to dS, in accordance with step a) above.

Referring to FIG. 1B, each surface sample dSi is associated with a hemisphere HEMi, tangent to the surface sample dSi at a contact point Pi.

Preferably, this contact point Pi corresponds to the top of the hemisphere HEMi. For the estimation of a scalar physical quantity at point M (such as an electrostatic potential, an acoustic pressure or the like), a single source Si is associated with the surface sample dSi. As will be seen below, in the case of the estimation of a vector quantity in a point of space M, we will rather assign three sources to each surface sample dSi.

Preferably, the hemisphere HE i is constructed as described below. During the aforementioned meshing step a), the surface of the obstacle OBS is evaluated, on the one hand, and, on the other hand, a number of surface samples dSi is chosen according to the desired precision of the estimation. of the physical quantity at point M. Thus, the surface of a sample dSi is given by S _D / N, where S _Q corresponds to the total surface of the obstacle and N corresponds to the chosen number of surface samples dSi.

The HEM hemisphere has the same area as the dSi sample. Thus, the radius Ri of the hemisphere is deduced from the expression:

2aR? = ^ o

¹ N

Each mesh represented by a surface sample dSi has, in the example described, a form of parallelogram, of center Pi corresponding to the point of intersection of the diagonals of this parallelogram. The hemisphere HE i is tangent to the surface sample dSi at this point Pi. Of course, the meshes can be of different shape, triangular or other. It is indicated in a general way that the point Pi corresponds to the barycenter of the mesh.

We thus define the position of the source Si (located in the center of the hemisphere HEMi). The distance separating the source S from the point of contact Pi corresponds to the radius Ri of the hemisphere HEMi and the line which passes through the points Pi and S is orthogonal to the mesh dSi.

In the example shown in FIG. 1A, the surface of a radiating element ER, corresponding for example to a wave generator, is further meshed. At each stitch of the 4/044

13

surface of the radiating element is associated with a surface sample dS'i, as will be seen below.

The matrix system that is shaped in step b) above corresponds to:

or the coefficients Vj (with j = l, 2, ..., N) of the first column matrix correspond to values each associated with a source, such as an electric charge (in the case of the estimation of a electric potential), or to a magnetic flux (in the case of the estimation of a magnetic potential), or even to a sound speed (in the case of the estimation of an acoustic pressure linked to the propagation of a sound wave); the coefficients V (Mi) (with i = l, 2, ..., N) of the second column matrix each correspond to a value of the physical quantity (an electrical or magnetic potential or a pressure) to be estimated at a point Mi space; the interaction matrix F comprises coefficients Ci, j whose general expression is given by:

It will thus be understood that the coefficients of the matrix F are interaction coefficients which depend on the distance separating each point of space i from a source Sj associated with a mesh dSj.

In the case of the propagation of an electric wave, the coefficients Ci, j, V _j and V (Mι), respectively of the interaction matrix of the first and of the second column matrix, are given by:

Vj --- qj [3]

where: ε ₀ is a dielectric constant,

MjS _j is a distance measured in algebraic value, - qj corresponds to an electric charge characterizing a source Sj, and - Ui corresponds to an electric potential at point Mi.

In the case of the propagation of a magnetic wave, the expression of these coefficients is as follows:

[4]

N (Mi) = 0i OR :

- μ ₀ corresponds to the magnetic peimédbiiicë of the medium where the point Mi is located, - ψj corresponds to the magnetic flux associated with the source s ₃ ; θ corresponds to the magnetic potential at point i.

In the context of the propagation of an ultrasonic wave, these coefficients are given by:

[5]

^V J = ^V J

V (Mi) = _Pi

in which: - i ² = -l. ω is the pulsation of the sound wave; - p is the density of the medium in which point i is located; the vector V _j corresponds to the sound speed coming from the source Sj; k corresponds to the wave vector of the sound wave; and

Pi corresponds to the sound pressure generated by the propagation of the ultrasonic wave at point Mi. In the expression of the coefficients cι, _j . the term dSj corresponds to the surface of the sample associated with the source Sj. Preferably, the mesh of a surface within the meaning of step a) of the method according to the invention is chosen so that each mesh has the same surface dS = dS ₁ = dS ₂ = ... = dSj.

It is noted in particular in the expression of the coefficients cι, j that they depend on the scalar product between the wave vector and the vector MjS _j . Thus, for ultrasonic waves, account is taken of a phase shift between the paths which link each source Sj to a point in three-dimensional space M, this phase shift being due to a path difference between the rays departing from each source and arriving at point M (as shown in Figure 4A). In particular, it will be understood that the angle of incidence of such a radius is taken into account in the expression of the coefficients of the interaction matrix F.

Of course, within the framework of the propagation of an electromagnetic wave of high frequency, therefore of short wavelength, which differs from the electrostatic or magnetostatic framework above, one takes account of the term of propagation expîk.rj in the expression of the interaction matrix, with respect to the geometry of the problem to be solved, as in the context of the propagation of an ultrasonic wave above (relation [5]).

Thus, the matrix system of equation [1] makes it possible to estimate, from an interaction matrix F and a 17

vector comprising the values V associated with the sources Sj, the coefficients of a vector (column matrix) comprising the values of physical quantity V (Mi) at the points of the space Mi.

To determine the values of the sources V _j , the following matrix system is applied:

OR: the coefficients of the interaction matrix F are expressed by with the indices i and j correspond respectively to the i ^th row and the j ^th column of the interaction matrix F. This interaction matrix comprises, for the determination of the values associated with the sources Vj, N rows and N columns, by recalling that N is the total number of meshes on the surface of the obstacle; the points i correspond to the top of the hemispheres HEMi of FIG. 1B.

The implementation of step c) of the method within the meaning of the present invention corresponds to calculating a boundary condition for the points Pi, of known properties, as will be seen below. The matrix system of equation [6] then becomes

OR :

F ^"1 corresponds to the inverse of the interaction matrix F; and the values V (Pi) are predetermined, as a function of the abovementioned boundary conditions.

The source values Vj are thus determined.

From the estimation of these source values Vj, we can calculate the scalar physical quantity at any point M in three-dimensional space, from the relation:

To obtain this expression of the scalar quantity V (M), the interaction matrix F can have only one line of coefficients Cj, with:

Cj = f (MSj), but always has N columns.

Referring again to FIG. 1A, it will be understood that the surface of the obstacle OBS receiving the wave that emits 4/044790

19

the radiating element ER acts, itself, as an active surface re-emitting a secondary wave (for example by reflection). Each source Si represents a contribution to the emission of this secondary wave.

In addition, to take into account both the presence of the main wave and the presence of the secondary wave at point M, the contribution of the main wave and the contribution of the secondary wave at point M are estimated. by the matrix system:

or

F 'is the interaction matrix between the surface of the radiating element and the point M; v'j (with j = l, 2, 3, ..., N ') is the value of the sources allocated to each surface sample dS'j of the radiating element, N' being the total number of meshes chosen for the active surface of the radiating element ER.

The coefficients of the matrix F 'are also a function of the distance MS'j, where S'j are the sources assigned to each surface sample dS'j of the radiating element.

According to an advantageous characteristic, the values of the sources of the obstacle Vj are determined according to the values of the sources of the radiating element Vj, which are themselves calculated as will be seen below with reference to FIGS. 4Λ, 413, 5A and 53.

We now refer to FIG. 2A, in which three sources are assigned to each surface sample dS, with a view to estimating a vector physical physical quantity N (M), at a point M of three-dimensional space.

Indeed, it will be understood that to estimate the vector quantity, by its three coordinates in the space x, y and z, the number of equations to be solved in relation to the previous matrix system must be multiplied by three. Thus, the matrix F ^"1 of the relation [7] must comprise three times more rows than previously. The interaction matrix F must, itself, comprise three times more columns than previously and, for this purpose, one advantageously provides for three sources per mesh when it is a question of determining the coordinates in one three-dimensional space of a vector N (M).

Referring to FIG. 2B, the three sources SAi, SBi, SCi, allocated to a surface sample dS are of respective positions determined as indicated below. As shown in FIG. 2B, the three sources SAi, SBi, SCi are coplanar and the plane comprising these three sources also comprises the base of the hemisphere HEMi. The hemisphere HEMi is constructed as indicated above (of the same surface as the surface of the mesh), with however the center of the hemisphere which here corresponds to the barycenter of the three sources SAi, SBi and SCi.

By "center of the hemisphere" is meant the center of the disc which constitutes the base of the hemisphere.

The three sources which are attributed to the surface sample dSi are arranged at the vertices of an equilateral triangle whose barycenter Gi corresponds to the center of the hemisphere. Preferably, each source SAi, SBi and SCi is arranged on the middle of a radius Ri of the hemisphere. Thus, the lines which connect the barycenter Gi to each source are angularly spaced by 120 °.

Referring to FIG. 2C, the angular orientation of the triangles formed by the source triplets is chosen randomly, from one surface sample to another. Advantageously, this avoids overperiodicity artifacts in the estimation of the vector quantity at point M, which could result from the choice of the same angular orientation of these triangles.

With reference to the different types of waves indicated above, the vector quantity N (M) to be estimated can be:

- an electric field, in the context of the propagation of an electric wave;

- a magnetic field, in the context of the propagation of a magnetic wave; and - a speed of sound at point M, within the framework of the propagation of ultrasonic waves. To determine the values associated with each source SAi, SBi, SCi, the matrix system is shaped according to the following relationship:

We note, in particular, that the interaction matrix Fy is of dimensions 3Nx3N, where N is the total number of surface samples. The interaction matrix is expressed here by the relation:

The coefficients of this matrix are expressed by

^ σV'J ^~ - uï "^J;! ^LJ j [13] with σ = A, B, ci = 1, 2, ..., N j = 1. 2, ..., N u = x, y . z.

and are also a function of a distance separating the contact point Pi from one of the sources Sσj (σ = A, B or C) of a triplet associated with a surface sample dS _j .

By inversion of the interaction matrix Fy, the values vσ _j associated with each source Sσj are thus determined, by applying boundary conditions to the values of the vector N at points Pi. These boundary conditions impose a value of the vector V, according to its three coordinates V _x (Pι), V _y (Pi) and V _z (Pi).

Once these source values v _j are thus determined, the expression of the vector is easily calculated

N at any point M in space, by the relation:

N (M) = N _x (M) x + N _y (M) y + N ₂ (M) z [14]

VχW) ≈ ∑f _x [(d {M, Sσ _j ) J. vσ _j

J = 1, .., N σ = A, B, C σ = A, B, C

V _Z (M) = ∑f_ [{d (M, Sσ _j ))]. vσ _j j = l N σ = A, B, C

x, y and z correspond to unit vectors carried by the x, y and z axes of three-dimensional space.

Thus, the interaction matrix Fy, when applied to any point M in space, ultimately comprises only three lines each associated with a coordinate of space x, y or z.

For different types of waves, the values of the sources vσ _j are, as before, an electric charge for an electric wave, a magnetic flux for a magnetic wave, a speed of sound for an ultrasonic wave.

More precisely, the coefficients of the interaction matrix F _y are determined from the relations [3], [4] and [5] above, specifying however that:

N (M) = -grâd [N (M)] [15]

V (M) being the scalar quantity previously calculated by equation [8]. Thus, for the estimation of a vector quantity N at point M and for the wave types mentioned above by way of example (electric, magnetic and ultrasonic), the coefficients of the interaction matrix F are _there inversely proportional to the square of a distance separating each source from point M, while for the estimation of a scalar quantity V at a point M in space, the coefficients of the interaction matrix F are simply inversely proportional to this distance. Each distance implies one of the sources of a triplet of a surface sample and a point M in space. The interaction matrix Fy then has 3Ν columns when it is a question of taking three sources per surface sample, while the interaction matrix F for the estimation of the scalar quantity only had N columns since only one source per surface sample was required.

More generally, we assign one source per sample when we know the boundary conditions for a scalar quantity and three sources per sample when we rather know the boundary conditions for a vector quantity.

Reference is now made to FIG. 3A to describe, by way of illustration, an application of the method according to the invention to the estimation of an electrical potential at a point M of the three-dimensional space, located between two plates of a capacitor . The armatures of this capacitor are brought to respective potentials VI and V2. The implementation of step a) first consists of meshing the respective surfaces of the two reinforcements. In the example shown in figuxe 3A, only two meshes have been shown for each frame, simply by way of illustration.

The application of step b) consists in formatting the matrix system involving the interaction matrix F and the column vector comprising the values of the sources Si to S ₄ . The multiplication of these two matrices makes it possible to obtain a column vector comprising the values of the potential at one or more points M in space.

The implementation of step c) of the method according to the invention consists in applying the matrix system to the contact points of the hemispheres Pi to P ₄ , of each surface sample dSi to dS ₄ . The following relationship results:

[16]

with V (Pι) = V (P ₂ ) = Vi V (P ₃ ) = V (P ₄ ) ≈ V ₂ Vi = qi, v ₂ ≈ q ₂ , v ₃ = q ₃ , v ₄ q Here, the boundary condition requires that the value of the potential at the contact points P_ and P ₂ corresponds to the potential VI of the first armature. Likewise, the electrical potential at the contact points P ₃ and P ₄ corresponds to the electrical potential of the second armature V2. By inversion of the interaction matrix applied to the point of contact Pi, one determines the values of the sources Vi which correspond, as expressed in relation [16], to electric charges qi.

The coefficients of the interaction matrix are

2 ^ ₀ ^p i ^s j perfectly known, since the positions of the sources Sj and the positions of the contact points P are determined beforehand, as shown in FIG. 1B.

The expression of the electrical potential V (M) at point M between the two plates is finally given by the expression:

We now refer to FIG. 3B in which the same armatures have been represented as in FIG. 3A, with substantially the same mesh, but with the aim, here, of estimating a vector quantity corresponding to the electric field E (M), at point M of three-dimensional space, We can apply relations [11] to [15] to estimate the value of the electric field at point M, with, in relation [13]:

with σ = A, B, C u = x, y, zi = 1, 2, 3, 4 j = 1, 2, 3, 4.

However, the values of the electric field at the point of contact Pi remain to be determined in relation [11].

One then introduces a predetermined general law of the behavior of the field (in reflection, in transmission, or other) at the level of the surface of the obstacle (reinforcements in the example of the above-mentioned capacitor), to know the values of the sources Vσj.

For example, if the electric wave is totally reflected by the surface of an obstacle (for example one of the two reinforcements), the electric field at a contact point Pi is normal to the surface dSi and its components E _x and E _y are zero. By way of illustration, if the surface of the reinforcement were only represented by a single surface sample with three sources, the values of its sources vA, vB and vC would all be equal to each other at the same value + q. 29

On the contrary, if the reflection coefficient is practically zero at the surface dS _α , the component of the electric field E _z at the point Pi is zero, which corresponds well to the case where the field is substantially tangent to the surface dSi. Thus, by way of illustration, if the surface of the reinforcement were represented only by a single surface sample with three sources, the values of its sources vA, vB and vC would be respectively, for example, + q, + q and -2q. For example, in the context of the propagation of a magnetic wave, if the surface of an eddy current sensor (with a normal component of the magnetic field zero) was represented by a single surface sample, the magnetic fluxes of the three sources associated with this surface sample would be + φ, + φ and -2φ.

It is thus understood that with the three sources per sample dSi, it is possible to define, for example as a function of the weighting of each source, any orientation of the field at the surface of the obstacle.

Of course, this approach assumes that the reflection coefficient R of an obstacle is known beforehand. In particular, it may be advantageous to compare a simulation and an experimental measurement to detect, on the surface of an obstacle, inhomogeneities or impurities which correspond to points on the surface of this obstacle which do not check the values of the coefficient of reflection R imposed at each predetermined point Pi of the obstacle. It is thus possible to assign a predetermined value of the reflection coefficient to each point Pi of the surface of the obstacle. For this purpose, we introduce a matrix R which is representative of the reflection coefficient at each point Pi. For an interaction between a radiating element and an obstacle, we can thus express the matrix system of the relation [9] otherwise, it is that is, by giving a single expression from all sources in the system (both the obstacle and the radiating element), as shown below.

In what follows, it is indicated that:

F (P) is the interaction matrix of the OBS obstacle applied to the points Pi of the surface of the OBS obstacle;

F (P ') is the interaction matrix of the obstacle OBS applied to the points P'i of the surface of the radiating element ER;

F '(P) corresponds to the interaction matrix of the radiating element ER applied to the points Pi of the surface of the obstacle OBS;

F '(P') corresponds to the interaction matrix of the radiating element ER applied to the points Pi of the surface of the radiating element ER; - v 'corresponds to the column vector comprising the values of the sources S' i of the radiating element ER; and v corresponds to the column vector containing the values of the sources Si of the obstacle OBS. On the obstacle, the contribution of the wave emitted by the radiating element ER is expressed by:

N '(P) = F (P) .v' [19]

The contribution of the secondary wave returned by the OBS obstacle is expressed, by definition, by the relation:

V (P) = F (P) .v [20]

However, in the example shown in FIG. 4A, the secondary wave simply corresponds to a reflection of the main wave. What is expressed by the relationship:

N (P) = R N '(P) [21]

where R corresponds to a reflection matrix each coefficient of which represents the contribution to the emission, by reflection, of the secondary wave, by each source Si (or Sσi, within the framework of an estimation of a vector quantity) of the OBS obstacle.

From the three relations [19], [20] and [21], we deduce the expression of the vector column v comprising the values of the sources of the obstacle, from the vector column v comprising the values of the sources of the radiating element , by the relation:

v = [F (P)] ^{~ 1} . R. [F ^< (P)]. v ^< [22] Furthermore, for a fine estimation of the scalar or vector quantities at point H, in particular to take account of multiple reflections, it is advantageous to take into account the contribution of the radiation by the obstacle, at the level of the surface of the radiating element. ER. For this purpose, account is taken, in the estimation of the boundary conditions at the surface of the radiating element ER (at points PT) of the contribution of the radiation from the sources ST of the radiating element and of the contribution of the emission of the secondary wave by the sources Si of the obstacle, by the relation:

VTCP _≈ FCD _V + FCP _V ' _[23]

We can thus adjust, thanks to the relation [23], the source values S'i of the radiating element ER, taking into account the reflection of the obstacle OBS, according to the following relation:

V _T (P) '= , [F (P) Y ¹ . R. [F '(P)] + F' (P) ') .v' [10]

Thus, one simply imposes boundary conditions for the radiating element, to deduce therefrom the values of the sources v'i. In practice, we will preferentially proceed as follows:

- after meshing the surfaces, the position of the points P and P'i and of the sources Si and ST are determined;

- depending on the type of wave in play, the coefficients of the matrices F (P), F '(P), F (P') and F '(P') are determined; / 044790

33

- as a function of a law of reflection of the obstacle, the coefficients of the reflection matrix are determined as in the example “≈ given below for an ultrasonic wave; - as a function of boundary conditions on the radiating element (whose behavior is generally known for a given problem), the values of the vector Vτ (P ^r ) are determined at the points P'i of the surface of the radiating element and the values of the sources S'i of the radiating element are deduced therefrom by inversion of the relation [10]; the values of the sources S of the obstacle are also deduced therefrom by application of the relation [22];

- once the values of all the sources Si and S'i have been determined, we can apply the matrix system given by the relation [9] to any point M in space, by applying to this point M the interaction matrices F and F '(involving the position of point M and the positions of the respective sources Si and S'i).

Referring again to FIG. 4A, it is considered that the obstacle OBS simply represents an interface between two media Ml and M2, thus forming a diopter which can be plane, as represented in the example of FIG. 4A, but also curved or of any general shape. The reflection coefficients Ri associated with each point Pi depend, within the framework of the propagation of an ultrasonic or electromagnetic wave of high frequency, on the angle of incidence βi of the ray coming from the source Si, at the point of the three-dimensional space M. 4/044790

34

For an ultrasonic wave, the expression of the reflection coefficients Ri is given by:

OR :

Ci is the speed of sound in the medium Mi; c ₂ is the speed of sound in the medium M ₂ ;

- p _x is the density of the medium Mi;

- p ₂ is the density of the medium M ₂ .

In this expression [24], the term cosβi can simply be estimated as a function of the spatial coordinates of the point M and the point representing the source Si.

Referring now to FIG. 4B, the same estimate can be made for a point M located in the medium M ₂ . In this case, the wave received by point M is a wave transmitted by the obstacle OBS. In particular, we note that the sources of the radiating element ER are no longer active, due to the occultation of the radiating element ER by the obstacle OBS. In transmission, the reasoning applies as before with a boundary condition imposed on points Pi by the values of the transmission coefficients Ti associated with each point Pi. In the context of the propagation of an ultrasonic wave, each transmission coefficient Ti is given by the relation:

As indicated previously, the terms cosβi can be determined as a function of the respective coordinates of the sources Si and of the point M.

To estimate the values of sources Si of the obstacle OBS, one applies the relation [22] while replacing the reflection matrix R by the transmission matrix T:

v = [F (P)] - ¹ τ [F (P)] v '[26]

Within the framework of the propagation of an ultrasonic wave, the coefficients of the matrices R and T are estimated for each source Si and for each point Pi. In particular, each coefficient Ti, _j or Ri, _j of the matrix T or of the matrix R (where i corresponds to the i ^th row and j corresponds to the j ^th column) is expressed as a function of an angle? i _j between a normal to the surface of the obstacle at point Pi and a passing line by the point Pi and by a source S _j . One can thus write, in a general way, the two relations expressing the values of the coefficients of the matrices T and R by the following respective relations:

Ry = f _r (cos? Ij) [28]

where f _t is given by the relation [25] and f _r is given by the relation [24].

More generally, with reference to FIGS. 4A and 4B, it is indicated that, if we consider the obstacle as a solid material representing a medium M2 distinct from a medium Ml in which the main wave initially propagated:

- for a reflection of the main wave on the obstacle as medium M2 (the surface of the obstacle forming a diopter between the mediums Ml and M2), the hemispheres HEMi are oriented towards the outside of the obstacle (figure 4A); - for a transmission of the main wave in the obstacle, the HEMi hemispheres are oriented towards the interior of the obstacle (Figure 4B).

We now refer to FIG. 5A to describe the case of a plane obstacle OBS of finite dimensions, excited by a radiating element ER, inclined at a predetermined angle relative to the obstacle OBS. As indicated previously, for an ultrasonic wave, the inclination of the radiating element will be taken into account to calculate the contribution of the wave emitted by the radiating element at point M. In addition, in a particularly advantageous manner, a surface which includes the surface of the obstacle (FIG. 5A). For a section of the space delimited by the radiating element, on the one hand, and the obstacle, on the other hand (FIG. 5A), we can consider three types of sources: the sources S 'i of the radiating element ER, sources SOi, which return the secondary wave, by reflection from the obstacle OBS, as a function of a certain reflection coefficient R of the obstacle; and - sources SSi, which do not return a secondary wave and to which a zero reflection coefficient can be assigned if the obstacle separates two media of identical indices. In this case, these sources SSi are considered as "extinct" in the section of the aforementioned space and are not taken into account in the calculations of the physical quantity at point M in FIG. 5A. On the other hand, these sources SSi can be active by reflection of the main wave if the obstacle OBS separates two media of different indices.

Furthermore, to estimate the scalar or vector quantities associated with a point M of a half-space delimited by the surface including the obstacle OBS (on the right of FIG. 5B), we consider: the sources SOT of the obstacle, active by transmission of the main wave, and sources SSi, to which we now assign a transmission coefficient equal to 1 if the obstacle separates two media with the same indices. These sources SSi finally behave (with near angles of incidence) like the sources S'i of the radiating element ER. The sources S'i of the radiating element can then be "switched off" for the calculation of the physical quantities in this half-space. To calculate the values v 'of the sources S'i of the radiating element (from which are deduced the values v of the sources of the obstacle according to the xelations [22] and [26J), one will simply apply boundary conditions to the points of the active surface of the radiating element ER. For example, for an ultrasonic wave propagation, it can be indicated that the sound velocities at the points of the surface of the radiating element ER are perpendicular to this surface and of modules v ₀ equal to each other.

In general, it is indicated that the three-dimensional space can thus be divided by interfaces delimiting environments of distinct properties, each interface representing an obstacle within the meaning of the present invention. We then calculate the physical quantities in each slice of the space. For example, in the context of studying a heterostructure (with several interfaces), the above method can be applied for successive sections of space by considering two interfaces: one representing a "radiating element" in the sense of FIGS. 4A and 5A, for example by transmission of a received wave, and the other representing an obstacle receiving the transmitted wave. Advantageously, account is taken, for each slice of the space, of the contributions of all the interfaces as expressed by the relations [10] and [22].

However, in a preferred practical embodiment, in particular for programming the simulation of an interaction, we will advantageously consider all the obstacles of the entire space around a point M and we will impose a condition on the position of the point M with respect to each source present in space.

Preferably, with reference to FIG. 7A, the scalar product SM.r is tested at each iteration with respect to a source S, for example in the form:

where r is the vector connecting the source S to the point of contact P of the half sphere with the surface element dS considered, in the case where only one source is provided per hemisphere. In the case where a triplet of sources SI, S2, S3 is rather provided, the base of the vector r is preferably located at the barycenter of the three sources SI, S2, S3. Moreover, in the case of three sources per surface sample, the calculation of the scalar product concerns each source Si of the triplet SI, S2, S3.

Typically, if cos # is positive, account is taken of the contribution of this source S in the estimation of the interaction.

On the other hand, if cos # is negative, we assign a zero value (scalar or vector) to this source S in the estimation of the interaction.

As a variant of the calculation of the dot product above, one can calculate an "altitude" of the point M. In this case, the test relates to a quantity of the type: S .r

Of course, other types of tests are possible. For example, in the case of a calculation with respect to the angle θ, one can choose this angle in a chosen opening cone, or other.

Finally, this approach advantageously makes it possible to systematize any configuration of the sources relative to the observation point M, by simply introducing an additional test step, at each iteration on a source S, of the position of this source S relative to the point M, as indicated above.

This approach proves to be particularly advantageous for surfaces to be meshed which are relatively complex, in particular when the observation point M is likely to be located in a gray area with respect to certain sources, as shown in FIG. 7B. In this FIG. 7B, the half-spheres associated with the sources in the shadow zone of the observation point M have been represented in dotted lines, and for which, therefore, the contribution is fixed as being zero in the estimated interaction . Of course, in the case where a source is screened by a sampled surface, although the scalar product SM.r associated with this source remains positive, a second test determines whether the vector SM crosses a sampled surface or not. In if so, this source is considered to be inactive specifically for the point M region.

Thus, in more general terms, the method within the meaning of the invention preferably provides at least one additional step, for each surface sample, of testing the value of a scalar product between:

a first vector r normal to the surface sample and directed towards the vertex P of the hemisphere, as shown in FIG. 7A, and

- a second vector SM drawn between a source S associated with this hemisphere and the point M which is located in the observation region, distinguishing, in particular: - the case where this scalar product is less than a predetermined threshold and the contribution of this source is not taken into account, and the case where this scalar product is greater than a predetermined threshold and the contribution of this source is effectively taken into account.

In the example above where we consider the angle θ between these two vectors, the aforementioned predetermined threshold is of course the value 0 and we simply distinguish the cases where the dot product is positive or negative.

Of course, this choice is not limiting so that for a heterostructure with several parallel diopters, one can still advantageously consider successive semi-spaces, as described above with reference to FIGS. 4A and 4B.

The simulation of FIG. 5C corresponds, for an ultrasonic wave, to the situation of FIGS. 5A and 5B taking into account: the contribution of the emission of the main wave by the radiating element ER; the contribution of the reflection of this main wave by the obstacle; and the contribution of the main wave transmission by the obstacle.

The level lines in FIG. 5C correspond to different levels of sound pressure. The radiating element ER is placed 10 mm from the obstacle OBS and inclined by 20 ° relative to the latter. We notice in particular interference fringes in a zone between the obstacle OBS and the radiating element ER. Such a simulation can advantageously indicate an ideal position for an ultrasonic sensor. These ultrasonic sensors usually include a transducer as an active radiating element and a detector for measuring the ultrasonic waves received. The simulation of FIG. 5C can thus also indicate the ideal shape of an ultrasonic sensor, according to the desired applications, for a given shape of obstacle.

The simulation of FIG. 5C was carried out by means of a matrix calculation programmed using the calculation software.

© MATLAB. The total number of meshes chosen for the obstacle and for the radiating element (here, a few hundred in all) is then optimized: on the one hand, to limit the duration of the calculations; and on the other hand, so that the size of the meshes remains lower than half a wavelength, so as to check the criterion of Rayleigh.

It is however indicated that, as the elements to be meshed in the implementation of the method according to the invention are simply surfaces, the calculation times are much shorter than those necessary in the implementation of a type calculation process "by finite elements".

The present invention can thus be manifested by the implementation of a succession of instructions for a computer program product stored in the memory of a hard disk or on a removable medium and taking place as follows: choice of a mesh pitch in particular as a function of the wavelength of the main wave;

determination of the coordinates of the sources Si and / or S'i and of the contact points Pi and / or PT; choice of a type of wave in play and calculation of the coefficients of the interaction matrices applied to the points Pi and / or PT by the implementation of a matrix calculation software; choice of a law of reflection and / or transmission of the surface of the obstacle and calculation of the coefficients of the reflection and / or transmission matrices; - calculation of the values of the sources Si and / or S'i; and calculation of scalar or vector quantities at any point in three-dimensional space.

As such, the present invention also relates to such a computer program product, stored in a central unit memory or on a removable medium suitable for cooperating with a reader of this central unit, and comprising in particular instructions for setting up implements the method according to the invention.

Of course, the present invention is not limited to the embodiment described above by way of example; it extends to other variants.

Thus, it will be understood that, even if in the above commented figures we represent both the surface of an obstacle and the surface of a radiating element, the present invention also applies to the estimation of physical quantities in as part of a wave interacting with an obstacle and emitted in the far field. In this context, it is not necessary to materialize the surface of a radiating element to be meshed and the relations

[8] and [14] above are sufficient to determine the interaction between this wave and the obstacle.

We have indicated above equations for calculating scalar or vector quantities at a point M in space, for electromagnetic, or even acoustic waves. Of course, these quantities can be estimated for other types of waves, in particular for thermal waves, electromagnetic waves involving radio frequency antennas, or the like.

Of course, the present invention is not limited to an application for non-destructive testing, but to any type of application, in particular in medical imaging, for example for the study of microsystems implementing acoustic microscopy with movable mirrors.

Interactions between a wave and a single obstacle have been described above. Of course, the present invention applies to an interaction with several obstacles. To this end, it is simply necessary to mesh the surfaces of these obstacles and to add their contribution for the estimation of a vector or scalar quantity at any point in space. Likewise, as indicated above, the surface of the obstacle OBS can be flat, or even curved, or of any complex shape.

Thus, in the context of a wave interacting with several obstacles in space, a simulation equivalent to that shown in FIG. 5C would make it possible to position sensors and / or radiating elements as a function of the configuration of these obstacles, in particular for an application to determining the position of loudspeakers in a partitioned passenger compartment, such as a passenger compartment of a motor vehicle.

The three-dimensional space can be divided into a plurality of regions, as described above with reference in Figures 4A, 4B, 5A and 5B. However, for a surface of one of said regions to be considered as an obstacle of a main wave, becoming active by emission of a secondary wave, the incidence of the main wave on this surface must preferably remain lower or equal to 90 °.

Claims

claims

1. Method for evaluating a physical quantity associated with an interaction between a wave and an obstacle, in a region of three-dimensional space, in which: a) a plurality of surface samples (dSi) is determined, by mesh at least one part represents the surface of an obstacle receiving a main wave and emitting, in response, a secondary wave, and each surface sample is assigned at least one source (Si) emitting an elementary wave representing a contribution to said secondary wave, b) forming a matrix system comprising: an interaction matrix (F (M)), invertible, applied to a given region (M) of space and comprising a number of columns corresponding to a total number from sources, a first column matrix of which each coefficient (v ^.) is associated with a source (Si) and characterizes the elementary wave which it emits, and a second column matrix, obtained by a multiplication of the first re matrix column by the interaction matrix and whose coefficients are values of a physical quantity (V (M)) representative of the wave emitted by all the sources in said given region (M), c) for estimate the coefficients of the first column matrix (v ^), assign selected physical quantity values (V (Pi)) to predetermined points (Pi), each associated with a surface sample (dSi), said selected values ( V (Pi)) being stored in the second matrix column, and this second column matrix is multiplied by the inverse of the interaction matrix applied to said predetermined points (Fi), d) to evaluate said physical quantity (V (M)) representing the wave emitted by the set of sources in a given region (M) of three-dimensional space, the interaction matrix is applied to said given region (M) and this interaction matrix is multiplied by the first column matrix comprising the coefficients estimated at step c).

2. Method according to claim 1, in which, to evaluate a physical quantity representative of an interaction between an element radiating a main wave and an obstacle receiving this main wave,

- in step a), a plurality of surface samples (dS'i) are also determined by mesh, together representing an active surface of the radiating element the main wave, and each is assigned sample of the active surface at least one source (S'i) emitting an elementary wave representing a contribution to said main wave,

- steps b), c) and d) are also applied to the samples of the active surface, and - said physical quantity (V (M)) representing the interaction between the radiating element and the obstacle is evaluated in a given region (M) of three-dimensional space, taking into account the contribution, in said given region

(M), of the main wave emitted by all the sources of the active surface and the contribution of the wave secondary emitted by all sources of the obstacle surface.

3. Method according to one of claims 1 and 2, wherein each coefficient of the interaction matrix, applied to a given region of space, is representative of an interaction between a source and said given region and the value of each coefficient is a function of a distance between a source and said given region.

4. Method according to one of claims 1 to 3, wherein the interaction matrix applied, in step c), to said predetermined points (Pi), comprises a number of lines corresponding to a total number of predetermined points ( Pi).

5. Method according to one of claims 1 to 4, wherein the physical quantity to be evaluated is a scalar quantity (V (Pi)) and, in step a), a single source is assigned to each surface sample.

6. Method according to claim 5, in which the interaction matrix (F (M)) applied, in step d), to a region of space (M), comprises a line.

7. Method according to one of claims 5 and 6, wherein each predetermined point (Pi) associated with a surface sample (dSi) corresponds to a contact point between this surface sample (dSi) and a hemisphere whose surface is equal to the area of this surface sample, and center corresponding to a position of the source (Si) which is assigned to this surface sample.

8. Method according to one of claims 5 to 7, in which:

- the main wave is an electric wave,

the coefficients of the first column matrix are electric charge values each associated with a source, and

- the coefficients of the second column matrix are values of electrical potential.

9. Method according to one of claims 5 to 7, in which:

- the main wave is a magnetic wave,

- the coefficients of the first column matrix are values of magnetic flux each associated with a source, and - the coefficients of the second column matrix are values of magnetic potential.

10. Method according to one of claims 5 to 7, in which: - the main wave is a sound wave,

the coefficients of the first column matrix are sound speed values each associated with a source, and

- the coefficients of the second column matrix are sound pressure values.

11. Method according to one of claims 1 to 4, in which the physical quantity to be evaluated is a vector quantity (V (Pi)) expressed by its three coordinates in three-dimensional space, and one attributes, to the step a), three sources (SAi, SBi, SCi) to each surface sample (dSi).

12. Method according to claim 11, in which the interaction matrix (F _γ (M)) applied, in step d), to a region of space (M), comprises a line for each coordinate (X , Y, Z) of space.

13. Method according to one of claims 11 and 12, in which: - the three sources assigned to each surface sample are substantially in the same plane and - each predetermined point (Pi) associated with a surface sample (dSi) corresponds at a point of contact between this sample and a hemisphere whose surface is equal to the surface of this sample, and of center corresponding to the position of a barycenter of the three sources.

14. The method of claim 13, wherein the three sources of the same surface sample substantially form an equilateral triangle, and the triangles of the surface samples are oriented substantially randomly with respect to each other.

15. Method according to one of claims 11 to 14, in which: - the main wave is an electric wave,

- the coefficients of the first column matrix are values of electric charge each associated with a source, and - the coefficients of the second column matrix are values of coordinates of an electric field.

16. Method according to one of claims 11 to 14, in which: - the main wave is a magnetic wave,

the coefficients of the first column matrix are magnetic flux values each associated with a source, and

- the coefficients of the second column matrix are values of coordinates of a magnetic field.

17. Method according to one of claims 11 to 14, in which:

- the main wave is a sound wave, - the coefficients of the first column matrix are values of speed of sound each associated with a source, and

- the coefficients of the second column matrix are values of coordinates of an acoustic speed.

18. Method according to one of the preceding claims, in which, to estimate the contribution of the secondary wave in said region given in step d), said values of physical quantity (V (Pi)) chosen in step c) are a function of a predetermined reflection coefficient and / or transmission of the main wave by each sample of the surface of the obstacle.

19. The method of claim 18, taken in combination with one of claims 6 and 12, wherein the secondary wave corresponds to a reflection of the main wave on the obstacle and the hemisphere is oriented outward. of the obstacle.

20. The method of claim 18, taken in combination with one of claims 6 and 12, wherein the secondary wave corresponds to a transmission of the main wave in the obstacle and the hemisphere is oriented inward. of the obstacle.

21. Method according to one of claims 19 and 20, in which, in step c), the values (v'i) associated with the sources (S'i) of the radiating element (ER) are determined and formats at least: - a first interaction matrix (F (P)) representing the contribution of the sources of the obstacle to the predetermined points on the surface of the obstacle (Pi),

a second interaction matrix (F ′ (P)) representing the contribution of the sources of the radiating element to the predetermined points on the surface of the obstacle (Pi),

- a reflection (R) or transmission (T) matrix, the coefficients of which represent reflection or transmission coefficients at each predetermined point (Pi) of the obstacle, to determine the values of the sources of the obstacle (Vi ) as a function of the values of the sources of the radiating element (v'i) and a multiplication of the first and second interaction matrices and of the reflection or transmission matrix.

22. The method of claim 21, wherein, in step c), the values (v'i) associated with the sources (S'i) of the radiating element (ER) are determined by taking into account the reception of the secondary wave by the radiating element (ER) and by further shaping: - a third interaction matrix (F (P ')) representing the contribution of the sources of the obstacle to the predetermined points of the surface of the radiating element (P'i), and a fourth interaction matrix (F '(P')) representing the contribution of the sources of the radiating element at predetermined points on the surface of the radiating element (P'i ).

23. Method according to one of claims 19 to 22, wherein the surface of one obstacle corresponds to an interface between two distinct media of a heterostructure.

24. Method according to one of the preceding claims, in which the main wave is a sound wave and the coefficients of the interaction matrix are each a function of an angle of incidence of an elementary wave coming from a source. in said given region (M).

25. Method according to one of claims 7 and 13, in which, for each surface sample, the value of a scalar product is tested between: - a first vector (r) normal to the surface sample and directed towards the apex (P) of the hemisphere (fig. 7A), and

- a second vector (SM) drawn between a source (S) associated with this hemisphere and said given region (M), distinguishing: the case where this scalar product is less than a predetermined threshold and the contribution of this source is not taken into account, and the case where this dot product is greater than a predetermined threshold and the contribution of this source is effectively taken into account.

26. Method according to one of the preceding claims, in which the main wave is a sound wave and, in step a), a total number of surface samples (dSi) is chosen substantially as a function of a length. of the sound wave to verify the Rayleigh criterion.

27. Method according to one of the preceding claims, in which a plurality of values of the physical quantity estimated in step d), obtained for a plurality of regions of space, are compared, to select a candidate region for the arrangement. of a radiating element intended to interact with the obstacle.

28. Method according to one of claims 2 to 27, wherein the radiating element is a sensor, for non-destructive testing, intended to analyze an object forming an obstacle of the main wave.

29. Computer program product, stored in a central unit memory or on a removable medium suitable for cooperating with a reader of this central unit, characterized in that it includes instructions for implementing the method according to the one of the preceding claims.