Classifications

• • G—PHYSICS
  • G06—COMPUTING; CALCULATING OR COUNTING
  • G06F—ELECTRIC DIGITAL DATA PROCESSING
  • G06F30/00—Computer-aided design [CAD]
  • G06F30/20—Design optimisation, verification or simulation
  • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]

Definitions

• 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.
• a so-called "finite element” modeling method 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 ⁇ > ⁇ 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.
• the physical quantity sought such as an electrostatic potential or a pressure value
• This border can be fictitious. Boundary conditions are imposed there. The potential is therefore unknown within the same domain.
• 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.
• 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.
• 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,
• 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.
• 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,
• radioactive 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.
• the physical quantity to be evaluated is a scalar quantity and, in step a), a unique source is assigned to each surface sample.
• 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.
• 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.
• 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.
• 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.
• 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.
• the overall properties of the obstacle are known, in particular in transmission and / or in reflection.
• this sensor being intended to analyze a target forming an obstacle of the main wave.
• 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.
• 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.
• 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. 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. 1A schematically represents the respective surfaces of a radiating element ER emitting a wave and of an obstacle OBS receiving this wave, meshed
• 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
• FIG. 4A represents, like FIGS. 1A and 2A, an interaction between a radiating element ER and an obstacle OBS, to evaluate a physical quantity
• FIG. 4B complementary to FIG.
• 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 ⁇ ;
• 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.
• 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.
• each surface sample dSi is associated with a hemisphere HEMi, tangent to the surface sample dSi at a contact point Pi.
• this contact point Pi corresponds to the top of the hemisphere HEMi.
• 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.
• the hemisphere HE i is constructed as described below.
• 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.
• 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.
• the radius Ri of the hemisphere is deduced from the expression:
• 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.
• 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.
• the surface of a radiating element ER corresponding for example to a wave generator, is further meshed.
• the surface of a radiating element ER corresponding for example to a wave generator
• step b) The matrix system that is shaped in step b) above corresponds to:
• the interaction matrix F comprises coefficients Ci, j whose general expression is given by:
• 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.
• ⁇ 0 is a dielectric constant
• MjS j is a distance measured in algebraic value
• - qj corresponds to an electric charge characterizing a source Sj
• - Ui corresponds to an electric potential at point Mi.
• - ⁇ 0 corresponds to the magnetic peimédbiiic ⁇ of the medium where the point Mi is located, - ⁇ j corresponds to the magnetic flux associated with the source s 3 ; ⁇ corresponds to the magnetic potential at point i.
• ⁇ 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
• Pi corresponds to the sound pressure generated by the propagation of the ultrasonic wave at point Mi.
• the term dSj corresponds to the surface of the sample associated with the source Sj.
• 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.
• 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
• 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.
• the interaction matrix F can have only one line of coefficients Cj, with:
• 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.
• F ' is the interaction matrix between the surface of the radiating element and the point M;
• 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.
• 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. 4 ⁇ , 413, 5A and 53.
• 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.
• 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).
• the three sources SAi, SBi, SCi, allocated to a surface sample dS are of respective positions determined as indicated below.
• 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.
• 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.
• each source SAi, SBi and SCi is arranged on the middle of a radius Ri of the hemisphere.
• the lines which connect the barycenter Gi to each source are angularly spaced by 120 °.
• the angular orientation of the triangles formed by the source triplets is chosen randomly, from one surface sample to another.
• 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.
• the vector quantity N (M) to be estimated can be:
• the matrix system is shaped according to the following relationship:
• 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 values v ⁇ j associated with each source S ⁇ j 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 (P ⁇ ), V y (Pi) and V z (Pi).
• N (M) N x (M) x + N y (M) y + N 2 (M) z [14]
• V Z (M) ⁇ f_ [ ⁇ d (M, S ⁇ j ))].
• x, y and z correspond to unit vectors carried by the x, y and z axes of three-dimensional space.
• 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.
• the values of the sources v ⁇ 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.
• V (M) being the scalar quantity previously calculated by equation [8].
• N the coefficients of the interaction matrix F are there inversely proportional to the square of a distance separating each source from point M
• V 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 3 ⁇ 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.
• 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.
• 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.
• 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 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.
• the electrical potential at the contact points P 3 and P 4 corresponds to the electrical potential of the second armature V2.
• 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.
• 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
• the reflection coefficient is practically zero at the surface dS ⁇
• 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.
• the values of its sources vA, vB and vC would be respectively, for example, + q, + q and -2q.
• the magnetic fluxes of the three sources associated with this surface sample would be + ⁇ , + ⁇ and -2 ⁇ .
• this approach assumes that the reflection coefficient R of an obstacle is known beforehand.
• a matrix R which is representative of the reflection coefficient at each point Pi.
• 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.
• the contribution of the wave emitted by the radiating element ER is expressed by:
• V (P) F (P) .v [20]
• the secondary wave simply corresponds to a reflection of the main wave. What is expressed by the relationship:
• 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 S ⁇ i, within the framework of an estimation of a vector quantity) of the OBS obstacle.
• V T (P) ' , [F (P) Y 1 . R. [F '(P)] + F' (P) ') .v' [10]
• 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 V ⁇ (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];
• 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 ⁇ i of the ray coming from the source Si, at the point of the three-dimensional space M. 4/044790
• Ci is the speed of sound in the medium Mi
• c 2 is the speed of sound in the medium M 2 ;
• the wave received by point M is a wave transmitted by the obstacle OBS.
• the sources of the radiating element ER are no longer active, due to the occultation of the radiating element ER by the obstacle OBS.
• 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.
• each transmission coefficient Ti is given by the relation:
• cos ⁇ i can be determined as a function of the respective coordinates of the sources Si and of the point M.
• 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 .
• 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).
• FIG. 5A 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.
• 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.
• a surface which includes the surface of the obstacle FIG. 5A.
• 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.
• 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.
• these sources SSi can be active by reflection of the main wave if the obstacle OBS separates two media of different indices.
• 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.
• 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.
• account is taken, for each slice of the space, of the contributions of all the interfaces as expressed by the relations [10] and [22].
• the scalar product SM.r is tested at each iteration with respect to a source S, for example in the form:
• 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.
• the base of the vector r is preferably located at the barycenter of the three sources SI, S2, S3.
• the calculation of the scalar product concerns each source Si of the triplet SI, S2, S3.
• the test relates to a quantity of the type: S .r
• 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.
• 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 .
• 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.
• 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:
• the aforementioned predetermined threshold is of course the value 0 and we simply distinguish the cases where the dot product is positive or negative.
• 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.
• 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.
• 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;
• 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.
• 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.
• 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.
• the surface of the obstacle OBS can be flat, or even curved, or of any complex shape.
• 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.
• the incidence of the main wave on this surface must preferably remain lower or equal to 90 °.

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