Computer-implemented method for identifying mechanical properties by coupled correlation of images and mechanical modelling

Abstract

A computer-implemented method for identifying mechanical parameters of an object subjected to mechanical stress is provided. The method comprises a step of acquiring, by an imaging means, images of the object taken before and during the application of the mechanical stress, three steps of calculating the effects due to the stress carried out either on the basis of the modeling of the recorded images or on the basis of a theoretical mechanical modeling of the stress, a step of defining a functional equal to the difference between the two models and a last step of minimizing said functional so that the experimental model is as close as possible to the theoretical mechanical model. Additional measurements make it possible to refine the method.

Claims

1. A computer-implemented method for identifying at least one mechanical parameter called a “target parameter” of an object subjected to a mechanical stress, wherein said computer-implemented method comprises the following steps: Step 1: acquiring, by an imaging means, at least two digital images of the object taken before and during an application of the mechanical stress and measuring a scale factor of the object; Step 2: calculating with a computer a first functional T.sub.CIN(U.sub.CIN) corresponding to a correlation of the at least two digital images depending on a displacement field U.sub.CIN represented using a first kinematic base, said displacement field U.sub.CIN being measured at any point of the object under stress between the at least two digital images of the object under load and without load; Step 3: calculating with the computer a calculated displacement field U.sub.CAL at any point of the object; Step 4: calculating with the computer a second functional T.sub.CAL(U.sub.CAL,{p},{q}) on a basis of the calculated displacement field U.sub.CAL represented using a second kinematic base, this second functional corresponding to a variational formulation of a mechanical model of the stress depending on a geometry of the object, forces applied, boundary conditions, at least a target parameter {p} and predetermined mechanical parameters {q}; Step 5: calculating with the computer a third functional T.sub.PAR(U.sub.CIN,U.sub.CAL) in a form of a quadratic norm, equal to a difference between U.sub.CIN and U.sub.CAL; and Step 6: minimizing with the computer, with respect to U.sub.CIN, U.sub.CAL and {p}, a total functional T.sub.TOT(U.sub.CIN,U.sub.CAL,{p},{q}) comprising at least the terms:
T.sub.TOT(U.sub.CIN,U.sub.CAL,{p},{q})=αT.sub.CIN(U.sub.CIN)+βT.sub.CAL(U.sub.CAL,{p},{q})γT.sub.PAR(U.sub.CIN,U.sub.CAL) α,β and γ being three non-zero weighting coefficient, wherein the step of minimizing with the computer identifies the “target parameter” that relates to at least one of the following: a mechanical property of a constitutive material of the object, one or more phases of the object, a geometry of the object, and/or quantities of the object; and wherein the imaging means comprises at least one of the following: a video camera, a still camera, a scanning electron microscope, an atomic force microscope, a tomography apparatus, an X-ray tomography apparatus, a magnetic resonance tomography apparatus, and/or an optical coherence tomography apparatus.

2. The computer-implemented method for identifying at least one mechanical parameter as claimed in claim 1, wherein when a behavior of the object is subjected to a time-dependent stress, the second functional T.sub.CAL(U.sub.CAL, {p}, {q}) is dependent on determined times.

3. The computer-implemented method for identifying at least one mechanical parameter as claimed in claim 1, wherein step 1 of the method comprises additional measurements F.sub.MES of forces, times or temperatures, step 3 of the method comprises evaluations F.sub.CAL corresponding to said additional measurements, step 4 is followed by a step 4bis of calculating a fourth functional T.sub.FOR(F.sub.CAL,F.sub.MES) proportional to a quadratic deviation between these quantities and a total functional T.sub.TOT(U.sub.CIN,U.sub.CAL,{p},{q}) of step 5 is equal to:
T.sub.TOT(U.sub.CIN,U.sub.CAL,{p},{q})=αT.sub.CIN(U.sub.CIN)+βT.sub.CAL(U.sub.CAL,{p},{q})+γT.sub.PAR(U.sub.CIN,U.sub.CAL)+χT.sub.FOR(F.sub.CAL({p},{q}), F.sub.MES) χ being a fourth weighting coefficient, said weighting coefficient χ is adjusted according to uncertainties associated with various quantities involved in the functionals, and/or according to a condition number of a problem tangent to the minimization of the functional T.sub.TOT.

4. The computer-implemented method for identifying at least one mechanical parameter as claimed in claim 1, wherein the minimization of the total functional T.sub.TOT is carried out by iterative method.

5. The computer-implemented method for identifying at least one mechanical parameter as claimed in claim 1, wherein the minimization of the total functional T.sub.TOT is carried out by iterative method, requiring the calculation of a gradient of T.sub.TOT.

6. The computer-implemented method for identifying at least one mechanical parameter as claimed in claim 1, wherein the first kinematic base is identical to the second kinematic base.

7. The computer-implemented method for identifying at least one mechanical parameter as claimed in claim 1, wherein a measurement uncertainty is estimated by a Hessian of the functional T.sub.TOT with convergence by having a measurement of acquisition noise available.

8. The computer-implemented method for identifying at least one mechanical parameter as claimed in claim 1, wherein the first kinematic base or the second kinematic base is produced on a finite element mesh.

9. A computer device suitable for identifying at least one mechanical parameter as claimed in claim 1.

10. A computer-readable medium with program for executing the method as claimed in claim 1.

11. A mechanical test system configured to identify at least one mechanical parameter called a “target parameter” of an object subjected to a mechanical stress, wherein said the mechanical test system comprises: an imaging device configured to acquire at least two digital images of the object taken before and during an application of the mechanical stress and measuring a scale factor of the object; a computer configured to calculate a first functional T.sub.CIN(U.sub.CIN) corresponding to a correlation of the at least two digital images depending on a displacement field U.sub.CIN represented using a first kinematic base, said displacement field U.sub.CIN being measured at any point of the object under stress between the at least two digital images of the object under load and without load; the computer further configured to calculate a calculated displacement field U.sub.CAL at any point of the object; the computer further configured to calculate a second functional T.sub.CAL(U.sub.CAL,{p},{q}) on a basis of the calculated displacement field U.sub.CAL represented using a second kinematic base, this second functional corresponding to a variational formulation of a mechanical model of the stress depending on a geometry of the object, forces applied, boundary conditions, at least a target parameter {p} and predetermined mechanical parameters {q}; the computer further configured to calculate a third functional T.sub.PAR(U.sub.CIN,U.sub.CAL) in a form of a quadratic norm, equal to a difference between U.sub.CIN and U.sub.CAL; and the computer further configured to calculate a minimization of, with respect to U.sub.CIN, U.sub.CAL and {p}, a total functional T.sub.TOT(U.sub.CIN,U.sub.CAL,{p},{q}) comprising at least the terms:
T.sub.TOT(U.sub.CIN,U.sub.CAL,{p},{q})=αT.sub.CIN(U.sub.CIN)βT.sub.CAL(U.sub.CAL,{p},{q})γT.sub.PAR(U.sub.CIN,U.sub.CAL) α,β and γ being three non-zero weighting coefficients, wherein the step of minimizing with the computer identifies the “target parameter” that relates to at least one of the following: a mechanical property of a constitutive material of the object, one or more phases of the object, a geometry of the object, and/or quantities of the object; and wherein the imaging device comprises at least one of the following: a video camera, a still camera, a scanning electron microscope, an atomic force microscope, a tomography apparatus, an X-ray tomography apparatus, a magnetic resonance tomography apparatus, and/or an optical coherence tomography apparatus.

12. A non-transitory computer-readable medium having instructions for execution by a computer for identifying at least one mechanical parameter called a “target parameter” of an object subjected to a mechanical stress, wherein said non-transitory computer-readable medium instructions comprising: Step 1: acquiring, by an imaging means, at least two digital images of the object taken before and during an application of the mechanical stress and measuring a scale factor of the object; Step 2: calculating a first functional T.sub.CIN(U.sub.CIN) corresponding to a correlation of the at least two digital images depending on a displacement field U.sub.CIN represented using a first kinematic base, said displacement field U.sub.CIN being measured at any point of the object under stress between the at least two digital images of the object under load and without load; Step 3: calculating a calculated displacement field U.sub.CAL at any point of the object; Step 4: calculating a second functional T.sub.CAL(U.sub.CAL,{p},{q}) on a basis of the calculated displacement field U.sub.CAL represented using a second kinematic base, this second functional corresponding to a variational formulation of a mechanical model of the stress depending on a geometry of the object, forces applied, boundary conditions, at least a target parameter {p} and predetermined mechanical parameters {q}; Step 5: calculating a third functional T.sub.PAR(U.sub.CIN,U.sub.CAL) in a form of a quadratic norm, equal to a difference between U.sub.CIN and U.sub.CAL; and Step 6: minimizing, with respect to U.sub.CIN, U.sub.CAL and {p}, a total functional T.sub.TOT(U.sub.CIN,U.sub.CAL,{p},{q}) comprising at least the terms:
T.sub.TOT(U.sub.CIN,U.sub.CAL,{p},{q})=αT.sub.CIN(U.sub.CIN)+βT.sub.CAL(U.sub.CAL,{p},{q})+γT.sub.PAR(U.sub.CIN,U.sub.CAL) α,α and γ being three non-zero weighting coefficients, wherein the step of minimizing with the computer identifies the “target parameter” that relates to at least one of the following: a mechanical property of a constitutive material of the object, one or more phases of the object, a geometry of the object, and/or quantities of the object; and wherein the imaging means comprises at least one of the following: a video camera, a still camera, a scanning electron microscope, an atomic force microscope, a tomography apparatus, an X-ray tomography apparatus, a magnetic resonance tomography apparatus, and/or an optical coherence tomography apparatus.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention will be better understood and other advantages will become apparent on reading the description that follows, which is provided without limitation, and by virtue of the appended figures, in which:

(2) FIG. 1, already described, shows the various steps in the representation of a stress in a sample using monitoring by means of the digital image correlation method known as CIN (DIC);

(3) FIG. 2, already described, shows the various steps in the representation of a stress in a sample by means of the modeling method;

(4) FIG. 3, already described, shows the various steps in the optimization of the target parameters {p} such as the mechanical properties, the geometry or the boundary conditions by means of the iterative method according to the prior art;

(5) FIG. 4 shows the various steps in the optimization of the target parameters by means of the iterative global method according to the invention.

DETAILED DESCRIPTION

(6) By way of example, FIG. 4 shows the various steps in the optimization of the target parameters by means of the iterative global method according to the invention. If the method according to the invention is compared with the method according to the prior art of FIG. 3, the main change between the two methods is the position of the calculation of the kinematics, in particular with regard to the boundary conditions. In the method according to the invention, the modeling also intervenes in these kinematics which are therefore no longer constant but form part of the optimization loop.

(7) The computer-implemented method for identifying at least one mechanical parameter called “target parameter” of a constitutive material of a test piece or of a part subjected to a known mechanical stress according to the invention comprises the steps described below.

(8) A first step comprises acquiring digital images of the object taken before, during and, for certain applications, after the application of the mechanical stress, by an imaging means, and measuring the scale factor of the object. Indeed, knowledge of the scale of the sample is essential if it is desired to identify mechanical properties that have a physical dimension.

(9) The imaging means may be any device used alone or in combination that makes it possible to obtain at least one image of the object. The images obtained by said means may be optical images obtained in various ranges of wavelengths well known to those skilled in the art.

(10) Second, a first functional T.sub.CIN is calculated on the basis of the images acquired using the chosen method. This first functional corresponds to the correlation of the digital images on the basis of the displacement field U.sub.CIN represented using a first relevant kinematic base, for example, on a finite element mesh.

(11) Conventionally, this first functional is the sum, over the region being studied, of the quadratic difference between the reference image and one or more corrected distorted images of the displacement field, but other criteria such as intercorrelation or the joint entropy of information may be chosen.

(12) This first step of the method may include additional measurements F.sub.MES such as force, time or temperature measurements.

(13) A third step consists in calculating the calculated displacement field U.sub.CAL represented using a second kinematic base, this second functional corresponding to the variational formulation of a mechanical model of the stress depending on the geometry of the object, the forces applied, the boundary conditions, at least the target parameters {p} and predetermined mechanical parameters {q}.

(14) Indeed, the same mechanical test may be modeled numerically, either by means of the finite element method, or by means of other techniques, to calculate the displacement field which is then denoted by U.sub.CAL(x).

(15) A fourth step consists in calculating a second functional T.sub.CAL(U, {p}, {q}) depending on the mechanical behavior of the one or more materials, the geometry of the part, the boundary conditions potentially including the applied forces and the time or times in question gathered together in the form of target {p} or predetermined {q} parameters and a set of nominal values for the target parameters. This fourth step of the method may include evaluations of force, time or temperature F.sub.CAL corresponding to additional measurements F.sub.MES, if they are available, making it possible to formulate an additional functional T.sub.FOR (F.sub.CAL, F.sub.MES) proportional to the quadratic difference of these last two quantities, potentially weighted by the inverse of the variances of the measurements.

(16) In a fifth step, a third functional T.sub.PAR(U.sub.CAL, U.sub.CIN) is introduced in the form of the quadratic norm of the difference between U.sub.CIN and U.sub.CAL. The two displacement fields merge, along with the other physical quantities measured and calculated if the target parameters are well identified and the predetermined parameters are appropriate, as well as the model used.

(17) Thus, the principle of the proposed identification is to minimize, in a last step, the weighted sum T.sub.TOT of these three or four functionals:
T.sub.TOT(U.sub.CIN,U.sub.CAL,{p},{q})=αT.sub.CIN(U.sub.CIN)+βT.sub.CAL(U.sub.CALV,{p},{q})+γT.sub.PAR(U.sub.CIN,U.sub.CAL)+χT.sub.FOR(F.sub.CAL({p},{q}),F.sub.MES) with respect to the two displacement fields U.sub.CAL and U.sub.CIN, as well as to the target parameters {p}. This functional T.sub.TOT is reduced to its first three terms in the case where the method does not include additional measurements.

(18) It should be noted that the problem may remain ill-posed if the range of stresses, the geometry, or even the definition of the images are unsuitable. In this circumstance, not all of the target parameters can be measured. A Tikhonov regularization, which corresponds to a penalization of the deviation between identified and expected parameters, may then be necessary to allow the problem to be solved numerically. The solution obtained should then be judged using its own uncertainty, for example, by considering the effect of measurement noise characterized beforehand on the minimization of T.sub.TOT, and without taking the Tikhonov regularization into account.

(19) The weighting coefficients (α,β, γ, χ) make it possible to give greater or lesser importance to the various terms according to the uncertainties associated with the quantities involved in the functionals and/or according to the condition number of the problem tangent to the minimization of the functional T.sub.TOT. It should be noted that any of the arbitrarily chosen weights may be set to 1.

(20) If the variational expression of the mechanical model is not directly accessible, it should be noted that the minimization of the functional T.sub.PAR with respect to U.sub.CAL is simply expressed for example in a finite element code by a linear elastic connection giving rise at each node to a nodal force proportional to the deviation between U.sub.CAL and U.sub.CIN. Thus, with respect to finite element modeling performed using a current professional computer code, able to include arbitrarily complex constitutive laws, the proposed formulation of T.sub.TOT in its minimization with respect to U.sub.CAL simply requires the introduction of an additional linear elastic connection at each node of the mesh. The solution obtained will be exactly that which minimizes the total functional with fixed U.sub.CIN, {p} and {q}. By alternating the minimization steps with respect to different subsets of unknowns, it is possible to reach the target minimization if the problem is well posed.

(21) This minimization of the functional T.sub.TOT may be performed, for example, by means of a Newton-Raphson method, via successive linearizations and corrections.

(22) Advantageously, the Hessian of the functional T.sub.TOT with convergence makes it possible to estimate the measurement uncertainty if a measurement of the acquisition noise is available, for example, via repeated acquisitions without stresses before performing the mechanical test. In particular, the well-posed character of the problem corresponding to strictly positive eigenvalues and in this case the condition number corresponding to the spectral radius of the Hessian may be appreciated. Otherwise, a Tikhonov regularization may be proposed.

(23) Advantageously, these elements also make it possible to validate or invalidate the model. Specifically: the displacement field U.sub.CIN makes it possible to estimate the residual field of the image correlation, that is to say the difference between the distorted and corrected images of the U.sub.CIN displacement field and the reference image; the modeling makes it possible to validate that the constitutive law and the equilibrium conditions are satisfied; any additional measurements which may be available are compared with those resulting from the modeling; the two displacement fields, one close to the measurement, U.sub.CIN, and the other close to the model, U.sub.CAL, are combined within one and the same functional gauging the consistency of the two approaches.

(24) Thus, each of the functionals used provides its own validation. Conversely, residuals that are too large to be compatible with acquisition noise signal model or measurement errors and provide indications as to how to enrich the interpretive model or to identify unanticipated measurement artifacts.

(25) Advantageously, the coupling of the various terms of the functional makes it possible to compensate for the ill-posed or ill-conditioned character of this or that functional. For example, a region of low contrast or insufficient lighting may not allow the measurement of U.sub.CIN using the functional T.sub.CIN alone. The functional T.sub.PAR may then compensate for the lack of information by calculation.

(26) Symmetrically, when the constitutive law or the geometrical non-linearities induce a loss of stability or uniqueness of the mechanical problem solution then the coupling functional T.sub.PAR may make it possible to restore the well-posed character of the problem and to follow via the model the same bifurcation branch.

(27) With convergence, the total functional reaches its minimum for a displacement field U.sub.CIN, a distinct calculated displacement field U.sub.CAL, values of forces, times or temperature or other physical quantities of the model F.sub.CAL and an estimate of the target parameters {p} for the identification proper. These parameters may be parameters of the material, relate to one or more phases, or the geometry of the object or other quantities (e.g. boundary conditions).

(28) The method is implemented by computer, thus making it possible to adapt a computer device for the identification of at least one mechanical parameter according to the method described above.

(29) Consequently, the method according to the invention may be implemented essentially by numerical calculation means which are perfectly achievable using current computing tools, but which in practice make it possible to analyze a wider range of materials, or with less expensive hardware, in particular in relation to the quality of the acquisition hardware for a given end result.