The main objective of this thesis is to propose and test some shape optimization techniques to identify and reconstruct deposits at the shell side of conductive tubes in steam generators using signals from eddy current coils. This problem is motivated by non-destructive testing applications in the nuclear power industry where the deposit clogging the cooling circuit may affect power productivity and structural safety. We consider in a first part an axisymmetric case for which we set the model by establishing a 2-D differential equation describing the eddy current phenomenon, which enable us to simulate the impedance measurements as the observed signals to be used in the inversion. To speed up numerical simulations, we discuss the behavior of the solution of the eddy current problem and build artificial boundary conditions, in particular by explicitly constructing DtN operators, to truncate the domain of the problem. In the deposit reconstruction, we adapt two different methods according to two distinct kinds of deposits. The first kind of deposit has relatively low conductivity (about 1e4 S/m). We apply the shape optimization method which consists in expliciting the signal derivative due to a shape perturbation of the deposit domain and to build the gradient by using the adjoint state with respect to the derivative and the cost functional. While for the second kind of deposit with high conductivity (5.8e7 S/m) but in the form of thin layer (in micrometers), the previous method encounter a high numerical cost due to the tiny size of the mesh used to model the layer. To overcome this difficulty, we build an adapted asymptotic model by appropriately selecting the the family of effective transmissions conditions on the interface between the deposit and the tube. The name of the asymptotic model is due to the fact that the effective transmissions conditions are derived from the asymptotic expansion of the solution with respect to a small parameter "delta" characterizing the thickness of the thin layer and the conductivity behavior. Then the inverse problem consists in reconstructing the parameters representing the layer thickness of the deposit. For both of the two approaches, we validate numerically the direct and inverse problems. In a second part we complement this work by extending the above methods to the 3-D case for a non-axisymmetric configuration. This is motivated by either non axisymmetric deposits or the existence of non axisymmetric components like support plates of steam generator tubes.
Authors
- Bibliographic Reference
- Zixian Jiang. Some inversion methods applied to non-destructive testings of steam generator via eddy current probe. Analysis of PDEs [math.AP]. Ecole Polytechnique X, 2014. English. ⟨NNT : ⟩. ⟨pastel-00943613⟩
- HAL Collection
- ['Ecole Polytechnique', 'PASTEL - ParisTech', 'ParisTech', 'CNRS - Centre national de la recherche scientifique', 'CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions', 'Centre de mathématiques appliquées (CMAP)', 'Polytechnique', 'Département de mathématiques appliquées', 'Centre de Mathématiques Appliquées', 'Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes', "Thèses du Centre de Mathématiques Appliquées de l'École polytechnique"]
- HAL Identifier
- 943613
- Institution
- École polytechnique
- Laboratory
- Centre de Mathématiques Appliquées - Ecole Polytechnique
- Published in
- France
Table of Contents
- Introduction 10
- I Eddy current inspection with axisymmetric deposits on tubes 16
- Simulation of eddy current probe 18
- Axisymmetric model 20
- Asymptotic behavior for large r 21
- Radial cut-off for eddy current simulations 23
- DtN operator and cut-off in the longitudinal direction 25
- Analysis of the non-selfadjoint eigenvalue problem 26
- Spectral decomposition of the DtN operator 29
- On the analysis of spectral error truncation 33
- Numerical test 36
- Error of domain cut-off in the radial direction 37
- Error introduced by the DtN maps 38
- Appendix: Some properties of the weighted spaces 43
- Proof of Lemma 1.1.1 43
- Proof of Lemma 1.1.6 45
- Proof of Lemma 1.2.1 46
- Appendix: Proof of Proposition 1.1.5 48
- Appendix: Local boundary conditions for radial cut-off 51
- Dirichlet boundary condition 51
- Robin boundary condition 53
- Appendix: 1-D Calibration 54
- Identification of lowly-conductive deposits 58
- Modeling of ECT signal for axisymmetric configurations 59
- Shape derivative of the impedance measurements 62
- Shape and material derivatives of the solution 62
- Shape derivative of the impedance 68
- Expression of the impedance shape derivative using the adjoint state 70
- Shape reconstruction of deposits using a gradient method 72
- Objective function 72
- Regularization of the descent direction 73
- Inversion algorithm 74
- Numerical tests 74
- On the reconstruction of the deposit conductivity and permeability 81
- The cost functional derivative withe respect to the conductivity 81
- Derivative with respect to the magnetic permeability 82
- Numerical tests 83
- On the reconstruction of the shape and physical parameters 85
- II Asymptotic models for axisymmetric thin copper layers 88
- Some asymptotic models for thin and highly conducting deposits 90
- Settings for asymptotic models 91
- Rescaled in-layer eddy current model 91
- Taylor developments for u+ 92
- Transmission conditions between (or ) and u 94
- Procedure for obtaining approximate transmission conditions Zm,n between u 95
- Asymptotic models for deposits with constant thickness 95
- Rescaling of the in-layer problem 95
- Transmission conditions between (or ) and u 97
- Computing algorithm for the rescaled in-layer field (or ) 99
- Computation of some approximate transmission conditions Zm,n 101
- Numerical tests for 1-D models 111
- Reconstruction of deposit thin layers via asymptotic models 120
- Asymptotic models for deposits with variable layer thickness 121
- Formal derivation of approximate transmission conditions Z1,0 121
- The asymptotic model of order 0 122
- Formal derivation of approximate transmission conditions Z1,1 123
- Mixed formulation for the asymptotic model using Z1,1 125
- Numerical validation of the 2-D asymptotic models 129
- Approximation of the impedance measurements 130
- Numerical tests on impedance measurements 132
- Thickness reconstruction via asymptotic model using Z1,0 133
- Derivative of the solution with respect to a thickness increment 133
- Adjoint state and derivative of the impedance measurements 133
- Thickness reconstruction by minimizing a least square cost functional 134
- Thickness reconstruction via asymptotic model using Z1,1 135
- Derivative of solution with respect to a thickness increment 135
- Adjoint state and derivative of the impedance measurements 136
- Thickness reconstruction by minimizing a least square functional 137
- Numerical tests 138
- Parameterized thin layers 138
- Reconstruction of arbitary thin layers 139
- III Extension to the 3-D case 142
- Eddy current tomography using shape optimization 144
- Formulation via vector potentials 145
- Problem for vector potential with Coulomb gauge condition 145
- Variational formulation of the eddy current problem via vector potentials 147
- Deposit reconstruction via shape optimization 148
- Shape and material derivatives of the solution 148
- Shape derivative of the impedance measurements 154
- Expression of the impedance shape derivative using the adjoint state 157
- Shape derivative for a least square cost functional 160
- Validation for axisymmetric configurations 161
- Appendices 161
- Differential identities 161
- Proof of Lemma 5.2.6 161
- Conclusion and perspectives 166
- Bibliography 168
