Mathematical Modeling for Corrosion Detection in Pipelines
Modèles Mathématiques pour l'Inspection Nondestructive des Pipelines
Résumé
The inverse problem of corrosion detection consists of the determination of the corrosion damage of an inaccessible part of the surface of a specimen when the available data are on the accessible part.
Difficulties of this inverse problem result from its inherent ill-posedness and nonlinearity. Most of the techniques for detecting the corrosion are based on iterative algorithms:least-square algorithms and Newton-type iteration schemes. In these methods, one needs tremendous computational costs and time
to get a close image to the true solution, since these iterative algorithms may not converge to an approximate solution.
The purpose of this work is to design a direct (non-iterative) technique for detecting corrosion in pipelines from voltage-to-current observations. Our new algorithm is of MUSIC-type (multiple signal classification) and is based on an accurate asymptotic representation formula for the steady state current perturbations.
Following an asymptotic formalism, we develop in this thesis new non-iterative methods to address the inverse problem of identifying an internal corrosive part of small Hausdorff measure in a pipeline by (i) electrical impedance, (ii) vibration analysis, and (iii) ultrasonic waves. Our new algorithms are based on accurate asymptotic representation formulae for the data.
In the first chapter, we establish an asymptotic representation formula for the steady state currents caused by internal corrosive parts of small Hausdorff measures. Based on this formula we design
a non-iterative method of MUSIC (multiple signal classification) type for localizing the corrosive parts from voltage-to-curren observations.
The vibration behavior of structures can be characterized in terms
of resonance frequencies and mode shapes which describe properties of the tested object in a global way but do not in general provide information about structural details. Our aim in the second
chapter is to develop a simple method to address the inverse problem of identifying an internal corrosive part of small Hausdorff measure in a pipeline by vibration analysis. The viability of our reconstruction method is documented by a variety of numerical results from synthetic, noiseless and noisy data.
In the third chapter, we develop three closely-related methods to address the inverse problem of identifying a collection of disjoint internal corrosive parts of small Hausdorff measures in pipelines from exterior ultrasonic boundary measurements. Our approaches also allow us to determine the actual number of corrosive parts present, as well as make use of one or multiple ultrasonic waves.
In the fourth chapter, we consider the problem of determining the boundary perturbations of an object from far-field electric or acoustic measurements. Assuming that the unknown object boundary is a small perturbation of a circle, we develop a linearized relation between the far-field data that result from fixed Dirichlet boundary conditions, entering as parameters, and the shape of the object, entering as variables. This relation is used to find the Fourier coefficients of the perturbation of the shape and makes use of an expansion of the Dirichlet-to-Neumann operator.
Difficulties of this inverse problem result from its inherent ill-posedness and nonlinearity. Most of the techniques for detecting the corrosion are based on iterative algorithms:least-square algorithms and Newton-type iteration schemes. In these methods, one needs tremendous computational costs and time
to get a close image to the true solution, since these iterative algorithms may not converge to an approximate solution.
The purpose of this work is to design a direct (non-iterative) technique for detecting corrosion in pipelines from voltage-to-current observations. Our new algorithm is of MUSIC-type (multiple signal classification) and is based on an accurate asymptotic representation formula for the steady state current perturbations.
Following an asymptotic formalism, we develop in this thesis new non-iterative methods to address the inverse problem of identifying an internal corrosive part of small Hausdorff measure in a pipeline by (i) electrical impedance, (ii) vibration analysis, and (iii) ultrasonic waves. Our new algorithms are based on accurate asymptotic representation formulae for the data.
In the first chapter, we establish an asymptotic representation formula for the steady state currents caused by internal corrosive parts of small Hausdorff measures. Based on this formula we design
a non-iterative method of MUSIC (multiple signal classification) type for localizing the corrosive parts from voltage-to-curren observations.
The vibration behavior of structures can be characterized in terms
of resonance frequencies and mode shapes which describe properties of the tested object in a global way but do not in general provide information about structural details. Our aim in the second
chapter is to develop a simple method to address the inverse problem of identifying an internal corrosive part of small Hausdorff measure in a pipeline by vibration analysis. The viability of our reconstruction method is documented by a variety of numerical results from synthetic, noiseless and noisy data.
In the third chapter, we develop three closely-related methods to address the inverse problem of identifying a collection of disjoint internal corrosive parts of small Hausdorff measures in pipelines from exterior ultrasonic boundary measurements. Our approaches also allow us to determine the actual number of corrosive parts present, as well as make use of one or multiple ultrasonic waves.
In the fourth chapter, we consider the problem of determining the boundary perturbations of an object from far-field electric or acoustic measurements. Assuming that the unknown object boundary is a small perturbation of a circle, we develop a linearized relation between the far-field data that result from fixed Dirichlet boundary conditions, entering as parameters, and the shape of the object, entering as variables. This relation is used to find the Fourier coefficients of the perturbation of the shape and makes use of an expansion of the Dirichlet-to-Neumann operator.
Dans les trois premiers chapitres de ce manuscrit de thèse, On propose trois nouvelles méthodes pour l'identification et la localisation des corrosions internes dans les pipelines. La première est par impédance électrique, la deuxième est par ondes guidées ultrasoniques et la troisième est par ultrasons.
On jette les bases mathématiques de ces différentes méthodes et on présente quelques tests numériques qui montrent leur efficacité.
Notre approche rentre dans la stratégie asymptotique développée au CMAP pour la résolution des problèmes inverses d'une manière robuste et stable. On exploite l'existence d'un petit paramètre (la mesure de Hausdorff de la partie corrosive) pour extraire des données la localisation de la partie corrosive et estimer son étendue. Le tout, d'abord, à travers des formules asymptotiques des mesures dépendantes du petit paramètre, rigoureusement établies à l'aide de la méthode des équations intégrales, et ensuite, par le biais de nouveaux algorithmes non-itératifs d'inversion. La plupart de ces algorithmes sont de type MUSIC (multiple signalclassification).
Le dernier chapitre est indépendant des trois premiers. il est consacré à la reconstruction de la forme d'un objet perturbé connaissant le champ lointain électrique ou acoustique. On développe pour le cas acoustique et électrique une relation linéarisée entre le champ lointain, résultant des données sur le bord de conditions de Dirichlet comme paramètre, et la forme de la structure perturbée comme variable. Cette relation nous ouvre la voie à la reconstruction
des coefficients de Fourier de la perturbation et nous aide à la reconstruction des coefficients de Fourier de la perturbation ce qui nous mène à formuler un développement asymptotique complet de
l'opérateur Dirichlet-Neumann.
On jette les bases mathématiques de ces différentes méthodes et on présente quelques tests numériques qui montrent leur efficacité.
Notre approche rentre dans la stratégie asymptotique développée au CMAP pour la résolution des problèmes inverses d'une manière robuste et stable. On exploite l'existence d'un petit paramètre (la mesure de Hausdorff de la partie corrosive) pour extraire des données la localisation de la partie corrosive et estimer son étendue. Le tout, d'abord, à travers des formules asymptotiques des mesures dépendantes du petit paramètre, rigoureusement établies à l'aide de la méthode des équations intégrales, et ensuite, par le biais de nouveaux algorithmes non-itératifs d'inversion. La plupart de ces algorithmes sont de type MUSIC (multiple signalclassification).
Le dernier chapitre est indépendant des trois premiers. il est consacré à la reconstruction de la forme d'un objet perturbé connaissant le champ lointain électrique ou acoustique. On développe pour le cas acoustique et électrique une relation linéarisée entre le champ lointain, résultant des données sur le bord de conditions de Dirichlet comme paramètre, et la forme de la structure perturbée comme variable. Cette relation nous ouvre la voie à la reconstruction
des coefficients de Fourier de la perturbation et nous aide à la reconstruction des coefficients de Fourier de la perturbation ce qui nous mène à formuler un développement asymptotique complet de
l'opérateur Dirichlet-Neumann.