In a layered thermal conductor, the inaccessible interface could be dam- aged by mechanical solicitation, chemical infiltration, aging. In this case, the original thermal properties of the specimen are modified. The defect occurs typically in form of delamination. The present paper deals with nondestructive evaluation of interface ther- mal conductance h from the knowledge of the surface temperature when the specimen is heated in some controlled way. The goal is achieved by expanding h in powers of the thickness of the upper layer. The mathematical analysis of the model produces exact formulas for the first coefficients of h which are tested on simulated and real data. The evaluation of interface flaws comes from reliable approximation of h.

A hidden defect D affecting the interface in a layered specimen is evaluated from temperature maps collected on its top side. An explicit formula that approximates thermal conductance of the damaged interface is derived by means of first order Thin Plate Approximation. This formula is used to obtain a geometrical description of D from experimental data.

This paper deals with the solution of an inverse problem for the heat equation aimed at nondestructive evaluation of fractures, emerging on the accessible surface of a slab, by means of Active Thermography. In real life, this surface is heated with a laser and its temperature is measured for a time interval by means of an infrared camera. A fundamental step in iterative inversion methods is the numerical solution of the underlying direct mathematical model. Usually, this step requires specific techniques in order to limit an abnormal use of memory resources and computing time due to excessively fine meshes necessary to follow a very thin fracture in the domain. Our contribution to this problem consists in decomposing the temperature of the damaged specimen as a sum of a term (with known analytical form) due to an infinite virtual fracture and the solution of an initial boundary value problem for the heat equation on one side of the fracture (i.e. on a rectangular domain). The depth of the fracture is a variable parameter in the boundary conditions that must be estimated from additional data (usually, measurements of the surface temperature). We apply our method to the detection of simulated cracks in concrete and steel specimens.

A composite specimen, made of two slabs and an interface A is heated through one of its sides S, in order to evaluate the thermal conductance H of A. The direct model consists of a system of Initial Boundary Value Problems completed by suitable transmission conditions. Thanks to the properties of multilayer diffusion, we reduce the problem to the slab between A and S only. In this case evaluating the thermal resistance of A means to identify a coefficient in a Robin boundary condition. We evaluate H numerically by means of Thin Plate Approximation.

Longitudinal defects of the internal coated surface of a metal pipe can be evaluated in a fast, precise and cheap way from thermal measurements on the external surface. In this paper, we study two classes of real situations in which the thickness of the coating is much smaller than the thickness of the metal tube: the transportation of potable water and crude oil. A very precise and stable reconstruction of damages is obtained by means of perturbation methods. To do this, first we translate a composite (coating-plus-tube) boundary value problem in a virtual one on the metallic part only. The information about possible damages is now included in the deviations delta h of the effective heat transfer coefficient from a known background value. Finally, we determine delta h by means of Thin Plate Approximation. (C) 2017 Elsevier Ltd. All rights reserved.

Viene studiata l'evoluzione nel tempo del degrado del rivestimento interno di un tubo metallico al cui interno scorre un fluido. Il degrado si manifesta in tagli longitudinali del rivestimento. Per questo motivo il problema puo' essere posto e risolto (per mezzo di Thin Plate Approximation) in due dimensioni. We deal with the mathematical model of the incremental degradation of the internal coating (e.g. a polymeric material) of a metallic pipe in which a fluid flows relatively fast. The fluid drags solid impurities so that longitudinal scratches, inaccessible to any direct inspection procedure, are produced on the coating. Time evolution of this kind of defects can be reconstructed from the knowledge of a sequence of temperature maps of the external surface. The time-varying orthogonal section of this damaged interface is determined as a function of time and polar angle through the identification of a suitable effective heat transfer coefficient by means of Thin Plate Approximation.

We derive a rule for the reconstruction of the internal heat transfer coefficient hint of a pipe, from temperature maps collected on the external face. The pipe is subjected to internal heating by connecting two electrodes to the external surface. To estimate hint we apply the perturbation theory to a thin plate approximation of a boundary value problem for the stationary heat equation.[object Object]

Infrared thermography is widely used in non-destructive testing and in the non-destructive evaluation of subsurface defects in several materials. The detection and reconstruction (location and shape) of a defect inside a material from thermal data requires the solution of an inverse heat conduction problem. Here the problem is tackled by the thin-plate approximation of the investigated domain. A number of physical quantities must be known for the reconstruction procedure to be successful: some relating to the material (thermal conductivity, heat capacity, density), usually known, and others relating to the heating process. This paper proposes procedures for accurately measuring the latter, whose importance is often not given due consideration. Those procedures allow us to accurately measure the heat flux distribution produced by the sources on the heated surface, and the heat exchange coefficient at the remaining surfaces, and are easily applicable in 'on field' situations.

We derive and test a formal explicit approximated rule for the reconstruction of a damaged inaccessible portion of the boundary of a thin conductor from thermal data collected on the opposite accessible face.

In this work, we find and test a new approximated formula (based on the thin plate approximation), for recovering small, unknown damages on the inaccessible surface of a thin conducting (aluminium) plate. We solve this inverse problem from a controlled heat flux and a sequence of temperature maps on the accessible front boundary of our sample. We heat the front boundary by means of a sinusoidal flux. In the meanwhile, we take a sequence of temperature maps of the same side by means of an infrared camera. This procedure is called active infrared thermography. The solution of the heat equation on the accessible boundary of the damaged sample simulates the collection of data. We use domain derivative to linearize the boundary value problem for heat equation. Then, Fourier analysis on the periodic component of solutions leads us to an elliptic BVP. Finally, we apply perturbation theory in order to find out an approximation of the damage. Numerical tests obtained with synthetic data are encouraging. The solution of the heat equation on the accessible boundary of the damaged sample simulates the collection of data by means of the infrared camera. We use domain derivative to linearize the BVP for heat equation. Then, Fourier analysis on the periodic component of solutions leads us to an elliptic BVP. Finally, we apply the perturbation theory in (Formula presented.) (a is the thickness of our sample) in order to find out an approximation of the damage. Numerical tests obtained with synthetic data are encouraging.

I consider a thin metallic plate whose top side is inaccessible and in contact with an aggressive environment (a corroding fluid, hard particles hitting the boundary, ...). On first approximation, heat exchange between metal and fluid follows linear Newtons cooling lawat least as long as the inaccessible side is not damaged. I assume that deviations from Newton's law are modelled by means of a nonlinear perturbative term h. On the other hand, I am able to heat the conductor and take temperature maps of the accessible side (Active Infrared Thermography). My goal is to recover the nonlinear perturbation of the exchange law on the inaccessible side. The problem is stated as an inverse ill-posed problem for the heat equation with nonlinear boundary conditions. I prove that the nonlinear term is identified by one Cauchy data set and produce approximated solutions by means of optimization. In conclusion, I try to drive mathematical modelling as far as possible: although I use advanced results in mathematical analysis (theory of nonlinear boundary value problems, domain derivative), I apply them immediately to technical problems.

Here we give mathematical support to the choice of the Helmholtz versus the Laplace equation for the formulation of an inverse problem in infrared thermography. Such a choice accounts for the values of physical parameters like the thermal conductivity and the Biot number. We check the effectiveness of the corresponding boundary value problems and compare the condition numbers. The Helmholtz choice is confirmed to be usually better, but there are cases in which Laplace (less expensive) works well.

A thin conducting plate has an inaccessible side in contact with aggressive external agents. On the other side, we are able to heat the plate and take temperature maps in laboratory conditions. Detecting and evaluating damages on the inaccessible side from thermal data requires the solution of a nonlinear inverse problem for the heat equation in presence of suitable boundary conditions for heat equation. We carry on this task using domain derivative and integral formulation of the corresponding boundary value problem. Under non restrictive hypothesis, we find explicit regularized schemes for Fourier reconstruction of damages.

A thin plate ? has an inaccessible side in contact with aggressive external agents. On the other side we are able to heat the plate and take temperature maps (thermal data) in laboratory conditions. Detecting and evaluating damages on the inaccessible side from thermal data requires the solution of a nonlinear inverse problem for the heat equation. To do this, it is extremely important to assign correct boundary conditions, in particular on the inaccessible boundary of ?. In several cases the boundary conditions must take account of heat exchange between ? and the environment. Here we discuss, from the quantitative point of view, the relation between the physical constants of the system (conductivity, width of the plate, ...) and the heat transfer through the boundary of ?.

Stationary thermography can be used for investigating the functional form of a nonlinear cooling lawthat describes heat exchanges through an inaccessible part of the boundary of a conductor. In this paper, we obtain a logarithmic stability estimate for the associated nonlinear inverse problem. This stability estimate is obtained from the convergence and sensitivity analysis of a finite difference method for the numerical solution of the Cauchy problem for Laplace's equation, based on the Störmer-Verlet scheme.

Nondestructive evaluation of hidden surface damage by means of stationary thermographic methods requires the construction of approximated solutions of a boundary identification problem for an elliptic equation. In this paper, we describe and test a regularized reconstruction algorithm based on the linearization of this class of inverse problems. The problem is reduced to an infinite linear system whose coefficients come from the Fourier discretization of the Robin boundary value problem for Laplace's equation.

We consider a thin metallic plate whose top side is inaccessible and in contact with a corroding fluid. Heat exchange between metal and fluid follows linear Newton's cooling law as long as the inaccessible side is not damaged. We assume that the effects of corrosion are modeled by means of a nonlinear perturbation in the exchange law. On the other hand, we are able to heat the conductor and take temperature maps of the accessible side. Our goal is to recover the nonlinear perturbation of the exchange law on the top side from thermal data collected on the opposite one (thermal imaging). In this paper we use a stationary model, i.e., the temperature inside the plate is assumed to fulfill Laplace's equation. Hence, our problem is stated as an inverse ill-posed problem for Laplace's equation with nonlinear boundary conditions. We study identifiability and local Lipschitz stability. In particular, we prove that the nonlinear term is identified by one Cauchy data set. Moreover, we produce approximated solutions by means of an optimizational method.

This paper deals with the construction of effective heuristic methods for investigating the effect of corrosion on the inaccesible side of a thin metallic plate. In particular, we derive explicit formulas for the Thin Plate Approximation of the positive functions $\gamma$ and $\theta$ that, in our model, describe energy dispersion and material loss respectively.

We consider a model for detecting corrosion on the (inaccessible) conducting top side of a metallic plate. We suppose that the effects of corrosion attack consist in material loss. The perturbation so induced in the geometry of the plate is described by a positive function $\theta$. We prove that a suitable data set, collected on the bottom side of the plate, identifies $\theta$ uniquely.