The inversion (= reconstruction) scheme, called the dynamical probe method,
is applied to active thermography to identify unknown inclusions in an
heat conductor and their physical properties. We assume the physical
properties of the inclusions and heat conductor are isotropic and
homogeneous. The measured data for the active thermography are the so-called Neumann
to Dirichlet map. By defining some indicator function via the measured
data, the identification is done by looking at the behavior of the indicator
function. The underlying analysis is the short time asymptotic of
fundamental solution of the heat equation with discontinuous coefficients.
The convergence of Levenberg–Marquardt method is discussed for the inverse problem to reconstruct the storage modulus and loss modulus for the so-called scalar model by a single interior measurement. The scalar model is the most simplest model for data analysis used as the modeling partial differential equation in the diagnosing modality called the magnetic resonance elastography which is used to diagnose for instance lever cancer. The convergence of the method is proved by showing that the measurement map which maps the above unknown moduli to the measured data satisfies the so-called the tangential cone condition. The argument of the proof is quite general and in principle can be applied to any similar inverse problem to reconstruct the unknown coefficients of the model equation given as a partial differential equation of divergence form by one single interior measurement. The performance of the method is numerically tested for the two-layered piecewise homogeneous scalar models in a rectangular domain and a circular domain.
Consider the interior transmission problem arising in inverse boundary value problems for the diffusion equation with discontinuous diffusion coefficients.
We prove the unique solvability of the interior transmission problem by constructing its Green function.
First, we construct a local parametrix for the interior transmission problem near the boundary in the Laplace domain, by using the theory of pseudo-differential operators with a large parameter.
Second, by carefully analyzing the analyticity of the local parametrix in the Laplace domain and estimating it there, a local parametrix for the original parabolic interior transmission problem is obtained via the inverse Laplace transform.
Finally, using a partition of unity, we patch all the local parametrices and the fundamental solution of the diffusion equation to generate a global parametrix for the parabolic interior transmission problem and then compensate it to get the Green function by the Levi method.
The uniqueness of the Green function is justified by using the duality argument, and then the unique solvability of the interior transmission problem is concluded.
We would like to emphasize that the Green function for the parabolic interior transmission problem is constructed for the first time in this paper.
It can be applied for active thermography and diffuse optical tomography modeled by diffusion equations to identify an unknown inclusion and its physical property.
We concerned with the asymptotic analysis for the dynamical probe method which is a reconstruction scheme to identify an anomaly inside a heat conductor from the Neumann-to-Dirichlet map. In this paper an inclusion was considered as an anomaly and we succeeded giving an improved asymptotic behavior of the indicator function defined in terms of the Neumann-to-Dirichlet map to identify not only the location of the inclusion but also some of its physical properties simultaneously. The two major improvements made for analyzing the asymptotic behavior of the indicator function are as follows. Firstly, we can know the distance to the boundary of unknown inclusion as we probe it from its outside. This improvement can avoid overshooting the boundary points as much as possible if we probe it from outside the inclusion numerically. Secondly, we can know the value of heat conductivity of inclusion as we probe close to the inclusion even without touching it.
A seven-layers parabolic model with Stephan–Boltzmann interface conditions and Robin boundary conditions is mathematically formulated to describe the heat transfer process in environment-three layers clothing-air gap-body system.
Based on this model, the solution to the corresponding inverse problem of simultaneous determination of triple fabric layers thickness is given in this paper, which satisfies the thermal safety requirements of human skin.
By implementing a stable finite difference scheme, the thermal burn injuries on the skin of the body can be predicted.
Then a kind of stochastic method, named as particle swarm optimization (PSO) algorithm, is developed to numerically solve the inverse problem.
Numerical results indicate that the formulation of the model and proposed algorithm for solving the corresponding inverse problem are effective.
Hence, the results in this paper will provide scientific supports for designing and manufacturing thermal protective clothing (TPC).