Simultaneous determination of two unknown thermal coefficients through a mushy zone model with an overspecified convective boundary condition

The simultaneous determination of two unknown thermal coefficients for a semi-infinite material under a phase-change process with a mushy zone according to the Solomon-Wilson-Alexiades model is considered. The material is assumed to be initially liquid at its melting temperature and it is considered that the solidification process begins due to a heat flux imposed at the fixed face. The associated free boundary value problem is overspecified with a convective boundary condition with the aim of the simultaneous determination of the temperature of the solid region, one of the two free boundaries of the mushy zone and two thermal coefficients among the latent heat by unit mass, the thermal conductivity, the mass density, the specific heat and the two coefficients that characterize the mushy zone. The another free boundary of the mushy zone, the bulk temperature and the heat flux and heat transfer coefficients at the fixed face are assumed to be known. According to the choice of the unknown thermal coefficients, fifteen phase-change problems arise. The study of all of them is presented and explicit formulae for the unknowns are given, beside necessary and sufficient conditions on data in order to obtain them. Formulae for the unknown thermal coefficients, with their corresponding restrictions on data, are summarized in a table.Comment: 27 pages, 1 Table, 1 Appendi

Numerical identification of a nonlinear diffusion law via regularization in Hilbert scales

We consider the reconstruction of a diffusion coefficient in a quasilinear elliptic problem from a single measurement of overspecified Neumann and Dirichlet data. The uniqueness for this parameter identification problem has been established by Cannon and we therefore focus on the stable solution in the presence of data noise. For this, we utilize a reformulation of the inverse problem as a linear ill-posed operator equation with perturbed data and operators. We are able to explicitly characterize the mapping properties of the corresponding operators which allow us to apply regularization in Hilbert scales. We can then prove convergence and convergence rates of the regularized reconstructions under very mild assumptions on the exact parameter. These are, in fact, already needed for the analysis of the forward problem and no additional source conditions are required. Numerical tests are presented to illustrate the theoretical statements.Comment: 17 pages, 2 figure

The enclosure method for the heat equation

This paper shows how the enclosure method which was originally introduced for elliptic equations can be applied to inverse initial boundary value problems for parabolic equations. For the purpose a prototype of inverse initial boundary value problems whose governing equation is the heat equation is considered. An explicit method to extract an approximation of the value of the support function at a given direction of unknown discontinuity embedded in a heat conductive body from the temperature for a suitable heat flux on the lateral boundary for a fixed observation time is given.Comment: 12pages. This is the final versio

An inverse source problem for the heat equation and the enclosure method

An inverse source problem for the heat equation is considered. Extraction formulae for information about the time and location when and where the unknown source of the equation firstly appeared are given from a single lateral boundary measurement. New roles of the plane progressive wave solutions or their complex versions for the backward heat equation are given.Comment: 23page

The Principle of Strain Reconstruction Tomography: Determination of Quench Strain Distribution from Diffraction Measurements

Evaluation of residual elastic strain within the bulk of engineering components or natural objects is a challenging task, since in general it requires mapping a six-component tensor quantity in three dimensions. A further challenge concerns the interpretation of finite resolution data in a way that is commensurate and non-contradictory with respect to continuum deformation models. A practical solution for this problem, if it is ever to be found, must include efficient measurement interpretation and data reduction techniques. In the present note we describe the principle of strain tomography by high energy X-ray diffraction, i.e. of reconstruction of the higher dimensional distribution of strain within an object from reduced dimension measurements; and illustrate the application of this principle to a simple case of reconstruction of an axisymmetric residual strain state induced in a cylindrical sample by quenching. The underlying principle of the analysis method presented in this paper can be readily generalised to more complex situations.Comment: 10 pages, 6 figure

The Principle of Strain Reconstruction Tomography: Determination of Quench Strain Distribution from Diffraction Measurements

Evaluation of residual elastic strain within the bulk of engineering components or natural objects is a challenging task, since in general it requires mapping a six-component tensor quantity in three dimensions. A further challenge concerns the interpretation of finite resolution data in a way that is commensurate and non-contradictory with respect to continuum deformation models. A practical solution for this problem, if it is ever to be found, must include efficient measurement interpretation and data reduction techniques. In the present note we describe the principle of strain tomography by high energy X-ray diffraction, i.e. of reconstruction of the higher dimensional distribution of strain within an object from reduced dimension measurements; and illustrate the application of this principle to a simple case of reconstruction of an axisymmetric residual strain state induced in a cylindrical sample by quenching. The underlying principle of the analysis method presented in this paper can be readily generalised to more complex situations.Comment: 10 pages, 6 figure