Thermal Tomography with Unknown Boundary

Abstract A common occurrence in many practical systems is that the desired result is known or given, but the conditions needed for achieving this result are not known. This situation leads to inverse problems, which are of particular interest in thermal processes. For instance, the temperature cycle to which a component must be subjected in order to obtain desired characteristics in a manufacturing system, such as heat treatment or plastic thermoforming, is prescribed. However, the necessary boundary and initial conditions are not known and must be determined by solving the inverse problem. Similarly, an inverse solution may be needed to complete a given physical problem by determining the unknown boundary conditions. Solutions thus obtained are not unique and optimization is generally needed to obtain results within a small region of uncertainty. This review paper discusses several inverse problems that arise in a variety of practical processes and presents some of the approaches that may be used to solve them and obtain acceptable and realistic results. Optimization methods that may be used for reducing the error are presented. A few examples are given to illustrate the applicability of these methods and the challenges that must be addressed in solving inverse problems. These examples include the heat treatment process, unknown wall temperature distribution in a furnace, and transport in a plume or jet involving the determination of the strength and location of the heat source by employing a few selected data points downstream. Optimization of the positioning of the data points is used to minimize the number of samples needed for accurate predictions.

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Estimating the Wall Heat Flux of Unsteady Conjugated Forced Convection Between Two Corotating Disks Using an Inverse Solution Scheme

Journal of Heat Transfer ◽

10.1115/1.2976788 ◽

2008 ◽

Vol 130 (12) ◽

Cited By ~ 1

Author(s):

David T. W. Lin ◽

Hung Yi Li ◽

Wei Mon Yan

Keyword(s):

Heat Flux ◽

Forced Convection ◽

Boundary Surface ◽

Temperature Sensors ◽

Inverse Solution ◽

Wall Heat Flux ◽

Unknown Boundary ◽

Measured Temperature ◽

Solution Scheme ◽

Good Agreement

An inverse solution scheme based on the conjugate gradient method with the minimization of the object function is presented for estimating the unknown wall heat flux of conjugated forced convection flows between two corotating disks from temperature measurements acquired within the flow field. The validity of the proposed approach is demonstrated via the estimation of three time- and space-dependent heat flux profiles. A good agreement is observed between the estimated results and the exact solution in every case. In general, the accuracy of the estimated results is found to improve as the temperature sensors are moved closer to the unknown boundary surface and the error in the measured temperature data is reduced.

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Photogrammetric Process to Monitor Stress Fields Inside Structural Systems

Sensors ◽

10.3390/s21124023 ◽

2021 ◽

Vol 21 (12) ◽

pp. 4023

Author(s):

Leonardo M. Honório ◽

Milena F. Pinto ◽

Maicon J. Hillesheim ◽

Francisco C. de Araújo ◽

Alexandre B. Santos ◽

...

Keyword(s):

Boundary Element ◽

Real Time ◽

Stress Fields ◽

Structural Systems ◽

Unknown Boundary ◽

Stress Distributions ◽

Displacement Fields ◽

Time Stress ◽

Beam Surface ◽

Measured Displacement

This research employs displacement fields photogrammetrically captured on the surface of a solid or structure to estimate real-time stress distributions it undergoes during a given loading period. The displacement fields are determined based on a series of images taken from the solid surface while it experiences deformation. Image displacements are used to estimate the deformations in the plane of the beam surface, and Poisson’s Method is subsequently applied to reconstruct these surfaces, at a given time, by extracting triangular meshes from the corresponding points clouds. With the aid of the measured displacement fields, the Boundary Element Method (BEM) is considered to evaluate stress values throughout the solid. Herein, the unknown boundary forces must be additionally calculated. As the photogrammetrically reconstructed deformed surfaces may be defined by several million points, the boundary displacement values of boundary-element models having a convenient number of nodes are determined based on an optimized displacement surface that best fits the real measured data. The results showed the effectiveness and potential application of the proposed methodology in several tasks to determine real-time stress distributions in structures.

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Extending the unified transform: curvilinear polygons and variable coefficient PDEs

IMA Journal of Numerical Analysis ◽

10.1093/imanum/dry085 ◽

2018 ◽

Vol 40 (2) ◽

pp. 976-1004 ◽

Cited By ~ 9

Author(s):

Matthew J Colbrook

Keyword(s):

Fourier Transforms ◽

Elliptic Boundary ◽

Neumann Boundary ◽

Integral Transforms ◽

Variable Coefficients ◽

Central Component ◽

Unknown Boundary ◽

Boundary Values ◽

Global Relation ◽

Chebyshev Interpolation

Abstract We provide the first significant extension of the unified transform (also known as the Fokas method) applied to elliptic boundary value problems, namely, we extend the method to curvilinear polygons and partial differential equations (PDEs) with variable coefficients. This is used to solve the generalized Dirichlet-to-Neumann map. The central component of the unified transform is the coupling of certain integral transforms of the given boundary data and of the unknown boundary values. This has become known as the global relation and, in the case of constant coefficient PDEs, simply links the Fourier transforms of the Dirichlet and Neumann boundary values. We extend the global relation to PDEs with variable coefficients and to domains with curved boundaries. Furthermore, we provide a natural choice of global relations for separable PDEs. These generalizations are numerically implemented using a method based on Chebyshev interpolation for efficient and accurate computation of the integral transforms that appear in the global relation. Extensive numerical examples are provided, demonstrating that the method presented in this paper is both accurate and fast, yielding exponential convergence for sufficiently smooth solutions. Furthermore, the method is straightforward to use, involving just the construction of a simple linear system from the integral transforms, and does not require knowledge of Green’s functions of the PDE. Details on the implementation are discussed at length.

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