The purpose of this study is to determine the possibility of an accurate assessment of the spatial distribution of material properties such as conductivities or impedances from boundary measurements when the governing partial differential equation is a Laplacian.
A series of numerical experiments were carefully performed. The results were analyzed and compared.
The results to date show that while the optimization procedure is able to obtain spatial distributions of the conductivity k that reduce the cost function significantly, the resulting conductivity k is still significantly different from the target (or real) distribution sought. While the normal fluxes recovered are very close to the prescribed ones, the tangential fluxes can differ considerably.
At this point, it is not clear why rigorous mathematical proofs yield results of convergence and uniqueness, while in practice, accurate distributions of the conductivity k seem to be elusive. One possible explanation is that the spatial influence of conductivities decreases exponentially with distance. Thus, many different conductivities inside a domain could give rise to very similar (infinitely close) boundary measurements.
This implies that the estimation of field conductivities (or generally field data) from boundary data is far more difficult than previously assumed when the governing partial differential equation in the domain is a Laplacian. This has consequences for material parameter assessments (e.g. for routine maintenance checks of structures), electrical impedance tomography, and many other applications.
This is the first time such a finding has been reported in this context.
Löhner, R. and Antil, H. (2020), "Determination of volumetric material data from boundary measurements: Revisiting Calderon’s problem", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/HFF-12-2019-0931Download as .RIS
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