Interpolation Technique in Solving Second Rank Polarization Tensor with Error Analysis
DOI:
https://doi.org/10.11113/mjfas.v21n3.4143Keywords:
Perturbation theory, error analysis, polarization tensor, linear element integration.Abstract
The Polarization Tensor (PT) serves as a fundamental property used to characterize the shape, size, and orientation of geometric shapes. Widely employed in engineering, particularly in electrical and magnetic domains for various objects with diverse metallic properties, PT has emerged as a valuable tool for researchers. Hence, this paper aims to provide a derivation of error analysis concerning high gradient problem of the first order PT. The derivation of error analysis is based on the numerical integration scheme which is Gaussian Quadrature. The numerical approximation of first order PT includes some errors due to computational approximations. Therefore, it is crucial to develop a method for estimating these errors to understand real-time inaccuracies, rather than just looking at errors after the computation is complete. The perturbation theory is used as a fundamental key for the derivation and representation of error in the PT problem. The derivation of the error analysis implemented the concept of Taylor series expansion of a function. Notably, our findings confirm that the computed PT error aligns with perturbation theory expectations. Our analysis prioritizes assessing the relative error in both data and computed solutions over norm-based evaluations, thus obviating dependency on problem-specific condition numbers. Future research directions include introducing variability in conductivity scenarios, which will further describe the robustness of algorithms developed in this paper.
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