On the computation of analytic sensitivities of eigenpairs in isogeometric analysis

TL;DR

This paper presents a method for efficiently computing higher-order sensitivities of eigenpairs in isogeometric analysis, enabling precise analysis of shape deformations in resonating structures like electromagnetic cavities.

Contribution

It introduces a systematic approach to differentiate system matrices in isogeometric analysis for eigenvalue sensitivities, including higher-order derivatives, applicable to shape deformation studies.

Findings

Enables closed-form computation of higher-order sensitivities

Applicable to shape morphing and manufacturing uncertainty analysis

Demonstrated on electromagnetic cavity eigenvalue problems

Abstract

The eigenmodes of resonating structures, e.g., electromagnetic cavities, are sensitive to deformations of their shape. In order to compute the sensitivities of the eigenpair with respect to a scalar parameter, we state the Laplacian and Maxwellian eigenvalue problems and discretize the models using isogeometric analysis. Since we require the derivatives of the system matrices, we differentiate the system matrices for each setting considering the appropriate function spaces for geometry and solution. This approach allows for a straightforward computation of arbitrary higher order sensitivities in a closed-form. In our work, we demonstrate the application in a setting of small geometric deformations, e.g., for the investigation of manufacturing uncertainties of electromagnetic cavities, as well as in an eigenvalue tracking along a shape morphing.