Functional maps regularization for high quality mesh morphing applied to shape registration in computed tomography

Abstract We propose an efficient surface mesh morphing method using the functional maps framework combined with Tikhonov regularization. This approach allows us to obtain high-quality meshes suitable for physical simulations. Our target application is solving partial differential equations (PDEs) on industrial components observed via computed tomography, with an automatic boundary condition assignment. We show how the morphing of a reference mesh can be formulated directly in the functional (spectral) domain, without requiring predefined correspondences or symmetry assumptions. First, the functional maps framework provides a robust way to establish continuous and orientation-preserving correspondences between non-rigid shapes, ensuring an accurate alignment between the ideal CAD model, form Computer Aided Design (CAD), and the real-world scanned part. The morphing process is further refined using Tikhonov regularization, ensuring accurate mesh correspondence while preserving mesh quality, which is essential for solving partial differential equations. Afterward we present numerical results on turbine blade cores in aeronautics, which feature complex geometries with cooling channels that may vary due to manufacturing variations and operational constraints. Finally, we demonstrate the effectiveness of our method through a numerical computation by calculating the Laplace-Beltrami spectrum on the morphed mesh using a linear finite element method. This approach significantly improves simulation accuracy by incorporating real geometric variations, making it highly suitable for industrial applications.