Structure-preserving invariant interpolation schemes for invertible second-order tensors

Tensor interpolation is an essential step for tensor data analysis in various fields of application and scientific disciplines. In the present work, novel interpolation schemes for general, that is, symmetric or non-symmetric, invertible square tensors are proposed. Critically, the proposed schemes rely on a combined polar and spectral decomposition of the tensor data (Formula presented.), followed by an individual interpolation of the two rotation tensors (Formula presented.) and (Formula presented.) and the positive definite diagonal eigenvalue tensor (Formula presented.) resulting from this decomposition. Two different schemes are considered for a consistent rotation interpolation within the special orthogonal group (Formula presented.), either based on relative rotation vectors or quaternions. For eigenvalue interpolation, three different schemes, either based on the logarithmic weighted average, moving least squares or logarithmic moving least squares, are considered. It is demonstrated that the proposed interpolation procedure preserves the structure of a tensor, that is, (Formula presented.) and (Formula presented.) remain orthogonal tensors and (Formula presented.) remains a positive definite diagonal tensor during interpolation, as well as scaling and rotational invariance (objectivity). Based on selected numerical examples considering the interpolation of either symmetric or non-symmetric tensors, the proposed schemes are compared to existing approaches such as Euclidean, Log-Euclidean, Cholesky and Log-Cholesky interpolation. In contrast to these existing methods, the proposed interpolation schemes result in smooth and monotonic evolutions of tensor invariants such as determinant, trace, fractional anisotropy (FA), and Hilbert's anisotropy (HA). Moreover, a consistent spatial convergence behavior is confirmed for first- and second-order realizations of the proposed schemes. The present work is mainly motivated by the frequently occurring necessity for remeshing or mesh adaptivity when applying the finite element method to complex problems of nonlinear continuum mechanics with inelastic constitutive behavior, which requires the consistent interpolation of tensor-valued history data for the transfer between different meshes. However, the proposed schemes are very general in nature and suitable for the interpolation of general invertible second-order square tensors independent of the specific application.