English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We introduce a new method, the Local Monge Parametrizations (LMP) method, toapproximate tensor fields on general surfaces given by a collection of localparametrizations, e.g.~as in finite element or NURBS surface representations.Our goal is to use this method to solve numerically tensor-valued partialdifferential equations (PDE) on surfaces. Previous methods use scalarpotentials to numerically describe vector fields on surfaces, at the expense ofrequiring higher-order derivatives of the approximated fields and limited tosimply connected surfaces, or represent tangential tensor fields as tensorfields in 3D subjected to constraints, thus increasing the essential number ofdegrees of freedom. In contrast, the LMP method uses an optimal number ofdegrees of freedom to represent a tensor, is general with regards to thetopology of the surface, and does not increase the order of the PDEs governingthe tensor fields. The main idea is to construct maps between the elementparametrizations and a local Monge parametrization around each node. We testthe LMP method by approximating in a least-squares sense different vector andtensor fields on simply connected and genus-1 surfaces. Furthermore, we applythe LMP method to two physical models on surfaces, involving a tension-drivenflow (vector-valued PDE) and nematic ordering (tensor-valued PDE). The LMPmethod thus solves the long-standing problem of the interpolation of tensors ongeneral surfaces with an optimal number of degrees of freedom.

Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations