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

Abstract

We introduce a new method, the Local Monge Parametrizations (LMP) method, to approximate tensor fields on general surfaces given by a collection of local parametrizations, e.g. as in finite element or NURBS surface representations. Our goal is to use this method to solve numerically tensor-valued partial differential equations (PDE) on surfaces. Previous methods use scalar potentials to numerically describe vector fields on surfaces, at the expense of requiring higher-order derivatives of the approximated fields and limited to simply connected surfaces, or represent tangential tensor fields as tensor fields in 3D subjected to constraints, thus increasing the essential number of degrees of freedom. In contrast, the LMP method uses an optimal number of degrees of freedom to represent a tensor, is general with regards to the topology of the surface, and does not increase the order of the PDEs governing the tensor fields. The main idea is to construct maps between the element parametrizations and a local Monge parametrization around each node. We test the LMP method by approximating in a least-squares sense different vector and tensor fields on simply connected and genus-1 surfaces. Furthermore, we apply the LMP method to two physical models on surfaces, involving a tension-driven flow (vector-valued PDE) and nematic ordering (tensor-valued PDE). The LMP method thus solves the long-standing problem of the interpolation of tensors on general surfaces with an optimal number of degrees of freedom.

A numerical approach for fluid deformable surfaces

“…The computational examples can provide benchmark problems for other numerical approaches, which can be extended to the considered model, e.g. Nitschke et al (2017), Olshanskii et al (2018), Torres-Sanchez, Santos-Olivan & Arroyo (2020), Lederer et al (2019). They also form the basis for more complex models, which include coupling with concentration fields for proteins and dependency of on concentration in lipid bilayers, or coupling with liquid crystal theory as in Nitschke et al (2019 a ) for Erickson-Leslie type models or with Landau-de Gennes theory on surfaces (Nitschke et al 2019 b ) for Beris-Edwards type models, which also can be extended by active contributions to model, e.g.…”

Vorticity‐stream function approaches are inappropriate to solve the surface Navier‐Stokes equation on a torus

“…Numerical approaches can be found in [2] using discrete exterior calculus (DEC) and in [3,[7][8][9] using surface finite elements for each component of an extended velocity field in the embedding space and a penalization or Lagrange parameter for the normal component. A third approach considers a local Monge parametrizations [10]. However, non of these approaches has yet been applied to the complex problem of an active liquid crystal, considered in [1].…”

Fluid deformable surfaces

“…The concept is extended to general surface vector and tensor fields in Nestler, Nitschke & Voigt (2019). Another approach to solve surface vector- and tensor-valued PDEs is considered in Torres-Sánchez, Santos-Olivan & Arroyo (2019 b ), which is based on a local Monge parameterization and does not required any additional degrees of freedom in the normal direction. Numerical analysis results needed for both approaches, and moreover a comparison in terms of stability, computational cost and implementational effort, do not yet exist.…”