Shape spaces are fundamental in a variety of applications including image registration, morphing, matching, interpolation, and shape optimization. In this work, we consider two-dimensional shapes represented by triangular meshes of a given connectivity. We show that the collection of admissible configurations representable by such meshes form a smooth manifold. For this manifold of planar triangular meshes we propose a geodesically complete Riemannian metric. It is a distinguishing feature of this metric that it preserves the mesh connectivity and prevents the mesh from degrading along geodesic curves. We detail a symplectic numerical integrator for the geodesic equation in its Hamiltonian formulation. Numerical experiments show that the proposed metric keeps the cell aspect ratios bounded away from zero and thus avoids mesh degradation along arbitrarily long geodesic curves.
We propose a novel discrete concept for the total generalized variation (TGV), which has originally been derived to reduce the staircasing effect in classical total variation (TV) regularization, in image denoising problems. We describe discrete, second-order TGV for piecewise constant functions on triangular meshes, thus allowing the TGV functional to be applied to more general data structures than pixel images, and in particular in the context of finite element discretizations. Particular attention is given to the description of the kernel of the TGV functional, which, in the continuous sett…
We present a high-order entropy stable discontinuous Galerkin (ESDG) method for the two dimensional shallow water equations (SWE) on curved triangular meshes. The presented scheme preserves a semi-discrete entropy inequality and remains well-balanced for continuous bathymetry profiles. We provide numerical experiments which confirm the high-order accuracy and theoretical properties of the scheme, and compare the presented scheme to an entropy stable scheme based on simplicial summation-by-parts (SBP) finite difference operators. Finally, we report the computational performance of an implementa…
Discretization of the equations of Viscous Plastic and Elastic Viscous Plastic (EVP) sea ice dynamics on triangular meshes can be done by placing discrete velocities at vertices, cells or edges. Since there are more cells and edges than vertices, the cell- and edge-based discretizations simulate more linear kinematic features at the same mesh than the vertex discretization. However, the discretization based on cell and edge velocities suffer from kernels in the strain rate or stress divergence operators and need either special strain rate computations as proposed here for cell velocities, or s…
We consider in this work the numerical resolution of a 2D shallow water system with a Coriolis effect and bottom friction stresses on unstructured meshes by a new Finite Volume Characteristics (FVC) scheme, which has been introduced in the preliminary works that will be cited below. Our main goal is to extend this approach to 2D unstructured formalism while preserving the physical and mathematical properties of the system, including the C-property. First, we present our extension by preserving the advantages of the finite volume discretization such as conservation property and the method of ch…
A highly efficient gas-kinetic scheme with fourth-order accuracy in both space and time is developed for the Navier-Stokes equations on triangular meshes. The scheme combines an efficient correction procedure via reconstruction (CPR) framework with a robust gas-kinetic flux formula, which computes both the flux and its time-derivative. The availability of the flux time-derivative makes it straightforward to adopt an efficient two-stage temporal discretization to achieve fourth-order time accuracy. In addition, through the gas-kinetic evolution model, the inviscid and viscous fluxes are coupled…
The Modified Direct Method (MDM) is an iterative mesh smoothing method for smoothing planar and surface meshes, which is developed from the non-iterative smoothing method originated by Balendran [1]. When smooth planar meshes, the performance of the MDM is effectively identical to that of Laplacian smoothing, for triangular and quadrilateral meshes; however, the MDM outperforms Laplacian smoothing for tri-quad meshes. When smooth surface meshes, for trian-gular, quadrilateral and quad-dominant mixed meshes, the mean quality(MQ) of all mesh elements always increases and the mean square error (M…
Linear Kinematic Features (LKFs) are found everywhere in the Arctic sea ice cover. They are strongly localized deformations often associated with the formation of leads and pressure ridges. Viscous-plastic sea ice models start to generate LKFs at high spatial grid resolution, typically with a grid spacing below 5 km. Besides grid spacing, other aspects of a numerical implementation, such as discretization details, may affect the number and definition of simulated LKFs. To explore these effects, the solutions of sea ice models with different grid spacings, mesh types, and numerical discretizati…
This work describes a concise algorithm for the generation of triangular meshes with the help of standard adaptive finite element methods. We demonstrate that a generic adaptive finite element solver can be repurposed into a triangular mesh generator if a robust mesh smoothing algorithm is applied between the mesh refinement steps. We present an implementation of the mesh generator and demonstrate the resulting meshes via examples.
To meet the demand for complex geometries and high resolutions of small-scale flow structures, a two-stage fourth-order subcell finite volume (SCFV) method combining the gas-kinetic solver (GKS) with subcell techniques for compressible flows over (unstructured) triangular meshes was developed to improve the compactness and efficiency. Compared to the fourth-order GKS-based traditional finite volume (FV) method, the proposed method realizes compactness effectively by subdividing each cell into a set of subcells or control volumes (CVs) and selecting only face-neighboring cells for high-order co…
Trimpack is a library of routines written in Fortran that allow to create unstructured triangular meshes in any domain and with an user-defined size distribution. The user must write a program that uses the elements of the library as if it were a mathematical tool. First, the domain must be defined, using point-defined boundaries, which the user provides. The library internally uses splines to mesh the boundaries with the node distribution function provided by the user. Several meshing methods are available, from simple Dalaunay mesh creation from a point cloud, an incremental Steiner-type alg…
Mesh adaptivity is a useful tool for efficient solution to partial differential equations in very complex geometries. In the present paper we discuss the use of polygonal mesh refinement in order to tackle two common issues: first, adaptively refine a provided good quality polygonal mesh preserving quality, second, improve the quality of a coarse poor quality polygonal mesh during the refinement process on very complex domains. For finite element methods and triangular meshes, convergence of a posteriori mesh refinement algorithms and optimality properties have been widely investigated, wherea…
We address here the discretization of the momentum convection operator for fluid flow simulations on 2D triangular and quadrangular meshes and 3D polyhedral meshes containing hexahedra, tetrahedra, prisms and pyramids. The finite volume scheme that we use for the full Euler equations is based on a staggered discretization: the density unknowns are associated with a primal mesh, whereas the velocity unknowns are associated with a "fictive" dual mesh. Accordingly, the convection operator of the mass balance equation is derived on the primal mesh, while the the convection operator of the momentum…
We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end, we extend the variational time discretization of geodesic calculus presented in Rumpf and Wirth (2015), which just requires an approximation of the squared Riemannian distance that is typically easy to compute. First we obtain first order discrete covariant derivatives via a Schild's ladder type discretization of parallel transport. Second order discrete covariant derivatives are then computed as n…
