Finite element simulation of three-dimensional free-surface flow problems

An adaptive finite element algorithm is described for the stable solution of three-dimensional free-surface-flow problems based primarily on the use of node movement. The algorithm also includes a discrete remeshing procedure which enhances its accuracy and robustness. The spatial discretisation allows an isoparametric piecewise-quadratic approximation of the domain geometry for accurate resolution of the curved free surface. The technique is illustrated through an implementation for surface-tension-dominated viscous flows modelled in terms of the Stokes equations with suitable boundary conditions on the deforming free surface. Two three-dimensional test problems are used to demonstrate the performance of the method: a liquid bridge problem and the formation of a fluid droplet

Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes

We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes. High order piecewise polynomials are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Our numerical method belongs to the category of direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry directly during the computation of the numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method, in which the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed with a second order TVD finite volume scheme. The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated. Finally, an application inspired by Inertial Confinement Fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure

Forman-Ricci flow for change detection in large dynamic data sets

We present a viable solution to the challenging question of change detection in complex networks inferred from large dynamic data sets. Building on Forman's discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on peer-to-peer networks. Furthermore we adapt the classical Ricci flow that already proved to be a powerful tool in image processing and graphics, to the case of undirected and weighted networks. The application of the proposed method on peer-to-peer networks yields insights into topological properties and the structure of their underlying data.Comment: Conference paper, accepted at ICICS 2016. (Updated version

Computational Geometry Column 42

A compendium of thirty previously published open problems in computational geometry is presented.Comment: 7 pages; 72 reference

Conforming restricted Delaunay mesh generation for piecewise smooth complexes

A Frontal-Delaunay refinement algorithm for mesh generation in piecewise smooth domains is described. Built using a restricted Delaunay framework, this new algorithm combines a number of novel features, including: (i) an unweighted, conforming restricted Delaunay representation for domains specified as a (non-manifold) collection of piecewise smooth surface patches and curve segments, (ii) a protection strategy for domains containing curve segments that subtend sharply acute angles, and (iii) a new class of off-centre refinement rules designed to achieve high-quality point-placement along embedded curve features. Experimental comparisons show that the new Frontal-Delaunay algorithm outperforms a classical (statically weighted) restricted Delaunay-refinement technique for a number of three-dimensional benchmark problems.Comment: To appear at the 25th International Meshing Roundtabl