Homogeneous crystallization of the Lennard-Jones liquid. Structural analysis based on Delaunay simplices method

The crystallization process of a simple liquid upon slow cooling has been modeled by the Monte-Carlo method. The model contains 10,000 Lennard-Jones atoms in the model box with periodic boundary conditions. The model structure is investigated at different stages of crystallization using Delaunay simplices. The simplex belonging to one or another particular crystal structure was determined by the shape of the given simplex taking into account the shape of its neighboring simplices. Simplices typical of the fcc and hcp crystal structures, as well as of polytetrahedral aggregates, not typical of crystals, were studied. The analysis has shown that the “precursors” of a hcp structure are strongly dominating over the “precursors” of a fcc structure in liquid phase before the beginning of crystallization. When crystallization starts, small embryos of the fcc structure are observed; the simplices peculiar to hcp are present at that in great amount, but they are distributed over the sample more uniformly. As crystallization proceeds, the portion of the fcc phase grows faster than hcp. However, no unified crystal appears in our case of slow cooling of the model. A complex polycrystalline structure containing crystalline regions with multiple twinning, pentagonal prisms and elements of icosahedral structures arises instead.     

A study of gridless method with dynamic clouds of points for solving unsteady CFD problems in aerodynamics

When solving unsteady computational fluid dynamics problems in aerodynamics with a gridless method, a cloud of points is usually required to be regenerated due to its accommodation to moving boundaries. In order to handle this problem conveniently, a fast dynamic cloud method based on Delaunay graph mapping strategy is proposed in this paper. A dynamic cloud method makes use of algebraic mapping principles and therefore points can be accurately redistributed in the flow field without any iteration. In this way, the structure of the gridless clouds is not necessarily changed so that the clouds regeneration can be avoided successfully. The spatial derivatives of the mathematical modeling of the flow are directly determined by using weighted least‐squares method in each cloud of points, and then numerical fluxes can be obtained. A dual time‐stepping method is further implemented to advance the two‐dimensional Euler equations in arbitrary Lagarangian–Eulerian formulation in time. Finally, unsteady transonic flows over two different oscillating airfoils are simulated with the above method and results obtained are in good agreement with the experimental data. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical simulations of the steady Navier–Stokes equations using adaptive meshing schemes

In this paper, we consider an adaptive meshing scheme for solution of the steady incompressible Navier–Stokes equations by finite element discretization. The mesh refinement and optimization are performed based on an algorithm that combines the so‐called conforming centroidal Voronoi Delaunay triangulations (CfCVDTs) and residual‐type local a posteriori error estimators. Numerical experiments in the two‐dimensional space for various examples are presented with quadratic finite elements used for the velocity field and linear finite elements for the pressure. The results show that our meshing scheme can equally distribute the errors over all elements in some optimal way and keep the triangles very well shaped as well at all levels of refinement. In addition, the convergence rates achieved are close to the best obtainable. Extension of this approach to three‐dimensional cases is also discussed and the main challenge is the efficient implementation of three‐dimensional CfCVDT generation that is still under development. Copyright © 2007 John Wiley & Sons, Ltd.     

Triangulations in CGAL

This paper presents the main algorithmic and design choices that have been made to implement triangulations in the computational geometry algorithms library .     

An efficient adaptive analysis procedure for node-based smoothed point interpolation method (NS-PIM)

This paper presents an efficient adaptive analysis procedure being able to operate in the framework of the node-based smoothed point interpolation method (NS-PIM). The NS-PIM uses three-node triangular cells and is very easy to be implemented, which make it an ideal candidate for adaptive analysis. In the present adaptive procedure, a new error indicator is devised for NS-PIM settings; two ways are proposed to calculate the local critical value; a simple h-type local refinement scheme is adopted and Delaunay technology is used for regenerating optimal new mesh. A number of typical numerical examples involving stress concentration and solution singularities have been tested. The results demonstrate that the present procedure achieves much higher convergence rate results compared to the uniform refinement, and can obtain upper bound solution in strain energy.     

The flow complex: A data structure for geometric modeling

We study a special case of the critical point (Morse) theory of distance functions namely, the gradient flow associated with the distance function to a finite point set in . The fixed points of this flow are exactly the critical points of the distance function. Our main result is a mathematical characterization and algorithms to compute the stable manifolds, i.e., the inflow regions, of the fixed points. It turns out that the stable manifolds form a polyhedral complex that shares many properties with the Delaunay triangulation of the same point set. We call the latter complex the flow complex of the point set. The flow complex is suited for geometric modeling tasks like surface reconstruction.     

Efficiently navigating a random Delaunay triangulation

Planar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. However, whilst a number of algorithms and existence proofs have been proposed, very little analysis is available for the properties of the paths generated and the computational resources required to generate them under a random distribution hypothesis for the input. In this paper we analyse a new deterministic planar navigation algorithm with constant spanning ratio (w.r.t the Euclidean distance) which follows vertex adjacencies in the Delaunay triangulation. We call this strategy cone walk. We prove that given n uniform points in a smooth convex domain of unit area, and for any start point z and query point q; cone walk applied to z and q will access at most sites with complexity with probability tending to 1 as n goes to infinity. We additionally show that in this model, cone walk is ‐memoryless with high probability for any pair of start and query point in the domain, for any positive ξ. We take special care throughout to ensure our bounds are valid even when the query points are arbitrarily close to the border. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 95–136, 2016     

Anisotropic unstructured mesh adaption for flow simulations

Three new ideas for anisotropic adaption of unstructured triangular grids are presented, with particular emphasis on fluid flow computations. © 1997 John Wiley & Sons, Ltd.     

Manifold Reconstruction in Arbitrary Dimensions Using Witness Complexes

It is a well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, under mild sampling conditions. In this paper, we prove that these results do not extend to higher-dimensional manifolds, even under strong sampling conditions such as uniform point density. On the positive side, we show how the sets of witnesses and landmarks can be enriched, so that the nice relations that exist between restricted Delaunay triangulation and witness complex hold on higher-dimensional manifolds as well. We derive from our structural results an algorithm that reconstructs manifolds of any arbitrary dimension or co-dimension at different scales. The algorithm combines a farthest-point refinement scheme with a vertex pumping strategy. It is very simple conceptually, and it does not require the input point sample to be sparse. Its running time is bounded by c(d)n ², where n is the size of the input point cloud, and c(d) is a constant depending solely (yet exponentially) on the dimension d of the ambient space. Although this running time makes our reconstruction algorithm rather theoretical, recent work has shown that a variant of our approach can be made tractable in arbitrary dimensions, by building upon the results of this paper. This work was done while S.Y. Oudot was a post-doctoral fellow at Stanford University. His email there is no longer valid.