Dimensional Polyhedra Research Articles

An improved scheme for the interface reconstruction and curvature approximation for flow simulations of two immiscible fluids

A new scheme is proposed for the approximation of the curvature field in the context of two-phase flows. Direct computation of the curvature from the volume fraction distribution produces so-called spurious currents, which are numerical non-physical velocities especially in the vicinity of the interface. The proposed method computes the curvature from the signed distance function. We have implemented the method inside OpenFOAM® framework (an open source C++ toolbox for computational fluid dynamics simulations based on finite volume discretization). The surface tension force has been formulated into the sharp surface force (SSF). The implementation is applicable for general mesh topologies (two or three dimensional polyhedra). Different validation test cases have been performed; static air bubble in water, droplet undergoing linear shear, rising bubble into viscous fluid, and the impact of a viscous droplet on a thin cylindrical fiber. The proposed method shows an accurate calculation of the curvature and a significant reduction of spurious currents by several orders of magnitude. Such reduction of spurious currents allows accurate simulations of surface tension dominated flows such as micro-droplets, or liquid–liquid separation.

Classification of the congruence classes of $$A_n^5$$ (n ⩾ 6) with 2-torsion free homology

In this paper, we classify the congruence classes of ${\boldsymbol~F}^5_{n(2)}$-polyhedra, i.e., $(n-1)$-connected, at most $(n+5)$-dimensional polyhedra with 2-torsion free homology. The proof relies on the matrix problem technique which was developed in the classification of representations of algebras and applied to the homotopy theory by Baues and Drozd (1999).

SHAPES OF POLYHEDRA, MIXED VOLUMES AND HYPERBOLIC GEOMETRY

We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to .

Stable homotopy classification of A n 4 -polyhedra with 2- torsion free homology

In this paper, we study the stable homotopy types of $\mathbf{F}^4_{n(2)}$-polyhedra, i.e., $(n-1)$-connected, at most $(n+4)$-dimensional polyhedra with 2-torsion free homologies. We are able to classify the indecomposable $\mathbf{F}^4_{n(2)}$-polyhedra. The proof relies on the matrix problem technique which was developed in the classification of representaions of algebras and applied to homotopy theory by Baues and Drozd.

The classification of 2 and 3 torsion free polyhedra

We classify the stable homotopy types of (n−1)-connected, (n+k)-dimensional polyhedra with 2 and 3-torsion free homologies for k ≤ 6. The technique used is matrix problem (bimodule categories) which is given by Drozd.

Poincaré duality and Serre fibrations

We prove that a proper map f : M m → N n between manifolds is a Serre fibration if it has the homotopy lifting property for ( m − n)-dimensional polyhedra, where n is close to m 2 .

Local instability of orbits in polygonal and polyhedral billiards

We classify when local instability of orbits of closeby points can occur for billiards in two dimensional polygons, for billiards inside three dimensional polyhedra and for geodesic flows on surfaces of three dimensional polyhedra. We sharpen a theorem of Boldrighini, Keane and Marchetti. We show that polygonal and polyhedral billiards have zero topological entropy. We also prove that billiards in polygons are positive expansive when restricted to the set of non-periodic points. The methods used are elementary geometry and symbolic dynamics.

The homotopy classification of ( n − 1)-connected ( n + 3)-dimensional polyhedra, n ≥ 4

The homotopy classification of ( n − 1)-connected ( n + 3)-dimensional polyhedra, n ≥ 4

Recognising polytopical cell complexes and constructing projection polyhedra

A simple cell complex C in Euclidean d -space E d is a covering of E d by finitely many convex j -dimensional polyhedra (the j -faces of C), each of which is in the closure of exactly d-j +1 d -faces of C . An algorithm that recognises when C is the projection of the set of faces bounding some convex polyhedron P ( C ) in E d+1 , and that constructs P ( C ) provided its existence is outlined. The method is optimal at least for d =2. No complexity results were previously known for both problems. The results have applications in statics, to the recognition of Voronoi diagrams, and to planar point-location,

Basic Commutators and Minimal Massey Products

The purpose of this paper is to continue the investigation into Massey products defined on two dimensional polyhedra, initiated in [13]. It will be shown that for many such spaces there is a hyperbolic model which can be used to study Massey products. More precisely, Massey products may be interpreted as intersections of geodesies in the Poincaré model. These elements are called minimal Massey products and are the analogue of Massey products over a system considered in Porter's paper. They enjoy the property of being uniquely defined (without indeterminacy) and of being multilinear and natural. Minimal products also satisfy symmetry properties generalising the symmetry properties enjoyed by cup products.A device which will be useful in the proof of the main theorem, 7.4, is the introduction of a class of complexes called basic complexes. These generalise the notion of a surface and each one houses a standard copy of a Massey product.

Three Dimensional Nets and Polyhedra and Structure in Nature is a Strategy for Design

<i>Three Dimensional Nets and Polyhedra</i> and <i>Structure in Nature is a Strategy for Design</i>

A Method of Combining Fixed Points

It is now well known that in the category of finite polyhedra the fixed point property is not preserved by the operations of suspension, Cartesian product, adjunction along a segment, and join. Thus far none of the examples given have involved polyhedra of dimension 2. It is shown in this paper that two fixed points $x$ and $y$ of a self-map of a polyhedron $K$ can be combined in a certain way if a certain criterion is satisfied by the $f$-image of a path from $x$ to $y$. Several corollaries follow, one of which is that if $K$ is a finite simply connected $2$-polyhedron with no local separating points, ${H_2}(K) \ne 0$, and $K$ has a $2$-simplex $\sigma$ such that ${\pi _1}(K - \operatorname {Int} \sigma ,z)$ is cyclic, then $K$ fails to have the fixed point property. This eliminates many $2$-dimensional polyhedra from consideration as examples.