Arbitrary Triangulations Research Articles

Numerical wave propagation for the triangular P1 DG– P2 finite element pair

The f-plane and β-plane wave propagation properties are examined for discretisations of the linearised rotating shallow-water equations using the P1 DG – P2 finite element pair on arbitrary triangulations in planar geometry. A discrete Helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations. In the f-plane (planar geometry, Coriolis force independent of space) case, this decomposition is used to obtain decoupled equations for the geostrophic modes, the inertia-gravity modes, and the inertial oscillations. As has been noticed previously, the geostrophic modes are steady. The Helmholtz decomposition is used to show that the resulting inertia-gravity wave equation is third-order accurate in space. In general the P1 DG – P2 finite element pair is second-order accurate, so this leads to very accurate wave propagation. It is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency f, and which do not propagate. A restriction of the P1 DG velocity space is proposed in which these modes are not present, leading to a finite element discretisation which is completely free of spurious modes. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised P1 DG – P2 equations in the β-plane (planar geometry, Coriolis force linear in space) case resulting in a Rossby wave equation which is also third-order accurate. This means that the dispersion relation for the wave propagation is very accurate; an illustration of this is provided by a numerical dispersion analysis in the case of a triangulation consisting of equilateral triangles.

Lattice gas simulations of dynamical geometry in two dimensions

We present a hydrodynamic lattice gas model for two-dimensional flows on curved surfaces with dynamical geometry. This model is an extension to two dimensions of the dynamical geometry lattice gas model previously studied in one dimension. We expand upon a variation of the two-dimensional flat space Frisch-Hasslacher-Pomeau (FHP) model created by Frisch [Phys. Rev. Lett. 56, 1505 (1986)] and independently by Wolfram, and modified by Boghosian [Philos. Trans. R. Soc. London, Ser. A 360, 333 (2002)]. We define a hydrodynamic lattice gas model on an arbitrary triangulation whose flat space limit is the FHP model. Rules that change the geometry are constructed using the Pachner moves, which alter the triangulation but not the topology. We present results on the growth of the number of triangles as a function of time. Simulations show that the number of triangles grows with time as t(1/3), in agreement with a mean-field prediction. We also present preliminary results on the distribution of curvature for a typical triangulation in these simulations.

MORPHING OF TRIANGULAR MESHES IN SHAPE SPACE

We present a novel approach to morph between two isometric poses of the same non-rigid object given as triangular meshes. We model the morphs as linear interpolations in a suitable shape space $\mathcal{S}$. For triangulated 3D polygons, we prove that interpolating linearly in this shape space corresponds to the most isometric morph in $\mathbb{R}^3$. We then extend this shape space to arbitrary triangulations in 3D using a heuristic approach and show the practical use of the approach using experiments. Furthermore, we discuss a modified shape space that is useful for isometric skeleton morphing. All of the newly presented approaches solve the morphing problem without the need to solve a minimization problem.

Inductive dimensions modulo simplicial complexes and ANR-compacta

New dimension functions G-dim and R-dim, where G is a class of finite simplicial complexes and R is a class of ANR-compacta, are introduced. Their definitions are based on the theorem on partitions and on the theorem on inessential mappings to cubes, respectively. If R is a class of compact polyhedra, then for its arbitrary triangulation τ , we have Rτ dim X = Rdim X for an arbitrary normal space X. To investigate the dimension function R-dim we apply results of extension theory. Internal properties of this dimension function are similar to those of the Lebesgue dimension. The following inequality Rdim X ≤ dim X holds for an arbitrary class R. We discuss the following Question: When R-dim X < ∞⇒ dim X < ∞?

Piecewise Linear Orthogonal Approximation

We derive Sobolev-type inner products with respect to which hat functions on arbitrary triangulations of domains in $\mathbb{R}^d$ are orthogonal. Compared with linear interpolation, the resulting approximation schemes yield superior accuracy at little extra cost.

Lagrange interpolation set along linear piecewise algebraic curves

This paper discusses the Lagrange interpolation problem in continuous bivariate spline spaces over regular triangulations. By using the so-called Lagrange interpolation set along piecewise algebraic curves, we develop a new approach of constructing the interpolation set for continuous spline spaces. We show the property of this set on star region, and construct the interpo- lation set for continuous bivariate spline spaces over arbitrary triangulations. The construction only depends on the number of points on the piecewise algebraic curve in each cell.

Frieze patterns for punctured discs

We construct frieze patterns of type DN with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram of type DN, we show that the numbers in the pattern can be interpreted as specialisations of cluster variables in the corresponding Fomin-Zelevinsky cluster algebra. This is generalised to arbitrary triangulations in an appendix by Hugh Thomas.

Semiclassical limit of 4-dimensional spin foam models

We study the semiclassical properties of the Riemannian spin foam models with Immirzi parameter that are constructed via coherent states. We show that in the semiclassical limit the quantum spin foam amplitudes of an arbitrary triangulation are exponentially suppressed, if the face spins do not correspond to a discrete geometry. When they do arise from a geometry, the amplitudes reduce to the exponential of i times the Regge action. Remarkably, the dependence on the Immirzi parameter disappears in this limit.

단순 다각형 계층구조에서의 삼각화와 경비가능충분집합

본 논문은 화랑문제 분야에 관한 것으로, 다각형의 계층구조에서 경비충분집합에 될 수 있는 기하학적인 요소들에 관해 다루었다. 경비충분 집합이 될 수 있는 기하학적인 요소로 다각형의 삼각화를 고려하였고, 다각형의 삼각화한 대각선분에 대해 완전가시성으로 양쪽을 다 감시할 경우 경비충분집합이 되는 삼각형의 부류가 볼록 다각형, 단변단조 다각형, 소용돌이 다각형임을 보였고, 그 외의 별모양 다각형, 단조 다각형, 완전외부가시성 다각형에서는 경비충분집합이 되지 못함을 보였다. In this paper, we consider a characterization of a Guard Sufficiency Set(GSS) in the hierarchy of simple polygons. we propose the diagonals of a arbitrary triangulation of a polygon as a GSS when guards see the diagonals with completely visibility and both sides of the diagonal. we show that this can be a GSS in convex polygons, unimodal polygons, spiral polygons but this can not be a GSS in star-shaped polygons, monotone polygons, completely external visible polygons.

A variational formulation in fracture mechanics

A variational approach to linear elasticity problems is considered. The family of variational principles is proposed based on the linear theory of elasticity and the method of integrodifferential relations. The idea of this approach is that the constitutive relation is specified by an integral equality instead of the local Hooke’s law and the modified boundary value problem is reduced to the minimization of a nonnegative functional over all admissible displacements and equilibrium stresses. The conditions of decomposition on two separated problems with respect to displacements and stresses are found for the variational problems formulated and the relation between the approach under consideration and the minimum principles for potential and complementary energies is shown. The effective local and integral criteria of solution quality are proposed. A numerical algorithm based on the piecewise polynomial approximations of displacement and stress fields over an arbitrary domain triangulation are worked out to obtained numerical solutions and estimate their convergence rates. Numerical results for 2D linear elasticity problems with cracks are presented and discussed.

On structure of bivariate spline space of lower degree over arbitrary triangulation

The structure of bivariate spline space over arbitrary triangulation is complicated because the dimension of a multivariate spline space depends not only on the topological property of the triangulation but also on its geometric property. A new vertex coding method to a triangulation is introduced in this paper to further study structure of the spline spaces. The upper bound of the dimension of spline spaces over triangulation given by L.L. Schumaker is slightly improved via the new vertex coding method. The structure of multivariate spline spaces S 2 1 ( ▵ ) and S 3 1 ( ▵ ) over arbitrary triangulation are studied via the method of smoothness cofactor and the structure matrix of multivariate spline ring by Luo and Wang. A kind of sufficient conditions on judging non-singularity of the S 2 1 ( ▵ ) and S 3 1 ( ▵ ) spaces over arbitrary triangulation is given, which only depends on the topological property of the triangulation. From the sufficient conditions, a triangulation strategy is presented at the end of the paper. The strategy ensures that the constructed triangulation is non-singular (or generic) for S 2 1 and S 3 1 .

Topological low-temperature limit of Z(2) spin-gauge theory

We study Z(2) lattice gauge theory on triangulations of compact d-manifolds. We reformulate the theory algebraically, describing it in terms of the structure constants of a bidimensional vector space H equipped with algebra and coalgebra structures, and prove that in the low-temperature limit H reduces to a Hopf algebra, in which case the theory becomes equivalent to a topological field theory. The degeneracy of the ground state is shown to be a topological invariant. This fact is used to compute the zeroth- and first-order terms in the low-temperature expansion of Z for arbitrary triangulations. In finite temperatures, the algebraic reformulation gives rise to new duality relations among classical spin models, related to changes of basis of H.

Nöther-type theorem of piecewise algebraic curves on triangulation

The piecewise algebraic curve is a kind generalization of the classical algebraic curve. Nother-type theorem of piecewise algebraic curves on the cross-cut partition is very important to construct the Lagrange interpolation sets for a bivariate spline space. In this paper, using the properties of bivariate splines, the Nother-type theorem of piecewise algebraic curves on the arbitrary triangulation is presented.

SU‐FF‐I‐63: Temporal Constrained Registration of Arbitrary Surface Contours for Use in 4D Radiation Therapy

Purpose: To quantify the validity of the temporal interpolation provided by a nonlinear uncorrelated contour based path length minimizing deformable registration algorithm in 4D radiation therapy. Method and Materials: The anatomical surfaces are represented as triangulated meshes. The registration algorithm is implemented by defining a norm between the triangulated meshes, based on the algorithms of Glaunès and Vaillant. This algorithm matches two surfaces as opposed to image intensities or specific points. Additionally, it optimizes the path between them allowing for a temporal interpolation. The registration is defined as the flow of an unlabeled 3D point set to an arbitrary target surface that minimizes both the norm between the target and registered surface and the non‐linear path length between the source and registered surface. Ground truth surfaces were generated from 4D lung datasets using the semi‐automatic mesh segmentation routine from a research version of Pinnacle, with expert manual correction. Results: To test this algorithm we compared its ability to interpolate intermediate shapes with a standard linear interpolation technique. We found that this algorithm performs comparably.The similarity metric used was an Euclidean distance error metric between the ground truth surface and a surface interpolated between a previous and subsequent ground truth surface. It was calculated between the interpolated and truth vertexes and their closest points on the truth and interpolated surfaces, respectively. This metric is independent of the interpolation and surface triangulation used, and thus is applicable to other interpolation algorithms. Conclusion: This work shows that this general, path length minimizing deformable registration algorithm based interpolation performs similarly to linear interpolation in approximately correlated, intrafractional datasets. This algorithm has the advantage of utilizing surfaces with arbitrary triangulations without point correspondences. We further discovered that this specific algorithm exhibited numerical instabilities that require further investigation.

Optimal Lagrange interpolation by quartic [formula omitted] splines on triangulations

We develop a local Lagrange interpolation scheme for quartic C 1 splines on triangulations. Given an arbitrary triangulation Δ , we decompose Δ into pairs of neighboring triangles and add “diagonals” to some of these pairs. Only in exceptional cases, a few triangles are split. Based on this simple refinement of Δ , we describe an algorithm for constructing Lagrange interpolation points such that the interpolation method is local, stable and has optimal approximation order. The complexity for computing the interpolating splines is linear in the number of triangles. For the local Lagrange interpolation methods known in the literature, about half of the triangles have to be split.

Automatic Shape Control of Triangular B-Splines of Arbitrary Topology

Triangular B-splines are powerful and flexible in modeling a broader class of geometric objects defined over arbitrary, non-rectangular domains. Despite their great potential and advantages in theory, practical techniques and computational tools with triangular B-splines are less-developed. This is mainly because users have to handle a large number of irregularly distributed control points over arbitrary triangulation. In this paper, an automatic and efficient method is proposed to generate visually pleasing, high-quality triangular B-splines of arbitrary topology. The experimental results on several real datasets show that triangular B-splines are powerful and effective in both theory and practice.

Structure and dimension of multivariate spline space of lower degree on arbitrary triangulation

In this paper, we discuss the structure of multivariate spline spaces on arbitrary triangulation by using the methods and results of smoothing cofactor and generator basis of modules. On the base of analyzing the algebraic and geometric results about singularity of S 2 1 ( Δ MS ) , we build the structure of triangulation and give some useful geometric conditions such that S μ + 1 μ ( Δ ) space is singular, and we obtain an algebraic condition which is necessary and sufficient for the singularity of S μ + 1 μ ( Δ ) spaces as well as their dimension formulae. Moreover, the structure matrix of spline spaces over any given partition is defined, which has been used to discuss the structure of S 3 1 ( Δ ) and S 2 1 ( Δ ) spaces over arbitrary triangulation and to prove the nonsingularity of S 3 1 ( Δ ) spaces. This partially settles a conjecture on the singularity of spline spaces in Wang et al., [Multivariate Spline and its Applications, Kluwer Press, Dordrecht, 2002; Academic Press, Beijing, 1994 (in Chinese)]. Meanwhile, the dimension formulae of S 3 1 ( Δ ) , S 2 1 ( Δ ) spaces and the dimension formulae of S μ + 1 μ ( Δ ) ( μ ⩾ 1 ) spaces are also given in this paper.

Triangulations and the Hajós Conjecture

The Hajós Conjecture was disproved in 1979 by Catlin. Recently, Thomassen showed that there are many ways that Hajós conjecture can go wrong. On the other hand, he observed that locally planar graphs and triangulations of the projective plane and the torus satisfy Hajós Conjecture, and he conjectured that the same holds for arbitrary triangulations of closed surfaces. In this note we disprove the conjecture and show that there are different reasons why the Hajós Conjecture fails also for triangulations.

Bidimensional interpolation operators of finite element type and degree of exactness two

For a given arbitrary triangulation of \(\mathbb R^2\), we construct an interpolating operator which is exact for the polynomials in two variables of total degree \(\leq 2\). This operator is local, in the sense that the information around an interpolation node is taken from a small region around this point. We study the remainder of the interpolation formula.

Lagrange Interpolation by Bivariate C1-Splines with Optimal Approximation Order

We develop the first local Lagrange interpolation scheme for C1-splines of degree q≥3 on arbitrary triangulations. For doing this, we use a fast coloring algorithm to subdivide about half of the triangles by a Clough–Tocher split in an appropriate way. Based on this coloring, we choose interpolation points such that the corresponding fundamental splines have local support. The interpolating splines yield optimal approximation order and can be computed with linear complexity. Numerical examples with a large number of interpolation points show that our method works efficiently.