Set Of Triangles Research Articles

Phase-field modeling and linearly energy-stable Runge–Kutta algorithm of colloidal crystals on curved surfaces

In this work, we develop an efficient and unconditionally energy-stable numerical method for solving a generalized H−1-gradient flow based Swift–Hohenberg equation for colloidal crystal on surfaces. By introducing an appropriate truncated nonlinear potential, we construct linear and temporally second-order accurate scheme based on the linear convex splitting and implicit-explicit Runge–Kutta approaches. The surface is discretized by a set of triangles. The spatial approximation is performed based on the finite volume type method. During every time step, our task simply involves solving a set of elliptic-type equations that have constant coefficients. We analytically prove the energy dissipation law and unique solvability. Numerical examples indicate that the proposed scheme not only works well on various surfaces but also has desired accuracy and energy stability. We also provide the MATLAB code for generating triangular mesh on the curved surface in the link: http://github.com/yang521/MATLAB-distmesh.

DR-Occluder: Generating Occluders Using Differentiable Rendering

The target of the occluder is to use very few faces to maintain similar occlusion properties of the original 3D model. In this paper, we present DR-Occluder, a novel coarse-to-fine framework for occluder generation that leverages differentiable rendering to optimize a triangle set to an occluder. Unlike prior work, which has not utilized differentiable rendering for this task, our approach provides the ability to optimize a 3D shape to defined targets. Given a 3D model as input, our method first projects it to silhouette images, which are then processed by a convolution network to output a group of vertex offsets. These offsets are used to transform a group of distributed triangles into a preliminary occluder, which is further optimized by differentiable rendering. Finally, triangles whose area is smaller than a threshold are removed to obtain the final occluder. Our extensive experiments demonstrate that DR-Occluder significantly outperforms state-of-the-art methods in terms of occlusion quality. Furthermore, we compare the performance of our method with other approaches in a commercial engine, providing compelling evidence of its effectiveness.

A novel approach to compute the spatial gradients of enriching functions in the X-FEM with a hybrid representation of cracks

The eXtended Finite Element Method (X-FEM) is a versatile technique to model discontinuities by enriching the trial functions with a prior solution. In the X-FEM, a crack can be explicitly represented by a set of triangles or implicit signed distances, i.e., level set functions, of the points of interest from the crack surface and the crack front. In the explicit representations, it is crucial to accurately evaluate surface normal, conormal, and tangent vectors along crack fronts for computations of the gradients of the enriching functions. The solution is very sensitive to these directional vectors, especially for non-planar 3D cracks. We here propose a novel approach to compute these gradients without evaluating the directional vectors. Level set functions are first set up in a hexahedral grid independent of the background mesh in the X-FEM. We can thus implement the Weight Essentially Non-Oscillatory (WENO) scheme to compute the gradients of level sets. The gradients of the enriching functions at any integration point can therefore be computed by interpolations and chain rules. We compare the implementation procedures of the explicit representation and the proposed hybrid representation in detail. A three-dimensional lens crack problem is studied to demonstrate the accuracy of the proposed method, especially for coarse meshes.

Groth’s algorithm to detect the possible presence of landmines using changes in the reflection of plants

Abstract. Post-conflict reconstruction includes the removal of land mines and remnants of war. The CSIR conducted field experiments to determine the impact of TNT in the soil on plants as a possible means to detect landmines. All the leaf clip readings were done using a spectrometer to determine reflection and absorption of light at one micrometre intervals between 350 to 2500 micrometre. Laboratory analysis such as UPLC qTOF MS indicated that the TNT have an influence on the plants. Several indices such as Modified Red Edge Normalized Difference Vegetation Index (mNDVI705), Red Edge Position (REP) and Moisture Stress Index (MSI) did not show any significant differences between control plants and experimental plants with different TNT concentrations. Groth’s pattern matching algorithm designed to match several photographs of the same part of the universe. A set of triangles using dominant stars are created for each photograph and matched using an error band. If the selected triangles from the two photographs fit within the error band, then they are from the same section of the universe. Bands for the Pléiades instruments were simulated using the data from the spectrometer for each plant. The reflectance value of the band and the normalised midpoint wavelength of each Pléiades band were used to construct the triangles. The control plant triangle is then matched with the experimental plant and if the triangles do not match, then the effect of TNT on the plant is significant. The initial results with the control plants and experimental plants are positive.

Triangle Influence Supersets for Fast Distance Computation

AbstractWe present an acceleration structure to efficiently query the Signed Distance Field (SDF) of volumes represented by triangle meshes. The method is based on a discretization of space. In each node, we store the triangles defining the SDF behaviour in that region. Consequently, we reduce the cost of the nearest triangle search, prioritizing query performance, while avoiding approximations of the field. We propose a method to conservatively compute the set of triangles influencing each node. Given a node, each triangle defines a region of space such that all points inside it are closer to a point in the node than the triangle is. This property is used to build the SDF acceleration structure. We do not need to explicitly compute these regions, which is crucial to the performance of our approach. We prove the correctness of the proposed method and compare it to similar approaches, confirming that our method produces faster query times than other exact methods.

Introducing isodynamic points for binary forms and their ratios

The isodynamic points of a plane triangle are known to be the only pair of its centers invariant under the action of the Möbius group {mathcal {M}} on the set of triangles, Kimberling (Encyclopedia of Triangle Centers, http://faculty.evansville.edu/ck6/encyclopedia). Generalizing this classical result, we introduce below the isodynamic map associating to a univariate polynomial of degree dge 3 with at most double roots a polynomial of degree (at most) 2d-4 such that this map commutes with the action of the Möbius group {mathcal {M}} on the zero loci of the initial polynomial and its image. The roots of the image polynomial will be called the isodynamic points of the preimage polynomial. Our construction naturally extends from univariate polynomials to binary forms and further to their ratios.

On asymptotic packing of geometric graphs

A set of geometric graphs is geometric-packable if it can be asymptotically packed into every sequence of drawings of the complete graph K n . For example, the set of geometric triangles is geometric-packable due to the existence of Steiner Triple Systems. When G is the 4-cycle (or 4-cycle with a chord), we show that the set of plane drawings of G is geometric-packable. In contrast, the analogous statement is false when G is nearly any other planar Hamiltonian graph (with at most 3 possible exceptions). A convex geometric graph is convex-packable if it can be asymptotically packed into the convex drawings of the complete graphs. For each planar Hamiltonian graph G , we determine whether or not a plane G is convex-packable. Many of our proofs explicitly construct these packings; in these cases, the packings exhibit a symmetry that mirrors the vertex transitivity of K n .

A strengthening of the spectral chromatic critical edge theorem: Books and theta graphs

AbstractA graph is color‐critical if it contains an edge whose removal reduces its chromatic number. Let be the Turán graph with vertices and parts. Given a graph , let be the Turán number of . Simonovits' chromatic critical edge theorem states that if is color‐critical with , then there exists an such that and the Turán graph is the only extremal graph provided . Nikiforov proved a spectral chromatic critical edge theorem. It asserts that if is color‐critical and , then there exists an (which is exponential with ) such that and is the only extremal graph provided , where is the spectral radius of and . In addition, if is either a complete graph or an odd cycle, then is linear with . A book graph is a set of triangles sharing a common edge and a theta graph is a graph which consists of two vertices connected by three internally disjoint paths with length one, two, and . Notice that both and are color‐critical. In this article, we prove that if , then contains a book with unless . Similarly, we prove that if , then contains a theta graph with for odd and for even unless . Our results imply that in the spectral chromatic critical edge theorem is linear with for book graphs and theta graphs. Our result for book graphs can be viewed as a spectral version of an Erdős conjecture (1962) stating that every ‐vertex graph with contains a book graph with Moreover, our result for theta graphs yields that every graph with contains a cycle of length for each . This is related to an open question by Nikiforov (2008) which asks for the maximum such that every graph of large enough order with contains a cycle of length for every .

Fast Parallel Evaluation of Exact Geometric Predicates on GPUs

This paper accelerates the exact evaluation of large numbers of 3D geometric predicates with an algorithm whose work is partitioned between the CPU and the GPU on a high-performance computer to exploit the relative strengths of each. The test algorithm computes all the red–blue intersections between a set of red 3D triangles and another set of blue 3D triangles. A sequence of filters is employed that progressively eliminates more and more red–blue pairs that do not intersect, finally leaving only the actual intersections. Initially, a uniform grid is constructed on the GPU to identify pairs of nearby triangles. Then, these pairs are tested for intersection with single-precision interval arithmetic on the GPU. The ambiguous cases are next filtered with double-precision interval arithmetic on the multi-core CPU, and finally the hard cases are re-evaluated in parallel on the CPU using arbitrary-precision rational numbers. The parallel speedup for the whole algorithm was up to 414 times. It took only 1.17 s to find the 18M intersections between two datasets containing a total of 14M triangles. The intersection computation was sped up by up to 1936 times. The techniques that gave this excellent performance should be useful for parallelizing other geometric algorithms in fields such as CAD, GIS, and 3D modeling.

Hyperbolic generalized triangle groups, property (T) and finite simple quotients

We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdan's property (T). Some of them are significantly shorter than the previously known shortest examples. Moreover, we show that some of those hyperbolic Kazhdan groups possess finite simple quotient groups of arbitrarily large rank; they constitute the first-known specimens combining those properties. All the hyperbolic groups we consider are non-positively curved k $k$ -fold generalized triangle groups, that is, groups that possess a simplicial action on a CAT(0) triangle complex, which is sharply transitive on the set of triangles, and such that edge-stabilizers are cyclic of order k $k$ . Appendices A, B and C are provided separately as supplementary material with the published article.

CALCULATED EVALUATION OF THE DEGREE OF SUBSOLIDUS STRUCTURE COMPLEXITY OF THE PHYSICO-CHEMICAL THREE-COMPONENT SYSTEMS

The study of state diagrams of multicomponent physicochemical systems is the most science-intensive task of materials science. Without knowledge about the structure of the state diagrams of such systems, it is impossible for technologists to predict the phase composition of materials during their production and use of products from them; the systematic analysis of the results of experimental studies to optimize the properties of the materials being developed becomes much more complicated. In refractory technology, the defining stage of production is solid-phase sintering, which makes the information on the subsolidus structure of state diagrams of physicochemical systems represented by a set of components in accordance with the planned phase composition of materials particularly important. Three-component systems, the simple components of which are refractory oxides, constitute the physicochemical basis of most mass-produced refractories and their subsolidus structure is quite clearly displayed in the concentration triangle of the system by a set of triangles, the vertices of which are the points of the composition of the compounds. The study is devoted to the establishment of an analytical relationship between the number of double and triple compounds and the number of all possible segments of connecting lines between the points of the composition of the connections, as well as the points of intersection of the compounds between themselves. During the research, the general principles of systems analysis, logical methods and terminology of physicochemical analysis of multicomponent systems, as well as information on elementary mathematics from the sections on numerical series, the basics of combinatorics and algebra were used. The corresponding analytical expressions make it possible to calculate the quantitative classification characteristics in the taxonomy of multicomponent systems by the degree of complexity of the structure of their subsolidus regions, in particular, when comparing the complexity of studies of three-component oxide systems and their typification. The formulas obtained were tested for calculations using examples of specific oxide systems. The research results allow one to obtain important quantitative characteristics for assessing the degree of complexity of the subsolidus structure of three-component systems.

A Block-Based Triangle Counting Algorithm on Heterogeneous Environments

Triangle counting is a fundamental building block in graph algorithms. In this article, we propose a block-based triangle counting algorithm to reduce data movement during both sequential and parallel execution. Our block-based formulation makes the algorithm naturally suitable for heterogeneous architectures. The problem of partitioning the adjacency matrix of a graph is well-studied. Our task decomposition goes one step further: it partitions the set of triangles in the graph. By streaming these small tasks to compute resources, we can solve problems that do not fit on a device. We demonstrate the effectiveness of our approach by providing an implementation on a compute node with multiple sockets, cores and GPUs. The current state-of-the-art in triangle enumeration processes the Friendster graph in 2.1 seconds, not including data copy time between CPU and GPU. Using that metric, our approach is 20 percent faster. When copy times are included, our algorithm takes 3.2 seconds. This is 5.6 times faster than the fastest published CPU-only time.

A Numerical Method for Computing Double Integrals with Variable Upper Limits

We propose and justify a numerical method for computing the double integral with variable upper limits that leads to the variableness of the region of integration. Imposition of simple variables as functions for upper limits provides the form of triangles of integration region and variable in the external limit of integral leads to a continuous set of similar triangles. A variable grid is overlaid on the integration region. We consider three cases of changes of the grid for the division of the integration region into elementary volumes. The first is only the size of the imposed grid changes with the change of variable of the external upper limit. The second case is the number of division elements changes with the change of the external upper limit variable. In the third case, the grid size and the number of division elements change after fixing their multiplication. In these cases, the formulas for computing double integrals are obtained based on the application of cubatures in the internal region of integration and performing triangulation division along the variable boundary. The error of the method is determined by expanding the double integral into the Taylor series using Barrow’s theorem. Test of efficiency and reliability of the obtained formulas of the numerical method for three cases of ways of the division of integration region is carried out on examples of the double integration of sufficiently simple functions. Analysis of the obtained results shows that the smallest absolute and relative errors are obtained in the case of an increase of the number of division elements changes when the increase of variable of the external upper limit and the grid size is fixed.

Construction of LDPC convolutional codes via difference triangle sets

In this paper, a construction of (n,k,delta ) LDPC convolutional codes over arbitrary finite fields, which generalizes the work of Robinson and Bernstein and the later work of Tong is provided. The sets of integers forming a (k, w)-(weak) difference triangle set are used as supports of some columns of the sliding parity-check matrix of an (n,k,delta ) convolutional code, where nin {mathbb {N}}, n>k. The parameters of the convolutional code are related to the parameters of the underlying difference triangle set. In particular, a relation between the free distance of the code and w is established as well as a relation between the degree of the code and the scope of the difference triangle set. Moreover, we show that some conditions on the weak difference triangle set ensure that the Tanner graph associated to the sliding parity-check matrix of the convolutional code is free from 2ell -cycles not satisfying the full rank condition over any finite field. Finally, we relax these conditions and provide a lower bound on the field size, depending on the parity of ell , that is sufficient to still avoid 2ell -cycles. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets.

High Order Godunov Type Multimesh Method for 3d Impact Problems of Elastoplastic Media

A numerical method for calculating the three-dimensional processes of impact interaction of elastoplastic bodies under large displacements and deformations based on the multi mesh sharp interface method and modified Godunov scheme is presented. To integrate the equations of dynamics of an elastoplastic medium, the principle of splitting in space and in physical processes is used. The solutions of the Riemann problem for first and second order accuracy for compact stencil for an elastic medium in the case of an arbitrary stress state are obtained and presented, which are used at the “predictor” step of the Godunov scheme. A modification of the scheme is described that allows one to obtain solutions in smoothness domains with a second order of accuracy on a compact stencil for moving Eulerian-Lagrangian grids. Modification is performed by converging the areas of influence of the differential and difference problems for the Riemann’s solver. The “corrector” step remains unchanged for both the first and second order accuracy schemes. Three types of difference grids are used. The first – a moving surface grid – consists of a continuous set of triangles that limit and accompany the movement of bodies; the size and number of triangles in the process of deformation and movement of the body can change. The second – a regular fixed Eulerian grid – is limited to a surface grid; separately built for each body; integration of equations takes place on this grid; the number of cells in this grid can change as the body moves. The third grid is a set of local Eulerian-Lagrangian grids attached to each moving triangle of the surface from the side of the bodies and allowing obtain the parameters on the boundary and contact surfaces. The values of the underdetermined parameters in cell’s centers near the contact boundaries on all types of grids are interpolated. Comparison of the obtained solutions with the known solutions by the Eulerian-Lagrangian and Lagrangian methods, as well as with experimental data, shows the efficiency and sufficient accuracy of the presented three-dimensional methodology.

Weighted Triangle-free 2-matching Problem with Edge-disjoint Forbidden Triangles

The weighted \({\mathcal {T}}\)-free 2-matching problem is the following problem: given an undirected graph G, a weight function on its edge set, and a set \({\mathcal {T}}\) of triangles in G, find a maximum weight 2-matching containing no triangle in \({\mathcal {T}}\). When \({\mathcal {T}}\) is the set of all triangles in G, this problem is known as the weighted triangle-free 2-matching problem, which is a long-standing open problem. A main contribution of this paper is to give the first polynomial-time algorithm for the weighted \({\mathcal {T}}\)-free 2-matching problem under the assumption that \({\mathcal {T}}\) is a set of edge-disjoint triangles. In our algorithm, a key ingredient is to give an extended formulation representing the solution set, that is, we introduce new variables and represent the convex hull of the feasible solutions as a projection of another polytope in a higher dimensional space. Although our extended formulation has exponentially many inequalities, we show that the separation problem can be solved in polynomial time, which leads to a polynomial-time algorithm for the weighted \({\mathcal {T}}\)-free 2-matching problem.

Shapes of hyperbolic triangles and once-punctured torus groups

Let Delta be a hyperbolic triangle with a fixed area varphi . We prove that for all but countably many varphi , generic choices of Delta have the property that the group generated by the pi -rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all varphi in (0,pi ){setminus }mathbb {Q}pi , a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space mathfrak {C}_theta of singular hyperbolic metrics on a torus with a single cone point of angle theta =2(pi -varphi ), and answer an analogous question for the holonomy map rho _xi of such a hyperbolic structure xi . In an appendix by Gao, concrete examples of theta and xi in mathfrak {C}_theta are given where the image of each rho _xi is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds.

Copy Move Forgery Detection based on double matching

Copy Move is a technique widespreadly used in digital image tampering, meaning Copy Move Forgery Detection (CMFD) is still a significant research. In this paper, a novel CMFD method is proposed, including double matching process and region localizing process. In double matching process, the first matching is conducted on Delaunay triangles consisting of Local Intensity Order Pattern (LIOP) keypoints, to find the approximate location of suspicious regions. In order to find sufficient keypoint pairs, the existing set of matching triangles is expanded by adding their neighbors iteratively, covering the whole tampered regions, and the second matching with a looser threshold is conducted on the vertices. In the region localizing process, considering the case of multiple copies, Density-Based Spatial Clustering of Applications with Noise (DBSCAN) is used to classify the keypoint pairs described in a new model. Experimental results indicate that the proposed method, with good robustness, outperforms some state-of-the-art methods.

On Tuza’s conjecture for triangulations and graphs with small treewidth

Tuza (1981) conjectured that the size τ(G) of a minimum set of edges that intersects every triangle of a graph G is at most twice the size ν(G) of a maximum set of edge-disjoint triangles of G. In this paper we present three results regarding Tuza’s Conjecture. We verify it for graphs with treewidth at most 6; we show that τ(G)≤32ν(G) for every planar triangulation G different from K4; and that τ(G)≤95ν(G)+15 if G is a maximal graph with treewidth 3. Our first result strengthens a result of Tuza, implying that τ(G)≤2ν(G) for every K8-free chordal graph G.

Packing Arc-Disjoint Cycles in Tournaments

A tournament is a directed graph in which there is a single arc between every pair of distinct vertices. Given a tournament T on n vertices, we explore the classical and parameterized complexity of the problems of determining if T has a cycle packing (a set of pairwise arc-disjoint cycles) of size k and a triangle packing (a set of pairwise arc-disjoint triangles) of size k. We refer to these problems as Arc-disjoint Cycles in Tournaments (ACT) and Arc-disjoint Triangles in Tournaments (ATT), respectively. Although the maximization version of ACT can be seen as the dual of the well-studied problem of finding a minimum feedback arc set (a set of arcs whose deletion results in an acyclic graph) in tournaments, surprisingly no algorithmic results seem to exist for ACT. We first show that ACT and ATT are both NP-complete. Then, we show that the problem of determining if a tournament has a cycle packing and a feedback arc set of the same size is NP-complete. Next, we prove that ACT is fixed-parameter tractable via a $$2^{\mathcal {O}(k \log k)} n^{\mathcal {O}(1)}$$ -time algorithm and admits a kernel with $$\mathcal {O}(k)$$ vertices. Then, we show that ATT too has a kernel with $$\mathcal {O}(k)$$ vertices and can be solved in $$2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}$$ time. Afterwards, we describe polynomial-time algorithms for ACT and ATT when the input tournament has a feedback arc set that is a matching. We also prove that ACT and ATT cannot be solved in $$2^{o(\sqrt{n})} n^{\mathcal {O}(1)}$$ time under the exponential-time hypothesis.