Reference Triangle Research Articles

Efficient entropy-stable discontinuous spectral-element methods using tensor-product summation-by-parts operators on triangles and tetrahedra

We present a new class of efficient and robust discontinuous spectral-element methods of arbitrary order for nonlinear hyperbolic systems of conservation laws on curved triangular and tetrahedral unstructured grids. Such discretizations employ a recently introduced family of sparse tensor-product summation-by-parts (SBP) operators in collapsed coordinates within an entropy-conservative modal formulation, which is rendered entropy stable when a dissipative numerical flux is used at element interfaces. The proposed algorithms exploit the structure of such SBP operators alongside that of the Proriol–Koornwinder–Dubiner polynomial basis used to represent the numerical solution on the reference triangle or tetrahedron, and a weight-adjusted approximation is employed in order to efficiently invert the local mass matrix for curvilinear elements. Using such techniques, we obtain an improvement in time complexity from O(p2d) to O(pd+1) relative to existing entropy-stable formulations using multidimensional SBP operators not possessing such a tensor-product structure, where p is the polynomial degree of the approximation and d is the number of spatial dimensions. The number of required entropy-conservative two-point flux evaluations between pairs of quadrature nodes is accordingly reduced by a factor ranging from 1.56 at p=2 to 4.57 at p=10 for triangles, and from 1.88 at p=2 to 10.99 at p=10 for tetrahedra. Through numerical experiments involving smooth solutions to the compressible Euler equations on isoparametric triangular and tetrahedral grids, the proposed methods using tensor-product SBP operators are shown to exhibit similar levels of accuracy for a given mesh and polynomial degree to those using multidimensional operators based on symmetric quadrature rules, with both approaches achieving order p+1 convergence with respect to the element size in the presence of interface dissipation as well as exponential convergence with respect to the polynomial degree. Furthermore, both operator families are shown to give rise to entropy-stable schemes which exhibit excellent robustness for test problems characteristic of under-resolved turbulence simulations. Such results suggest that the algorithmic advantages resulting from the use of tensor-product operators are obtained without compromising accuracy or robustness, enabling the efficient extension of the benefits of entropy stability to higher polynomial degrees than previously considered for triangular and tetrahedral elements.

Adaptive range-based localization algorithm based on trilateration and reference node selection for outdoor wireless sensor networks

Locating the nodes of outdoor wireless sensor networks (WSNs) using (tri)lateration with a low-cost ranging technique, such as the received signal strength indicator (RSSI), often results in inaccurate location estimates. This can mostly be explained by the combined effect of distance estimate errors and localization geometry, both of which are subject to the reference nodes used. To develop techniques for reducing localization error, the distance estimate errors and localization geometry must be analyzed and taken into account. To address these challenges, this paper aims to seek ways to improve the quality of range-based trilateration localization for WSN nodes in varying outdoor conditions. Based on simulations, we analyze the effects of ranging error and localization geometry on localization error. For that purpose, we introduce a simple measure to evaluate the geometry of reference triangle (GRT). To improve localization accuracy and precision, we propose an adaptive range-based localization (ARBL) algorithm that is based on trilateration and reference node selection. In ARBL, the GRT values are calculated for each 3-combination of a preselected reference node set, based on which the combinations are selected. The algorithm exploits these reference node 3-combinations aiming to find the best ones at a given time using a selection criteria that is based on ranging error and localization geometry. The simulation and experimental results indicate that the proposed algorithm reduces localization error considerably. This shows that it is possible to achieve sufficient localization accuracy using range-based trilateration localization, even based on the RSSI in challenging outdoor conditions, by employing applicable techniques and information.

Alternative proposal of the high-order Gauss quadrature for reference triangle in the generalized finite element method

In this paper we propose a new distribution of Gaussian points to compute the weak form integrals of the Generalized Finite Element Method (GFEM). The relevance of this new distribution is the possibility of evaluating the integrals of oscillatory functions inside the reference triangle in an alternate way. A simple scheme of relocation of the quadrature points allows to improve the efficacy of the method. A wave propagation problem is solved with the proposed technique and its performance is compared to conventional and other existing proposals. In addition, we propose a new distribution of Gaussian points to the tetrahedral reference element

Conceptualizing Workarounds: Meanings and Manifestations in Information Systems Research

We reviewed papers in core IS outlets that defined the term workaround or presented an example of a workaround. In the analysis, we used Ogden and Richard's triangle of reference as a theoretical f ...

A high‐order FEM with exact geometry description for the Laplacian on implicitly defined surfaces

SummaryIn this paper, a high‐order finite element method for partial differential equations on smooth surfaces is proposed. The surface is defined as the intersection of a rectangular cuboid and an implicitly defined surface. Therefore, the surface of interest may not be closed. The main novel contribution in this work is the incorporation of an exact geometry description of surfaces with boundary into the finite element method. To this end, a piecewise planar triangulation is mapped onto the surface of interest by making use of the implicit surface definition. The mapping uses predefined search directions and can, therefore, be tailored to consider boundaries. High‐order hierarchical shape functions are utilized for the field approximation. They are defined on a reference triangle in the usual way. The proposed method is easy to implement and bypasses the need for a high‐order geometry description. Furthermore, due to the exact geometry, the imposition of Dirichlet boundary conditions, source terms, and mesh refinement are easy to carry out.

On [formula omitted]-robust saturation for [formula omitted]-AFEM

For the Poisson problem in two dimensions, we consider the standard adaptive finite element loop solve, estimate, mark, refine, with estimate being implemented using the p-robust equilibrated flux estimator, and, mark being Dörfler marking. As a refinement strategy we employ p-refinement. We investigate the question by which amount the local polynomial degree on any marked patch has to be incremented in order to achieve a p-independent error reduction. We show that the analysis can be transferred from the patches to a reference triangle, and therein we provide clear-cut computational evidence that any increment proportional to the polynomial degree (for any fixed proportionality constant) yields the desired reduction. The resulting adaptive method can be turned into an instance optimal hp-adaptive method by the addition of a coarsening routine.

SYMBOLIC INTERACTIONS:TOWARDS A COGNITIVE SCIENTIFIC THEORY OF MEANING IN HUMAN TECHNOLOGY INTERACTION

Information technology has perpetuated the role of symbolism in everyday life practice, through its reliance on sign systems for its creation and operation. Increasingly attention has been placed on applying semiotic techniques to analyze user interface design and usability. Surprisingly, although the move towards symbolic interaction has been one of the most striking components of the digital shift, it has proven difficult to build bridges between semiotics and HTI-design thinking. In this article we argue that the problems in linking semiotic analysis of human technology interaction with modern HTI-design paradigms such as usability or user experience arise from a theoretical gap between the paradigms of semiotics, human cognition and thinking. Consequently, it is necessary to reformulate principal insights of semiotics such as the triangle of reference, by replacing the intuitive concept of thought, with modern philosophical and psychological concepts of human thinking. This allows the unification of usability research based on cognitive research with the analysis of signs in modern semiotics. It is possible to unify the conceptual analysis of signification and semiosis with conceptual and empirical work typical to modern human technology interaction research and design, thus making semiotics an essential tool within the swiftly developing paradigms of interaction design.

On the stability of the boundary trace of the polynomial [formula omitted]-projection on triangles and tetrahedra

For the reference triangle or tetrahedron T, we study the stability properties of the L2(T)-projection ΠN onto the space of polynomials of degree N. We show ‖ΠNu‖L2(∂T)2≤C‖u‖L2(T)‖u‖H1(T). This implies optimal convergence rates for the approximation error ‖u−ΠNu‖L2(∂T) for all u∈Hk(T), k>1/2.

A generalized Seebach’s theorem

We present a generalized version of Seebach’s theorem considering the shapes of Cevian triangles with respect to points not only inside but also outside the reference triangle \(ABC\). The generalized version claims that for each positively oriented triangle \(A_1B_1C_1\) we have one to three points in the plane with positively oriented Cevian triangles directly similar to \(A_1B_1C_1\). With one exception the same is true for negatively oriented Cevian triangles. An information on trilinear coordinates of these points is provided. As an application we consider points having Cevian triangles directly similar to the reference triangle. In the positively oriented case besides the obvious candidate, the centroid, we found two additional such points in obtuse triangles: the intersections of the de Longchamps line with the circumcircle. As another application we consider points with equilateral Cevian triangles and present an example of a triangle in which the Kimberling triangle center \(X_{370}\) is not constructible by ruler and compass.

Measuring Data Registration Based on Triangle Constraint

This paper studied the fine registration techniques of measuring data of complex product and proposed a new registration method based on triangle constraint. Firstly, the paper made a research on specifying the proper measuring reference points and the setting of reference coordinate system, and afterwards proposed a fine registration algorithm of measuring data, Which is about realizing the fusion of reference coordinate system based on the congruent judgment of reference triangle and fulfilling precise match of data with measuring error on the basis of controlling the rotation of data blocks and minimizing the registration error.

Computational bases for [formula omitted] and [formula omitted] on triangles

In this article, we derive computational bases for Raviart–Thomas (RT) and Brezzi–Douglas–Marini (BDM) (vector) approximation spaces on a triangulation of a domain in R2. The basis functions, defined on the reference triangle, have a Lagrangian property. The continuity of the normal component of the approximation across the edges in the triangulation is satisfied by the use of the Piola transformation and the Lagrangian property of the basis functions. A numerical example is given demonstrating the approximation property of the bases.

Quantification of large and localized deformation in granular materials

Quantifying large deformation in granular assemblies using concepts originating from continuum mechanics is a challenging task because of (1) the discontinuous nature of granular displacement, which does not allow the definition of a continuum measure of deformation, and (2) the almost inevitable shear band localization. These problems exist in both real-world granular materials and their numerical idealizations using particle-based simulations. In this work a new method is developed in order to address these issues. Instead of creating a meshed equivalent continuum for quantifying small engineering strains, the new method performs independent random queries on the velocity gradient characteristics of arbitrary sub-domains in the assembly through the novel concept of overlapping reference triangles, thus, enabling rigorous handling of large deformations which are usually associated with localization. The proposed method is illustrated and validated by discrete element method (DEM) simulation of a biaxial compressive test, in which apparent shear banding takes place. The homogenized deformation quantifications based on the new method match the estimations from the imposed boundary conditions. The numerical examples are also applied to (1) quantifying the heterogeneous distribution of deformation over the specimen, (2) visualizing the nucleation process of shear bands, and (3) characterizing shear flow patterns in shear bands. An investigation on the effects of the reference triangle sizes yields some inspiring and practically significant results.

New adaptive modal and DTV filtering routines for the DG method on triangular grids applied to the Euler equations

In this paper, we continue our investigations started in Meister et al. (Numer. Methods Partial Differ. Equ. 2011) on the development of adaptive filtering techniques for DG methods on unstructured triangular grids solving hyperbolic conservation laws. While in our former work the focus was put on scalar conservation laws in order to study the principle mechanisms of the proposed techniques, new results are now presented concerning the application of adaptive modal and DTV filtering to the Euler equations of gas dynamics. To demonstrate the applicability of this approach for practically relevant problems, we perform experiments on standard test cases such as shock-vortex interactions, a double mach reflection as well as flow through a wind tunnel containing a forward facing step. The constructed modal filter is based on a specific spectral viscosity formulation on triangular grids. In this context, this paper gives a new result on the energy dissipation of the involved damping step which is based on an interesting estimate on weighted L 2-norms on the reference triangle. With respect to adaptive modal filtering, we investigate the behaviour of two different shock indicators which are new candidates for elementwise modal filtering in the DG context. Furthermore, we propose a novel adaptive strategy for the original DTV iteration as well as the modified DTV filter on subtriangle graphs constructed in Meister et al. (2011). For both filtering techniques, convergence studies away from the discontinuities of the exact entropy solution are additionally carried out for a scalar nonlinear test case yielding an improvement on the former results in Meister et al. (2011).

Application of the control volume mixed finite element method to a triangular discretization

SUMMARY A two-dimensional control volume mixed finite element method is applied to the elliptic equation. Discretization of the computational domain is based in triangular elements. Shape functions and test functions are formulated on the basis of an equilateral reference triangle with unit edges. A pressure support based on the linear interpolation of elemental edge pressures is used in this formulation. Comparisons are made between results from the standard mixed finite element method and this control volume mixed finite element method. Published 2011. This article is a US Government work and is in the public domain in the USA.

An as-rigid-as-possible approach to sensor network localization

We present a novel approach to localization of sensors in a network given a subset of noisy inter-sensor distances. The algorithm is based on “stitching” together local structures by solving an optimization problem requiring the structures to fit together in an “As-Rigid-As-Possible” manner, hence the name ARAP. The local structures consist of reference “patches” and reference triangles, both obtained from inter-sensor distances. We elaborate on the relationship between the ARAP algorithm and other state-of-the-art algorithms, and provide experimental results demonstrating that ARAP is significantly less sensitive to sparse connectivity and measurement noise. We also show how ARAP may be distributed.

The Fermat point configuration

In this article a theorem similar to Napoleon’s theorem is established for the Fermat point configuration of a triangle. Areal coordinates are used throughout, with ABC as triangle of reference. For a full account of how to define and use these coordinates see Bradley [1]. Alternative synthetic proofs, generously supplied by the referee, are also included. The presentation of the paper with its emphasis on coordinates was designed in part to show that an algebraic treatment of the Fermat point configuration is possible, as well as the more familiar synthetic or complex number treatments.

Global shape versus local feature: An angle illusion

We have shown previously that the precision of angle judgments depends strongly on the global stimulus configuration: discrimination thresholds for angles that form part of isosceles triangles are up to 3 times lower than for those that form part of scalene triangles [Kennedy, G. J., Orbach, H. S., & Loffler, G. (2006). Effects of global shape on angle discrimination. Vision Research, 46(8–9), 1530–1539]. Here, we investigated whether or not the perceived size of an angle (accuracy) is also affected by the overall shape of which it forms a part. Observers compared the relative sizes of angles contained in isosceles triangles with those of angles in scalene triangles and points of subjective equality were determined. For a reference angle of 60°, angles embedded in isosceles triangles were judged to be on average 14° larger than angles embedded in scalene triangles. This result is largely independent of the reference angle, triangle orientation and triangle size. Moreover, the effect is present whether or not triangles of different shapes enclose the same area, whether or not the side of the triangle opposite the angle is present and whether the triangle is outlined or defined by dots at its vertexes. In sum, our results provide evidence for a novel illusion where an angle embedded in an isosceles triangle is judged substantially larger than the same angle embedded in a scalene triangle. This finding demonstrates that mechanisms for computing angles are sensitive to the context within which angles are presented.

Twenty-one points on the nine-point circle

The nine points for which the nine-point circle is named are the vertices of three triangles: the medial, orthic, and Euler triangles of a reference triangle ABC. All nine of the vertices lie on the circle, which Dan Pedoe [1] proclaims ‘the first really exciting one to appear in any course on elementary geometry’. An online summary of properties of the circle, which is also called the Euler circle, is given at MathWorld [2].

Asymptotic properties of some triangulations of the sphere

In this paper we analyse a method for triangulating the sphere originally proposed by Baumgardner and Frederickson in 1985. The method is essentially a refinement procedure for arbitrary spherical triangles that fit into a hemisphere. Refinement is carried out by dividing each triangle into four by introducing the midpoints of the edges as new vertices and connecting them in the usual ‘red’ way. We show that this process can be described by a sequence of piecewise smooth mappings from a reference triangle onto the spherical triangle. We then prove that the whole sequence of mappings is uniformly bi-Lipschitz and converges uniformly to a non-smooth parameterization of the spherical triangle, recovering the Baumgardner and Frederickson spherical barycentric coordinates. We also prove that the sequence of triangulations is quasi-uniform, that is, areas of triangles and lengths of the edges are roughly the same at each refinement level. Some numerical experiments confirm the theoretical results.

Gold-plated mode of CP-violation in decays of Bc meson from QCD sum rules

A model-independent method based on the triangle ideology is implemented to extract the CKM-matrix angle γ in the decays of the doubly heavy long-lived meson Bc. We analyse a colour structure of diagrams and conditions to reconstruct two reference triangles by tagging the flavour and CP eigenstates of mesons in the fixed exclusive channels. The characteristic branching ratios are evaluated in the framework of QCD sum rules.