Triangular Domain Research Articles

Aliasing errors due to quadratic nonlinearities on triangular spectral /hp element discretisations

In this paper, consideration is given to how aliasing errors, introduced when evaluating nonlinear products, inexactly affect the solution of Galerkin spectral/hp element polynomial discretisations on triangles. A theoretical discussion is presented of how aliasing errors are introduced by a collocation projection onto a set of quadrature points insufficient for exact integration, and consider interpolation projections to geometrically symmetric ollocation points. The discussion is corroborated by numerica examples that elucidate the key features. The study is first motivated with a review of aliasing errors introduced in one-dimensional spectral-element methods (these results extend naturally to tensor-product quadrilaterals and hexahedra.) Within triangular domains two commonly used expansions are a hierarchical, or modal, expansion based on a rotationally non-symmetric collapsed-coordinate system, and a Lagrange expansion based on a set of rotationally symmetric nodal points. Whilst both expansions span the same polynomial space, the construction of the two bases numerically motivates a different set of collocation points for use in the collocation projection of a nonlinear product. The purpose of this paper is to compare these two collocation projections. The analysis and results show that aliasing errors produced using a collocation projection on the rotationally non-symmetric, collapsed-coordinate system are significantly smaller than those for a collocation projection using the rotationally symmetric nodal points. In the case of the collapsed coordinate projection, if the Gaussian quadrature order employed is less than half the polynomial order of the integrand, then it is possible for the aliasing error to modify the constant mode of the expansion and therefore affect the conservation property of the approximation. However, the use of a collocation projection onto a polynomial expansion associated with a set of rotationally symmetric nodal points within the triangle is always observed to be non-conservative. Nevertheless, the rotationally symmetric collocation will maintain the overall symmetry of the triangular region, which is not typically the case when a collapsed coordinate quadrature projection is used.

Computing orthogonal polynomials on a triangle by degree raising

We give an algorithm for computing orthogonal polynomials over triangular domains in Bernstein–Bézier form which uses only the operator of degree raising and its adjoint. This completely avoids the need to choose an orthogonal basis (or tight frame) for the orthogonal polynomials of a given degree, and hence the difficulties inherent in that approach. The results are valid for Jacobi polynomials on a simplex, and show the close relationship between the Bernstein form of Jacobi polynomials, Hahn polynomials and degree raising.

On the Bernstein--Bézier form of Jacobi polynomials on a simplex

Here, we give a simple proof of a new representation for orthogonal polynomials over triangular domains which overcomes the need to make symmetry destroying choices to obtain an orthogonal basis fo...

Boundary reduction technique and triangular differential quadrature domain decomposition method for polygonal region

The differential quadrature method (DQM) is an attractive numerical method with high efficiency and accuracy. But the conventional DQM is limited in its application to regular regions by using functional values along a mesh line to approximate derivatives. To deal with problems on irregular geometric domains, coordinate transformation has to be conducted. The triangular differential quadrature method (TDQM) proposed by Zhong [Zhong H. Triangular differential quadrature. Commun Numer Meth Eng 2000; 16:401–8; Zhong H. Triangular differential quadrature and its application to elastostatic analysis of Reissner plates. Int J Solids Struct 2001; 38:2821–32], avoid the coordinate transformation. In this paper, the domain decomposition method (DDM) is used for the elliptical boundary problems on a pentagonal region. In every sub-domain, we solve the partial differential equations with TDQM. With boundary reduction technique, the functional values on internal points can be eliminated. The system of equations, which satisfied by the boundary points, can be obtained. Numerical results show that triangular differential quadrature domain decomposition method (TDQDDM) is easy and effective for treating the problems on irregular region.

Dual functionals of said-bézier type generalized ball bases over triangle domain and their application

A family of Said-Bezier type generalized Ball (SBGB) bases and surfaces with a parameter H over triangular domain is introduced, which unifies Bezier surface and Said-Ball surface and includes several intermediate surfaces. To convert different bases and surfaces, the dual functionals of bases are presented. As an application of dual functionals, the subdivision formulas for surfaces are established.

On the Bernstein–Bézier form of Jacobi polynomials on a simplex

Here, we give a simple proof of a new representation for orthogonal polynomials over triangular domains which overcomes the need to make symmetry destroying choices to obtain an orthogonal basis for polynomials of fixed degree by employing redundancy. A formula valid for simplices with Jacobi weights is given, and we exhibit its symmetries by using the Bernstein–Bézier form. From it, we obtain the matrix representing the orthogonal projection onto the space of orthogonal polynomials of fixed degree with respect to the Bernstein basis. The entries of this projection matrix are given explicitly by a multivariate analogue of the F 2 3 hypergeometric function. Along the way we show that a polynomial is a Jacobi polynomial if and only if its Bernstein basis coefficients are a Hahn polynomial. We then discuss the application of these results to surface smoothing problems under linear constraints.

Inverse theorems for Bernstein-Chlodowsky type polynomials

In this paper we establish two inverse theorems for Bernstein-Chlodowsky type polynomials of two variables in a rectangular and a triangular domain.

山岳表面の等高線に基づいた曲線メッシュの生成

The need for ways to obtain highly accurate 3D shape data from existing 2D contours is increasingly necessary in some applications, such as in disaster prevention, environmental analysis, and creating construction plans. In this paper, we propose an algorithm that reconstructs a curve mesh model of mountain with triangular and quadrilateral domains that is generated from discrete contours data by using constrained Delaunay triangulation method. To prove the effectiveness of our method, we interpolated the curve mesh into free-form surface model with G1 continuity and compared the accuracy and quantity of data with other methods.

On Approximation for Functions of Two Variables on a Triangular Domain

The problem of approximation of continuous functions of two variables by Bernstein-Chlodowsky polynomials on a triangular domain is studied. Moreover, the problem of weighted approximations of continuous functions of two variables by a sequences of linear positive operators is discussed.

Correction to “On the Calculation of Potential Integrals for Linear Source Distributions on Triangular Domains”

Correction to “On the Calculation of Potential Integrals for Linear Source Distributions on Triangular Domains”

CONSTRAINED QUADRILATERAL MESHES OF BOUNDED SIZE

We introduce a new algorithm to convert triangular meshes of polygonal regions, with or without holes, into strictly convex quadrilateral meshes of small bounded size. Our algorithm includes all vertices of the triangular mesh in the quadrilateral mesh, but may add extra vertices (called Steiner points). We show that if the input triangular mesh has t triangles, our algorithm produces a mesh with at most [Formula: see text] quadrilaterals by adding at most t+2 Steiner points, one of which may be placed outside the triangular mesh domain. We also describe an extension of our algorithm to convert constrained triangular meshes into constrained quadrilateral ones. We show that if the input constrained triangular mesh has t triangles and its dual graph has h connected components, the resulting constrained quadrilateral mesh has at most [Formula: see text] quadrilaterals and at most t+3h Steiner points, one of which may be placed outside the triangular mesh domain. Examples of meshes generated by our algorithm, and an evaluation of the quality of these meshes with respect to a quadrilateral shape quality criterion are presented as well.

Jacobi‐weighted orthogonal polynomials on triangular domains

We construct Jacobi‐weighted orthogonal polynomials , on the triangular domain T. We show that these polynomials over the triangular domain T satisfy the following properties: , r = 0, 1, …, n, and for r ≠ s. And hence, , n = 0, 1, 2, …, r = 0, 1, …, n form an orthogonal system over the triangular domain T with respect to the Jacobi weight function. These Jacobi‐weighted orthogonal polynomials on triangular domains are given in Bernstein basis form and thus preserve many properties of the Bernstein polynomial basis.

Generalized Fourier transform on an arbitrary triangular domain

In this paper, we construct generalized Fourier transform on an arbitrary triangular domain via barycentric coordinates and PDE approach. We start with a second-order elliptic differential operator for an arbitrary triangle which has the so-called generalized sine (TSin) and generalized cosine (TCos) systems as eigenfunctions. The orthogonality and completeness of the systems are then proved. Some essential convergence properties of the generalized Fourier series are discussed. Error estimates are obtained in Sobolev norms. Especially, the generalized Fourier transforms for some elementary polynomials and their convergence are investigated.

Boundary conditions and variable ground state entropy for the antiferromagnetic Ising model on a triangular lattice

The ground state entropy density of the antiferromagnetic Ising model on a triangular lattice is considered in the infinite volume limit as a function of boundary conditions on a finite triangular domain. The ground states of this domain do not map to a dimer covering and so cannot be classified into string sectors. A parametrized boundary condition is identified that allows the entropy density to be tuned to values between nondegeneracy and maximal degeneracy. The results are compared to those for a rectangular periodic domain for which the ground states can be classified into string sectors.

Unsteady natural convection in a triangular enclosure induced by surface cooling

This study considers the unsteady natural convection in a triangular domain induced by a constant cooling at the water surface. The study consists of two parts: a scaling analysis and numerical simulations. The scaling analysis for small bottom slopes reveals that three laminar flow regimes, namely conductive, transitional and convective flow regimes are possible depending on the Rayleigh number. Proper scales have been established to quantify the transient flow development in each of these flow regimes. The numerical simulation has verified the scaling predictions.

Differential quadrature domain decomposition method for problems on a triangular domain

AbstractThe differential quadrature method (DQM) has been studied for years and it has been shown by many researchers that the DQM is an attractive numerical method with high efficiency and accuracy. The conventional DQM is mostly effective for one‐dimensional and multidimensional problems with geometrically regular domains. But to deal with problems on a triangular domain, we will meet difficulties. In this article we will study how to solve problems on a triangular domain by using DQM combined with the domain decomposition method (DDM). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

Disintegration in high-grade titania slags: low temperature oxidation reactions of ferro-pseudobrookite

The predominant phase in commercial high-grade solidified titania slags is a compound with the pseudobrookite or M3 O5 structure in which a substantial proportion of the titanium is in the trivalent oxidation state. Cooling of the slag in air results in the oxidation of the pseudobrookite to form phases that are associated with the disintegration of the solid slag. The presence of excessive amounts of fine-grained oxidised material renders it unsuitable for subsequent fluidised bed chlorination for the production of TiO2 pigment. Depending on the degree of reduction, the pseudobrookite phase has a composition with variable iron content. In high-grade slag, the stoichiometry of this phase is approximating that of (Fe0.27 Mg0.07 Al0.04 Ti3+ 1.35Ti4+1.35)O5. Single crystal structure analysis revealed no significant change in the room temperature structure at temperatures of 250°C and 350°C in air. At temperatures above 700°C, the oxidation of the slag is associated with the formation of mainly anatase, rutile, and oxidised M3O 5. No discernible disintegration takes place during this reaction. Between ~550°C and 700°C, the reaction is a slow one involving the formation of anatase and oxidised M3O5. Very little disintegration is associated with this reaction. Below ~550°C, the dominant reaction changes to a fast reaction involving the formation of a single disordered phase (MOx), related to both M3 O5 and anatase, and this results in extensive disintegration. This oxidation takes place in two steps. Initially, large cracks form parallel to the a-crystallographic axis direction in the precursor M3 O5 crystals. Triangular domains of the oxidised phase having a composition MOx form on the crack surfaces. Secondary cracks form along the c-axis directions, dissecting the domains. The cracks originating from adjacent domains join up to segment the crystal into small rectangular fragments as small as 10μm in size.

Volterra Kernel Identification Using Triangular Wavelets

The Nblterra series provides a convenient framework for the representation of nonlinear dynamical systems. One of the main drawbacks of this approach, however, is the large number of terns that are often needed to represent Wblterra kernels. In this paper we present an approach whereby wavelets are used to obtain low-order estimates of first-order and second-order blterra kernels. Several constructions of tensorproduct wavelets have been employed for some%blterra kernel approximations. In this paper, a triangular wavelet basis is constructed for the representation of the triangular fonn of the second-order kernel. These wavelets are piecewise-constant, orthonormal, and are supported over the triangular domain over which the second-order kernel is defined. The well-known Haar wavelet is used concurrently for the identification of the first-order kernel. This kernel identification algorithm is demonstrated on a prototypical nonlinear oscillator. It is shown that accurate kemel estimates can be obtained in terms of a relatively small number of wavelet coefficients. It is also demonstrated that, for this particular system, the derived Volterra model is valid for input amplitudes below a specified bound. When the input amplitude exceeds this threshold, higher-order kernels are needed to adequately describe the system dynamics. Thus, the approach taken in this paper is applicable to a large class of nonlinear systems provided that the input excitation is sufficiently bounded.

Improved Quadrature Formulas for Boundary Integral Equations With Conducting or Dielectric Edge Singularities

In this paper we derive new two-dimensional (2-D) quadrature formulas for the discretization of boundary integral equations in the presence of conducting or dielectric edges. The proposed formulas allow us to exactly integrate polynomials of degree less than or equal to five, multiplied by an algebraic singular factor that diverges along one side of the triangular integration domain. This is the kind of singularity that occurs when physical edges are present in both conducting and dielectric bodies. Numerical tests are performed on the presented formulas, in order to validate the achieved improvement in accuracy, and examples are given of their application to the determination of radar cross-section of 3-D metallic objects.

The historical functional linear model

AbstractThe authors develop a functional linear model in which the values at time t of a sample of curves yi (t) are explained in a feed‐forward sense by the values of covariate curves xi(s) observed at times s ±.t. They give special attention to the case s ± [t — δ, t], where the lag parameter δ is estimated from the data. They use the finite element method to estimate the bivariate parameter regression function β(s, t), which is defined on the triangular domain s ± t. They apply their model to the problem of predicting the acceleration of the lower lip during speech on the basis of electromyographical recordings from a muscle depressing the lip. They also provide simulation results to guide the calibration of the fitting process.