On the non-degeneracy property of the longest-edge trisection of triangles

The longest-edge (LE) trisection of a triangle t is obtained by joining the two equally spaced points of the longest-edge of t with the opposite vertex. In this paper we prove that for any given triangle t with smallest interior angle τ>0, if the minimum interior angle of the three triangles obtained by the LE-trisection of t into three new triangles is denoted by τ₁, then τ₁?τ/c₁, where . Moreover, we show empirical evidence on the non-degeneracy property of the triangular meshes obtained by iterative application of the LE-trisection of triangles. If τ_n denotes the minimum angle of the triangles obtained after n iterative applications of the LE-trisection, then τ_n>τ/c where c is a positive constant independent of n. An experimental estimate of c≈6.7052025350 is provided.     

A SET OF SYMMETRIC QUADRATURE RULES ON TRIANGLES AND TETRAHEDRA

We present a program for computing symmetric quadrature rules on triangles and tetrahedra. A set of rules are obtained by using this program. Quadrature rules up to order 21 on triangles and up to order 14 on tetrahedra have been obtained which are useful for use in finite element computations. All rules presented here have positive weights with points lying within the integration domain.     

The construction of numerical integration rules of degree three for product regions

Numerical integration formulas in n-dimensional Euclidean space of degree three are discussed. In this paper, for the product regions a method is presented to construct numerical integration formulas of degree three with 2n real points and positive weights. The presented problem is a little different from those dealt with by other authors. All the corresponding one-dimensional integrals can be different from each other and they are also nonsymmetrical. In this paper an n-dimensional numerical integration problem is turned into n one-dimensional moment problems, which simplifies the construction process. Some explicit numerical formulas are given. Furthermore, a more generalized numerical integration problem is considered, which will shed light on the final solution to the third degree numerical integration problem.     

Finite element approximation with numerical integration for differential eigenvalue problems

Error estimates of the finite element method with numerical integration for differential eigenvalue problems are presented. More specifically, refined results on the eigenvalue dependence for the eigenvalue and eigenfunction errors are proved. The theoretical results are illustrated by numerical experiments for a model problem.     

A comparison of numerical integration rules for the meshless local Petrov–Galerkin method

The meshless local Petrov–Galerkin (MLPG) method is a mesh-free procedure for solving partial differential equations. However, the benefit in avoiding the mesh construction and refinement is counterbalanced by the use of complicated non polynomial shape functions with subsequent difficulties, and a potentially large cost, when implementing numerical integration schemes. In this paper we describe and compare some numerical quadrature rules with the aim at preserving the MLPG solution accuracy and at the same time reducing its computational cost.     

Error bounds for quasi-Monte Carlo integration with uniform point sets

We establish new error bounds for quasi-Monte Carlo integration for node sets with a special kind of uniformity property. The methods of proving these error bounds work for arbitrary probability spaces. Only the bounds in terms of the modulus of continuity of the integrand require also the structure of a metric space.     

On the asymptotic integration of nonlinear differential equations

The aim of the present paper is twofold. Firstly, the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations: the class of second order equations with Bihari-like nonlinearity. Secondly, some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations.     

Parameter extension for combined hybrid finite element methods and application to plate bending problems

Based on a weighted average of the modified Hellinger-Reissner principle and its dual, the combined hybrid finite element (CHFE) method was originally proposed with a combination parameter limited in the interval (0, 1). In actual computation this parameter plays an important role in adjusting the energy error of discretization models. In this paper, a novel expression of the combined hybrid variational form is used to show the relationship between the resultant method and some Galerkin/least-squares stabilized finite scheme for plate bending problems. The choice of combination parameter is then extended to (−∞, 0) ? (0, 1). Existence, uniqueness and convergence of the solution of discrete schemes are proved, and the advantage of the parameter extension in computation is discussed. As an application, improvement of Adini’s rectangular element by the CHFE approach is performed.     

A fourth degree integration formula for the n-dimensional simplex

A fourth degree integration formula is given for the n-dimensional simplex for all n2, which is invariant under the group G of all affine transformations of T_n onto itself. The formula contains (n²+5n+6)/2 nodes.     

Cost-effective modeling strategies for the analysis of spliced steel connections

The paper presents several cost-effective modeling strategies that can be used by structural engineers in practice to determine the stresses in the spliced members. The computational efficiency and the modeling effort required for the several modeling options are also discussed. The deformation mechanisms and load transfer for several types of connections are illustrated. Optimization techniques are also presented to economize the computer time for connections with large number of bolts. Results are presented to compare the accuracy of several modeling strategies commonly used in practice. It is shown that for eccentric connections, the flexural bending largely affect the maximum tensile and compressive stresses within the joint. The difference may reach up to 54%. Finally, Experimental comparisons are made with the numerical procedures for typical connection model.     

Some algorithms for numerical quadrature using the derivatives of the integrand in the integration interval

Some quadrature formulae using the derivatives of the integrand are discussed. As special cases are obtained generalizations of both the ordinary and the modified Romberg algorithms. In all cases the error terms are expressed in terms of Bernoulli polynomials and functions.     

Quasi-Monte Carlo methods for integration of functions with dominating mixed smoothness in arbitrary dimension

In a celebrated construction, Chen and Skriganov gave explicit examples of point sets achieving the best possible _L2$L_2$-norm of the discrepancy function. We consider the discrepancy function of the Chen–Skriganov point sets in Besov spaces with dominating mixed smoothness and show that they also achieve the best possible rate in this setting. The proof uses a b$b$-adic generalization of the Haar system and corresponding characterizations of the Besov space norm. Results for further function spaces and integration errors are concluded.     

On the practical application of the modified romberg algorithm

Experience gained in the use of the modified Romberg algorithm is reported. An alternative form of the algorithm based on a cosine transformation of Euler-MacLaurin's and Euler's second formula is discussed. Some examples where the error estimate incorporated in the algorithm partly fails are given. Finally an ALGOL procedure combining the algorithm discussed in a previous paper of the author [1] and the modification discussed in this paper is described.     

Stability and accuracy of DQ-based step-by-step integration methods for structural dynamics

In this paper, a new DQ-based compact step-by-step integration method is proposed. Analytical proof of stability is presented. The method is unconditionally stable and not affected by algorithmic damping. Besides, sixth-order convergence can be achieved. A classical nonlinear model is studied as example application. Compared to other similar procedures, this new method provides accurate results, even if the step size is relatively large.     

On numerical improvement of Gauss–Legendre quadrature rules

Among all integration rules with n points, it is well-known that n-point Gauss–Legendre quadrature rule∫₋₁¹f(x) dx∑_i=1ⁿw_if(x_i)has the highest possible precision degree and is analytically exact for polynomials of degree at most 2n−1, where nodes x_i are zeros of Legendre polynomial P_n(x), and w_i's are corresponding weights.In this paper we are going to estimate numerical values of nodes x_i and weights w_i so that the absolute error of introduced quadrature rule is less than a preassigned tolerance ε₀, say ε₀=10⁻⁸, for monomial functionsf(x)=x^j, j=0,1,…,2n+1.(Two monomials more than precision degree of Gauss–Legendre quadrature rules.) We also consider some conditions under which the new rules act, numerically, more accurate than the corresponding Gauss–Legendre rules. Some examples are given to show the numerical superiority of presented rules.     

Exact integration formulas for the finite volume element method on simplicial meshes

This article considers the technological aspects of the finite volume element method for the numerical solution of partial differential equations on simplicial grids in two and three dimensions. We derive new classes of integration formulas for the exact integration of generic monomials of barycentric coordinates over different types of fundamental shapes corresponding to a barycentric dual mesh. These integration formulas constitute an essential component for the development of high‐order accurate finite volume element schemes. Numerical examples are presented that illustrate the validity of the technology. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

On the numerical evaluation of Cauchy transforms

In this paper we consider simple methods for the reconstruction of the Cauchy transform over a curve when an explicit parametrization of the latter is not provided. The methods consist of replacing the parametrization of the curve by piecewise polynomial interpolation followed by the use of Newton-Cotes type formulae for the integration. The order of convergence of the resulting quadrature is higher than would be expected on the basis of considerations involving just interpolation theory, provided that the Cauchy transform is evaluated at known nodes on the curve. These results allow the calculation of the Cauchy transform at other points with the same accuracy if this scheme is followed by an interpolatory formula of sufficiently high accuracy.     

Explicit through-thickness integration schemes for geometric nonlinear analysis of laminated composite shells

Degenerated shell elements were found to be attractive in solving homogeneous shell problems. Direct extension of the same to layered shells becomes computationally inefficient as, in the computation of element matrices, 3-D numerical integration in each layer and summation over the layers have to be carried out. In order to make the formulation efficient, explicit through-thickness schemes have been devised for linear problems. The present paper deals with the extension of the same to geometric nonlinear problems with options of small and large rotations. The explicit through-thickness integration becomes possible due to the assumption on the variation of inverse Jacobian through the thickness. Depending on the assumptions, three different schemes under large and small rotation cases have been presented and their relative numerical accuracy and computational efficiency have been evaluated. It has been observed that there is no sacrifice on the numerical accuracy due to the assumptions leading to the explicit through-thickness integration, but at the same time, there is considerable saving in the computational time. The computational efficiency improves as the number of layers in the laminate increases. The small rotation formulation with the assumption of linear variation of Jacobian inverse across the thickness and based on further approximation regarding certain submatrices is seen to be computationally efficient, as applied to geometric nonlinear layered shell problems.