Precise integration methods based on Lagrange piecewise interpolation polynomials

This paper introduces two new types of precise integration methods for dynamic response analysis of structures, namely, the integral formula method and the homogenized initial system method. The applied loading vectors in the two algorithms are simulated by the Lagrange piecewise interpolation polynomials based on the zeros of the first Chebyshev polynomial. Developed on the basis of the integral formula and the Lagrange piecewise interpolation polynomial and combined with the recurrence relationship of some key parameters in the integral computation suggested in this paper with the solving process of linear algebraic equations, the integral formula method has been set up. On the basis of the Lagrange piecewise interpolation polynomial, and transforming the non‐homogenous initial system into the homogeneous dynamic system, the homogenized initial system method without dimensional expanding is presented; this homogenized initial system method avoids the matrix inversion operation and is a general homogenized high‐precision direct integration scheme. The accuracy of the presented time integration schemes is studied and is compared with those of other commonly used schemes; the presented time integration schemes have arbitrary order of accuracy, wider application and are less time consuming. Two numerical examples are also presented to demonstrate the applicability of these new methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Approximating three-dimensional steady-state potential flow problems using two-dimensional complex polynomials

A new advance in the Complex Variable Boundary Element Method, or CVBEM, is its extension to three-dimension (3D). This advance breaks down the barrier of limiting CVBEM models to two-dimensional (2D) problems, and also opens the door to solving 3D potential problems with other 2D numerical analogs. In this paper, a 3D analog is developed using 2D basis functions of the complex analytic polynomial type. Thus, 2D complex polynomials are being applied to 3D potential problems. This new advance may be of interest to those involved in applied mathematics, complex variables, boundary elements, and numerical solution of partial differential equations of the Laplace of Poisson type.     

Using Lagrange principle for solving two-dimensional integral equation with a positive kernel

This article is devoted to a Lagrange principle application to an inverse problem of a two-dimensional integral equation of the first kind with a positive kernel. To tackle the ill-posedness of this problem, a new numerical method is developed. The optimal and regularization properties of this method are proved. Moreover, a pseudo-optimal error of the proposed method is considered. The efficiency and applicability of this method are demonstrated in a numerical example of an image deblurring problem with noisy data.     

Noisy interpolation of sparse polynomials in finite fields

We consider a polynomial analogue of the hidden number problem introduced by Boneh and Venkatesan, namely the sparse polynomial noisy interpolation problem of recovering an unknown polynomial f(X) ∈ [X] with at most w non-zero terms from approximate values of f(t) at polynomially many points t ∈ selected uniformly at random. We extend the polynomial time algorithm of the first author for polynomials f(X) of sufficiently small degree to polynomials of almost arbitrary degree. Our result is based on a combination of some number theory tools such as bounds of exponential sums and the number of solutions of congruences with the lattice reduction technique. The new idea is motivated by Waring's problem and uses a recent bound on exponential sums of Cochrane, Pinner, and Rosenhouse.     

周期函数的Lagrange型插值逼近

求文构造了一类Lagrange型插值三角多项式．并给出以高阶模为阶的逼近偏差估计．     

Simple instability criteria for a family of complex‐coefficient polynomials

Abstract

In this paper, the A[r, R) stability and A[r, R) instability are introduced. The instability of a family of complex‐coefficient polynomials is investigated. Simple instability criteria are derived to guarantee the instability of such polynomials. Finally, a numerical example is provided to illustrate the main results.     

Fundamental solutions for variable density two-dimensional elastodynamic problems

Fundamental solutions in the form of free-space Green's functions are developed for a class of two-dimensional, variable density elastodynamic problems. These fundamental solutions are evaluated by means of a coordinate transformation based on conformal mapping in conjunction with wave decomposition, which allows for both vertical and lateral heterogeneities, and can be used within the context of a boundary integral equation formulation analogous to that originally proposed by Cruse and Rizzo (J Math Anal Appl 22 (1968) 244). Finally, a numerical example serves to illustrate the methodology developed herein.     

Avoiding negative elastic moduli when using Lagrange interpolation for material grading in finite element analysis

Polynomial interpolations, one of the most common interpolations used in finite element methods (FEMs), are a workhorse of many FEM codes. These interpolations are readily available for all kinds of elements, and using them for modeling the variation of elastic moduli in graded elements is thus both convenient and natural. Yet, like all polynomial interpolations, they can be prone to oscillations that can result in regions of negative elastic modulus in the element, even with only positive nodal values of elastic moduli. The result of these negative modulus regions, even if the region is small, can be unexpected singularities in the solution. This defeats the purpose of using polynomial interpolations for capturing material grading in the element. We demonstrate the issue using three-node quadratic Lagrange interpolations of material grading in otherwise isoparametric p-type elements and show how to avoid this problem.     

Mixed interpolation in primitive variable finite element formulations for incompressible flow

This paper shows that mixed interpolation is required because of the nature of the pressure terms in the equations. These lead to finite element equations which in many circumstances do not uniquely determine the pressure, if the same interpolations are used for pressure and velocities. Several new combinations of pressure and velocity interpolation are analysed with the aid of a novel diagrammatic technique. In particular we consider some very interesting combinations in which, over nearly all the flow, the pressure and velocity are approximated on elements by biquadratic polynomials which are continuous across element boundaries. The theory of this paper is shown to be in complete agreement with numerical experiment.     

A meshless local natural neighbour interpolation method for analysis of two-dimensional piezoelectric structures

A novel meshless method applied to solve two-dimensional piezoelectric structures is presented and discussed in this paper. It is called meshless local natural neighbour interpolation (MLNNI) method, which is derived from the generalized meshless local Petrov–Galerkin (MLPG) method as a special case. In the present method, nodal points are spread on the analysed domain and each node is surrounded by a polygonal sub-domain, which can be conveniently constructed with Delaunay tessellations. The spatial variation of the displacements and the electric potential are interpolated by the natural neighbour interpolation. As the shape functions so constructed possess the delta function property, the essential boundary conditions can be imposed by directly substituting the corresponding terms in the system of equations. Furthermore, the usage of three-node triangular FEM shape functions as test functions reduces the order of integrands involved in domain integrals. Numerical examples are presented at the end to demonstrate the applicability and accuracy of the present approach in analysing two-dimensional piezoelectric structures.     

Solution of the two-dimensional wave equation by using wave polynomials

The paper demonstrates a specific power-series-expansion technique to solve approximately the two-dimensional wave equation. As solving functions (Trefftz functions) so-called wave polynomials are used. The presented method is useful for a finite body of certain shape geometry. Recurrent formulas for the wave polynomials and their derivatives are obtained in the Cartesian and polar coordinate system. The accuracy of the method is discussed and some examples are shown.     

A complex variable solution of two-dimensional heat conduction of composites reinforced with periodic arrays of cylindrically orthotropic fibers

Problems of two-dimensional steady-state heat conduction for composites with doubly periodic arrays of cylindrically orthotropic fibers are dealt with. A new complex variable method is presented by introducing an appropriate coordinate transformation to convert the governing differential equation into a harmonic one, and the eigenfunction expansions of the field variables in a unit cell are derived. Then by using a generalized variational functional which absorbs the periodicity condition, an eigenfunction expansion–variational method based on a unit cell is developed to solve such problems. A convergence analysis and a comparison with finite element calculations are conducted to demonstrate the correctness and efficiency of the present method. A discussion is made about the effects of the cylindrical orthotropy of the fiber and the existence of the isotropic core in the fiber on the effective conductivity of the composite. An engineering equivalent parameter, which reflects the overall influence of the thermal conductivities of the matrix and fibers as well as the interfacial characteristic on the effective thermal conductivity of the composite, is found. It is shown that the present first-order approximation of the effective thermal conductivity of the composite can be written in a unified formula for different microstructural characteristics and possesses a good engineering accuracy.     

Solution of a two-dimensional heat-conduction problem for a geometrically complex domain by an integrointerpolation method

A methodology is proposed for the construction of an algorithm to solve heat transfer problems for spatial domains of complex geometric shape.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 56, No. 3, pp. 464–471, March, 1989.     

复杂地质曲面三维插值—逼近拟合构造方法

针对水利水电工程多源地质数据的特点,充分考虑了地质精度要求、曲面连续性和数据存储量等多方面的均衡,提出并实现了基于NURBS(non-uniform rational B-splines,非均匀有理样条曲线)技术的复杂地质曲面插值—逼近拟合构造方法。该方法对于工程关键区域集中且均匀分布的原始数据,采用NURBS蒙皮插值方法,使曲面严格通过这些数据点;对于周边区域分布离散的数据,采用NURBS逼近拟合方法,使曲面在给定精度下充分逼近原始数据;最后对整体曲面的地质结构合理性、几何性和精度进行检查分析和调整。实例表明,该方法所构造的地质曲面能满足地质工程师的实际需要,并能为进一步的三维地质建模提供基础。     

A point interpolation mesh free method for static and frequency analysis of two-dimensional piezoelectric structures

 A mesh free method called point interpolation method (PIM) is presented for static and mode-frequency analysis of two-dimensional piezoelectric structures. In the present method, the problem domain and its boundaries are represented by a set of properly scattered nodes. The displacements and the electric potential of a point are interpolated by the values of nodes in its local support domain using shape functions derived based on a point interpolation scheme. Techniques are discussed to surmount the singularity of the moment matrix. Variational principle together with linear constitutive piezoelectric equations is used to establish a set of system equations for arbitrary-shaped piezoelectric structures. These equations are assembled for all quadrature points and solved for displacements and electric potentials. A polynomial PIM program has been developed in MATLAB with matrix triangularization algorithm (MTA), which automatically performs a proper node enclosure and a proper basis selection. Examples are also presented to demonstrate the accuracy and stability of the present method and their results are compared with the conventional FEM results from ABAQUS as well as the analytical or experimental ones. Received: 6 February 2002 / Accepted: 5 August 2002     

Two-dimensional elastodynamics by the time-domain boundary element method: Lagrange interpolation strategy in time integration

This work presents a time-truncation scheme, based on the Lagrange interpolation polynomial, for the solution of two-dimensional elastodynamics by the time-domain boundary element method. The reduction in the number of assembled matrices, maintaining a compromise between accuracy and computational economy and efficiency, is the main purpose of the present work. In order to verify the accuracy of the proposed formulation, two examples are presented and discussed.     

A dynamic programming-based heuristic for the variable sized two-dimensional bin packing problem

This paper addresses a variable sized two-dimensional bin packing problem. We propose two heuristics, H1 and H2, stemming from the dynamic programming idea by aggregating states to avoid the explosion in the number of states. These algorithms are elaborated for different purposes: H1 builds a general packing plan for items, while H2 can provide solutions by considering a variety of customer demands, such as guillotine cutting style and rotation of items. The performance of both algorithms is evaluated based on randomly generated instances reported in the literature by comparing them with the lower bounds and optimal solutions for identical bins. Computational results show that the average gaps are 8.97% and 13.41%, respectively, for H1 and H2 compared with lower bounds, and 5.26% and 6.26% compared with optimal solutions for identical bins. We also found that we can save 6.67% of space, on average, by considering variable sized bins instead of a bin packing problem with identical bins.     

The equations of Lagrange written for a non-material volume

Summary The Lagrange equations are extended with respect to a non-material volume which instantaneously coincides with some material volume of a continuous body. The surface of the non-material volume is allowed to move at a velocity which is different from the velocity of the material surface. The non-material volume thus represents an arbitrarily moving control volume in the terminology of fluid mechanics. The extension of the Lagrange equations to a control volume is derived by using the method of fictitious particles. Within a continuum mechanics based framework, it is assumed that, the instantaneous positions of both, the original particles included in the material volume, and the fictitious particles included in the control volume, are given as function of their positions in the respective reference configurations, of a set of time-dependent generalized coordinates, and of time. The corresonding spatial formulations are also assumed to be available. Imagining that the fictitious particles do transport the density of kinetic energy of the original particles, the partial derivatives of the total kinetic energy included in the material volume with respect to generalized coordinates and velocities are related to the respective partial derivatives of the total kinetic energy contained in the control volume. Hence follow the Lagrange equations for a control volume by substituting the above relations into the classical formulations for a material volume. In the present paper, holonomic problems are considered. The correction terms in the newly derived version of the Lagrange equations contain the flux of kinetic energy appearing to be transported through the surface of the control volume. This flux comes into the play in the form of properly formulated partial derivatives. Our version of the Lagrange equations is tested using the rocket equation and a folded falling string as illustrative examples.     

Numerical method for solving heat-conduction problems for two-dimensional bodies of complex shape

A finite-difference scheme is described for a curvilinear orthogonal net which permits the use of a single algorithm for calculating bodies of various shapes.Notation x, y independent variables - u, v orthogonal coordinates - F(w)=F(u + iv) function of a complex variable - g(u,v)= F(w)/w Jacobian of transformation from (u,v) to (x,y) - thermal conductivity - c volumetric heat capacity - Q heat release per unit volume - T temperature - f value of temperature on boundary of region - time - L, L₁, L₂ differential operators - (u,v) solution of differential problem in canonical region - ^j, ₁ ^j , ₂ ^j , t^J, t ₁ ^j , t ₂ ^j network functions in canonical region - ^j, t^*j solutions of difference problems using rectangular and orthogonal nets respectively - {u_i, v_k} rectangular net in canonical region G - {x_i,k, y_i, k} orthogonal net in given region G^* - u_i, v_k dimensions of cell of rectangular net - u_i,v _i,k dimensions of cell of orthogonal net - h, maximum dimension of cell for rectangular and orthogonal nets respectively - ₁, ₂, difference operators for rectangular and orthogonal nets - A, B, C, D, A^*, B^*, C^*, D^* coefficients of difference scheme for rectangular net - D, Ã, B coefficients of difference scheme for orthogonal net Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 3, pp. 503–509, March, 1981.