任意形状孔口单边裂纹问题的边界配位解法

本文提出了一组复应力函数,采用边界配位方法对不同形状孔口(包括圆、椭圆、矩形及菱形孔口)的单边裂纹平板的应力强度因子进行了计算.计算结果表明,对长度和宽度远大于孔口和裂纹几何尺寸的试件,配位法与用其他方法所得的无限大板含圆或椭圆孔边裂纹问题的解符合得很好.同时,对其他孔口问题,特别是有限大板情形,本文给出了一系列计算结果.本文所提出的函数及计算过程可以应用于任意形状孔口单边裂纹平板的计算.     

Solution of some two-dimensional problems of elasticity

Using the method of boundary elements, we obtain numerical solutions of two-dimensional (plane deformation) boundary-value problems on the elastic equilibrium of infinite and finite homogeneous isotropic bodies having elliptic holes with cracks and cuts of finite length. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.     

Hexahedral finite element with an open crack

A method of creating the stiffness matrix of a hexahedral eight-node finite element with a single, nonpropagating, transverse, one-edge crack at half of its length is presented in this paper. The crack was modelled by adding an additional flexibility matrix to that of the noncracked element. The terms of the additional matrix have been calculated by use of the laws of fracture mechanics. Employing the elaborated element a numerical test has been worked out, the results of which are compared with the data of analytical solutions accessible in the literature, and a high conformity with them has been obtained. The element presented in the paper may be applied to the static and dynamic analysis of different types of structural elements with material defects in the form of cracks. The described method of creating the stiffness matrix of the element allows to create different kinds of finite elements with cracks provided that the stress intensity factors for a given type of crack are known.     

Elliptic systems of partial differential equations and the finite element method

An existence and uniqueness theorem is established for finite element solutions of elliptic systems of partial differential equations. To establish this result, an extension of Gårding's inequality is obtained which is valid for functions that do not necessarily vanish on the boundary of the region. To accomplish this extension, a stronger ellipticity condition, called very strong ellipticity, is defined with various necessary and sufficient conditions given.     

Global existence and large-time behavior of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force

In this paper, we consider the equations of Magnetohydrodynamics with Coulomb force which is of hyperbolic–parabolic–elliptic mixed type. By constructing the approximate solutions to the modified system with an artificial pressure term added, global existence of finite energy weak solutions is established via the weak convergence method. More careful argument has been paid to overcome the new difficulty arising from the Poisson term of Coulomb force in two dimensions when the adiabatic exponent is close to one. We also investigate the large-time behavior of such weak solutions after discussing the regularity and uniqueness of solutions to the stationary problem.     

Special finite elements including stress concentration effects of a hole

Special finite elements are developed for efficient evaluation of stress concentration around a hole in complex structures. The complex variable formulation is used to derive a special set of stress functions which embody the stress concentration effects of a hole. The stress functions in combination with an independent displacement field assumed along the element boundary are used to construct the special elements with the hybrid displacement finite element method. Several numerical examples are presented to show that the used of special finite elements to model critical regions around a hole, together with conventional finite elements to model other regions away from the hole, is not only very convenient but also highly accurate.     

The weighted Ritz‐Galerkin method for elliptic boundary value problems on unbounded domains

Recently Babus?ka‐Oh introduced the method of auxiliary mapping (MAM) which efficiently handles elliptic boundary value problems containing singularities. In this paper, a special weighted residue method, the Weighted Ritz‐Galerkin Method (WRGM), is investigated by introducing special weight functions. Together with this method, MAM is modified to yield highly accurate finite element solutions to general elliptic boundary value problems on the exterior of bounded domains at low cost. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 301–326, 2003.     

Computer-based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace's,Poisson's,and the elasticity equations

In this article we address the problem of the existence of superconvergence points for finite element solutions of systems of linear elliptic equations. Our approach is quite different from all other studies of superconvergence. We prove that the existence of superconvergence points can be guaranteed by a numerical algorithm, which employs a finite number of operations (provided that there is no roundoff-error). By employing this approach, we can reproduce all known results on superconvergence of finite element solutions for linear elliptic problems and we can obtain many new results. Here, in particular, we address the problem of the superconvergence points for the gradient of finite element solutions of Laplace's and Poisson's equations and we show that the sets of superconvergence points are very different for these two cases. We also study the superconvergence of the components of the gradient of the displacement, the strain and stress for finite element solutions of the equations of elasticity. For Laplace's and Poisson's equations (resp. the equations of elasticity), we consider meshes of triangular as well as square elements of degree p, 1 ? p ? 7 (resp. 1 ? p ? 4). For the meshes of triangular elements we investigate the effect of the geometry of the mesh by considering four mesh patterns that typically occur in practical meshes, while in the case of square elements, we study the effect of the element type (tensor-product, serendipity, or other). © 1996 John Wiley & Sons, Inc.     

A unified approach to a posteriori error estimation using element residual methods

Summary This paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations. It is shown that, by solving appropriate local residual type problems, one can obtain upper bounds on the error in the energy norm. Moreover, in the special case of adaptiveh-p finite element analysis, the estimator will also give a realistic estimate of the error. A key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems. The work extends and combines several existing methods to the case of fullh-p finite element approximation on possibly irregular meshes with, elements of non-uniform degree. As a special case, the analysis proves a conjecture made by Bank and Weiser [Some A Posteriori Error Estimators for Elliptic Partial Differential Equations, Math. Comput.44, 283–301 (1985)].     

THE LOWER APPROXIMATION OF EIGENVALUE BY LUMPED MASS FINITE ELEMENT METHOD

In the present paper, we investigate properties of lumped mass finite element method (LFEM hereinafter) eigenvalues of elliptic problems. We propose an equivalent formulation of LFEM and prove that LFEM eigenvalues are smaller than the standard finite element method (SFEM hereinafter) eigenvalues. It is shown, for model eigenvalue problems with uniform meshes, that LFEM eigenvalues are not greater than exact solutions and that they are increasing functions of the number of elements of the triangulation, and numerical examples show that this result equally holds for general problems.     

Uniform convergence of finite volume element method with Crouzeix–Raviart element for non-self-adjoint and indefinite elliptic problems

In this paper, we consider the finite volume element method based on the Crouzeix–Raviart element and prove the existence, uniqueness and uniform convergence of the finite volume element approximations for the non-self-adjoint and indefinite elliptic problems under minimal elliptic regularity assumption.     

p－version有限元的快速高精度算法

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Free bending vibrations of an isotropic plate with free holes

A solution of the problem of the free bending vibrations of a thin finite isotropic plate with stiffly clamped outer edge is obtained by using functions of a complex variable. Results of numerical investigations are presented for an elliptic plate with acircular hole.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 67–70, 1989.     

Three positive solutions for nonlinear elliptic equations in finite strip with hole

In this paper, we study the effect of the domain on the multiplicity of positive solutions for the nonlinear elliptic equation. Using the Ekeland variational principle and the center mass function, we prove that there are at least three positive solutions for the nonlinear elliptic equation in finite strip with holes.     

SUPERCONVERGENCE OF LEAST-SQUARES MIXED FINITE ELEMENTS FOR ELLIPTIC PROBLEMS ON TRIANGULATION

In this paper, we present the least-squares mixed finite element method and investigate superconvergence phenomena for the second order elliptic boundary-value problems over triangulations. On the basis of the L~2-projection and some mixed finite element projections, we obtain the superconvergence result of least-squares mixed finite     

Development and Analysis of Higher Order Finite Volume Methods over Rectangles for Elliptic Equations

Currently used finite volume methods are essentially low order methods. In this paper, we present a systematic way to derive higher order finite volume schemes from higher order mixed finite element methods. Mostly for convenience but sometimes from necessity, our procedure starts from the hybridization of the mixed method. It then approximates the inner product of vector functions by an appropriate, critical quadrature rule; this allows the elimination of the flux and Lagrange multiplier parameters so as to obtain equations in the scalar variable, which will define the finite volume method. Following this derivation with different mixed finite element spaces leads to a variety of finite volume schemes. In particular, we restrict ourselves to finite volume methods posed over rectangular partitions and begin by studying an efficient second-order finite volume method based on the Brezzi–Douglas–Fortin–Marini space of index two. Then, we present a general global analysis of the difference between the solution of the underlying mixed finite element method and its related finite volume method. Then, we derive finite volume methods of all orders from the Raviart–Thomas two-dimensional rectangular elements; we also find finite volume methods to associate with BDFM ₂ three-dimensional rectangles. In each case, we obtain optimal error estimates for both the scalar variable and the recovered flux.     

任意形状孔口双边裂纹平板的应力强度因子计算

本文采用Muskhclishvili弹性力学的复变函数和边界配位方法对不同形状孔口双边裂纹问题进行了研究,计算了圆孔、椭圆孔、矩形孔、菱形孔等不同形状孔口双边裂纹,以及Ⅰ型和复合型等不同类型断裂试件的应力强度因子,本文方法简单方便,精度较高,与某些已有计算结果的问题比较,本文方法所得的结果是令人满意的.同时,本方法可以应用于不同几何形状和加载条件下的孔口双边裂纹有限大板的计算,是解这一类问题的一致有效方法.     

Radiation and Scattering by Complex Conformal Antennas on a Circular Cylinder

A technique for characterizing and designing complex conformal antennas flush-mounted on a singly-curved surface is presented. This approach is based on the hybrid finite element–boundary integral (FE–BI) method. A related method was proposed in the past utilizing cylindrical-shell finite element and roof-top rectangular basis functions for the boundary integral. Although that method proved very powerful for analyzing cylindrical–rectangular patch arrays flush-mounted to a circular cylinder, the requirement for uniform meshing in the aperture ultimately limited its usefulness. In this present formulation, tetrahedral elements are used to expand the volumetric electric fields while similar basis functions are used for the boundary integral. The curvature of the aperture is explicitly included via the use of the circular cylinder dyadic Green's function. After presentation of the formulation and validation using several well-understood examples, an example is presented that illustrates the capabilities of this method for modeling complex conformal antennas heretofore not examined by rigorous methods due to inherent limitations of the various published methods.     

CBS constants for multilevel splitting of graph-Laplacian and application to preconditioning of discontinuous Galerkin systems

The goal of this work is to derive and justify a multilevel preconditioner of optimal arithmetic complexity for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. Our approach is based on the following simple idea given in [R.D. Lazarov, P.S. Vassilevski, L.T. Zikatanov, Multilevel preconditioning of second order elliptic discontinuous Galerkin problems, Preprint, 2005]. The finite element space of piece-wise polynomials, discontinuous on the partition , is projected onto the space of piece-wise constant functions on the same partition that constitutes the largest space in the multilevel method. The discontinuous Galerkin finite element system on this space is associated to the so-called “graph-Laplacian”. In 2-D this is a sparse M-matrix with -1 as off diagonal entries and nonnegative row sums. Under the assumption that the finest partition is a result of multilevel refinement of a given coarse mesh, we develop the concept of hierarchical splitting of the unknowns. Then using local analysis we derive estimates for the constants in the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality, which are uniform with respect to the levels. This measure of the angle between the spaces of the splitting was used by Axelsson and Vassilevski in [Algebraic multilevel preconditioning methods II, SIAM J. Numer. Anal. 27 (1990) 1569–1590] to construct an algebraic multilevel iteration (AMLI) for finite element systems. The main contribution in this paper is a construction of a splitting that produces new estimates for the CBS constant for graph-Laplacian. As a result we have a preconditioner for the system of the discontinuous Galerkin finite element method of optimal arithmetic complexity.     

Frequency response analysis of laminated composite beams

Conclusions The modeling of laminated composite beams has been derived systematically from the three-dimensional elasticity relations. The correctness of the solution found by using the present finite element model is verified by comparison with the results obtained by analytical solutions and other results presented in the literature. Numerical results indicate that the present technique can given accurate results for frequency response analysis for laminated composite beams. Loss factors of structures obtained by the method of complex eigenvalues and the direct frequency response method exhibit very good agreement. Optimum design of a laminated composite beam by the finite element method and the method of experiment planning has been successfully presented.Published in Mekhanika Kompozitnykh Materialov, Vol. 30, No. 5, pp. 664–674, September–October, 1994.