九参数广义协调元的收敛性

众知周所,解板弯曲问题的Zienkiewicz不协调三次元只对特殊的单元剖分才收敛.但由于这种元采用单元顶点的函数值及二个一阶导数值作为节点参数,计算简单,总体自由度少,所以相继出现一些对Zienkiewicz元的改进形式,使之对任意剖分均收敛,如拟协调元,TRUNC元,Specht元.对这些元的分析见[7—9].最近龙     

Polynomial of Degree Four Interpolation on Triangles

A method for constructing the $C^1$ piecewise polynomial surface of degree four on triangles is presented in this paper. On every triangle, only nine interpolation conditions, which are function values and first partial derivatives at the vertices of the triangle, are needed for constructing the surface.     

Stability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains

In this paper we show that the h-p spectral element method developed in [3,8,9] applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska-Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable. An erratum to this article is available at .     

Graves triads in the geometry of the triangle

A Graves triad is a cyclic triad of triangles, each circumscribing the next, thus forming a Pappus configuration. In the Euclidean geometry of the triangle the points of contact of the inscribed and escribed circles, the intersections of internal and external angle bisectors with opposite sides, and the feet of altitudes give rise to sixteen Graves triads, in four sets of four each. Each set of four is associated with a rectangular hyperbola through the vertices, orthocentre, in-centre, Gergonne point, and Nagel point, or their external analogues, and having its centre at a Feuerbach point.     

基于层次T网格的分片孔斯曲面重构

本文提出一种基于任意层次T网格的多项式(PHT)样条空间$S(3,3,1,1,T)$的一个新的曲面重构算法.该算法由分片插值于层次T网格上每个小矩形单元对应4个顶点的16个参数的孔斯曲面形式给出.对于一个给定的T网格和相应基点处的几何信息(函数值,两个一阶偏导数和混合导数值),可得到与$S(3,3,1,1,T)$的PHT样条曲面相同的结果,且曲面表达形式更简单,同时,在离散数据点的曲面拟合中,我们给出了自适应的曲面加细算法.数值算例显示,该自适应算法能够有效的拟合离散数据点.     

On random perturbations of Hamiltonian systems with many degrees of freedom

We consider a class of random perturbations of Hamiltonian systems with many degrees of freedom. We assume that the perturbations consist of two components: a larger one which preserves the energy and destroys all other first integrals, and a smaller one which is a non-degenerate white noise type process. Under these assumptions, we show that the long time behavior of such a perturbed system is described by a diffusion process on a graph corresponding to the Hamiltonian of the system. The graph is homeomorphic to the set of all connected components of the level sets of the Hamiltonian. We calculate the differential operators which govern the process inside the edges of the graph and the gluing conditions at the vertices.     

A quadrilateral Morley element for biharmonic equations

In this paper, we propose a Morley-type finite element for quadrilateral meshes to solve biharmonic problems. For each quadrilateral $Q$ , the finite element space is defined by the span of $P_2(Q)$ plus two functions in $P_3(Q)$ . Each of the cubic polynomials is the product of a pair of equations of opposite edges and the equation of the bimedian between them. The degrees of freedom consist of the values at vertices and integrals of normal derivatives over edges. Optimal orders of convergence are proved both in discrete $H^2$ and $H^1$ seminorms. Several numerical tests confirm the convergence analysis.     

Quad mesh generation for k-sided faces and hex mesh generation for trivalent polyhedra

Two mesh generation methods are presented in this paper: one for quadrilateral meshing of faces with convex vertices but an arbitrary number of sides, the other hexahedral meshing of polyhedra with convex edges, vertices connected to three edges and an arbitrary number of faces each of which has at least three sides. Efficient mapping methods for determining nodal positions are developed based on an extension to transfinite mapping and it is shown that these are identical to those obtained by iterative isoparametric smoothing.     

双参数13参四边形板元

本文提出了一个双参数13参四边形板元, 其中一套参数按照广义分片检验要求选取, 另一套真正节点参数按照使离散系统更易解的要求选取,通过数值方法将一套节点参数线性过渡到另一套, 并且证明了该元的收敛性, 给出算例加以验证.     

Calculation of Stresses and Consistent Tangent Moduli from Automatic Differentiation of Hyerelastic Strain Energy Functions Through the Use of Hyper Dual Numbers

Many materials as e.g. engineering rubbers, polymers and soft biological tissues are often described by hyperelastic strain energy functions. For their finite element implementation the stresses and consistent tangent moduli are required and obtained mainly in terms of the first and second derivative of the strain energy function. Depending on its mathematical complexity in particular for anisotropic media the analytic derivatives may be expensive to be calculated or implemented. Then numerical approaches may be a useful alternative reducing the development time. Often-used classical finite difference schemes are however quite sensitive with respect to perturbation values and they result in a poor accuracy. The complex-step derivative approximation does never suffer from round-off errors, cf. [1], [2], but it can only provide first derivatives. A method which also provides higher order derivatives is based on hyper dual numbers [3]. This method is independent on the choice of perturbation values and does thus neither suffer from round-off errors nor from approximation errors. Therefore, here we make use of hyper dual numbers and propose a numerical scheme for the calculation of stresses and tangent moduli which are almost identical to the analytic ones. Its uncomplicated implementation and accuracy is illustrated by some representative numerical examples. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes

We consider an algorithm called FEMWARP for warping triangular and tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a two- or three-dimensional domain defined by a boundary mesh (segments in one dimension or triangles in two dimensions) that has a volume mesh (triangles in two dimensions or tetrahedra in three dimensions) in its interior. It also takes as input a prescribed movement of the boundary mesh. It computes as output updated positions of the vertices of the volume mesh. The first step of the algorithm is to determine from the initial mesh a set of local weights for each interior vertex that describes each interior vertex in terms of the positions of its neighbors. These weights are computed using a finite element stiffness matrix. After a boundary transformation is applied, a linear system of equations based upon the weights is solved to determine the final positions of the interior vertices. The FEMWARP algorithm has been considered in the previous literature (e.g., in a 2001 paper by Baker). FEMWARP has been successful in computing deformed meshes for certain applications. However, sometimes FEMWARP reverses elements; this is our main concern in this paper. We analyze the causes for this undesirable behavior and propose several techniques to make the method more robust against reversals. The most successful of the proposed methods includes combining FEMWARP with an optimization-based untangler.     

Averaging of gradient in the space of linear triangular and bilinear rectangular finite elements

An averaging method for the second-order approximation of the values of the gradient of an arbitrary smooth function u = u(x ₁, x ₂) at the vertices of a regular triangulation T _h composed both of rectangles and triangles is presented. The method assumes that only the interpolant Π _h [u] of u in the finite element space of the linear triangular and bilinear rectangular finite elements from T _h is known. A complete analysis of this method is an extension of the complete analysis concerning the finite element spaces of linear triangular elements from [Dalík J., Averaging of directional derivatives in vertices of nonobtuse regular triangulations, Numer. Math., 2010, 116(4), 619–644]. The second-order approximation of the gradient is extended from the vertices to the whole domain and applied to the a posteriori error estimates of the finite element solutions of the planar elliptic boundary-value problems of second order. Numerical illustrations of the accuracy of the averaging method and of the quality of the a posteriori error estimates are also presented.     

二阶椭圆问题基于泡函数的简化的稳定化二阶混合有限元格式

本文研究二阶椭圆方程基于泡函数的稳定化的二阶混合有限元格式,通过消去泡函数导出一种自由度很少的简化的稳定化的二阶混合有限元格式, 误差分析表明消去泡函数的简化格式与带有泡函数的格式具有相同的精度而可以节省6N_p个自由度(其中N_p三角形剖分中的顶点数目).       

On Narrow Hexagonal Graphs with a 3-Homogeneous Suborbit

A connected graph of girth m 3 is called a polygonal graph if it contains a set of m-gons such that every path of length two is contained in a unique element of the set. In this paper we investigate polygonal graphs of girth 6 or more having automorphism groups which are transitive on the vertices and such that the vertex stabilizers are 3-homogeneous on adjacent vertices. We previously showed that the study of such graphs divides naturally into a number of substantial subcases. Here we analyze one of these cases and characterize the k-valent polygonal graphs of girth 6 which have automorphism groups transitive on vertices, which preserve the set of special hexagons, and which have a suborbit of size k – 1 at distance three from a given vertex.     

Vibration analysis of quasi-symmetric structures

An efficient computational procedure is presented for the free vibration analysis of structures with unsymmetric geometry. The procedure is based on approximating the unsymmetric vibrational response of the structure by a linear combination of a few symmetric and antisymmetric modes (global approximation vectors), each obtained using approximately half the degrees of freedom of the original model. The three key elements of the procedure are: (a) use of mixed finite element models having independent shape functions for the internal forces (stress resultants) and generalized displacements, with the internal forces allowed to be discontinuous at interelement boundaries, (b) operator splitting, or additive decomposition of the different arrays in the governing finite element equations to delineate the contributions to the symmetric and antisymmetric response vectors, and (c) use of a reduction method through successive application of the finite element method and the classical Bubnov-Galerkin technique. The finite element method is first used to generate a few symmetric and antisymmetric global approximation response vectors. Then, the classical Bubnov-Galerkin technique is used to substantially reduce the size of the eigenvalue problem.

An initial set of global approximation vectors is selected to be a few symmetric and antisymmetric eigenvectors, and their various-order derivatives with respect to a tracing parameter identifying all the correction terms to the symmetric (and antisymmetric) eigenvectors. A modified (improved) set of approximation vectors is obtained by using the inverse iteration procedure. The effectiveness of the proposed procedure is demonstrated by means of a numerical example.     

Some sample path properties of a random walk on the cube

Some properties of the set of vertices not visited by a random walk on the cube are considered. The asymptotic distribution of the first timeQ this set is empty is derived. The distribution of the number of vertices not visited is found for times nearEQ. Next the first time all unvisited vertices are at least some distanced apart is explored. Finally the expected time taken by the path to come within a distanced of all points is calculated. These results are compared to similar results for random allocations.     

A positive cell vertex Godunov scheme for a Beeler–Reuter based model of cardiac electrical activity

The monodomain model is a widely used model in electrocardiology to simulate the propagation of electrical potential in the myocardium. In this paper, we investigate a positive nonlinear control volume finite element scheme, based on Godunov's flux approximation of the diffusion term, for the monodomain model coupled to a physiological ionic model (the Beeler–Reuter model) and using an anisotropic diffusion tensor. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in conforming finite element methods. The diffusion term which involves an anisotropic tensor is discretized on a dual mesh using the diffusion fluxes provided by the conforming finite element reconstruction on the primal mesh and the other terms are discretized by means of an upwind finite volume method on the dual mesh. The scheme ensures the validity of the discrete maximum principle without any restriction on the transmissibility coefficients. By using a compactness argument, we obtain the convergence of the discrete solution and as a consequence, we get the existence of a weak solution of the original model. Finally, we illustrate by numerical simulations that the proposed scheme successfully removes nonphysical oscillations in the propagation of the wavefront and maintains conduction velocity close to physiological values.     

Three‐dimensional quadratic nonconforming brick element

A new nonconforming brick element with quadratic convergence for the energy norm is introduced. The nonconforming element consists of on a cube [?1,1]³, and 14 degree of freedom (DOF). Two types of DOF are introduced. One consists of the value at the eight vertices and six face‐centroids and the other consists of the value at the eight vertices and the integration value of six faces. Error estimates of optimal order are derived in both broken energy and norms for second‐order elliptic problems. If a genuine hexahedron, which is not a parallelepiped, is included in the partition, the proposed element is also convergent, but with a lower order. Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 158–174, 2014     

An algorithm for finding the essential sets of arcs of certain graphs

Given a graph and a weight function which associates to each path an element of a Q-semiring, an essential set of arcs is defined as the complement of a maximal set of arcs which can be removed from the graph without changing the weight of the optimal paths for any pair of vertices. Conditions are given under which a graph admits a unique set of essential arcs and an algorithm is proposed to test for this condition and find this set of arcs.     

Lines in hypergraphs

One of the De Bruijn-Erd?s theorems deals with finite hypergraphs where every two vertices belong to precisely one hyperedge. It asserts that, except in the perverse case where a single hyperedge equals the whole vertex set, the number of hyperedges is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of simply described families, near-pencils and finite projective planes. Chen and Chvátal proposed to define the line uv in a 3-uniform hypergraph as the set of vertices that consists of u, v, and all w such that {u;v;w} is a hyperedge. With this definition, the De Bruijn-Erd?s theorem is easily seen to be equivalent to the following statement: If no four vertices in a 3-uniform hypergraph carry two or three hyperedges, then, except in the perverse case where one of the lines equals the whole vertex set, the number of lines is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of two simply described families. Our main result generalizes this statement by allowing any four vertices to carry three hyperedges (but keeping two forbidden): the conclusion remains the same except that a third simply described family, complements of Steiner triple systems, appears in the extremal case.