Approximation of functionally graded plates with non-conforming finite elements

In this paper rectangular plates made of functionally graded materials (FGMs) are studied. A two-constituent material distribution through the thickness is considered, varying with a simple power rule of mixture. The equations governing the FGM plates are determined using a variational formulation arising from the Reissner–Mindlin theory. To approximate the problem a simple locking-free Discontinuous Galerkin finite element of non-conforming type is used, choosing a piecewise linear non-conforming approximation for both rotations and transversal displacement. Several numerical simulations are carried out in order to show the capability of the proposed element to capture the properties of plates of various gradings, subjected to thermo-mechanical loads.     

Superconvergence for triangular finite elements

Based on two classes of the orthogonal expansions in a triangle, superconvergence of m-degree triangular finite element solution (for evenm) and its average gradient (for oddm) at symmetric points for a second order elliptic problem are studied. There are no other superconvergence points independent of the coefficients of elliptic equation. Project supported by the National Natural Science Foundation of China (Grant No. 19331021).     

Optimal rectangular MITC finite elements for Reissner–Mindlin plates

In recent years a family of finite elements named mixed interpolated tensorial components (MITC) has been introduced for the numerical approximation of Reissner–Mindlin plates. The elements have been proved to be locking free. In this article, we consider the MITC rectangular finite elements and show that it is possible to reduce the number of internal degrees of freedom in the approximation of the rotation field without losing order of convergence. Our mathematical analysis is carried out combining some results for the Stokes problem with the special features of the MITC finite elements. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 575–585, 1997     

Using infinite series to calculate the centre of mass of triangular plates

We present an alternative and pedagogical method to calculate the centre of mass of homogeneous triangular plates by using scaling, symmetry and geometric infinite series. This work also aims to better understand problems that involve concepts of centre of mass of discrete and continuous systems.     

Methods of finite element analysis of arbitrary buckling forms of sandwich plates and shells

Two statements of the problem of arbitrary buckling forms (BFs) (including synphasic, antiphasic, mixed flexural, flexural-shear, and shear forms in the tangential directions) of general-form sandwich shells and two schemes of its solution by the FEM are given. The first of the schemes is based on the use of refined linear equations for determination of the precritical stress-strain state and linearized equations of neutral equilibrium with all parametric addends necessary to determine the critical loads and reveal the possible BFs. The second one uses the general geometrically nonlinear relations of elasticity theory for investigation of the whole deformation process up to buckling in terms of a modified incremental (stepwise) statement of the problem. Examples of solution of particular problems are given.Center for Study of Dynamics and Stability, Tupolev Kazan' State Technical University, Kazan', Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 473–486, July–August, 2000.     

Primal-dual variational problems by boundary and finite elements

In this paper the primal-dual (or mixed) formulation is studied for self-adjoint elliptic problems coupled with a boundary integral equation. It is shown that, after introducing a suitable complementary variational principle, the problem is reduced to finding a stationarity point of a constrained functional. Some numerical examples are reported for a second-order differential equation on unbounded domains.     

The obstacle problem for beams and plates

The numerical solution of the obstacle problem for beams and plates by means of variational inequalities and finite elements is examined. Algorithms for the solution of the discrete problem are discussed and particular attention is paid to different methods of approximating the constraint. The results of some numerical experiments for beams and plates are included.     

A nodal coupling of finite and boundary elements

This article presents and analyzes a simple method for the exterior Laplace equation through the coupling of finite and boundary element methods. The main novelty is the use of a smooth parametric artificial boundary where boundary elements fit without effort together with a straight approximate triangulation in the bounded area, with the coupling done only in nodes. A numerically integrated version of the algorithm is also analyzed. Finally, an isoparametric variant with higher order is proposed. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 555–570, 2003     

Finite differences on triangular grids

We define a finite differences method for triangular grids and we show how to link it to a finite element method. From this new point of view we then analyze properties of the solution and convergence. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 567–579, 1998     

Triangular finite elements of HCT type and classC ρ

Let τ be some triangulation of a planar polygonal domain Ω. Given a smooth functionu, we construct piecewise polynomial functionsv∈C ^ρ(Ω) of degreen=3 ρ for ρ odd, andn=3ρ+1 for ρ even on a subtriangulation τ₃ of τ. The latter is obtained by subdividing eachT∈ρ into three triangles, andv/T is a composite triangular finite element, generalizing the classicalC ¹ cubic Hsieh-Clough-Tocher (HCT) triangular scheme. The functionv interpolates the derivatives ofu up to order ρ at the vertices of τ. Polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements of this type.     

A stability theory of sandwich structural elements 1. Analysis of the current state and a refined classification of buckling forms

The main stages of development of the stability theory of sandwich structural elements are considered. The mechanism of their stability loss is revealed using the experimental data and theoretical solutions obtained on the basis of refined statements of problems. A classification of all possible forms of stability loss is given within the limits of continuum representation of load-bearing layers and the core of these structures.Center for Study of Dynamics and Stability, Tupolev Kazan State Technical University, Kazan, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 707–716, November–December, 1999.     

A mixed finite element for the stokes problem using quadrilateral elements

The object of this paper is to complete the results obtained in [3] by showing that the new mixed finite element that we have constructed in [3] also works for quadrilateral elements and to compare this method with the standard finite volume method. Estimates of optimal order are derived for both the new mixed finite element and an associated finite volume method.     

Multilevel preconditioners for discretizations of the biharmonic equation by rectangular finite elements

Recently, some new multilevel preconditioners for solving elliptic finite element discretizations by iterative methods have been proposed. They are based on appropriate splittings of the finite element spaces under consideration, and may be analyzed within the framework of additive Schwarz schemes. In this paper we discuss some multilevel methods for discretizations of the fourth-order biharmonic problem by rectangular elements and derive optimal estimates for the condition numbers of the preconditioned linear systems. For the Bogner–Fox–Schmit rectangle, the generalization of the Bramble–Pasciak–Xu method is discussed. As a byproduct, an optimal multilevel preconditioner for nonconforming discretizations by Adini elements is also derived.     

Pointwise supercloseness of pentahedral finite elements

This article derives the weak estimate of the first type for pentahedral finite elements over uniform partitions of the domain for the Poisson equation. The estimate for the W^1,1‐seminorm of the discrete derivative Green's function is also given. Using these two estimates, we obtain the pointwise supercloseness of derivatives of the pentahedral finite element approximation and the interpolant to the true solution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Approximation by quadrilateral finite elements

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order in and order in is that the given space of functions on the reference element contain all polynomial functions of total degree at most . In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree . We show, by means of a counterexample, that this latter condition is also necessary. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

     

Tunneling of the hard‐core model on finite triangular lattices

We consider the hard‐core model on finite triangular lattices with Metropolis dynamics. Under suitable conditions on the triangular lattice sizes, this interacting particle system has 3 maximum‐occupancy configurations and we investigate its high‐fugacity behavior by studying tunneling times, that is, the first hitting times between these maximum‐occupancy configurations, and the mixing time. The proof method relies on the analysis of the corresponding state space using geometrical and combinatorial properties of the hard‐core configurations on finite triangular lattices, in combination with known results for first hitting times of Metropolis Markov chains in the equivalent zero‐temperature limit. In particular, we show how the order of magnitude of the expected tunneling times depends on the triangular lattice sizes in the low‐temperature regime and prove the asymptotic exponentiality of the rescaled tunneling time leveraging the intrinsic symmetry of the state space.     

Supercharacters for finite groups of triangular type

We construct a supercharacter theory for the finite groups of triangular type. Its special case is the supercharacter theory for algebra groups of Diaconis and Isaacs. The supercharacter analog of the Kirillov formula for irreducible characters of unipotent groups is proved.     

On equilibrium finite elements in three-dimensional case

The space of divergence-free functions with vanishing normal flux on the boundary is approximated by subspaces of finite elements that have the same property. The easiest way of generating basis functions in these subspaces is considered.