A lumped mass finite element method for vibration analysis of elastic plate-plate structures

The fully discrete lumped mass finite element method is proposed for vibration analysis of elastic plate-plate structures. In the space directions, the longitudinal displacements on plates are discretized by conforming linear elements, and the transverse displacements are discretized by the Morley element. By means of the second order central difference for discretizing the time derivative and the technique of lumped masses, a fully discrete lumped mass finite element method is obtained, and two approaches to choosing the initial functions are also introduced. The error analysis for the method in the energy norm is established, and some numerical examples are included to validate the theoretical analysis.     

Adini‐Q1‐P3 FEM for general elastic multi‐structure problems

An Adini‐Q₁‐P₃ finite element method is introduced to solve general elastic multi‐structure problems, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and rotational angles on rods are discretized by conforming linear (bilinear or trilinear) elements, and transverse displacements on plates and rods are discretized by Adini elements and Hermite elements of third order, respectively. The unique solvability and optimal error estimates in the energy norm are established for the discrete method, whose numerical performance is illustrated by some numerical examples. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1092–1112, 2011     

SBFEM elements for thin-walled composite beams with arbitrary layup

The scaled boundary finite element method (SBFEM) is a semi-analytical method in which only the boundary is discretized. The results on the boundary are scaled into the domain with respect to a scaling center which must be “visible” from the whole boundary. For beam-like problems the scaling center can be selected at infinity and only the cross-section is discretized. Two new elements for thin-walled beams have been developed on the basis of the first order shear deformation theory. The beam sections are considered to be multilayered laminate plates with arbitrary layup. The arbitrary cross-section is discretized with beam elements of Timoshenko type. Using the virtual work principle gives the SBFEM equation, which is a system of differential equations of a gyroscopic type. The solution is calculated using the matrix exponential function. The elements have been tested and compared with a finite element model and they give good results. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the convergence of nonnested multigrid methods with nested spaces on coarse grids

Multigrid methods for discretized partial differential problems using nonnested conforming and nonconforming finite elements are here defined in the general setting. The coarse‐grid corrections of these multigrid methods make use of different finite element spaces from those on the finest grid. In general, the finite element spaces on the finest grid are nonnested, while the spaces are nested on the coarse grids. An abstract convergence theory is developed for these multigrid methods for differential problems without full elliptic regularity. This theory applies to multigrid methods of nonnested conforming and nonconforming finite elements with the coarse‐grid corrections established on nested conforming finite element spaces. Uniform convergence rates (independent of the number of grid levels) are obtained for both the V and W‐cycle methods with one smoothing on all coarse grids and with a sufficiently large number of smoothings solely on the finest grid. In some cases, these uniform rates are attained even with one smoothing on all grids. The present theory also applies to multigrid methods for discretized partial differential problems using mixed finite element methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 265–284, 2000     

An auxiliary space preconditioner for linear elasticity based on the generalized finite element method

We construct and analyze a preconditioner of the linear elasticity system discretized by conforming linear finite elements in the framework of the auxiliary space method. The auxiliary space preconditioner is based on two auxiliary spaces corresponding to discretizations of the scalar Poisson equation by linear finite elements and the generalized finite element method. Copyright © 2011 John Wiley & Sons, Ltd.     

SOME ESTIMATES WITH NONCONFORMING ELEMENTS IN DOMAIN DECOMPOSITION ANALYSIS

1.IntroductionFOrsimplicityoftheexposition,weconsidertheellipticboundaryvalueproblemonaboundedopenpolygonaldomainfiCEZwhereItiswell--knownthat(1.1)hasauniquesolutionueH'(fl)(of.[7,15,16]).Supposethatfib~{e}isaquajsi--uniformmeshoffi,i.e.,fibsatisfieswhere…     

A mixed differential quadrature and finite element free vibration and buckling analysis of thick beams on two-parameter elastic foundations

As a first endeavor, a mixed differential quadrature (DQ) and finite element (FE) method for boundary value structural problems in the context of free vibration and buckling analysis of thick beams supported on two-parameter elastic foundations is presented. The formulations are based on the two-dimensional theory of elasticity. The problem domain along axial direction is discretized using finite elements. The resulting system of equations and the related boundary conditions are discretized in the thickness direction and in strong-form using DQM. The method benefits from low computational efforts of the DQ in conjunction with the effectiveness of the FE method in general geometry and systematic boundary treatment resulting in highly accurate and fast convergence behavior solution. The boundary conditions at the top and bottom surface of the beams are implemented accurately. The presented formulations provide an effective analysis tool for beams free of shear locking. Comparisons are made with results from elasticity solutions as well as higher-order beam theory.     

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.     

Conforming and nonconforming harmonic finite elements

ABSTRACT

Instead of using the full polynomial space, a conforming and a nonconforming finite element methods are designed where only harmonic polynomials (a much smaller space) are employed in the computation. The conforming quadratic harmonic polynomial finite element is defined only on a special triangular grid. The nonconforming quadratic harmonic finite element is defined on general triangular grids. The optimal order of convergence is proved for both finite element methods, and confirmed by numerical computations. In addition, numerical comparisons with the standard conforming and nonconforming finite elements are presented.     

Extended multiscale FEM for design of beams and frames with complex topology

A numerical method for design of beams and frames with complex topology is proposed. The method is based on extended multi-scale finite element method where beam finite elements are used on coarse scale and continuum elements on fine scale. A procedure for calculation of multi-scale base functions, up-scaling and downscaling techniques is proposed by using a modified version of window method that is used in computational homogenization. Coarse scale finite element is embedded into a frame of a material that is representing surrounding structure in a sense of mechanical properties. Results show that this method can capture displacements, shear deformations and local stress-strain gradients with significantly reduced computational time and memory comparing to full scale continuum model. Moreover, this method includes a special hybrid finite elements for precise modelling of structural joints. Hence, the proposed method has a potential application in large scale 2D and 3D structural analysis of non-standard beams and frames where spatial interaction between structural elements is important.     

杂交有限元的区域分解法

近几年来,由于并行计算机的迅速发展,求解椭圆型微分方程的区城分解法又引起了人们的重视.这一方法的基本思想是把求解区域分解成许多子区域,每个子区域上用一台计算机求解.这种方法适应并行机的需要,是用并行机解大型椭圆型偏微分方程的一     

On the finite element approximation of a cascade flow problem

Summary The finite element analysis of a cascade flow problem with a given velocity circulation round profiles is presented. The nonlinear problem for the stream function with nonstandard boundary conditions is discretized by conforming linear triangular elements. We deal with the properties of the discrete problem and study the convergence of the method both for polygonal and nonpolygonal domains, including the effect of numerical integration.     

Non-Conforming Domain Decomposition with Hybrid Method

We present a non-conforming domain decomposition technique for solving elliptic problems with the finite element method. Functions in the finite element space associated with this method may be discontinuous on the boundary of subdomains. The sizes of the finite meshes, the kinds of elements and the kinds of interpolation functions may be different in different subdomains. So, this method is more convenient and more efficient than the conforming domain decomposition method. We prove that the solution obtained by this method has the same convergence rate as by the conforming method, and both the condition number and the order of the capacitance matrix are much lower than those in the conforming case.     

A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.     

三角形单元协调与非协调位移的能量正交关系

基于组合稳定化变分原理,周天孝提出的组合杂交法是绝对收敛和稳定的,它给出了一种系统化的增强应力/应变方法,并建立了一簇低阶仿射等价的n-cube(n=2,3)单元。本文论证了单元上应力插值为线性,位移插值为协调线性部分和非协调二次部分之和的三角形组合杂交单元其协调部分与非协调部分的能量正交关系,进而得到此三角形单元刚度矩阵等同于协调的三角形线性元刚度矩阵,即非协调部分无应变增强特性。     

Mixed finite element models for free vibrations of thin-walled beams

Simple mixed finite element models are developed for the free vibration analysis of curved thin-walled beams with arbitrary open cross section. The analytical formulation is based on a Vlasov's type thin-walled beam theory which includes the effects of flexural-torsional coupling, and the additional effects of transverse shear deformation and rotary inertia. The fundamental unknowns consist of seven internal forces and seven generalized displacements of the beam. The element characteristic arrays are obtained by using a perturbed Lagrangian-mixed variational principle. Only C⁰ continuity is required for the generalized displacements. The internal forces and the Lagrange multiplier are allowed to be discontinuous at interelement boundaries.

Numerical results are presented to demonstrate the high accuracy and effectiveness of the elements developed. The standard of comparison is taken to be the solutions obtained by using two-dimensional plate/shell models for the beams.     

Approximation of general arch problems by straight beam elements

Summary In this paper, we approximate the solution of a problem of a general arch by a nonconforming method using straight beam elements and taking into account numerical integration. Compatibility conditions which have to be satisfied at the mesh points are given. These conditions ensure for this method the same order of convergence as usual conforming finite element methods.     

关于高精度12参矩形板元的分析

It is proved that the so-called a set of 12-parameter rectangular plate elements with high accuracy constructed by using double set parameter method and undetermined method are, in fact, the same one; the real shape function space is nothing but the Adini's element's, which has nothing to do with the other high degree terms and leads to a new method for constructing the high accuracy plate elements. This fact has never been seen for other conventional and unconventional, conforming and nonconforming rectangular plate elements, such as Quasi-conforming elements, generalized conforming elements and other double set parameter finite elements. Moreover, such kind of rectangular elements can not be constructed by the conventional finite element methods.     

A note on the determination of influence lines and surfaces using finite elements

This paper presents a simple yet general procedure to determine influence lines and surfaces for frames, beams, trusses, and plates. The method is based on the application of the Muller-Breslau principle and finite elements. It can be easily implemented with most commercial finite element codes.     

Controllability method for acoustic scattering with spectral elements

We formulate the Helmholtz equation as an exact controllability problem for the time-dependent wave equation. The problem is then discretized in time domain with central finite difference scheme and in space domain with spectral elements. This approach leads to high accuracy in spatial discretization. Moreover, the spectral element method results in diagonal mass matrices, which makes the time integration of the wave equation highly efficient. After discretization, the exact controllability problem is reformulated as a least-squares problem, which is solved by the conjugate gradient method. We illustrate the method with some numerical experiments, which demonstrate the significant improvements in efficiency due to the higher order spectral elements. For a given accuracy, the controllability technique with spectral element method requires fewer computational operations than with conventional finite element method. In addition, by using higher order polynomial basis the influence of the pollution effect is reduced.