The finite element solution of a class of elastica problems

A finite element procedure for the analysis of an inextensional elastica bent through frictionless supports is presented. Element displacements are expressed in terms of cubic Hermite polynomials with nodal displacements and derivatives being determined to minimise the strain energy. It is shown that the axial force at any point is proportional to the square of the bending moment, which enables member incremental stiffness matrices to be expressed in a similar form to those used in stability analysis. The iterative procedure proposed for solving the nonlinear stiffness equations may readily be incorporated into any continuous beam program by the modification and addition of very few statements. An example is considered which indicates that accurate solutions may be obtained to a wide range of practical problems using the proposed technique.     

A finite element solution of diffraction problems in unbounded domains

In this paper, the method introduced in [3] is extended and applied to diffraction problems of acoustics and hydrodynamics. The problems dealt with are linear elliptic and may involve non-constant coefficients; they are set in unbounded domains. The method uses an integral representation formula with a regular kernel which allows an equivalent problem to be set in an interior annular closed region; it is shown how irregular frequencies are avoided. A variational formulation and its finite element discretization are detailed. Some numerical results are shown which support the validity of the technique and corroborate the theoretical analysis.     

Application of the moving finite element method to moving boundary Stefan problems

The moving finite element method is modified to be an effective solution for a version of the classical ice-water Stefan problem. Also, the cylindrical Stefan problem is numerically solved and compared with an existing perturbation solution.     

Application of a Galerkin finite element method to atmospheric transport problems

Numerical simulation of the movement of a contaminant within the atmosphere presents difficulties due to (a) The multi-dimensionality of the problem; (b) The fact that the horizontal transport is usually convection dominated; (c) The boundary conditions are mixed; (d) Both slow and fast atmospheric chemical reactions can be important. In this study, numerical experiments using a Crank-Nicolson Galerkin finite element method to solve the time-dependent partial differential equations demonstrate the applicability and accuracy of this method for the variety of conditions encountered in atmospheric pollutant modeling. The Crank-Nicolson Galerkin method using piecewise linear, piecewise cubic Hermite polynomials, and upwind finite elements is shown to accurately model the pure convection of initial wave forms. Numerical results studying the interactions of convection, diffusion, chemical reaction, pollutant removal, and the effects of contaminant emission source strength, source location and multiple sources are also presented.     

Application of patched elements based on a previous finite element solution in plane elasticity problems

A new approach for displacement recovery in the finite element method (FEM) is presented. It is connected with the Ritz method, applied in some previous authors works. The interpolation functions pass through some nodal positions previously obtained by FEM. The functions are approximated by Lagrangian polynomials of third degree. Unknown multipliers are introduced at a certain number of nodes. In order to determine them, the minimum potential energy principle is applied. The numerical results, compared with those obtained by direct application of various finite element types, are usually closer to the exact ones. An attempt to construct a simple error indicator is made.     

Vectorizable preconditioners for mixed finite element solution of second-order elliptic problems

We compare the performance of the CG (conjugate gradient) method, the point-wise incomplete factorization preconditioned CG method, and the block-incomplete factorization preconditioned CG method for solving problems arising in mixed finite element discretization of second order elliptic differential equations. The robustness and vectorizable properties of these methods are illustrated by a large set of numerical tests.     

An effective multilayered sandwich beam-link finite element for solution of the electro-thermo-structural problems

The first part of this contribution deals with deriving the effective matrices of the multilayered sandwich 2D-link/beam finite element made of FGM (functionally graded material) with variation of material properties and rectangular cross-section for solution of weak-coupled electric–thermal–structural problems. The variation of effective material properties is caused by both continuous longitudinal and layer-wise symmetric transversal variation of the constituent’s volume fractions and constituent’s material properties. The second part of this contribution is completed by numerical validation, which documents the high accuracy and effectiveness of our proposed electro-thermo-structural composite (FGMs) link/beam finite element.     

A semi-numerical approach to the buckling and post-buckling solution of plate

In this semi-numerical approach to buckling of plates, a combination of polynomial and trigonometric functions are used as displacement functions in the Rayleigh-Ritz method. It is shown that a variety of loading and boundary conditions can be handled using simple variation of the trigonometric function proposed here. A two-dimensional plate buckling problem is therefore reduced to selecting one of the set of trigonometric function shown. The buckling coefficient values are then computed as eigenvalues of the stiffness and geometric matrix pair. These values compare well with available analytical and numerical approach solutions. The approach can also be extended to post buckling analysis using the eigenvectors found.     

Application of multi-scale finite element methods to the solution of the Fokker–Planck equation

This paper presents an application of multi-scale finite element methods to the solution of the multi-dimensional Fokker–Planck equation. The Fokker–Planck, or forward Kolmogorov, equation is a degenerate convective–diffusion equation arising in Markov-Process theory. It governs the evolution of the transition probability density function of the response of a broad class of dynamical systems driven by Gaussian noise, and completely describes the response process. Analytical solutions for the Fokker–Planck equation have been developed for only a limited number of low-dimensional systems, leading to a large body of approximation theory. One such approach successfully applied to the solution of these problems in the past is the finite element method, though for systems of dimension three or less. In this paper, a multi-scale finite element method is applied to the Fokker–Planck equation in an effort to develop a formulation that can yield higher accuracy on cruder spatial discretizations, thus reducing the computational overhead associated with large scale problems that arise in higher dimensions.     

An analysis of dual formulations for the finite element solution of two-body contact problems

This paper examines the convergence properties of dual finite element formulations of the two-dimensional frictionless two-body contact problem under the assumption of infinitesimal kinematics. The centerpiece of the proposed analysis is the well-known Babuška–Brezzi condition, suitably adapted to the present problem. It is demonstrated for certain canonical geometries that several widely used methods that employ pressure or force interpolations derived from the discretizations of both surfaces violate the Babuška–Brezzi condition, thus producing increasingly oscillatory solutions under mesh refinement. Alternative algorithms are proposed that circumvent this difficulty and are shown to yield convergent solutions.     

有限元网格划分相关问题分析研究

以有限元分析软件HyperMesh为计算平台,分析两个简单模型的静态受力产生的变形和应力,阐述有限元网格不同划分方法对计算精度的影响。     

Application of the finite element method to machinery

The finite element method has given the designer of machinery components and structures a new tool for analysis of stresses, temperatures and dynamic vibrations.In the following some applications of the tool to different problems within this field are shown (by some examples), using the SESAM-69 complex.Three-dimensional solids as well as three-dimensional plate and bar structures are treated, and the results are compared to analytical or experimental solutions where such are known.These examples should verify the FEM as very attractive to the modem machinery-designer.     

Meaningful existence of finite element solutions to off-limit problems

Problems for which the variational principle of minimum potential energy breaks down, and are therefore formally unsuitable for a finite element analysis, are shown to possess, nevertheless, a useful discrete solution.     

A parallel finite element solution method

New parallel computer architectures have revolutionized the design of computer algorithms, and promise to have significant influence on algorithms for structural engineering computations. In this paper, a parallel finite element solution method is presented. The solution method proposed does not require the formation of global system equations, but computes directly the element distortions, as opposed to solving a system of nodal equations. An element or substructure is mapped on to a processor of an MIMD multiprocessing system. Each processor stores only the information relevant to the element or substructure for which the processor represents. The finite element computations can be performed in parallel, in that a processor generates the local stiffness, computes the element distortions and determines the stress-strain characteristics for the element or substructure associated with the processor.     

An implicit and incremental formulation for the solution of elastoplastic problems by the finite element method

In this paper, a new approach is developed to solve elastoplastic problems by the finite element method. This approach involves two steps: (1) A mechanical formulation using the principle of virtual work and an implicit incremental form of the constitutive equations. This form is obtained by an approximate integration of the flow rules over the increment and includes the yield criterion itself. (2) A resolution algorithm to solve the nonlinear equations obtained by the mechanical formulation: Two resolution algorithms based on the Newton-Raphson method are proposed and compared. As the mechanical formulation is no more bound to the resolution algorithm, the results obtained by these two algorithms are the same and are path independent. Two numerical examples are presented: A thick cylinder under an internal pressure and a tensile sample. The numerical results obtained by the presented approach are compared with those obtained by the classical I.S.M. The comparison shows that the accuracy of the results does not vary when the load increment size increases as in the I.S.M. For a given accuracy this method requires about 15 times less computer time than the I.S.M. for the same memory space. This approach is easy to implement in a program based on the I.S.M. and has been extended to compressible and viscoplastic materials.     

Computing the null space of finite element problems

We present a method for computing the null space of finite element models, including models with equality constraints. The method is purely algebraic; it requires access to the element matrices, but not to the geometry or material properties of the model.Theoretical considerations show that under certain conditions, both the amount of computation and the amount of memory required by our method scale linearly with model size; memory scales linearly but computation scales quadratically with the dimension of the null space. Our experiments confirm this: the method scales extremely well on 3-dimensional model problems. In general, large industrial models do not satisfy all the conditions that the theoretical results assume; however, experimentally the method performs well and outperforms an established method on industrial models, including models with many equality constraints.The accuracy of the computed null vectors is acceptable, but the method is usually less accurate than a more naive (and computationally much more expensive) method.     

Application of the finite element method to off-shore structures

The merits of finite element methods in connection with the design of offshore structures are discussed using the Ekofisk Tank as an example. Various design approaches and procedures are evaluated on the same basis and some of the experiences gained are mentioned. The conclusion is drawn that the finite element method is a very powerful tool in the evaluation of new designs, but to be successful, it requires good planning and experienced personnel.The procedures presently applied for determination of wave loads do not represent a very realistic modelling of the actual storm conditions. To show what improvements can be made in this field, an outline of how the irregular wave system may be formulated and how the calculated loads may be applied in the stress analysis is given. The procedure is described with reference to a semisubmersible drilling platform but the principle may be applied to any offshore structure.     

Applications of nonlinear finite element method to contact problems and paper handling problems

A model that predicts the paper behavior during the printing process at various time intervals is proposed. The tendency of wrinkling can therefore be observed. This model will accommodate any paper topology induced by moisture effects or otherwise, and can analyze virtually any roll geometry and roll combinations.Nonlinear Finite Element Method is applied to the contact problem of two layered cylinders compressing a paper with arbitrary initial shape. The analysis assumed plane strain condition, the paper can therefore be treated as a beam. The result from the contact problem is applied to predict the behavior of the paper as it is feeding through the roll system. The Total Lagrange Formulation with Piola-Kirchoff Stress and Green-Lagrange strained is used. The quasi-static solution at different time intervals is presented. It was found that they agreed favorably with experiments.     

On the automatic solution of nonlinear finite element equations

An algorithm for the automatic incremental solution of nonlinear finite element equations in static analysis is presented. The procedure is designed to calculate the pre- and post-buckling/collapse response of general structures. Also, eigensolutions for calculating the linearized buckling response are discussed. The algorithms have been implemented and various experiences with the techniques are given.     

Galerkin's finite element formulation of the second-order boundary-value problems

A Galerkin's finite element approach based on weighted-residual formulation is presented to find approximate solutions to obstacle, unilateral and contact second-order boundary-value problems. The approach utilizes a piece-wise linear approximations utilizing linear Langrange polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of this scheme in comparison to collocation, finite-difference and spline methods.