Showing papers on "Smoothed finite element method published in 1972"

Variational crimes in the finite element method

Abstract: Publisher Summary This chapter focuses on variational crimes in the finite element method. The finite element method is nearly a special case of the Rayleigh-Ritz technique. The convenience and effectiveness of the finite element technique is regarded as conclusively established; it has brought a revolution in the calculations of structural mechanics, and other applications are rapidly developing. The chapter reviews the modifications of the Ritz procedure which have been made to achieve an efficient finite element system. On a regular mesh one could regard the system of Ritz-finite element equations KQ = F as a finite difference scheme, and then the patch test would be equivalent to the formal consistency of the difference equations with the correct differential equation. The chapter discusses the convergence theory for non-conforming elements. It is impossible for a polynomial to satisfy a condition like u = 0 on a general curved boundary. Therefore, some alteration in the boundary condition will be necessary. The most important possibility is to change the domain.

Finite Element Methods in Continuum Mechanics

Abstract: Publisher Summary The chapter presents a brief introduction to the different finite element formulations for linear elastic solids and discusses similar formulations for several other field problems. The chapter presents detailed illustrations for several typical finite element formulations. In the finite element formulation, displacement and stress fields are assumed to be continuous within each discrete element. This formulation calls for modified variational principles for which the continuity or equilibrium conditions along the interelement boundaries are introduced as conditions of constraint and appropriate boundary variables are used as the corresponding Lagrangian multipliers. The chapter presents the several variational principles and the corresponding models used in the finite element formulation. The large majority of the existing finite element formulations are based on the assumed displacement approach. The chapter discusses equilibrium problems of linear elastic solids. There are several other problems in solid mechanics, which can be formulated by means of variational principles and hence can be solved by finite element methods. The finite element methods have also been extended to nonlinear problems resulting from elastic-plastic material properties or from large deflections or finite strains.

Finite element formulation by variational principles with relaxed continuity requirements

Abstract: Publisher Summary This chapter focuses on finite element formulation by variational principles with relaxed continuity requirements. The finite element method has long been recognized as an extension of the well-known Ritz procedure for constructing approximate solutions to the governing variational principle associated by a given boundary value problem. The method was originally used in the analysis of solid mechanics problems and several alternative variational principles in elasticity have been employed in the finite element formulation. As in the finite element method a continuum is subdivided by finite element mesh, it is possible to modify the variational principles by allowing discontinuous fields at the interelement boundaries and hence, to create the so-called hybrid models in finite element analyses. The chapter reviews various finite element formulations for two typical partial differential equations. The first one involves the harmonic equation which has its application in many continuum mechanics problems typical of which are torsion of prismatic bars, deflection of stretched membranes, heat conduction, and potential flow. The second one is associated with the bi-harmonic equation which is associated, for example, with the bending of thin plates.

Finite element analysis of slow non-newtonian channel flow

Abstract: A finite element method is applied to isothermal slow channel flow of power-law fluids. The fully developed flow is normal to the channel cross section. The method and results are compared with a finite difference method for rectangular channels and with exact solutions for the Newtonian case. An advantage of finite element methods is the flexibility of the mesh of elements approximating the continuum, chosen to suit the particular problem. Arbitrary boundary shapes can be handled as illustrated by a rectangular channel with rounded corners.

Solution of transient groundwater flow problems by the finite element method

Abstract: The derivation of the basic equations of a finite element method for transient groundwater flow problems is simplified when the approximation of the time derivative by a finite difference equation is made before the introduction of the variational principle. Existing programs for steady state problems can easily be extended to the transient case in this way, and a stable numerical procedure can be obtained.

Dual analysis by finite elements : linear and non linear applications

Abstract: : (FORCES)), STRESSES, DEFORMATION, STRAIN(MECHANICS), STABILITY, EQUILIBRIUM(PHYSIOLOGY), FORCE(MECHANICS), NUMERICAL ANALYSISCOMPUTER AIDED ANALYSIS, SHELL THEORY, FINITE ELEMENT ANALYSISThe report presents the results obtained during the aforementioned period concerning the linear and non linear applications of the dual analysis technique. New formulation for finite element derivation are proposed including equilibrium, conforming and hybrid elements. A new method of solution of the structural problem is based on the stress function formulation. The slab analogies are used to discover additional finite element families. A useful complementary energy principle is derived for large deformations. The stability criterion is obtained from the second variation of the total energy. A non linear shell theory is presented which avoids the difficulties of defining the constitutive equations. A computer program for linear applications is documented. (Author)

Geometrical non-linear analysis of structures by finite elements

Abstract: The paper shows a comprehensive analysis of geometrically non linear structural problems by the finite element method. The theoretical approach is based on a variational principle stating the incremental equilibrium through the stationarity of a functional that could be defined as the incremental total potential energy. The analysis is carried out in two distinct phases: first a prediction of the behaviour of the structure subjected to an increment of load then a correction by means of Newton Raphson method of the results obtained in the previous incremental step. The approach makes it possible to determine the complete load deflection curve either in the prebuckling region or in the postbuckling one and to find out the critical load taking into account the deflection prior to buckling (non linearized buckling analysis).

Nonlinear acoustic response analysis of plates using the finite element method.

Abstract: The paper describes a perturbation technique to perform the nonlinear response analysis of plate structures under random acoustic excitation. In the analysis, use is made of a conforming plate element together with a nonlinear plate stiffness element which is dependent on the modal response of the structure. Applying these elements and the associated consistent mass matrices, the equivalent linear eigen matrix of the complete plate is organized. The eigen solution and the following modal spectral computation completes the iteration cycle. The iteration process is repeated until apparent convergent data are acquired. A flow diagram and a numerical example are included which illustrate the application of the method to practical problems.

Finite element analysis of perturbed compressible flow

Abstract: The finite element representation of the linearized equations governing the steady compressible flow of an isentropic perfect gas is considered. The fully non-linear system of equations is linearized on the basis of small perturbation theory. The finite element matrix equations for an arbitrary polygonal element are generated by the method of weighted residuals: Galerkin's criterion is used. As an example, a triangular element in two-dimensional flow is treated in detail and numerical result for a sample problem are given.

Finite Element Method for Wood Mechanics

Abstract: A three-dimensional finite element method for general anisotropy materials is presented. The method uses a parallelepiped element and includes the directional properties of the materials, making it easily applicable to problems in wood mechanics. By varying the stiffness and allowing some elements to assume zero stiffness, a variety of shapes can be studied, including those with cracks and voids. The limitations of the method are primarily related to the available computer storage and to the fitting of rectangular-sided elements to curved boundaries. Examples are presented to indicate the types of problems this technique is capable of solving. These examples include wood mechanics problems of compression, both parallel and at angles to the ring and grain, tension of a member with holes, and torsion.

A finite element analogue of the modified Rayleigh‐Ritz method for vibration problems

Abstract: In this paper finite element method for natural vibration problems, based on a modified Rayleigh-Ritz method is presented. Numerical results of a few simple examples indicate that the method is promising.