Showing papers on "Extended finite element method published in 1968"

Application of finite elements to the solution of Helmholtz's equation

Abstract: A novel method, that of finite elements, for the solution of Helmholtz's equation is suggested. Various 2- and 3-dimensional problems are solved using this method, and the results are compared with more conventional techniques, particularly the finite-difference method, which it may be regarded to supersede. The ease with which various boundary conditions may be handled is discussed and illustrated. Nonhomogeneous configurations present no difficulty, nor do they require any special formulation. There is considerable scope for the further development of the technique, which has, until now, been applied mainly to the solution of Laplace or Poisson equations.

Matrix bandwidth minimization

Abstract: In the analysis of a structural problem by the finite element method, a large order stiffness matrix is created which describes mathematically the inter-connectivity of the system The structure is defined in three dimensional space by discrete points called nodes Each node is represented by its coordinates in the space The nodes are then connected by the various finite elements that the particular computer program may utilize (ie, bar members, rectangular or triangular panels, three dimensional tetrahedrons, etc)

Accuracy and Convergence of Finite Element Approximations

Abstract: : The paper reports on a theoretical investigation of the convergence properties of several finite element approximations in current use and assesses the magnitude of the principal errors resulting from their use for certain classes of structural problems. The method is based on classical order of error analyses commonly used to evaluate finite difference methods. Through the use of the Taylor series differential or partial differential equations are found which represent the convergence and principal error characteristics of the finite element equations. These resulting equations are then compared with known equations governing the continuum, and the error terms are evaluated for selected problems. Finite elements for bar, beam, plane stress, and plate bending problems are studied as well as the use of Straight or curved elements to approximate curved beams. The results of the study provide basic information on the effect of interelement compatibility, unequal size elements, discrepancies in triangular element approximations, flat element approximations to curved structures, and the number of elements required for a desired degree of accuracy.

Finite Element Analysis of Fluid Flows

Abstract: : The finite element method is applied to several simple cases of steady flow of a perfect, incompressible fluid. It is shown that the finite element representation accurately reflects the behavior of the classical flow equations. Finite elements form the basis for a versatile analysis procedure applicable to problems in several different fields. The earliest applications were to problems in structural mechanics. In recent years, nonstructural problems also have been treated by this method. The finite element method represents an approximate procedure for satisfying the problem in terms of its variational formulation. In structural mechanics this is generally accomplished by determining displacement fields based on satisfying the minimum potential energy theorem. Consequently, finite elements furnish a useful alternative scheme for applying the well-known Ritz method. For nonstructural problems, it is essential that the appropriate variational expressions be known beforehand. For the flow problems taken up in this paper, such expressions are well known. The governing matrix equation for the assemblage of elements is based on the properties derived for a single typical element. These properties, in turn, depend on assuming a mathematical form for the primary unknown of the problem and then satisfying the variational principle. For the elasticity problem, the unknowns are the displacements, while for the perfect incompressible fluid, either the velocity potential or the stream function may be used. Of great interest is that structural and nonstructural elements may often be identical in shape and may be represented by similar mathematical expressions. By way of illustration, the problems taken up in this paper were solved using the new ASTRA structural program developed at The Boeing Company. Finally, it should be pointed out that the major difference between the elasticity and fluid flow problems lies in the boundary conditions to be satisfied.

Least squares smoothing of experimental data using finite elements

Abstract: A program for interpreting Moire fringe data using a digital computer is described. The finite element technique overcomes certain intrinsic defects of existing methods which use polynomials, and promises to give better results, especially where stress concentrations are present. An experimental example compares the technique with the polynomial technique.

Dynamic Analysis of Shells Using Doubly-Curved Finite Elements

Abstract: : This paper discusses the theoretical development of a computer program for the static and dynamic analysis of general shell structures. The theory is based on the finite element concept and uses both the generalized finite element method and the direct stiffness method to form the pertinent equations. The treatment of shell surface geometry, the displacement functions and elemental degrees of freedom, and the modification of the generalized stiffness method required for the implementation of the triangular element are described. The correlation of both theoretical and experimental results with those obtained by the present method are shown along with idealizations required for accurate results. Static and dynamic solution results are compared,

Use of Fourier series in the finite element method.

Abstract: P IAN1'2 has demonstrated the use of the energy method in deriving element stiffness matrices in connection with the displacement method of structural analysis. His procedure is based on the representation of an element displacement function in terms of m undetermined coefficients where the number m may be larger than the number of generalized displacements n.2 When m is larger than n, it is possible to satisfy not only stress equilibrium and boundary displacement continuity but also to maintain slope continuity along the normal directions of the element edges. Having the element displacement function, the total energy of the element! may be evaluated, and then the condition of the minimum potential energy yields the stiffness matrix. The aforementioned procedure is illustrated in Ref. 2 by determining the stiffness matrix for a rectangular plate under plane stress conditions. The displacement function is chosen in the form of a power series as follows : u = ai +

A finite element method for the plastic bending analysis of structures

Abstract: : A finite element technique is presented for the plastic analysis of structures subjected to out-of-plane bending, alone or in combination with in- plane membrane stresses. The method makes uses of a linear matrix equation of finite element analysis, formulated to include the effect of initial strains. Application of the procedure is made to beam and arch structures in the presence of both types of nonlinearity, and to rectangular plates for which material nonlinearity alone is present.

A finite-difference method for the solution of one-dimensional non-stationary problems in magneto-hydrodynamics☆

Abstract: IN theoretical investigations of a number of applied problems in magneto-hydra dynamics (various types of MHD-generators, problems of astrophysics etc.) there is particular interest in the study of interaction processes between a compressible electrically conducting gas and a magnetic field for arbitrary Reynolds numbers Rem and the magnetic interaction parameters R, = W/&p, where H is the magnetic field strength and p the pressure. In this case and in physical experiments an important role is played by the investigation of mathematical models which take into account mainly the non-linear relations between the non-stationary processes of magneto-hydrodynamics. In the one-dimensional approximation numerical methods also enable us not only to study the quantitative sides of the processes, but also to establish a number of new qualitative regularities. Thus the use of numerical methods for equations of magnetohydrodynamics, taking into account complex non-linear dissipative processes, has made possible the solution of a number of actual physical problems 11-61. In [6l a new physical phenomenon is described, the so-called T-layer effect a high-temperature, electricallyconducting, self-sustaining layer of gas, arising at a definite part of the mass due to Joule heating.

The application of dynamic relaxation to the finite element method of structural analysis.

Abstract: : The report is a study in depth of the method of Dynamic Relaxation, a matrix iterative method for the solution of simultaneous linear equations, and used principally in problems of structural analysis under static stress conditions. Whereas earlier investigators have restricted themselves almost exclusively to the finite difference formulation in space of both the equations of motion and the constitutive relationships, the present study formulates an alternative approach using finite elements in space. The mathematical basis for the finite element approach in space is developed, and followed by an analysis of convergence based upon a transformation into a standard eigenvalue problem for the error vector, intimately associated with the conditioning of the equations. Optimum convergence is studied and comparisons made with other iterative methods. Dynamic Relaxation is demonstrated by applying it to plane stress problems of statically loaded plates having discontinuities in the form of circular, elliptical and filleted square holes. (Author)

A new finite element model for solid continua

Abstract: : The proposed finite element model is based on separate assumptions of interior and interelement displacements and on the assumed boundary tractions of each individual element. The associated variational functional for this model is presented. This method has the same merits of the assumed stress method in that a compatibility displacement function at the interelement boundary can be easily constructed. In the plate bending problems, the matrix to be inverted to obtain the element stiffness matrix by the present method is, in general, of smaller order than that of the assumed stress method when the same order of approximation is used in both methods. (Author)

Computation of stresses and strains in dams made of local materials, earth slopes, and their foundations, by the finite element method

Abstract: The experinece of the application of the finite element method shows that this is the most efficient and convenient of the methods used at the present time for the computation of stresses in slopes and earth dams.

Stress analysis by finite elements

Abstract: THE METHOD OF FINITE ELEMENTS IS OUTLINED FOR DETERMINATION OF STRESS AND DISPLACEMENT FIELDS. THE CHIEF FEATURES ARE ILLUSTRATED AND ADVANTAGES OF THE TECHNIQUE ARE ILLUSTRATED BY SEVERAL EXAMPLES. THE THEORY IS PRESENTED OF THE FINITE ELEMENTS METHOD. THIS METHOD IS NOT AS WIDELY APPLICABLE AS FINITE DIFFERENCES, BECAUSE ONLY THOSE PROBLEMS WHICH HAVE AN EQUIVALENT VARIATIONAL FORMULATION MAY BE ATTACKED BY FINITE ELEMENTS. ONE DISADVANTAGE IS THAT STRESSES ARE AVERAGED ACROSS ANY ONE ELEMENT. IT IS NEVER CLEAR WHAT POINT IN THE TRIANGULAR ELEMENT CORRESPONDS TO THE AVERAGE STRESS STATE. ALL PROBLEMS WERE RUN ON AN IBM 7094 DIGITAL COMPUTER. IT IS ANTICIPATED THAT THE METHOD OF FINITE ELEMENTS WILL BECOME A STANDARD TOOL FOR ELASTIC ANALYSIS WITHIN THE NEXT FEW YEARS.