Showing papers on "Finite element method published in 1974"

Concepts and Applications of Finite Element Analysis

Abstract: Notation. Introduction. One-Dimensional Elements, Computational Procedures. Basic Elements. Formulation Techniques: Variational Methods. Formulation Techniques: Galerkin and Other Weighted Residual Methods. Isoparametric Elements. Isoparametric Triangles and Tetrahedra. Coordinate Transformation and Selected Analysis Options. Error, Error Estimation, and Convergence. Modeling Considerations and Software Use. Finite Elements in Structural Dynamics and Vibrations. Heat Transfer and Selected Fluid Problems. Constaints: Penalty Forms, Locking, and Constraint Counting. Solid of Revolution. Plate Bending. Shells. Nonlinearity: An Introduction. Stress Stiffness and Buckling. Appendix A: Matrices: Selected Definition and Manipulations. Appendix B: Simultaneous Algebraic Equations. Appendix C: Eigenvalues and Eigenvectors. References. Index.

A stiffness derivative finite element technique for determination of crack tip stress intensity factors

Abstract: A finite element technique for determination of elastic crack tip stress intensity factors is presented. The method, based on the energy release rate, requires no special crack tip elements. Further, the solution for only a single crack length is required, and the crack is 'advanced' by moving nodal points rather than by removing nodal tractions at the crack tip and performing a second analysis. The promising straightforward extension of the method to general three-dimensional crack configurations is presented and contrasted with the practical impossibility of conventional energy methods.

Local and global smoothing of discontinuous finite element functions using a least squares method

Abstract: The concepts and potential advantages of local and global least squares smoothing of discontinuous finite element functions are introduced. The relationship between local smoothing and the ‘reduced’ integration' technique is established. Examples are presented to illustrate the application of the two smoothing techniques to the finite element stresses from several structural analysis problems. The paper concludes with some practical recommendations for discontinuous finite element function smoothing.

Statistical Identification of Structures

Abstract: A method is formulated for systematically using experimental measurements of the natural frequencies and mode shapes of a structure to modify stiffness and mass characteristics of a finite element model. Throughout the modification process, which does not require complete data, the finite element model remains consistent. An additional feature is that the engineer's confidence in the modeling of the various finite elements is quantified and incorporated into the revision procedure. Examples demonstrate the convergence and versatility of the method.

Interior estimates for Ritz-Galerkin methods

Abstract: Interior a priori error estimates in Sobolev norms are derived from interior RitzGalerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on both uniform and nonuniform meshes. It is shown that the error in an interior domain 2 can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error in a weaker norm over a slightly larger domain which measures the effects from outside of the domain Q. Additional results are given in the case when the subspaces are defined on a uniform mesh. Applications to specific boundary value problems are given. 0. Introduction. There are presently many methods which are available for computing approximate solutions of elliptic boundary value problems which may be classified as Ritz-Galerkin type methods. Many of these methods differ from each other in some respects (for example, in how they treat the boundary conditions) but have much in common in that they have what may be called "interior Ritz-Galerkin equations" which are the same. Here we shall be concerned with finding interior estimates for the rate of convergence for such a class of methods which are consequences of these interior equations. Let us briefly describe, in a special case, the type of question we wish to consider. Let &2 be a bounded domain in RN with boundary M2 and consider, for simplicity, the problem of finding an approximate solution of a boundary value problem (0.1) \u =f in Q2, (0.2) Au= g on U2, where A is some boundary operator. Suppose now that we are given a one-parameter family of finite-dimensional subspaces Sh (0 < h < 1) of an appropriate Hilbert space in which u lies and that, for each h, we have computed an approximate solution Uh c Sh to u using some Ritz-Galerkin type method. Here we have in mind, for example, methods such as the "engineer's" finite element method [8], [22], the Aubin-Babuska penalty method [2], [4], the methods of Nitsche [12], [13] or the Received October 15, 1973. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. Copyright i 1974, American Mathematical Society

Representation of singularities with isoparametric finite elements

Abstract: The development of a generalized quadrilateral finite element that includes a singular point at a corner node is presented. Inter-element conformability is maintained so that monotone convergence is preserved. The global-local concept of finite elements is used to formulate the complete set of equations. Examples of crack tip singularities are given.

The solution of viscous incompressible jet and free-surface flows using finite-element methods

Abstract: : The authors discuss the creation of a finite element program suitable for solving incompressible, viscous free surface problems in steady axisymmetric or plane flows. For convenience in extending program capability to non-Newtonian flow, non-zero Reynolds numbers, and transient flow, a Galerkin formulation of the governing equations is chosen, rather than an extremum principle. The resulting program is used to solve the Newtonian die-swell problem for creeping jets free of surface tension constraints. The authors conclude that a Newtonian jet expands about 13%, in substantial agreement with experiments made with both small finite Reynolds numbers and small ratios of surface tension to viscous forces. The solutions to the related stick-slip problem and the tube inlet problem, both of which also contain stress singularities, are also given. (Modified author abstract)

Automatic triangulation of arbitrary planar domains for the finite element method

Abstract: The object of this paper is to describe a new algorithm for the semi-automatic triangulation of arbitrary, multiply connected planar domains. The strategy is based upon a modification of a finite element mesh genration algorithm recently developed. 1 The scheme is designed for maximum flexibility and is capable of generating meshes of triangular elements for the decomposition of virtually any multiply connected planar domain. Moreover, the desired density of elements in various regions of the problem domain is specified by the user, thus allowing him to obtain a mesh decomposition appropriate to the physical loading and/or boundary conditions of the particular problem at hand. Several examples are presented to illustrate the applicability of the algorithm. An extension of the algorithm to the triangulation of shell structures is indicated.

Using the Finite Element Method to Determine Temperature Distributions in Orthogonal Machining

Abstract: Temperature distributions for typical cases of orthogonal machining with a continuous chip were obtained numerically by solving the steady two-dimensional energy equation using the finite element method. The distribution of heat sources in both the primary and secondary zones was calculated from the strain-rate and flow stress distributions. Strain, strain-rate and velocity distributions were calculated from deformed grid patterns obtained from quick-stop experiments. Flow stress was considered as a function of strain, strain-rate and temperature. The chip, workpiece and tool (actual shape and size) were treated as one system and material properties such as density, specific heat and thermal conductivity were considered as functions of temperature.

Application of quadratic isoparametric finite elements in linear fracture mechanics

Abstract: Quadratic isoparametri c elements are shown to embody 1//r singularity f or c alculating s tress i ntensity f actors of elastic f racture mechanics. The singularity is obtained by placing the mid-side node on any side at the quarter p oint. Figure 1 shows the 2-dimensional, 8-noded quadrilateral (a) and 6-noded triangle (b), isoparametric e lements with the mid-side nodes near the crack tip at the quarter nodes. Figure 2 shows the S-dimensional elements w ith the mid-side node near the crack edge at the quarter p oints. The local s trains in these e lements vary as 1//r throughout the element. In the 3-D case, the strains along the crack edge are non-singular. A very important feature of these elements is that they satisfy the necessary requirements for convergence [I] in their singular form as well as in their non-singular form. They, therefore, pass the patch test [2], possess rigid body motion (R.B.M.), constant strain modes, interelement compatibility, and continuity of displacements. In contrast, other special crack tip elements [3,4], do not possess rigid body motion modes and do not pass the patch test, thus making their use in the problems cited below questionable. The existence of rigid body motion and constant strain modes in the proposed isoparametric elements allows the calculation of stress intensity factors for thermal gradients in 2- and 3-D problems and in problems where symmetry about the crack cannot be invoked (R.B.M. exists). In addition, since these elements are part of the element library of most general purpose programs, their use in linear fracture mechanics is very tractable. The element formulation in its non-singular form is well documented ([I], pp. 103-154). The element in the singular form is formulated exactly in the same manner except for a restriction on the location of the nodal points. In summary, the element is formulated by mapping its geometry from the cartesian space into a unit curvilinear space using special quadratic functions [i]. The same functions, in the curvilinear space, are used to interpolate the displacements within the element, hence the name isoparametric. In order to achieve the required singularity, the Jacobian of transformation [J], from the cartesian to the curvilinear space, is made singular by placing the mid-side nodes near the crack tip at the quarter points. The singularity occurs only at the crack tip point. It can be easily shown, for example, that for the rectangular form of the case in Figure la, the strain in the local x-direction along the line i-2, is given by

Improved Two-Dimensional Finite Element

Abstract: Plane elements based on assumed displacement fields are reviewed A plane element based on the assumed-stress hybrid principle is found to model bending action as well as any of the displacement-based elements Contrary to many displacement-based elements, the hybrid is valid for arbitrary quadrilateral shape A modification is introduced to remove the orientation-dependence of the element and improve its stress prediction The element formulation is detailed so that it may be easily adopted by interested users Numerical examples suggest that among plane quadrilaterals having four nodes and eight degrees-of-freedom, the modified hybrid is the most effective element currently available

Finite-Element Solution of the Eddy-Current Problem in Magnetic Structures

Abstract: Analysis of the eddy-currentproblem in magnetic structures by the method of Finite-elements is presented. The linear diffusion equation representing the appropriate energy functional is described. The field region is discretised by triangular Finite-elements and the solution to the field problem is obtained by minimizing the energy functional with respect to each of the vertex values of the vector potential. Expressions for the magnetic field, electric field and eddy-current losses are presented. The method is applied to a few cases of engineering interest and compared with results of classical analysis and tests.

Transient two‐dimensional heat conduction problems solved by the finite element method

Abstract: A finite element weighted residual process has been used to solve transient linear and non-linear two-dimensional heat conduction problems. Rectangular prisms in a space-time domain were used as the finite elements. The weighting function was equal to the shape function defining the dependent variable approximation. The results are compared in tables with analytical, as well as other numerical data. The finite element method compared favourably with these results. It was found to be stable, convergent to the exact solution, easily programmed, and computationally fast. Finally, the method does not require constant parameters over the entire solution domain.

The stability inLq of theL2-projection into finite element function spaces

Abstract: The stability ofL 2-projection into many standard finite element spaces as a map intoL q , 1?q??, is demonstrated. TheL 2-projection ofu?L q is shown to be a quasi-optimal approximation ofu inL q .

Flow analysis network (FAN)—A method for solving flow problems in polymer processing

Abstract: A finite element method is proposed for solving two dimensional flow problems in complex geometrical configurations commonly encountered in polymer processing. The method is applicable to flow in relatively narrow gaps of variable thickness and any desired shape. It was developed for analyzing flow in injection molding dies and certain extrusion dies. The fluid can be any non-Newtonian fluid which is incompressible, inelastic, and time independent. The flow field is divided into an Eulerian mesh of cells. Around each node, located at the center of the cell, a local flow analysis is made. The analysis around all nodes results in a set of linear algebraic equations with the pressures at the nodes as unknowns. The simultaneous solution of these equations results in the required pressure distribution, from which the flow rate distribution is obtained. Solution for the isothermal Newtonian flow problem is obtained by a one-time solution of the equations, whereas solution of a non-Newtonian problem requires iterative solution of the equations.

Automatic masters for eigenvalue economization

Abstract: In dynamic analyses using the finite element method very large matrices are derived. The size of the matrices is usually reduced using an eigenvalue economization method. This paper describes an automatic technique for selecting the optimum variables to be kept as masters using the economization method. Some examples are presented and these lead to an empirical statement about expected accuracy.

Stresses in nearly incompressible materials by finite elements with application to the calculation of excess pore pressures

Abstract: A number of problems are analysed by the displacement method to assess the stress accuracy at very low compressibilities. ‘Parabolic’ isoparametric elements are used. It is found that the mean stress becomes grossly in error at the centre and edges of each element as the compressibility is reduced whereas the deviatoric stress components do not. All stress components retain good accuracy at the ‘reduced’ integration sampling points (2 × 2 Gauss). ‘Exact’ integration yields a similar stress distribution to ‘reduced’ but the mean stress is grossly in error at the integration points (3 × 3 Gauss). Exceptions, however, occur. These findings are interpreted, and a rule for predetermining whether or not accurate stresses can be obtained at the integrating points is suggested. Thus it is shown that the displacement method is suitable for analysing materials which for practical purposes are incompressible. A procedure is then presented for analysing porous media-both linear and non-linear-by separating the stiffness into ‘effective’ and ‘pore fluid’ components. This allows excess pore pressure to be calculated explicitly. Applications to saturated soils are given which make use of the findings of the first part of the paper.

A second order finite element method for the one‐dimensional Stefan problem

Abstract: We describe a finite element method for the one-dimensional Stefan problem The elements are quadrilaterals of the space-time plane which are determined at each time-step in relation with the position of the free boundary The method appears as a generalization of the classical Crank-Nicolson scheme, since it is identical to this scheme in the case of rectangular elements; it has the advantage of providing a simple and accurate determination of the free boundary Numerical experiments show that the order of accuracy is equal to 2

Vibrational response of sonar transducers using piezoelectric finite elements

Abstract: The finite element method is applied to the vibrational analysis of electromechanical sonar transducers of arbitrary geometry Three‐dimensional hexahedral finite elements which include the effects of piezoelectric coupling are formulated, and the solution of the resulting coupled electroelastic equations of motion is presented The vibrational response of a particular transducer element is computed, and comparisons with experimental mesurements are made The calculated deformations of the transducer are presented in the form of computer generated displays A data reduction scheme is also utilized to clarify the physical meaning of the transducer response

Finite element bending analysis of multilayer plates

Abstract: The development of a general quadratic multilayer plate element is presented for the analysis of arbitrarily layered curved plates. In the formulation, each layer of the multilayer plate can have different orthotropic properties and can deform locally. Examples of bending problems are presented which demonstrate the applicability of the formulation.

Steady state flow in rigid networks of fractures

Abstract: The finite element has been used to develop two numerical methods of calculating the flow characteristics of rigid networks of planar fractures. One method uses triangular elements to investigate details of laminar flow in fractures of irregular cross section combined with that of a permeable rock matrix. The other method uses line elements and is designed only for flow in networks of planar fractures in an impermeable matrix. As an example of the application of the finite element approach the line element method was used to develop a series of dimensionless graphs that characterize seepage in idealized fracture systems beneath dams. These methods treat two-dimensional flow in the laminar regime for networks of fractures of arbitrary orientation and aperture distribution.