Showing papers in "International Journal for Numerical Methods in Engineering in 1981"

A uniform strain hexahedron and quadrilateral with orthogonal hourglass control

Abstract: The treatment of zero energy modes which arise due to one-point integration of first-order isoparametric finite elements is addressed A method for precisely isolating these modes for arbitrary geometry is developed Two hourglass control schemes, viscous and elastic, are presented and examined In addition, a convenient one-point integration scheme which analytically integrates the element volume and uniform strain modes is presented

Two‐dimensional stress intensity factor computations using the boundary element method

Abstract: A multidomain boundary element formulation for the analysis of general two-dimensional plane strain/stress crack problems is presented. The numerical results were accurate and efficient. The analyses were performed using traction singular quater-point boundary elements on each side of the crack tip(s) with and without transition elements. Traction singular quarter-point boundary elements contain the correct √r displacement and 1/√r traction variations at the crack tip. Transition elements are appended to the traction singular elements to model the √r displacement variation. The 1/√r traction singularity is not represented with these elements. Current research studies for the crack propagation analysis of quasi-static and fatigue fracture problems are discussed.

Finite element analysis of deformation of strain‐softening materials

Abstract: A finite element analysis of strain-softening materials is presented in which the shear band of prescribed thickness is assumed to exist within elements where maximal stress intensity is reached. The incremental stiffness matrix of the element is derived including shear band deformation. Examples presented in the Paper demonstrate that the load-displacement curve and the displacement field are not sensitive to the mesh size used in the solution.

Numerical results from large deflection beam and frame problems analysed by means of elliptic integrals

Abstract: Numerical evaluations of elliptic integral solutions of some large deflection beam and frame problems are presented. The values are given in tabular form with up to six significant figures. The numerical technique used for evaluating the elliptic integrals is described.

Improved iteration strategy for nonlinear structures

Abstract: A solution strategy for the analysis of nonlinear structures is described. The strategy is a simple extension of existing Newton-type procedures, and can easily be incorporated into existing computer programs. Earlier work which contributed to the development of the strategy is reviewed and the theory of the procedure is presented. Six examples, covering several different types of structural behaviour are described. These examples suggest that the strategy is remarkably stable and efficient.

The boundary element method applied to the analysis of two‐dimensional nonlinear sloshing problems

Abstract: This paper deals with an application of the boundary element method to the analysis of nonlinear sloshing problems, namely nonlinear oscillations of a liquid in a container subjected to forced oscillations. First, the problem is formulated mathematically as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions, assuming the fluid to be inviscid and incompressible and the flow to be irrotational. Next, the governing equation (Laplace equation) and boundary conditions, except the dynamic boundary condition on the free surface, are transformed into an integral equation by employing the Galerkin method. Two dynamic boundary condition is reduced to a weighted residual equation by employing the Galerkin method. Two equations thus obtained are discretized by the use of the finite element method spacewise and the finite difference method timewise. Collocation method is employed for the discretization of the integral equation. Due to the nonlinearity of the problem, the incremental method is used for the numerical analysis. Numerical results obtained by the present boundary element method are compared with those obtained by the conventional finite element method and also with existing analytical solutions of the nonlinear theory. Good agreements are obtained, and this indicates the availability of the boundary element method as a numerical technique for nonlinear free surface fluid problems.

A general two‐dimensional, graphical finite element preprocessor utilizing discrete transfinite mappings

Abstract: Finite element preprocessor programs have been developed in recent years in order to expedite the task of data preparation. A key decision in the design of these programs is the choice of a method for automated mesh generation. Three popular methods (Laplacian, isoparametric and transfinite mappings) are compared. Transfinite mappings with boundary information represented in discrete form are found to possess distinct advantages for the task of mesh generation. A two-dimensional preprocessor program utilizing the discrete transfinite mappings is described. Interactive computer graphics techniques are used extensively to facilitate data preparation and display. The geometry-generating routines are general, and may be used in any finite element application. A specialized graphical ‘attribute editor’ for structural mechanics problems is also described. This editor provides an efficient method of specifying boundary conditions, material properties, loads, etc.

Some computational aspects of elastic-plastic large strain analysis

Abstract: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.

`Infinite domain' elements

Abstract: A parametric element is formulated which enables the economic modelling of ‘infinite domain’ type problmes. A typical problem is an opening in a stress field in an infinite medium, either in two or three dimensions. The strategy is to model around the opening with two or three layers of conventional isoparametric finite elements and surround these with a single layer of ‘infinite domain’ elements. Several sample problems has been analysed for circular, square and spherical openings in infinite media, and the results compared with either theoretical or boundary element solutions which include the ‘infinite’ boundary in their solution technique. Copyright

Mixed models and reduced/selective integration displacement models for nonlinear analysis of curved beams

Abstract: Simple mixed models are developed for use in the geometrically nonlinear analysis of deep arches. A total Lagrangian description of the arch deformation is used, the analytical formulation being based on a form of the nonlinear deep arch theory with the effects of transverse shear deformation included. The fundamental unknowns comprise the six internal forces and generalized displacements of the arch, and the element characteristic arrays are obtained by using Hellinger-Reissner mixed variational principle. The polynomial interpolation functions employed in approximating the forces are one degree lower than those used in approximating the displacements, and the forces are discontinuous at the interelement boundaries. Attention is given to the equivalence between the mixed models developed herein and displacement models based on reduced integration of both the transverse shear and extensional energy terms. The advantages of mixed models over equivalent displacement models are summarized. Numerical results are presented to demonstrate the high accuracy and effectiveness of the mixed models developed and to permit a comparison of their performance with that of other mixed models reported in the literature.

Dynamic stress concentration studies by boundary integrals and Laplace transform

Abstract: The dynamic stress field and its concentrations around holes of arbitrary shape in infinitely extended bodies under plane stress or plane strain conditions are numerically determined. The material may be linear elastic or viscoelastic, while the dynamic load consists of plane compressional waves of harmonic or general transient nature. The method consists of applying the Laplace transform with respect to time to the governing equations of motion and formulating and solving the problem numerically in the transfomed domain by the boundary integral equation method. The stress field can then be obtaind by a numerical inversion of the trasformed solution. The correspondence principle is invoked for the case of viscoelastic material behavious. The method is simplified for the case of harmonic waves where no numerical inversion is involved.

A general formulation for coupled thermal flow of metals using finite elements

Abstract: A finite element formulation to deal with the flow of metals coupled with thermal effects in presented. The deformation process of the metal is treated using the visco-plastic flow approach and the solution technique for the coupled problem implies a simultaneous solution for velocities and temperatures. Some aspects of the numerical solution of the problem are given and in the last part of the paper some examples of steady-state extrusion and rolling problems showing the applicability of the method are shown.

A finite element solution for the two‐dimensional elastic contact problems with friction

Abstract: A numerical procedure developed for solving the two-dimensional elastic contact problems with friction is presented. This is a generalization of a procedure developed by Francavilla and Zienkiewicz to include frictional effects under proportionate loading. The method uses the flexibility matrix obtained by inversion of condensed stiffness matrix formed by eliminating all the nodes except those where contact is likely to take place and those with external forces. Compatibility of displacements for both normal and tangential directions is applied to those nodes which do not slip. However, for the nodes which slip, compatibility of displacements is applied for normal direction only and slip condition is applied in the tangential direction. The technique has been applied to several problems and very good results have been obtained. The number of iterations needed are very small.

Techniques for thermal sensitivity analysis

Abstract: Techniques for computing the sensitivity of temperatures to changes in design variables needed for designing thermal protection systems are considered. It is shown that the choice of the most efficient technique depends on the ratio of number of temperature constraints to the number of design variables, as well as on the thermal analysis method employed. The analysis is specialized to the case of a structure modeled by finite elements, and an example of an insulation panel is used to demonstrate the techniques.

The application of quasi‐Newton methods in fluid mechanics

Abstract: The use of quasi-Newton methods is studied for the solution of the nonlinear finite element equations that arise in the analysis of incompressible fluid flow. An effective procedure for the use of Broyden’s method in finite element analysis is presented. The quasi-Newton method is compared with the commonly employed successive substitution and Newton-Raphson procedures, and it is concluded that the use of Broyden‘s method can constitute an effective solution strategy.

Plane isoparametric hybrid‐stress elements: Invariance and optimal sampling

Abstract: Formulation and applications of the hybrid-stress finite element model to plane elasticity problems are examined. Conditions for invariance of the element stiffness are established for two-dimensional problems, the results of which are easily extended to three-dimensional cases. Next, the hybrid-stress functional for a 3-D continuum is manipulated into a more convenient form in which the location of optimal stress/strain sampling points can be identified. To illustrate these concepts, 4- and 8-node plane isoparametric hybrid-stress elements which are invariant and of correct rank are developed and compared with existing hybrid-stress elements. For a 4-node element, lack of invariance is shown to lead to spurious zero energy modes under appropriate element rotation. Alternative 8-node elements are considered, and the best invariant element is shown to be one in which the stress compatibility equations are invoked. Results are also presented which demonstrate the validity of the optimal sampling points, the effects of reduced orders of numerical integration, and the behaviour of the elements for nearly incompressible materials.

Computation of limit loads

Abstract: For computing the collapse state of a perfect plastic continuum we suggest a mixed finite element approximation to the infinite dimensional mathematical programming problem of limit analysis. The convergence of the limit load to the exact solution is discussed, as well as solution methods for the discrete problem. Finally the method is applied to a classical problem in plane strain.

The use of time domain diakoptics in time discrete models of fields

Abstract: The principles of diakoptics are extended to time domain solutions and the technique is applied to the solution of fields by introducing space approximating polynomials. Since transmission-line networks provide time-discrete models suitable for exact numerical analysis, the topic is approached by tearing these transmission-line models and then showing how reconnection is made in the time domain.

A boundary integral equation method for plate flexure with conditions inside the domain

Abstract: A method for solving boundary value problems for thin plate flexure as described by Kirchhoff's theory is proposed. An integral formulation leads to a system of boundary integral equations involving values of deflection, slope, bending moment and transverse shear force along the edge. A discretization leading to a matrix formulation is proposed. To solve problems with inner conditions in the plate domain, an elimination of boundary unknowns proves successful. The degenerate case where the boundary is free (which leads to a non-invertible matrix) is investigated. Three examples illustrate the efficiency of the method.

Infinite elements with 1/rn type decay

Abstract: This paper presents a method of developing a family of 1/rn type infinite elements for the analysis of problems definite in unbounded media. The proposed method is a direct extension of the conventional finite element method. The resulting improper integrals are integrated exactly over the infinite element domains. Two numerical examples in elastic half-space static problems are investigated to illustrate the applicability and accuracy of the method. The use of the proposed infinite elements yields excellent results and preserves all the advantages of the finite element method.

Cubic spline boundary elements

Abstract: The boundary integral method is formulated and applied using cubic spline interpolation along the boundary for both the geometry and the primary variables. The cubic spline interpolation has continuous first and second derivatives between elements, thus allowing the accurate calculation of derivative dependent functions (on the boundary) such as velocity in potential flow. The spline functions also smooth the geometry and can represent curved sections with fewer nodes. The results of numerical experiments indicate that the accuracy of the boundary integral equation method is improved for a given number of elements by using cubic spline interpolation. It is, however, necessary to use numerical quadrature. The quadrature slows calculation and/or degrades the accuracy. The numerical experiments indicate that most problems run faster for a given accuracy using linear interpolation. There seems to be a class of problems, however, which requires higher order interpolation and/or continuous derivatives for which the cubic spline interpolation works much better than linear interpolation.

C1 quintic interpolation over triangles: Two explicit representations

Abstract: Two explicit representation of a C1 quintic interpolant over triangles are given. These representations are derived by generalization of Coons' methods and Bernstein-Bezier methods, respectively.

Calculation of sensitivity derivatives in thermal problems by finite differences

Abstract: The optimum design of a structure subject to temperature constraints is considered When mathematical optimization techniques are used, derivatives of the temperature constraints with respect to the design variables are usually required In the case of large aerospace structures, such as the Space Shuttle, the computation of these derivatives can become prohibitively expensive Analytical methods and a finite difference approach have been considered in studies conducted to improve the efficiency of the calculation of the derivatives The present investigation explores two possibilities for enhancing the effectiveness of the finite difference approach One procedure involves the simultaneous solution of temperatures and derivatives The second procedure makes use of the optimum selection of the magnitude of the perturbations of the design variables to achieve maximum accuracy

Structural analogies for scalar field problems

Abstract: This paper describes how general-purpose finite element structural analysis computer programs can be used, without modification, to solve various scalar field equations such as the wave equation, the Helmholtz equation, Laplace's equation, Poisson's equation, the heat equation and the telegraph equation, as well as mixed field problems (such as coupled structural-acoustic problems) which involve these equations