Alf Samuelsson

Implicit and dynamic explicit solutions of blade forging using the finite element method

Abstract Blade forging has been studied and solved by two elastic-plastic finite element codes, NIKE2D and DYNA2D, as a two-dimensional plane-strain problem. The main difference between the two codes is that the first one is implicit while the second is dynamic explicit. The efficiency of implicit and explicit solutions regarding blade forging has been studied. The initial position of the circular preform was optimized so that the corner spaces of the dies become filled and the pressure on the dies become as small as possible thus satisfying the design condition of flashless forging. The grading of the finite element mesh for the perform has been studied and been found to be of great importance.

Further discussion on four-node isoparametric quadrilateral elements in plane bending

The aim of the study is: first—to show that the finite element method for plane bending is not sensitive to the distorted geometry of isoparametric quadrilateral elements, second—to close out further investigations on irregular elements based on Wilson incompatible functions employing ad hoc methods to force elements to pass the patch test, regardless of the type of the variational formulation used for the problem. A systematic review of papers dealing with finite four-node plane elements with two degrees of freedom at each node is presented. Selected results of the benchmark example of the mesh distortion test used by all cited authors are compared suggesting that no improvement has been made in more than two decades of intensive development of new elements and methods in the field. An attempt has been made to localize the pathological behaviour of irregular elements under consideration by calculating the areas of the elements after deformation for different values of the distortion parameter. It made possible to observe the mass density of the whole deformed system. The tip deflections at the lower and upper edge at the free end of the cantilever beam were depicted as well, illustrating unexpected physical phenomena of the longitudinal fibers deviating from being parallel. The horizontal displacement components at two common nodes of the adjacent elements have been compared with the analytical solution of the problem for the varying distortion parameter. The resulting horizontal displacement distribution along the common side of the elements has been shown to be the source of unreal behavior of all classes of irregular elements based on Wilson incompatible functions. Increasing the order of the interpolation functions for the displacement field, i.e. application of subparametric elements, makes the finite element method insensitive to element distortion. Copyright

Solution of quasi‐static, force‐driven problems by means of a dynamic‐explicit approach and an adaptive loading procedure

Argues that the dynamic‐explicit approach has in recent years been successfully applied to the solution of various quasi‐static, elastic‐plastic problems, especially in the metal forming area. A condition for the success has, however, been that the problems have been displacement‐driven. The solution of similar force‐driven problems, using this approach, has been shown to be much more complicated and computationally time consuming because of the difficulties in controlling the unphysical dynamic forces. Describes a project aiming to develop a methodology by which a force‐driven problem can be analysed with similar computational effort as a corresponding displacement‐driven one. To this end an adaptive loading procedure has been developed, in which the loading rate is controlled by a prescribed velocity norm. Presents several examples in order to exhibit the merits of the proposed procedure.

ADAPTIVE MODELLING OF PLATES IN BENDING AND SHEAR

In this adaptive finite element approach we start off with a Germain-Kirchhoff (GK) solution for the plate and end up close to the Reissner-Mindlin (RM) solution, see Wendt [1].

Finite element adaptivity in dynamics and elastoplasticity

The paper discusses adaptive procedures for FE-analysis of elastomechanic problems. Error indicators and estimators are based on interpolation theory or on post-processing measures. For the mesh generation and regeneration an isoline technique is utilized by which the error can be reasonably uniformly distributed by interpolation theory.

Distortion measures and inverse mapping for isoparametric 8τnode plane finite elements with curved boundaries

Utilizing systematically differential geometry the paper describes a method which substantially improves results obtained by Yuan et al. (1994), though the same technique is used in both articles. An 8-node isoparametric element with curved boundaries is analysed as an object of differential geometry. Inverse transformations between normal (geodesic) co-ordinates and natural (isoparametric) co-ordinates are derived in terms of a Taylor series which is convergent and does not need many terms to give an excellent approximation of the element shape with four curved sides. The concept of local normal co-ordinates results in the definition of distortion measures of a plane element. It is shown, by exploring the theory of geodesic curves, that the distortion parameters of a chord quadrilateral, spanned on the corner nodes of the 8-node element with curved boundaries, are the basic distortion measures for this 8-node element. Thus, significant reduction of the number of these parameters, from 12 to 4, from previous works is obtained. For the purpose of the finite element method, which is very sensitive to a shape of quadrilateral elements, only basic deviation measures from a regular form of a plane element are of interest. The distortion measures due to curvatures of sides seem to be of secondary significance in the analysis if straight sides of the chord quadrilateral and curved boundaries are isomorphic. The mathematical analysis used is quite general and relies strongly on differential geometry. The results are independent of co-ordinate systems. The meaning of element distortion measures is suggested. This analysis can be extended to curved surfaces in R3.

Computational and experimental modeling of stretch forming operations

The paper reports selected results from a study of the stretch forming process. Sheet materials with a wide range of mechanical properties have been tested, including a carbon steel, 70/30 brass, 18/8 stainless steel, and an aluminum-magnesium alloy. A comprehensive comparison of numerical simulations with experimental results shows quantitative agreement for important process factors such as strain distributions and punch force. The parameter variation study demonstrates considerable influence of the material parameters and the friction coefficient on the results. The strain distributions are found to be most sensitive to parameter variations. The experiments were performed with a cylindrical punch and a firmly clamped blank. The sheet was lubricated with various oils. Strains were measured with an etched grid. The numerical calculations were based on a finite element procedure developed for simulation of arbitrarily shaped 3-D forming operations. The emphasis was put on the proper description of large strain and large rotation plasticity. Relatively simple hardening and friction models used appear to be satisfactory and requiring reasonable computational time. The elastic-plastic description, with Hill’s anisotropic model, was used as this allowed for elastic effects, unloading, and multistage processes.

Verification of an explicit finite-element code for the simulation of the press forming of rectangular boxes of coated sheet steels

Abstract The industrial aim for shorter lead times in the development process for new products has encouraged a fast development of finite-element procedures for the simulation of sheet-metal forming. These procedures can shorten the design stage for new sheet-forming tools and the try-out period. In order to reach these goals robust calculation procedures and extensive verification of results are required. This paper presents a detailed validation of one promising type of finite-element code, namely the explicit code DYNA3D. Experiments for verification were performed on a zinc-coated sheet steel with a hot-dip galvanized coating. Stretch forming and deep drawing of cylindrical cups were performed. Rectangular boxes were formed from rectangular blanks and blanks with cut corners. Dies both with and without draw beads were used. Punch forces, flange draw-in and strain distributions were measured. The pressings with cylindrical shape were used to determine coefficients of friction with a fitting procedure based on comparison of data from pressings and from the DYNA3D calculations. These tribological data and constitutive data of the steels were used in the simulation of pressings of rectangular boxes. The calculations with DYNA3D gave a good description of flange draw-in and the strain distributions in the pressings. In most of the cases studied the punch-force curves were well reproduced. It is concluded that the present code is well suited for the simulation of sheet-forming operations.

On application of differential geometry to computational mechanics

Developments in the fields of computational science—the finite element method—and mathematical foundations of continuum mechanics result in many new algorithms which give solutions to very complicated, complex, large scaled engineering problems. Recently, the differential geometry, a modern tool of mathematics, has been used more widely in the domain of the finite element method. Its advantage in defining geometry of elements [13–15] or modeling mechanical features of engineering problems under consideration [4–7] is its global character which includes also insight into a local behavior. This fact comes from the nature of a manifold and its bundle structure, which is the main element of the differential geometry. Manifolds are generalized spaces, topological spaces. By attaching a fiber structure to each base point of a manifold, it locally resembles the usual real vector spaces; e.g. R3. The properties of a differential manifold M are independent of a chosen coordinate system. It is equivalent to say, that there exists smooth or Cr differentiable atlases which are compatible. In this paper a short survey of applications of differential geometry to engineering problems in the domain of the finite element method is presented together with a few new ideas. The properties of geodesic curves have been used by Yuan et al. [13–15], in defining distortion measures and inverse mappings for isoparametric quadrilateral hybrid stress four- and eight-node elements in R2. The notion of plane or space curves is one of the elementary ones in the theory of differential geometry, because the concept of a manifold comes from the generalization of a curve or a surface in R3. Further, the real global nature of differential geometry, has been used by Simo et al. [4,6,7]. A geometrically exact beam finite strain formulation is defined. The mechanical basis of such a nonlinear model can be found in the mathematical foundation of elasticity [18]. An abstract infinite dimensional manifold of mappings, a configuration space, is constructed which permits an exact linearization of algorithms, locally. A similar approach is used by Pacoste [5] for beam elements in instability problems. Special attention is focused on quadrilateral hybrid stress membrane elements with curved boundaries which belong to a series of isoparametric elements developed by Yuan et al. [14]. The distortion measures are redefined for eight-node isoparametric elements in R2 for which geodesic coordinates are used as local coordinates.

The consistent strain method in finite element plasticity

Abstract An algorithm for finite element analysis of problems in elastoplasticity with continuous stress and strain approximation is presented. By a global iteration procedure, equilibrium is preserved at the nodes in a weak sense, and the local constitutive relation between stresses and strains is satisfied. A high order numerical integration is used to achieve a good quality stiffness matrix and to evaluate the boundary between elastic and plastic regions in the case of partly plastic elements.