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A synopsis is presented of the numerical finite element methodology currently in use at the Institute for Computer Applications (ICA) for the simulation of industrial forming processes. The development of the method is based on the inelastic properties of the material with an extension towards the inclusion of elastic effects and accounts for the thermal phenomena occurring in the course of the deformation. An essential constituent of the computational procedure is the treatment of the unsteady contact developing between the workpiece material and the tool during forming, and of the associated friction phenomena. Automatic mesh generation and variable discretization adaptable to the development of the numerical solution are of importance for industrial applications. These aspects are presented and discussed. Furthermore, solution techniques for thermomechanically coupled problems are considered and investigated with respect to their numerical properties. Application to industrial forming processes is demonstrated by means of three‐dimensional hot rolling and of superplastic sheet forming.

Addresses problems in mechanics and physics involving two or more coupled variables of different nature, or a number of distinct domains which interact. For these kinds of problems, considers numerical solution by the coupling of operators appertaining to the individual participating phenomena, or defined in the domains. Reviews the co‐operation of distinct discretized operators in connection with the integration of temporal evolution processes, and the iterative treatment of stationary equations of state. The specification of subtasks complies with the demand for an independent treatment on different processing units arising in parallel computation. Physical subtasks refer to problems of different field variables interacting on the continuum level; their number is usually small. Fine granularity may be achieved by separating the problem region into subdomains which communicate via the boundaries. In multiphysics simulations operators are preferably combined such that subdomains are processed in parallel on different units, while physical phenomena are processed sequentially in the subdomain.

The purpose of the present study is to explore the incomplete substitution of the simplex triangular finite element by either of two models: one evolving out as part of the element flexibility, and the other as part of the element stiffness.

The elastic energy stored in each of the units under stress or strain decides on stiffer and weaker responses. The pertaining Rayleigh quotient in terms of the flexibility matrices allows bounding the distance of the spring cell models to the finite element in dependence of the triangle configuration.

Despite a superiority of the flexibility cell concept observed in computations, the study reveals constellations of shape and stressing of the triangle that favour the stiffness concept. The latter is seen to behave stiffer than its flexibility counterpart and produces results more distant to the finite element in most cases.

The difference between the stiffness and the flexibility approach to spring cells is investigated for triangular elements in dependence of the geometrical configuration under specific conditions of stressing. This suffices to refute an exclusive superiority of the flexibility concept although largely true.

The results of the investigation appear useful in deciding between the spring cell models depending on the case of a spring lattice application.

The flexibility approach to the spring cell is not widely known yet. This cell model deserves a study on performance and comparison to the different, more common stiffness cell model.

The employment of spring cell substitutes for the numerical analysis of solids and structures in place of finite elements has occasioned research on the subject with regard to both, the applicability of existing approaches and the advancement of concepts. This paper aims to explore in the context of linear elasticity the substitution of the simplex tetrahedral element in space and the triangle in the plane by corresponding spring cells deduced on a flexibility basis using the natural formalism.

The natural formalism is characterized by the homogeneous definition of strain and stress along the lines connecting nodes of the simplex tetrahedron and the triangle. The elastic compliance involves quantities along the prospective spring directions and offers itself for the transition to the spring cell. The diagonal entities are interpreted immediately as spring flexibilities, the off-diagonal terms account for the completeness of the substitution. In addition to the isotropic elastic material, the concept is discussed for anisotropic elasticity in the plane.

The natural point of view establishes the spring cell as part of the continuum element. The simplest configuration of pin-joined bars discards all geometrical and physical cross effects. The approach is attracting by its transparent simplicity, revealing deficiencies of the spring cell and identifying directly conditions for the complete substitution of the finite element.

The spring cell counterparts of the tetrahedral- and the triangular finite elements allow employment in problems in three and two dimensions. However, the deficient nature of the approximation requires attention in the design of the discretization lattice such that the conditions of complete finite element substitution are approached as close as possible.

Apart from plane geometries, triangular spring cells have been assembled to lattice models of space structures such as membrane shells and similar. Tetrahedral cells have been used, in modelling plates and shell structures exhibiting bending stiffness.

The natural formalism of simplex finite elements in three and two dimensions is used for defining spring cells on a flexibility basis and exploring their properties. This is a novel approach to spring cells and an original employment of the natural concept in isotropic and anisotropic elasticity.

An intended numerical analysis of solids and structures by spring cell substitutes in place of finite elements has occasioned considerable research on the subject. This paper aims to expose two alternative concepts evolving out of Argyris’ natural approach to the simplex triangular element. One is based on an approximation of the element flexibility and the other approximates the stiffness with coincidence at the ideal conditions of complete substitution.

Characteristic of the natural formalism is the homogeneous definition of strain and stress along the sides of the triangular element. The associated elastic compliance offers itself for the transition to the spring cell. The diagonal entities are interpreted immediately as springs along the element sides, and the off-diagonal terms account for the completeness of the substitution. In addition to the flexibility concept, the spring cell is deduced alternatively from the element’s natural stiffness. The difference in the flexibility result lies in the calculatory cross-sectional areas of the elastic bar members.

From the natural point of view, the spring cell evolves out of the continuum element to the desired degree of substitution. The simplest configuration of pin-joined bars discards all geometrical and physical cross effects. The approach is attractive because of its transparent simplicity.

The difference between the stiffness and the flexibility approach to spring cells is demonstrated for triangular elements that suit the problems lying in plane stress or plane strain. More general states of stress and strain involve spring cell counterparts of the tetrahedral finite element.

Apart from plane geometries, triangular spring cells are assembled to lattice models of space structures, such as membrane shells and similar.

The natural formalism of simplex finite elements is used for deducing spring cells in two variants and exploring their properties. This is a novel approach to spring cells and an original employment of the natural concept.

Under restriction of an isotropic elastic response of deformed lattice, develops a covariant theory of finite elastoplasticity in principal axes of a pair of deformation tensors. In material description, the tensor pair consists of the plastic deformation tensor and the total deformation Cauchy‐Green tensor. Applies the proposed theory to elastoplastic membrane shells, whose references and current configurations can be arbitrary space‐curved surfaces. Pressure‐insensitive von Mises yield criterion with isotropic hardening and a quadratic form of the strain energy function given in terms of elastic principal stretches are considered as a model problem. Through an explicit enforcement of the plane stress condition we arrive at a reduced two‐dimensional problem representation, which is set in the membrane tangent plane. Numerical implementation of the presented theory relies crucially on the operator split methodology to simplify the state update computation. Presents a set of numerical examples in order to illustrate the performance of the presented methodology and indicate possible applications in the area of sheet metal forming.

New contact boundary modelling is achieved with a basic set of 2 and 3 dimension contact primitives. Contact constraints are originally introduced in the variational equations and associated Newton—Raphson scheme via an external penalty formulation using primitive equations. Consequently, penalty part of external load vector and tangent stiffness matrices are developed for all contact primitives. In this way, contact prescribed boundary displacements are also taken into account. Contact treatment is then completed with Newton—Raphson elements for elastic and plastic regularized friction constitutive models. In this paper, the process is extended to elastoplastic models. Finally, we propose a self acting procedure with contact algorithms (interiority, sliding and contact loss) and related subroutines for implementation in finite element framework. We illustrate these developments by means of two‐dimensional open die forging and three‐dimensional plate coining typical benchmarks with reference to bulk elastoplastic and viscoplastic constitutive models.

An isothermal, steady‐state, hot‐strip rolling problem for rigid‐plastic, strain rate sensitive and incompressible materials with coulomb‐siebel friction contact conditions is considered. The corresponding penalty variational statement is given and a convergent finite element ‐ secant‐modulus method is presented. Two‐dimensional example problems are solved and comparisons with the available experimental and numerical results are made. The influence of the friction coefficient on the contact velocities, stresses and equivalent strain rates is illustrated.

The paper describes developments in the numerical analysis of metal forming processes mainly motivated by industrial applications. It deals with a complete consideration of the unsteady contact developing between the material and the die, the regeneration of the finite element mesh during the course of the calculation, and with the simulation of superplastic forming processes. In particular, an approach relating both the contact pressure and the friction force to the motion of the material relative to the die surface leads to a convenient computational procedure and to a smooth numerical behaviour under friction. The topological part of the contact algorithm appears well‐suited also for the redefinition of the discretization mesh. As a selected application, superplastic forming is considered in conclusion. Industrial practice requires the adjustment of the forming pressure to a prescribed value of the maximum rate of deformation in the material.

This paper gives a review of the finite element techniques (FE) applied in the area of material processing. The latest trends in metal forming, non‐metal forming and powder metallurgy are briefly discussed. The range of applications of finite elements on the subjects is extremely wide and cannot be presented in a single paper; therefore the aim of the paper is to give FE users only an encyclopaedic view of the different possibilities that exist today in the various fields mentioned above. An appendix included at the end of the paper presents a bibliography on finite element applications in material processing for the last five years, and more than 1100 references are listed.