Institute of Fundamental Technological Research

Shape and non-shape optimization is carried out for metal forming processes. This means a unified treatment of both shape parameters and other process parameters which are assumed to be design variables. An optimization algorithm makes use of the results of the analysis problem and of the sensitivity parameters obtained as a byproduct of the basic solution, in the context of the direct differentiation method. The shape sensitivity stage is formulated within the domain parametrization approach. Two alternative mappings are proposed to obtain the required derivatives with respect to the shape parameters. The behaviour of different functional considered and the effect of the boundary conditions on the optimal design are discussed.

Algorithms, Boundary conditions, Computer simulation, Finite difference method, Friction, Mathematical models, Optimization, Perturbation techniques, Sensitivity analysis, Domain parameterization, Material derivations (MDA), Optimization algorithms, Process parameters, Metal forming

A new finite element to analyze problems of anisotropic hyperelasticity is presented. The constitutive equations are derived by means of the energy method, which leads to the stress measure conjugate to the logarithmic strain. Equilibrium equation are integrated in the current configuration. Multiplicative instead of additive - decomposition of the time derivative of a strain tensor function is applied as a crucial step that makes possible the formulation for anisotropic hyperelastic materials. Unlike previous known anisotropic large deformation models, the one here presented assures the energy conservation while using the anisotropic elastic constants and the logarithmic strain measure. It is underlined that for the first time a model including all these features is presented. Some numerical examples are shown to illustrate the results obtained with this model and to compare them with other known anisotropic models.

Anisotropic material, Constitutive behaviour, Elastic material, Finite element method, Logarithmic strain measure

On the basis of a recently developed method which allows the use of linear elements for metal forming simulation within the flow approach, sensitivity analysis is carried out. Aiming at large, industrial problems, attention is focused on the explicit version, which is considered more effective for such problems, although implicit time integration is possible as well. By time step splitting a stabilization sub-matrix is obtained, which allows the use of equal interpolation for velocity and pressure. Specifically, linear triangles and tetrahedra have been used, which are easily generated by automatic meshers. Sensitivity analysis is carried out by the Direct Differentiation Method, with which similar analyses have been performed by the author for the flow approach within a direct solution scheme.

Algorithms, Computer simulation, Differential equations, Integration, Interpolation, Pressure, Sensitivity analysis, Velocity, Direct differential method, Implicit time integration, Incompressible linear elements, Linear triangles, Tetrahedra, Time step splitting, Metal forming

Simulation of transient and stationary metal-forming processes is presented in the context of a mixed velocity-pressure formulation with a rigid-viscoplastic material model. A fractional step technique is used to obtain in a natural way a discrete Laplacian in the pressure-pressure entries of the system matrix which, as is known, introduces the stabilizing effect and allows for the use of equal order interpolating functions for both discretized variables, in spite of the Babuska-Brezzi condition. The method is suitable for both implicit and explicit time integration schemes. In particular, the explicit version (forward Euler with mass lumping) is developed. The discretization is carried out for linear triangles and tetrahedra which most easily can make use of automatic mesh generators and result in a simple code. Attention is focused on the stabilizing effect as a function of material and numerical parameters. The technique is illustrated with extrusion and upsetting simulation and compared with a known pressure stabilization method.

Algorithms, Computer simulation, Integration, Interpolation, Matrix algebra, Viscoplasticity, Babuska-Brezzi condition, Fractional step techniques, Split algorithms, Metal forming

The coupled deformation and heat flow problems are simultaneously solved for metal-forming processes within the framework of the flow approach. Heat generation by plastic deformation and temperature dependence of the material properties are accounted for. The model foresees a big range of values for the Peclet number by introducing an upwind technique. As a byproduct, sensitivity analysis is carried out by the direct differentiation method. The effect of material or interface properties and numerical parameters involved in the upwind scheme on the solution can thus be assessed. As a practical application, direct and inverse extrusion processes are simulated.

Approximation theory, Galerkin methods, Interfaces (materials), Mathematical models, Metal extrusion, Optimization, Plastic deformation, Problem solving, Sensitivity analysis, Thermal effects, Direct differentiation method, Heat flow, Thermo-mechanical modelling, Computer simulation

A simple element to model frictional contact in steady state metal forming processes is presented together with the sensitivity analysis to the friction coefficient in a Coulomb friction law. The interest of such model arises from the analysis of rolling processes and a two dimensional approach to cutting problems, where the contact zone is to be determined, however a stationary state is present in most part of the operation. The flow approach proves to be an adequate method to handle efficiently this situation. The contact elements impose a restriction in the velocity component normal to the boundary and a tangential friction force opposite to the velocity. The parts of the boundary which are not in closed contact are treated as free surfaces, which must fulfill the condition of being streamlines. Sensitivity analysis with respect to the friction coefficient is performed by the Direct Differentiation Method (DDM). The effect of variations in this parameter is discussed for the simulation of an extrusion and a cutting problem.

Computer simulation, Extrusion, Friction, Metal cutting, Rolling, Sensitivity analysis, Contact zone, Coulomb friction law, Direct differentiation method, Flow approach, Friction coefficient, Frictional contact, Steady state metal forming processes, Metal forming

The flow formulation for metal forming analysis based on a rigid-viscoplastic material rncdel is considered. Specifically, sensitivity evaluation techniques are discussed for different solution variables with respect to variations in parameters entering the constitutive (and other) equations such as material constants, imposed velocities or friction coefficient. A method to avoid spurious pressure modes is introduced which allows to use Ql/Ql elements and thus to accurately calculate pressures, their sensitivities and friction forces. In addition, one pressure unknown for each node is available in this method, thus yielding a finer discretisation for this variable.

Computer aided analysis, Friction, Sensitivity analysis, Velocity, Viscoelasticity, Metal forming processes, Parameter sensitivity, Metal forming

Shape sensitivity analysis is presented for metal-forming processes described in terms of the flow approach [1]. The direct differentiation method is employed to derive the sensitivity expressions. The continuum approach is adopted so that both the equilibrium equations and response functional are differentiated before discretization. The necessary derivatives with respect to shape design variables are calculated making use of the framework already available for isoparametric elements (control volume approach). In finite element implementation a system of equations is obtained which has the same system matrix as the equilibrium problem. The right-hand side is calculated by partially differentiating the internal and external forces with respect to the design parameters. The procedure is illustrated by calculating sensitivities of some independent and dependent variables with respect to the die angle in an extrusion problem and to the roll radius in a plane rolling.

Calculations, Continuum mechanics, Differentiation (calculus), Finite element method, Matrix algebra, Metal extrusion, Metal forming, Rolling, Die angle, Direct differentiation method, Discretization, Equilibrium equations, Response functional, Roll radius, Shape sensitivity analysis, Sensitivity analysis

The general problem of sizing, material and loading parameter sensitivity of non‐linear systems is presented. Both kinematic and path‐dependent material non‐linearities are considered; non‐linear sensitivity path is traced by an incremental solution strategy. The variational approach employed is quite general and can be employed for studying sensitivity of various path‐dependent highly non‐linear phenomena. Both the direct differentiation method (DDM) and adjoint system method (ASM) are discussed in the context of continuum and finite element mechanics. The merits of using the consistent tangent matrix and the necessity of accumulation of design derivatives of stresses and internal parameters are indicated. Aspects of sensitivity problems in metal forming are also discussed. A number of examples illustrate the paper.

Direct differentiation method, Adjoint system method, Metal forming

This paper deals with numerical aspects of nonlinear static analysis of a guyed mast of 649 m height. A concept of a new structure constructed of solid bars instead of tubes is analyzed. Due to decrease of member diameters the wind load declines, thus the horizontal displacements are smaller. A few structural variants of the proposed mast are calculated.

guyed mast, nonlinear analysis, tendons