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Uses a rigid viscoplastic formulation to simulate hot and cold forging processes. The finite element solution uses mixed methods in which the independent variables can be velocities, pressures and deviatoric stresses. Uses interface elements both in the mechanical and the thermal analysis, to take into account the effects of contact and friction, thermal conductivity of lubricants and heat generated by friction. The code developed includes an adaptive mesh refinement, triggered by an error estimator based on energy norms evaluated from nodal stress values, recovered from a local continuous polynomial expansion, and those given by the numerical solution. Assesses the code developed, using experimental results.

Describes a hybrid formulation solution based on variational techniques, where unknown functions are combined with a set of Fourier trigonometric expansions. The unknown functions refer to the toroidal shell distortion in the longitudinal direction when the shell is submitted to generalised in‐plane forces in the linear‐elastic stress field. This set of functions is solved from a system of differential equations, having calculated the eigenvalues and corresponding eigenvectors in the complex field from the system equations matrix. For this purpose the MAPLE^® mathematical solver was used. This module led to an analytical solution, which is exact for each Fourier term used in the global solution. The final results agree very well with finite element method or other solutions obtained using a total expansion of trigonometric functions in the longitudinal and meridional directions.

An optimisation method for design of intermediate die shapes needed in some forging operations is presented. The basic problem consists of finding an optimal two‐step forging sequence by automatically designing the shape of the preforming tools. The optimisation problem is defined based on an inverse formulation. The objective function of the optimisation problem is a function describing the quality of the obtained part by measuring the die underfill. The finite element method is used to simulate the forging problem. The optimisation method is based on a modified sequential unconstrained minimisation technique and a gradient method. The sensitivity‐dependent algorithm requires computing the derivatives of the objective function with respect to the design variables defining the preform shapes. A direct differentiation method has been developed for this purpose. The optimisation scheme is demonstrated with two axisymmetric forging examples in which optimal preform dies are obtained.