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

The performance of path-following techniques depends on so-called constraint equations, which are used in the augmented incremental systems. It is common practice in the literature, that these equations are explicitly defined following the arbitrary geometrical or physical ideas. In the present paper, their role is discussed in order to establish a reliable method of augmentation. It is shown that the explicitly defined constraint equation is not necessary. The implicit formulation is proposed, which is based on the rank analysis performed on the rectangular matrix of the homogeneous incremental system. The proposed method allows to solve problems, where the well-known continuation schemes fail. The theoretical considerations are illustrated by the numerical examples.

In certain problems of loading of elastic-perfectly plastic thin sheets a continuous displacement solution may not exist. The evolution of plastic zone is then connected with the evolution of discontinuity lines in both velocity and displacement fields. In the present paper it is assumed that in the presence of discontinuity lines the localized plastic zones start to proceed. A numerical study of decohesion within thin elastic-plastic sheets is conducted to total collapse. It is shown, that the localized plastic flow may develop simultaneously with the diffuse plastic zones. The structural softening caused by decohesive cracks is coupled with a complex elasto-plastic deformation process, where the previously developed diffuse plastic zones are subjected to unloading. The post-critical analysis is performed using a new reliable algorithm of a continuation method. The algorithm is based on a rank analysis of the rectangular matrix of the homogeneous set of incremental equations.

In this paper, it is demonstrated that incremental problems of nonlinear potential and nonpotential systems, including sensitivity problems, can be uniformly treated within the analysis of a homogeneous set of equations. Regular and critical states are considered. It is shown that rank analysis of a rectangular matrix of a homogeneous set of incremental equations reveals all possible problems associated with singularity conditions. When considering nonlinear design modification problems, it is necessary to use derivatives of the secant and tangent stiffness matrices. A direct approach to differentiation of stiffness matrices on a finite element level in sensitivity problems is also presented. Simple illustrative examples are discussed.

design sensitivity, nonlinear problems, incremental analysis

This paper is concerned with rank analysis of rectangular matrix of a homogeneous set of incremental equations regarded as an element of continuation method. The rank analysis is based on a known feature that every rectangular matrix can be transformed into the matrix of echelon form. By inspection of the rank, correct control parameters are chosen and this allows not only for rounding limit and turning points but also for branch‐switching near bifurcation points.

rank analysis, rectangular matrix

IN THE ELASTO-PLASTIC analysis of disks within the small strain theory, a continuous displacement solution may not exist and discontinuities in both velocity and displacement may occur within hyperbolic stress regimes or along transition lines between elliptic and hyperbolic regimes. To study elasto-plastic behaviour in the presence of discontinuity lines, it is assumed that an additional constitutive relation exists between displacement discontinuity and interface traction along a stationary discontinuity line. The general formulation is illustrated by a solution for an axisymmetric disk, using both Tresca and Huber-Mises yield conditions. It is demonstrated how the solution evolves from brittle to ductile response depending on disk thickness.