Jean-Loup Chenot

An inverse analysis using a finite element model for identification of rheological parameters

Abstract In order to reduce the discrepency between experimental and numerical development, a parameter automatic identification procedure from rheological test is formulated as an inverse problem. The direct model which permits to simulate the large strain behaviour during the rheological test is a Finite Element Code. The inverse problem is formulated as finding a set of rheological parameters starting from a known constitutive equation. The goal is to compute the parameter vector which minimizes an objective function representing, in the least square sense, the difference between experimental and numerical data. The high nonlinearity of the problem to be solved, requires the use of an accurate evaluation of the sensitivity matrix by analytical differentiation of governing equations with respect to the parameters. Thus the optimisation algorithm is strongly coupled with the finite element simulation. This method, namely a Computer Aided Rheology (CAR) methodology is possible in principle for all tests able to be simulated. This paper concerns the thermoviscoplastic deformation during torsion and tension tests.

OPTIMAL DESIGN FOR NON-STEADY-STATE METAL FORMING PROCESSES—I. SHAPE OPTIMIZATION METHOD

We suggest a shape optimization method for a non-linear and non-steady-state metal forming problem. It consists in optimizing the initial shape of the part as well as the shape of the preform tool during a two-step forging operation, for which the shape of the second operation is known. Shapes are described using spline functions and optimal parameter values of the splines are searched in order to produce, at the end of the forging sequence, a part with a prescribed geometric accuracy, optimal metallurgical properties and for a minimal production cost. The finite element method, including numerous remeshing operations, is used for the simulation of the process. We suggest using a least-squares-type algorithm for the unconstrained optimization method (based on external penalty) for which we describe the calculation of the derivatives of the objective function. We show that it can reduce to calculations which are equivalent to the derivative calculations of steady-state processes and to evolution equations. Therefore, the computational cost of such an optimization is quite reasonable, even for complex forging processes. Lastly, in order to reduce the errors due to the numerous remeshings during the simulation, we introduce error estimation and adaptive remeshing methods with respect to the calculation of derivatives.

3-D finite element modelling of the forging process with automatic remeshing

Abstract The purpose of this paper is to illustrate the interest of using a finite element program to simulate the forging process of industrial parts. The forge3 code, which can simulate hot forging of industrial parts, is presented: thermo-mechanical formulation, numerical resolution. It is well known that, in an updated lagrangian approach using a convective mesh, the degeneracy of the mesh occurs very rapidly and stops the simulation. An automatic mesh generation procedure for 3-D complex geometries has been developed with which it is possible to create the initial mesh of the billet as well as to remesh it after its degeneracy. This technique enables to simulate the whole forging process of complex industrial parts using quadratic tetrahedra. In order to show the effectiveness of the method, the example of the forging of a tripod has been computed. The simulation results show that the computation can be carried out using the described remeshing procedure and that it can be applied with success to even more complex geometries.

Inverse problems in finite element simulation of metal forming processes

Focuses on the inverse problems arising from the simulation of forming processes. Considers two sets of problems: parameter identification and shape optimization. Both are solved using an optimization method for the minimization of a suitable objective function. The convergence and convergence rate of the method depend on the accuracy of the derivatives of this function. The sensitivity analysis is based on a discrete approach, e.g. the differentiation of the discrete problem equations. Describes the method for non‐linear, non‐steady‐state‐forming problems involving contact evolution. First, it is applied to the parameter identification and to the torsion test. It shows good convergence properties and proves to be very efficient for the identification of the material behaviour. Then, it is applied to the tool shape optimization in forging for a two‐step process. A few iterations of the inverse method make it possible to suggest a suitable shape for the preforming tools.

A plane-strain elastoplastic finite-element model for cold rolling of thin strip

Abstract In cold rolling of thin strip, elastic roll deformation is a prominent phenomenon which may indeed govern the whole process. Analysis of the literature suggests a number of methods to solve this coupled problem; for the most severe operations, the coupling technique is more important than the precision of the computation of stress and strain. To perform as general an analysis as possible, a completely coupled finite element model is formulated, meshing a global strip-roll system with internal interface with sliding and friction. The model is two-dimensional and only analyzes roll flattening. The basic equations and numerical formulation are described. Application to several kinds of rolling passes is examined (temper rolling, thin foil rolling) with special emphasis on roll deformed shape and behaviour of metal in the roll gap (sliding/sticking zones, elastic/plastic zones).

NUMERICAL FORMULATIONS AND ALGORITHMS FOR SOLVING CONTACT PROBLEMS IN METAL FORMING SIMULATION

The problem of contact between a part and a tool or between two deforming bodies is analysed in view of metal-forming processes. For simplicity, only the case of viscoplastic materials is considered. The velocity formulation is presented. The contact with a rigid tool is considered first and various forms of contact formulations are described: nodal contact, integral or discrete formulation with a penalty or with a Lagrange multiplier method. The time integration is considered which results in various explicit or implicit contact formulations. A special attention is paid to the contact between two deforming bodies or to the self-contact when a fold is generated in the work-piece. Some 2-D and 3-D application examples illustrate the effectiveness of the proposed formulations and algorithms. Copyright

An overview of numerical modelling techniques

The mechanical equations for large deformations occurring in metal forming processes are recalled. The finite element approaches for viscoplastic or for elastic viscoplastic materials are presented briefly. Different forms of the virtual work equation for viscoplastic or elastoplastic materials, in dynamic or quasi-static processes, are reviewed. The finite element discretisation is summarised and different time integration schemes are analysed, using a compact symbolic notation. The problem of meshing, remeshing and ALE formulation is mentioned and the space-time finite element method is briefly compared with the more classical approaches.

Error estimators for viscoplastic materials: application to forming processes

The analysis of error estimation is addressed in the framework of viscoplasticity problems, this is to say, of incompressible and non‐linear materials. Firstly, Zienkiewicz—Zhu (Z2) type error estimators are studied. They are based on the comparison between the finite element solution and a continuous solution which is computed by smoothing technique. From numerical examples, it is shown that the choice of a finite difference smoothing method (Orkisz’ method) improves the precision and the efficiency of this type of estimator. Then a Δ estimator is introduced. It makes it possible to take into account the fact that the smoothed solution does not verify the balance equations. On the other hand, it leads us to introduce estimators for the velocity error according to the L2 and L∞norms, since in metal forming this error is as important as the energy error. These estimators are applied to an industrial problem of extrusion, demonstrating all the potential of the adaptive remeshing method for forming processes.

Numerical and Physical Modeling of Polymer Crystallization

Abstract In this first paper, we have revisited Avramis model and cast its basic equations into a differential system. This system is integrated numerically, which avoids unnecessary simplifying assumptions generally used in order to get analytical expressions. This allows us to introduce the variations of nucleation and growth parameters as a function of processing ones (temperature, cooling rate, shear rate, etc.). Our analysis shows that it is necessary to take into account the variation of the initial number of potential nuclei with temperature, which was usually ignored. Finally, an outpout of our calculations is the size distribution of the morphological entities, i. e., a quantitative information on microstructure.

Numerical treatment of contact and friction in FE simulation of forming processes

Abstract The problem of contact between a part and a tool or between two deforming bodies is analysed in view of metal forming processes. For simplicity, only the case of viscoplastic materials is considered. The velocity formulation is presented, with different time integration schemes. The contact with a rigid tool is considered first and various forms of contact formulations are described: nodal contact, integral formulation with a penalty or with a Lagrange multiplier method. Time integration is considered which results in various explicit or implicit contact formulations. Special attention is paid to the contact between two deforming bodies or to the self contact when a fold is generated in the workpiece.