Keywords: sensitivity analysis, large deformations, elastoplasticity, finite element method, direct differentiation method.

Influence of a discontinuous nature of the elastoplastic systems response at large strain onto their sensitivity with respect to a design parameter is considered in the paper. It is discussed in the framework of the finite element modelling using the direct differentiation method for the sensitivity response calculation. It is shown that sensitivity response cannot be a smooth function on its overall path. This fact is confirmed by some examples where the sensitivities have been calculated by the presented direct differentiation procedure and the corresponding results have been compared to the results obtained by the central finite difference method.

Large strain elastoplastic behaviour of systems is considered within a standard theoretical approach that assumes additiveness of deformation rates and hypoelastic characterisation of the elastic response using the rotation-neutralised quantities, e.g. [1]. Thus the sensitivity analysis is carried out on a different theoretical basis as it is presented in other papers addressing the discussed subject (see e.g. [2,3,4]). Using the direct differentiation method only some essential parts of the sensitivity equation derivation are presented. In spite the fact that a discretization of a system by finite elements is taken into account, a differentiation process is performed on the governing equations of continuous form. The resulting sensitivity equation could then be discretized for the further numerical formulation. This approach, the so-called a continuous one, has also been applied in reference [4].

The sensitivity equation is linear with respect to the incremental sensitivity of the basic response variable, i.e. sensitivity of the incremental displacement vector field. Therefore, the sensitivity variable can be calculated without iterative procedure. Even more, the coefficient matrix of the equation is exactly the same as the algorithmic tangent matrix used at the last iteration in the solution of the primal problem, i.e. in evaluation of the system response. The solution of the sensitivity problem requires thus only a little part of the computer time needed for the primal analysis of the system.

The sensitivity calculations in the case of elastoplastic systems are fully characterized by the material transition from the elastic to the elastoplastic state, and vice-versa. Because of this at some points on the response path of such systems the sensitivity does not exist, or is not unique. The sensitivities of the problem dependent quantities are therefore not smooth functions of the design variables even in a continuous formulation. This may become a serious problem in the context of approximate discrete analyses, where integration over the time interval is performed incrementally, and the space domain is discretized by finite elements. Actually, in continuous formulations non-uniqueness of the sensitivities is likely to be present, but it is unlikely to appear when a numerical integration of the constitutive equations using finite time increments is considered. In addition, because of the domain discretization the corresponding sensitivity response on the overall response path will not be smooth, and it may experience jumps that are not consistent with the natural response of the system. This is no doubt an inconvenience that may bother effectiveness, or even reliability, of the solution algorithms based on the sensitivity data.

In view of this inconvenient aspect the derivation of the sensitivity expressions is given to an extent from which the most relevant terms for a discontinuous sensitivity response could be noticed. These terms are the consistent constitutive moduli with their numerical values depending on the material state, which is established during the solution of the primal problem at integration points of the numerical model. By using a consistent tangent operator in the integration a sufficient degree of objectivity of the material transition is attained, however, it is demonstrated that the time incrementation, but above all the space discretization, introduce some new effects with a significant impact on the numerically computed sensitivities.

The discussed issues are analysed by considering three examples. The sensitivity calculations are carried out by the presented direct differentiation method, and in order to enable a comparison also by a more time consuming central finite difference method

1

T. Rojc, "On a mixed approach to the finite element solution of large strain elasto-plastic problems" in "Proceedings of the 4th International Conference on Computational Plasticity", Owen, D. R. J., Oñate, E., Hinton, E. (Editors), Pineridge Press, Swansea, UK, 95-106, 1995.

2

L. Fourment, T. Balan and J. L. Chenot, "Optimal design for non-steady-state metal forming processes�II. Application of shape optimization in forging", Int. J. Numer. Meth. Engng., 39, 51�65, 1996. doi:10.1002/(SICI)1097-0207(19960115)39:1<51::AID-NME845>3.3.CO;2-R

3

J.C. Gelin, O Ghouati, "An inverse solution procedure for material parameters identification in large plastic deformations", Communications in Numerical Methods in Engineering, 12, March, 161-173, 1996. doi:10.1002/(SICI)1099-0887(199603)12:3<161::AID-CNM961>3.0.CO;2-E

4

N. Zabaras, Y. Bao, A. Srikanth and W. G. Frazier, "A continuum Lagrangian sensitivity for metal forming processes with applications to die design problems", International Journal for Numerical Methods in Engineering; 48, 679-720, 2000. doi:10.1002/(SICI)1097-0207(20000620)48:5<679::AID-NME895>3.0.CO;2-U

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £122 +P&P)