Keywords: virtual prototype, real time deformation, non-linear mechanical model, Karhunen-Loève expansion, enriched method.

Mechanical industries use simulators to test assembly or maintenance manual operations, and thus optimize the mechanical design, maintenance and training of operators. This virtual prototyping permits, during the design phase, an operator to check in real time the feasibility of assembling two parts, or to check the access to engine parts [1]. Some manual operations require the handling of rigid parts [1], others induce more or less large deformations of the handled part [2].

The manipulation of deformable parts has been studied by the medical industry [3] and the graphics industry [4], but these models are not always respected for continuum mechanics, and are not adapted for the non-linear mechanical industrial problems.

Our applications focus on flexible parts, like an automobile hose. Here, the mechanical problem is considered with geometrical and material non linearities. To solve this kind of problem, we propose to use the classic finite element method (FEM). But these computations can not be feasible in real time for running a simulator using deformable parts. For some applications, specific strategies are then employed [5] but are not generalizable.

We propose a methodology composed of two phases, the training phase then the immersion phase. Before the phase of real time immersion, we start a representative calculation campaign of possible handlings, we focus on this training phase in this paper. Thereafter, the real time phase will be obtained by the superposition of the deformation modes obtained from the training phase.

Industrial problems solved by the FEM lead to large models, so we interested in methods of model reduction. In order to manage the training phase, we present two methods to reduce the model dimension, an a posteriori method then an a priori method [6,7,8].

The a posteriori approach is interesting in terms of reduction of the data volume necessary for the future real time phase. The a priori approach is an adaptive strategy, with enrichment of the modal basis during the campaign. The size of this basis increases the cost of the campaign calculation when the number of degrees of freedom and the number of modes is important. It was observed for the application of the hose. Thus, during the campaign, the reduction of this basis is necessary, when it reaches a certain limit, with the Karhunen-Loève expansion.

This a priori approach provides results interesting for the management of the handling of an automobile hose. However this training phase is still to be optimised by a technique of hyper-reduction [9]. The resolution of the system will be done then on a reduced number of degrees of freedom.

Thereafter, the immersion in real time phase will be realised by interpolation from the results of the training phase.

1

W. Son, K. Kim, N.M. Amato, J.C. Trinckle, "A Generalized Framework for Interactive Dynamic Simulation for Multirigid Bodies", IEEE Transactions on Systems Man and Cybernetics Part B Cybernetics, 34(2):912-924, 2004. doi:10.1109/TSMCB.2003.818434

2

F. Krause; J. Neumann, "Haptic interaction with non-rigid materials for assembly and dissassembly in product development", CIRP-Annals-Manufacturing-Technology; 50 (1) : 81-84, 2001. doi:10.1016/S0007-8506(07)62076-9

3

G. Picinbono, H. Delingette, and N. Ayache, "Non-Linear Anisotropic Elasticity for Real-Time Surgery Simulation", Graphical Models, 65(5):305-321, 2003. doi:10.1016/S1524-0703(03)00045-6

4

P.G. Kry, D.L. James and D.K. Pai, "EigenSkin: real Time Large Deformation Character Skinning in Hardware", ACM SIGGRAPH Symposium on Computer Animation, 153-160, 2002.

5

A. Mikchevitch, J.C. Léon and A. Gouskov, "Realistic force simulation in path planning for virtual assembly of flexible beam parts", Virtual Concept, Biarritz, 2003.

6

K. Karhunen, "Zur Spektraltheorie stochasticher Prozesse", Annales Academiae Scientiarum Fennicae, 37, 1946.

7

P. Krysl, S. Lall, J.E. Marsden, "Dimensional Model Reduction in Non-linear Finite Element Dynamics of Solids and Structures", International Journal for Numerical Methods in Engineering, 2000. doi:10.1002/nme.167

8

F. Druesne, J.L. Dulong, P. Villon, "Real time simulation of non linear mechanical model", Virtual Concept, Biarritz, 2005.

9

D. Ryckelynck, "A priori hyperreduction method: an adaptive approach", Journal of computational physics 202 346-366, 2005. doi:10.1016/j.jcp.2004.07.015

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