Keywords: virtual prototype, contact, interactive simulation, reduced response surface, enrichment method, Karhunen-Loève expansion.

In an industrial context, the virtual assembly of mechanical components can be interesting during the design phase to simulate a manual operation of maintenance. This operation can be realised with interactive simulator. So it is necessary to calculate in real time the contact due to the assembly. But the analysis in real time of a complex non-linear mechanical model by using the finite element method (FEM) is not possible. So the simulation is divided in two steps: a learning step and a real time step. During the first one, a campaign of possible load cases with the FEM is calculated. This campaign is representative of the different possible manipulations of the virtual prototype. The second step is done by interpolation on these results. This paper focuses on the learning step to calculate the campaign as quickly as possible and to obtain a reduced response surface [1].

To reduce the number of degrees of freedom (DOF) n, an enrichment method [2] is used to solve the quasi-static equilibrium. The displacement is decomposed into a basis Phi of few m modes, and coefficients a. In this way, a reduced tangent matrix (m * m) and a reduced residual )m * 1) can be defined. During each Newton Raphson iteration, the cost of the inversion of the reduced tangent matrix is very smaller than the inversion of the classical tangent matrix (n * n) because m<<n. To describe non-linearities, the a priori initial basis has to be enriched, and so its size increases progressively during the campaign. In order to reduce this size, we apply the Karhunen-Loève Expansion (KLE) on the coefficients a.

The model reduction method is applied to the crushing of a half cylinder on an non-deformable planar structure. The campaign is composed of load cases which are successive crush levels. The enrichment method is faster than the classical FEM by about 9%. The stored data size was reduced from 4500 DOF to 201 modes. Moreover, enrichment with the KLE is better than enrichment without the KLE. The CPU time is divided by 2 compared with classical FEM, as a result of the basis size control. At the end of the campaign, 10 modes are obtained, compared with 4500 DOF.

So this paper shows that the model reduction method allows a reduced response surface to be quickly built compared with the classical FEM. In a next step, our method will be compared with hyperreduction [3]. The optimisation of the campaign, which is regular on our model, will also be necessary for more complex problems.

1

F. Druesne, J.L. Dulong, P. Villon, "Model reduction applied to simulation of mechanical behaviour for flexible parts", The 8th International Conference on Computational Structures Technology, 2006. doi:10.4203/ccp.83.218

2

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

3

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

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