A Priori Reduction method and A Priori Hyper-Reduction method for a non linear problem

HAL Id: hal-02958311

https://hal.archives-ouvertes.fr/hal-02958311

Submitted on 5 Oct 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A Priori Reduction method and A Priori

Hyper-Reduction method for a non linear problem

J Dulong, Frédéric Druesne, P Villon

To cite this version:

J Dulong, Frédéric Druesne, P Villon. A Priori Reduction method and A Priori Hyper-Reduction method for a non linear problem. ECCM 2010 IV European Conference on Computational Mechanics, 2010, Paris, France. �hal-02958311�

ECCM 2010

IV European Conference on Computational Mechanics Palais des Congrès, Paris, France, May 16-21, 2010

A Priori Reduction method and A Priori Hyper-Reduction method for a non linear problem

J.L. Dulong, F. Druesne, P. Villon

Laboratoire Roberval, Université de Technologie de Compiègne, France, {jean-luc.dulong, frederic.druesne, pierre.villon}@utc.fr

The A Priori Hyper-Reduction (APHR) method was developed for few years by D.Ryckelynck [2]. Based on an extremely reduced finite element model, this method aims to perform extremely fast finite element calculations. This work aims to study some aspects of its numerical behavior.

Methods

In statics, the standard Finite Element Method (FEref) results in a system of equations which is solved by a Newton Raphson (NR) procedure. After calculating the tangent stiffness matrix KT and the equation residual r, the displacement increment du is obtained by the linear system K_Tdu=r .

A first way to reduced the system size [1] is to project the displacement on a base Φ, known a priori : u=a . The reduced displacement a is now calculated by equation also projected on this base (1), following a Galerkin approach. This method will be called A Priori Reduction (APR).





The previous equation results in a reduced stiffness matrix, which can be inverted rapidly, but a large matrix product is necessary between KT and Φ . To reduce this last cost, Ryckelynck has proposed [2] to assemble KT only on few degrees of freedoms, choosen correctly, and determined by a sparse identity matrix P. This method was called APHR. The resulting iterative equation (2) comes from the minimization of the residual following [3].





Results

Some tests were led on a short embedded beam, loaded at its extremity (figure 1a). The mechanical problem is non linear due to neohookean material, and large displacements. Calculations was led with FEAP software. An important point is that an adaptative time increment was used allowing the method to increase the time increment if possible.

Figure 2a shows that time CPU increases when loading force increases. Mainly, for low force, APR and APHR are faster than FEref, but the speed up is reduce (about 1.6 ). On the contrary, for high loading force, FEref curve is characterized by a slope change : for maximum force, the speed up of APR and APHR is about 5. The change in FEref behavior is due to limit in NR iterations (figure 2b) : if 5 FEref iterations cannot solve the system, then time increment has to be reduced. On the contrary, APR and APHR have the possibility to increase the number of iterations to reach solution. And then,

1

time increment can stay large : it can even increase. This behavior can be explained by the fact than, with FEref method, only one search direction is used to solved the system, determined by KT. With APR and APHR, several search directions are used, determined by base Φ size.

Comparing the two A Priori methods, APHR is slightly faster than APR but speed up is low.

Indeed, time to assemble and calculate reduced stiffness matrix is decreased compared to APR, but a larger base Φ is required to compensate for the lack of information due to reduced mesh (figure 1b).

So the time to adapt the base becomes preponderant during resolution.

Figure 1: Complete (a) and reduced (b) meshes

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

0 50 100 150 200 250 300

Force (N)

CPU time (s)

FE ref APR APHR

0 200 400 600 800 1000

3 4 5 6 7

Number of iterations / time increment

Number of time increments

FE ref APR APHR

Figure 2: Methods CPU time (a) and number of iterations (b) Conclusion

The speed up due to APHR compared to APR is low. The main part of the speed up is due to the numerical behavior of FEref limited in number of iterations. The A Priori methods are superior when non linearity is large.

References

[1] 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.

[2] D. Ryckelynck, A priori hyperreduction method: an adaptive approach, Journal of computational physics, Vol. 202, pp 346-366, 2005.

[3] B. Lefevre, F. Druesne, J.L. Dulong, P. Villon, Different formulations for model reduction to simulate the crush of a mechanical part, International Review of Mechanical Engineering, 2009 ;3(2) : 162-170.

2