# Benchmarks made in the PLEIADES platform

## Fuel performance codes based on the Cast3M finite element solver

Numerous performances assessments were made within the PLEIADES platform : replacing fortran implementations by their MFront counterparts led to significant improvements, from $$30\%$$ to $$50\%$$ of the total computational times of some fuel performance codes developed in the platform.

This improvements were mainly due to fact that the behaviour integration schemes changed from explicit Runge-Kutta schemes to implicit ones. The main benefit of MFront was to grant users an easier access to the implicit schemes.

## Cyrano3 fuel performance code

Relying on the specific modelling hypotheses supported by this code, namely axisymmetrical generalised plane stress and axisymmetrical generalised plane strain, highly specialised and efficient implementations of mechanical behaviours were developed in Cyrano3 fuel performance code (see Thouvenin et al. 2010) for both isotropic and orthotropic materials : numerical integration boils down to solving a scalar non linear equation in both cases and provides the consistent tangent operator.

The figure below compares the total computational times of a native implementation of a cladding behaviour to its equivalent MFront implementation: the last one appears to be competitive with the native implementation (the average computational time using the MFront implementation is sightly lower than the average computational time using the native implementation).

# Benchmarks based on the Code-Aster finite element solver

Some benchmarks comparing the implementation generated through MFront to the native implementation provided by the Code-Aster finite element solver. Graphical illustrations shows that the results obtained with both implementations are indistinguishable.
Behaviour and test description Algorithm Total computational times (Code-Aster vs MFront) Graphical illustration
Visco-plastic and damaging for steel (see Mustata and Hayhurst 2005; EDF 2012) - 3D Notched specimen implying large deformation Implicit $$17mn 43s$$ vs $$7mn 58s$$
Damaging for concrete (see Mazars, Hamon, and Grange 2014; EDF 2013b), 3D beam bending Default $$45mn$$ vs $$63mn$$
Generic Single crystal viscoplasticity (see Méric and Cailletaud 1991; EDF 2013a), 3D aggregate, 300 grains Implicit $$28mn$$ vs $$24mn$$
FCC single crystal viscoplasticity (see Monnet, Naamane, and Devincre 2011 EDF (2013a)) , 2D specimen with displacement boundary conditions from EBSD experiment Ìmplicit $$33m54s$$ vs $$29m30s$$
FCC homogeneized polycrystals 30 grains (see Berveiller and Zaoui 1978; EDF 2013a), unit testing Runge-Kutta 4/5 $$9s67$$ vs $$8s22$$
Anisotropic creep with phase transformation, 3D pipe (see EDF 2013c) Implicit $$180s$$ vs $$171s$$

Developers of the Code-Aster general purpose finite element solver, made independent extensive tests, comparing their own native implementations to the ones generated with MFront, generally using an implicit scheme in both cases. Without discussing the very details of each test performed, several general conclusions can be drawn:

• native implementations offers superior performances in the case of simple explicit behaviours (Mazars damaging behaviour (see Mazars, Hamon, and Grange 2014)) in the case of isotropic behaviours that can be reduce to one scalar equations (see Chaboche and Cailletaud 1996). For explicit behaviours, the difference will be reduced by the development of an optimised treatment of MFront behaviours. In the second case, the difference can be explained by the fact that the Code-Aster implementations uses the Brent algorithm (see Brent 1973) which clearly outperforms the standard Newton method. The availability of this algorithm in MFront is planed.
• for more complex behaviours, MFront implementation are on par or outperforms the native implementations.

For a given behaviour, the development time was found significantly lower with MFront.

# References

Berveiller, M., and A. Zaoui. 1978. “An Extension of the Self-Consistent Scheme to Plastically-Flowing Polycrystals.” Journal of the Mechanics and Physics of Solids 26 (5–6): 325–44. doi:10.1016/0022-5096(78)90003-0.

Brent, Richard P. 1973. Algorithms for Minimization Without Derivatives. Dover Publications.

Chaboche, J.L., and G. Cailletaud. 1996. “Integration Methods for Complex Plastic Constitutive Equations.” Computer Method in Applied Mechanics and Engineering 133: 125–55.

EDF. 2012. “Comportement Viscoplastique Avec Endommagement de Hayhurst.” Référence du Code Aster R5.03.13 révision : 8886. EDF-R&D/AMA. http://www.code-aster.org.

———. 2013a. “Comportements élastoviscoplastiques Mono et Polycristallins.” Référence du Code Aster R5.03.11 révision : 10623. EDF-R&D/AMA. http://www.code-aster.org.

———. 2013b. “Modèle d’endommagement de Mazars.” Référence du Code Aster R7.01.08 révision : 10461. EDF-R&D/AMA. http://www.code-aster.org.

———. 2013c. “Modèle de Comportement élasto-Visqueux META_LEMA_ANI Avec Prise En Compte de La Métallurgie Pour Les Tubes de Gaine Du Crayon Combustible.” Documentation du Code-Aster R4.04.05. EDF-R&D/AMA. http://www.code-aster.org.

Mazars, Jacky, François Hamon, and Stéphane Grange. 2014. “A New 3D Damage Model for Concrete Under Monotonic, Cyclic and Dynamic Loadings.” Materials and Structures, October, 1–15. doi:10.1617/s11527-014-0439-8.

Méric, L., and Georges Cailletaud. 1991. “Single Crystal Modelling for Structural Calculations.” Journal of Engineering Material and Technology 113 (January): 171–82.

Monnet, G., S. Naamane, and B. Devincre. 2011. “Orowan Strengthening at Low Temperatures in Bcc Materials Studied by Dislocation Dynamics Simulations.” Acta Materialia 59 (2): 451–61. doi:10.1016/j.actamat.2010.09.039.

Mustata, R., and D. R. Hayhurst. 2005. “Creep Constitutive Equations for a 0.5Cr 0.5 Mo 0.25V Ferritic Steel in the Temperature Range 565 °C - 675 °C.” International Journal of Pressure Vessels and Piping 82 (5): 363–72. doi:10.1016/j.ijpvp.2004.11.002.

Thouvenin, Gilles, Daniel Baron, Nathalie Largenton, Rodrigue Largenton, and Philippe Thevenin. 2010. “EDF CYRANO3 Code, Recent Innovations.” In LWR Fuel Performance Meeting/TopFuel/WRFPM. Orlando, Florida, USA.