Writing portable behaviour implementation with MFront

Thomas Helfer

2014

Current interfaces to mechanical behaviours

umat for the CEA’s Cast3M finite element solver
aster for the EDF’s Code-Aster finite element solver
cyrano for the Cyrano3 fuel performance code.
zmat for the Zebulon finite elemnet solver developed by the École des Mines ParisTech (France), Onera, NW Numerics & Modeling, Inc (USA) and various academic partners.

Strains

Solver	Convention
TFEL	\(\left(\epsilon^{\mathrm{to}}_{xx},\epsilon^{\mathrm{to}}_{yy},\epsilon^{\mathrm{to}}_{zz},\sqrt{2}\,\epsilon^{\mathrm{to}}_{xy},\sqrt{2}\,\epsilon^{\mathrm{to}}_{xy},\sqrt{2}\,\epsilon^{\mathrm{to}}_{yz}\right)\)
Cast3M	\(\left(\epsilon^{\mathrm{to}}_{xx},\epsilon^{\mathrm{to}}_{yy},\epsilon^{\mathrm{to}}_{zz},2\,\epsilon^{\mathrm{to}}_{xy},2\,\epsilon^{\mathrm{to}}_{xy},2\,\epsilon^{\mathrm{to}}_{yz}\right)\)
Code-Aster	\(\left(\epsilon^{\mathrm{to}}_{xx},\epsilon^{\mathrm{to}}_{yy},\epsilon^{\mathrm{to}}_{zz},\sqrt{2}\,\epsilon^{\mathrm{to}}_{xy},\sqrt{2}\,\epsilon^{\mathrm{to}}_{xy},\sqrt{2}\,\epsilon^{\mathrm{to}}_{yz}\right)\)
Zebulon	\(\left(\epsilon^{\mathrm{to}}_{xx},\epsilon^{\mathrm{to}}_{yy},\epsilon^{\mathrm{to}}_{zz},\sqrt{2}\,\epsilon^{\mathrm{to}}_{xy},\sqrt{2}\,\epsilon^{\mathrm{to}}_{yz},\sqrt{2}\,\epsilon^{\mathrm{to}}_{xz}\right)\)

1D case

Solver	Convention
Cast3M	\(\left(\epsilon^{\mathrm{to}}_{rr},\epsilon^{\mathrm{to}}_{zz},\epsilon^{\mathrm{to}}_{\theta}\right)\)
Code-Aster	\(\left(\epsilon^{\mathrm{to}}_{rr},\epsilon^{\mathrm{to}}_{zz},\epsilon^{\mathrm{to}}_{\theta}\right)\)
Cyrano3	\(\left(\epsilon^{\mathrm{to}}_{rr},\epsilon^{\mathrm{to}}_{\theta},\epsilon^{\mathrm{to}}_{zz}\right)\)

Stresses

Solver	Convention
TFEL	\(\left(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sqrt{2}\,\sigma_{xy},\sqrt{2}\,\sigma_{xy},\sqrt{2}\,\sigma_{yz}\right)\)
Cast3M	\(\left(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{xy},\sigma_{xy},\sigma_{yz}\right)\)
Code-Aster	\(\left(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sqrt{2}\,\sigma_{xy},\sqrt{2}\,\sigma_{xy},\sqrt{2}\,\sigma_{yz}\right)\)
Zebulon	\(\left(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sqrt{2}\,\sigma_{xy},\sqrt{2}\,\sigma_{yz},\sqrt{2}\,\sigma_{xz}\right)\)

Solver	Convention
Cast3M	\(\left(\sigma_{rr},\sigma_{zz},\sigma_{\theta}\right)\)
Code-Aster	\(\left(\sigma_{rr},\sigma_{zz},\sigma_{\theta}\right)\)
Cyrano3	\(\left(\sigma_{rr},\sigma_{\theta},\sigma_{zz}\right)\)

Convertions

Convertions are handled by the interface
Internal state variables are stored using TFEL conventions
- In the material frame (for orthotropic behaviours)
- Except for ZeBuLoN

Orthotropy

Recommended definition for the orthotropic axes for pipes

behaviour and orthotropic axes definitions must be consistent
no general purpose solver allow uniform definition of the orthotropic axes for all modelling hypotheses
- example of pipes

Supported modelling hypotheses

Solver	Convention
Cast3M	Axisymmetrical generalised plane strain, plane strain, generalised plane strain, plane stress, axisymmetrical, tridimensional
Code-Aster	Plane strain, plane stress (1), axisymmetrical, tridimensional
Zebulon	\(1D\), \(2D\), \(3D\) (2)
Cyrano	Axisymmetrical generalised plane strain, axisymmetrical generalised plane stress (3)

Can be supported by the behaviour or by the solver (De Borst)
Modelling hypothesis is not accessible within the behaviour. Plane stress is supported by the finite element (additional ddl)
The default.

Plane stress

Support for plane stress and axisymmetrical generalised plane stress shall be explicitely provided
- for performance reasons
See the MFront documentation

Prediction operator

|:———:|:——-:| |Cast3M | No (1) | |Code-Aster | Elastic, secant, tangent | |Zebulon | No | |Cyrano | No |

Expected in PLEIADES version (2015 ?)

Consistent tangent operator

Expected in PLEIADES version (2015 ?)
By default.

Finite strains

for standard laws, use the Miehe logarithmic strains framework:
- available in Cast3M (through MFront), Code-Aster, ZeBuLoN
- can easily inserted in Cyrano (paper under review)
avoid using Cast3M’s PASAPAS GRANDS_DEPLACEMENTS option

Consistent tangent operator in finite strain

|:———:|:——-:| |Cast3M | No | |Code-Aster | \({\displaystyle \frac{\displaystyle \partial \underline{\tau}}{\displaystyle \partial \Delta\, \underset{\tilde{}}{\mathbf{F}}}}\) (1) | |Zebulon | \(C^{T}\) (2) |

MFront generates convertion between consistent tangent operators : use the most natural one for your formalism

\(\Delta\, \underset{\tilde{}}{\mathbf{F}} = \left.\underset{\tilde{}}{\mathbf{F}}\right|_{t+\Delta\,t}.\left(\left.\underset{\tilde{}}{\mathbf{F}}\right|_{t}\right)^{-1}\)
The only finite strain formulation supported by MFront is the Updated_Lagrangian.