`TFEL/MFront`

(15/12/2017)Versions 3.1 and 3.0.3 of `TFEL`

, `MFront`

and `MTest`

has been released on the 15th December, 2017.

A detailed version of the release notes is available here.

From a user point of view, TFEL 3.1 brings many important features:

- The gallery is a list of detailed documentations about implementing clean, efficient and reliable mechanical behaviour using
`MFront`

that will be updated as TFEL and MFront evolve. In each case, the integration algorithm is fully described. The gallery currently contains \(17\) entries covering finite strain elasticity, finite strain viscoplasticity, damage, plasticity and viscoplasticity. The gallery. Those implementations are also gathered in a`cmake`

projects called`MFrontGallery`

which can be cloned on github: https://github.com/thelfer/MFrontGallery - The integration of the
`TFEL/NUMODIS`

library which provides many functionalities associated with single crystal description coming from the`NUMODIS`

project. Those functionalities are available in`MFront`

through many new keywords that simplify to a great extent the way small and finite strain single crystal behaviours can be written. See the following page for a detailled description: http://tfel.sourceforge.net/singlecrystal.html - The
`DDIF2`

behaviour is used in CEA fuel performances codes to describe cracking of the nuclear fuel. The`DDIF2`

brick allows this model to be easily coupled to the viscplastic behaviour of the fuel material and can be used as easily as the`StandardElasticity`

brick. See this page for details: http://tfel.sourceforge.net/ddif2.html - An important work has been devoted to enhance the overall numerical reproducibility and stability of the algorithms of
`TFEL/Math`

. It is now much safer to use the`-ffast-math`

option in`gcc`

and`clang`

which can drastically increase the efficiency of the generated code. This option is automatically added when using the`--obuild=level2`

option of`mfront`

. See the release notes for details. - Support for the Hershey/Hosford yield criteria which generalizes the Mises criteria. Indeed the Hershey/Hosford yield criteria lies between the Mises criteria and the Tresca criteria. Various functions have been introduced to compute the equivalent stress associated with this criteria, its first and second derivatives. See this page for details: http://tfel.sourceforge.net/tensors.html
- Support for the Barlat yield criteria which generalizes the Hill criteria. Various functions have been introduced to compute the equivalent stress associated with this criteria, its first and second derivatives. See this page for details: http://tfel.sourceforge.net/tensors.html
- Abritrary non linear constraints can now be defined in
`MTest`

. - Better support of the Windows platform. Using the
`cmake`

generator, using`mfront`

on windows with the`Microsoft`

compilers is easy and closely follows the existing practices on other operating systems. - Improved installation options. Two versions, or the same version in different flavours (compiled in debug mode, compiled with different compilers), of
`TFEL/MFront`

can now be installed in parallel. See http://tfel.sourceforge.net/install.html for details.

Two new interfaces have been introduced in `MFront`

:

- A native interface for the
`CalculiX`

solver. Here native is used to distinguish this interface from the`Abaqus/Standard`

interface which can also be used within`CalculiX`

. This interface can be used with`CalculiX`

2.13. - An interface for the
`ANSYS`

`APDL`

solver. The latter is still experimental.

`@NumericallyComputedJacobianBlocks`

keywordComputing the jacobian of the implicit system is the most difficult part of implementing a behaviour. Computing the jacobian by finite difference is interesting but significantly decreases the performances of the behaviour and can be (very) sensitive to the choice of the numerical perturbation.

The `@NumericallyComputedJacobianBlocks`

keyword is used select a list of jacobian blocks that have to be computed numerically. This is more efficient than computing the whole jacobian numerically. Combined with the ability to compare the jacobian to a numerical approximation, the user now has the ability to build the jacobian incrementally, block by block and checks at each steps that their analytical expressions are correct.

Portability is a convincing sign of software quality and maintainability:

- usage of functionalities specific to operating systems are well identified.
- it demonstrates that the code is not dependant of the system libraries, such as the
`C`

or`C++`

libraries.

`TFEL`

has been tested successfully on a various flavours of `LinuX`

and `BSD`

systems (including `FreeBSD`

and `OpenBSD`

). The first ones are mostly build on `gcc`

, `libstdc++`

and the `glibc`

. The second ones are build on `clang`

and `libc++`

.

`TFEL`

can be build on `Windows`

in a wide variety of configurations and compilers:

- native ports can be build using the
`Visual Studio`

(2015 and 2017) or`MingGW`

compilers. `TFEL`

can be build in the`Cygwin`

environment.

`TFEL`

have reported to build successfully in the `Windows Subsystem for LinuX`

(`WSL`

) environment.

Although not officially supported, more exotic systems, such as `OpenSolaris`

and `Haiku`

, have also been tested successfully. The `Minix`

operating systems provides a pre-release of `clang`

`3.4`

that fails to compile `TFEL`

.

Version 3.1 has been tested using the following compilers:

`gcc`

on various`POSIX`

systems: versions 4.7, 4.8, 4.9, 5.1, 5.2, 6.1, 6.2, 7.1, 7.2`clang`

on various`POSIX`

systems: versions 3.5, 3.6, 3.7, 3.8, 3.9, 4.0, 5.0`intel`

. The only tested version is the 2018 version on`LinuX`

. Intel compilers 15 and 16 are known not to work due to a bug in the EDG front-end that can't parse a syntax mandatory for the expression template engine. The same bug affects the Blitz++ library (see http://dsec.pku.edu.cn/~mendl/blitz/manual/blitz11.html). Version 2017 shall work but were not tested.`Visual Studio`

The only supported versions are the 2015 and 2017 versions. Previous versions do not provide a suitable`C++11`

support.`PGI`

compiler (NVIDIA): version 17.10 on`LinuX`

`MinGW`

has been tested successfully in a wide variety of configurations/versions, including the version delivered with`Cast3M 2017`

.

Compiler and options | Success ratio | Test time |
---|---|---|

`gcc 4.9.2` |
100% tests passed | 681.19 sec |

`gcc 4.9.2+fast-math` |
100% tests passed | 572.48 sec |

`clang 3.5` |
100% tests passed | 662.50 sec |

`clang 3.5+libstcxx` |
99% tests passed | 572.18 sec |

`clang 5.0` |
100% tests passed | 662.50 sec |

`icpc 2018` |
100% tests passed | 511.08 sec |

`PGI 17.10` |
99% tests passed | 662.61 sec |

Concerning the `PGI`

compilers, performances may be affected by the fact that this compiler generates huge shared libraries (three to ten times larger than other compilers).

This is mainly a bug fix version of the `3.0`

series. Detailed release notes are available here. There are no known regressions.

The Green yield criterion is based on the definition of an equivalent stress \({\sigma_{\mathrm{eq}}}\) defined as follows: \[ {\sigma_{\mathrm{eq}}}=\sqrt{{{{\displaystyle \frac{\displaystyle 3}{\displaystyle 2}}}}\,C\,{\underline{s}}\,\colon\,{\underline{s}}-F\,{{\mathrm{tr}{{\left({\underline{\sigma}}\right)}}}}^{2}} \] where \({\underline{s}}\) is the deviatoric stress tensor: \[ {\underline{s}}={\underline{\sigma}}-{{{\displaystyle \frac{\displaystyle 1}{\displaystyle 3}}}}\,{{\mathrm{tr}{{\left({\underline{\sigma}}\right)}}}}\,{\underline{I}} \]

A new entry in the gallery shows how to build a perfect plastic behaviour based on this equivalent stress. The implementation is available here: https://github.com/thelfer/MFrontGallery/blob/master/generic-behaviours/plasticity/GreenPerfectPlasticity.mfront

Functions to compute the Barlat equivalent stress and its first and second derivatives are now available in `TFEL/Material`

.

The Barlat equivalent stress is defined as follows (See Barlat et al. (2005)): \[ {\sigma_{\mathrm{eq}}}^{B}= \sqrt[a]{ \frac{1}{4}\left( \sum_{i=0}^{3} \sum_{j=0}^{3} {{\left|s'_{i}-s''_{j}\right|}}^{a} \right) } \]

where \(s'_{i}\) and \(s''_{i}\) are the eigenvalues of two transformed stresses \({\underline{s}}'\) and \({\underline{s}}''\) by two linear transformation \({\underline{\underline{\mathbf{L}}}}'\) and \({\underline{\underline{\mathbf{L}}}}''\): \[ \left\{ \begin{aligned} {\underline{s}}' &= {\underline{\underline{\mathbf{L'}}}} \,\colon\,{\underline{\sigma}}\\ {\underline{s}}'' &= {\underline{\underline{\mathbf{L''}}}}\,\colon\,{\underline{\sigma}}\\ \end{aligned} \right. \]

The linear transformations \({\underline{\underline{\mathbf{L}}}}'\) and \({\underline{\underline{\mathbf{L}}}}''\) are defined by \(9\) coefficients (each) which describe the material orthotropy. There are defined through auxiliary linear transformations \({\underline{\underline{\mathbf{C}}}}'\) and \({\underline{\underline{\mathbf{C}}}}''\) as follows: \[ \begin{aligned} {\underline{\underline{\mathbf{L}}}}' &={\underline{\underline{\mathbf{C}}}}'\,\colon\,{\underline{\underline{\mathbf{M}}}} \\ {\underline{\underline{\mathbf{L}}}}''&={\underline{\underline{\mathbf{C}}}}''\,\colon\,{\underline{\underline{\mathbf{M}}}} \end{aligned} \] where \({\underline{\underline{\mathbf{M}}}}\) is the transformation of the stress to its deviator: \[ {\underline{\underline{\mathbf{M}}}}={\underline{\underline{\mathbf{I}}}}-{{{\displaystyle \frac{\displaystyle 1}{\displaystyle 3}}}}{\underline{I}}\,\otimes\,{\underline{I}} \]

The linear transformations of the deaviator stress \({\underline{\underline{\mathbf{C}}}}'\) and \({\underline{\underline{\mathbf{C}}}}''\), are defined as follows: \[ {\underline{\underline{\mathbf{C}}}}'= {{{\displaystyle \frac{\displaystyle 1}{\displaystyle 3}}}}\, \begin{pmatrix} 0 & -c'_{12} & -c'_{13} & 0 & 0 & 0 \\ -c'_{21} & 0 & -c'_{23} & 0 & 0 & 0 \\ -c'_{31} & -c'_{32} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c'_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & c'_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & c'_{66} \\ \end{pmatrix} \quad \text{and} \quad {\underline{\underline{\mathbf{C}}}}''= \begin{pmatrix} 0 & -c''_{12} & -c''_{13} & 0 & 0 & 0 \\ -c''_{21} & 0 & -c''_{23} & 0 & 0 & 0 \\ -c''_{31} & -c''_{32} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & c''_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & c''_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & c''_{66} \\ \end{pmatrix} \]

When all the coefficients \(c'_{ji}\) and \(c''_{ji}\) are equal to \(1\), the Barlat equivalent stress reduces to the Hosford equivalent stress.

The following function has been implemented:

`computeBarlatStress`

: return the Barlat equivalent stress`computeBarlatStressNormal`

: return a tuple containg the Barlat equivalent stress and its first derivative (the normal)`computeBarlatStressSecondDerivative`

: return a tuple containg the Barlat equivalent stress, its first derivative (the normal) and the second derivative.

The implementation of those functions are greatly inspired by the work of Scherzinger (see Scherzinger (2017)). In particular, great care is given to avoid overflows in the computations of the Barlat stress.

Those functions have two template parameters:

- the type of symmetric tensors used for the stress tensor (automatically deduced, but required if the second parameter is specified).
- the eigen solver to be used.

The following example computes the Hosford equivalent stress, its normal and second derivative:

```
const auto l1 = makeBarlatLinearTransformation<N,double>(-0.069888,0.936408,
0.079143,1.003060,
0.524741,1.363180,
1.023770,1.069060,
0.954322);
const auto l2 = makeBarlatLinearTransformation<N,double>(-0.981171,0.476741,
0.575316,0.866827,
1.145010,-0.079294,
1.051660,1.147100,
1.404620);
stress seq;
Stensor n;
Stensor4 dn;
std::tie(seq,n,dn) = computeBarlatStressSecondDerivative(s,l1,l2,a,seps);
```

In this example, `s`

is the stress tensor, `a`

is the Hosford exponent, `seps`

is a numerical parameter used to detect when two eigenvalues are equal.

If `C++-17`

is available, the previous code can be made much more readable:

```
const auto l1 = makeBarlatLinearTransformation<N,double>(-0.069888,0.936408,
0.079143,1.003060,
0.524741,1.363180,
1.023770,1.069060,
0.954322);
const auto l2 = makeBarlatLinearTransformation<N,double>(-0.981171,0.476741,
0.575316,0.866827,
1.145010,-0.079294,
1.051660,1.147100,
1.404620);
const auto [seq,n,dn] = computeBarlatStressSecondDerivative(s,l1,l2,a,seps);
```

The header `TFEL/Material/Hosford.hxx`

introduces three functions which are meant to compute the Hosford equivalent stress and its first and second derivatives. *This header is automatically included by MFront*

The Hosford equivalent stress is defined by: \[ {\sigma_{\mathrm{eq}}}=\sqrt{{{{\displaystyle \frac{\displaystyle 1}{\displaystyle 2}}}}{{\left({{\left|\sigma_{1}-\sigma_{2}\right|}}^{2}+{{\left|\sigma_{1}-\sigma_{3}\right|}}^{2}+{{\left|\sigma_{2}-\sigma_{3}\right|}}^{2}\right)}}} \] where \(s_{1}\), \(s_{2}\) and \(s_{3}\) are the eigenvalues of the stress.

Therefore, when \(a\) goes to infinity, the Hosford stress reduces to the Tresca stress. When \(n = 2\) the Hosford stress reduces to the von Mises stress.

The following function has been implemented:

`computeHosfordStress`

: return the Hosford equivalent stress`computeHosfordStressNormal`

: return a tuple containg the Hosford equivalent stress and its first derivative (the normal)`computeHosfordStressSecondDerivative`

: return a tuple containg the Hosford equivalent stress, its first derivative (the normal) and the second derivative.

The following example computes the Hosford equivalent stress, its normal and second derivative:

```
stress seq;
Stensor n;
Stensor4 dn;
std::tie(seq,n,dn) = computeHosfordStressSecondDerivative(s,a,seps);
```

In this example, `s`

is the stress tensor, `a`

is the Hosford exponent, `seps`

is a numerical parameter used to detect when two eigenvalues are equal.

If `C++-17`

is available, the previous code can be made much more readable:

`const auto [seq,n,dn] = computeHosfordStressSecondDerivative(s,a,seps);`

`TFEL`

`3.0.2`

(25/10/2017)Version 3.0.2 of `TFEL`

, `MFront`

and `MTest`

has been released on the 25th October, 2017.

This is mainly a bug fix version of the `3.0`

series. Detailed release notes are available here. There are no known regressions.

Thanks to Guillaume Michal, University of Wollongong (NSW, Australia), several implementation of behaviours suitable for the description of mild steel are now available:

- Three versions of the Johnson-Cook behaviour:
`JohnsonCook_s`

: which describes strain hardening but does not describe rate effects,`JohnsonCook_ssr`

: which describes both strain hardening and rate effects.`JohnsonCook_ssrt`

: which describes both strain hardening, rate effects and adiabatic heating.- The Rusinek-Klepaczko law as described in: "Constitutive relations in 3-D for a wide range of strain rates and temperatures - Application to mild steels". A. Rusinek, R. Zaera and J. R. Klepaczko. Int. J. of Solids and Structures, Vol. 44, 5611-5634, 2007.

`Code_Aster`

Thanks to M. Abbas, `MFront`

finite strain behaviours can now be used in the total lagragian framework in `Code_Aster`

(called `GROT_GDEP`

). First tests confirm that the robustness and the effiency of this framework are much better than with the `SIMO_MIEHE`

framework.

`MFront`

in the `Windows`

`Subsystem`

for `LinuX`

environment (3/08/2017)After `Visual Studio`

, `Mingw`

and `Cygwin`

, there is a new way to get `MFront`

working on `Windows`

!

Rafal Brzegowy have successfully compiled and tested `MFront`

using the `Windows`

`Subsystem`

for `LinuX`

(WSL). He was able to use `MFront`

generated behaviours with `CalculiX`

delivered by the `bConverged`

suite.

All tests worked, except some tests related to the `long double`

support in WSL.

`TFEL`

and `MFront`

now have their official twitter account !

We will use it to spread various events and developments make in `TFEL`

and `MFront`

.

`MFront`

and `Salome-Meca`

(1/08/2017)Thanks to Jordi Alberich, a spanish introduction to `MFront`

and `Salome-Meca`

is available here:

http://tfel.sourceforge.net/tutorial-spanish.html

The talks of the third MFront Users Day are available here:

https://github.com/thelfer/tfel-doc

`Cast3M`

2017 has been released.

A binary version of TFEL compiled for Cast3M 2017 is now available for download on sourceforge:

https://sourceforge.net/projects/tfel/files/

`MTest`

Arbitrary non linear constraints can be now imposed in `MTest`

using `@NonLinearConstraint`

keyword.

Abritray non linear constraints can be used to:

- Impose a constant stress triaxiality.
- Impose a constant first Piola-Kirchoff stress in a creep test described in finite strain.

A constraint \(c\) is imposed by introducing a Lagrange multiplier \(\lambda\).

Consider a small strain elastic behaviour characterised by its free energy \(\Psi\). In the only loading is the constraint \(c\), the solution satisfies: \[ \underset{\underline{\varepsilon},\lambda}{\min}\Psi-\lambda\,c \]

In this case, the constraint \(c\) is equivalent to the following imposed stress:

\[ -\lambda\,{{\displaystyle \frac{\displaystyle \partial c}{\displaystyle \partial \underline{\varepsilon}}}} \]

If the constraint is \(\sigma_{xx}-\sigma_{0}\), where \(\sigma_{0}\) is a constant value, the previous equation shows that imposing this constraint *is not equivalent* to imposing an uniaxial stress state \(\left(\sigma_{xx}\,0\,0\,0\,0\,0\right)\).

The implementation of an an isotropic viscoplastic behaviour with several kinematic variables following the Amstrong-Frederic evolution law is described here

The behaviour is described by a standard split of the strain \({{\underline{\varepsilon}}^{\mathrm{to}}}\) in an elastic and a plastic parts, respectively denoted \({{\underline{\varepsilon}}^{\mathrm{el}}}\) and \({{\underline{\varepsilon}}^{\mathrm{vis}}}\):

\[ {{\underline{\varepsilon}}^{\mathrm{to}}}={{\underline{\varepsilon}}^{\mathrm{el}}}+{{\underline{\varepsilon}}^{\mathrm{vis}}}\]

The stress \({\underline{\sigma}}\) is related to the the elastic strain \({{\underline{\varepsilon}}^{\mathrm{el}}}\) by a the standard Hooke behaviour:

\[ {\underline{\sigma}}= \lambda\,{{\mathrm{tr}{{\left({{\underline{\varepsilon}}^{\mathrm{el}}}\right)}}}}\,{\underline{I}}+2\,\mu\,{{\underline{\varepsilon}}^{\mathrm{el}}}\]

The viscoplastic behaviour follows a standard viscoplastic behaviour: \[ {{\underline{\dot{\varepsilon}}}^{\mathrm{vis}}}=\left\langle{{{\displaystyle \frac{\displaystyle F}{\displaystyle K}}}}\right\rangle^{m}\,{\underline{n}}=\dot{p}\,{\underline{n}} \]

where \(F\) is the yield surface defined below, \(<.>\) is Macaulay brackets, \({\underline{n}}\) is the normal to \(F\) with respect to the stress and \(p\) is the equivalent plastic strain.

The yield surface is defined by: \[ F{{\left({\underline{\sigma}},{\underline{X}}_{i},p\right)}}={{\left({\underline{\sigma}}-\sum_{i=1}^{N}{\underline{X}}_{i}\right)}}_{\mathrm{eq}}-R{{\left(p\right)}}=s^{e}_{\mathrm{eq}}-R{{\left(p\right)}} \]

where:

- \(R{{\left(p\right)}}\) describes the isotropic hardening as a function of the equivalent viscoplastic strain \(p\).
- the \(N\) tensors \({\underline{X}}_{i}\) (i) are backstresses describing the kinematic hardening.
- \({{\left(.\right)}}_{\mathrm{eq}}\) is the Von Mises norm.

We have introduced an effective deviatoric stress \({\underline{s}}^{e}\) defined by: \[ {\underline{s}}^{e}={\underline{s}}-\sum_{i=1}^{N}{\underline{X}}_{i} \] where \({\underline{s}}\) is the deviatoric part of the stress.

The normal is then given by: \[ {\underline{n}}={{\displaystyle \frac{\displaystyle \partial F}{\displaystyle \partial {\underline{\sigma}}}}}={{{\displaystyle \frac{\displaystyle 3}{\displaystyle 2}}}}\,{{{\displaystyle \frac{\displaystyle {\underline{s}}^{e}}{\displaystyle s^{e}_{\mathrm{eq}}}}}} \]

The isotropic hardening is defined by: \[ R{{\left(p\right)}}=R_{\infty} + {{\left(R_{0}-R_{\infty}\right)}}\,\exp{{\left(-b\,p\right)}} \]

\[ {\underline{X}}_{i}={{{\displaystyle \frac{\displaystyle 2}{\displaystyle 3}}}}\,C_{i}\,{\underline{a}}_{i} \]

The evolution of the kinematic variables \({\underline{a}}_{i}\) follows the Armstrong-Frederic rule:

\[ {\underline{\dot{a}}}_{i}={{\underline{\dot{\varepsilon}}}^{\mathrm{vis}}}-g[i]\,{\underline{a}}_{i}\,\dot{p}=\dot{p}\,{{\left({\underline{n}}-g[i]\,{\underline{a}}_{i}\right)}} \]

CEA and EDF are pleased to announce that the third MFront users meeting will take place on May 30th 2017 and will be organized by CEA DEC/SESC at CEA Saclay in the DIGITEO building (building 565, room 34).

Researchers and engineers willing to present their works are welcome.

They may contact the organizers for information at: .

Registration by sending an email at the previous address before May 12th is required to ensure proper organization.

`tfel-plot`

projectThe `tfel-plot`

project is meant to create:

- a
`C++11`

library for generating`2D`

plots based on the Qt library (version 4 or version 5). - a drop-in replacement of
`gnuplot`

called`tplot`

.

This project is based on:

- The
`TFEL`

libraries. - The
`Qt`

framework .

Compared to `gnuplot`

, we wanted to introduced the following new features:

- The ability to select a curve and modify its properties (color, line width, line style, etc..) interactively.
- The ability to use
`tplot`

from the command line. - A widget that can be used in every
`Qt`

code. - The ability to load
`C`

functions in shared libraries, such a the ones generated by the`MFront`

code generator. - Usage of some features introduced by the
`TFEL`

project such as:- kriging interpolation.
- the ability to differentiate functions using the
`diff`

operator.

- Support for themes.

More details can be found on the dedicated `github`

page:

https://github.com/thelfer/tfel-plot

A new example has been added in the gallery here.

This example describes a simple orthotropic behaviour.

The behaviour is described by a standard split of the strain \({{\underline{\varepsilon}}^{\mathrm{to}}}\) in an elastic and a plastic parts, respectively denoted \({{\underline{\varepsilon}}^{\mathrm{el}}}\) and \({{\underline{\varepsilon}}^{\mathrm{p}}}\):

\[ {{\underline{\varepsilon}}^{\mathrm{to}}}={{\underline{\varepsilon}}^{\mathrm{el}}}+{{\underline{\varepsilon}}^{\mathrm{p}}}\]

The stress \({\underline{\sigma}}\) is related to the the elastic strain \({{\underline{\varepsilon}}^{\mathrm{el}}}\) by a the orthotropic elastic stiffness \({\underline{\underline{\mathbf{D}}}}\):

\[ {\underline{\sigma}}= {\underline{\underline{\mathbf{D}}}}\,\colon\,{{\underline{\varepsilon}}^{\mathrm{el}}}\]

The plastic part of the behaviour is described by the following yield surface: \[ f{{\left({\sigma_{H}},p\right)}} = {\sigma_{H}}-\sigma_{0}-R\,p \]

where \({\sigma_{H}}\) is the Hill stress defined below, \(p\) is the equivalent plastic strain. \(\sigma_{0}\) is the yield stress and \(R\) is the hardening slope.

The Hill stress \({\sigma_{H}}\) is defined using the fourth order Hill tensor \(H\): \[ {\sigma_{H}}=\sqrt{{\underline{\sigma}}\,\colon\,{\underline{\underline{\mathbf{H}}}}\colon\,{\underline{\sigma}}} \]

The plastic flow is assumed to be associated, so the flow direction \({\underline{n}}\) is given by \({{\displaystyle \frac{\displaystyle \partial f}{\displaystyle \partial {\underline{\sigma}}}}}\):

\[ {\underline{n}} = {{\displaystyle \frac{\displaystyle \partial f}{\displaystyle \partial {\underline{\sigma}}}}} = {{{\displaystyle \frac{\displaystyle 1}{\displaystyle {\sigma_{H}}}}}}\,{\underline{\underline{\mathbf{H}}}}\,\colon\,{\underline{\sigma}}\]

The default eigen solver for symmetric tensors used in `TFEL`

is based on analitical computations of the eigen values and eigen vectors. Such computations are more efficient but less accurate than the iterative Jacobi algorithm (see (Kopp 2008; Kopp 2017)).

With the courtesy of Joachim Kopp, we have introduced the following algorithms:

- Jacobi
- QL with implicit shifts
- Cuppen
- Analytical
- Hybrid
- Householder reduction

The implementation of Joachim Kopp have been updated for `C++-11`

and make more generic (support of all the floatting point numbers, different types of matrix/vector objects). The algorithms have been put in a separate namespace called `fses`

(Fast Symmetric Eigen Solver) and is independant of the rest of `TFEL`

.

We have also introduced the Jacobi implementation of the `Geometric`

`Tools`

library (see (Eberly 2016; Eberly 2017)).

Those algorithms are available in 3D. For 2D symmetric tensors, we fall back to some default algorithm as described below.

Name | Algorithm in 3D | Algorithm in 2D |
---|---|---|

`TFELEIGENSOLVER` |
Analytical (TFEL) | Analytical (TFEL) |

`FSESJACOBIEIGENSOLVER` |
Jacobi | Analytical (FSES) |

`FSESQLEIGENSOLVER` |
QL with implicit shifts | Analytical (FSES) |

`FSESCUPPENEIGENSOLVER` |
Cuppen's Divide & Conquer | Analytical (FSES) |

`FSESANALYTICALEIGENSOLVER` |
Analytical | Analytical (FSES) |

`FSESHYBRIDEIGENSOLVER` |
Hybrid | Analytical (FSES) |

`GTESYMMETRICQREIGENSOLVER` |
Symmetric QR | Analytical (TFEL) |

The various eigen solvers available are enumerated in Table 1.

The eigen solver is passed as a template argument of the `computeEigenValues`

or the `computeEigenVectors`

methods as illustrated in the code below:

```
tmatrix<3u,3u,real> m2;
tvector<3u,real> vp2;
std::tie(vp,m)=s.computeEigenVectors<Stensor::GTESYMMETRICQREIGENSOLVER>();
```

Algorithm | Failure ratio | \(\Delta_{\infty}\) | Times (ns) | Time ratio |
---|---|---|---|---|

`TFELEIGENSOLVER` |
0.000632 | 7.75e-14 | 252663338 | 1 |

`GTESYMMETRICQREIGENSOLVER` |
0 | 2.06e-15 | 525845499 | 2.08 |

`FSESJACOBIEIGENSOLVER` |
0 | 1.05e-15 | 489507133 | 1.94 |

`FSESQLEIGENSOLVER` |
0.000422 | 3.30e-15 | 367599140 | 1.45 |

`FSESCUPPENEIGENSOLVER` |
0.020174 | 5.79e-15 | 374190684 | 1.48 |

`FSESHYBRIDEIGENSOLVER` |
0.090065 | 3.53e-10 | 154911762 | 0.61 |

`FSESANALYTICALEIGENSOLVER` |
0.110399 | 1.09e-09 | 157613994 | 0.62 |

We have compared the available algorithm on \(10^{6}\) random symmetric tensors whose components are in \([-1:1]\).

For a given symmetric tensor, we consider that the computation of the eigenvalues and eigenvectors failed if: \[ \Delta_{\infty}=\max_{i\in[1,2,3]}\left\|{\underline{s}}\,\cdot\,\vec{v}_{i}-\lambda_{i}\,\vec{v}_{i}\right\|>10\,\varepsilon \] where \(\varepsilon\) is the accuracy of the floatting point considered.

The results of those tests are reported on Table 2:

- The standard eigen solver available in previous versions of
`TFEL`

offers a very interesting compromise between accuracy and numerical efficiency. - If very accurate results are required, the
`FSESJACOBIEIGENSOLVER`

eigen solver is a good choice.

`TFEL`

for `FreeBSD`

(20/01/2017)Thanks to the work of Pedro F. Giffuni , an official port of `TFEL/MFront`

is available for `FreeBSD`

:

http://www.freshports.org/science/tfel/

To install an executable package, you can now simply do:

`pkg install tfel-mfront`

Alternativel, to build and install `TFEL`

from the ports tree, one can do:

```
cd /usr/ports/science/tfel/
make
make install
```

The implementation of the Ogden behaviour is now described in depth in the following page:

http://tfel.sourceforge.net/ogden.html

This behaviour is interesting as it highlights the features introduced in `TFEL-3.0`

for computing isotropic functions of symmetric tensors.

This page uses the formal developments detailled in:

http://tfel.sourceforge.net/hyperelasticity.html

Concerning hyperviscoelasticity, a page describing a generic implementation is available here:

http://tfel.sourceforge.net/hyperviscoelasticity.html

If you have particular wishes on behaviours implementation that you would like to see treated, do not hesitate to send a message at .

A page referencing examples of well written mechanical behaviours has been created here:

http://tfel.sourceforge.net/gallery.html

For each behaviours, we will try to provide tutorial-like pages explaining the implementations details (usage of tensorial objects, special functions of the `TFEL`

library, choice of the algorithms, and so on...)

The first attempt is an hyperelastic behaviour already available in `Code_Aster`

: the Signorini behaviour. You can find the page here:

http://tfel.sourceforge.net/signorini.html

This is still under review, so corrections and feed-backs would be greatly appreciated. The following behaviours are planned to be addressed:

- the Ogden hyperelastic behaviour which will show how we can compute eigenvalues, eigenvectors and isotropic functions of symmetric tensors.
- hyperviscoelasticity (a very small extension of hyperelasticity, indeed)
- simple plastic behaviours

If you have particular wishes on behaviours implementation that you would like to see treated, do not hesitate to send a message at .

From a user point of view, TFEL 3.0 brings many game-changing features:

- New mechanical behaviour interfaces to several finite element solvers (
`Europlexus`

,`Abaqus/Standard`

,`Abaqus/Explicit`

,`CalculiX`

) - The support of behaviours bricks.
- The ability to simulate pipes using a rigourous finite strain framework in
`MTest`

. - The introduction of various accelerations algorithms used for solving the mechanical equilibrium when the consistent tangent operator is not available.
- The development of a stable API (application programming interface) in
`C++`

and`python`

which allow building external tools upon`MFront`

and`MTest`

. This point is illustrated by the development by EDF MMC in the Material Ageing Plateform of a new identification tool which is particularly interesting.

Many improvements for mechanical behaviours have also been made:

- Definition and automatic computation of elastic material properties from external MFront files.
- Support for the automatic computation of the thermal expansion and swelling in the mechanical behaviour.
- Better support for orthotropic behaviours with the notion of orthotropic convention which affects various keywords (
`@ComputeStiffnessTensor`

,`@ComputeThermalExpansion`

,`@Swelling`

,`@AxialGrowth`

,`@HillTensor`

etc..). - An initial support of non local mechanical behaviours.
- Time steps management from the mechanical behaviour.
- Consistent tangent operator support in the Cast3M interface.
- Easier computations of isotropic functions of symmetric tensors and their derivatives.
- New material properties interfaces for various programming languages (
`fortran03`

,`java`

,`GNU/Octave`

).

A detailed version of the release notes is available here.

`UnilateralMazars`

(1 June 2016)A new behaviour implementation has been submitted by F. Hamon (EDF R&D AMA). This behaviour describes the damaging behaviour of concrete with unilateral effects as described in a dedicated paper:

A new 3D damage model for concrete under monotonic, cyclic and dynamic loadings. Jacky Mazars, François Hamon and Stéphane Grange. Materials and Structures ISSN 1359-5997 DOI 10.1617/s11527-014-0439-8

This implementation can be found in the current development sources of MFront.

The Abaqus Explicit interface is becoming quite usable du to the extensive testing efforts of D. Deloison (Airbus). Here is an example a punching test (This test was also modelled using Abaqus Standard).

`@Brick "StandardElasticity";`

Behaviour bricks will be one of the most important new feature of `TFEL 3.0`

.

A dedicated page has been created here

`EUROPLEXUS`

interface (27 May 2016)An interface for the EUROPLEXUS explicit finite element solver has been developed.

EUROPLEXUS (EPX) is a simulation software dedicated to the analysis of fast transient phenomena involving structures and fluids in interaction.

See the dedicated page for more information.

`tfel-doc`

`github`

repository. Talks of the Second MFront Users Day (27 May 2016)A `github`

repository has been set up to store various documents describing TFEL and MFront usage. The talks of the first and second MFront Users Days are available there:

https://github.com/thelfer/tfel-doc

`github`

repository (27 May 2016)The `subversion`

repository used by CEA and EDF are now synchronized with a public `githbub`

repository:

https://github.com/thelfer/tfel

All the branches, commit description and history of TFEL are available. This repository is read-only. Its purpose is to integrate TFEL in continous-integration projects which depends on TFEL.

`mfront`

module (May 2016)A new `python`

module has been introduced to analyse `MFront`

files.

An overview of the module is available here.

Here is a typical usage example:

```
import mfront
dsl = mfront.getDSL("Chaboche.mfront")
dsl.setInterfaces(["aster"])
dsl.analyseFile("Chaboche.mfront",[])
# file description
fd = dsl.getFileDescription()
print("file author: ", fd.authorName)
print("file date: ", fd.date)
print("file descrption:\n", fd.description)
# targets information
tgt = dsl.getTargetsDescription()
# loop over (to be) generated libraries
for l in tgt:
print(l)
```

`Cast3M`

2016 has been released.

This version allow even better integration with `MFront`

and can now directly be used to make direct calls to `MFront`

libraries for material properties (mechanical behaviours can be used since Cast3M 2015).

This syntax is now officially supported:

```
Ty = 'TABLE' ;
Ty.'LIB_LOI' = 'libCastemM5.so' ;
Ty.'FCT_LOI' = 'M5_YoungModulus' ;
Ty.'VARIABLES' = 'MOTS' 'T';
mo = 'MODELISER' m 'MECANIQUE' 'ELASTIQUE';
ma = 'MATERIAU' mo 'YOUN' Ty 'NU' 0.3;
```

A binary version of TFEL compiled for Cast3M 2016 is now available for download on sourceforge:

https://sourceforge.net/projects/tfel/files/

CEA and EDF are pleased to announce that the second MFront users meeting will take place on May 20th 2016 and will be organized by EDF R&D at the EDF Lab Paris Saclay location.

Researchers and engineers willing to present their works are welcome. They may contact the organizers for information at: .

This users day will take place on Friday, May 20th, 2016 at EDF Lab Paris-Saclay (access map).

Registration is required to ensure proper organization. See the dedicated form on the Code_Aster website: http://www.code-aster.org/spip.php?article906

The current development version now includes an experimental interface to the `Abaqus`

solver through the `umat`

subroutine.

The following results shows the results obtained on notched beam under a cyclic loading with an isotropic hardening plastic beahviour implemented with `mfront`

:

The `mfront`

results can be compared to the results obtained using `Abaqus`

in-built model on the following figure:

The `Abaqus`

interface is still in its early stage of developments. A full description of its usage and current abilities can be found in the associated documentation.

Feed-back from users would be greatly welcomed.

An IMSIA seminar about MFront will be held on Januar, 27 2016. Here is the official announcement (in french):

Séminaire IMSIA : Implémentation de lois de comportement mécanique à l’aide du générateur de code MFront

Créé dans le cadre de la simulation des éléments combustibles nucléaire au sein d’une plate-forme logicielle nommée PLEIADES, MFront est un générateur de code, distribué en open-source [1], qui vise à permettre aux ingénieurs/chercheurs d’implémenter de manière simple et rapide des lois de comportements mécaniques de manière indépendante du code cible (EF ou FTT) [2,3].

Ce dernier point permet d’échanger les lois MFront entre différents partenaires, universitaires ou industriels. Le lien vers les codes cible se fait via la notion d’interface. A l’heure actuelle, des interfaces existent pour les codes Cast3M, Code-Aster, Abaqus, ZeBuLoN, AMITEX_FFT et d’autres codes métiers, et sont en cours de développement pour d’autres codes tels Europlexus, …

Ce séminaire proposera une description des fonctionnalités de MFront (lois en transformations infinitésimales et en grandes transformations, modèles de zones cohésives) et commentera plusieurs exemples d’applications, discutera des performances numériques obtenues et soulèvera la question de la portabilité des connaissances matériau. Nous montrerons qu’une démarche cohérente allant des expérimentations aux codes de calcul est nécessaire.

[1] http://tfel.sourceforge.net

[2] Introducing the open-source mfront code generator: Application to mechanical behaviours and material knowledge management within the PLEIADES fuel element modelling platform. Thomas Helfer, Bruno Michel, Jean-Michel Proix, Maxime Salvo, Jérôme Sercombe, Michel Casella, Computers & Mathematics with Applications, Volume 70, Issue 5, September 2015, Pages 994-1023, ISSN 0898-1221, http://dx.doi.org/10.1016/j.camwa.2015.06.027.

[3] Implantation de lois de comportement mécanique à l’aide de MFront : simplicité, efficacité, robustesse et portabilité. T. Helfer, J.M. Proix, O. Fandeur. 12ème Colloque National en Calcul des Structures 18-22 Mai 2015, Giens (Var)

A poster describing the use of `MFront`

in EDF `Cyrano3`

fuel performance code has been presented at the LWR Fuel Performance Meeting 2015 (13 - 17 September 2015, Zurich, Switzerland):

Version 2.0.3 is mostly a bug-fix release:

- Fix of a minor bug in the
`CxxTokenizer`

class which was appears when using the`clang`

`libc++`

standard library. This prevented many`MTest`

tests to work on Mac OS X. - Fix of Ticket #9
- Fix of Ticket #10
- Fix of Ticket #11
- The class
`ExternalBehaviourDescription`

was introduced - The
`AxialGrowth`

entry was added to the glossary

A full description of the 2.0.3 release can be found here (in french).

`mfront-doc`

(19 August 2015)`mfront-doc`

allows the user to extract the documentation out of `MFront`

file. `mfront-doc`

generates files in `pandoc`

markdown format. Those files can be processed using `pandoc`

and be converted to one of the many file format supported by `pandoc`

, including `LaTeX`

, `html`

or various Word processor formats: Microsoft Word docx, OpenOffice/LibreOffice ODT.

`mfront-doc`

is developped in the 3.0.x branche of `TFEL`

. A overview of the `mfront-doc`

functionalities can be found here.

New documentation pages were added to describe the `MTest`

and `MFront`

keywords:

`MFront`

keywords, sorted by domain specific languages:`MTest`

keywords

The current development version of `MFront`

includes two new interfaces for material properties:

`java`

designed for the`java`

language.`octave`

designed for`GNU Octave`

which is a high-level interpreted language, primarily intended for numerical computations.

Here is an example of a `GNU Octave`

session used to compute the Young Modulus of uranium-plutonium carbide \(UPuC\) for various porosities over a range of temperatures:

```
octave:1> T=[300:100:1500]
T =
300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500
octave:2> y01=UPuC_YoungModulus(T,0.1)
y01 =
1.7025e+11 1.6888e+11 1.6752e+11 1.6616e+11 1.6480e+11 1.6344e+11 1.6207e+11 1.6071e+11 1.5935e+11 1.5799e+11 1.5662e+11 1.5526e+11 1.5390e+11
octave:3> y02=UPuC_YoungModulus(T,0.2)
y02 =
1.1853e+11 1.1758e+11 1.1663e+11 1.1568e+11 1.1474e+11 1.1379e+11 1.1284e+11 1.1189e+11 1.1094e+11 1.0999e+11 1.0905e+11 1.0810e+11 1.0715e+11
```

The number of papers in which `MFront`

is used is increasing. A dedicated page has been added here.

If you publish papers which refers to `MFront`

, please consider contributing to this page.

The first paper dedicated to `MFront`

, written by Thomas Helfer, Bruno Michel, Jean-Michel Proix, Maxime Salvo, Jérôme Sercombe, and Michel Casella, has been accepted Computers and Mathematics with Applications. The paper is available online on the sciencedirect website:

http://www.sciencedirect.com/science/article/pii/S0898122115003132

The PLEIADES software environment is devoted to the thermomechanical simulation of nuclear fuel elements behaviour under irradiation. This platform is co-developed in the framework of a research cooperative program between Électricité de France (EDF), AREVA and the French Atomic Energy Commission (CEA). As many thermomechanical solvers are used within the platform, one of the PLEAIADES’s main challenge is to propose a unified software environment for capitalisation of material knowledge coming from research and development programs on various nuclear systems.

This paper introduces a tool called mfront which is basically a code generator based on C++ (Stroustrup and Eberhardt, 2004). Domain specific languages are provided which were designed to simplify the implementations of new material properties, mechanical behaviours and simple material models. mfront was recently released under the GPL open-source licence and is available on its web site: http://tfel.sourceforge.net/.

The authors hope that it will prove useful for researchers and engineers, in particular in the field of solid mechanics. mfront interfaces generate code specific to each solver and language considered.

In this paper, after a general overview of mfront functionalities, a particular focus is made on mechanical behaviours which are by essence more complex and may have significant impact on the numerical performances of mechanical simulations. mfront users can describe all kinds of mechanical phenomena, such as viscoplasticity, plasticity and damage, for various types of mechanical behaviour (small strain or finite strain behaviour, cohesive zone models). Performance benchmarks, performed using the Code-Aster finite element solver, show that the code generated using mfront is in most cases on par or better than the behaviour implementations written in fortran natively available in this solver. The material knowledge management strategy that was set up within the PLEIADES platform is briefly discussed. A material database named sirius proposes a rigorous material verification workflow.

We illustrate the use of mfront through two case of studies: a simple FFC single crystal viscoplastic behaviour and the implementation of a recent behaviour for the fuel material which describes various phenomena: fuel cracking, plasticity and viscoplasticity.

`Cast3M`

2015 has been released.

This release allow direct call to `MFront`

libraries for mechanical behaviours. The following syntax of the `MODELISER`

operator is now officially supported:

```
mod1 = 'MODELISER' s1 'MECANIQUE' 'ELASTIQUE' 'ISOTROPE'
'NON_LINEAIRE' 'UTILISATEUR'
'LIB_LOI' 'src/libUmatBehaviour.so'
'FCT_LOI' 'umatnorton'
'C_MATERIAU' coel2D
'C_VARINTER' stav2D
'PARA_LOI' para2D
'CONS' M;
```

See the dedicated page for more information.

`Salome-Meca`

2015.1 is out (1O April 2015)`Salome-Meca`

2015.1, which combines the `Salome`

platform and `Code-Aster`

, has been released on 1O April 2015. This version is the first to include a pre-packaged version of `TFEL`

(version 2.0.1).

`Salome-Meca`

and `Code_Aster`

Users Day (26 March 2015)A presentation of `MFront`

was done during the `Salome-Meca`

and `Code_Aster`

Users Day 2015 user meeting at EDF Lab Clamart.

Slides can be found here.

`MFront`

user meeting (6 Februar 2015)The first `MFront`

user meeting was held in Cadarache on Februar,6 2015. 27 participants from CEA, EDF, Areva and CNRS could discuss and comment about their use of MFront.

Various subjects were discussed:

- Introduction to the user meeting
- New functionalities introduced in
`TFEL`

/`MFront`

2.0 - Feed-back from users:
- Feed-back from the
`Code-Aster`

development team - Feed-back from the
`Cyrano3`

development team

- Feed-back from the
- Modèle de couplage Fluage/Endommagement/Réaction Alcali Granulat (RAG) du béton
- Modélisation à différentes échelles du comportement mécanique du dioxyde d'uranium
- Writing portable behaviour with
`MFront`

- On the road to
`TFEL 3.x`

Those talks are available here

`MFront`

user meeting (12 December 2014)We are glad to announce that the first `MFront`

user meeting will be held in Cadarache on Februar,6 2015. Everyone is invited but a registration must be performed before Januar, 16 2015 ().

Various subjects are already planned:

- New functionalities introduced in
`TFEL`

/`MFront`

2.0 - Feed-back from users:
- Feed-back from the
`Code-Aster`

development team - Feed-back from the
`Cyrano3`

development team

- Feed-back from the
`MFront`

usage for concrete modelling- Examples of
`MFront`

usage in finite strain analyses - Application of
`MFront`

to nuclear fuel modelling - How to write portable behaviour implementations ?
- Conclusions

Other talks are welcomed.

`MFront`

at the `Cast3M`

user meeting (4 December 2014)The `Cast3M`

user meeting was held in Paris on November 28, 2014. Jérémy Hure had a talk about the application of `MFront`

in finite strain elasto-plasticity. This talk is available here.

`AMITEX_FFTP`

has its own website (4 December 2014)`AMITEX_FFTP`

has now its own dedicated webiste.

The main purpose of `AMITEX_FFTP`

is to implement an efficient distributed mechanical solver based on Fast Fourier Transform. `AMITEX_FFTP`

is developped by CEA in the Departement of Nuclear Material.

`AMITEX_FFTP`

is distributed under a free license for research and education purpose.

`TFEL`

version 2.0.1 is now available. This is mainly a bug-fix release after version 2.0.0.

This version is meant to be used in Code-Aster version 12.3 that will be released in January 2015.

A new contact address has been created: .

This address can be used to contact directly the developers of `TFEL`

and `MFront`

for specific issues. However, if your issue may interest a broader audience, you may want to send a post to the TFEL users mailing lists: .

`MFront`

talk at Materiaux 2014 Montpellier (24 November 2014)An `MFront`

talk was given at Materiaux 2014. Slides (in french) are available here.

`Windows 64bits`

and `Cast3M 2014`

(18 November 2014)A beta version of tfel-2.0.1 for `Windows 64bits`

and `Cast3M`

`2014`

has been released. A binary installer is provided here.

Installing this version requires a functional installation of `Cast3M`

\(2014\) (which shall be patched to call external libraries) and the `MSYS`

shell (It is recommended not to install `mingw`

compilers along with the `MSYS`

shell as `Cast3M`

provides its own version of those compilers).

Installation instructions of those requirements are available here.

Any feedback would be gratefully acknowledge.

**Note:** The binary provided requires the `mingw`

libraries delivered with `Cast3M`

`2014`

.

**Note:** A standalone version of tfel-2.0.1 will be provided shortly.

`MFront`

behaviours can now be used in `AMITEX_FFTP`

(24 October 2014)`AMITEX_FFTP`

is a massively parallel mechanical solver based on FFT developed at CEA. `MFront`

behaviours can be used in `AMITEX_FFTP`

through the `UMAT`

interface introduced by the `Cast3M`

finite element solver.

`TFEL/MFront`

on `Cast3M`

web site (15 October 2014)A page dedicated to `MFront`

is now available on the `Cast3M`

web site.

Here is the official announcement by Jean-Paul DEFFAIN (in French):

Bonjour,

Une version libre de MFront est désormais officiellement disponible sous licence GPL et CECILL-A.

Cette version 2.0 permet entre autres de générer des lois de comportements en transformations infinitésimales et en grandes transformations ainsi que des modèles de zones cohésives. Les lois générées sont utilisables dans les codes aux éléments finis Cast3M, Code-Aster, ZeBuLoN, l’ensemble des applications développées dans la plateforme PLEIADES, notamment Cyrano3, et le solveur FFT de la plate-forme. Des interfaces vers d'autres codes peuvent être rajoutées.

Un projet dédié a été crée sur sourceforge (http://sourceforge.net/projects/tfel) et fournit :

- un site dédié (http://tfel.sourceforge.net)
- un espace de téléchargement (http://sourceforge.net/projects/tfel/files) qui permet d'accéder à la version 2.0
- les listes de diffusion tfel-announce et tfel-discuss (http://sourceforge.net/p/tfel/tfel)
- un forum (http://sourceforge.net/p/tfel/discussion)
- un outil de demande d'évolution et de déclaration de bugs par tickets (http://sourceforge.net/p/tfel/tickets)
Pour les personnes souhaitant contribuer au développement, le dépôt subversion est accessible sur le serveur:

https://svn-pleiades.cea.fr/SVN/TFEL

L’accès à ce dépôt est ouvert à tous mais nécessite l’ouverture d’un compte spécifique sur demande au .

Nous remercions chaleureusement tous ceux qui ont contribué à cette version et invitons toutes les personnes intéressées à se joindre au développement de MFront.

Jean-Paul DEFFAIN

Chef du programme SIMU

Commissariat à l'Énergie Atomique

Barlat, F., H. Aretz, J. W. Yoon, M. E. Karabin, J. C. Brem, and R. E. Dick. 2005. “Linear Transfomation-Based Anisotropic Yield Functions.” *International Journal of Plasticity* 21 (5): 1009–39. doi:10.1016/j.ijplas.2004.06.004.

Eberly, David. 2016. “A Robust Eigensolver for 3 × 3 Symmetric Matrices.” https://www.geometrictools.com/Documentation/RobustEigenSymmetric3x3.pdf.

———. 2017. “Geometric Tools.” http://www.geometrictools.com/.

Kopp, Joachim. 2008. “Efficient Numerical Diagonalization of Hermitian 3x3 Matrices.” *International Journal of Modern Physics C* 19 (3): 523–48. doi:10.1142/S0129183108012303.

———. 2017. “Numerical Diagonalization of 3x3 Matrices.” https://www.mpi-hd.mpg.de/personalhomes/globes/3x3/.

Scherzinger, W. M. 2017. “A Return Mapping Algorithm for Isotropic and Anisotropic Plasticity Models Using a Line Search Method.” *Computer Methods in Applied Mechanics and Engineering* 317 (April): 526–53. doi:10.1016/j.cma.2016.11.026.