3. Operands#

3.1. general operands#

3.1.1. Tag OPTION#

This keyword makes it possible to determine the type of standard to use:

“ENER_RELA” for the energy standard (FEM and X- FEM).
“DEPL_RELA” for the \({L}^{2}\) displacement standard (FEM and X- FEM).
“LAGR_RELA” for contact pressure standard \({L}^{2}\) (FEM only).

This keyword also determines the type of field corresponding to the CHAM_GD keyword:

the stress field at Gauss points in the case OPTION =” ENER_RELA “.
The field of movement at the nodes in cases OPTION =” DEPL_RELA “and OPTION =” LAGR_RELA “.

3.1.2. Operand MODELE#

The name of the model on which the option is calculated. It must be the same model that was used to perform the mechanical calculation from which the field given by the keyword CHAM_GD comes from.

3.1.3. Operand GROUP_MA#

The GROUP_MA operand allows you to specify the cell groups for which the energy calculations or \({L}^{2}\) standards will be carried out.

3.2. Operands for OPTION = “ENER_RELA”#

The “ENER_RELA” option makes it possible to estimate the difference between the stress field obtained by the finite element calculation \({\sigma }_{h}\) and the reference stress field \(\sigma\).

For each group of elements in the list given by operand GROUP_MA, the macro command POST_ERREUR calculates:

The elastic energy of the field difference \({\sigma }_{h}-\sigma\)

\(\frac{1}{2}{\int }_{{\Omega }_{i}}({\sigma }_{h}-\sigma )\mathrm{:}{D}^{-1}\mathrm{:}({\sigma }_{h}-\sigma )\mathit{dV}\),

where \({\Omega }_{i}\) is the domain obtained by concatenating all the cells in the group of elements in question and \(D\) is the Hooke tensor.

The elastic energy of the reference stress field \(\sigma\)

\(\frac{1}{2}{\int }_{{\Omega }_{i}}\sigma \mathrm{:}{D}^{-1}\mathrm{:}\sigma \mathit{dV}\).

Finally, the relative error in terms of the energy norm is obtained by:

\(e=\sqrt{\frac{\sum _{i}\frac{1}{2}{\int }_{{\Omega }_{i}}({\sigma }_{h}-\sigma )\mathrm{:}{D}^{-1}\mathrm{:}({\sigma }_{h}-\sigma )\mathit{dV}}{\sum _{i}\frac{1}{2}{\int }_{{\Omega }_{i}}\sigma \mathrm{:}{D}^{-1}\mathrm{:}\sigma \mathit{dV}}}\),

where the sum is taken over all the groups of cells.

3.2.1. Operand CHAM_GD#

Constraint field \({\sigma }_{h}\) extracted from a finite element calculation result.

3.2.2. Operand CHAM_MATER#

The name of the material field to be used for energy calculations. It is recommended that it be the same model that was used to perform the mechanical calculation from which the field given by the CHAM_GD keyword comes.

3.2.3. Operand DEFORMATION#

This keyword makes it possible to define the hypotheses used for the calculation of deformations (cf. [U4.51.11], §4.5). The only value allowed is” PETIT “, which corresponds to small movements and deformations.

3.2.4. Operands SIXX, SIYY, SIZZ, SIXY,, SIXZ, and SIYZ#

These keywords make it possible to define the \(\sigma\) reference constraint field components in the form of formula objects. These operands are optional because all the components that are not specified are set to zero.

The value of each keyword is a list of formulas to be mapped to the list of mesh groups specified by the GROUP_MA operand.

A common source of error is not to fill in SIZZpour a plane problem, forgetting that, in the general case (i.e. non-zero Poisson’s ratio), the \({\sigma }_{\mathit{zz}}\) component of the stress tensor \(\sigma\) is zero only zero in the case of plane constraints (C_ PLAN).

3.2.5. example#

Calculation of the error in terms of the energy norm for a crack opening in pure mode I, for a flat problem. Note that component \({\sigma }_{\mathit{zz}}\) is null for this problem and is therefore not specified.

# extracting the constraint field from the result data structure

Scal= CREA_CHAMP (OPERATION =' EXTR ',

TYPE_CHAM =' ELGA_SIEF_R ',

RESULTAT = UTOT,

NOM_CHAM =' SIEF_ELGA ',

NUME_ORDRE =1)

# calculation of the error in terms of the energy norm

tab NRJ = POST_ERREUR (OPTION =' ENER_RELA ',

CHAM_GD =Scale,

MODELE = MODELK,

DEFORMATION =' PETIT ',

CHAM_MATER = CHMA,

GROUP_MA =' SURF ',

SIXX = SXX,

SIYY = SYY,

SIXY = SXY,

)

)

3.2.6. Table produced#

The table produced contains, for each cell in the cell group, the energy of the difference field \({\sigma }_{h}-\sigma\) and the energy of the reference field \(\sigma\). It also contains the sum over all cell groups of the energy of the field difference \({\sigma }_{h}-\sigma\), the sum over all the cell groups of the energy of the reference field \(\sigma\), and the relative error in terms of the energy norm.

GROUP_MA	DIFFERENCE		REFERENCE	ERREUR RELATIVE
SURF	1.53608E-09	3.50518E-06
TOTAL	1.53608E-09	3.50518E-06	2.09340E-02

3.3. Operands for OPTION = “DEPL_RELA”#

The “DEPL_RELA” option makes it possible to estimate the difference between the displacement field obtained by the finite element calculation \({u}_{h}\) and the reference displacement field \(u\).

For each group of elements in the list given by operand GROUP_MA, the macro command POST_ERREUR calculates:

Standard \({L}^{2}\) of the \({u}_{h}-u\) difference field

\(\sqrt{{\int }_{{\Omega }_{i}}{\parallel {u}_{h}-u\parallel }^{2}\mathit{dV}}\),

where \({\Omega }_{i}\) is the domain obtained by concatenating all the cells in the group of elements in question.

The \({L}^{2}\) standard of the \(u\) reference displacement field

\(\sqrt{{\int }_{{\Omega }_{i}}{\parallel u\parallel }^{2}\mathit{dV}}\).

Finally, the relative error in terms of the \({L}^{2}\) displacement norm is obtained by:

\(e=\sqrt{\frac{\sum _{i}{\int }_{{\Omega }_{i}}{\parallel {u}_{h}-u\parallel }^{2}\mathit{dV}}{\sum _{i}{\int }_{{\Omega }_{i}}{\parallel u\parallel }^{2}\mathit{dV}}}\),

where the sum is taken over all the groups of cells.

3.3.1. Operand CHAM_GD#

The displacement field \({u}_{h}\) extracted from a finite element calculation result.

3.3.2. DX, DY, and DZ operands#

These keywords make it possible to define the reference displacement field components \(u\) in the form of formula objects. These operands are optional because all the components that are not specified are set to zero.

The value of each keyword is a list of formulas to be mapped to the list of mesh groups specified by the GROUP_MA operand.

3.3.3. example#

Calculation of the error in terms of the \({L}^{2}\) displacement standard for a crack opening in pure mode I, for a plane problem.

# extraction of the displacement field from the result data structure

Ucal= CREA_CHAMP (OPERATION =' EXTR ',

TYPE_CHAM =' NOEU_DEPL_R ',

RESULTAT = UTOT,

NOM_CHAM =' DEPL ',

NUME_ORDRE =1)

# calculation of the error in terms of the L2 norm of the displacement

tabL2= POST_ERREUR (OPTION =' DEPL_RELA ',

CHAM_GD =Ucal,

MODELE = MODELK,

GROUP_MA =' SURF ',

DX=U1,

DY=U2)

3.3.4. Table produced#

The table produced contains, for each mesh in the mesh group, the norm \({L}^{2}\) of the difference field \({u}_{h}-u\) and the norm \({L}^{2}\) of the reference field \(u\). It also contains the sum over all the cell groups of the standard \({L}^{2}\) of the difference field \({u}_{h}-u\), the sum over all the cell groups of the norm \({L}^{2}\) of the reference field \(u\), and the relative error in terms of the norm \({L}^{2}\) of the displacement.

GROUP_MA	DIFFERENCE		REFERENCE	ERREUR RELATIVE
SURF	1.14688E-09	7.60569E-06
TOTAL	1.14688E-09	7.60569E-06	1.50793E-04

3.4. Operands for OPTION = “LAGR_RELA”#

The option “DEPL_RELA” makes it possible to estimate the difference between the contact pressure obtained by the finite element calculation \({\lambda }_{h}\) and the reference pressure field \(\lambda\).

For each group of elements in the list given by operand GROUP_MA, the macro command POST_ERREUR calculates:

Standard \({L}^{2}\) of the \({\lambda }_{h}-\lambda\) difference field

\(\sqrt{{\int }_{{\Gamma }_{i}}{({\lambda }_{h}-\lambda )}^{2}\mathit{dS}}\),

where \({\Gamma }_{i}\) is the domain obtained by concatenating all the cells in the group of elements in question.

The \({L}^{2}\) standard of the \(\lambda\) reference displacement field

\(\sqrt{{\int }_{{\Gamma }_{i}}{\lambda }^{2}\mathit{dS}}\).

Finally, the relative error in terms of contact pressure standard \({L}^{2}\) is obtained by:

\(e=\sqrt{\frac{\sum _{i}{\int }_{{\Gamma }_{i}}{({\lambda }_{h}-\lambda )}^{2}\mathit{dS}}{\sum _{i}{\int }_{{\Gamma }_{i}}{\lambda }^{2}\mathit{dS}}}\),

where the sum is taken over all the groups of cells.

3.4.1. Operand CHAM_GD#

Contact pressure field \({\lambda }_{h}\) extracted from a finite element calculation result.

3.4.2. Operand LAGS_C#

This keyword allows you to define the reference contact pressure \(\lambda\) in the form of a formula object.

The keyword value is a list of formulas to be mapped to the list of mesh groups specified by the GROUP_MA operand.

3.4.3. example#

Calculation of the error in terms of the pressure norm \({L}^{2}\) for the case of the inclusion of two crowns.

# definition of analytical contact pressure

PRES = FORMULE (NOM_PARA =( 'X', 'Y'), VALE ='-pres_cont* EXP (0.1) ')

# extraction of the displacement field from the result data structure

Ucal= CREA_CHAMP (OPERATION =' EXTR ',

TYPE_CHAM =' NOEU_DEPL_R ',

RESULTAT = RESU1,

NOM_CHAM =' DEPL ',

NUME_ORDRE =1)

# calculation of the error in terms of the L2 contact pressure norm

tabL2= POST_ERREUR (OPTION =” LAGR_RELA “,

CHAM_GD =Ucal,

MODELE =ME,

GROUP_MA =”S2R2”,

LAGS_C = PRES)

3.4.4. Table produced#

The table produced contains, for each mesh in the mesh group, the norm \({L}^{2}\) of the difference field \({\lambda }_{h}-\lambda\) and the norm \({L}^{2}\) of the reference field \(\lambda\). It also contains the sum over all the mesh groups of the standard \({L}^{2}\) of the difference field \({\lambda }_{h}-\lambda\), the sum over all the cell groups of the norm \({L}^{2}\) of the reference field \(\lambda\), and the relative error in terms of the \({L}^{2}\) contact pressure norm.

GROUP_MA	DIFFERENCE		REFERENCE	ERREUR RELATIVE
S2R2	1.24905E+03	1.79688E+05
TOTAL	1.24905E+03	1.79688E+05	6.95124E-03