7. Implanting shell elements in Code_Aster#
7.1. Description#
These elements (with names MEC3QU9H and MEC3TR7H) are based on cells QUAD9 and TRIA7 which have curved geometry [R3.07.04].
7.2. Use#
These elements are used as follows:
AFFE_MODELE (MODELISATION: “COQUE_3D”…) for the triangle and the quadrangle.
Routine INI080 is used for standard numerical integration calculations.
AFFE_CARA_ELEM (COQUE :( EPAISSEUR :'EP'
ANGL_REP: (« « ) COEF_RIGI_DRZ: “CTOR”)
to introduce shell characteristics.
ELAS: (e:YOUNG NU: \(\nu\) ALPHA: \(\alpha\).. RHO: \(\rho\)..)
For a thermo-elastic isotropic behavior that is homogeneous in thickness we use the keyword ELAS in DEFI_MATERIAU where we define the coefficients \(E\), Young’s modulus, \(\nu\), Poisson’s ratio, \(\alpha\), thermal expansion coefficient,, thermal expansion coefficient and \(\rho\), the density.
AFFE_CHAR_MECA (DDL_IMPO: (
EX:.. OF:.. DZ:.. DRX:.. DRY:.. DRZ:.. degree of freedom of the plate in the global coordinate system.
FORCE_COQUE: (FX:.. FY:.. FZ:.. MX:.. MY:.. MY:..)
These are the surface forces on plate elements. These efforts can be given in the global frame or in the user frame defined by ANGL_REP.
FORCE_NODALE: (FX:.. FY:.. FZ:.. MX:.. MY:.. MY:..)
These are shell efforts in the global benchmark.
7.3. Geometric nonlinear « elasticity » calculation#
The calculation requires the following user instructions:
COMPORTEMENT: (RELATION: 'elas'
COQUE_NCOU: 1 (or more)
DEFORMATION: 'GROT_GDEP')
Numerical integration into thickness is based on a multi-layer approach with 3 integration points per layer. This is the approach currently used in hardware nonlinear [R3.07.04]. The post-processing options SIGM_ELNO for constraints and VARI_ELNO for internal variables (here zero) are activated by default at the convergence of each archived step.
7.4. Implantation#
The options FULL_MECA, RIGI_MECA_TANG, and RAPH_MECA are already active in the mec3qu9h.cata and mec3tr7h.cata elementary catalogs for hardware nonlinearity. They direct the calculation to /fort/te0414.f, then to /fort/vdxnlr.f to calculate and store, among other things, the tangent symmetric stiffness matrix in the address corresponding to mode MMATUURPMATUUR.
For geometric nonlinear analysis, the calculation of the tangent stiffness matrix is directed to the new routine VDGNLR. This matrix is not symmetric and must be stored in the address corresponding to the MMATUNS PMATUNS mode.
The two symmetric and non-symmetric local modes are defined at the same time, at the output of the elementary catalogs. The tangent matrix of non-symmetric stiffness in geometric nonlinear is stored at the address reserved for a non-symmetric matrix. On the other hand, if it is a question of material nonlinearity in small deformations, the entire tangent stiffness matrix is stored at the address corresponding to the non-symmetric mode. The lower triangular portion is duplicated from the upper triangular portion. Hardware nonlinear calculation in small deformations therefore also takes place in non-symmetric.
7.4.1. Modification of TE0414#
The calculation is directed to /fort/vdgnlr.f when behavior type COMPORTEMENT is true, i.e. when the problem is geometric nonlinear.
7.4.2. Addition of a routine VDGNLR#
Depending on the option, the /fort/vdgnlr.f routine has the role of:
Options: FULL_MECA and RAPH_MECA:
Calculate the 6 components of the state of local Cauchy stresses (confused with the state of Piola-Kirchhoff stresses of the second kind) at the points of normal numerical integration and the nodal vector of internal forces. They are stored in local modes ECONTPG PCONTPR and MVECTUR PVECTUR respectively. As an additional remark, it should be noted that SIEF_ELGA/SIGM_ELGA after the resolution is of the PK2 type while SIEF_ELNO/SIGM_ELNO in post-processing is of the Cauchy type.
Options: FULL_MECA and RIGI_MECA_TANG:
Calculate and store the non-symmetric stiffness tangent matrix in the mode MMATUNS PMATUNS .
7.5. Linear buckling calculation#
Option RIGI_MECA_GE, inactive until now, is activated in the elementary catalogs mec3qu9h.cata and mec3tr7h.cata.
The new TE0402 is dedicated to the calculation of the geometric rigidity matrix due to the initial stresses for Euler buckling. The plane states of the local Cauchy constraints (zero component \({S}_{\text{nn}}\)) are recovered at the points of normal Gauss numerical integration. These stress states must be obtained by post-processing with calculation option SIEF_ELGA following a linear analysis (mode ECONTPG PCONTRR).
In Euler buckling analysis, Cauchy \(\left[\sigma \right]\) constraints can be confused with the second species Piola-Kirchhoff \(\left[S\right]\) constraints. That is why we will keep the \(\left[S\right]\) rating.
The stiffness matrix of the initial stresses can be broken down into a symmetric classical part and a non-symmetric non-classical part. The first is calculated according to the global stress tensor \(3\mathrm{\times }3\), unlike the second, which is calculated according to the local stress vector \(5\mathrm{\times }1\).
Since the current algorithm for solving the eigenvalue problem [R5.06.01] only considers symmetric matrices, we force the symmetry of the non-classical part of the geometric matrix before storing the upper triangular part of the entire matrix in the MMATUUR PMATUUR mode.