3. Modeling, loading#

To assign the Fourier axisymmetric finite elements to the mesh, the AFFE_MODELE operator is used in the following way:

mo = AFFE_MODELE (MAILLAGE = my,

AFFE = _F (TOUT = “OUI”, PHENOMENE = “MECANIQUE” or “THERMIQUE”, MODELISATION = “AXIS_FOURIER”) );

The decomposition of the load into Fourier series must have been done beforehand by the user

Be \(f=\sum _{l=0}^{N}\left[(\begin{array}{ccc}\mathrm{cos}l\theta & & 0\\ & \mathrm{cos}l\theta & \\ 0& & -\mathrm{sin}l\theta \end{array}){F}_{l}^{s}(r,z)+(\begin{array}{ccc}\mathrm{sin}l\theta & & 0\\ & \mathrm{sin}l\theta & \\ 0& & \mathrm{cos}l\theta \end{array}){F}_{l}^{a}(r,z)\right]\)

with \({F}_{l}^{s(a)}=({f}_{rl}^{\mathrm{ss}(a)},{f}_{zl}^{s(a)},{f}_{\theta l}^{s(a)})\)

The charges \({F}_{l}^{s}\) and \({F}_{l}^{a}\) are introduced harmonic by harmonic and type by type by the operator AFFE_CHAR_MECA. The mode or type is not specified at this level.

Example: a loading with distributed pressure in symmetric mode 1 and in pure torsional mode (antisymmetric mode 0) is assumed.

We will write:

ch1sym = AFFE_CHAR_MECA (Template= mo,

PRES_REP = _F (GROUP_MA = “grma”, PRES = p));

ch0anti = AFFE_CHAR_MECA (Template = mo,

FORCE_NODALE = _F (FZ = f, NOEUD = “N1”)));

Dirichlet-type boundary conditions will be introduced in a separate load:

chdir = AFFE_CHAR_MECA (Template= mo,

DDL_IMPO = _F (GROUP_NO = “big”, DX=0., DY=0., DZ=0.,));

The loads allowed by Fourier elements are:

in elasticity:

Elements	Nature of load	Keyword AFFE_CHAR_MECA
TRIA3 - TRIA6 QUAD4 - QUAD8 - QUAD9	Volume forces Rotation Gravity Point forces	FORCE_INTERNE ROTATION PESANTEUR FORCE_NODALE
SEG2 - SEG3	Pressure Surface forces	PRES_REP FORCE_CONTOUR

in thermal:

Elements	Nature of load	Keyword AFFE_CHAR_THER
Surface	Heat Source	SOURCE
Edge	Normal flow imposed Exchange	FLUX_REP ECHANGE

3.1. Special case of gravity#

We consider the cylinder below whose shaded part corresponds to the meshed part on which an axisymmetric Fourier calculation is performed (plane \(\mathrm{rZ}\)).

This cylinder is subject to its own weight \(P\) directed along the \(-\text{OX}\) axis.

\(\theta\)

The load to be applied here is \(\overrightarrow{P}=({P}_{r},{P}_{z},{P}_{\theta })\)

\(\theta =0\)

\(\theta\)

With \({P}_{r}=-g\mathrm{cos}\theta\) \({P}_{z}=0\) \({P}_{\theta }=g\mathrm{sin}\theta\)

\(\theta =\frac{3}{2}\pi\)

We check that for any value of \(\theta\) we have a loading R with amplitude \(g>0\) directed downwards.

The desired load is therefore \(g(-\mathrm{1,0},-1)\) according to the Fourier mode decomposition (symmetric harmonic and cosine series development for the radial component). \(\theta =\pi\)

\(\theta\)

Attention, the direction is under the assumption that \(P\) is directed along the axis

\(-\text{OX}\). For another orientation of \(P\), the calculation must be adapted. R

\(X(r)\)

\(Y(Z)\)

\(Z(\theta )\)

\(\overrightarrow{P}\)

However, since the loading direction vector is normalized in AFFE_CHAR_MECA, it is necessary to multiply by this norm to obtain the desired load.

The load to be introduced in AFFE_CHAR_MECA behind the PESANTEURest keyword is therefore \(\sqrt{2}g(-\mathrm{1,0},-1)\).

In MACRO_ELAS_MULT, this load will be assigned to symmetric harmonic 1 (cosine series expansion for the radial component).