3. Operands#
3.1. Keyword INIT_ALEA#
♦ INIT_ALEA = neither [I]
The INIT_ALEA keyword initializes the seed of random sequences used to define random fields. Two consecutive calculations with the same initialization then produce the same result.
3.2. Keyword MEDIANE#
♦ MEDIANE = med [R]
Keyword to define the median value of the log-normal random field. In general, the median value is associated with the*best-estimate* value.
3.3. Keyword COEF_VARI#
♦ COEF_VARI = cov [R]
Keyword to define the coefficient of variation for random fields. The coefficient of variation is defined as the ratio between the standard deviation and the mean of the random field. In the case of log-normal law fields, logarithmic standard deviation \(\mathrm{\beta }\) is linked to the coefficient of variation by the formula \(\mathit{cov}=\sqrt{({\mathrm{exp}}^{({\mathrm{\beta }}^{2})}-1)}\).
3.4. Keywords LONG_CORR_X, LONG_CORR_Y, and LONG_CORR_Z#
♦ | LONG_CORR_X = lCx [R]
| LONG_CORR_Y = Lcy [R]
| LONG_CORR_Z = lCz [R]
Keyword to define the correlation length of random fields according to the \(X\) direction (if specified).
The definition of correlation lengths is that of Vanmarcke: \({L}_{c}={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}R(u)\mathit{du}\) where \(R(u)\) is the correlation function for the variable \(u\) (here: the distance in the direction \(X\)). The correlation function is a simple \(R(u)={\mathrm{exp}}^{(-u/(0.5{L}_{c}))}\) exponential in each direction.
LONG_CORR_Y and LONG_CORR_Z are analogous to those in LONG_CORR_X for the \(Y\) and \(Z\) directions.
3.5. Keywords NB_TERM_X, X_ MINI and X_ MAXI#
These keywords are mandatory if LONG_CORR_X is entered.
♦ NB_TERM_X = Nx [I]
The number of terms to remember for the Karhunen-Loève decomposition according to the \(X\) direction. The number of terms defines the number of eigenfunctions and therefore the small fluctuations of the variable parameter. As the random field is generated on unit domains \([\mathrm{0,1}]\), you have to choose the number of terms in relation to the size of the domain and the discretization. Otherwise, it is recommended to take Nbt equal to the domain extension divided by the mesh size (here (xmax-xmin) /dx).
♦ X_ MINI = xmin [I]
The min coordinate according to the domain extension in the \(X\) direction.
♦ X_ MAXI = xmax [I]
The max coordinate according to the extension of the domain in the \(X\) direction.
The last two keywords define the extension of the domain on which random fields must be generated (bounding volume).
3.6. Keywords NB_TERM_Y, Y_ MINIet Y_ MAXI#
These keywords are mandatory if LONG_CORR_Yestrenseigné.
3.7. Keywords NB_TERM_Z, Z_ MINIet Z_ MAXI#
These keywords are mandatory if LONG_CORR_Zestrenseigné.
3.8. Keywords NB_TERMou PRECISION to reduce the model#
◊ NB_TERM =/Nbt [I]
◊ PRECISION =/prec (min=0.0, max=1.0) [R]
These optional keywords can be used to reduce the model in the case of 2D or 3D fields.
The mandatory keywords NB_TERM_X, NB_TERM_Yet NB_TERM_Zpermettent define the number of terms for the Karhunen-Loève (KL) representation to be calculated for each component (X and/or Y and/or Z). In the case of a 2D random field, we have two components, and for the 3D, we have 3 components. The total energy for this representation of KL is calculated as the sum of all combinations of eigenvalue products. For example, in the case of a 2D field along the X and Y directions with Nx and Ny eigenfunctions respectively, the KL representation includes Nxy=Nx*Ny terms.
In order to retain a reduced number of terms, while maintaining a good representation of the fields, the cross-terms are classified according to their contribution to the total energy of decomposition. The model can then be reduced by using the Nbt terms that are the most energetic. For the example above, you need Nbt < Nxy.
If PRECISIONest is entered, then prec* 100% of the most energetic terms are retained among all the calculated terms.
If NB_TERM is entered, then the NB_TERMles with the most energy are retained among the total calculated terms.