4. Implemented in Code_Aster#
4.1. Study of the stochastic convergence of the numerical model#
4.1.1. Transitional case#
The convergence of the stochastic solution must be studied in relation to the number \(n\) of modes and the number \({n}_{s}\) of Monte Carlo simulations. As the stochastic solution is a second-order process (by hypothesis, cf. [§3.4.1.2]), its convergence can be analyzed by studying applications such as:
\({\mid \parallel {\ddot{Z}}_{j}^{n}\parallel \mid }^{2}={\int }_{0}^{T}E{\left\{{\ddot{Z}}_{j}^{n}(t)\right\}}^{2}\text{dt}\), eq 4.1.1-1
where is a second-order stochastic process indexed by and to value in \(R\) representing the acceleration of the \(j\) th degree of freedom of the structure.
In the context of Monte Carlo simulations, this norm is estimated for \(n\) fixed from a set of \({n}_{s}\) random realizations by approximating with
\({\text{conv}}_{j}(n,{n}_{S}{)}^{2}={\int }_{0}^{T}{(\frac{1}{{n}_{S}}\sum _{i=1}^{{n}_{S}}{\ddot{Z}}_{j}^{n}(t;{\theta }_{i}))}^{2}\text{dt}\) eq 4.1.1-2
The stochastic convergence of the model is thus analyzed according to the dimension of the reduced model (i.e. the \(n\) mode number of the eigenspace of the average finite element model onto which the stochastic nonlinear dynamic system was projected in paragraph [§2.2]) and the number \({n}_{s}\) of Monte Carlo simulations by studying the function.
4.1.2. Harmonic case#
Convergence in the case of a transitory resolution can be transposed directly in the case of a harmonic resolution, with the standard:
\({\mid \parallel {Z}_{j}^{n}\parallel \mid }^{2}={\int }_{{\omega }_{1}}^{{\omega }_{2}}E{\left\{{Z}_{j}^{n}(\omega )\right\}}^{2}d\omega\), eq 4.1.2-1
4.2. Choice of dispersion parameters#
To use the method, dispersion parameters \(\delta\) must be set. Two approaches can be*a priori* used to set the value of these parameters.
The first approach is to identify the value of the \(\delta\) parameters for a given structure or for a structure class using appropriate methods. For this, it is possible to use experimental results of the dynamic responses of the structure. It is also possible to use numerical simulations built using a parametric approach to uncertainties. In the latter case, it should be noted that only errors in the model data are taken into account, since modeling errors cannot be taken into account by the parametric approach.
The second approach consists in not setting a priori a fixed value of the parameters \(\mathrm{\delta }\) but in making them vary within a given range (only 3 scalars to vary for the mass, stiffness and damping matrices on the non-parametric part in comparison with the very large number of parameters to be varied simultaneously in a classical parametric study). This approach makes it possible to conduct a global analysis of sensitivity to uncertainties. In the case of the absence of objective information on which dispersion parameters to choose, it is preferable to use such an approach. The proposed non-parametric method then appears to be a robust and simple approach for analysing sensitivity to uncertainties.
4.3. Main steps#
The implementation in Code_Aster is composed of three main steps: the construction of the average reduced matrix model, the generation of the realizations of the response seen as a stochastic process, and finally the statistical post-processing of these realizations. The last two steps in fact constitute the direct Monte Carlo numerical simulation method.
Step 1: Construction of the average reduced matrix model
The average reduced matrix model is built using a classical sequence of operators depending on the precise analysis carried out, the main ones of which can be: ASSE_MATRICE, CALC_MODES, MODE_STATIQUE, CALC_CHAR_SEISME,, PROJ_BASE,…
Step 2: generating the achievements of the transient response
The \({n}_{s}\) achievements of the stochastic transient response are calculated in a loop in Python language composed of:
Generation of :math:`p`*th realizations of generalized random mass, stiffness and damping matrices by Gene_matr_alea (doc. [U4.36.06]). These matrices are not diagonal and therefore require full storage.
Generation of the :math:`p`*th realizations of the random variables of the parameters of nonlinearities by Gene_ VARI_alea (doc. [U4.36.07]).
Calculation of the :math:`p`*th realization \({Q}^{n}(t;p)\) or \({Q}^{n}(\omega ;p)\) solution of the stochastic matrix system \(s\). This realization is the solution of the classical matrix system in which the matrices and the second members are the realizations previously generated. The calculation is therefore performed by dyna_tran_modal or DYNA_LINE_HARM (with matr_asse_ GENE_R and vect_asse_ GENE as input).
1-Extraction of the temporal observations of the predefined physical degrees of freedom (for example :math:`{ddot{Z}}_{{}_{i}}^{n}(t;p)` or :math:`{Y}_{j}^{n}(omega ;p)`, but also possibly the displacement fields, speed, constraints, etc.)*via recu_function (after a REST_GENE_PHYS for \({Y}_{j}^{n}(\omega ;p)\)).
2-Calculation of the corresponding spectra (by calc_function (SPEC_OSCI) for \(\omega \to {S}_{j}(\xi ,\omega ;p)\) and calc_function (MODULE) for \({Y}_{j}^{n}(\omega ;p)\) ).
Evaluation,*via calc_function keywords COMB or PUISSANCE or ENVELOPPE, of contributions to the estimators of means, second-order moments, max. and min. extreme sample values for standardized spectra:
\({\stackrel{ˆ}{m}}_{\mathrm{1j}}(x,w;p)={S}_{j}(x,w;p)+{\stackrel{ˆ}{m}}_{\mathrm{1j}}(x,w;p-1)\), \({\stackrel{ˆ}{m}}_{\mathrm{2j}}(x,w;p)={S}_{j}(x,w;p{)}^{2}+{\stackrel{ˆ}{m}}_{\mathrm{2j}}(x,w;p-1)\),
\({\stackrel{ˆ}{S}}_{j\text{,max}}(x,w;p)=\text{Max}\left\{{S}_{j}(x,w;p),{\stackrel{ˆ}{S}}_{j\text{,max}}(x,w;p-1)\right\}\), \({\stackrel{ˆ}{S}}_{j\text{,min}}(x,w;p)=\text{Min}\left\{{S}_{j}(x,w;p),{\stackrel{ˆ}{S}}_{j\text{,min}}(x,w;p-1)\right\}\).
Step 3: statistical post-processing
Means, standard deviations, max. and min. extreme sample values for normalized spectra can be evaluated*via* calc_function (COMB):
\({m}_{\mathrm{1j}}(x,w)=\frac{1}{{n}_{s}}{\stackrel{ˆ}{m}}_{\mathrm{1j}}(x,w;{n}_{s})\), \({m}_{\mathrm{2j}}(x,w)=\frac{1}{{n}_{s}}{\stackrel{ˆ}{m}}_{\mathrm{2j}}(x,w;{n}_{s})\).
Confidence intervals can then be drawn from extreme sample values or limits obtained by Chebychev cf. [§3.4.4].
In the transitory case, an example is given by a test case of a flexure plate with shock nonlinearities, cf. doc. [V5.06.001] [bib1]. More details are given in the doc. [U2.08.05] [bib2].
4.4. Numerical efficiency of the non-parametric approach#
The non-parametric approach is more economical in terms of calculation time than a purely parametric approach in which the parameters of geometry, materials, etc. are random variables.
In the purely parametric approach, the finite element model depends on uncertain parameters. For each Monte Carlo simulation, the finite element model is different. It is therefore necessary, for each simulation, to calculate the elementary matrices, to perform the assemblies, to pass to relative coordinates, to solve the problem to the eigenvalues, to project on a modal basis, to solve the reduced system and to return to the physical base and then to relative coordinates.
In the non-parametric approach, only the reduced system is different at each simulation. It is therefore simply necessary, for each simulation, to solve the reduced system and return to the physical base and then to relative coordinates. In particular, the resolution of the problem at the eigenvalues of the mean finite element model is done once and for all, before the Monte Carlo simulations.
The resulting gain in computing time is variable, but it can be significant. As a first approximation, this gain in calculation time depends on the ratio between the time CPU required to solve the eigenvalues and the time CPU necessary to solve the reduced system. The greater this ratio, the more advantageous the non-parametric approach is compared to the purely parametric approach. In particular, the gain in calculation time can be very significant for structures with a very large number of degrees of freedom and a small-dimensional modal base.