PCEg Setting Procedure
4.1
Method
As previously said, the UQ analysis has been made using a PCEg method. Three input variables have been considered (µ, σ, ψ) and, since we had not enough information to choose a particular PDF, the input variables have been modeled by uniform probability distributions, see Fig.(3.11).
Because of this, following the Askey scheme (Tab.(2.1)), the polyno-mial basis used to construct the output functions is a Legendre basis. The analyzed output quantities are those defined in the previous section (see Tab.(3.4)); thus, considering three time steps of the simulation, the total number of output functions is 120. Each output function is the polyno-mial constructed by DAKOTA using the PCEg. For each response function DAKOTA is able to compute:
• statistical parameters as the mean value, the standard deviation, the skewness and the kurtosis;
• the discrete CDF, using a set of probability levels chosen by the user (in our case:[0.01 0.05 0.25 0.50 0.75 0.95 0.99]) ;
The technique used to cut the unbounded response function (Eq.(2.2)) is the tensor product approach. Since no sources of anisotropy have been considered, the three dimensions of the problem, one for each input variables, have been treated in the same way and with the same number of quadrature points. Using this technique the total number of terms presented in each response function can be evaluated by Eq.(2.8).
4.2
Choice of Quadrature Order
The quadrature order has been decided varying the number of quadrature abscissas until the convergence of the output values has been reached. We started from a quadrature order of 3 and we stopped at a quadrature order of 7.
The change in the quadrature order produces a variation in the number of deterministic runs that the code makes in order to compute the PCEg coefficients, as reported in Tab.(4.1).
Quadrature Order Total number of integration points
3 27
4 64
5 125
6 216
7 343
Table 4.1: Total number of quadrature points associated to each quadrature order.
According to Eq.(2.7), Tab.(4.2) shows the maximum polynomial order presents in the multivariate term and the total number of coefficients that
Quadrature Order Max. Pol. Order Num. PCEg coeff. 3 P2 27 4 P3 64 5 P4 125 6 P5 216 7 P6 343
Table 4.2: Variation of the maximum polynomial order and of the response coefficients number as a function of the quadrature order.
Plotting the CDFs (Fig.(4.1)) of a generic output function (i.e. the mean value of the parcels’ diameter in the first stripe), the effect of the quadrature order can be studied observing how the response values change by varying the number of quadrature points.
0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 CDF µ Quadrature Order 3 Quadrature Order 4 Quadrature Order 5 Quadrature Order 6 Quadrature Order 7
(a) CDFs releated to the mean value of the parcels diameters present in Stripe 1-Air at time 10600s. It is possible to observe that, varying the quadrature orderd, the CDFs are approssimatly costant. 1.2975 1.298 1.2985 1.299 0.4726 0.4728 0.473 0.4732 CDF µ Quadrature Order 3 Quadrature Order 4 Quadrature Order 5 Quadrature Order 6 Quadrature Order 7 (b) Zoom of the CDFs.
By the previous figures it appears that, even with the lowest quadrature order, good results are reached in term of convergence of the solution. This can be better quantified also considering the mean value of the response func-tion associated with the mean diameter of the parcels present in Stripe 1-Air (see Tab.(4.3)).
Quadrature Order Mean value [φ]
3 1.1223
4 1.1206
5 1.1207
6 1.1210
7 1.1207
Table 4.3: Variation of the mean value of parcels diameter as a function of the quadrature order.
In fact, starting from order 3, the response values do not significantly change and the error committed, passing from a quadrature order of 3 to a quadrature order of 7, is approximately 0.001%.
However, we have chosen a quadrature order of 7, since a total time of less than 2 hours is required to run all the simulations and this can be considered adequate for our work (see Tab.(4.4)).
Quadrature Order Time [s]
3 806
4 1620
5 3433
6 4152
7 5337
Since the Legendre polynomials are defined in the range [-1,+1], the quadrature abscissas must be conveniently weighted. The weights associ-ated to the 7 quadrature points are reported in Tab.(4.5):
Quadrature points Weights
1 (Lowest value) -0.9491079123 2 -0.7415311856 3 -0.4058451514 4 0 5 0.4058451514 6 0.7415311856 7 (Upper value) 0.9491079123
Fig.(4.2) shows the 3D computational grid built by DAKOTA accord-ing to the previous settaccord-ings. Each point represent an input tern used by DAKOTA in order to run the deterministic simulations.
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 µ σ ψ
(a) 3D grid used to evaluate the integration points for a quadrature order of 7
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 µ σ
(b) View of the (µ, σ) plane of the grid. Figure 4.2: Sketch of the quadrature grid.
Thus, each response function is written in the form of a polynomial (Eq.(2.2)) composed by 343 coefficients (Eq.(2.8)) and the multivariate basis Ψj(ξ) associated to each term is a combination of 1D Legendre polynomials up to a maximum order of 6. Fig.(4.3) shows the coefficients and the respec-tive multivariate basis of the first 28 terms of the polynomial which is the response function of the mean diameter of the parcels in Stripe 1-Air.
Figure 4.3: Image of a polynomial response function present in the output file
of DAKOTA. For each coefficient, the Pi are the unidimensional Legendre
polynomials which are mixed together in order to generate a multivariate basis.
