3.2 Methodology
3.2.3 Demonstration of GPDE
In the next two subsections, GPDE is demonstrated first for the case of three uniformly refined mesh levels (Section 3.2.3.1), then for five adaptively refined mesh levels (Chapter 3.2.3.2) using the heat transfer model from Section 2.4. The geometry is shown in Figure 3.2 withnℓ = 9numbered output locations for reference in the results (and a typical adaptively refined mesh). Discretization errors will be computed for each location.
Figure 3.2:Turbine disc thermal model discretization error estimation locations (thermocouple positions).
3.2.3.1 GPDE for uniform refinement compared to Richardson extrapolation
Mesh settings were iteratively modified to achieve the refinement ratiorx≈1.62(rt≈2.62) which results in the three mesh level settings forhandEtlisted in Table 3.1. FE solutions for each mesh were computed and outputs were extracted at the 9 thermocouple locations at timet= 2010seconds4. Then, GP models were fit to the three solutions as a function ofh, which are shown in Figure 3.3.
Table 3.1:Uniform refinement discretization error study settings for element sizehand adaptive time step error limit Et.
Mesh Level units 1 2 3
Characteristic element size,h mm 3.04 4.91 7.96 Temporal error limit,Et K 0.73 1.91 5
Figure 3.3:GP fit of temperature as a function of average element size [mm] for uniformly refined meshes at locations P1-P9 shown in Figure 3.2. The red point isfˆath= 0.
4This time was chosen since it occurs during the transient maneuver of the square cycle (Figure 2.3), where errors are near their worst.
In cases where the GPs in Figure 3.3 are less well-defined in the extrapolation regionh→ 0(e.g., P1, P6, P7, P9), it is observed that the estimate forfˆ(red point) drifts towards the mean of the three mesh level values fkand its standard deviation grows toward the standard deviation of the three valuesfk. This ‘saturation’ is most obvious and complete in the case of P1, which fails to find a good fit due to lack of a clear trend. Although tighter uncertainty bounds is preferred, this ‘failure’ mechanism is a reasonable default when there is no clear convergence towardfˆ. In the RE/GCI approach, such behavior precludes the use of the estimate altogether.
The GPDE estimate Eq. 3.6 was computed from these results at mesh level 2 and compared to the RE- based GCI (computed using the same meshes) and ϵhr in Figure 3.4. The RE computations did not achieve agreement between the observed and formal order of convergence5due to challenges with achieving refinement in the asymptotic range (location P8 is at a relatively thin structure which prevents uniform refinement; P1 is near boundary conditions that induced high temperature gradients). The recovery based solution (ϵhr) results in a substantially error than either the GP or GCI approach, which may supports the guidance in [16, 46] that recovery based discretization error estimators are not useful in magnitude for VVUQ. Finally, the comparison between GCI and GPDE is not perfect, but since the GCI results were not within the asymptotic range, it was decided they were satisfactory enough to carry forward with the adaptively refined mesh study. Future work should consider applying this approach to simpler geometry in order to demonstrate observed order of accuracy.
0 0.5 1 1.5 2 2.5 3 3.5
P1 P2 P3 P4 P5 P6 P7 P8 P9
1σError [K]
Verification Error Method Comparison (Fs=1)
GCI Uniform GP Uniform σ_hr Uniform
Figure 3.4: Comparison of three discretization error estimators for uniformly refined mesh: RE-based GCI, GP, and the recovery-based error estimator standard deviationσhr. These represent 1σerrors (Fs= 1).
3.2.3.2 GPDE for adaptive refinement
Next, to perform the adaptive refinement discretization study, five cases were defined. Based on SME experience, an upper bound ofEx = 5K and lower bound ofEx = 1K were chosen, resulting in a spatial refinement ratio rx ≈1.5and temporal refinement ratiort =r2x ≈2.24. The settings forExandEtare listed in Table 3.2. A minimum ofEt= 0.45K was set to avoid excessive time steps. The FE solutions were obtained at the locations P1-P9 and GP models fit as a function ofEx. These GPs are are shown in Figure 3.5 along with the estimatedfˆ
5Thus, the GCI approach suggests that a safety factor ofFs = 3is applied when using the results for UQ studies. However, for this comparisons below no safety factor is applied (Fs= 1).
(red dot), obtained by extrapolation toEx= 0.
Table 3.2:Adaptive refinement discretization error study with adaptive error limit settings.
Mesh Level units 1 2 3 4 5
Spatial error limit,Ex K 1 1.50 2.24 3.34 5 Temporal error limit,Et K 0.45 1.01 2.25 5.03 11.25
Figure 3.5: GP fit of temperature as a function of adaptive refinement accuracy [K] for adaptively refined meshes at locations P1-P9 shown in Figure 3.2. The red point isfˆatEx= 0.
The GP bias (bk = ˆf−fk), GP standard deviation (σfˆ), and the GPDE standard deviation (σhg) are tabulated in the first three rows of Table 3.3. The third mesh level (k = 3) is chosen for the purpose of demonstration in this study (in practical situations the finest mesh may not be chosen in order to balance runtime and accuracy).
For comparison, the table includes the standard deviation and range offkfor the five mesh levels. At most of the locations (except P6), the GP’s standard deviation is smaller than the 5-solution standard deviation (σfˆ<Std[fk]), indicating that the extrapolation has not reached saturation.
Table 3.3:GP discretization error study results for adaptive refinement (in Kelvin, att= 2010sec.).
Location→ 1 2 3 4 5 6 7 8 9
GP mean bias,b3= ˆf−f3 1.84 0.18 0.85 2.52 1.15 0.11 0.18 0.89 0.12 GP std. dev.,σfˆ 1.31 0.49 0.87 1.01 0.55 0.19 0.69 0.33 0.36 GPDE std. dev.,σhg 2.26 0.52 1.21 2.71 1.28 0.22 0.72 0.95 0.38 5-solution std. dev. 1.66 1.08 0.98 1.47 0.98 0.19 0.88 0.65 0.39
5-solution range 4.87 3.00 2.88 4.24 2.80 0.45 2.16 1.78 1.15
3.2.3.3 Comparison of GPDE for uniform and adaptive refinement
The uniform and adaptive refinement study results are compared in Figure 3.6. The adaptive results (Figure 3.5) show a higher level of non-monotonicity relative to uniform refinement (Figure 3.3), as one might expect due to local refinement effects. This also results in larger GPDE than the uniform case at several locations, as Figure 3.6 highlights. These results are again compared toϵhr, which again show its potential limitations as an estimator for use in UQ. However, as is shown in the next section, due to its availability throughout the FE solution domain (space and time), it will be used to obtain the additional uncertainty due to parameter dependence.
0 0.5 1 1.5 2 2.5 3
P1 P2 P3 P4 P5 P6 P7 P8 P9
1σError [K]
Verification Error Method Comparison (Fs=1)
GP Uniform GP Adaptive σ_hr Uniform σ_hr Adaptive
Figure 3.6:Comparison of uniform and adaptive refinement discretization errors for GP and the recovery-based error estimator standard deviationσhr. These represent 1σerrors (Fs= 1).
The medium (mid-level) meshes from the uniform and adaptive refinement studies are shown in Figure 3.7.
While the adapted mesh errors are higher in some locations, the number of elements is substantially smaller. This reflects the trade-off between analysis speed and accuracy that the user must choose between. The adaptively refined mesh errors could be reduced by choosing mesh level 1 or 2 (rather than 3).
(a) (b)
Figure 3.7:Comparison of (a) uniformly refined and (b) adaptively refined meshes. The medium (middle) mesh level is shown for each from the two studies.