Guaranteed characterization of thermal conductivity and diffusivity in presence of model uncertainty

Figures & data

Figure 1. Model of the set-up: each of the five layers is modeled using a quadrupole to introduce the convection exchange on the rear side; the arrows stand for the direction of the heat flow; the relative position of the output thermocouple is also represented.

Table 1. Prior domains of the nuisance parameters

i	qi
1	R1	[1.8, 4] × 10^−5 m^2K W^−1
2		[3.38, 7] × 10^−4 s^1/2
3	R	[2.4, 2.6] × 10^−5 m^2K W^−1
4	R5	[1.8, 4.3] × 10^−5 m^2K W^−1
5		[9, 9.4] × 10^−1
6	ξ	[0.3, 0.7]
7	h	[5, 10] W m^−2 K^−1

Figure 2. The sets to be characterized: the solution set , its projection Π on and a section Σ(q*) corresponding to a point value .

Figure 3. Model output (O) and (+) and pattern imposed by the interval data [y_i] (rectangles) as functions of f_i.

Figure 4. Estimate of the set Σ(q²) obtained with h=10: inner approximation (black) and outer approximation (union of the black and grey boxes). The least-square estimate and its ellipsoidal confidence region are also depicted.

Figure 5. Inner approximation (black) and outer approximation Π (union of black and grey boxes) of the set Π obtained by PROJECT for ϵ=0.01 in 82 s.

Table 2. Influence on the uncertainty on the parameters of interest of the uncertainty associated with each nuisance parameter

i	qi		[si]
1	R1	38%	[1, 1.17]	[2.63, 3.08]
2		39%	[1, 1.48]	[2.56, 3.80]
3	R	41%	[1, 1.45]	[2.44, 3.54]
4	R5	4%	[1.89, 5.31]	[47.25, 132.75]
5		2%	[1.83, 7.18]	[91.5, 359]
6	ξ	40%	[1, 1.33]	[2.5, 3.33]
7	h	33%	[1, 1.55]	[3.03, 4.70]