Reconstruction of the heat transfer coefficient at the interface of a bi-material

Figures & data

Figure 1. Schematic representation of the space bicomponent domain $\mathrm{\Omega }={\mathrm{\Omega }}_1\cup {\mathrm{\Omega }}_2$ in two-dimensions.

Figure 2. (a) The normalized objective functionalal $\overline{J}\left[{\phi }^n\right]$, (b) the normalized accuracy error $\overline{E}\left[{\phi }^n\right]$, and (c) numerical solutions of $\phi \left(t\right)$ with initial guess ${\phi }^0\left(t\right)=0.5A$ and $\kappa =1$, for noise $p\in \{0,1,3\}$, for Example 1.

Figure 3. Numerical solutions of $\phi \left(t\right)$ for various values of the smoothing parameter $\kappa \in \{0,10,{10}^2\}$, with initial guess ${\phi }^0\left(t\right)=0.8A$, for (a) p=0, (b) p=1 and (c) p=3 noise, for Example 1.

Figure 4. The normalized objective functionalal $\overline{J}\left[{\phi }^n\right]$ for (a) $\kappa =0$ and (b) $\kappa ={10}^3$, and the normalized accuracy error $\overline{E}\left[{\phi }^n\right]$ for (c) $\kappa =0$ and (d) $\kappa ={10}^3$, for $p\in \{0,3,5\}$ noise, for Example 2.

Figure 5. Variation of the normalized accuracy error $\overline{E}\left[{\phi }^n\right]$, at the stopping iteration number $n_s$, with respect to κ, for $p\in \{0,3,5\}$ noise, for Example 2.

Figure 6. Retrieved solutions using various values of κ for (a) p=0, (b) p=3 and (c) p=5 noise, for Example 2.

Figure 7. (a) The normalized objective functionalal $\overline{J}\left[{\phi }^n\right]$ and (b) the normalized accuracy error $\overline{E}\left[{\phi }^n\right]$ for $\kappa =1$, for $p\in \{0,3,5\}$ noise, for Example 3.

Figure 8. The variation of the normalized accuracy error $\overline{E}\left[{\phi }^n\right]$ with the smoothing parameter κ, for $p\in \{0,3,5\}$ noise, for Example 3.

Figure 9. (a) The exact $\phi (y,t)$ given by (45(45) ${\phi }^{\star }(y,t)=0.5A\frac{y}{L_y}+0.5A\frac{t}{T}+0.1A,(y,t)\in [0,L_y]\times [0,T],$(45) ) and the numerical solutions obtained with (b) $\kappa =0$ and (c) $\kappa ={10}^2$ after 40 iterations, for exact data (p=0), for Example 3.

Figure 10. The numerical solutions of $\phi (y,t)$ for p=3 noise: (a) $\kappa =0$, (b) $\kappa ={10}^2$, and for p=5 noise: (c) $\kappa =0$, (d) $\kappa ={10}^2$, for Example 3.

Figure 11. The variation of the normalized accuracy error $\overline{E}\left[{\phi }^n\right]$, at the corresponding stopping iteration numbers, with the number of measurement points $N_m$, for $\kappa \in \{0,{10}^2\}$ and p=5 noise, for Example 3.

Figure 12. The variation of the normalized accuracy error $\overline{E}\left[{\phi }^n\right]$ with the smoothing parameter κ, for $p\in \{0,3,5\}$ noise, for Example 4.

Figure 13. (a) The exact $\phi (y,t)$ given by (46(46) ${\phi }^{\star }(y,t)=0.5A\cdot \sin \left(\frac{2\pi y}{L_y}\right)+A,(y,t)\in [0,L_y]\times [0,T],$(46) ) and the numerical solutions obtained with (b) $\kappa =0$ and (c) $\kappa ={10}^3$ after 30 iterations, for exact data (p=0), for Example 4.

Figure 14. The numerical solutions of $\phi (y,t)$ for p=3 noise: (a) $\kappa =0$, (b) $\kappa ={10}^3$, and for p=5 noise: (c) $\kappa =0$, (d) $\kappa ={10}^3$, after $n_s$ iterations, for Example 4.

Figure 15. The exact and numerical solutions of $\phi (y,10)$ for (a) $\kappa =0$ and (b) $\kappa ={10}^3$ after $n_s$ iterations, for $p\in \{3,5\}$ noise, for Example 4.