Lipschitz stability estimate and reconstruction of Lamé parameters in linear elasticity

Figures & data

Figure 1. Simulation results for Example 1: History of the cost function J and the $L^{\mathrm{\infty }}$-norm of $J^{\mathrm{\prime }}$ in the case of $\epsilon =0.0$ and $\rho =0.0$.

Figure 2. Simulation results for Example 1: History of the cost function J and the $L^{\mathrm{\infty }}$-norm of $J^{\mathrm{\prime }}$ in the case of $\epsilon =0.03$ and $\rho =0.00001$.

Figure 3. Simulation results for Example 1: History of the cost function J and the $L^{\mathrm{\infty }}$-norm of $J^{\mathrm{\prime }}$ in the case of $\epsilon =0.05$ and $\rho =0.00001$.

Table 1. Simulation results for Example 1: Reconstruction of constant Lamé parameters.

	$({\lambda }_i,{\mu }_i)$	$({\lambda }_c,{\mu }_c)$	$({\lambda }_e,{\mu }_e)$	${\displaystyle \frac{|{\lambda }_c-{\lambda }_e|}{\left|{\lambda }_e\right|}}$	${\displaystyle \frac{|{\mu }_c-{\mu }_e|}{\left|{\mu }_e\right|}}$
$ϵ=0.0,\rho =0.0$	(1,1)	(2.9999, 6.9999)	(3,7)	3.33e-07	1.42e-07
$ϵ=0.03,\rho =0.00001$	(1,1)	(2.6564, 6.8989)	(3,7)	0.1145	0.0144
$ϵ=0.05,\rho =0.00001$	(1,1)	( 2.6527,6.8869)	(3,7)	0.1157	0.0161

Figure 4. Simulation results for Example 2: The exact Lamé parameters.

Figure 5. Simulation results for Example 2: The computed Lamé parameters ($ϵ=0.0$ and $\rho =0.0$).

Figure 6. Simulation results for Example 2: The computed Lamé parameters ($ϵ=0.03$ and $\rho =0.0001$).

Figure 7. Simulation results for Example 2: Error for λ and μ ($ϵ=0.0$ and $\rho =0.0$).

Figure 8. Simulation results for Example 2: Error for λ and μ ($ϵ=0.03$ and $\rho =0.0001$).

Figure 9. Simulation results for Example 2: History of the cost function J and the $L^{\mathrm{\infty }}$-norm of $J^{\mathrm{\prime }}$ in the $ϵ=0.0$ and $\rho =0.0$.

Figure 10. Simulation results for Example 2: History of the cost function J and the $L^{\mathrm{\infty }}$-norm of $J^{\mathrm{\prime }}$ in the case of $ϵ=0.03$ and $\rho =0.0001$.

Figure 11. Simulation results for Example 3: The exact Lamé parameters.

Figure 12. Simulation results for Example 3: The computed Lamé parameters ($ϵ=0.0$ and $\rho =0.0$).

Figure 13. Simulation results for Example 3: The computed Lamé parameters ($ϵ=0.03$ and $\rho =0.0001$).

Figure 14. Simulation results for Example 3: Error for λ and μ ($ϵ=0.0$ and $\rho =0.0$).

Figure 15. Simulation results for Example 3: Error for λ and μ ($ϵ=0.03$ and $\rho =0.0001$).

Figure 16. Simulation results for Example 3: History of the cost function J and the $L^{\mathrm{\infty }}$-norm of $J^{\mathrm{\prime }}$ ($ϵ=0.0$ and $\rho =0.0$).

Figure 17. Simulation results for Example 3: History of the cost function J and the $L^{\mathrm{\infty }}$-norm of $J^{\mathrm{\prime }}$ ($ϵ=0.03$ and $\rho =0.0001$).