Identification of thermal conductivity for orthotropic FGMs by DT-DRBEM and L-M algorithm

Figures & data

Table 1. The fundamental operations of the DTM.

Original function	Transformed function
$x\left(\tau \right)=f\left(\tau \right)+g\left(\tau \right)$	$X\left(m\right)=F\left(m\right)+G\left(m\right)$
$x\left(\tau \right)={\displaystyle \frac{\mathrm{d}f\left(\tau \right)}{\mathrm{d}\tau }}$	$X\left(m\right)=\left(m+1\right)F\left(m+1\right)$
$x\left(\tau \right)={\displaystyle \frac{{\mathrm{d}}^{n}f\left(\tau \right)}{\mathrm{d}{\tau }^n}}$	$X\left(m\right)={\displaystyle \frac{\left(m+n\right)!}{m!}}F\left(m+n\right)$
$x\left(\tau \right)=\exp \left(\lambda \tau \right)$	$X\left(m\right)={\displaystyle \frac{{\lambda }^m}{m!}}$
$x\left(\tau \right)={\tau }^r$	$X\left(m\right)=\delta \left(m-r\right)=\left\{\begin{array}{l}1,m=r\\ 0,otherwise\end{array}\right.$
$x\left(\tau \right)=f\left(\tau \right)g\left(\tau \right)$	$X\left(m\right)={\sum }_{n=0}^{m}F\left(n\right)G\left(m-n\right)$

Figure 1. Computational models of BEM and FEM. (a) BEM 1, (b) BEM 2, (c) BEM 3, (d) FEM.

Table 2. The detailed boundary element models.

Model	Boundary elements	Boundary nodes	Internal nodes	Total nodes
(a)	20	20	16	36
(b)	32	32	49	81
(c)	40	40	81	121

Figure 2. The temperature distribution for different computational models. (a) BEM 1, (b) BEM 2, (c) BEM 3, (d) FEM.

Table 3. The comparison of DT-DRBEM and FD-DRBEM.

	DT-DRBEM			FD-DRBEM
Nodes	$\mathrm{\Delta }\tau =0.05$	$\mathrm{\Delta }\tau =0.01$	$\mathrm{\Delta }\tau =0.005$	$\mathrm{\Delta }\tau =0.05$	$\mathrm{\Delta }\tau =0.01$	$\mathrm{\Delta }\tau =0.005$
(0.2, 0.2)	11.590287	11.590287	11.590287	11.585383	11.590287	11.590287
(0.2, 0.6)	16.069455	16.069455	16.069455	16.066344	16.069454	16.069454
(0.4, 0.8)	17.726167	17.726167	17.726167	17.720308	17.726167	17.726167
(0.6, 0.4)	12.369837	12.369837	12.369837	12.370289	12.369837	12.369837
(0.8, 0.6)	13.699417	13.699417	13.699417	13.697171	13.699412	13.699412

Table 4. The comparison of the L-M and the CGM.

	Estimated value		Iteration number		Computation time(s)
Initial value	L-M	CGM	L-M	CGM	L-M	CGM
$a_1=1.0$, $a_2=0.05$	$a_1=2.000$, $a_2=0.100$	$a_1=1.990$, $a_2=0.107$	7	79	50	8863
$a_3=0.5$, $a_4=0.1$	$a_3=1.000$, $a_4=0.200$	$a_3=0.995$, $a_4=0.206$
$a_1=1.0$, $a_2=1.0$	$a_1=2.000$, $a_2=0.100$	$a_1=1.998$, $a_2=0.100$	7	105	56	11232
$a_3=1.0$, $a_4=1.0$	$a_3=1.000$, $a_4=0.200$	$a_3=1.004$, $a_4=0.198$
$a_1=3.0$, $a_2=0.15$	$a_1=2.000$, $a_2=0.100$	$a_1=1.994$, $a_2=0.103$	6	154	41	17810
$a_3=1.5$, $a_4=0.3$	$a_3=1.000$, $a_4=0.200$	$a_3=1.002$, $a_4=0.200$

Figure 3. DRBEM computational model.

Figure 4. The results for different initial guesses. (a) ${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}={\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}/6$, (b) ${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}={\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}/3$, (c) ${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}=1.5{\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}$, (d) ${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}=2{\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}$_.

Table 5. The inverse results for different initial guesses.

		${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}={\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}/6$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}={\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}/3$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}=1.5{\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}=2{\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}$
	${\mathit{a}}_{\mathrm{e}\mathrm{a}\mathrm{x}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$
$a_1$	1.000000	1.000000	0.999969	0.999913	1.000000
$a_2$	0.300000	0.300001	0.300004	0.300055	0.300000
$a_3$	2.000000	2.000000	1.999989	2.000141	2.000002
$a_4$	0.200000	0.200000	0.199998	0.199954	0.200000

Figure 5. The results for different number of measurement points. (a) 3 measurement points, (b) 8 measurement points, (c) 12 measurement points, (d) 17 measurement points.

Table 6. The inverse results for different measurement points.

		$M=3$	$M=8$	$M=12$	$M=17$
	${\mathit{a}}_{\mathrm{e}\mathrm{a}\mathrm{x}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$
$a_1$	1.000000	1.349656	0.999995	1.000000	1.000001
$a_2$	0.300000	0.363770	0.300001	0.299998	0.299999
$a_3$	2.000000	1.535150	1.999987	1.999968	2.000000
$a_4$	0.200000	0.504574	0.200006	0.200015	0.200000

Table 7. The inverse results for different measurement errors.

		$\sigma =1\mathrm{\%}$		$\sigma =3\mathrm{\%}$		$\sigma =5\mathrm{\%}$
	${\mathit{a}}_{\mathrm{e}\mathrm{a}\mathrm{x}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	$E_{\mathrm{r}\mathrm{e}\mathrm{l}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	$E_{\mathrm{r}\mathrm{e}\mathrm{l}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	$E_{\mathrm{r}\mathrm{e}\mathrm{l}}$
$a_1$	1.000000	1.012957	1.30%	0.965499	3.45%	0.952461	4.75%
$a_2$	0.300000	0.290590	3.14%	0.310721	3.57%	0.312060	4.02%
$a_3$	2.000000	1.981359	0.93%	1.926160	3.69%	1.908671	4.57%
$a_4$	0.200000	0.205280	2.64%	0.208039	4.02%	0.215146	7.57%

Figure 6. DRBEM computational model.

Figure 7. The results for different initial guesses. (a) ${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}={\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}/6$, (b) ${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}={\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}/3$, (c) ${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}=1.5{\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}$, (d) ${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}=2{\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}$_.

Table 8. The inverse results for different initial guesses.

		${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}={\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}/6$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}={\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}/3$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}=1.5{\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{i}}=2{\mathit{a}}_{\mathrm{e}\mathrm{x}\mathrm{a}}$
	${\mathit{a}}_{\mathrm{e}\mathrm{a}\mathrm{x}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$
$a_1$	2.000000	1.999925	2.000002	2.000349	2.000536
$a_2$	0.500000	0.499965	0.499998	0.500031	0.500087
$a_3$	1.000000	1.000024	1.000000	0.999942	0.999908
$a_4$	0.400000	0.399968	0.400001	0.400367	0.400720
$a_5$	0.200000	0.200026	0.200000	0.199715	0.199373

Figure 8. The results for different number of measurement points. (a) 3 measurement points, (b) 7 measurement points, (c) 10 measurement points, (d) 14 measurement points.

Table 9. The inverse results for different number of measurement points.

		$M=3$	$M=7$	$M=10$	$M=14$
	${\mathit{a}}_{\mathrm{e}\mathrm{a}\mathrm{x}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$
$a_1$	2.000000	2.272941	2.000054	2.000293	2.000020
$a_2$	0.500000	0.569939	0.500029	0.499945	0.500004
$a_3$	1.000000	1.240204	1.000027	1.000066	1.000006
$a_4$	0.400000	0.184487	0.399926	0.399976	0.399990
$a_5$	0.200000	0.552493	0.200051	0.200085	0.200013

Table 10. The inverse results for different measurement errors.

		$\sigma =1\mathrm{\%}$		$\sigma =3\mathrm{\%}$		$\sigma =5\mathrm{\%}$
	${\mathit{a}}_{\mathrm{e}\mathrm{a}\mathrm{x}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	$E_{\mathrm{r}\mathrm{e}\mathrm{l}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	$E_{\mathrm{r}\mathrm{e}\mathrm{l}}$	${\mathit{a}}_{\mathrm{i}\mathrm{n}\mathrm{v}}$	$E_{\mathrm{r}\mathrm{e}\mathrm{l}}$
$a_1$	2.000000	1.959837	2.01%	2.076655	3.83%	2.062476	3.12%
$a_2$	0.500000	0.504104	0.82%	0.471220	5.76%	0.455133	8.97%
$a_3$	1.000000	1.010316	1.03%	1.004463	0.45%	1.028090	2.81%
$a_4$	0.400000	0.384563	3.86%	0.390555	2.36%	0.395801	1.05%
$a_5$	0.200000	0.207147	3.57%	0.218355	9.18%	0.245349	22.7%