The inverse determination of the thermal contact resistance components of unidirectionally reinforced composite

Figures & data

Figure 1. A unidirectional reinforced fibrous composite with fibre arrangement according to a square array for the imperfect thermal contact between fibre and matrix: (a) general view, (b) formulation of a nondimensional boundary value problem in a repeated element.

Figure 2. The effective thermal conductivity as a function of radius of fibres in the matrix for different values of the ratio of thermal conductivity fibres to the matrix, F, and for different values of resistance number, γ.

Table 1. The value of resistance number γ for E = 0.7, for the different values of F for known temperature in M3 points in the matrix and for known λ_z/λ_m.

Datas		Results
		N3 = 10				N3 = 8		N3 = 6		N3 = 5
					Iter		Iter		Iter		Iter
0.1	0.518357	100.00	7.86E−06	3.16E−04	1096	100	130	100.00	1612	0.0073	425
	0.509996	10.00	8.77E−06	3.59E−04	489	10	479	10.00	1097	0.0170	282
	0.474349	1.00	1.07E−05	4.44E−04	54	1	1271	1.00	239	0.0005	1675
	0.446793	0.10	1.16E−05	4.85E−04	12	0.1	24	0.10	71	0.1020	132
	0.442058	0.01	1.17E−05	4.91E−04	93	0.01	113	0.01	180	0.0371	498
1	0.994556	100.00	1.61E−08	1.84E−06	1748	100	1703	100.00	1213	0.0002	1421
	0.949967	10.00	9.78E−08	3.46E−06	1431	10	1369	10.00	1428	0.0001	1604
	0.723159	1.00	3.97E−06	1.56E−04	11	1	12	1.00	14	0.9415	1909
	0.493099	0.10	1.02E−05	4.25E−04	11	0.1	11	0.10	12	0.1002	150
	0.447123	0.01	1.16E−05	4.84E−04	615	0.01	65	0.01	217	0.0130	909
5	1.687394	100.00	1.09E−05	7.80E−04	17	100	17	100.00	17	99.6320	17
	1.639267	10.00	8.42E−06	6.12E−04	14	10	14	10.00	14	0.0003	1394
	1.307356	1.00	2.22E−07	4.27E−05	11	1	11	1.00	12	0.9996	9
	0.672712	0.10	5.50E−06	2.23E−04	10	0.1	11	0.10	11	0.1000	8
	0.469287	0.01	1.09E−05	4.56E−04	11	0.01	20	0.01	40	0.0113	283
10	1.921272	100.00	1.70E−05	1.21E−03	17	100	17	100.00	17	99.6425	219
	1.886550	10.00	1.49E−05	1.07E−03	14	10	14	10.00	14	9.9915	418
	1.613496	1.00	3.90E−06	3.31E−04	11	1	11	1.00	12	0.9998	9
	0.850962	0.10	2.00E−06	7.72E−05	10	0.1	8	0.10	11	0.1000	9
	0.496224	0.01	1.01E−05	4.22E−04	185	0.01	46	0.01	213	0.0105	57

Table 2. The value of resistance number γ for E = 0.7, for the different values of F for known temperature in M3 points on the upper boundary and for known λ_z/λ_m.

Datas		Results
		N3 = 10				N3 = 8		N3 = 6		N3 = 5
					Iter		Iter		Iter		Iter
0.1	0.518357	100.00	7.86E−06	3.16E−04	16	100.00	17	100.00	17	98.71	16
	0.509996	10.00	8.77E−06	3.59E−04	13	10.00	13	10.00	14	10.04	18
	0.474349	1.00	1.07E−05	4.44E−04	13	1.00	13	1.00	13	0.99	216
	0.446793	0.10	1.16E−05	4.85E−04	12	0.10	12	0.10	12	0.07	18
	0.442058	0.01	1.17E−05	4.91E−04	12	0.01	12	0.01	12	0.02	119
1	0.994556	100.00	1.61E−08	1.84E−06	20	100.00	20	100.00	20	98.91	20
	0.949967	10.00	9.78E−08	3.46E−06	16	10.00	16	10.00	16	9.98	16
	0.723159	1.00	3.97E−06	1.56E−04	13	1.00	13	1.00	13	1.00	13
	0.493099	0.10	1.02E−05	4.25E−04	11	0.10	11	0.10	11	0.10	11
	0.447123	0.01	1.16E−05	4.84E−04	9	0.01	9	0.01	9	0.01	211
5	1.687394	100.00	1.09E−05	7.80E−04	20	100.00	20	100.00	20	98.82	24
	1.639267	10.00	8.42E−06	6.12E−04	17	10.00	17	10.00	17	9.98	17
	1.307356	1.00	2.22E−07	4.27E−05	13	1.00	13	1.00	13	1.00	12
	0.672712	0.10	5.52E−06	2.24E−04	11	0.10	11	0.10	11	0.10	11
	0.469287	0.01	1.09E−05	4.56E−04	8	0.01	8	0.01	8	0.01	8
10	1.921272	100.00	1.70E−05	1.21E−03	20	100.00	20	100.00	20	98.48	20
	1.886550	10.00	1.49E−05	1.07E−03	17	10.00	17	10.00	17	9.98	17
	1.613496	1.00	3.90E−06	3.31E−04	13	1.00	13	1.00	13	1.00	12
	0.850962	0.10	2.01E−06	7.73E−05	11	0.10	11	0.10	11	0.10	11
	0.496224	0.01	1.01E−05	4.22E−04	8	0.01	8	0.01	8	0.01	168

Table 3. The value of resistance number γ for E = 0.7, for the different values of F for known temperature in N3 = 10 points in the matrix and for known λ_z/λ_m for disturbed data ΔT = 0.1%.

Datas		Results
N3 = 10, ΔT = 0.1%
					Iter
0.1	0.518364	46.4075	1.03E−02	2.57E−01	216
	0.510004	11.5292	7.00E−03	1.70E−01	1424
	0.474358	1.0646	5.91E−03	1.44E−01	12
	0.446802	0.1170	4.89E−03	1.01E−01	103
	0.442067	0.0162	3.12E−03	6.04E−02	818
1	0.994556	88.4183	1.08E−03	2.81E−02	131
	0.949967	10.4162	5.63E−03	1.39E−01	2230
	0.723163	0.9863	1.18E−02	2.88E−01	26
	0.493107	0.1003	5.04E−03	1.26E−01	12
	0.447132	0.0091	5.65E−03	1.43E−01	12
5	1.687399	148.0763	1.78E−02	4.35E−01	1423
	1.639269	10.1771	2.97E−03	7.63E−02	1258
	1.307353	0.9948	1.45E−02	3.58E−01	401
	0.672717	0.0997	1.11E−02	2.79E−01	12
	0.469296	0.0095	6.47E−03	1.62E−01	9
10	1.921281	363.9451	5.34E−03	1.33E−01	973
	1.886557	10.4811	1.51E−02	3.69E−01	1743
	1.613494	1.0004	1.52E−02	3.78E−01	14
	0.850965	0.0998	8.91E−03	2.06E−01	143
	0.496232	0.0098	9.49E−03	2.51E−01	210

Table 4. The value of resistance number γ for E = 0.7, for the different values of F for known temperature in N3 = 10 points on the upper boundary and for known λ_z/λ_m for disturbed data ΔT = 0.1%.

Datas		Results
N3 = 10, ΔT = 0.1%
					Iter
0.1	0.518364	114.9104	1.42E−04	1.70E−03	10
	0.510004	10.1615	3.71E−05	1.94E−03	17
	0.474358	1.0169	7.84E−05	5.70E−03	13
	0.446802	0.0893	1.64E−04	5.50E−03	12
	0.442067	0.0150	7.73E−05	3.09E−03	10
1	0.994556	97.2524	1.98E−05	1.13E−03	407
	0.949967	10.0536	2.69E−05	2.73E−03	16
	0.723163	0.9993	1.04E−04	1.25E−03	14
	0.493107	0.1003	8.88E−05	3.16E−03	12
	0.447132	0.0092	2.10E−04	6.23E−03	9
5	1.687399	101.5508	1.26E−04	1.75E−03	352
	1.639269	10.1046	6.54E−05	3.87E−03	18
	1.307353	0.9984	1.09E−04	6.09E−03	14
	0.672717	0.0999	2.63E−05	2.49E−03	12
	0.469296	0.0099	1.04E−04	3.08E−03	9
10	1.921281	–	–	–	–
	1.886557	10.0029	1.44E−04	2.44E−03	18
	1.613494	1.0007	2.29E−05	1.60E−03	14
	0.850965	0.1001	2.11E−05	3.48E−03	11
	0.496232	0.0100	5.58E−05	3.61E−03	9

Table 5. The impact of the number of collocation points on the value of the resistance number and the maximum error of fulfilling the boundary conditions at control points.

		,,			,,
3	2	10	7.45E−03	7.96E−02	10	7.13E−04	6.64E−03
4	3	9.99976725	1.64E−03	2.85E−02	10.0000025	1.43E−04	2.14E−03
5	4	9.99994733	3.76E−04	9.42E−03	9.99999798	2.90E−05	6.16E−04
6	5	1.8799939	8.31E−04	6.81E−02	10.0000039	6.17E−06	1.75E−04
7	6	10	7.45E−03	7.96E−02	10.0000846	1.92E−07	1.01E−05
8	7	–	–	–	10.188653	5.11E−03	4.26E−02
		,,			,,
3	2	10	8.58E−03	6.58E−02	10	1.17E−02	8.47E−02
4	3	9.99999937	1.58E−03	1.80E−02	9.99999919	2.12E−03	2.18E−02
5	4	9.99998335	2.77E−04	3.85E−03	10.0000016	3.60E−04	4.03E−03
6	5	10.0000116	4.60E−05	6.10E−04	9.99999056	5.56E−05	3.35E−04
7	6	10.0000139	8.41E−06	6.12E−04	9.99998628	1.49E−05	1.07E−03
8	7	9.17472511	3.35E−03	6.44E−02	10.2050555	1.81E−04	6.01E−03

Table 6. Convergence of the Levenberg–Marquardt method for the four test examples.

	,			,		,		,
Iter			Iter
1	1.68E−03	1.02022	1	8.66E−01	1.21766	1.17E−01	1.9160	7.57E−02	2.0771
2	2.23E−03	1.04077	2	1.32E−01	1.19667	4.62E−02	3.4278	1.64E−02	3.7247
3	3.05E−03	1.06168	3	1.30E−01	2.00207	9.10E−03	5.6657	2.53E−03	6.0614
4	4.02E−03	1.08302	4	1.13E−01	3.82916	8.75E−04	8.1147	2.05E−04	8.4486
5	5.06E−03	1.10482	5	1.36E−02	6.36371	2.23E−05	9.6437	3.70E−06	9.7593
84	1.88E−04	9.15282	6	1.76E−03	8.67281	2.66E−08	9.9873	2.04E−09	9.9942
85	6.36E−06	9.90740	7	2.56E−05	9.82066	4.33E−14	10.0000	6.85E−16	10.0000
86	1.09E−09	9.99865	8	8.23E−09	9.99668	3.53E−24	10.0000	8.93E−29	10.0000
87	8.38E−17	10.0	9	1.48E−15	10.0
88	4.07E−24	10.0	10	2.41E−23	10.0