Model order reduction for numerical simulation of particle transport based on numerical integration approaches

Figures & data

Figure 1. Modelling plasma reactors with linearization and splitting techniques.

Table 1. Numerical errors of the linearization techniques 1)–4) for the non-linearity $\alpha =3$.

$n_{\mathrm{i}\mathrm{t}}$	$n_{\mathrm{i}\mathrm{n}\mathrm{t}}$	$n_{\mathrm{t}\mathrm{s}}$	1)	2)	3)	4)
1	1	1000	9.8475e-2	9.8475e-2	9.8475e-2	9.8475e-2
1	10	100	9.8475e-2	9.8475e-2	9.8475e-2	9.8475e-2
1	100	10	9.8475e-2	9.8475e-2	9.8475e-2	9.8475e-2
2	1	1000	9.0918e-2	1.5120e-1	1.5081e-1	1.1983e-1
2	10	100	3.5313e-3	6.5402e-3	6.4061e-3	4.9586e-3
2	100	10	4.0558e-4	6.5628e-4	5.3055e-4	4.6807e-4
3	1	1000	1.2493e-2	6.5628e-4	2.2099e-2	1.7378e-2
3	10	100	1.5572e-4	3.4215e-4	3.3263e-4	2.4289e-4
3	100	10	1.8933e-6	3.3227e-6	2.4834e-6	2.1881e-6
4	1	1000	6.2378e-3	1.7962e-2	1.7332e-2	1.0961e-2
4	10	100	9.9185e-5	1.2001e-4	7.9856e-5	1.0473e-5
4	100	10	9.0573e-5	9.0986e-5	9.0857e-5	1.1228e-7
5	1	1000	6.3537e-4	2.0411e-3	1.8416e-3	1.1300e-3
5	10	100	7.1710e-6	8.1401e-6	6.8926e-6	2.2160e-7
5	100	10	4.9658e-7	4.9905e-7	8.3813e-7	1.6957e-7
6	1	1000	4.1674e-4	1.3140e-3	5.6985e-4	4.5031e-4
6	10	100	9.5261e-5	1.0045e-4	1.0032e-4	1.0809e-7
6	100	10	9.0569e-5	9.0972e-5	9.0879e-5	1.0153e-7

Notes: $n_{\mathrm{i}\mathrm{t}}$ – Number of iterations.

$n_{\mathrm{i}\mathrm{n}\mathrm{t}}$ – Number of intervals.

$n_{\mathrm{t}\mathrm{s}}$ – Number of time-steps per interval.

Table 2. Numerical errors of the linearization techniques 5)–7) for the non-linearity $\alpha =3$.

$n_{\mathrm{i}\mathrm{t}}$	$n_{\mathrm{i}\mathrm{n}\mathrm{t}}$	$n_{\mathrm{t}\mathrm{s}}$	5)	6)	7)
1	1	1000	9.8475e-2	9.8475e-2	9.8475e-2
1	10 (520)	100	9.8475e-2	9.8475e-2	9.8475e-2
1	100 (506)	10	9.8475e-2	9.8475e-2	9.8475e-2
2	1	1000	1.0035e-1	1.1983e-1	9.5693e-2
2	10 (520)	100	4.0352e-3	4.9725e-3	3.9457e-3
2	100 (506)	10	4.4931e-4	4.8199e-4	5.4264e-4
3	1	1000	1.4154e-2	1.7378e-2	1.3401e-2
3	10 (520)	100	1.8682e-4	2.4377e-4	1.8378e-4
3	100 (506)	10	2.1183e-6	1.8343e-6	2.5603e-6
4	1	1000	7.7274e-3	1.0961e-2	7.3968e-3
4	10 (520)	100	1.0074e-4	1.0918e-5	2.8216e-4
4	100 (506)	10	8.1542e-5	4.1890e-7	1.8985e-4
5	1	1000	8.0592e-4	1.1300e-3	8.7664e-4
5	10 (520)	100	7.1293e-6	2.9535e-7	2.0267e-5
5	100 (506)	10	3.6889e-7	6.7839e-7	8.0301e-7
6	1	1000	5.0661e-4	4.5031e-4	9.5049e-4
6	10 (520)	100	9.5053e-5	3.9646e-7	2.7745e-4
6	100 (506)	10	8.1536e-5	4.0604e-7	1.8985e-4

Notes: $n_{\mathrm{i}\mathrm{t}}$ – Number of iterations.

$n_{\mathrm{i}\mathrm{n}\mathrm{t}}$ – Number of intervals (in brackets for 6).

$n_{\mathrm{t}\mathrm{s}}$ – Number of time-steps per interval.

Figure 2. The numerical errors of the different linearization techniques.

Table 3. Numerical errors of the linearization techniques 1)–4) for the non-linearity $\alpha =9$.

$n_{\mathrm{i}\mathrm{t}}$	$n_{\mathrm{i}\mathrm{n}\mathrm{t}}$	$n_{\mathrm{t}\mathrm{s}}$	1)	2)	3)	4)
1	1	1000	3.0525e-2	3.0525e-2	3.0525e-2	3.0525e-2
1	10	100	3.0525e-2	3.0525e-2	3.0525e-2	3.0525e-2
1	100	10	3.0525e-2	3.0525e-2	3.0525e-2	3.0525e-2
2	1	1000	4.2549e-2	7.3916e-2	7.3137e-2	5.6628e-2
2	10	100	1.5040e-3	5.8300e-3	5.6046e-3	3.4490e-3
2	100	10	2.2588e-4	5.3110e-4	3.5869e-4	2.9221e-4
3	1	1000	1.3013e-3	4.3157e-3	4.2140e-3	2.7381e-3
3	10	100	1.5103e-4	7.5462e-4	7.1752e-4	4.2603e-4
3	100	10	3.2169e-6	8.3987e-6	5.3383e-6	4.2744e-6
4	1	1000	1.6604e-3	7.0663e-3	6.0327e-3	3.5399e-3
4	10	100	1.2930e-4	2.5234e-4	4.6314e-5	3.5944e-5
4	100	10	1.1783e-4	1.1985e-4	1.1941e-4	3.7769e-7
5	1	1000	8.6945e-5	2.9361e-4	1.1841e-4	8.5853e-5
5	10	100	2.5020e-5	3.7781e-5	2.0100e-5	2.7185e-6
5	100	10	1.9681e-6	2.0042e-6	2.8301e-6	4.1358e-7
6	1	1000	3.9150e-4	8.2547e-4	3.3717e-4	8.9143e-5
6	10	100	1.2548e-4	1.5344e-4	1.5146e-4	4.9597e-7
6	100	10	1.1782e-4	1.1977e-4	1.1958e-4	3.2660e-7

Notes: $n_{\mathrm{i}\mathrm{t}}$ – Number of iterations.

$n_{\mathrm{i}\mathrm{n}\mathrm{t}}$ – Number of intervals.

$n_{\mathrm{t}\mathrm{s}}$ – Number of time-steps per interval.

Figure 3. The numerical errors of the different linearization techniques.

Table 4. Numerical errors of the optimal linearization technique 4).

$n_{\mathrm{i}\mathrm{t}}$	$n_{\mathrm{i}\mathrm{n}\mathrm{t}}$	$n_{\mathrm{t}\mathrm{s}}$	4)
1	1	1000	8.0852e-2
1	10	100	8.0852e-2
1	100	10	8.0852e-2
2	1	1000	5.1843e-3
2	10	100	4.6017e-4
2	100	10	4.5785e-5
3	1	1000	4.5400e-4
3	10	100	4.0729e-6
3	100	10	4.0599e-8
4	1	1000	1.4512e-5
4	10	100	1.3147e-8
4	100	10	2.1281e-11
5	1	1000	7.6057e-7
5	10	100	5.6436e-11
5	100	10	1.3809e-11
6	1	1000	1.6326e-8
6	10	100	8.0876e-12
6	100	10	7.9682e-12

Table 5. Physical parameters.

Pressure in the chamber	$p={9.810}^{-2}-2$ mbar
Precursor temperature	$T_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{r}}=71.5deg\mathrm{C}$
Velocity (argon gas)	$v=30.0$ cm${\hspace{0.17em}}^3$/min
Inflow velocity of the precursor gas	$v_{\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}}=0.60$ cm${\hspace{0.17em}}^3/$min

Figure 4. x-axis: Time scale, y-axis: concentration, numerical solution and exact solution coincide in the same line.

Figure 5. Spatial domain of the PE-CVD apparatus.

Table 6. Computed and experimental fitted parameters with UG simulations.

Power [W]	Bias [V]	$R_{\mathrm{C}}$ at material 1	$R_{\mathrm{T}\mathrm{i}}$ at material 2	Ratio (C:Ti) (numerical)
300	0	${0.110}^{-4}$	${2010}^{-4}$	$5.5:1.5=3.6$
600	0	${1.010}^{-4}$	${2010}^{-4}$	$4.4:1.5=2.93$
900	0	${1.510}^{-4}$	${2010}^{-4}$	$3.8:1.5=2.53$
300	–10	${2.810}^{-4}$	${2010}^{-4}$	$3.1:1.5=2.066$
600	–10	${1.010}^{-15}$	${2010}^{-4}$	$5.7:1.5=3.8$
900	–10	${1.010}^{-15}$	${6010}^{-4}$	$5.7:05=11.4$

Figure 6. Two inflow sources $x_{\mathrm{T}\mathrm{i}},y_{\mathrm{T}\mathrm{i}}=(35,190)$ and $x_{\mathrm{T}\mathrm{i}},y_{\mathrm{T}\mathrm{i}}=(215,190)$ with perpendicular velocity and 100 time-steps, with the ratio between C and Ti equal to 3.6.

Figure 7. Two inflow sources $x_{\mathrm{T}i},y_{\mathrm{T}\mathrm{i}}=(35,190)$ and $x_{\mathrm{T}\mathrm{i}},y_{\mathrm{T}\mathrm{i}}=(215,190)$ with perpendicular velocity and 150 time-steps, with the ratio between C and Ti equal to 3.6.

Figure 8. Deposition rates in the case of two point sources, x = 35,215, y = 190, with perpendicular velocity and 150 time-steps, with the ratio between C (+ line) and Ti (x line) equal to 3.6.