Chapter 6. Concluding Remarks
6.1 Summary
In this dissertation, a general parallel three-dimensional electrostatic particle-in-cell scheme with finite element method (PIC-FEM) using unstructured mesh is proposed and verified. A multi-level graph-partitioning technique is used to dynamically decompose the computational domain to improve the parallel performance during runtime. Completed parallelized PIC-FEM code is used to simulate several important physical problems, including field emission, DC/RF gas discharge and DC/RF magnetron plasmas. In brief summary, the major achievements in the present dissertation can be listed as follows:
1. A parallelized three-dimensional electrostatic Poisson’s equation solver using Galerkin finite element method with an unstructured mesh is developed and validated. Study of parallel performance of the parallelized PIC-FEM code is performed on the HP-IA64 clusters. With subdomain-by-subdomain scheme for parallel conjugate gradient method, parallel efficiency can reach 84% at 32 processors of HP PC clusters at NCHC. This code coupled with PAMR was used to accurately and efficiently simulate field emission from emitter with complicated geometry without considering space-charge effects, as demonstrated in Chapter 5.
2. A parallelized three-dimensional vector potential magnetostatic Poisson’s equation solver using Galerkin finite element method with an unstructured mesh is developed and validated. Study of parallel performance of the parallelized
PIC-FEM code is performed on the HP-IA64 clusters. With subdomain-by-subdomain scheme for parallel conjugate gradient method, parallel efficiency can reach 75% at 32 processors of HP PC clusters at NCHC.
This code was used to simulate the magnetic field around permanent magnets or coils for magnetron plasma simulation as demonstrated in Chapter 5.
3. A general parallelized three-dimensional PIC-FEM code is developed and validated. This PIC-FEM code integrates the parallelized Poisson’s equation solver with the PIC and Monte Carlo collision (MCC) schemes on an unstructured tetrahedral mesh. Charged particles can be traced either cell-by-cell on an unstructured mesh. This is achieved using leap-frog time-integration method and Boris rotational scheme when magnetic field is involved. Charge assignment and force (field) interpolation between charged particles and grid points is implemented using the same interpolation function originated from the FEM. In addition, dynamic domain decomposition (DDD) with weighting based on number of particles is used to balance the workload among processors during runtime. Study of parallel performance of the parallelized PIC-FEM code is performed on the HP-IA64 clusters. Results using DDD for a typical RF gas discharge show that parallel efficiency can reach 83% at 32 processors.
4. Completed parallelized PIC-FEM code was used to simulate several important problems to demonstrate its superior capability in handling practical problems.
These problems include field emission from a silicon tip under external electric field, two typical three-dimensional DC and RF gas-discharge plasmas, and two typical three-dimensional DC and RF magnetron plasmas with permanent magnets. Results are either compared well with experiments or demonstrate the correct physical pictures as expected.
6.2 Recommendation for Future Study
In the present dissertation, we have developed and tested a parallelized three-dimensional PIC-FEM code using an unstructured mesh on memory-distributed parallel machines. We have also applied this code to simulate several important physical problems. Based on the viewpoints of further improving this PIC-FEM code, several possible directions of research are recommended for the future study and are summarized as follows:
1. To implement a better preconditioner for parallel conjugate gradient method for solving the Poisson’s equation more efficiently to shorten the runtime and improve the speedup at higher number of processors.
2. To incorporate a Maxwell’s equation solver that uses edge-based finite element method into the present PIC-FEM code to further extend its applicability in plasma related simulation, such as ICP, ECR and microwave plasmas.
3. To incorporate a simulation module that can model realistic external circuits into the present PIC-FEM code, which are often coupled to a RF-type gas discharge.
4. To extend the database of collision data for other types of plasma, such as methane with hydrogen, which are very important in growing carbon nanotubes.
5. To incorporate a parallelized DSMC (direct simulation Monte Carlo) module into the PIC-FEM code to consider the neutral transport self-consistently that is
very important in some plasma flow, such as magnetron plasma.
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Table 1. Main excellent features of a field emission display
1. Thin panel thickness (~2mm) 2. Self-emissive
3. Distortion free image
4. Wide viewing angle (~170∘)
5. Quick response in the order of µs by controlling with analog or digital without active elements
6. Tolerance to environment as high as that of receiving tubes 7. Free from the terrestrial magnetic effect
8. Free from the changes in the ambient magnetism 9. Quick start of operation
10. Less dead space of images
11. Low power consumption display device
12. Good stable characteristics in severe environmental conditions
Table 2. Time breakdown and speedup of Poisson’s equation solver at the different number of processors
Processor No. 1 2 4 8 16 32
Total time (seconds) 138.17 79.17 42.53 14.78 8.21 5.13 CG solver time (%)
Matrix assembling time (%) Communication time (%)
98.8 0.44 N/A
99.1 0.36 4.45
94.33 0.32 28.1
76.79 0.47 34.5
85.14 0.42 35.32
94.54 0.31
37
Speedup 1 1.74 3.25 9.35 16.83 26.93
Table 3. Evolution of simulation parameters at different levels of mesh refinement.
(EMAX is the local maximum electric field strength at the surface of CNT field emitter).
Refinement Level Number of nodes Number of elements EMAX (V/nm) 0
1 2 3 4 5 6
7006 (7006) 22750 (24892) 34927 (38896) 44080 (47984) 51638 (55488) 61241 (59279) 67173
27814 (27814) 110218 (121064) 175254 (196378) 225156 (245975) 264259 (284766) 313092 (306368) 345307
8.218482 (8.21848) 10.20636 (10.20257) 11.50804 (11.50135) 11.54894 (11.51166) 11.32366 (11.32647) 11.32303 (11.32665) 11.32324
* Numbers in the parentheses represent numerical data obtained using a posteriori error estimator with prescribed global relative error εpre=0.0003.
Table 4. Evolution of simulation parameters at different levels of mesh refinement.
(BMAX is the local maximum magnetic field strength at the center of magnet arrays).
Refinement Level Number of nodes Number of elements BMAX (T) 0
1 2 3 4 5 6
7845 38364 54355 70773 98743 108415 108840
46953 228201 319482 414616 574237 629268 631711
0.869774 0.870979 0.870808 0.871388 0.871401 0.871556 0.871553
Table 5. The important geometrical parameters of CNT triode- and tetrode-type field
Table 6. Characteristics of device performance for different focus types.
Focus type Emission current from tip (A)
Gate current (A)
Anode current (A)
Spot diameter at anode
(μm)
Without focus 2.48E-05 ~0 2.48E-05 528.68
Magnetic focus (Bz= 0.2 T) 2.48E-05 ~0 2.48E-05 296.99 Magnetic focus (Bz= 0.35 T) 2.48E-05 ~0 2.48E-05 52.01 Electrostatic focus (Vf= -5 V) 5.47E-06 2.39E-06 3.08E-06 154.76 Electrostatic focus (Vf= 0 V) 5.69E-06 1.35E-06 4.35E-06 226.26 Electrostatic focus (Vf= 5 V) 6.50E-06 1.56E-06 4.93E-06 210.20
Figure 1.1 Representation of the parameter space in plasma etching. The key internal plasma properties (middle) are the bridge between externally controlled variables (top) and the figures of merit (bottom).
Figure 2.1 Element equation from a typical element (e) are used for each element in the mesh.
Figure 2.2 A three-Dimensional C -linear standard tetrahedral element. 0
Figure 2.3 A three-Dimensional C -linear standard hexahedral element. 0 (1,1,-1)
(1, -1,-1) (1,-1,1)
(-1,-1,1) (-1,1,1)
(-1,1,-1) 1 (-1,-1,-1)
(1,1,1)
ξ ω
η
Figure 2.4 (a) Vertex-based. (b) edge-based (c) element-based partition of 4 × 3 mesh into two sub-regions.
(a)
(b)
(c)
Ω
2Ω
1Ω
2Ω
1Ω
1Ω
2Figure 2.5 An L-shape domain subdivided into three sub-domains.
Ω
2Γ
12Ω
3Ω
1Ω
2Γ
13Γ
12Figure 2.6 Sketch of graph and mesh.
Figure 2.7. Flowchart of the parallel mesh refinement module.
Figure 2.8 Flowchart of the coupled PPES-PAMR method.
Figure 2.9 The flowchart of parallel FEM.
(a)
L (mm)
Potential(Volts)
0 5 10 15 20
0 2000 4000 6000 8000 10000
simulation analytical solution
simulation analytical solution