應用非結構性網格之平行化三維PIC-FEM程式的研究與發展

Abstract

本論文研究目的主要是發展與驗證一平行化三維粒子式Particle-In-Cell (PIC-FEM)和蒙地卡羅法程式，並使用multi-level圖形切割技術於三維非結構性網格的動態區域切割法。此程式可應用於許多重要物理問題上的模擬，包括場發射元件、直流氬氣放電電漿、射頻氬氣放電電漿、直流氬氣磁控式電漿和射頻氬氣磁控式電漿。 本研究主要可分為三部分。第一部份：使用Galerkin有限元素法來分別離散靜電Poisson方程式來求解三維靜電場問題以及靜磁向量Poisson方程式來求解三維靜磁場問題。當平行電腦叢集數目大於10時，程式使用的平行化conjugate gradient method來求解矩陣問題；反之，當平行電腦叢集數目小於10時，則使用直接矩陣法MUMPS來求解矩陣問題。再者，將以上所發展的平行化靜電和靜磁程式結合平行可調適網格再切割模組(parallel adaptive mesh refinement module, PAMR)；因此，當計算區域出現位勢場變化較大處，計算網格將會自動被切割以獲得更正確的數值解。兩者程式平行效率測試於國家高速電腦提供的32台HP-IA64平行電腦叢集上進行，結果顯示靜電和靜磁程式的平行化效率於32台電腦腦叢集下分別可達到84%和75%。 論文的第二部份：主要是結合第一部份發展的三維平行化靜電和靜磁程式及三維平行PIC-FEM程式於三維非結構性四面體網格上。PIC-FEM程式使用蛙跳法與Boris法來計算粒子運動方程式。因計算網格的不同，粒子追蹤法可利用非結構性網格關係來追蹤運動粒子軌跡。帶電粒子和計算網格點之間的權重函數為有限元素法中的形狀函數。再者，使用動態區域切割法來平均分散平行電腦間的工作量以改善平行化效率。最後模擬一近似一維的直流氬氣放電電漿和射頻氬氣放電電漿來驗展程式的正確性，並模擬三維射頻氬氣放電電漿來測試程式平行效率，測試結果顯示，如使用粒子數目為動態區域切割法的權重，於32台HP-IA64電腦腦叢集下還可達到83%平行效率。 論文最後部份，為展現三維平行PIC-FEM程式在許多重要物理問題上優秀的模擬能力，其應用問題包括三維奈米碳管式場發射元件與矽式場發射元件、三維直流氬氣放電電漿、三維射頻氬氣放電電漿、三維直流氬氣磁控式電漿和射頻氬氣磁控式電漿。

A general parallel three-dimensional electrostatic particle-in-cell scheme with finite element method (PIC-FEM) using an unstructured mesh is proposed and verified in this dissertation. 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 this thesis, research is divided into three different phases. In the first phase, a parallelized three-dimensional electrostatic Poisson’s equation solver using Galerkin finite element method using an unstructured mesh is developed and validated. In addition, a parallelized three-dimensional vector potential magnetostatic Poisson’s equation solver is developed and validated. Furthermore, these two solvers are coupled, respectively, with a parallel adaptive mesh refinement (PAMR) module, to automatically improve the resolution of solution near where the property gradient is large. In both solvers, resulting algebraic equations are solved using either the parallel conjugate gradient method with a subdomain-by-subdomain scheme for more processors (>10) or the direct sparse matrix solver for fewer processors (<10). Parallel speedup test for solvers using parallel conjugate gradient method is performed on a HP-IA64 cluster system up to 32 processors at NCHC of Taiwan. Results show that parallel efficiency can reach 84% and 75% at 32 processors for the electrostatic Poisson’s equation solver and magnetostatic vector Poisson’s equation solver, respectively. In the second phase, a general parallel three-dimensional PIC-FEM code is developed and validated. This PIC-FEM code integrates the parallelized Poisson’s equation solver, developed in the first phase, with the PIC and Monte Carlo collision (MCC) schemes on an unstructured tetrahedral mesh. Charged particles are 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 show that parallel efficiency can reach 83% at 32 processors. In the third phase, the parallelized PIC-FEM code is used to simulate several important problems to demonstrate its superior capability in handling practical problems. These problems include field emission from a CNT /silicon based emitter under external electric field and magnetic field, two typical three-dimensional DC and RF gas discharge plasmas, and two typical three-dimensional DC and RF magnetron plasmas with permanent magnets.