利用非結構性網格之平行化電漿流體模型程式的開發、 驗證及應用

Development, Validation and Application of a Parallel Plasma Fluid Modeling Code Using Finite-Volume Method With an Unstructured Grid

陳冠霖(Kuan-Lin Chen)

指導教授 ： 吳宗信

國立交通大學/工學院/機械工程系所/博士(2019年)

摘要

本論文旨在開發與驗證一套利用非結構性網格之平行化電漿流體模型程式，並利用文獻中低壓與常壓之電漿源結果為範例進行程式驗證，並應用於電容耦合式基礎電漿物理的模擬與分析。為因應不同的環境條件與應用領域，所研發的程式主要可以分為兩大電漿流體模型，包括採用1）局部場假設 (Local Field Approximation, LFA) ，適用於稍低壓至常壓下之電漿流體模型；2）局部平均能量假設 (Local Mean Energy Approximation, LMEA) ，於較低壓至常壓下之電漿流體模型。首先，在採用局部場假設之電漿流體模型中，因其傳輸係數與化學反應速率為降電場（reduced field）之函數，故不需求解電子能量方程式。因假設離子與背景氣體分子達到熱平衡，而不需要求解其能量方程式。對電子與離子的動量方程式則均以飄移-擴散假設 (Drift-Diffusion Approximation) 來進行處理，並帶入連續方程式來進行求解得到電子與離子密度分佈。蒲松方程式 (Poisson’s equation) 則是用來求解瞬時電位分佈以求得瞬時電場分佈。另外，在局部平均能量假設之電漿流體模型中，由於傳輸係數與化學反應速率為電子能量的函數，因此必須求解電子的能量方程式來得到電子溫度分佈。而對於離子在低壓的環境下受鞘層電場加速而導致離子溫度於鞘層區域遠高於背景溫度；故本程式提供三種不同離子溫度模型, 第一種為假設與背景溫度相同、第二種為使用半經驗公式來求解、第三種則為直接求解離子能量方程式。對電子與離子的動量方程式則是以飄移-擴散假設來進行處理，並帶入連續方程式來進行求解得到電子與離子密度分佈，但於較低壓情況下離子慣性力影響很大，故本程式在為了捕捉離子慣性力的影響時，可採用直接求解離子動量方程式。同樣的蒲松方程式則是用來求解瞬時電位分佈以求得瞬時電場分佈。最後對於中性氣體密度分佈的求解，則是忽略對流的效應而只考慮擴散行為來進行簡化。
在數值方法的部分，我們採用有限體積分法離散在電漿流體模型中所有的偏微分方程式；另採用Scharfetter-Gummel 算則來處理飄移-擴散通量，以及HLL (Harten-Lax-van Leer) 近似黎曼算則 (Approximate Riemann Solver) 來處理離子相關方程式的對流項。對於擴散項在非結構性網格的情況下，我們採用空間泰勒展開式去處理非正交網格所帶來的影響，並採用最小平方法來計所有算體中心物理性質的梯度。為了克服計算電漿流體模型時間步階過小的問題，本論文則是於必要時採用求解半隱式蒲松方程式來獲得較大的時間步階。並以平行化generalized minimal residual method (GMRES)的矩陣疊代方法，搭配平行化additive Schwarz method (ASM)前提(precondition)方式，進行數值解析所有離散方程式。而在平行化的部分是利用區域分割方法 (domain decomposition)，均勻分配每顆電腦的計算量；在程式撰寫方面則採用 ultraMPP (ultra-fast Massive Parallel Platform) 平行計算平台，並藉由分散式叢集電腦來進行運算。
首先，在局部場假設程式之開發方面，則以常壓流注放電電漿 (streamer discharge) 與常壓表面介質放電電漿 (surface dielectric barrier discharge) 進行驗證。接著，在以低壓下之一維氬氣電容耦合式電漿 (capacitively coupled plasma) 、一維氦氣電容耦合式電漿模擬與二維軸對稱GEC reference cell的氬氣電容耦合式電漿等題目，分別根據文獻資料考慮飄移-擴散假設或直接求解離子動量方程式並採用局部平均能量近似來進行程式之開發與驗證。其結果顯示在採用不同電漿流體模型之模擬結果與文獻實驗與模擬數據相比均有相當的一致性。最後，利用已開發的程式進行應用，包括：1）離子在氬氣電容耦合式正電漿中不同壓力與電源頻率情形下，離子慣性力的重要性分析；2）電容耦合式正電漿(氬氣)、負電漿(四氟化碳)模擬中，離子動量方程式之聲速模型，對不同通量重建數值方法 (flux reconstruction scheme) 的影響分析；3) 氬氣電容耦合式電漿並展示增加離子能量方程式及使用非結構性網格之模擬結果。
論文最後將總結本論文的研究成果並提出對未來研究方向的建議。

關鍵字

電漿流體模型 ； 流注放電電漿 ； 介電質放電電漿 ； 非結構性有限體積法

並列摘要

This dissertation aims to develop and validate a parallel plasma fluid modeling code with the cell-centered finite-volume method using an unstructured grid, and to apply to simulate many low-pressure and atmospheric-pressure plasma sources. There are two major approaches applied for the current fluid code, which include: The local field approximation (LFA) and the local mean energy approximation (LMEA). Firstly, for the LFA, there is no need to solve the electron energy density equation since the transport coefficients and the reaction rates are functions of the reduced field only. The ion temperature is generally the same as that of the background gas assuming the thermal equilibrium is maintained between ions and background gas. The momentum equations for electrons and ions are often simplified using the drift-diffusion approximation (DDA) and are combined with the continuity equations to form the number density equations of electrons and ions, respectively. The Poisson's equation is used to solve the instantaneous potential and the electric field self-consistently. Secondly, for the LMEA, in the low-pressure region, the transport coefficients and the reaction rates related to electron are functions of the electron energy. Therefore, the electron energy density equation must be solved to obtain the temporal and spatial distributions of electron temperature. (local mean energy approximation). Under very low-pressure condition, the ions can be accelerated by the large electric field in the sheath and become much more energetic than the background gas. Therefore, the developed fluid code provides three different kinds of ion temperature models. The first assumes that the ion temperature is the same as the background temperature, the second employs the semi-empirical formula for calculating the ion temperature, and the third is to directly solve the ion energy density equation. For the momentum equation of the electron, the treatment is similar to that of LFA. However, under the very low-pressure environment, the full ion momentum equations are solved directly. The Poisson's equation is used to solve the instantaneous potential and the electric field self-consistently. Finally, the continuity and momentum equations for the background gas is solved without considering the convection effect caused by the flow field.
For the numerical methods, the cell-centered finite-volume method is employed to discretize all governing equations in the fluid model. The Scharfetter-Gummel scheme is adopted to handle the drift-diffusion flux, and the approximate Riemann solver named HLL is used to handle the convection term of the ion equations. For the diffusion terms in the case of an unstructured grid, the spatial Taylor expansion is used to deal with the effects of non-orthogonality of the mesh, and the cell-center gradient of plasma properties is calculated using the least-square method. To overcome the restriction of the time step size in the simulation, the semi-implicit approach for solving the Poisson’s equation is used to obtain a larger time step size without the restriction of dielectric relaxation. All discretized equations are solved one by one self-consistently using the iterative parallel generalized minimal residual method (GMRES) in conjunction with the parallel additive Schwarz method (ASM) as the preconditioner. For the parallelization of this plasma fluid modeling code, the domain decomposition method is used to distribute the computational load evenly among different processors of a PC cluster. For the programming of the fluid code, the ultraMPP (ultra-fast Massive Parallel Platform), which is a commercial parallel computing platform, is used to greatly alleviate the programming effort.
Firstly, the fluid code using the LFA is developed and verified by using a 2D-axisymmetric streamer discharge and a 2D surface dielectric barrier discharge under atmospheric-pressure conditions. In addition, the argon capacitively coupled plasma (CCP), the helium CCP and the argon CPP using Gaseous Electronic Conference (GEC) reference cell under low-pressure condition are used to develop and verify the code by using the DDA and the full ion momentum equations. The results show that the simulation results using different plasma fluid models are in good agreement with the published experimental and simulation data whenever is available.
Secondly, the developed fluid code is used: (1) to understand the importance of ion inertia force with varying pressure and frequency in an argon CCP; (2) to investigate the effect of acoustic speed model on various reconstruction schemes in the electropositive (argon) and electronegative (CF4) CCPs with the original and reformulated ion-related modeling equations when the HLL flux scheme is used; (3) to simulate an two-dimensional axisymmetric argon CCP considering the ion energy equation using an unstructured grid.
Thirdly, major findings of this thesis are summarized with some recommendations of the future work at the end.

並列關鍵字

plasma fluid model ； streamer discharge ； capacitively coupled plasma ； unstructured finite-volume method ； dielectric barrier discharge

參考文獻

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