在電漿流體模型中利用時間多重尺度演算法之發展與驗證及其應用

Development, Validation and Application of a Temporal Multi-scale Algorithm for Efficient Fluid Modeling of Gas Discharges

古必任(Bi-Ren Gu)

指導教授 ： 吳宗信

國立交通大學/工學院/機械工程系所/博士(2014年)

https://doi.org/10.6842/NCTU.2014.01119

摘要

本論文主要研究目的為在電漿流體模型中利用時間多重尺度的演算法(Temporal Multi-scale Algorithm, TMA)。時間多重尺度演算法能夠有效率的耦合平行化的電漿流體模型(Lin et al., 2012a)以及平行化的氣流場模型(Hu et al., 2011)，這兩個模型都是使用有限體積法離散方程式。此演算法可以將耦合電漿及流場的多重尺度計算的計算時間進行縮減。本論文主要分為三個主要部分：第一個部分介紹電漿流體模型、氣流場模型以及時間多重尺度演算法。第二個部分則利用一維氦氣電漿介電質放電來驗證時間多重尺度法。第三個部分利用二維氬氣電容耦合式電漿作為一個應用的例證。
在論文第一部分，時間多重尺度演算法可以使用在一維及二維的電漿模擬之中。我們利用電中性氣體物種的化學反應時間特性是否隨外部電源的震盪頻率變化來做為區分，將這些物種分成「快速」跟「緩慢」的電中性氣體物種。在此演算法當中，我們將模擬分成兩個部分：(1)所有物種的電漿流體模型模擬以及(2)不同時間特性的「緩慢」電中性氣體物種質量守恆方程式模擬。在這兩個部分之間我們會互相交換經過計算得到的時間平均的物種濃度及不同類型化學反應的化學反應速率或是物種濃度源項。第一個部分因為電子運動的侷限，模擬的時間步階為10-10秒；第二個部分則可以針對一維的擴散-反應特性以及二維的對流-擴散-反應特性將時間步階放大來加速達到穩態解。
在論文第二部分，在一維氦氣電漿介電質放電的模擬中，我們以提供25千赫茲、峰值電壓值為6,000伏特、波形為正弦波電源所驅動的常壓電漿源為例，來展示時間多重尺度演算法對於加速電漿流體模型模擬的潛力。與此相比較的例證為利用單一時間步階進行計算的標竿例證。在此使用的電漿化學組為36個物種及121條化學反應式的含雜質常壓氦氣介電質放電電漿，其雜質包含了百萬分之25的氮氣、百萬分之10的氧氣及百萬分之1的水氣。在一維的結果當中，在所有的空間平均物種濃度的相對誤差都在百分之1以內的前提之下，使用時間多重尺度演算法例證的計算時間只有標竿例證的百分之四(25倍加速)，此時的模擬條件為使用5個初始電漿週期、5個補完電漿週期以及4個重複階段。若再將精準度的要求放鬆到百分之44，在使用2個初始電漿週期、2個補完電將週期以及2個重複階段的情況下可以達到92倍的加速。
在論文第三部分，在二維氬氣電容耦合電漿當中，我們以提供頻率60百萬赫茲、電源功率200瓦、波形為正弦波的電源所驅動的氣壓500毫托氬氣電容耦合電漿源為例證，來驗證結合電漿流體模型與氣體模型之耦合數值演算法。我們在「國家實驗研究院高速網路與計算中心」的叢集式電腦「御風者」中使用24個處理器，在使用計算格點14,427的網格、耦合電漿流體模型與氣流場模型10次與實際物理時間達到10秒的條件下，計算時間為14.4個小時。結果顯示僅需要4-5次耦合即可達到電漿以及氣流場的擬穩態解，故模擬計算時間僅需前述之一半。在例證的結果之中，電漿對氣流場加熱的效果使特定觀察點的氣流場溫度從未耦合的凱氏406度在耦合之後升溫到442度。結果顯示電漿中氬氣離子的焦耳熱(joule heating)對於加熱背景氣體的貢獻度以及重要性。在最後本研究針對氬氣電容耦合電漿進行詳細的參數研究，背景氣體壓力區間為500毫托到3.5托、電源供應器提供的功率區間為50到1,000瓦、頻率區間為13.56到60百萬赫茲。
本論文的最後總結了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 ； 時間多重尺度演算法 ； 氦氣 ； 氬氣 ； 介電質放電電漿 ； 電容耦合電漿 ； 電漿

並列摘要

A temporal multi-scale algorithms (TMA) for efficient fluid modeling of gas discharges are proposed in this thesis. TMA, which is an efficient hybrid numerical algorithm, combines a parallel plasmas fluid modeling (Lin et al., 2012a) and a parallel gas flow solver (Hu et al., 2011), which both employed cell-centered finite-volume method. This algorithm intends to greatly reduce the computational time of multidimensional modeling gas discharges to an acceptable runtime, considering effect of mutual interaction between gas flow and gas discharge. The thesis is divided into three major parts, which are described in the following in turn.
In the first phase, temporal multi-scale algorithms are proposed for one- and two-dimensional gas discharges. We classify the neutral species into the “fast” and “slow” neutral species by using the reactive characteristics of species as compared to oscillating discharge. In the algorithm, we separate the simulation into two parts, which include fluid modeling for all species and diffusion-reaction (1D) or convection-diffusion-reaction (2D) with different time scales for slow species. Proper two-way coupling between these two parts is required to seamlessly integrate the mutual integration in an efficient manner. For the former the time step size is constrained by electron motion which is on the order of 10-10 s and for the latter the time step size can be greatly enlarged based on diffusion (1D) or convection (2D) time scale for reaching a steady state of gas flow.
In the second phase, a one-dimensional gas discharge modeling is used to validate the proposed TMA. A helium dielectric barrier discharge (DBD) driven by a power source with a frequency of 25 kHz is used as an example to demonstrate the superior capability of TMA in accelerating fluid modeling simulation, while maintaining essentially the same accuracy of solution as compared to the lengthy benchmarking fluid modeling using single time-scale approach. The plasma chemistry considers 36 species and 121 reaction channels, which include some impurities such as nitrogen (25 ppm), oxygen (10 ppm) and water vapor (1 ppm), in addition to the helium itself. The one-dimensional results show that the runtime using the TMA can be dramatically reduced down to 4% (25 times faster) with a relative difference of spatially averaged number densities generally less than 1% for all species between the TMA and the benchmarking cases when 5 initial cycles, 5 supplementary cycles and 4 repeated stages are used. Further reduction of the accuracy requirements down to 44% for some specific species can lead to 92 times of speedup with the use of 2 initial cycles, 2 supplementary cycles and 2 repeated stages.
In the third phase, two-dimensional axisymmetric gas discharge modeling is used to demonstrate the efficiency of the proposed 2D TMA in practice. An argon capacitively coupled plasma (CCP) (0.5 torr) driven by a power source having a frequency of 60 MHz with a constant power absorption of 200 W is used as an example to demonstrate the use of TMA through effective coupling fluid modeling and gas flow solvers. 14,427 computational cells and 24 cores of ALPS PC cluster at National Center for High-Performance Computing, Taiwan, are used throughout the study. It takes 14.4 hours for 10 two-way couplings, which can reach physical time of 10 s for gas flow field. Only 4-5 iterative couplings are enough to reach a quasi-steady state for gas discharge and a steady state for gas flow. The results show that plasma heats the gas flow and, for example, the gas temperature rises from 406 to 442 K at the edge of the substrate in the plasma bulk region, in which the former is the temperature without coupling. It shows the importance of heating of the background gas mainly caused by the Joule heating of argon ions in the discharge. At the end, a detailed parametric study of thermal-fluid and plasma properties is performed using the same CCP under various test conditions of pressure (0.5-3.5 torr), power driving frequency (13.56-100 MHz) and plasma power absorption (50-1,000 W).
Major findings of the thesis and recommendations for future work are outlined at the end of the thesis.

並列關鍵字

plasma fluid model ； temporal multi-scale algorithm ； helium ； argon ； dielectric barrier discharge ； capacitively coupled plasma ； plasma ； discharge

參考文獻

Anghel, S.D., Simon, A., and Frentiu, T. (2005). Characterization of a very low power argon CCP. J. Anal. At. Spectrom. 20, 966–973.

Balay, S., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., and Zhang, H. (2011). PETSc library, http://www.mcs.anl.gov/petsc.

Bird, R.B., Stewart, W.E., and Lightfoot, E.N. (2007). Transport Phenomena (John Wiley & Sons).

Camara, N., and Zekentes, K. (2002). Study of the reactive ion etching of 6H-SiC and 4H-SiC in SF6/Ar plasmas by optical emission spectroscopy and laser interferometry. Solid-State Electron. 64, 1959–1963.

Cercignani, C. (1987). The Boltzmann Equation and Its Applications (Springer New York).