臺灣博碩士論文加值系統

在一油滴抽取的過程中，在多孔隙介質內的油滴因為毛細作用的抗力而被困在毛細管中不能輕易被抽出，因此在抽取過程中會額外施加一振動頻率，使油滴產生共振抵抗毛細作用的抗力，因此提升油滴抽取的效率。而本研究即以在抽取過程中最低能量消耗為目標建構一最佳化控制問題，並調整額外施加之振動頻率與外力以達使油滴可抽出之最小距離。
為解決此一最佳化控制問題，本研究使用擬譜法最佳化控制解決。擬譜法最佳化控制為一數值分析方法，其過程首先為離散化原函數簡化原函數的複雜度，並選取適當之節點逼近，其後在加入微分轉換與積分轉換處理函數中的微分積分項，最後再加入可容許誤差值彌補離散化過程後的微小偏差，再經過一連串的轉換後，便能使原問題轉變為一非線性問題，再利用GAMS、MATLAB等軟體便能得到此一問題之解答。

In the process of extracting oil, it is common that there is still a large amount of oil remaining in the crust because of the tension force and resistance caused by the capillary in the porous media. To eliminate the trapped oil in the porous media, a excitation frequency is applied and causes oil blobs resonance. Because of the resonance, oil blobs can easily cancel out the resistance of capillarity and tension force. In this research, an optimal control problem is constructed in order to minimize the energy consumption by adjusting the excitation frequency and extracting force. Pseudospectral method, which is a numerical method specializing in solving partial differential equation, is applied to simplify the origin function and transforms the origin optimal control problem into a nonlinear program. Therefore, this problem can eventually be solved by GAMS.

誌謝 I
中文摘要 II
ABSTRACT III
目錄 IV
圖目錄 VI
表目錄 VI
Chapter1 緒論 1
1.1 研究動機 1
1.2 研究目的 1
1.3 章節說明 2
Chapter2 文獻回顧 3
2.1 油滴運動 3
2.2 擬譜法 3
2.2.1 譜方法 3
2.2.2 運用擬譜法於最佳化控制問題 4
Chapter3 研究方法 5
3.1 擬譜法(Pseudospectral method) 5
3.1.1 離散化原函數 6
3.1.2 選取節點 10
3.1.3 產生微分矩陣與積分權重 11
3.1.4 加入容許誤差 13
Chapter4 模型介紹 15
4.1 最佳化問題變數意義 15
4.2 均勻管之最佳化模式 16
4.3 漸變管之最佳化模式 17
4.4 以擬譜法最佳化控制分析問題 19
Chapter5 結果與討論 23
5.1 參數設定 23
5.2 結果 23
Chapter6 結論與建議 26
參考文獻 27

[1].石賢棟(2013)，「Mixed Jacobi-Fourier Spectral Method for Fisher Equation」，上海師範大學計算數學系碩士班碩士論文。
[2].馬石莊(2011)，「應用偏微分方程與科學計算講義」。
[3].陳佑航(2008)，「以第一原理含時密度泛函理論研究研究雙原子分子在超短牆場雷射下的多光子效應及量子最佳控制理論」,台灣大學物理系碩士班碩士論文。
[4].Garg, Divya, et al, 2009, “An overview of three pseudospectral methods for the numerical solution of optimal control problems.” Advances in the Astronautical Sciences 135.1.
[5].I. M. Ross and M. Karpenko, 2012, “A Review of Pseudospectral Optimal Control: From Theory to Flight”, Annual Reviews in Control, Vol.36, pp.182-197,2012.
[6].Kamron Saniee, 2007, “ A Simple Expression for Multivariate Lagrange Interpolation”.
[7].Markus Hilpert, 2006, “Capillary-induced resonance of blobs in porous media: Analytical solutions, Lattice-Boltzmann modeling, and blob mobilization”, Journal of Colloid and Interface Science.
[8].M. Hilpert, G. H. Jirka and E. J. Plate, 2000, “Capillarity-induced resonance of oil blobs in capillary tubes and porous media”, Geophysics, 2000.
[9].Qi Gong, I. Michael Ross and Fariba Fahroo, 2009, “Pseudospectral optimal control on arbitrary grids”, AAS/AIAA Astrodynamics Specialist Conference.
[10].Qi Gong, Wei Kang and I. Michael Ross, 2006, “A Pseudospectral Method for the Optimal Control of Constrained Feedback Linearizable Systems”, IEEE Transactions On Automatic Control, VOL 51, 50.7.
[11].Qi. Gong, M. Ross, W. Kang, F. Fahroo, 2008, “Connections between the convector mapping theorem and convergence of pseudospectral methods for optimal control”, Computational Optimization and Applications, Vol. 41, pp. 307-335.
[12].Shao-Yiu Hsu, Joseph Katz and Markus Hilpert, 2012, “Theoretical and experimental study of resonance of blobs in porous media”, GEOPHSICS, VOL. 77, NO. 5.
[13].Shao-Yiu Hsu and Markus Hilpert, 2007, “An Ordinary Differential Equation for an oil Blob Excited by a Seismic Wave”.
[14].Solomonoff, A., and E. Turkel, 1989, “Global properties of pseudospectral methods.” Journal of Computational Physics 81.2.
[15].W. Kang, Q. Gong and I. M. Ross, 2007, “On the convergence of nonlinear optimal control using pseudospectral methods for feedback linearizable systems, International Journal of Robust and Nonlinear Control, Vol. 17, Issue 14, pp. 1251-1277.