臺灣博碩士論文加值系統

滑坡是危及生命並造成財產損失的一種極大的自然災害，地下水的存在和變化是導致邊坡失穩的主要原因，因此分析地下水土壤滲流機制，計算邊坡的穩定性，對確保生產建設和人民財產安全有著重要的實際意義。本研究利用無網格法中的Trefftz法求解地下水滲流問題，首次推導出在補注和蒸發條件下自由液面完整T型基底函數。並結合飽和與非飽和土層Trefftz基底對地下水滲流機制與邊坡穩定進行分析研究。論文主要進行了以下幾個方面研究：
（1）以Dupuit-Boussinesq方程作為自由液面控制方程，通過分離變數法分別推導出補注和蒸發情況時，穩態和暫態自由液面在第一類邊界和第二類邊界條件下Trefftz的完整T型基底函數；並通過引入特徵長度來降低系統的病態性。
（2）根據推導的自由液面Trefftz的完整T型基底，通過matlab程式設計求解不規則區域穩態均質第一類邊界條件、規則區域穩態均質混合邊界條件、不規則區域暫態均質第一類和混合邊界條件下的自由液面，並與解析解進行對比驗證。兩者的最大絕對誤差在 以下。計算結果證明了Trefftz法求解自由液面的可行性與正確性。
（3）通過自由液面Trefftz基底計算出補注和蒸發情況下的自由液面，在此基礎上，採用飽和與非飽和土層Trefftz基底分別求解兩種土層的滲流， 研究成果顯示；地下水得到補注時，水力坡降從高水位一側向低水位一側遞減，非飽和土層中土體孔隙水壓從上向下增加；當地下水蒸發時，結果與補注時相反。
（4）自由含水層中的地下水和大氣間存在水量交換，根據交換水量以及邊界條件，利用自由液面Trefftz基底，計算出三維空間的自由液面。將所求出的自由液面作為飽和土層的頂部邊界，並採用三維飽和土Trefftz基底計算自由含水層在補注和蒸發條件下的三維穩態滲流問題。研究成果顯示；四周為定水頭邊界條件，在補注情況下，自由液面有上凸的現象，水流從中心向四周由垂直運動逐漸變為水準運動。在蒸發條件下，結果與補注時相反。
（5）針對Trefftz時空配點法求解的暫態地下水滲流問題，結合STABL軟體，對庫水位降落聯合降雨入滲情況下的邊坡進行即時穩定性分析。研究成果顯示；庫岸邊坡在降雨入滲條件下，坡內水力坡降越大，邊坡越不穩定。

Landslide is a great natural disaster that is life-threatening and causes property loss. The existence and change of groundwater level is the main cause of slope instability. Therefore, analyzing the seepage mechanism of groundwater and calculating the stability of slopes have important practical significance for ensuring production and construction and people's property safety.In this paper, the Trefftz method wich is a kind of meshless method is used to solve the groundwater seepage problem. The T-complete basis function
of the free surface is derived under the condition of after-teeming and evaporation. The Combination of the Trefftz basis function for saturated and unsaturated soil are adopted to solve the seepage mechanism and slope stability of groundwater. This findings and conclusion remarks are listed as follows.
(1)The governing equation of the free surface is the Dupuit-Boussinesq equation. The proposed method is to derive the general solutions using the method of separation of variables in the case of recharge and evaporation. The Dirichlet and Neumann boundary conditions for steady-state and transient free surface problems are considered. Furthermore, we adopt the characteristic length to reduce the ill-conditioned system.
(2)Several numerical examples including different boundary conditions in steady-state and transient free surface problems are carried out to verify the proposed method .The free surface with homogenous soil under different boundary conditions is then compared with the analytical solution. The results show that the maximum absolute error can reach the order of 10-10. It is found that the proposed method may provide promising numerical results for seepage problems with the free surface.
(3)Groundwater from unsaturated soil layer through the free surface into the in the process of the stability of saturated soil seepage its flow rate constant.The free surface of the continuous seepage condition is solved by The T-complete basis function of the free surface, and the soil layer is divided into the saturated soil layer and the unsaturated soil layer by the free liquid surface. This method reduces the difficulty of solving the continuous seepage problem. Research results showed that the groundwater is charging, the hydraulic gradient decreases from the high potential energy to the low side, and the pore water pressure of the unsaturated soil increases from the top to the bottom in the unsaturated soil layer; when the groundwater evaporates, the result is opposite to that of the charging.
(4)According to the water exchange between the groundwater and the atmosphere in the free aquifer and the boundary conditions, the free liquid surface in the three-dimensional space was calculated by using The T-complete basis function of the free surface.Useing the The T-complete basis function of the three-dimensional Laplacian calculate the steady-state seepage flow of the saturated soil layer in the three-dimensional free aquifer under the conditions of recharge and evaporation and the free surface are used as the top boundary. The research results show that the surrounding is Dirichlet boundary conditions. In the case of charging, the free liquid surface has a convex phenomenon, and the water flow gradually changes from vertical to horizontal flow from the center to the periphery. Under evaporation conditions, the results are the opposite of when charged.
(5)The Trefftz space-time collocation method is used to solve the transient state seepage problem under the changing water level and rainfall infiltration condition. The calculated transient state seepage uses STABL software to analyze the real-time stability of the slope. The research results show that under the condition of rainfall infiltration, the greater the hydraulic gradient in the slope is, the more unstable it is.

目次
摘要 ......................................................I
Abstract ............................................... III
目次 ..................................................... V
圖次 .................................................. VIII
表次 ..................................................... X
第一章 緒論 .............................................. 1
1.1研究目的與意義 ......................................... 1
1.2國內外研究現狀 ......................................... 2
1.2.1地下水滲流研究現狀 ................................... 2
1.2.2Trefftz 方法發展與應用 ............................... 5
1.3本文研究內容 ........................................... 7
1.3.1 篇章結構 ........................................... 7
1.3.2 技術路線圖 ......................................... 9
第二章 滲流與邊坡穩定基本理論 ............................. 10
2.1飽和土滲流理論 ........................................ 10
2.1.1 Darcy定律 ......................................... 10
2.1.2 飽和土層滲流控制方程 ............................... 11
2.2 自由液面理論 ......................................... 14
2.2.1 Dupuit定律 ........................................ 14
2.2.2 Dupuit-Boussinesq方程 ............................. 16
2.3非飽和土理論 .......................................... 18
2.3.1 非飽和層吸力理論 ................................... 18
2.3.2 土水特徵曲線 ....................................... 19
2.3.3 含水量和滲透係數關係 ............................... 21
2.3.4 非飽和土滲流控制方程 ............................... 22
2.4邊坡穩定理論 ......................................... 25
2.4.1 影響邊坡穩定因素 ................................... 25
2.4.2 邊坡穩定分析方法 ................................... 25
2.5小結 ................................................. 28

第三章 Trefftz法求解地下水滲流理論 ........................ 29
3.1飽和土Trefftz基底 .................................... 29
3.1.1 二維飽和土Trefftz基底 .............................. 29
3.1.2 三維飽和土Trefftz基底 .............................. 29
3.2 自由液面Trefftz基底 .................................. 31
3.2.1 穩態基底推導 ....................................... 31
3.2.2 暫態基底推導 ....................................... 37
3.3 非飽和土Trefftz基底 .................................. 50
3.4 Trefftz配點法 ....................................... 50
3.4.1 特徵長度 .......................................... 51
3.4.2 Trefftz時空配點法求解流程 ...................................................... 52
3.4.3 小結 .............................................. 56
第四章 數值模式驗證 ...................................... 57
4.1穩態數值驗證 .......................................... 57
4.1.1 不規則區域均質土自由液面問題(一類邊界條件) ........... 57
4.1.2 規則區域均質土自由液面問題(混合邊界條件) ............. 59
4.1.3 階數與點數對誤差的影響 ............................. 62
4.2暫態數值驗證 .......................................... 64
4.2.1 不規則區域均質土自由液面問題(一類邊界條件) ........... 64
4.2.2 不規則區域均質土自由液面問題(混合邊界條件) ........... 69
4.2.3 階數與點數對誤差的影響 ............................. 74
4.3小結 ................................................. 76
第五章 地下水滲流案例分析 ................................. 78
5.1補注蒸發下飽和非飽土層滲流問題 ......................... 78
5.1.1 計算自由液面 ....................................... 78
5.1.2 計算飽和土非飽和土滲流 ............................. 81
5.1.3 結果分析 .......................................... 84
5.2 自由含水層三維穩態滲流問題 ............................ 85
5.2.1 計算三維自由液面 ................................... 85
5.2.2 三維飽和土滲流 ..................................... 87
5.2.3 结果分析 .......................................... 88
5.3地下水滲流對庫岸邊坡穩定影響 ........................... 90
5.3.1 計算模型及土體參數 ................................. 91
5.3.2 降雨入滲對邊坡穩定影響 ............................. 91
5.3.3 庫水位對邊坡穩定影響 ............................... 93
第六章結論與展望 ......................................... 96
6.1結論 ................................................. 96
6.2展望 ................................................. 97
參考文獻 ................................................ 98

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