應用細胞自動機於垂向二維地下水流與熱流偶合數值模式之建立

Abstract

摘要 數值模式的建置，可分為四大階段，分別是概念模式階段、數學模式階段、數值模式階段與數值程式階段。傳統上，在數值模式階段均是以各式各樣的數值方法進行離散動作，其中包含有限差分法(FDM)、有限元素法(FEM)等，數值模式是依據模擬區域、邊界條件等離散成一個矩陣方程式，最後再透過矩陣解法求得各結點之模擬值。傳統之矩陣解法效率雖高，然而其本質上並不適合平行運算，對於大量結點之運算，其龐大計算量無法透過日趨便宜的處理器分散處裡。因此本研究將以細胞自動機理論為基礎進行開發，細胞自動機特點之一為其網格形狀並不侷限於矩形網格，因此易於針對複雜形狀邊界之模擬。此外，細胞自動機理論本身即隱含平行特性，因此未來進一步擴張為平行運算後，對於大量細胞之運算可以大幅降低計算時間。 本研究的以細胞自動機為基礎，配合徐昇氏網格及守恆定律，針對水流與熱流偶合問題發展一個水流與熱流偶合數值模式。本研究透過九個驗證案例證實模式正確性。在水流熱流偶合案例中顯示，透過偶合模擬，熱流模擬結果會因為水流流動而產生對流效應，因此影響其溫度分佈；另外，相較於傳統地下水模式之定溫假設，偶合模式之水流模擬結果會因為溫度變化而產生不同之流動情形。此外，透過本研究結果發現，應用細胞自動機理論於數值模式之開發，可以採用一系列分散之方程組，而無須採用整合式的偏微分方程式，因此在全新問題之模式開發上可以大幅減少數學推導之繁瑣流程。

Abstract Development of a numerical model generally consists of four major steps; conceptual model proposing, mathematical formulation construction, numerical solution derivation and program development. The numerical solution derivation is the focus of this research the derivation usually applies different numerical schemes such as finite different method (FDM) or finite element method (FEM) to discretize the associated governing equations and resulting as matrix equations. A program is then developed to solve the matrix equations solving matrix equations is usually a sequentially based approach and is difficult to be parallelized. Inspired by previous discussion, this research develops a numerical model with groundwater flow and heat transport coupling by applying the concept of Cellular Automata. The relations between cells are defined by using Vonoroi Diagram; instead of solving the conventional partial differential equations that governing the groundwater flow and head transport, this proposed methodology solves a series of fundamental equations. Conservation of water and heat are the fundamental physical laws, several hypothesical cases were simulated to validate the accuracy of the model. In the groundwater flow and heat transport coupling cases, simulation results indeed shows that groundwater flow velocity affected the overall temperature distribution. Comparing the results with that of conventional groundwater flow simulation in a constant temperature condition, the spatial variation of temperature distribution also affects hydraulic head distribution. Beside the simulation validation, the proposed novel simulation model can extend its computational capacity by adding new physical law with minimal code modification.