徐昇氏網格應用於二維地下水流數值模式之建立

Abstract

數值模式已成為分析地下水相關問題之主要工具，而發展數值模式常用的數值方法有有限差分法、有限元素法與有限體積法等，有限差分法的優點在於簡單直接且計算效率高，然其缺點在於格網形狀僅可是矩形而缺乏彈性，有限元素法雖有網格形狀上的彈性，但其理論較複雜且計算效率較低，有限體積法之理論複雜度及且計算效率介於兩者之間，其優點為所得數值解符合質量守恆。因此若能延續有限差分法的簡單及高效率，克服矩形網格限制，整合有限體積法符合守恆條件的精神，則可發展出更簡單、精確、高效率及高應用彈性的數值模式。 本研究的目的在以徐昇氏網格為基礎，配合守恆定律，針對地下水流問題發展一新的地下水流數值模式。本研究藉由CGAL函式庫建立徐昇氏網格，並以各徐昇氏網格為控制體積，配合相關之方程式進行守恆運算，再以迭代方式進行數值求解。本研究以十一個案例驗證模式之正確性，及展現模式應用的彈性如局部加密等。結果顯示本模式在各案例中均可符合質量守恆的要求。此外，配合徐昇氏網格之局部格網加密，可在增加少量計算時間的情形下，大幅提昇計算精度，有效提升模式應用彈性及計算效率。

Numerical model simulation technique has become a powerful tool for analyzing groundwater problems. Although several numerical methods, such as the finite difference method (FDM), the finite element method (FEM) and the finite volume method (FVM), are widely used for numerical model development the FDM has the advantage of easy implementation and high computation efficiency. However, because the FDM requires a rectangular grid, solving problems with irregular boundary is difficult, thus limiting its practical application. Unlike the FDM, grid shape of the FEM is flexible and easily applies to solve problems with irregular boundary. However, the FEM is more complicated and computationally less efficient than the FDM. Easy implementation and computational efficiency for the FVM lies between the FDM and FEM. The FVM advantage is its guaranteed solution to satisfy conservation conditions. Regarding the previous discussion, this research proposes a novel numerical method, easy to implement, with high computational efficiency and a flexible grid shape for high practical application. The current study adopts the grid for the proposed groundwater numerical mode that applies the Voronoi diagram and the Computation geometry algorithm library (CGAL) to generate the Voronoi grid in the model. For each Voronoi cell (grid), the mass conservation of water and other relevant physical laws form the relationship between a cell and its neighboring cells. The proposed model applies iteration methods for computing solutions cell by cell. Using the proposed model to verify model capability solves several hypothesis problems. Solving problems with point source demonstrates that applying the irregular Voronoi grid uses only one tenth of that using the rectangular grid for similar solution accuracy.