搭配調適網格加密功能之平行化Poisson-Boltzmann方程式有限元素數值分析研究

Abstract

本研究完成平行化三維Poisson-Boltzmann之程式建立，主要是利用有限元素法搭配Monotone iteration 處理非線性項的部份，搭配本實驗室發展的平行化調適網格模組，將需要加密的區域，做適當的加密。在程式的驗證上，主要是以模擬圓球體，圓柱體的電雙層電位分佈，在和解析解與文獻中發表的近似解做比較。,於邊界電位等於一的情形下，解析解，模擬解與近似解，三者的結果一致，由此可驗證程式的正確性。驗證完程式的正確性後，再將本程式與平行化調適網格做搭配後模擬兩顆帶電離子於無限遠空間內的電雙層電位分佈，將結果與歷年的文獻做比較，經歷五個階段的網格加密後，計算電位，電場值與作用力後，比較文獻上的資料，大約有0.06%的誤差。於後，模擬兩帶電離子於圓柱型電洞內的電雙層電位分佈，計算並比較作用力值後，與文獻上的資料大約有1%上的誤差，這些誤差推測是網格的緊密度不足所導致，但因電腦資源的問題，網格也無法再繼續加密下去。在文獻中，都是以二階以上的有限元素分析電雙層電位，在未來如能提高本程式的有限元素階數，相信可以獲得更有效率，更精準的分析。 此程式可用來解析電雙層內的電位分布，也可解析膠體間的交互作用力的分析。對於電滲流分析及薄膜效率分析上，都有不錯的應用。

A parallel Poisson Boltzmann equation solver using finite element method with adaptive mesh refinement is proposed. A Monotone iterative method was used to solve the nonlinear equation arising from the finite element discretization procedure. A 3D Poisson-Boltzmann equation was used to model the electric double layer field. And the solver will be using to simulate this phenomenon. First, in order to verify code accuracy we model the EDL potential distribution at sphere and cylinder, and compared with analytical solution and approximant solution. These results are all the same in small zeta potential. After verification, we couple with parallel adaptive mesh refinement to compute some simple cases such as two identical charged spherical particles and computer the interaction force. After five step refinement levels, we calculate the potential, electric field and interaction force, and compared with previously literature. It has 0.06% inaccuracy between our results and previously work. After this case, we apply our code to simulate the situation of two identical particles in a cylindrical pore, and compared with previously works. It also has 1% inaccuracy. The mesh distribution may influence the result, but we can not refine mesh after five level according to computing resources. In the previously works, most of them are use high order finite element method. If we can use high level mesh in future, we can have more accuracy result.