使用調適網格加密功能之有限元素平行化三維Poisson-Boltzmann Equation程式之發展與驗證

Abstract

探討微粒-微粒與微粒-平板之間的相互作用力在生物化學的領域是相當重要的。在假設電解液為平衡的狀態下，可以經由著名的Poisson-Boltzmann方程式電雙層理論得到帶電物體的勢能分佈。因此，我們決定開發以有限元素法三維平行化非線性Poisson-Boltzmann方程式程式(PPBS)搭配平行化調適網格加加密功能程式(PAMR)，並且完成驗證的工作。本論文之研究分成兩大主軸，分別敘述如下： 第一部分，以非線性Poisson-Boltzmann方程式採用非結構性四面體網格之葛勒金有限元素法來完成程式開發，包括了典型的一階與二階的形狀函數元素。因為使用了nodal quadrature的技巧，使得原先的牛頓法Jacobian矩陣僅剩下對角線，以此擬牛頓疊代法來處理非線性項矩陣的部份，有助於平行化程式的完成。接下來，則使用SBS的技巧搭配平行的共軛梯度法來處理線性矩陣方程組。完成的程式以兩個範例來驗證，第一個驗證的範例是帶電球體的勢能分佈，所得答案與解析解、近似解作一比較，結果相當正確。第二個驗證範例則是兩個帶電球體在圓柱孔內，結果顯示與先前所發表的論文結果相符。以上兩個驗證範例證明了平行化Poisson-Boltzmann方程式程式的開發完成。此外，平行化效能使用國家高速網路與計算中心的HP 叢集式電腦系統來做驗證，測試一個帶電球體在圓柱孔內的範例，結果顯示使用了32顆CPU時仍有76.2%的平行效能。在第一部份的最後，我們使用了二階形狀函數元素，所得結果證明了比一階形狀函數元素要來的準確。 第二部份，主要是發展一個以非結構性四面體網格為主，採用h-切割為基礎之分散式記憶體動態領域分解的PAMR程式，而資料結構則使用了較節省記憶體空間之cell-base方式來記錄網格的資訊，可以同時運用在node-base與cell-base的數值方法上。一般的步驟包括了一個分為八個網格的等向性切割，然後以非等向切割搭配網格品質控制將hanging node有效率地移除。我們測試PAMR在叢集式電腦上最多64顆CPU的平行化效能，結果呈現在32顆CPU時仍有N1.5的效能(N為CPU個數)。接著我們將PPBES與PAMR做一個結合，並使用a posteriori的誤差評估方法，驗證兩個帶電球體在圓柱孔內，結果證明了使用PAMR可以增加PPBES解的準確度。最後，利用PPBES-PAMR的程式模擬兩個帶電球體靠近帶電平板的相互影響力之分析，與實驗結果相比較，若考慮實驗本身的不確定因素與誤差，模擬結果與實驗結果有相當程度的符合，這是目前已知與實驗結果相比較之最佳模擬結果。

Understanding of the interactive particle-particle and particle-wall forces in colloidal systems (electrolytes) plays a very important role in bio-chemistry related research. By assuming equilibrium in the electrolytes, the potential distribution with a very thin electrical double layer near the charged object can be well described by the well-known Poission-Boltzmann equation. Thus, a parallelized 3-D nonlinear Poisson-Boltzmann equation solver (PPBES) using finite element method (FEM) with parallel adaptive mesh refinement (PAMR) is proposed and verified. In this thesis, the research is divided into two phases, which are described as follows. In the first phase, the nonlinear Poisson-Boltzmann equation is discretized using Galerkin finite element method with unstructured tetrahedral mesh. Interpolation within a typical element includes the first-order and second-order shape functions. Inexact Newton iterative scheme is used to solve the nonlinear matrix equation resulting from the FE discretization. Jacobian matrix resulting from the Newton iterative scheme is diagonalized using nodal quadrature, which further facilitates the easier parallel implementation. A parallel conjugate gradient (CG) method with a subdomain-by-subdomain (SBS) scheme is then used to solve the linear algebraic equation each iterative step. Completed code is verified using two typical examples. The first validated case is the potential distribution around a charged sphere. Excellent agreement of the simulation results with analytical (linearized case) and approximate (nonlinear case) solutions are obtained. The second validated case is the interaction between like-charged spheres within a cylindrical pore. Results show that the agreement between the present simulation and previous results are excellent. The above two typical simulations validate the present implementation of the PPBES. Further, the parallel performance is studied on a HP PC-cluster system at NCHC using a test case with a charged sphere confined in a cylindrical pore. Results show that 76.2% of parallel efficiency can be reached at processors of 32. Also FE discretization using the second-order shape function is demonstrated to be more accurate than using the first-order shape function at the end of this phase. In the second phase, an h-refinement based PAMR scheme for an unstructured tetrahedral mesh using dynamic domain decomposition on a memory-distributed machine is developed and tested in detail. A memory-saving cell-based data structure is designed such that the resulting mesh information can be readily utilized in both node- or cell-based numerical methods. The general procedures include isotropic refinement from one parent cell into eight child cells and then followed by anisotropic refinement, which effectively removes the hanging nodes, with a simple mesh-quality control scheme. Parallel performance of this PAMR is studied on a PC-cluster system up to 64 processors. Results show that the parallel speedup scales approximately as N1.5 up to 32 processors, where N is the number of processors. Then, procedure of coupling the PPBES with the PAMR using a posteriori error estimator is presented and verified using a test case with two like-charged spheres in a cylindrical pore. Results show that PAMR can systematically increase the solution accuracy of the PPBES. Finally, the coupled PPBES-PAMR code is used to simulate the interactive force between two like-charged spheres near a charged planar wall. Results are in excellent agreement with experimental data considering the experimental uncertainties, which is the first simulation in the literature to the best knowledge of the author.