表列演算法生成空間填充曲線的雙向鍊結串列式區塊結構自適應網格精緻化方法

Abstract

這篇論文提出了一種新的、創新的區塊結構自適應網格生成方法。目前的自適應網格生成方法通常復雜且不靈活，使得它們難以實施和定制特定應用程序。所提出的方法通過引入一個簡單而靈活的數據結構，使用雙向鏈表（ DLL）在計算過程中存儲各個塊的信息，從而解決了這些問題。自適應網格生成的一個關鍵挑戰是維護具有不同網格細化級別的塊之間的連通性。所提出的方法通過利用面向連續的希爾伯特空間填充曲線（ SFC）來解決這個挑戰。詳細描述了一個基於表格列表的算法，概述了生成 SFC 並根據特定應用程序的要求修改它的過程。計算塊的鄰域的傳統定義已經修改，以增強整體連通性和提高算法效率。這種修改允許更凖確地表示塊之間的關係，從而提高自適應網格生成過程的凖確性。所提出的方法已成功應用於開發一個名為 SADHANA 的自適應網格框架，該框架用於解決天體物理學中遇到的時

This dissertation presents a new and innovative approach towards the development of block-structured adaptive mesh generation. The current methodologies for adaptive mesh generation are often complex and inflexible, making them difficult to implement and customize for specific applications. The proposed approach addresses these issues by introducing a simple yet flexible data structure utilizing a Doubly Linked List (DLL) to store the information of individual blocks during computation.

One of the key challenges in adaptive mesh generation is maintaining connectivity between blocks with varying levels of mesh refinement. The proposed approach addresses this challenge by utilizing a face-continuous Hilbert Space Filling Curve (SFC) to maintain connectivity. A tabulated list-based algorithm is described in detail, outlining the process of generating the SFC and modifying it as per the requirements of the specific application.

The conventional definition of the neighborhood of a computational block has been modified to enhance the overall connectivity and to improve the efficiency of the algorithm. This modification allows for a more accurate representation of the relationship between blocks, which in turn improves the accuracy of the adaptive mesh generation process.

The proposed approach has been successfully applied in the development of an adaptive mesh framework called SADHANA, which is used to solve time-dependent hyperbolic equations encountered in astrophysics. The development of SADHANA was made possible through the support of Dr. Hsien Shang from the Computational Astrophysical Sciences (CompAS) group at the Institute of Astronomy and Astrophysics, Academia Sinica (ASIAA). This support included both financial funding and access to the computational facility, without which this research would not have been possible.

In conclusion, this dissertation focuses on the novel and existing concepts developed by the current PhD candidate, which collectively have become the foundation for the initial development of the SADHANA code. The proposed approach for adaptive mesh generation is simple, flexible, and effective, making it suitable for a wide range of applications. The success of the proposed approach in the development of SADHANA serves as a testament to its potential for future use in other fields.