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| 基于超穷映射的复杂域微分方程的谱元法 |
| Spectral Element Method Combined with Transfinite Mapping for Numerical Solution of Differential Equations in Complex Domains |
| 投稿时间：2024-09-18 修订日期：2024-11-23 |
| DOI： |
| 中文关键词: 谱元法 超穷映射 高精度 快速收敛 h-p型 |
| 英文关键词:Spectral Element Method(（SEM） Transfinite Mapping（TMT）, High Accuracy, Fast Convergence, h-p Version |
| 基金项目: |
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| 摘要点击次数: 23 |
| 全文下载次数: 0 |
| 中文摘要: |
| 本文结合谱元法（Spectral Element Method, SEM）与超穷映射技术（Transfinite Mapping Technique, TMT），探讨其在复杂域微分方程数值计算中的应用，并提出相应的算法实现方案。该方法的核心步骤包括正交基函数的选择与谱元法系数矩阵的构建，其中TMT用于复杂域边界的精确参数化。与传统有限元法相比，该方法在高阶谱元应用中能够采用更大的几何单元，并在复杂边界拟合方面表现出显著优势。通过地震荷载作用下重力坝结构的动力响应分析及自由移动边界渗流问题两个工程实例，系统验证了基于TMT的高阶谱元法的优越性与计算效率。研究表明，在处理自由表面或移动边界等复杂边值问题时，该方法能够以较低的计算成本实现与有限元法相当的计算精度。 |
| 英文摘要: |
| This study integrates the spectral element method (SEM) with the transfinite mapping technique (TMT) to address the numerical computation of differential equations in complex domains. A detailed numerical implementation of the algorithm is proposed, encompassing the selection of orthogonal basis functions and the construction of the SEM coefficient matrix. SEM leverages spectral methods as its theoretical foundation, while TMT enables the parameterization of complex domain boundaries. Compared to traditional finite element methods, this approach allows for the use of larger geometric elements and improved boundary fitting when employing high-order spectral elements. The efficacy and applicability of the proposed method are demonstrated through two engineering examples: the dynamic response of a gravity dam under seismic loads and a free-moving boundary seepage problem. Notably, for complex boundary value problems, such as those involving free surfaces or moving boundaries, the method achieves computational accuracy comparable to finite element methods with significantly reduced computational effort. |
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