求解移动边界问题的自适应四叉树-比例边界有限元方法

The numerical solutions for moving boundary problems are of significant scientific value in both theoretical fundamental research and engineering practice. In order to solve moving boundary problems more reliably and effectively, this project presents a new Adaptive Quadtree-Scaled Boundary Finite Element Method, by considering both 2D and 3D phase change heat transfer problems as research background. The advantage of the proposed method is: (1) The effective quadtree mesh is employed, and the scaled boundary method is used to build polygon elements, so that mesh distortion can be avoided as no mesh deformation is required, and no enrichment function is not required like in the XFEM neither, and this quadtree scaled boundary element is convenient to deal with 3D cases; (2) Adaptive computations in both space and time domain are employed to establish an automatic numerical model, which can satisfy a prescribed accuracy requirement at minimum computational cost. In addition, several new interpolations are employed in building scaled boundary elements to increase their accuracy in this project, and the performance of both explicit front-tracking and level set method will be tested in the simulation of interface evolution. The outcome of this fundamental research will also contribute to the development in the broader area of moving boundary problems, such as fluid-structure interaction and crack growth. It appears that currently there is very few work directly relative to this program. Because of its importance of research object and innovation of the proposed method, this project will have both important scientific value and good engineering application prospect.

移动边界问题的数值求解具有重要的理论研究和工程应用价值。为了更加高效和可靠地求解移动边界问题，本项目以二维/三维相变传热问题为研究背景，拟发展一种新的自适应四叉树-比例边界有限元方法，其显著优点为：（1）基于高效的四叉树网格，借助比例边界方法构造多边形单元，不需要移动网格，避免了网格畸变带来的精度和效率降低，也不需要构造XFEM富集函数，并且便于处理三维问题；（2）采用空间-时域自适应计算，借助四叉树网格加密和时域分段自适应算法，自动满足计算精度要求，具有较高的计算效率。此外，本项目还将采用基于新型插值函数的比例边界方法以提高单元精度，并探讨显式追踪和水平集方法在界面演化模拟中的表现。项目相关成果有望扩展应用于流体-结构相互作用、裂纹扩展等问题中。目前在移动边界问题研究中很少有类似本项目的直接相关报道。鉴于研究对象的重要性和算法的创新性，本项目将具有重要的科学价值和良好的工程应用前景。

移动边界问题的数值求解具有重要的理论研究和工程应用价值。为了更加高效和可靠地求解移动边界问题，本项目以二维/三维相变传热问题为研究背景，发展了一种新的自适应四叉树-比例边界有限元方法，其显著优点为：（1）基于高效的四叉树网格，借助比例边界方法构造多边形单元，不需要移动网格，避免了网格畸变带来的精度和效率降低，也不需要构造XFEM富集函数，并且便于处理三维问题；（2）采用空间-时域自适应计算，借助四叉树网格加密和时域分段自适应算法，自动满足计算精度要求，具有较高的计算效率。本项目发展了一种新型的基于单元状态的自适应加密准则，能够方便地应用于自适应计算中。计算结果所示，自适应计算通过加密很少的相变区域网格就能较大幅度地提高初始网格的计算精度，其计算时间仅为完全加密的28%。本项目所建立的计算模型应用于相变储能材料的分析和设计中，同时，将项目所提的自适应四叉树-比例边界元方法推广应用于二维/三维非均质材料等效传热/粘弹性参数预测中，能够方便地处理夹杂随机分布带来的材料边界处的网格变化。此外，为了拓展本项目研究成果的应用，发展了针对不确定性分析的模糊比例边界元方法，处理第三类边界条件的比例边界元方法，以及面向非线性问题应用的非局部损伤力学模型。项目研究成果在Numerical Heat Transfer, Part B，International Journal of Heat and Mass Transfer, Acta Mechanica Solida Sinica等SCI期刊发表，基于本项目申请获批国家自然基金委-欧盟委员会国际合作项目一项。

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