AI 中文总结
研究在移动界面模拟等中对演化边界区域重复椭圆求解效率低的问题，核心方法是为非拟合晶格格林函数方法开发静态-动态边界约简，主要贡献是实现并验证针对多种变化障碍物的更新策略。
AI 中文摘要
在移动界面模拟、设计和反应导航中，会出现对具有演化边界的区域进行重复椭圆求解的情况。即使固定笛卡尔网格可避免重新网格化，但为每个配置重建所有边界相互作用会限制重复求解的效率。我们为规定的移动平面区域上的非拟合晶格格林函数方法开发了一种静态-动态边界约简。该公式通过格林表示将问题简化为边界支撑的未知数。其构建顺序与常规相反，先离散笛卡尔算子再形成格林表示，避免了边界网格和奇异求积。该方法还分离了与固定几何相关的相互作用和受运动影响的相互作用，在整个模拟中重用不变部分，仅更新涉及变化边界的耦合。在真实界面交点处施加边界条件，重用晶格核数据，并通过快速正弦变换求解器重建内部场。主要贡献是一种针对平移、变形、出现和拓扑变化的障碍物的已实现且经过验证的更新策略。
英文摘要
Repeated elliptic solves on domains with evolving boundaries arise in moving-interface simulation, design, and reactive navigation. Even when a fixed Cartesian grid avoids remeshing, rebuilding all boundary interactions for every configuration can limit the efficiency of repeated solves. We develop a static--dynamic boundary reduction for an unfitted lattice Green's function method on prescribed moving planar domains. Like boundary integral and boundary element methods, the formulation reduces the problem to boundary-supported unknowns through a Green representation. Its construction, however, reverses the usual order: the Cartesian operator is discretized before the Green representation is formed, rather than representing the continuous problem first and then discretizing the boundary. This discretize-then-represent viewpoint avoids boundary meshes and singular quadrature. The method also separates interactions associated with stationary geometry from those affected by motion, reuses the invariant part throughout a simulation, and updates only couplings involving the changing boundary. Boundary conditions are imposed at true interface intersections, lattice-kernel data are reused, and the interior field is reconstructed by a fast sine-transform solver. The principal contribution is an implemented and validated update strategy for translating, deforming, appearing, and topology-changing obstacles.
