An unfitted boundary algebraic equation method with static-dynamic reduction for evolving implicit geometries
arXiv:2607.14808
The paper presents an unfitted lattice Green's function method with a static‑dynamic reduction that efficiently solves elliptic problems on domains with moving or changing boundaries by reusing the stationary part of the computation and updating only the boundary interactions.
Abstract
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.