This work proposes a novel variational approximation of partial differential equations on moving geometries determined by explicit boundary representations. The benefits of the proposed formulation are the ability to handle large displacements of explicitly represented domain boundaries without generating body-fitted meshes and remeshing techniques. For the space discretization, we use a background mesh and an unfitted method that relies on integration on cut cells only. We perform this intersection by using clipping algorithms. To deal with the mesh movement, we pullback the equations to a reference configuration (the spatial mesh at the initial time slab times the time interval) that is constant in time. This way, the geometrical intersection algorithm is only required in 3D, another key property of the proposed scheme. At the end of the time slab, we compute the deformed mesh, intersect the deformed boundary with the background mesh, and consider an exact transfer operator between meshes to compute jump terms in the time discontinuous Galerkin integration. The transfer is also computed using geometrical intersection algorithms. We demonstrate the applicability of the method to fluid problems around rotating (2D and 3D) geometries described by oriented boundary meshes. We also provide a set of numerical experiments that show the optimal convergence of the method.

翻译：本文提出了一种新颖的变分近似方法，用于求解由显式边界表示所确定的移动几何体上的偏微分方程。该公式的优越性在于能够处理显式表示区域边界的大位移问题，而无需生成贴体网格或采用重网格技术。在空间离散化方面，我们采用背景网格和一种非拟合方法，该方法仅依赖于切割单元上的积分运算。我们通过裁剪算法实现几何相交处理。为处理网格移动问题，我们将方程拉回至一个随时间不变的参考配置（初始时间片的空间网格与时间区间的乘积）。通过这种方式，几何相交算法仅需在三维空间中执行，这是本方案的另一关键特性。在时间片结束时，我们计算变形后的网格，将变形边界与背景网格进行相交处理，并采用网格间的精确传递算子来计算时间间断伽辽金积分中的跳跃项。该传递算子同样通过几何相交算法进行计算。我们通过面向边界网格描述的旋转几何体（二维与三维）周围的流体问题，验证了该方法的适用性。同时，我们提供了一系列数值实验，证明了该方法具有最优收敛性。