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：本文提出了一种新的变分近似方法，用于处理由显式边界表示确定的移动几何上的偏微分方程。该公式的优势在于能够处理显式表示域边界的大位移，而无需生成体拟合网格和重网格技术。对于空间离散化，我们使用背景网格和一种仅依赖于切割单元积分的非拟合方法。我们通过裁剪算法实现这种交集计算。为了处理网格移动，我们将方程拉回到一个在时间上恒定的参考构型（初始时间片上的空间网格乘以时间区间）。这样，几何交集算法仅在三维空间中被需要，这是所提方案的另一个关键特性。在时间片结束时，我们计算变形网格，将变形边界与背景网格相交，并考虑网格间的精确转移算子，以计算时间间断伽辽金积分中的跳跃项。转移过程也通过几何交集算法计算。我们展示了该方法在由定向边界网格描述的旋转几何（二维和三维）周围流体问题中的应用可行性。我们还提供了一组数值实验，展示了该方法的最优收敛性。