Space-time unfitted finite elements on moving explicit geometry representations

TL;DR

This work develops a space-time unfitted finite element method for PDEs on moving domains described by explicit boundary representations, avoiding 4D body-fitted meshes and remeshing. It pulls each time slab problem back to a time-extruded reference configuration, discretizes with tensor-product space-time FE spaces, and enforces boundary conditions via Nitsche while stabilizing cut cells with agfem. A key novelty is an exact inter-slab transfer mechanism based on 3D geometric intersections, enabling precise evaluation of time jumps without projecting between meshes. The approach is validated with hp-convergence tests in 2D/3D and applied to incompressible flows around rotating CAD-based geometries, demonstrating optimal convergence and robustness to large displacements. This framework eliminates remeshing bottlenecks and projection errors, paving the way for scalable simulations of moving interfaces and potential FSI applications, with future work on high-order deformation extensions and distributed-memory implementations.

Abstract

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.

Figures (11)

• Figure 1: Representation of the space-time domain $Q$ embedded within an artificial space-time domain $Q_\mathrm{art}$. The spacial domains $\Omega(t)$ and $\Omega_\mathrm{art}$ in 2D are extruded in the time dimension. The solution is computed in each time slab $J^n=(t^{n},t^{n+1})$ on the space-time domain $Q^n$, which is embedded in $Q_\mathrm{art}$.
• Figure 2: Representation of the deformation map $\pmb{\varphi}^n_h$ resulting from extending the variation of the surface map $\pmb D$ in the time slab domain $Q^n$ (or in the time slab artifical domain $Q^n_\mathrm{art}$). The undeformed configuration $\hat{Q}^n$ will be used for the fe analysis.
• Figure 3: Representation of the deformation map $\pmb{D}$ and the extended map $\pmb{\varphi}_h^n$ in the time slab $J^n=(t^{n-1},t^n)$. This example represents the external domain. The spatial deformation map $\pmb{D}(t): \mathcal{B}_h(0) \mapsto \mathcal{B}_h(t),\ t\in[0,T]$, is defined on the surface mesh, and the extended map $\pmb{\varphi}_h^n(t): \Omega^{n} \mapsto \tilde{\Omega}(t),\ t\in J^n$, is defined on the spatial domain. At $t^{n+1}$, the description of the inner boundary $\partial \Omega_I^{n+1}$ may differ across maps: $\pmb{\varphi}_h^n(t^{n+1})(\partial\Omega_I^{n}) \approx \pmb{D}(t^{n+1})( \mathcal{B} _{h}(0) )$, where $\partial\Omega_I^{n} = \pmb{D}(t^{n})( \mathcal{B}_h(0) )$. The approximation error decreases with the spatial discretization size $h$.
• Figure 4: Mesh sequence for solution transfer between time slabs. The solution obtained in $\bar{\mathcal{T}}_{}^{n-1}$ in (a) is then evaluated at $\bar{\mathcal{T}}_{}^{n}$ (c). Acting as a bridge, the intersected mesh $\bar{\mathcal{T}}_{\mathrm{int}}^{n}= \pmb{\varphi}^{n-1}_h (t^{n}) ( \bar{\mathcal{T}}^{n-1} )\cap \bar{\mathcal{T}}^{n} \cap \Omega ^{n}$ in (b) facilitates the evaluation by providing injective cell maps to both active meshes.
• Figure 5: Representation of the cell maps used in the time slab interface integration. The maps $\phi_{K_1}$ and $\phi_{K_2}$ send the reference element $\hat{K}$ to the physical space of $K_1\in\bar{\mathcal{T}}^{n-1}$ and $K_2\in\bar{\mathcal{T}}^{n}$, resp. The map $\phi_{K_\mathrm{int}}$ sends the reference element $\hat{K}_\mathrm{int}$ to the physical space of $K_\mathrm{int}\in\bar{\mathcal{T}}^{n-1}_\mathrm{int}$.