High order unfitted finite element discretizations for explicit boundary representations

TL;DR

The paper tackles PDEs on domains bounded by explicit, nonlinear boundary representations from CAD, where body-fitted meshing is impractical. It develops an automatic pipeline that combines a robust intersection algorithm between high-order Bézier surface representations and a background Cartesian mesh, nonlinear trimming, surface partitioning, and Bézier-based parametrization to form cut cells and their boundaries. By leveraging Stokes-based moment-fitting quadratures and least-squares Bézier parametrizations, the method achieves optimal hp-convergence for both geometry approximation and PDE solutions, demonstrated on analytic benchmarks and real CAD geometries. The approach is designed to be highly parallelizable (cell-wise) and suitable for integration into optimization and transient/FEM workflows, with future work targeting octree AMR, distributed memory, and FSI/transient extensions.

Abstract

When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer a significant advantage in dealing with complex geometries, eliminating the need for generating unstructured body-fitted meshes. However, current unfitted finite elements on nonlinear geometries are restricted to implicit (possibly high-order) level set geometries. In this work, we introduce a novel automatic computational pipeline to approximate solutions of partial differential equations on domains defined by explicit nonlinear boundary representations. For the geometrical discretization, we propose a novel algorithm to generate quadratures for the bulk and surface integration on nonlinear polytopes required to compute all the terms in unfitted finite element methods. The algorithm relies on a nonlinear triangulation of the boundary, a kd-tree refinement of the surface cells that simplify the nonlinear intersections of surface and background cells to simple cases that are diffeomorphically equivalent to linear intersections, robust polynomial root-finding algorithms and surface parameterization techniques. We prove the correctness of the proposed algorithm. We have successfully applied this algorithm to simulate partial differential equations with unfitted finite elements on nonlinear domains described by computer-aided design models, demonstrating the robustness of the geometric algorithm and showing high-order accuracy of the overall method.

Figures (13)

• Figure 1: Example of the embedded nonlinear domain in 2D. Figure (a) presents a nonlinear oriented skin mesh $\mathcal{B}$ embedded in an active mesh $\mathcal{T}$. The intersections, computed with the techniques proposed in this work, result in the two-level partitions $\mathcal{T}^\mathrm{cut}$ and $\mathcal{B}^\mathrm{cut}$ shown in (b). These partitions are utilized for integrating unfitted formulations. It is important to note that the intersections in 2D are points that can be represented exactly. However, the intersections in 3D are trimming curves that must be approximated in general.
• Figure 2: Representation of the surface $\mathcal{B}\doteq \partial \Omega$ (see (a)), its intersection $\mathcal{B}^{\mathrm{cut}}_K \doteq \mathcal{B} \cap K$ for a background cell $K \in \mathcal{T}$ (see (b)), and the domain interior of the cell $K^\mathrm{cut} \doteq K \cap \Omega$ (see (c)). In order to compute $\mathcal{B}^{\mathrm{cut}}_K$, we identify first the subset of cells $\mathcal{B}_K \subseteq \mathcal{B}$ touching $K$ (see cells with red edges in (a)). Next, for each triangle $F \in \mathcal{B}_K$, we compute the intersection of $F \cap K$ at the reference fe, i.e., we compute $\pmb{{\phi}}_F^{-1}(F \cap K)$. Finally, we intersect $K$ with the surface portion $\mathcal{B}^\mathrm{cut}_K$ to obtain $K^\mathrm{cut}$. It is worth to note that $\mathcal{B}^{\mathrm{cut}}_K \subset \partial K^\mathrm{cut}$.
• Figure 3: Definition of intersection points. The intersection of curves and planes (see (a)) is computed through univariate root-finding methods. The surface-plane-plane intersection points in (b) also represent the intersection of surface-ine intersections. The aa critical points in (c) are defined in the reference space of $F$. These points split the surface-plane intersection curves into monotonic curves in the reference space (see an analogous reference space in Fig. \ref{['fig:refinement-rules-a']}). Both computations, surface-plane-plane intersection points and aa critical points, require solving a bivariate root system.
• Figure 4: Refinement steps of Alg. \ref{['alg:aa-refinement']}. Step 1 is represented in (a) and (b). Steps 2-4 are represented in (c)-(e), resp. Red and blue dashed lines represent the $\hat{\gamma}_f^0$ and $\hat{\gamma}_{f^\prime}^0$-curves, resp., $f,f^\prime \in \Lambda^2(K)$. Interior black lines represent the proposed partition of $\hat{F}$ for a given invariant. Solid black lines are the edges of $\Lambda^1( \hat{F})$. There are multiple possible partitions, depending on the order the intersections are processed.
• Figure 5: Representation of clipping algorithm $F\cap K$. In (a), we start with a refined face $\hat{F}_R\in \mathrm{ref}_1(\hat{F})$ such that the $\hat{\alpha}$-curves are strictly monotonic with respect to the axes in the reference space and do not intersect in $\hat{F}_R\setminus \Lambda^0(\hat{F}_R)$. Here, the $\hat{\alpha}$-curves are $\hat{\alpha}_f \in C_f(\hat{F}_R)$ (red) and $\hat{\alpha}_{f^\prime} \in C_{f^\prime}(\hat{F}_R)$ (blue) for $f,f^\prime \in \Lambda^2(K)$. Then, in (b), we generate a partition $\mathrm{ref}_2(\hat{F}_R)$ through the $\hat{\alpha}$-curves, in which we classify the faces with respect to $K$. Finally, in (c), we restrict $\mathrm{ref}_2(\hat{F}_R)$ inside $K$ (see Alg. \ref{['alg:split-connect']}).

Theorems & Definitions (19)

• Proposition 3.3
• proof
• Proposition 3.4
• proof
• Proposition 3.5
• proof
• Theorem 3.6
• proof
• Remark 3.7
• Remark 3.8