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.

翻译：在科学与工业问题建模中，几何体通常由计算机辅助设计软件生成的显式边界表示进行建模。非拟合（亦称嵌入式或浸入式）有限元方法在处理复杂几何体时具有显著优势，无需生成非结构化的体拟合网格。然而，当前针对非线性几何体的非拟合有限元方法仅限于隐式（可能为高阶）水平集几何。本研究提出一种新型自动化计算流程，用于在由显式非线性边界表示定义的域上逼近偏微分方程的解。针对几何离散化，我们提出一种新颖算法，用于生成非线性多面体上的体域与表面积分求积规则，以计算非拟合有限元方法中的所有项。该算法基于边界非线性三角剖分、表面单元kd树细化（将表面与背景单元的非线性交集简化为与线性交集微分同胚的简单情形）、稳健多项式求根算法及曲面参数化技术。我们证明了所提算法的正确性。通过将本算法应用于计算机辅助设计模型描述的非线性域上非拟合有限元偏微分方程模拟，成功验证了该几何算法的鲁棒性，并展示了整体方法的高阶精度。