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：在科学和工业问题的建模中，几何体通常通过计算机辅助设计软件获取的显式边界表示进行建模。非拟合（也称为嵌入或浸入）有限元方法在处理复杂几何体方面具有显著优势，无需生成非结构化贴体网格。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）水平集几何体。在本工作中，我们提出了一种新颖的自动化计算流程，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，我们设计了一种新算法，用于生成非线性多面体上的体积分和面积分求积规则，这些积分为计算非拟合有限元方法中的所有项所必需。该算法基于边界的非线性三角剖分、表面单元的kd树细化（将表面单元与背景单元的非线性相交简化为微分同胚等价于线性相交的简单情形）、稳健的多项式求根算法以及表面参数化技术。我们证明了所提算法的正确性。我们已成功将该算法应用于使用非拟合有限元方法在由计算机辅助设计模型描述的非线性域上模拟偏微分方程，验证了几何算法的稳健性，并展示了整体方法的高阶精度。