The automated finite element analysis of complex CAD models using boundary-fitted meshes is rife with difficulties. Immersed finite element methods are intrinsically more robust but usually less accurate. In this work, we introduce an efficient, robust, high-order immersed finite element method for complex CAD models. Our approach relies on three adaptive structured grids: a geometry grid for representing the implicit geometry, a finite element grid for discretising physical fields and a quadrature grid for evaluating the finite element integrals. The geometry grid is a sparse VDB (Volumetric Dynamic B+ tree) grid that is highly refined close to physical domain boundaries. The finite element grid consists of a forest of octree grids distributed over several processors, and the quadrature grid in each finite element cell is an octree grid constructed in a bottom-up fashion. We discretise physical fields on the finite element grid using high-order Lagrange basis functions. The resolution of the quadrature grid ensures that finite element integrals are evaluated with sufficient accuracy and that any sub-grid geometric features, like small holes or corners, are resolved up to a desired resolution. The conceptual simplicity and modularity of our approach make it possible to reuse open-source libraries, i.e. openVDB and p4est for implementing the geometry and finite element grids, respectively, and BDDCML for iteratively solving the discrete systems of equations in parallel using domain decomposition. We demonstrate the efficiency and robustness of the proposed approach by solving the Poisson equation on domains given by complex CAD models and discretised with tens of millions of degrees of freedom.

翻译：采用边界拟合网格对复杂CAD模型进行自动化有限元分析存在诸多困难。浸入有限元方法本质上更为鲁棒，但通常精度较低。本文提出一种针对复杂CAD模型的高效、鲁棒、高阶浸入有限元方法。该方法基于三重自适应结构化网格：用于表征隐式几何的几何网格、用于离散物理场的有限元网格，以及用于计算有限元积分的积分网格。几何网格采用稀疏VDB（体素动态B+树）网格，在物理域边界附近进行高度细化。有限元网格由分布在多个处理器上的八叉树网格森林构成，每个有限元单元内的积分网格是以自底向上方式构建的八叉树网格。我们使用高阶拉格朗日基函数在有限元网格上离散物理场。积分网格的分辨率确保有限元积分能以足够精度计算，并能解析任何亚网格几何特征（如小孔或尖角），直至达到所需分辨率。该方法概念简洁、模块化程度高，使得复用开源库成为可能：例如分别采用openVDB和p4est实现几何网格与有限元网格，并利用BDDCML通过区域分解并行迭代求解离散方程组。通过在由复杂CAD模型给定、且离散后具有数千万自由度的求解域上求解泊松方程，我们验证了所提方法的高效性与鲁棒性。