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面临诸多困难。浸入式有限元方法本质上更具鲁棒性，但通常精度较低。本文针对复杂CAD模型提出一种高效、鲁棒的高阶浸入式有限元方法。该方法基于三种自适应结构化网格：用于表示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求积网格。几何网格采用稀疏VDB（体动态B+树）网格，在物理域边界附近高度细化；有限元网格由分布在多个处理器上的八叉树森林构成；每个有限元单元内的求积网格采用自底向上构建的八叉树结构。我们在有限元网格上使用高阶拉格朗日基函数离散物理场。求积网格的分辨率确保有限元积分的计算精度，并能够按预期精度解析小孔、尖角等子网格几何特征。该方法的概念简洁性与模块化特性使其能够复用开源库：采用openVDB与p4est分别实现几何网格与有限元网格，并借助BDDCML通过区域分解并行迭代求解离散方程组。最后，通过求解复杂CAD模型域上的泊松方程（离散自由度达数千万级），验证了所提方法的效率与鲁棒性。