In high-order and high-dimensional finite elements, ill-conditioned nodal distributions are often computationally cost-prohibitive. As a result, uniform distributions quickly fall apart. For tensor-product like elements, Gauss-Legendre-Lobatto (GLL) nodal distributions are often used as a substitute. Besides these, other efficient nodal distributions are difficult to create due to a desired symmetry within elements and conformity with neighboring elements. In this paper, we provide a general framework to construct symmetric, well-conditioned, cross-element compatible nodal distributions which can be used for high-order and high-dimensional finite elements. Starting from the inherent symmetries in any potential element, the framework is used to build up nodal groups in a structured and efficient manner utilizing the natural coordinates of each element, while ensuring nodes stay within the elements. By constructing constrained symmetry groups, the vertices, edges, and faces, of all elements are required to conform to their respective lower-dimensional distributions. Optimizing over these groups yields the desired optimized nodal distributions. We demonstrate the strength of this framework by creating and comparing optimized nodal distributions with GLL distributions (in elements such as the line, quadrilateral, and hexahedron), and its robustness by generating optimized nodal distributions for otherwise difficult elements (such as the triangle, tetrahedron, and triangular prism).

翻译：在高阶高维有限元中，病态节点分布通常会导致计算成本过高。因此，均匀分布难以适用。对于张量积型单元，常采用高斯-勒让德-洛巴托（GLL）节点分布作为替代方案。除此外，由于单元内部需要保持对称性且需与相邻单元保持一致性，难以构造其他高效的节点分布。本文提出一个通用框架，可构造适用于高阶高维有限元的对称、良态、跨单元兼容的节点分布。该框架从任意潜在单元的固有对称性出发，利用各单元的自然坐标，以结构化且高效的方式构建节点组，同时确保节点始终位于单元内部。通过构造受约束的对称群，所有单元的顶点、边和面均需满足其各自低维分布的约束。对这些群组进行优化即可得到所需的最优节点分布。我们通过构造并对比最优节点分布与GLL分布（在线段、四边形和六面体等单元中）验证了该框架的有效性，并通过对三角形、四面体和三棱柱等难处理单元生成最优节点分布证明了其鲁棒性。