Computational analysis with the finite element method requires geometrically accurate meshes. It is well known that high-order meshes can accurately capture curved surfaces with fewer degrees of freedom in comparison to low-order meshes. Existing techniques for high-order mesh generation typically output meshes with same polynomial order for all elements. However, high order elements away from curvilinear boundaries or interfaces increase the computational cost of the simulation without increasing geometric accuracy. In prior work, we have presented one such approach for generating body-fitted uniform-order meshes that takes a given mesh and morphs it to align with the surface of interest prescribed as the zero isocontour of a level-set function. We extend this method to generate mixed-order meshes such that curved surfaces of the domain are discretized with high-order elements, while low-order elements are used elsewhere. Numerical experiments demonstrate the robustness of the approach and show that it can be used to generate mixed-order meshes that are much more efficient than high uniform-order meshes. The proposed approach is purely algebraic, and extends to different types of elements (quadrilaterals/triangles/tetrahedron/hexahedra) in two- and three-dimensions.

翻译：有限元方法中的计算分析需要几何精确的网格。众所周知，与低阶网格相比，高阶网格能够以更少的自由度准确捕捉曲面。现有高阶网格生成技术通常为所有单元生成相同多项式阶数的网格。然而，在远离曲边界或界面的区域使用高阶单元会增加计算成本，而不会提高几何精度。在先前工作中，我们提出了一种生成体贴均匀阶网格的方法，该方法将给定网格变形以与指定为水平集函数零等值面的目标表面对齐。我们扩展该方法以生成混合阶网格，使域的曲面部分用高阶单元离散，而其他区域使用低阶单元。数值实验证明了该方法的鲁棒性，并表明其能生成比高阶均匀网格高效得多的混合阶网格。所提方法是纯代数方法，可推广至二维和三维中的不同单元类型（四边形/三角形/四面体/六面体）。