高阶过渡单元：xNy单元概念——第一部分：静力学
High order transition elements: The xNy-element concept -- Part I: Statics
AI总结：
本文提出兼容的xNy高阶过渡单元，基于超限映射耦合不同尺寸、形函数与多项式阶次的单元，实现独立h/p网格连接，并通过数值算例验证其收敛性能。
摘要：
先进的过渡单元在有限元法（FEM）许多需要局部网格细化的应用中至关重要。考虑到解中存在奇异性的问题，必须采用自适应hp细化过程。即使在今天，这仍是一项极具挑战性的任务，尤其是在仅使用四边形/六面体单元并因此遇到悬节点问题时。然而，与三角形/四面体单元相比，这些单元类型因精度更高而在计算力学中更受青睐。因此，我们提出了一种兼容的过渡单元——xNy单元，它具备耦合不同单元类型的能力。相邻单元可以具有不同的单元尺寸、形函数类型和多项式阶次。由此，可以组合独立细化的h网格和p网格。该方法基于超限映射概念，构成了对pNh单元概念的扩展/推广。通过若干数值算例，详细研究了收敛行为，并数值确定了渐近收敛速率。总体而言，研究发现所提出方法在局部网格细化过程中给出了非常有前景的结果。
英文摘要：
Advanced transition elements are of utmost importance in many applications of the finite element method (FEM) where a local mesh refinement is required. Considering problems that exhibit singularities in the solution, an adaptive hp-refinement procedure must be applied. Even today, this is a very demanding task especially if only quadrilateral/hexahedral elements are deployed and consequently the hanging nodes problem is encountered. These element types, are, however, favored in computational mechanics due to the improved accuracy compared to triangular/tetrahedral elements. Therefore, we propose a compatible transition element - xNy-element - which provides the capability of coupling different element types. The adjacent elements can exhibit different element sizes, shape function types, and polynomial orders. Thus, it is possible to combine independently refined h- and p-meshes. The approach is based on the transfinite mapping concept and constitutes an extension/generalization of the pNh-element concept. By means of several numerical examples, the convergence behavior is investigated in detail, and the asymptotic rates of convergence are determined numerically. Overall, it is found that the proposed approach provides very promising results for local mesh refinement procedures.
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