Worsey-Farin 细分上的离散弹性正合复形
Discrete Elasticity Exact Sequences on Worsey-Farin Splits
AI总结：
本文在三维 Worsey-Farin 细分上构造离散弹性正合复形，给出位移、应变、应力和载荷有限元空间及唯一可解自由度与交换投影，并得到无顶点或边自由度的最低阶强对称应力元。
摘要：
我们在三维 Worsey-Farin 细分上构造了协调有限元弹性复形。位移、应变、应力和载荷空间通过表示变形、不相容性和散度的微分算子在弹性复形中相连接。对于这些分量空间中的每一个，我们都给出了 Worsey-Farin 网格上相应的有限元空间。我们为这些有限元建立了唯一可解自由度，这也产生了光滑函数上的交换（上链）投影。这些复形中空间的一个显著特点是在网格的子单纯形处不存在外部超光滑性。值得注意的是，该复形产生了首个在三维中没有顶点或边自由度的（强）对称应力有限元。此外，最低阶应力空间仅使用分段线性函数，这是应力空间可行的最低多项式次数。
英文摘要：
We construct conforming finite element elasticity complexes on Worsey-Farin splits in three dimensions. Spaces for displacement, strain, stress, and the load are connected in the elasticity complex through the differential operators representing deformation, incompatibility, and divergence. For each of these component spaces, a corresponding finite element space on Worsey-Farin meshes is exhibited. Unisolvent degrees of freedom are developed for these finite elements, which also yields commuting (cochain) projections on smooth functions. A distinctive feature of the spaces in these complexes is the lack of extrinsic supersmoothness at subsimplices of the mesh. Notably, the complex yields the first (strongly) symmetric stress finite element with no vertex or edge degrees of freedom in three dimensions. Moreover, the lowest order stress space uses only piecewise linear functions which is the lowest feasible polynomial degree for the stress space.
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