论文标题
在H1上，H（Curl）和H（Sym Curl）有限元素用于基质值卷曲问题
On H1, H(curl) and H(sym Curl) finite elements for matrix-valued Curl problems
论文作者
论文摘要
在这项工作中，我们测试了由$ [\ mathit {h}^1]^{3 \ times 3} $，$ [\ mathit {h}（\ mathrm {curl}）的有限元子空间近似的矩阵值行为。 $ \ Mathit {h}（\ Mathrm {sym} \ Mathrm {curl}）$对于线性抽象的变分问题，连接到了放松的微态模型。引入了相应的有限元素的配方，然后是数值基准和我们的结论。松弛的微态连续体模型通过用卷曲替换自由能函数中的微阶段的完整梯度来降低经典微态模型的连续性假设。这将为微阶段提供更大的解决方案空间，即$ [\ Mathit {h}（\ Mathrm {curl}）]^3 $代替经典$ [\ Mathit {h}^1]^{3 \ times 3} $。仅采用卷曲的对称部分，可以进一步削弱微科的连续性条件。如最近的作品所示，MicroDistortion的新适当空间是$ \ Mathit {H}（\ Mathrm {Sym} \ Mathrm {Curl}）$。新引入的空间为松弛的微态连续体理论带来了一种新的差分复合物。
In this work we test the numerical behaviour of matrix-valued fields approximated by finite element subspaces of $[\mathit{H}^1]^{3\times 3}$, $[\mathit{H}(\mathrm{curl})]^3$ and $\mathit{H}(\mathrm{sym}\mathrm{Curl})$ for a linear abstract variational problem connected to the relaxed micromorphic model. The formulation of the corresponding finite elements is introduced, followed by numerical benchmarks and our conclusions. The relaxed micromorphic continuum model reduces the continuity assumptions of the classical micromorphic model by replacing the full gradient of the microdistortion in the free energy functional with the Curl. This results in a larger solution space for the microdistortion, namely $[\mathit{H}(\mathrm{curl})]^3$ in place of the classical $[\mathit{H}^1]^{3\times 3}$. The continuity conditions on the microdistortion can be further weakened by taking only the symmetric part of the Curl. As shown in recent works, the new appropriate space for the microdistortion is then $\mathit{H}(\mathrm{sym}\mathrm{Curl})$. The newly introduced space gives rise to a new differential complex for the relaxed micromorphic continuum theory.
