松弛微观模型的高阶Bernstein-Bézier和Nédélec有限元
Higher order Bernstein-Bézier and Nédélec finite elements for the relaxed micromorphic model
摘要：
松弛微观模型是一个广义连续模型，在空间$X = [H^1]^3 \times [H(\textrm{curl})]^3$中是适定的。因此，有限元公式的模型依赖于$H^1$ -符合子空间和Nédélec元素的离散解决方案相应的变分问题。这项工作适用于最近推出的多面体模板方法的建设Nédélec元素。这是结合Bernstein-Bézier多项式和对偶数来完成的，以便计算模型的hp-FEM解。伯恩斯坦-贝塞尔多项式允许在组装过程中的最佳复杂性，因为它们自然分解成单变量伯恩斯坦基函数。在这项工作中，这一特点是进一步扩大了使用对偶数，以计算其值和他们的衍生物同时进行。应用多面体模板方法来构造Nédélec基函数，使它们能够直接继承底层Bernstein-Bézier基的最佳复杂性。我们介绍了Bernstein-Bézier基及其对单变量Bernstein基函数的因式分解，通过对偶数自动微分的原理，以及基于Bernstein-Bézier多项式和多面体模板方法的Nédélec元素的详细构造。这是补充与相应的技术嵌入狄利克雷边界条件，强调一致的耦合条件。在放松的微观模型的例子中示出的元素的性能。
英文摘要：
The relaxed micromorphic model is a generalized continuum model that is well-posed in the space $X = [H^1]^3 \times [H(\textrm{curl})]^3$. Consequently, finite element formulations of the model rely on $H^1$-conforming subspaces and Nédélec elements for discrete solutions of the corresponding variational problem. This work applies the recently introduced polytopal template methodology for the construction of Nédélec elements. This is done in conjunction with Bernstein-Bézier polynomials and dual numbers in order to compute hp-FEM solutions of the model. Bernstein-Bézier polynomials allow for optimal complexity in the assembly procedure due to their natural factorization into univariate Bernstein base functions. In this work, this characteristic is further augmented by the use of dual numbers in order to compute their values and their derivatives simultaneously. The application of the polytopal template methodology for the construction of the Nédélec base functions allows them to directly inherit the optimal complexity of the underlying Bernstein-Bézier basis. We introduce the Bernstein-Bézier basis along with its factorization to univariate Bernstein base functions, the principle of automatic differentiation via dual numbers and a detailed construction of Nédélec elements based on Bernstein-Bézier polynomials with the polytopal template methodology. This is complemented with a corresponding technique to embed Dirichlet boundary conditions, with emphasis on the consistent coupling condition. The performance of the elements is shown in examples of the relaxed micromorphic model.
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