We present a numerical method, based on a tensor order parameter description of a nematic phase, that allows fully anisotropic elasticity. Our method thus extends the Landau-de Gennes Q-tensor theory to anisotropic phases. A microscopic model of the nematogen is introduced (the Maier-Saupe potential in the case discussed in this paper), combined with a constraint on eigenvalue bounds of Q. This ensures a physically valid order parameter Q (i.e., the eigenvalue bounds are maintained), while allowing for more general gradient energy densities that can include cubic nonlinearities, and therefore elastic anisotropy. We demonstrate the method in two specific two dimensional examples in which the Landau-de Gennes model including elastic anisotropy is known to fail, as well as in three dimensions for the cases of a hedgehog point defect, a disclination line, and a disclination ring. The details of the numerical implementation are also discussed.

我们提出了一种基于向列相的张量阶参数描述的数值方法，该方法允许完全各向异性的弹性。因此，我们的方法将Landau-de Gennes Q张量理论扩展到各向异性相。引入了线虫的微观模型（本文讨论的情况为Maier-Saupe势），并结合了Q的特征值范围的约束。这样可以确保物理上有效的订单参数Q（即保持特征值边界），同时允许更一般的梯度能量密度，其中可以包括立方非线性，因此也包括弹性各向异性。我们在两个特定的二维示例（其中包括弹性各向异性的Landau-de Gennes模型）已知失败的两个特定二维示例中，以及在刺猬点缺陷，旋错线和旋错环的情况下的三个维中证明了该方法。还讨论了数值实现的细节。