Bridging scales - from Quantum Mechanics to Continuum Mechanics. A Finite Element approach.

本提案考虑了电弹性问题的并行耦合量子力学（QM）-连续力学（CM）方法。尽管人们已经做出努力来弥合物质的不同描述，但许多问题尚未得到回答。首先，一个有效的有限元（FE）为基础的解决方案的密度泛函理论（DFT）的Kohn-Sham（KS）方程的方法将进一步发展。基于非局部赝势的有限元解的h-自适应性，以及结构优化过程中的网格变换和变形映射的制定是有待研究的主要课题。应该注意的是，到目前为止，还没有DFT方法的开源实现，它使用FE基础并提供hp细化能力。FE基由于其完备性、精细化的可能性以及基于区域分解的良好极化性质而在DFT理论中非常有吸引力。其次，QM量将与它们的CM对应物（例如，位移、变形梯度、Piola应力、极化等）相关。这将使用拉格朗日配置中的平均值来实现。为此，需要对KS方程的基于FE的解进行完全控制。然后，该程序进行测试的一个代表性的数值例子-弯曲的单壁碳纳米管。在CM方面，将利用表面增强连续理论来适当地捕获表面效应。应该指出的是，虽然有几个理论工作存在于这个问题上，没有数值的尝试，以检查其有效性的测试示例。最后，基于不同公式之间的对应关系，提出了一种并行耦合的QM-CM方法。耦合将以交错的方式实现，即QM和CM问题将通过它们之间的适当信息交换来迭代求解。将考虑石墨烯片中裂纹扩展的测试问题。作为本项目的长期目标，将开发电弹性问题的耦合策略。据我所知，没有QM-CM耦合方法能够处理电弹性问题。

The concurrently coupled Quantum Mechanics (QM) - Continuum Mechanics (CM) approach for electro-elastic problems is considered in this proposal. Despite the fact that efforts have been made to bridge different description of matter, many questions are yet to be answered. First, an efficient Finite Element (FE)-based solution approach to the Kohn-Sham (KS) equations of Density Functional Theory (DFT) will be further developed. The h-adaptivity in the FE-based solution with non-local pseudo-potentials, as well as the mesh transformation during the structural optimization and formulation of the deformation map are the main topics to be studied. It should be noted that until now there exists no open-source implementation of the DFT approach which uses a FE basis and provides hp-refinement capabilities. A FE basis is very attractive in the context of the DFT theory because of its completeness, refinement possibility as well as good polarization properties based on domain decomposition. Second, QM quantities will be related to their CM counterparts (e.g. displacements, deformation gradient, the Piola stress, polarization, etc). This will be achieved using averaging in the Lagrangian configuration. To that end the full control over a FE-based solution of the KS equations is required. The procedure is then to be tested on a representative numerical example - bending of a single wall carbon nanotube. On the CM side, the surface-enhanced continuum theory will be utilized to properly capture surface effects. It should be noted that although several theoretical works exist on this matter, no numerical attempts have been made to check their validity on test examples. Lastly, based on the correspondence between different formulations, a concurrently coupled QM-CM method will be proposed. Coupling will be achieved in a staggered way, i.e. QM and CM problems will be solved iteratively with a proper exchange of information between them. A test-problem of crack propagation in a graphene sheet will be considered. As a long term goal of the project, coupling strategies for electro-elastic problems will be developed. To the best of my knowledge, non of the QM-CM coupling method is capable to handle electro-elastic problems.