分子/连续耦合多尺度方法是微纳米力学研究中发展起来的一种重要数值方法，相较于经典的分子动力学和有限元等适用于特定尺度范围的模拟方法，它通过耦合不同尺度算法，以达到兼顾计算精度和计算效率的目的。至今为止，分子/连续耦合多尺度方法已经得到了广泛的研究和应用，但对于方法的一些关键问题，如分子/连续区随微观缺陷演化的自适应划分、从二维体系向三维扩展、大规模并行、提高迭代过程的计算效率等，仍有待进一步的探索和改进。本文以耦合分子/集团统计热力学（HMCST）方法为基础，针对多尺度耦合的自适应算法和能量极小化算法开展深入研究，以期提高其计算精度和计算效率，进一步拓展多尺度计算方法的可靠性和应用范围。

本文首先针对分子/集团统计热力学多尺度方法的理论框架和三种不同的表象方法：分子统计热力学（MST）、集团统计热力学（CST）以及HMCST方法，构建了多尺度计算流程，完善了HMCST通用并行计算程序包。

基于此，本文提出一种新的分子/连续耦合的自适应分区算法。该算法以不同区域能量相对误差作为切换准则，通过连续区向分子区进行逐渐切换的方式，实现了分子/连续区随位错扩展的自适应演化计算。相较之前文献中的方法，该算法无需进行复杂的网格细化等处理，计算简单高效；且可以方便地从二维拓展到三维问题。在此基础上，进一步提出自适应计算的并行求解方案，对于分子和连续表象区分别采用不同的并行分解方法，以保障并行效率。结合HMCST多尺度方法，将新的自适应算法应用于二维和三维纳米压入计算，结果表明，该自适应算法可以有效捕捉位错等缺陷的扩展过程，具有良好的可靠性和拓展性。

进一步，本文研究并提出了适用于多尺度耦合系统的能量极小化改进算法。通过三维纳米压入算例，比较了CG、LBFGS和FIRE三种不同能量极小化算法下HMCST和MST所需的计算代价，发现多尺度体系能量极小化存在不同尺度区域收敛步调不一致的问题，由此带来明显更多的迭代步数和更长的计算时间。对此，本文基于原子和结点振荡频率相协调的理论方法，提出了一种改进的FIRE算法，获得原子和结点的等效质量关系，可确定最优的计算参数。同时，提出修正加速参数和迭代格式两种用于FIRE算法的改进策略，进一步提高多尺度计算的能量极小化计算效率。自适应HMCST计算结果表明，改进型的FIRE算法，相比传统的CG和LBFGS算法，计算效率分别提高33和31倍。这一研究成果不仅指出了多尺度计算中能量极小化算法的重要性，也为进一步提高多尺度方法的计算效率和可靠性提供了重要参考。

Molecule/continuum coupling multiscale method is a type of important numerical simulation methods to study micro-nanoscale mechanical behaviors. It is proposed to balance the computational accuracy and efficiency through coupling algorithms on different scales compared with the simulation methods on a single scale such as the classical molecular dynamics (MD) and finite element method (FEM). So far, the molecule/continuum coupling multiscale method has been widely studied and applied in practice, yet there are some key issues, including adaptivity with the evolution of microscopic defects, extension from 2D to 3D systems, large-scale parallel computation and improvement of computational efficiency during iteration process, that still need to be further discussed and promoted. Based on the hybrid molecular/cluster statistical thermodynamics (HMCST) proposed by our group, this paper mainly focused on the adaptivity and energy minimization algorithm, in order to improve the accuracy and efficiency as well as promote the reliability and universality of multiscale simulations.

First of all, for the proposed molecular cluster statistical thermodynamics (MCST) multiscale framework including molecular statistical thermodynamics (MST), cluster statistical thermodynamics (CST) and HMCST with different representations, this paper set up the computational process and implement the homemade program package for multiscale simulations.

On this basis, a new adaptive algorithm for molecule/continuum coupling methods was put forward. An adaptive criterion based on the standard deviation of atomic energy in different regions was used. Moreover, we adopted a transition scheme to realize the switching from continuum to molecule regions with the propagation of dislocations. Compared with the methods in previous literature, it is not required to carry out the complex mesh refinement in this adaptive algorithm, thus the new algorithm is more simple; and it is also easy to extend from 2D to 3D systems. To further improve the efficiency, the scheme of parallel computations for multiscale simulations was put forward, in which different decomposition methods are used in molecule and continuum regions. This adaptive algorithm used in HMCST was applied to 2D and 3D nanoindentations. Compared with the results of MST and non-adaptive HMCST, the adaptive HMCST performs well in predicting the propagations of dislocations in crystals, thus it has high reliability and practicability.

Furthermore, an improved energy minimization algorithm was proposed for multiscale systems. By comparing the computational costs of three energy minimization algorithms: conjugate gradient (CG), limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) and fast inertial relaxation engine (FIRE), it was found that regions on different scales have distinct convergence paces, leading to increasing iteration steps and calculation time. Therefore, an improved FIRE algorithm was proposed to solve this problem. By harmonizing the oscillation frequencies of atoms and nodes, the relationship of their effective masses could be obtained to correct the step sizes of iterations. Subsequently, two schemes including modified acceleration parameters and iterative correction were put forward to further improve the FIRE algorithm. The HMCST with improved FIRE algorithm was then applied to simulations, and the efficiency of improved FIRE algorithm is increased by 33 and 31 times compared with the traditional CG and LBFGS algorithms, respectively. The findings in this work not only point out the importance of energy minimization algorithm in multiscale simulations, but also provide valuable insights and references for further improving the computational efficiency and fidelity of multiscale methods.