微纳米尺度力学的进一步发展，亟需突破分子模拟方法的计算规模瓶颈，以更高的效率研究微纳米尺度的关键力学问题。极值方法作为准静态分子模拟和多种多尺度模拟方法的核心算法，其计算效率决定了模拟所能达到的最大空间尺度。系统研究极值方法在分子模拟中的计算效率和可靠性，不仅可以明晰制约分子模拟计算效率的根本原因，也可以为发展高效的分子模拟新算法提供指导，对于进一步扩大分子模拟规模具有重要的研究意义和应用价值。本文针对多种纳米结构的准静态加载过程，深入研究了不同极值方法的计算效率和可靠性及其影响因素，并提出一系列加速算法和策略。主要研究工作包括：

（一）研究了初始构型、极值方法类型和收敛条件对准静态分子模拟计算效率和可靠性的影响。结果表明，每个加载步的初始构型不仅对计算效率产生显著影响，也会改变原子键的断裂以及微结构的演化。初始构型的获得与加载方式密切相关，对加载过程中原子势阱演化规律的研究表明，基于弹性变形的映射加载方式可以有效提高计算效率。通过考察六种不同类型的极值方法在结构驰豫和位错演化计算中的效率，发现共轭梯度（CG）法的综合性能最佳，BB方法在结构驰豫中收敛效率比有限内存的拟牛顿方法（Limited-memory BFGS, LBFGS）更高，而FIRE算法收敛性能很差。通过研究收敛准则和收敛阈值对模拟计算效率和可靠性的影响，指出不同收敛准则的特点，为具体应用中收敛参数的选择提供了指导和建议。

（二）研究了积分格式和监视器对FIRE方法在准静态分子模拟中收敛性能的影响。首先，采用不同积分格式的FIRE方法进行不同类型分子模拟的研究，发现向前欧拉（Forward Euler, FE）积分的计算效率最低；半隐式欧拉（Semi-implicit Euler, SE）积分采用激进的更新方式，加速效果最好，但是容易在位错演化中出现不收敛的情况；Velocity Verlet (VV)积分的计算效率略低于SE，但收敛性好于SE。因此，积分格式对FIRE的加速效果有显著影响，这一结论不同于文献中认为的没有影响的观点。其次，针对SE积分格式不收敛的问题，提出在FIRE方法中使用能量监视器，相比原有的功率监视器，采用能量监视器的SE积分格式能在保留高计算效率、低内存占用的前提下，又具有良好的收敛性。针对多个体系的计算结果表明，FIRE方法采用能量监视器和SE积分是进行一般准静态模拟的最佳选择，而采用功率监视器和SE积分则具有临界事件分析能力。最后，通过将FIRE方法应用到单晶铜三维纳米压入和石墨烯结构驰豫的模拟中，发现FIRE方法对于偏“硬”的金属材料力学性能计算有较高的计算效率，而对于偏“软”的材料其计算能力仍需要作进一步改进。

（三）结合材料的宏/微观力学行为和规律，提出多种加速准静态分子模拟的算法和策略。（1）基于均匀连续加载步之间每个原子对外载响应的相似性，提出惯性加速分子静力学方法。该方法通过学习上一次极小化过程中原子运动的经验，直接将加载后的初始构型映射到能量极小附近，可提高计算效率一个数量级以上。（2）针对简单剪切模型中刃型位错的形成和传播，提出位错加速演化算法，结果表明新算法中位错传播的计算量与模型尺寸无关。（3）针对纳米压入中的位错传播，提出分区域能量极小化算法，以较小代价完成位错演化，进一步提高了计算效率。（4）结合分子统计热力学方法（MST）和分子静力学方法，提出先势能极小化再自由能极小化的策略，有效提高了有效温度下的准静态分子模拟在复杂位错演化时的计算效率。

（四）系统研究了分子静力学（MS）和分子动力学（MD）在进行大规模纳米压入模拟时的计算效率和可靠性。从计算效率方面来看，MD的计算量与压入深度呈线性关系，而MS的计算量呈非线性增长，二者均存在局限性。压深较浅时，MS计算量小于MD，而当压深达到一定程度时，MS计算量大于MD。对MS计算中位错演化跟踪研究表明，位错环的长程传播是导致MS计算量非线性增长的主要原因。而MD采用固定弛豫时间来模拟准静态过程，计算过程不“关注”位错环的传播，在弹性变形和局部位错事件中过度驰豫而导致计算效率降低。位错定量计算的结果表明塑性区内位错密度的合理刻画是保证纳米压入结果可靠性的基本因素，可依此指导MS收敛精度和MD弛豫时间的选择，以保证模拟的可靠性。

（五）针对大规模纳米压入模拟提出两种加速算法。首先，结合MS和MD各自的优点，提出限定迭代次数的MS方法，在弹性变形和局部位错演化时，利用MS的优点实现快速收敛，当发生位错环的长程传播时利用MD的优点强行终止迭代，实现了计算量的准线性变化。其次，深入研究了分离位错环在纳米压入中对效率和可靠性的影响，提出消除分离位错环算法，将分离的位错环在临界位置直接抹除以代替长程传播，从而减轻塑性区与分离位错环相互作用并提高计算效率。结果表明新算法可以避免位错环长程传播过程，提高效率八倍以上，并且有效减弱了边界条件对纳米压入的影响；同时限制了非均匀变形区域的范围，为纳米压入多尺度模拟面临的非均匀变形区域快速演化的难题提供一种可行的解决方案。

Development of micro/nano mechanics requires new molecular simulation methods that could deal with lager systems in order to study some key mechanical problems with higher efficiency and fidelity. The minimization method, as the core procedure of molecular statics (MS) and multi-scale approaches, has been widely used in quasi-static mechanical analysis of micro/nano structures. Efficiency of the minimization method determines the maximum simulation scale in these applications. Therefore, investigating the efficiency and reliability of minimization methods in molecular simulations could not only uncover the mechanisms that govern the efficiency limit but also provide guidelines for developing new high-efficiency simulation methods. In this work, minimization methods with different algorithms are applied to serval typical mechanical problems in nanoscale. Factors that affect the efficiency and reliability of the algorithms are studied systematically. Based on the study, accelerated algorithms and strategies are proposed. The main results and conclusions are as follows:

(1) Influence of the initial configuration, optimization algorithm and convergence condition on the efficiency and fidelity of minimization method is investigated. It is found that initial configurations obtained from different loading operations have severe influence on the efficiency of minimization method in simulations of molecular system with bond-breaking events and dislocation evolutions. Analysis of energy landscape evolutions indicates that the mapping-style loading operation usually generates initial configurations very close to the final states, so it shows much higher efficiency than the direct loading operation. Minimization methods with six different optimization algorithms are applied to two typical molecular simulation cases: structural relaxation and dislocation propagation. The Barzilai-Borwein algorithm (BB) shows the fastest convergence in the case of structural relaxation, while the Limited-memory BFGS algorithm (LBFGS) is the fastest one in simulation of dislocation propagation. The FIRE algorithm shows poor efficiency in both cases which is not as fast as reported in other papers. By considering memory consumptions and convergence rate, the results suggest that the conjugate gradient (CG) algorithm should be the first choice in large-scale molecular simulations. Influence of convergence control algorithms on the efficiency and reliability of minimization method in molecular simulations are also studied. Based on the simulation results, practical guidelines for choosing convergence algorithm and conditions are discussed accordingly.

(2) Effects of integration formula and state monitor on the performance of FIRE method used in molecular simulations are studied. It is found that performance of the Forward Euler (FE) integration is always the worst in all the simulation cases. The Semi-implicit Euler (SE) integration with aggressive strategy shows the fastest convergence speed, while the Velocity Verlet (VV) integration is slightly slower than SE but can reach to lower energy states. The conclusion of integration-dependent performance of the FIRE method is different from that of the other reports. In order to ensure the convergence of SE integration, an energy monitor with less memory usage is proposed. Further simulation results indicate that SE integration with the energy monitor is the best choice for quasi-static molecular simulations. Applications of the FIRE method to three-dimensional nanoindentation and relaxation of warping graphene reveals that it has higher efficiency on relaxing “hard” materials but lower efficiency for “soft” materials.

(3) Accelerated algorithms and strategies based on macro mechanics knowledge are proposed for molecular simulation of several typical deformation modes: (i) Inertia accelerated molecular statics (IAMS) method is put forward based on characteristics of atomic motions at adjacent steps. By learning from previous minimization process, IAMS maps initial configuration to the state close to final energy minimum which let it be at least 10 times faster than ordinary MS method. (ii) Accelerated dislocation propagation method is designed for edge dislocation evolution in the simple shear model. In the new algorithm, computational cost of dislocation propagation is independent of model size. (iii) Partition minimization algorithm with and without mapping are built for the simulation of inital dislocation evolution in nanoindentation. Results show promising efficiency improvement by reducing the cost of dislocation propagation. (iv) Algorithm combining potential minimizing and free energy minimization is proposed to improve computational efficiency of molecular statistical thermodynamics (MST) method for quasi-static analysis of nanostructures at finite temperature.

(4) The comparative simulations of dislocation evolution in nanoindentation are carried out in terms of both molecular dynamics (MD) and molecular stataics (MS) methods, to explore what really govern the computational efficiency and fidelity in molecular simulations relevant to dislocation evolutions. It is found that although all simulations can present similar relationship between indentation force and depth, there still might be some significant differences in the simulated dislocation patterns and computational efficiency. Firstly, the MS simulations show more complicated dislocations. Secondly, the necessary computational effort of MS increases nonlinearly with indentation depth, compared to the linear dependence in MD simulations. More importantly, it is revealed that the time consumption of the minimization iteration is strongly dependent on the moving of dislocation loops and increases greatly when dislocation loops move long distances. Whereas MD simulations of complicated dislocations patterns may need less time cost but present immature dislocation evolutions, since the relaxation steps in MD simulations are fixed beforehand, regardless of the dislocation loops moving to equilibrium state or not. Dislocation quantitative analysis shows that the reasonable description of dislocation density within the plastic zone is the necessary factor for nanoindentation simulations, which can be used to help select the proper convergence threshold in MS or relaxation time in MD to trade off between efficiency and accuracy.

(5) Two accelerated algorithms are put forward particularly for molecular simulation of large-scale nanoindentation. Firstly, the limited-iteration molecular statics (LIMS) which combines advantages of MS and MD is proposed. Specifically, it takes advantage of MS with quick convergence for elastic deformation and localized dislocation events, and MD with limited iteration steps for dislocation loops propagation As a result, LIMS shows sub-linear computational cost in nanoindentation simulations. Secondly, the removing separated dislocation loop (RSDL) algorithm is constructed. By removing dislocation loops at a specific location, the long-range propagation of dislocation loops as well as non-physical interactions between the plastic zone and artificial boundaries is avoid in the RSDL algorithm, so the computation cost can be dramatically reduced. It shows 8 times higher efficiency by introducing the RSDL in MS simulations. Moreover, it is expected that the RSDL could be utilized in multi-scale methods to deal with growing atomic regions due to dislocation evolution in order to improve efficiency.