This paper provides a detailed comparison in a solids mechanics context of adaptive mesh refinement methods for all-quadrilateral and all-hexahedral meshes. The adaptive multigrid Local Defect Correction method and the well-known hierarchical h-adaptive refinement techniques are placed into a generic algorithmic setting for an objective numerical comparison. Such a comparison is of great interest as local multigrid AMR approaches are from now rarely employed to adaptively solve implicit systems in solid mechanics. However they present various interesting features mainly related to their intrinsic idea of partitioning the degrees of freedom on different mesh levels. For this study, we rely on a fully-automatic mesh refinement algorithm providing the desired refined mesh directly from the user-prescribed accuracy. The refinement process is driven by an a posteriori error estimator combined to mesh optimality criteria. In this study, the most efficient strategy based on mesh optimality criterion and refinement ratio is identified for all-quadrilateral and all-hexahedral finite elements meshes. The quality of refined meshes is finally appreciated in term of number of nodes but also through the verification of final solution's accuracy. A special attention is devoted to the fulfillment of local precisions which are of great importance from an engineering point of view. Numerical 2D and 3D experiments of different complexities revealing local phenomena enable to highlight the essential features of the considered mesh refinement methods within an elastostatic framework. This study points out the great potentialities of locally adaptive multigrid method, which clearly appears to be the most powerful strategy in terms of standard metrics of efficiency (dimension of systems to be solved, storage requirements, CPU time).

本文针对全四边形和全六面体网格在自适应网格细化方法的实体力学上下文中提供了详细的比较。自适应多网格局部缺陷校正方法和著名的分层h自适应精化技术被放入通用算法设置中，以进行客观的数值比较。由于从现在开始很少采用局部多网格AMR方法自适应地求解固体力学中的隐式系统，因此这种比较引起了人们的极大兴趣。但是，它们呈现出各种有趣的特征，这主要与它们在不同网格级别上划分自由度的内在想法有关。对于本研究，我们依靠一种全自动的网格细化算法，直接根据用户指定的精度提供所需的细化网格。精炼过程是由后验驱动的误差估计器结合到网格最优标准。在这项研究中，针对所有四边形和全部六面体有限元网格，确定了基于网格最优性准则和细化比的最有效策略。最终，可以根据节点数以及通过验证最终解决方案的准确性来提高精炼网格的质量。从工程的角度来看，要特别注意实现局部精度，这是非常重要的。数值2 D和3 D通过揭示局部现象的不同复杂性实验，可以在弹性静力学框架内突出显示所考虑的网格细化方法的基本特征。这项研究指出了本地自适应多网格方法的巨大潜力，就效率的标准度量标准（要解决的系统尺寸，存储要求，CPU时间）而言，本地自适应多网格方法显然是最强大的策略。