Point collocation methods are strong form approaches that can be applied to continuum mechanics problems and possess attractive features over weak form-based methods due to the absence of a mesh. While various adaptive strategies have been proposed to improve the accuracy of weak form-based methods, such techniques have received little attention for strong form-based methods. In this paper, combined rh-adaptivity, in which r- and h-adaptivities are adopted iteratively, is applied to the local maximum entropy point collocation method for the first time to solve linear elasticity problems. Material force residuals act as driving forces in r-adaptivity to relocate collocation points, reducing the error associated with a given point distribution. Physical equilibrium residuals are used as the error estimator in h-adaptivity to determine the insertion locations for new points, diminishing the error caused by inadequate degrees of freedom. Issues arising in mesh-based methods, such as mesh distortion and hanging nodes, are entirely absent from the proposed method. The paper introduces the approach for the first time and the study is therefore confined to 2D domains. Numerical examples are presented to demonstrate the performance of the proposed adaptive strategies, comparing convergence rates and computational costs using uniform refinement, pure r-, h- and combined rh-adaptivities.

点搭配方法是强形式方法，可以应用于连续介质力学问题，并且由于没有网格而具有比基于弱形式的方法有吸引力的特征。虽然已经提出了各种自适应策略来提高基于弱形式的方法的准确性，但这些技术对于基于强形式的方法却很少受到关注。在本文中，组合RH -adaptivity，其中[R -和ħ -adaptivities迭代通过，被施加到局部最大熵点配置法首次求解线性弹性的问题。物质力残差作为r 中的驱动力- 重新定位搭配点的适应性，减少与给定点分布相关的错误。物理平衡残差被用作h自适应中的误差估计器，以确定新点的插入位置，减少由自由度不足引起的误差。所提出的方法中完全不存在基于网格的方法中出现的问题，例如网格失真和悬挂节点。该论文首次介绍了该方法，因此该研究仅限于二维域。数值例子展示了所提出的自适应策略的性能，使用统一细化、纯r -、h - 和组合rh比较收敛速度和计算成本- 适应性。