ADAPTIVE CONSISTENT HIGH ORDER ELEMENT-FREE GALERKIN METHOD
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摘要: 近来提出的一致性高阶无单元伽辽金法通过导数修正技术大幅度减少了所需积分点数目,并能够精确地通过线性和二次分片试验,显著改善标准无单元伽辽金法的计算效率、精度和收敛性.本文在此基础之上,充分利用无单元法易于在局部区域添加节点的优势,发展了一致性高阶无单元伽辽金法的h型自适应分析方法.根据应变能密度梯度该方法自适应地确定需节点加密的区域,基于背景积分网格的局部多层细化要求生成新的计算节点,同时考虑了节点分布由密到疏渐进过渡的情形.采用相邻两次计算的应变能的相对误差作为自适应过程的停止准则,将所发展自适应无网格法应用于由几何外形、边界外载和体力等因素造成的应力集中问题的计算分析.数值结果表明,所发展方法能够自适应地对高应力梯度区域进行节点加密,自动给出合理的计算节点分布.与已有的标准无网格法的自适应分析相比,所发展方法在计算效率、精度和应力场光滑性等方面均展现出显著优势.与采用节点均匀分布的一致性高阶无单元伽辽金法相比,它大幅度地减少了计算节点数目,有效提高了一致性高阶无单元伽辽金法在分析应力集中等存在局部高梯度问题时的计算效率和求解精度.Abstract: The recently developed consistent high order element-free Galerkin (EFG) method not only dramatically reduces the number of quadrature points in domain integration but also accurately passes the linear and quadratic patch tests, and remarkably improves the computational efficiency, accuracy and convergence of the standard EFG methods.On this basis, this work presents the h-adaptive analysis for consistent high order EFG method by taking advantage of the convenience of the EFG method in adding approximation nodes locally.The proposed method adaptively determines the region which needs nodal refinement according to the gradient of the strain energy density.The generation of the new approximation nodes is based on the multi-level local mesh refinement of the background integration mesh.The gradual transition between the regions with and without nodal refinement is also considered.The relative error of the strain energy in two successive computation is adopted as the stop-criterion of the adaptive process.The proposed adaptive meshfree method is applied to the analysis of stress concentration caused by geometry, external boundary loads and body forces.Numerical results show that the developed method is able to refine the region with high stress gradient adaptively and to generate reasonable distribution of approximation nodes automatically.In comparison with the existing adaptive schemes of the standard EFG method, the proposed method shows remarkable advantages on computational efficiency, accuracy and the smoothness of the resulting stress fields.In comparison with the consistent high order EFG method using uniform nodal distribution, the proposed adaptive method dramatically reduces the number of computational nodes.As a consequence, it significantly improves the computational efficiency and accuracy of the consistent high order EFG method for the analysis of problems with local high gradients such as stress concentration.
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表 1 方板圆孔问题的两种自适应方法比较
Table 1. Comparison of the two adaptive methods for the plate with a hole problem
表 2 受压半无限平面问题的两种自适应方法比较
Table 2. Comparison of the two adaptive methods for the pressure-loaded half plane problem
表 3 变体力板问题两种自适应方法比较
Table 3. Comparison of the two adaptive methods for the plate with non-constant body force problem
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