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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图 8 方板圆孔问题的位移误差-节点数曲线
Figure 8. Displacement error-number of nodes curve of the plate with a hole problem
图 9 方板圆孔问题$\sigma _{yy} $应力场比较
Figure 9. Comparison of $\sigma _{yy} $ stress field of the plate with a hole problem
图 10 受压半无限平面问题示意图
Figure 10. Schematic diagram of the pressure-loaded half plane problem
图 12 受压半无限平面问题的位移误差-节点数曲线
Figure 12. Displacement error-number of nodes curve of the pressure-loaded half plane problem
图 13 受压半无限平面问题$\sigma _{xx}$应力场比较
Figure 13. Comparison of the $\sigma _{xx} $ stress field of the pressure-loaded half plane problem
图 14 变体力板示意图及初始节点分布
Figure 14. Schematic diagram of the plate with non-constant body force problem and its initial node distribution
图 15 变体力板问题的自适应结果
Figure 15. Adaptivity for the plate with non-constant body force problem
图 16 变体力板问题$\sigma _{xx} $应力场比较
Figure 16. Comparison of the $\sigma _{xx} $ stress field of the plate with non-constant body force problem
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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[1] 黄凯, 白鸿柏, 路纯红等. 无网格法及其在金属成形中的应用综述.锻压技术, 2016, 41(2):12-16 http://www.cnki.com.cn/Article/CJFDTOTAL-DYJE201602003.htmHuang Kai, Bai Hongbai, Lu Chunhong, et al. Summarization on meshfree method and its application in metal forming. Forming and Stamping Technology, 2016, 41(2):12-16(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-DYJE201602003.htm [2] Hah ZH, Youn SK. Eulerian analysis of bulk metal forming processes based on spline-based meshfree method. Finite Elements in Analysis and Design, 2015, 106:1-15 doi: 10.1016/j.finel.2015.07.004 [3] Muthu N, Falzon BG, Maiti SK, et al. Modified crack closure integral technique for extraction of SIFs in meshfree methods. Finite Elements in Analysis and Design, 2014, 78:25-39 doi: 10.1016/j.finel.2013.09.005 [4] 马文涛, 许艳, 马海龙. 修正的内部基扩充无网格法求解多裂纹应力强度因子. 工程力学, 2015, 32(10):18-24 http://www.cnki.com.cn/Article/CJFDTOTAL-GCLX201510004.htmMa Wentao, Xu Yan, Ma Hailong. Solving stress intensity factors of multiple cracks by using a modified intrinsic basis enriched meshless mehthod. Engineering Mechanics, 2015, 32(10):18-24(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-GCLX201510004.htm [5] Tanaka S, Suzuki H, Sadamoto S, et al. Analysis of cracked shear deformable plates by an effective meshfree plate formulation. Engineering Fracture Mechanics, 2015, 144:142-157 doi: 10.1016/j.engfracmech.2015.06.084 [6] Wang DD, Wu JC. An efficient nesting sub-domain gradient smoothing integration algorithm with quadratic exactness for Galerkin meshfree methods. Comput Methods Appl Mech Engrg, 2016, 298:485-519 doi: 10.1016/j.cma.2015.10.008 [7] Hillman M, Chen JS. An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics. Int J Numer Meth Engng, 2016, 107:603-630 doi: 10.1002/nme.v107.7 [8] Duan QL, Li XK, Zhang HW, et al. Second-order accurate derivatives and integration schemes for meshfree methods. International Journal for Numerical Methods in Engineering, 2012, 92(4):399-424 doi: 10.1002/nme.v92.4 [9] Duan QL, Gao X, Wang BB, et al. Consistent element-free Galerkin method. International Journal for Numerical Methods in Engineering, 2014, 99:79-101 doi: 10.1002/nme.v99.2 [10] 王冰冰, 高欣, 段庆林. 热传导的二阶一致1点积分无网格法.工程力学, 2014, 31(9):1-6 http://www.cnki.com.cn/Article/CJFDTOTAL-GCLX201409002.htmWang Bingbing, Gao Xin, Duan Qinglin. Meshfree method for heat conduction using quadratic consistent 1-point integration scheme. Engineering Mechanics, 2014, 31(9):1-6(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-GCLX201409002.htm [11] 王冰冰, 陈嵩涛, 段庆林. 动力分析的二阶一致无网格法. 应用力学学报, 2014, 31(3):353-358 http://www.cnki.com.cn/Article/CJFDTOTAL-YYLX201403010.htmWang Bingbing, Chen Songtao, Duan Qinglin. Quadratically consistent meshfree method for dynamics. Chinese Journal of Applied Mechanics, 2014, 31(3):353-358(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-YYLX201403010.htm [12] Duan Qinglin, Gao Xin, Wang Bingbing, et al. A four-point integration scheme with quadratic exactness for three-dimensional element-free Galerkin method based on variationally consistent formulation. Comput Methods Appl Mech Engrg, 2014, 280(10):84-116 http://cn.bing.com/academic/profile?id=18c4f2bb4b6ad3833626f7242726d4d7&encoded=0&v=paper_preview&mkt=zh-cn [13] Ortiz-Bernardin A, Hale JS, Cyron CJ. Volume-averaged nodal projection method for nearly-incompressible elasticity using meshfree and bubble basis functions. Comput Methods Appl Mech Engrg, 2015, 285:427-451 doi: 10.1016/j.cma.2014.11.018 [14] Ortiz-Bernardin A, Puso MA, Sukumar N. Improved robustness for nearly-incompressible large deformation meshfree simulations on Delaunay tessellations. Comput Methods Appl Mech Engrg, 2015, 293:348-374 doi: 10.1016/j.cma.2015.05.009 [15] 陈炎,曹树良,梁开洪等.结合自适应网格的基于特征线方程的分离算法.力学学报, 2009, 41(5):666-670 http://lxxb.cstam.org.cn/CN/abstract/abstract141847.shtmlChen Yan, Cao Shuliang, Liang Kaihong, et al. A new grid adaptive strategy combined characterized based split method. Chinese Journal of Theoretical and Applied Mechanics, 2009, 41(5):666-670(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract141847.shtml [16] 徐云,陈军,蔚喜军.多尺度材料模拟的自适应有限元方法研究. 力学学报, 2009, 41(5):722-729 http://lxxb.cstam.org.cn/CN/abstract/abstract141771.shtmlXu Yun, Chen Jun, Yu Xijun. A new adaptive finite element method for multiscale dynamic simulation. Chinese Journal of Theoretical and Applied Mechanics, 2009, 41(5):722-729(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract141771.shtml [17] Nguyen-Xuan H, Wu CT, Liu GR. An adaptive selective ES-FEM for plastic collapse analysis. European Journal of Mechanics A/Solids, 2016, 58:278-290 doi: 10.1016/j.euromechsol.2016.02.005 [18] Duarte CA, Oden JT. An h-p adaptive method using clouds. Comput Methods Appl Mech Engrg, 1996, 139:237-262 doi: 10.1016/S0045-7825(96)01085-7 [19] 刘欣,朱德懋,陆明万等.平面裂纹问题的h,p,hp型自适应无网格方法的研究.力学学报, 2000, 32(3):308-318 http://lxxb.cstam.org.cn/CN/abstract/abstract140880.shtmlLiu Xin, Zhu Demao, Lu Mingwan, et al. H,p,hp adaptive meshless method for plane crack problem. Acta Mechanica Sinica, 2000, 32(3):308-318(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract140880.shtml [20] Chung HJ, Belytschko T. An error estimate in the EFG method. Comp Mech, 1998, 21:91-100 doi: 10.1007/s004660050286 [21] Sethuraman R, Hormis JV. An error estimator and adaptivity based on basis coefficient-vector field of element-free Galerkin method. International Journal of Computational Methods, 2011, 8(1):91-118 doi: 10.1142/S0219876211002502 [22] He YQ, Yang HT, Deeks AJ. A node based error estimator for the element-free Galerkin(EFG) methods. International Journal of Computational Methods, 2014, 11(4):13500591-24 http://cn.bing.com/academic/profile?id=2d9375f58dd944b64ece2dd3b92d8254&encoded=0&v=paper_preview&mkt=zh-cn [23] Hiiussler-Combe U, Korn C. An adaptive approach with the element-free Galerkin method. Comput Methods Appl Mech Engrg, 1998, 162:203-222 doi: 10.1016/S0045-7825(97)00344-7 [24] Rabczuk T, Belytschko T. Adaptivity for structured meshfree particle methods in 2D and 3D. Int J Numer Meth Engng, 2005, 63:1559-1582 doi: 10.1002/(ISSN)1097-0207 [25] Liu GR, Tu ZH. An adaptive procedure based on background cells for meshless methods. Comput Methods Appl Mech Engrg, 2002, 191:1923-1943 doi: 10.1016/S0045-7825(01)00360-7 [26] 干年妃, 李光耀, 钟志华等.金属体积成型过程的自适应无网格方法.中国机械工程, 2006, 17(24):2612-2617 http://www.cnki.com.cn/Article/CJFDTOTAL-ZGJX200624018.htmGan Nianfei, Li Guangyao, Zhong Zhihua, et al. Adaptive meshfree method on bulk forming process. China Mechanical Engineering, 2006, 17(24):2612-2617(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-ZGJX200624018.htm [27] Luo YH, Hiiussler-Combe U. A gradient-based adaptation procedure and its implementation in the element-free Galerkin method. Int J Numer Meth Engng, 2003, 56:1335-1354 doi: 10.1002/nme.v56:9 [28] 张征,刘更,刘天祥.接触问题的自适应无网格伽辽金方法.中国机械工程, 2007, 18(23):2868-2873 http://www.cnki.com.cn/Article/CJFDTOTAL-ZGJX200723023.htmZhang Zheng, Liu Geng, Liu Tianxiang. An adaptive element-free method Galerkin model for contact problems. China Mechanical Engineering, 2007, 18(23):2868-2873(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-ZGJX200723023.htm [29] Metsis P, Lantzounis N, Papadrakakis M. A new hierarchical partition of unity formulation of EFG meshless methods. Comput Methods Appl Mech Engrg, 2015, 283:782-805 doi: 10.1016/j.cma.2014.10.016 [30] Belytschko T, Lu Y, Gu L. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2):229-256 doi: 10.1002/(ISSN)1097-0207 [31] Fernández-Méndez S, Huerta A. Imposing essential boundary conditions in mesh-free methods. Computer Methods in Applied Mechanics and Engineering, 2004, 193(12):1257-1275 http://cn.bing.com/academic/profile?id=0d8fc8049c5c0f9ba63ea36139841ac8&encoded=0&v=paper_preview&mkt=zh-cn [32] Duan QL, Belytschko T. Gradient and dilatational stabilizations for stress-point integration in the element-free Galerkin method. International Journal for Numerical Methods in Engineering, 2009, 77(6):776-798 doi: 10.1002/nme.v77:6 -
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