插值型无单元Galerkin比例边界法与有限元法的耦合在压电材料断裂分析中的应用
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摘要
插值型无单元Galerkin比例边界法是一种只需在边界上采用插值型无单元Galerkin法离散且无需基本解的半解析方法,能有效求解压电材料的断裂问题.为进一歩提高这种方法的适用性,该文提出了一种用于压电材料断裂分析的插值型无单元Galerkin比例边界法耦合有限元法(finite element method,FEM)的分析方法.裂纹周边一定范围的计算域采用插值型无单元Galerkin比例边界法离散,其余区域采用FEM离散.插值型无单元Galerkin比例边界法方程和FEM方程的耦合可利用界面两侧广义位移的连续条件方便地实现.最后,给出了两个数值算例验证了该文所提方法的有效性.
The interpolating element-free Galerkin scaled boundary method( IEFG-SBM) is a semianalytical method which only requires discretizing the boundary with the interpolating element-free Galerkin(EFG) method without fundamental solution. This method is very powerful to deal with fracture problems of piezoelectric materials. In order to further improve the applicability of the IEFGSBM,a coupled IEFG-SBM and finite element method( FEM) for fracture analysis of piezoelectric materials was developed. The IEFG-SBM was utilized to model the domain close to the crack tip and the FEM was employed in the remaining domain. Based on continuity conditions at the interface between the IEFG-SBM sub-domain and the FEM sub-domain,the coupled formula of the proposed method can be conveniently derived. Finally,2 numerical examples were presented to demonstrate the validity of the proposed method.
The interpolating element-free Galerkin scaled boundary method( IEFG-SBM) is a semianalytical method which only requires discretizing the boundary with the interpolating element-free Galerkin(EFG) method without fundamental solution. This method is very powerful to deal with fracture problems of piezoelectric materials. In order to further improve the applicability of the IEFGSBM,a coupled IEFG-SBM and finite element method( FEM) for fracture analysis of piezoelectric materials was developed. The IEFG-SBM was utilized to model the domain close to the crack tip and the FEM was employed in the remaining domain. Based on continuity conditions at the interface between the IEFG-SBM sub-domain and the FEM sub-domain,the coupled formula of the proposed method can be conveniently derived. Finally,2 numerical examples were presented to demonstrate the validity of the proposed method.
引文
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[14]殷德胜,尹栓,周宜红.裂缝分析的比例边界有限元与有限元耦合的虚拟结构面模型[J].计算力学学报,2014,31(6):735-741.(YIN Desheng,YIN Shuan,ZHOU Yihong. Coupled SBFEM and FEM for crack analysis based on virtual discontinuous surface method[J]. Chinese J ournal of Computational M echanics,2014,31(6):735-741.(in C hinese))
[15]YANG Z J,W ANG X F,YIN D S,et al. A non-matching finite element-scaled boundary finite element coupled method for linear elastic crack propagation modelling[J]. Computers&Structures,2015,153(C):126-136.
[16]张雄,刘岩.无网格法[M].北京:清华大学出版社,2004.(ZHANG Xiong,LIU Yan. Meshless M ethods[M]. Beijing:T singhua U niversity Press,2004.(in C hinese))
[17]W ANG B L,M AI Y W. A piezoelectric material strip with a crack perpendicular to its boundary surfaces[J]. International J ournal of Solids and Structures,2002,39(17):4501-4524.
[2]LI C,M AN H,SONG C M,et al. Fracture analysis of piezoelectric materials using the scaled boundary finite element method[J]. Engineering Fracture Mechanics,2013,97(1):52-71.
[3]LI Q H,CHEN S S,LUO X M. Steady heat conduction analyses using an interpolating elementfree G alerkin scaled boundary method[J]. Applied Mathematics and Computation,2017,300:103-115.
[4]陈莘莘,童谷生,万云.弹性力学问题的插值型无单元伽辽金比例边界法[J].中国科学:物理学力学天文学,2017,47(3):034601.(CHEN Shenshen,T ONG Gusheng,W AN Yun. An interpolating element-free G alerkin scaled boundary method for the elasticity problem[J]. Scientia Sinica:Physica,M echanica&Astronomica,2017,47(3):034601.(in C hinese))
[5]陈莘莘,王娟.压电裂纹的插值型无单元伽辽金比例边界法分析[J].机械工程学报,2017,53(6):53-59.(CHEN Shenshen,W ANG Juan. Analysis of interpolating element-free Galerkin scaled boundary method for piezoelectric cracks[J]. J ournal of Mechanical Engineering,2017,53(6):53-59.(in C hinese))
[6]王伟,伊士超,姚林泉.分析复合材料层合板弯曲和振动的一种有效无网格方法[J].应用数学和力学,2015,36(12):1274-1284.(W ANG W ei,YI Shichao,YAO Linquan. An effective meshfree method for bending and vibration analyses of laminated composite plates[J]. Applied Mathematics and M echanics,2015,36(12):1274-1284.(in C hinese))
[7]王峰,周宜红,郑保敬,等.基于滑动Kriging插值的M LPG法求解结构非耦合热应力问题[J].应用数学和力学,2016,37(11):1217-1227.(W ANG Feng,ZHOU Yihong,ZHENG Baojing,et al. A meshless local Petrov-G alerkin method based on the moving Kriging interpolation for structural uncoupled thermal stress analysis[J]. Applied Mathematics and Mechanics,2016,37(11):1217-1227.(in C hinese))
[8]肖毅华,张浩锋,平学成.无网格对称粒子法中两类热边界条件的处理[J].华东交通大学学报,2014,31(4):65-70.(XIAO Yihua,ZHANG Haofeng,PING Xuecheng. T reatments of two kinds of thermal boundary conditions in meshless symmetric particle method[J]. J ournal of East China J iaotong University,2014,31(4):65-70.(in C hinese))
[9]SONG C M. A matrix function solution for the scaled boundary finite element equation in statics[J]. Computer Methods in Applied Mechanics and Engineering,2004,193:2325-2356.
[10]DEEKS A J,W OLF J P. A virtual work derivation of the scaled boundary finite element method for elastostatics[J]. Computational Mechanics,2002,28(6):489-504.
[11]W ANG J F,SUN F X,CHENG Y M. An improved interpolating element-free Galerkin method w ith a nonsingular w eight function for tw o-dimensional potential problems[J]. Chinese Physics B,2012,21(9):090204.
[12]W ANG J F,W ANG J F,SUN F X,et al. An interpolating boundary element-free method with nonsingular w eight function for tw o-dimensional potential problems[J]. International J ournal of Computational M ethods,2013,10(6):1350043.
[13]LANCAST ER P,SALKAUSKAS K. Surface generated by moving least squares methods[J].M athematics of Computation,1981,37(155):141-158.
[14]殷德胜,尹栓,周宜红.裂缝分析的比例边界有限元与有限元耦合的虚拟结构面模型[J].计算力学学报,2014,31(6):735-741.(YIN Desheng,YIN Shuan,ZHOU Yihong. Coupled SBFEM and FEM for crack analysis based on virtual discontinuous surface method[J]. Chinese J ournal of Computational M echanics,2014,31(6):735-741.(in C hinese))
[15]YANG Z J,W ANG X F,YIN D S,et al. A non-matching finite element-scaled boundary finite element coupled method for linear elastic crack propagation modelling[J]. Computers&Structures,2015,153(C):126-136.
[16]张雄,刘岩.无网格法[M].北京:清华大学出版社,2004.(ZHANG Xiong,LIU Yan. Meshless M ethods[M]. Beijing:T singhua U niversity Press,2004.(in C hinese))
[17]W ANG B L,M AI Y W. A piezoelectric material strip with a crack perpendicular to its boundary surfaces[J]. International J ournal of Solids and Structures,2002,39(17):4501-4524.