模拟裂纹扩展的单位分解扩展无网格法
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摘要
单位分解扩展无网格法(PUEM)是一种求解不连续问题的新型无网格方法。其基于单位分解思想,通过在传统无网格法的近似函数中加入扩展项来反映由裂纹所产生的不连续位移场。详细描述了水平集方法,PUEM不连续近似函数的构造及控制方程的离散。针对裂纹扩展问题,提出了一种十分简单的水平集更新算法;讨论了不同的节点数、高斯积分阶次以及围线积分区域对应力强度因子计算结果的影响,并给出了合理的参数;模拟了边裂纹和中心裂纹的扩展问题,并与XFEM的数值结果进行了比较。数值算例表明,本文方法具有较高的计算精度,是模拟裂纹扩展非常有效的方法,具有广阔的应用前景。
The enriched meshless method based on the partition of unity(PUEM)is a new numerical method for modeling discontinuities.In PUEM,in order to represent the discontinuous displacement field around crack,enriched functions were added in the approximation of traditional meshless based on the ideas of partition of unity.The theory of level set,construction of discontinuous approximation function and discrete format of governing equation were introduced in detail.In view of crack growth,a simple updating algorithm of level set was presented.The impact for the computational results of stress intensity factors using different nodal numbers,Gaussian integral orders of background cells and integral domain of crack tip were discussed.Both growth examples including edge and centre crack were simulated by the combination of PUEM and the level set method.The results of numerical examples show PUEM has higher accuracy,and is an effective meshless method for crack propagation.
The enriched meshless method based on the partition of unity(PUEM)is a new numerical method for modeling discontinuities.In PUEM,in order to represent the discontinuous displacement field around crack,enriched functions were added in the approximation of traditional meshless based on the ideas of partition of unity.The theory of level set,construction of discontinuous approximation function and discrete format of governing equation were introduced in detail.In view of crack growth,a simple updating algorithm of level set was presented.The impact for the computational results of stress intensity factors using different nodal numbers,Gaussian integral orders of background cells and integral domain of crack tip were discussed.Both growth examples including edge and centre crack were simulated by the combination of PUEM and the level set method.The results of numerical examples show PUEM has higher accuracy,and is an effective meshless method for crack propagation.
引文
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[9]Nguyen V P,Rabczuk T,Bordas S,et al.Meshlessmethods:a review and computer implementation as-pects[J].Mathematics and Computers in Simulation,2008,79:763-813.
[10]Melenk J M,Bubuska I.The partition of the unity fi-nite element method:basic theory and application[J].Computer Methods in Application Mechanics andEngineering,1996,139:289-314.
[11]Duarte C A,Oden J T.An H-P adaptive method usingclouds[J].Computer Methods in Applied Mechanicsand Engineering,1996,139:237-262.
[12]Babuska T,Melenk J M.The partition of unity meth-od[J].International Journal for Numerical Methodin Engineering,1997,40:727-758.
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[14]Belytschko T,Lu Y Y,Gu L.Element free Galerkinmethods[J].International Journal for NumericalMethod in Engineering,1994,37:229-256.
[15]中国航空学会.应力强度因子手册[M].北京:科学出版社,1993.(Chinese Aviation Academa.Stress Inten-sity Factors Handbook[M].Beijing:Science Press,1993.(in Chinese)).
[2]Belytschko T,Krongauz Y,Fleming M,et al.Smoot-hing and accelerated computations in the element-freeGarlerkin method[J].Journal of Computational andApplied Mathematics,1996,74:111-126.
[3]Krongauz Y,Belytschko T.EFG approximation withdiscontinuous derivatives[J].International Journalfor Numerical Methods in Engineering,1998,41(7):1215-1233.
[4]Organ D,Fleming,Terry T,et al.Continuous mesh-less approximations for nonconvex bodies by diffrac-tion and transparency[J].Computational Mechanics,1996,18:225-235.
[5]Fleming M,Chu Y A,Moran B,et al.Enriched ele-ment free Galerkin methods for crack tip fields[J].International Journal for Numerical Methods inEngineering,1997,40:1483-1504.
[6]Ventura A,Xu J X,Belytschko T.A vector level setmethod and new discontinuity approximations forcrack growth by EFG[J].International Journal forNumerical Methods in Engineering,2002,54:923-944.
[7]Moes N,Dolbow J,Belytschko T.A finite elementmethod for crack growth without remeshing[J].In-ternational Journal for Numerical Methods in Engi-neering,1999,46:131-135.
[8]Rabczuk T,Zi G.A meshfree based on the local parti-tion of unity for cohesive cracks[J].ComputationalMechanics,2006,39(6):743-760.
[9]Nguyen V P,Rabczuk T,Bordas S,et al.Meshlessmethods:a review and computer implementation as-pects[J].Mathematics and Computers in Simulation,2008,79:763-813.
[10]Melenk J M,Bubuska I.The partition of the unity fi-nite element method:basic theory and application[J].Computer Methods in Application Mechanics andEngineering,1996,139:289-314.
[11]Duarte C A,Oden J T.An H-P adaptive method usingclouds[J].Computer Methods in Applied Mechanicsand Engineering,1996,139:237-262.
[12]Babuska T,Melenk J M.The partition of unity meth-od[J].International Journal for Numerical Methodin Engineering,1997,40:727-758.
[13]Duarte C A,Babuska I,Oden J T.Generalized finiteelement methods for three dimensional structure me-chanics problems[J].Computers&Structures,2000,77:215-232.
[14]Belytschko T,Lu Y Y,Gu L.Element free Galerkinmethods[J].International Journal for NumericalMethod in Engineering,1994,37:229-256.
[15]中国航空学会.应力强度因子手册[M].北京:科学出版社,1993.(Chinese Aviation Academa.Stress Inten-sity Factors Handbook[M].Beijing:Science Press,1993.(in Chinese)).