基于XFEM与自适应网格的非均质材料裂纹扩展模拟
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摘要
针对含有间断的非均匀材料的断裂问题,论文将虚节点多边形单元(virtual node polygonal element,VP)的形函数引入到扩展有限元(extended finite element method,XFEM)中,提出了一种基于四叉树结构的动态网格细化方法,该方法可对间断面附近单元实现可调控的多层级细化,特别是对于裂纹扩展问题,可实现裂尖附近单元的动态网格细化与粗化.基于以上网格细化方法,论文将之前论文中提出的VP-XFEM进行扩展,以研究非均质材料中的裂纹扩展问题.为验证算法的准确性与计算效率,针对含有孔洞及材料界面的断裂问题,论文给出了相应的算例.结果显示,与传统的XFEM相比,该方法能够获得更好的精度、收敛性以及计算效率.
Defects such as holes,inclusions and cracks greatly affect the integrity and performance of engineering structures.Accurate analysis of their influences on fracture parameters and crack growth is critical to ensure structural safety.In recent decades,numerical methods have been widely used to deal with the crack growth problem in heterogeneous materials.However,the application of numerical simulation is limited by the mesh-relevant issues associated with crack growth and material interfaces.The extended finite element method(XFEM)has gained wide attention since it was proposed.The feature that discontinuities can be independent of the mesh avoids the difficulty due to re-meshing,thus greatly simplifying the processing of meshes.Despite this,computational accuracy and efficiency have been more and more the main concerns for researchers.Here we propose an adaptive mesh refinement method to model the fracture problem of heterogeneous materials based on the quadtree structure.By introducing the shape functions of virtual node polygonal elements into the approximation of standard XFEM,a regulatable multi-level mesh refinement in the vicinity of discontinuities can be created.Especially for cracks,the dynamic mesh refinement and coarsening near the crack tips can be realized by the proposed method.With the local mesh refinement around discontinuities,the computational accuracy can be guaranteed without introducing too many new elements and nodes.Based on the above mesh refinement method,we extend the VP-XFEM proposed in a previous paper to study the crack propagation problem in heterogeneous materials.To verify the accuracy and computational efficiency of the algorithm,corresponding examples are given for the fracture problem involving holes and material interfaces.The results show that this method can achieve a good agreement in crack growth paths with the experiments.Besides,better accuracy and computational efficiency can be acquired compared with the traditional XFEM.The proposed method provides a suitable alternative to the XFEM,especially for engineering applications.
Defects such as holes,inclusions and cracks greatly affect the integrity and performance of engineering structures.Accurate analysis of their influences on fracture parameters and crack growth is critical to ensure structural safety.In recent decades,numerical methods have been widely used to deal with the crack growth problem in heterogeneous materials.However,the application of numerical simulation is limited by the mesh-relevant issues associated with crack growth and material interfaces.The extended finite element method(XFEM)has gained wide attention since it was proposed.The feature that discontinuities can be independent of the mesh avoids the difficulty due to re-meshing,thus greatly simplifying the processing of meshes.Despite this,computational accuracy and efficiency have been more and more the main concerns for researchers.Here we propose an adaptive mesh refinement method to model the fracture problem of heterogeneous materials based on the quadtree structure.By introducing the shape functions of virtual node polygonal elements into the approximation of standard XFEM,a regulatable multi-level mesh refinement in the vicinity of discontinuities can be created.Especially for cracks,the dynamic mesh refinement and coarsening near the crack tips can be realized by the proposed method.With the local mesh refinement around discontinuities,the computational accuracy can be guaranteed without introducing too many new elements and nodes.Based on the above mesh refinement method,we extend the VP-XFEM proposed in a previous paper to study the crack propagation problem in heterogeneous materials.To verify the accuracy and computational efficiency of the algorithm,corresponding examples are given for the fracture problem involving holes and material interfaces.The results show that this method can achieve a good agreement in crack growth paths with the experiments.Besides,better accuracy and computational efficiency can be acquired compared with the traditional XFEM.The proposed method provides a suitable alternative to the XFEM,especially for engineering applications.
引文
[1]Belytschko T,Lu Y Y,Gu L.Crack propagation by element-free Galerkin methods[J].Engineering Fracture Mechanics,1995,51(2):295-315.
[2]Portela A,Aliabadi M H,Rooke D P.The dual boundary element method:Effective implementation for crack problems[J].International Journal for Numerical Methods in Engineering,1992,33(6):1269-1287.
[3]Kikuchi M,Wada Y,Shintaku Y,Suga K,Li Y.Fatigue crack growth simulation in heterogeneous material using s-version FEM[J].International Journal of Fatigue,2014,58:47-55.
[4]Mo3s N,Dolbow J,Belytschko T.A finite element method for crack growth without remeshing[J].International Journal for Numerical Methods in Engineering,1999,46(1):131-150.
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[6]庄茁,柳占立,成斌斌,廖剑晖.扩展有限单元法[M].北京:清华大学出版社,2012.(Zhuang Z,Liu Z L,Cheng B B,Liao J H.The Extended Finite Element Method[M].Beijing:Tsinghua University Press,2012.(in Chinese))
[7]余天堂.扩展有限单元法:理论,应用及程序[M].北京:科学出版社,2014.(Yu T T.The Extended Finite Element Method-Theory,Application and Program[M].Beijing:Science Press,2014.(in Chinese))
[8]Sukumar N,Chopp D L,Mo3s N,Belytschko T.Modeling holes and inclusions by level sets in the extended finite-element method[J].Computer Methods in Applied Mechanics and Engineering,2001,190(46-47):6183-6200.
[9]Zhuang Z,Cheng B B.Equilibrium state of mode-Isub-interfacial crack growth in bi-materials[J].International Journal of Fracture,2011,170(1):27-36.
[10]Singh I V,Mishra B K,Bhattacharya S,Patil R U.The numerical simulation of fatigue crack growth using extended finite element method[J].International Journal of Fatigue,2012,36(1):109-119.
[11]Jiang S,Du C,Gu C,Chen X.XFEM analysis of the effects of voids,inclusions and other cracks on the dynamic stress intensity factor of a major crack[J].Fatigue&Fracture of Engineering Materials&Structures,2014,37(8):866-882.
[12]Kumar S,Singh I V,Mishra B K.A homogenized XFEM approach to simulate fatigue crack growth problems[J].Computers&Structures,2015,150:1-22.
[13]王志勇,马力,吴林志,于红军.基于扩展有限元法的颗粒增强复合材料静态及动态断裂行为研究[J].固体力学学报,2011,32(6):566-573.(Wang Z Y,Ma L,Wu L Z,Yu H J.Investigation of static and dynamic fracture behavior of particle-reinforced composite materials by the extended finite element method[J].Chinese Journal of Solid Mechanics,2011,32(6):566-573.(in Chinese))
[14]Teng Z H,Sun F,Wu S C,Zhang Z B,Chen T,Liao D M.An adaptively refined XFEM with virtual node polygonal elements for dynamic crack problems[J].Computational Mechanics,2018,62(5):1087-1106.
[15]唐旭海,吴圣川,郑超,张建海.基于虚节点的多边形有限元法[J].应用数学和力学,2009,30(10):1153-1164.(Tang X H,Wu S C,Zheng C,Zhang JH.A novel virtual node method for polygonal elements[J].Applied Mathematics and Mechanics(English Edition),2009,30(10):1233-1246.(in Chinese))
[16]吴圣川,吴玉程.ALOF-新一代三维疲劳裂纹扩展分析软件[J].计算机辅助工程,2011,20(01):136-140.(Wu S C,Wu Y C.ALOF:new 3Dfatigue crack propagation analysis software[J].Computer Aided Engineering,2011,20(01):136-140.(in Chinese))
[17]Zheng C,Tang X H,Zhang J H,Wu S C.A novel mesh-free poly-cell Galerkin method[J].Acta Mechanica Sinica,2009,25(4):517-527.
[18]Lee H,Krishnaswamy S.Quasi-static propagation of subinterfacial cracks[J].Journal of Applied Mechanics,2000,67(3):444-452.
[2]Portela A,Aliabadi M H,Rooke D P.The dual boundary element method:Effective implementation for crack problems[J].International Journal for Numerical Methods in Engineering,1992,33(6):1269-1287.
[3]Kikuchi M,Wada Y,Shintaku Y,Suga K,Li Y.Fatigue crack growth simulation in heterogeneous material using s-version FEM[J].International Journal of Fatigue,2014,58:47-55.
[4]Mo3s N,Dolbow J,Belytschko T.A finite element method for crack growth without remeshing[J].International Journal for Numerical Methods in Engineering,1999,46(1):131-150.
[5]Osher S,Sethian J A.Fronts propagating with curvature-dependent speed:algorithms based on Hamilton-Jacobi formulations[J].Journal of Computational Physics,1988,79(1):12-49.
[6]庄茁,柳占立,成斌斌,廖剑晖.扩展有限单元法[M].北京:清华大学出版社,2012.(Zhuang Z,Liu Z L,Cheng B B,Liao J H.The Extended Finite Element Method[M].Beijing:Tsinghua University Press,2012.(in Chinese))
[7]余天堂.扩展有限单元法:理论,应用及程序[M].北京:科学出版社,2014.(Yu T T.The Extended Finite Element Method-Theory,Application and Program[M].Beijing:Science Press,2014.(in Chinese))
[8]Sukumar N,Chopp D L,Mo3s N,Belytschko T.Modeling holes and inclusions by level sets in the extended finite-element method[J].Computer Methods in Applied Mechanics and Engineering,2001,190(46-47):6183-6200.
[9]Zhuang Z,Cheng B B.Equilibrium state of mode-Isub-interfacial crack growth in bi-materials[J].International Journal of Fracture,2011,170(1):27-36.
[10]Singh I V,Mishra B K,Bhattacharya S,Patil R U.The numerical simulation of fatigue crack growth using extended finite element method[J].International Journal of Fatigue,2012,36(1):109-119.
[11]Jiang S,Du C,Gu C,Chen X.XFEM analysis of the effects of voids,inclusions and other cracks on the dynamic stress intensity factor of a major crack[J].Fatigue&Fracture of Engineering Materials&Structures,2014,37(8):866-882.
[12]Kumar S,Singh I V,Mishra B K.A homogenized XFEM approach to simulate fatigue crack growth problems[J].Computers&Structures,2015,150:1-22.
[13]王志勇,马力,吴林志,于红军.基于扩展有限元法的颗粒增强复合材料静态及动态断裂行为研究[J].固体力学学报,2011,32(6):566-573.(Wang Z Y,Ma L,Wu L Z,Yu H J.Investigation of static and dynamic fracture behavior of particle-reinforced composite materials by the extended finite element method[J].Chinese Journal of Solid Mechanics,2011,32(6):566-573.(in Chinese))
[14]Teng Z H,Sun F,Wu S C,Zhang Z B,Chen T,Liao D M.An adaptively refined XFEM with virtual node polygonal elements for dynamic crack problems[J].Computational Mechanics,2018,62(5):1087-1106.
[15]唐旭海,吴圣川,郑超,张建海.基于虚节点的多边形有限元法[J].应用数学和力学,2009,30(10):1153-1164.(Tang X H,Wu S C,Zheng C,Zhang JH.A novel virtual node method for polygonal elements[J].Applied Mathematics and Mechanics(English Edition),2009,30(10):1233-1246.(in Chinese))
[16]吴圣川,吴玉程.ALOF-新一代三维疲劳裂纹扩展分析软件[J].计算机辅助工程,2011,20(01):136-140.(Wu S C,Wu Y C.ALOF:new 3Dfatigue crack propagation analysis software[J].Computer Aided Engineering,2011,20(01):136-140.(in Chinese))
[17]Zheng C,Tang X H,Zhang J H,Wu S C.A novel mesh-free poly-cell Galerkin method[J].Acta Mechanica Sinica,2009,25(4):517-527.
[18]Lee H,Krishnaswamy S.Quasi-static propagation of subinterfacial cracks[J].Journal of Applied Mechanics,2000,67(3):444-452.