Three Dimensional Improved E Xtended Finite Element Method(IXFEM) And Its Applications

Structures in engineering inevitably contain various defects,such as microscopic or macro-scopic cracks.These defects reduce the bearing capacity of the structure,and cracks may prop-agate over time,leading to failure accidents.Modelling three-dimensional crack propagation is a classical difficulty for finite element method(FEM).e Xtended Finite Element Method(XFEM(Belytschko 1999))becomes one of the most important advances for numerical sim-ulation of crack growth in the recent 30 years.Immediately after the method was proposed,it attracted broad attention.Around 2010,mainstream commercial software started to support XFEM as an alternative tool to FEM for crack analysis,signaling the acceptance of XFEM in CAE industries.Although unprecedented progress has been made on crack analyses since the invention of XFEM,several theoretical and practicable issues still remain.For examples,the XFEM module of the mainstream commercial software ABAQUS,ANSYS cannot support the crack tip enrichment technique in the crack growth analysis.The thesis aims to solve the associated problems in 3D static and dynamic crack growth analysis with XFEM,and to develop an efficient and accurate 3D crack growth analysis tool.In static and dynamic crack growth analysis,the following three challenges still remain for XFEM.Firstly,it is the lack of an efficient and high-accuracy geometry representation of the non-planar surface of 3D cracks.The spatial geometry representation is the unique difference and difficulty compared with 2D crack growth simulation.Secondly,in order to deliver optimal convergence in solving a singular problem,the well-known linear dependence issue,which results in slow or even non-convergence for an iterative solver,arises if XFEM uses geometrical refinement enrichment strategy with a fixed size of crack tip enrichment area during mesh refinement.Thirdly,energy consistency and mass concentration issues hinder,since the beginning of the GFEM/XFEM,a straightforward extension to dynamic problems.With a special emphasis on accurate geometries of the 3D crack geometry,we propose in this thesis a B-spline ruled surface method(BRSM),an analytical,mesh free crack representa-tion algorithm,offering high order smoothness in the surface and C~0smoothness between the surfaces.Compared with the mesh-based geometric description,the proposed method has no mesh-resolution dependence,maintains the geometric accuracy of the 3D crack front,avoids the ambiguity of the normal definition in the crack front and the crack surface,naturally elim-inates the numerical oscillation of stress intensity factors,and significantly improves the com-putational efficiency of level set values.As a whole,high efficiency and high accuracy of 3D crack growth simulation are achieved.In order to solve the problem of linear correlation and poor conditioning of the global stiffness matrix introduced by Partition of Unity approximation,the 2D improved extended finite element method(IXFEM)without extra degrees of freedom in tip enrichment is extended to 3D.The linear iterative convergence issue caused by the linear dependence is fundamentally overcome.The poor conditioning of the global stiffness matrix is completely avoided in the geometrical refinement,as a result,the optimal convergence rate for the singular problem is delivered.By eliminating the extra degrees of freedom in crack tip enrichment,solved is that the traditional XFEM cannot be directly extended from static crack simulation to dynamic crack analysis.The daunting issues of the so-called"null"critical time step and the energy consis-tency are fundamentally solved by IXFEM.The extension from the 3D static crack growth simulation to the 3D dynamic becomes the same natural as FEM.The three dimensional inter-action integration technique for dynamic problems is derived;the high accuracy of 3D dynamic stress intensity factors is assured.Finally,a C++code for 3D static and dynamic crack growth analyses is developed.The developed IXFEM shares the same shape functions except the local area around the crack.As such IXFEM is easy to implement and fits into,without a difficulty,a FE software architecture.The contributions of the thesis are as follows.(1)A B-spline ruled surface method(BRSM),which offers good geometrical accuracy,good numerical efficiency,and no mesh-resolution dependence.(2)An IXFEM for three-dimensional non-planar fatigue crack growth analysis.(3)An IXFEM for three-dimensional non-planar dynamic crack growth analysis.