Stable Generalized Finite Element Method For Interfacial Crack In Bimaterials And Its DOF-gathering Technique

The bi-material problem has important applications in piezoelectric materials,laminate composites,thermal barrier coatings,microelectronic devices,etc.Because of the non-smooth characteristics such as discontinuity and singularity,the Finite Element method(FEM)requires mesh matching and refinement operations for solving the crack problem,resulting in a heavy mesh generation effort.The Generalized FEM(GFEM)provides a successful solution to the cracking problem by using a simple,fixed mesh and improving the approximation accuracy by augmenting(also known as enrich)special functions that characterize the local features of the real solution.Most of the recent research focuses on homogeneous materials problem,and only some of the algorithm convergence studies have been conducted for the bi-material problem.Some important results of GFEM for homogeneous material crack problems,such as stability,robustness,theoretical analysis,and reduction of degree of freedoms,have not yet been addressed in bi-material crack problems.Compared with the homogeneous material problem,the bimaterial problem has more complex non-smooth features due to the difference of material coefficients,such as weak discontinuity and fibrillation singularity in addition to discontinuity and singularity,etc.The design and analysis of the related GFEM algorithms are very difficult and still in the initial stage.In this study,we construct and analyze a stable GFEM format for bi-material crack problems,namely Stable GFEM.Firstly,we design the enrichment function based on the non-smooth characteristics of the bi-material problem:the linear Heaviside function is used to mimic the discontinuity at the crack,the distance function is used to mimic the weak discontinuity at the uncracked interface,and a class of 8-fold singularity function is used to mimic the singularity of the exact solution.Secondly,in order to improve the condition number of SGFEM stiffness matrix,we stabilize SGFEM by changing the partition of unity(PU)function and applying local principal component analysis(LPCA).Numerical results show that the new SGFEM can achieve the optimal convergence order for the bi-material crack problem,the condition number of the stiffness matrix is of the same order as that of the finite element method,and neither the convergence nor the stability changes when the crack is close to the boundary of the cells,i.e.,the algorithm is robust.These results are new in the field of GFEM for bi-material crack problems.Theoretical analysis of GFEM for crack problems is difficult due to the simultaneous existence of singularity,discontinuity and weak discontinuity,the overlap of the three related enrichment functions,and the so-called blending element error of the GFEM in dealing with the distance function of weak discontinuity.These difficulties make the theoretical analysis of GFEM for bi-material cracking problem very difficult,and the GFEM analysis method for the related homogeneous material cracking problem cannot be directly applied to the bi-material problem with weak discontinuities.In this paper,we adopt the SGFEM framework to decouple the three types of enrichment functions,analyze the local approximation errors one by one,and then integrate the relevant results together by PU function to finally prove the optimal convergence order of the studied method.The theoretical results hold for the proposed consistent triangular and quadrilateral meshes,and also for the general PU function,which mainly satisfies the stability similar to the finite element shape function.Numerical experiments show that the 8-group geometry enriched SGFEM of this paper can reach the optimal convergence order O(h);its SCN can reach O(h-2),the same order as that of standard finite elements;its scaling condition number is robust to Γb and the relative positions of the grid lines are robust;also,the condition number of SGFEM is insensitive to the ratio of the material coefficients.In order to reduce the computational effort,we have developed a degree of freedom reduction technique for SGFEM,the so-called DOF-gathering technique.The ramp function is used to integrate the singularity enrichment of SGFEM,which greatly reduces the overall degrees of freedom of the stiffness matrix.The ramp function of the developing DOF-gathering SGFEM is constructed from the PU function,which is different from the ramp function of the homogenous material problem.The two common types of ramp functions can be unified to design algorithms and perform theoretical analysis.The optimal convergence of the proposed DOF-gathering SGFEM is proved.With the local partial principal component analysis technique and orthogonal processing,the number of conditions of the stiffness matrix is also of the same order as the finite element method.Numerical experiments show that the 8-group geometry enriched DOF-gathering SGFEM of this paper can reach the optimal convergence order O(h);its SCN can reach O(h-2),the same order as that of standard finite elements;its scaling condition number is robust to Γb and the relative positions of the grid lines are robust;also,the condition number of DOF-gathering SGFEM is insensitive to the ratio of the material coefficients.