h-Adaptive Extended Finite Element Method for Structural Optimization

Structural optimization has become a powerful tool to inspire engineers for more reasonable and economical designs during the past decades. Compared to previously developed material based approaches, the level set method for structural optimization is gaining popularity recently due to its flexibility in boundary representation and handling complex topological changes of structure. The structural boundary is tracked by an implicit level set function and its evolution is driven by boundary velocity which is derived from sensitivity analysis with the result of finite element analysis. However, conventional Finite Element Method (FEM) is troublesome in handling either moving boundaries between solid material and voids or topological changes, as the finite element meshes need to conform to the boundaries of structure resulting in the time-consuming remeshing process. With the advent of Extended Finite Element Method (X-FEM), shape functions of conventional FEM are extended with enrichment functions, which make X-FEM suitable for representing aforementioned moving boundaries, and usually fixed uniform meshes are employed so that mesh management difficulties can be avoided. However, to capture precision boundaries, denser meshes are desired, which to some extent decreases the efficiency of the X-FEM, and meanwhile finite element analysis is regarded as the most time-consuming process and conducted at each iterative step during the optimization. In the level set based structural optimization, boundaries are most concerned where denser meshes are most desired while the regions far away from boundaries only need coarser meshes. Therefore, to improve the efficiency of the X-FEM and shorten the optimization process, it is essential to adjust underlying meshes adequately. The solution scheme is that finite element meshes of higher resolution are distributed in the vicinity of the boundaries while meshes of relatively lower resolution are in the regions far away from the boundaries to significantly decrease the computational time while ensuring the accuracy. The motivation of this dissertation is to develop an efficient and accurate X-FEM scheme with adaptive meshes for structural optimization in the level set framework.;Based on previous studies on conventional X-FEM with fixed uniform meshes, h-Adaptive X-FEM is investigated and developed in both two and three-dimensions. Multilevel adaptive meshes are generated to fit but not necessary to conform to structural boundaries by the method of mesh coarsening to gradually remove unnecessary elements from initial fine uniform meshes. The underlying meshes are depicted by Quadtree (for 2D) or Octree (for 3D) representations which are suitable for managing multilevel data on the one hand and make the generation of adaptive meshes efficient on the other hand. In this thesis, adaptive meshes are restrained to 1-irregular meshes and elements with hanging nodes are produced during the process of mesh coarsening. The hanging nodes are treated as regular nodes with degrees of freedoms (DOFs) and shape functions which are modified to satisfy Partition of Unity (POU) property for each element. Quadrature for different types of elements is studied and special schemes should be adopted for elements crossed by boundaries and elements with hanging nodes where kinks would exist. As the X-FEM generalizes conventional FEM to handle structure whose boundaries are not necessarily covered by conforming meshes, the boundary conditions are possible inside the meshes which presents difficulties while imposing boundary conditions. On the one hand, the imposition of Neumann boundary conditions is not difficult because it only requires a modification of the integral domain from borders of elements to the boundaries insides elements. However, imposing Dirichlet boundary conditions is non-trivial. In this dissertation, Nitsche's method is employed to enforce Dirichlet boundary conditions. In order to verify the imposition of boundary conditions, accuracy, efficiency and convergent rate of the proposed X-FEM, 2D examples with theoretical solution are treated as benchmarks and 3D numerical examples are conducted by the comparison with solutions produced by ANSYS software.;The mean compliance problems of structural optimization are investigated by combining level set method and the proposed h-Adaptive X-FEM. Notably, the X-FEM is achieved on the adaptive meshes while the evolution of level set is conducted on the fine uniform meshes. The adaptive meshes are updated accordingly along with the propagation of structural boundaries at each optimization step, and meanwhile lots of mature algorithms can be used for level set evolution with uniform grids directly. Numerical examples both in 2D and 3D commonly used in literatures are treated as benchmarks especially two practical applications further verify the reliability of the proposed X-FEM.;Keywords: Finite element method, X-FEM, h-Adaptive X-FEM, Level set method, Structural optimization, Nitsche's method, Neumann boundary conditions, Dirichlet boundary conditions, Quadtree, Octree, Hanging nodes.