Study Of Topology Optimization For Geometrically Nonlinear Structures Using Separated Density Field Interpolation

Topology optimization based on nonlinear analysis has broad application prospects in the field of engineering, such as to design bi-stable actuation elements used for energy harvesting, compliant mechanisms that are widely applied in Micro Electronic Mechanical System (MEMS), and special structures employed in the precision manufacture as well as the biomedical technology.Currently almost all studies on topology optimization of structures with large deformation are based on the finite element method for the displacement field analysis and accordingly element-based density representation for the material density distribution. However, the challenge in performing reliable and robust topology optimization in the FE-based framework for problems involving large deformations lies in two-folds. Firstly, mesh distortion may seriously worsen the numerical performance of the displacement field analysis and thus cause difficulties for numerically solving. Secondly, local instabilities may occur in some small regions with low density values, which also cause convergence difficulties. Thus, the aim of this dissertation is to formulation a novel framework of topology optimization for geometrically nonlinear structures, which enable to overcome above numerical problems and perform effectively on topology analysis design even undergo large deformations.Based on the Element-free Galerkin method, an analysis-independent point-based density variable approach is proposed for topology optimization of geometrically nonlinear structures. Without limitation of "mesh", this optimization approach avoids the mesh distortion problem often encountered in the finite element analysis of large deformations. The continuous material density field of topology optimization problem is formulated on the basis of point-wise density variables description. This density field is constructed by a materially physical meaning-preserving interpolation with the density values of the design variables points, which can be freely positioned independently of the field points used in the displacement analysis.Since the traditional displacement-based criterion of convergence is infeasible in such circumstances, an energy criterion of convergence is suggested to resolve the well-known convergence difficulty, which would be usually encountered in low density regions. To predict and resolve the numerical instabilities problems which are usually occurred on the point-based design variables scheme, such as "islanding" phenomenon or "layering" phenomenon, a stability condition is given by analyzing the topology problem based on the mixed variation-concept. Consequently, the instabilities are settled by the proposed separated density filed framework.Numerical examples are given to demonstrate the effectiveness of the developed approach. It is shown that relatively clear optimal solutions can be achieved, without exhibiting numerical instabilities like the so-called "layering" or "islanding" phenomena even in large deformation cases. This study not only confirms the potential of the EFG method in topology optimization involving large deformations, but also provides a novel truly element-free topology optimization framework that can easily incorporate other meshless analysis methods for specific purposes.