A topology optimization method for geometrically nonlinear structures with meshless analysis and independent density field interpolation

Based on the element-free Galerkin (EFG) method, an analysis-independent density variable approach is proposed for topology optimization of geometrically nonlinear structures. This method eliminates the mesh distortion problem often encountered in the finite element analysis of large deformations. The topology optimization problem is formulated on the basis of point-wise description of the material density field. This density field is constructed by a physical meaning-preserving interpolation with the density values of the design variable points, which can be freely positioned independently of the field points used in the displacement analysis. An energy criterion of convergence is used to resolve the well-known convergence difficulty, which would be usually encountered in low density regions, where displacements oscillate severely during the optimization process. 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 topology optimization framework based on element-free discretization of displacement and density fields, which can also easily incorporate other meshless analysis methods for specific purposes.