| The main objective of this research is to develop various hybrid meshfree formulations as an enhancement of the standard meshfree formulation for certain classes of problems exhibiting roughness in the solution or locking difficulty due to constraint conditions.; An extended meshfree method, developed by coupling meshfree formulation with a particular solution as an enhancement of the meshfree solution, is first presented for Poisson problems. This approach is further extended to meshfree formulation for shear deformable plates, where a locking-free stabilized conforming nodal integration for a Mindlin-Reissner plate variational equation is also proposed. Numerical examples of Poisson problem as well as shear deformable plate problems demonstrate the effectiveness of the proposed method.; Next, a new approach to resolve numerical difficulties in meshfree shell formulation is presented. The limitations of meshfree approximation functions formulated using a parametric coordinate, as well as the singularity problem resulting from the imposition of reproducing conditions on shell surfaces using Cartesian coordinates, are addressed. A constrained reproducing kernel approximation is proposed to remove this singularity and to allow the construction of meshfree shape functions on shell surfaces using global Cartesian coordinates for arbitrary geometry. A new meshfree shell formulation is introduced using the proposed constrained reproducing kernel approximation, in conjunction with the stabilized conforming nodal integration, to achieve stability, accuracy, and efficiency in shell applications.; Another difficulty in the conventional meshfree formulation is associated with the material interface jump conditions in composite materials that cannot be properly approximated by the smooth meshfree shape functions with arbitrary point distribution. Here, an interface enriched reproducing kernel approximation is introduced to embed the discontinuity across the material interface in the meshfree shape functions for analysis of composite materials. Further, as the conventional meshfree formulation cannot properly handle the periodicity conditions that commonly occur in homogenization analysis of composites, an enhanced meshfree approximation is developed to correct this deficiency. Application of the proposed interface enriched reproducing kernel approximation with corrected periodic treatment to the homogenization of magnetostrictive particle filled elastomers is considered as a specific use of this method. |
