Progressive adaptivity formulation in Galerkin meshfree method

Progressive adaptivity formulation in the framework of meshfree methods is proposed and has been successfully applied to elastic and elastoplastic problems. Both improved efficiency and accuracy are achieved in linear and nonlinear problems by employing the proposed progressive adaptivity method.; The error indicator is an important component of adaptive analysis. The recovery approach for construction of the error indicator can be formulated with significant simplicity under meshfree framework since the meshfree shape functions can be arbitrarily smooth. In the proposed method, the smoothing procedures required in the recovery of strain and stress in finite element method are completely eliminated. The relative error, energy error density indication and incremental energy error density indication is computed by taking the difference between the recovery solution and numerical solution directly obtained from the discrete equilibrium equations. Numerical results show that the error indicator can effectively identify locations that require higher resolution in the discretization for elastic and elastoplastic problems.; State variables are calculated and updated incrementally at the integration points in the analysis of elastoplastic problems. State variable transfer is required in progressive adaptivity due to model refinement. Several transfer operators are proposed for transfer of displacement, stress, strain, internal variables, and contact information between the coarse and refined models. The equilibrium and consistency in constitutive equations are both considered during state variable transfer. In the transfer of state variables, the trial state variables are first transferred to the new quadrature points and then the return mapping is performed to satisfy plasticity evolution equations. To minimize interpolation error during large deformation computation, a two-step state variable transfer is proposed by first splitting integration cells for stress or strain transfer and then inserting new particles for displacement transfer. The proposed transfer operators are applied to elastoplastic problems using the updated Lagrangian formulation with multiplicative decomposition. The effectiveness of the progressive adaptivity method has been demonstrated in elastoplastic contact problems with applications to metal forming processes.