No commercial reproduction, distribution, display or performance rights in this work are provided.Description: There have been substantial developments in meshfree methods, particle methods, and generalized finite element methods since the mid 1990s. Meshfree methods particle methods updated Lagrangian maximum entropy tensile instability finite deformation severe plastic deformation (SPD) equal-channel angular extrusion (ECAE) multiscale modeling phase Transformation energy minimization homogenization nonconvex energy potential The results from both of these applications show that the proposed method brings new possibilities to the subject of computational solid mechanics that are not within the reach of traditional mesh-based and meshfree methods. In the second application, the aforementioned max-ent meshfree method outperforms FEM and FFT-based schemes in numerical relaxation of nonconvex energy potentials, which is essential in discovering the effective response and associated energy-minimizing microstructures and patterns.

the finite element method (FEM)) are limited due to severe mesh distortions. This scheme is especially suitable for ECAE as the latter falls into the category of severe plastic deformation processes where simulations using mesh-based methods (e.g. In ECAE, the aforementioned enhanced maximum-entropy scheme allows the stable simulation of large deformations at the macroscale. This thesis presents an extensive study of two applications - the modeling of equal-channel angular extrusion (ECAE) based on high-fidelity plasticity models, and the numerical relaxation of nonconvex energy potentials.

The improved local maximum-entropy approximation method is of a general construct and has a wide variety of applications. The proposed method achieves robust stability in the updated-Lagrangian setting and fully realizes the potential of meshfree methods in simulating large-deformation mechanics, as shown for benchmark problems of severe elastic and elastoplastic deformations. The proposed method offers an adaptive approximation that addresses the tensile instability which arises in updated-Lagrangian meshfree methods during severe, finite deformations. This thesis develops an enhanced meshfree method based on the local maximum-entropy (max-ent) approximation and explores its applications.