Truncated Weighted Singular Value Decomposition Solution of Tikhonov Regularized Softening Shear Band Problems in Finite Elements

ABSTRACT A Truncated Weighted Singular Value Decomposition (TWSVD) based approach is proposed for the Tikhonov regularized solution of the ill‐posed softening type nonlocal plasticity model. Tikhonov regularization provides a stable, smooth, and mesh‐independent solution of integral‐type nonlocal plasticity, but is computationally expensive for large‐scale problems, particularly in three‐dimensional models. The proposed solution technique offers an efficient method for the solution of the large‐scale Euler equation resulting from the Tikhonov regularization, which is significant for practical applications of softening problems. The TWSVD approach modifies the truncated SVD method to apply to the numerical integration method commonly used in finite element (FE) models. The mathematical formulation of the proposed approach is presented with a linear isotropic plasticity model, and a stress‐update algorithm is developed for the FE application of the approach. The solution algorithm is implemented in a three‐dimensional model with isoparametric brick elements. The solution approach is validated with a benchmark shear‐band softening problem. Results show that the TWSVD method can significantly reduce the computational time compared to conventional Tikhonov‐regularized solutions, without sacrificing accuracy, making it a practical and efficient solution tool for large‐scale systems in nonlocal plasticity models with softening behavior.