Chaemin Lee | Searcharxiv

arXiv2022

Preconditioning for finite element methods with strain smoothing

Strain smoothing methods such as the smoothed finite element methods (S-FEMs) and the strain-smoothed element~(SSE) method have successfully improved the performance of finite elements, and there have been numerous applications of them in finite element analysis. For the sake of efficient applications to large-scale problems, it is important to develop a mathematically and numerically well-elaborated iterative solver for the strain smoothing methods. In this paper, inspired by the spectral properties of the strain smoothing methods, we propose efficient ways of preconditioning for the methods. First, we analyze the spectrums of the stiffness matrices of the edge-based S-FEM and the SSE method. Then, we propose an improved two-level additive Schwarz preconditioner for the strain smoothing methods by modifying local solvers appropriately. For the sake of convenience of implementation, an alternative form of the preconditioner is also proposed by defining the coarse-scale operation in terms of the standard FEM. We verify our theoretical results through numerical experiments.

math.NA↗

arXiv2021

A variational framework for the strain-smoothed element method

This paper is devoted to a rigorous mathematical foundation for the convergence properties of the strain-smoothed element (SSE) method. The SSE method has demonstrated improved convergence behaviors compared to other strain smoothing methods through various numerical examples; however, there has been no theoretical evidence for the convergence behavior. A unique feature of the SSE method is the construction of smoothed strain fields within elements by fully unifying the strains of adjacent elements. Owing to this feature, convergence analysis is required, which is different from other existing strain smoothing methods. In this paper, we first propose a novel mixed variational principle wherein the SSE method can be interpreted as a Galerkin approximation of that. The proposed variational principle is a generalization of the well-known Hu--Washizu variational principle; thus, various existing strain smoothing methods can be expressed in terms of the proposed variational principle. With a unified view of the SSE method and other existing methods through the proposed variational principle, we analyze the convergence behavior of the SSE method and explain the reason for the improved performance compared to other methods. We also present numerical experiments that support our theoretical results.

math.NA↗