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.
Nonlinear Geotechnical Analysis Using a Polygonal Cell-Based Smoothed Finite Element Framework
College of Water Conservancy and Hydropower Engineering, Hohai University（河海大学水利水电学院）；PowerChina Kunming Engineering Corporation Limited（中国电建昆明工程有限公司）；College of Harbour, Coastal and Offshore Engineering, Hohai University（河海大学港口海岸与近海工程学院）
Elasto-plastic cell-based smoothed finite element method solving geotechnical problems
PowerChina Kunming Engineering Corporation Limited（中国水利水电昆明工程有限公司）；College of Water Conservancy and Hydropower Engineering, Hohai University（河海大学水利水电学院）
Displacement based finite element formulations over polygons: a comparison between Laplace interpolants, strain smoothing and scaled boundary polygon formulation
School of Civil & Environmental Engineering The University of New South Wales（新南威尔士大学土木与环境工程学院）
Convergence Analysis of Adaptive Finite Element Algorithms for a Regularized Variational Model of Quasi-Static Brittle Fracture in "Strain-Limiting" Elastic Solids
Academy of Scientific and Innovative Research, CSIR-Structural Engineering Research Center（印度科学研究所学院，CSIR结构工程研究中心）；Center for Scientific and Industrial Research（科学与工业研究中心）；Department of Mechanical Engineering, Indian Institute of Technology, Madras（印度马德拉斯理工学院机械工程系）；Faculté des Sciences, de la Technologie et de al Communication, University of Luxembourg（卢森堡大学科学与技术通信学院）
Variational inequalities of multilayer elastic systems with interlayer friction: existence and uniqueness of solution and convergence of numerical solution
