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.

本文致力于为应变平滑元素（SSE）方法的收敛特性建立严格的数学基础。通过各种数值示例，与其他应变平滑方法相比，SSE方法已证明改善了收敛性能。然而，目前尚无理论证据证​​明其收敛性。SSE方法的一个独特功能是通过完全统一相邻单元的应变来在单元内构造平滑的应变场。由于这一特征，需要进行收敛分析，这与其他现有的应变平滑方法不同。在本文中，我们首先提出了一种新颖的混合变分原理，其中SSE方法可以解释为该方法的Galerkin近似。提出的变分原理是对著名的Hu-Washizu变分原理的概括。因此，可以根据提出的变分原理来表达各种现有的应变平滑方法。通过提出的变分原理，对SSE方法和其他现有方法具有统一的看法，我们分析了SSE方法的收敛行为，并解释了与其他方法相比性能得以改善的原因。我们还提出了支持我们的理论结果的数值实验。我们分析了SSE方法的收敛行为，并解释了与其他方法相比性能提高的原因。我们还提出了支持我们的理论结果的数值实验。我们分析了SSE方法的收敛行为，并解释了与其他方法相比性能提高的原因。我们还提出了支持我们的理论结果的数值实验。