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<p>Accurate simulation of materials at micro-scales requires higher-order continuum theories, such as the Modified Couple Stress Theory (MCST), which inherently demands <i>C</i><sup>1</sup> continuity in the numerical solution. Constructing <i>C</i><sup>1</sup> discretizations within the classical finite element framework remains challenging and is often restricted to structured meshes. In contrast, the Virtual Element Method (VEM) provides a flexible framework for discretizing general unstructured polygonal meshes satisfying standard mesh-quality conditions, while naturally accommodating higher-order continuity requirements. Existing VEM formulations for higher-order theories typically rely on empirically tuned, problem-dependent stabilization terms, which compromise theoretical rigor and robustness. In this work, we propose a stabilization-free virtual element method (SFVEM) for the MCST that overcomes this limitation within a <i>C</i><sup>1</sup>-continuous setting. The key ingredient of the proposed formulation is an enhanced virtual element space that enables the explicit computability of higher-order moments, thereby allowing the construction of an L<sup>2</sup> projection operator for strains and curvatures. The elemental stiffness matrix is then computed directly from this projection, ensuring sufficient rank and numerical stability without introducing any artificial stabilization. A comprehensive set of numerical examples, including standard benchmark problems and comparisons with experimental data, demonstrates that the proposed method is robust and capable of accurately capturing size-dependent mechanical effects in micro-scale materials. These results indicate that the proposed SFVEM provides a reliable and efficient computational tool for the simulation of microstructured solids.</p>

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Eliminating empirical stabilization: a two-dimensional C1 stabilization-free virtual element method for modified couple stress theory

  • 期刊论文

  • 发表日期：

    2026年5月16日

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  • Ming-Jun Zhao,
  • Peng-Zhao Mo,
  • Song Cen,
  • Yan Shang,
  • Chen-Feng Li

摘要

Accurate simulation of materials at micro-scales requires higher-order continuum theories, such as the Modified Couple Stress Theory (MCST), which inherently demands C1 continuity in the numerical solution. Constructing C1 discretizations within the classical finite element framework remains challenging and is often restricted to structured meshes. In contrast, the Virtual Element Method (VEM) provides a flexible framework for discretizing general unstructured polygonal meshes satisfying standard mesh-quality conditions, while naturally accommodating higher-order continuity requirements. Existing VEM formulations for higher-order theories typically rely on empirically tuned, problem-dependent stabilization terms, which compromise theoretical rigor and robustness. In this work, we propose a stabilization-free virtual element method (SFVEM) for the MCST that overcomes this limitation within a C1-continuous setting. The key ingredient of the proposed formulation is an enhanced virtual element space that enables the explicit computability of higher-order moments, thereby allowing the construction of an L2 projection operator for strains and curvatures. The elemental stiffness matrix is then computed directly from this projection, ensuring sufficient rank and numerical stability without introducing any artificial stabilization. A comprehensive set of numerical examples, including standard benchmark problems and comparisons with experimental data, demonstrates that the proposed method is robust and capable of accurately capturing size-dependent mechanical effects in micro-scale materials. These results indicate that the proposed SFVEM provides a reliable and efficient computational tool for the simulation of microstructured solids.