Development of Locking-Free FE2 Scheme Using Hybrid Elements

  • Kumar, Sourav (Indian Institute of Technology Ropar)
  • Agrawal, Manish (Indian Institute of Technology Ropar)
  • Kumar, Navin (Indian Institute of Technology Ropar)

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Multiscale modeling plays a crucial role in accurately predicting the behavior of composite, biological, and numerous other natural materials that exhibit heterogeneity at the microscopic scale. The finite element method-based multiscale modeling (also known as FE2) is widely used to simulate the behavior of such materials [1]. In this approach, a representative volume element (RVE), which is sufficiently large to encompass the main characteristics of the material’s microstructure is associated with each integration point in the discretized macroscopic domain. Finally, the macroscopic boundary value problem is solved through an iterative process, where the microscopic boundary value problems are solved within each macroscopic iteration loop in a coupled manner. Literature [1,2] indicates that although the conventional FE2 method is flexible, it is susceptible to ’ locking’ in specific circumstances where the problem involves the bending of structures, near-incompressibility conditions, or analysis of thin structural geometries. In such cases, accurately capturing the mechanical response requires a higher number of degrees of freedom, which substantially increases computational cost and time. It is well known that hybrid elements are much less prone to volumetric, shear, and membrane locking compared with conventional finite elements [2]. However, their application has been limited to single-scale modeling and remains unexplored for multiscale modeling. These elements are formulated based on the two-field Hellinger-Reissner variational principle, where displacement and stress are interpolated independently. In this work, we propose a hybrid element-based FE2 scheme in the context of finite deformation, which, to the best of our knowledge, represents the first such attempt in the literature. We assess its performance against the conventional FE2 method through various numerical examples, demonstrating its superior computational efficiency. Notably, the hybrid FE2 approach requires approximately ten times fewer degrees of freedom to achieve convergence compared to conventional FE2, significantly reducing computational cost and time.