Modal analysis of biological structures based on the smoothed finite element methods

The smoothed finite element model exhibits a "softening effect," resulting in reduced stiffness compared to the standard finite element model. This study employs the smoothed finite element methods (S-FEMs) with automatically generated tetrahedral meshes to perform modal analysis of biological structures subjected to arbitrary dynamic forces. Various S-FEM models are developed, including Edge-based, Face-based, and Node-based cross-element smoothing domains within the tetrahedral mesh framework, referred to as ES/FS/NS-FEM-T4. Using the gradient smoothing technique, the process of obtaining the strain-displacement matrix requires only the value of the shape function, not the inverse of the shape function, and no mapping is required. Additionally, by incorporating a Taylor expansion term for the strain gradient within the node-based smoothing domain framework, we introduce a stable node-based smoothed finite element method (SNS-FEM). Furthermore, the Lanczos algorithm and the modal superposition technique are integrated into our S-FEM models to compute the transient response of bone structures within the human body. The results obtained from S-FEMs are evaluated against the standard finite element method with respect to accuracy, convergence, and computational efficiency.