An h-Version Adaptive Finite Element Scheme for Eigensolutions of Free Vibration of Three-Dimensional Cracked Elasticity Utilizing Element Subdivision-Based Error Estimator

The disturbance of microcracks in three-dimensional elasticity affects the inherent properties and performance of free vibration, and traditional theoretical methods have difficulty obtaining analytical solutions for eigensolutions of free vibration of elasticity containing microcracks, making it challenging to conduct complex dynamic response analysis induced by microcracks. This study developed a characterization method for microcrack damage in three-dimensional elasticity, and an h-version adaptive finite element (FE) scheme for eigensolutions of free vibration of three-dimensional cracked elasticity utilizing an element sub-division-based error estimator was proposed. Through the results of representative examples, the proposed method can obtain solutions of three-dimensional elasticity that meet the prespecified error tolerance and can provide fewer optimized elements than conventional methods that rely on denser meshes to obtain high-precision solutions $(\bar{\omega}_{k}^{h})$. To accurately capture the variation in the vibration mode of the domain around the crack, this proposed adaptive method automatically densifies meshes. The presence of cracks reduces the stiffness of the elasticity, thereby reducing the frequency values of each order; as the crack depth $(h_{c})$ and number increase, the stiffness of the elasticity deteriorates and decreases, thereby reducing the frequency.