Precise Integration Symplectic Analytical Singular Element for Cracks Analysis Under Transient Thermal Conduction

Abstract

Numerical modeling of mechanical behavior of cracks under transient thermal conduction involves solving an initial value problem (IVP) and two boundary value problems (BVPs). Both of the BVPs have a singularity issue. Drawbacks such as numerical error accumulation and high computational expense of existing numerical approaches should be overcome. This contribution intends to build a unified framework with highly efficiency and accuracy for the numerical modeling of cracks under thermal shock. The precise integration method (PIM) and the symplectic analytical singular element (SASE) have been demonstrated to be favorable alternatives for each problem, i.e., the PIM for solving the IVP and SASE for the BVP. However, it is found that these two methods cannot be combined directly. In order to incorporate the SASEs into the PIM, the existing SASEs are reformulated for the thermal shock cracks analysis. Details of the mathematical derivations are provided. The validity of the proposed method is demonstrated through numerical examples.