A Symplectic Analytical Singular Element for Steady-State Thermal Conduction With Singularities in Anisotropic Material

Abstract

Modeling of steady-state thermal conduction for crack and v-notch in anisotropic material remains challenging. Conventional numerical methods could bring significant error and the analytical solution should be used to improve the accuracy. In this study, crack and v-notch in anisotropic material are studied. The analytical symplectic eigen solutions are obtained for the first time and used to construct a new symplectic analytical singular element (SASE). The shape functions of the SASE are defined by the obtained eigen solutions (including higher order terms), hence the temperature as well as heat flux fields around the crack/notch tip can be described accurately. The formulation of the stiffness matrix of the SASE is then derived based on a variational principle with two kinds of variables. The nodal variable is transformed into temperature such that the proposed SASE can be connected with conventional finite elements (FE) directly without transition element. Structures of complex geometries and complicated boundary conditions can be analyzed numerically. The generalized flux intensity factors (GFIFs) can be calculated directly without any postprocessing. A few numerical examples are worked out and it is proven that the proposed method is effective for the discussed problem, and the structure can be analyzed accurately and efficiently.