Last modified: 2023-05-16
Abstract
In this paper, arbitrarily high-order line, triangular and tetrahedral elements are constructed based on Lagrange polynomial interpolation formulation and they are used in the zonal free element method analysis. The line element is used in the regular zone, and the triangular and tetrahedral elements are used in triangular and tetrahedral zones for 2D and 3D problems, respectively. A set of analytical expressions for computing the first- and second-order partial derivatives with respective to the global coordinates are derived for the constructed elements. The derived spatial derivatives can be directly substituted into the governing partial differential equations and related boundary conditions to set up the system of equations with physical variables at collocation nodes as unknowns.
The presented free element method in this paper is a type of collocation methods, in which line/triangular/tetrahedral elements are utilized at each collocation node to establish the solution scheme solving engineering problems. To solve the problem involving multiple materials or irregular geometries, a zonal technique is employed, in which the computational domain is divided into several zones according to material types and geometric characteristics. For nodes shared by two or more zones, the traction/flux equilibrium equations are used to constrain their variation across the zonal interfaces. The proposed method will be used to solve some challenging thermal and mechanical problems of 2D and 3D composite structures to demonstrate the robustness of the proposed method.