Piecewise hierarchical p-version curved shell finite element for heat conduction in laminated composites

Abstract This paper presents a nine-node three-dimensional curved shell finite element formulation for linear steady-state heat conduction in laminated composites where the temperature approximation for the laminate is piecewise hierarchical and is derived based on the P-version. The temperature approximation for the element is developed by first establishing a hierarchical temperature approximation for each lamina of the laminate and then by imposing inter-lamina continuity conditions of temperature at the interface between the laminas. The approximation functions and the nodal variables for a lamina are derived directly from the Lagrange family of interpolation functions of order pξ, pη and kPζ. This is accomplished by first establishing one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ, η and kζ directions for the equivalent (3, 3 and 1 node) configurations that correspond to pξ + 1, pη + 1 and kpζ + 1 equally spaced nodes in the ξ, η, and kζ directions and then taking their products. The nodal variables for the laminated shell element are derived from the nodal variables of the laminas and the inter-lamina continuity conditions of temperature. The element formulation ensures C0 continuity of temperature across inter-element as well as the inter-lamina boundaries.