Completely hierarchical axisymmetric shell element based on p-version for heat conduction in laminated composites

Abstract

A completely hierarchical p-version finite element formulation is presented for an axisymmetric shell element for heat conduction in laminated composites, where the element temperature field can be of arbitrary polynomial orders p ξ and p η in the longitudinal (ξ) and the transverse (η) directions of the shell element. The approximation functions and the corresponding nodal variables for the axisymmetric shell element are derived by first constructing the one-dimensional hierarchical approximation functions of orders p ξ and p η and the corresponding hierarchical nodal variable operators for each of the two directions ξ and η and then taking their product (sometimes called the tensor product). This procedure gives the approximation functions and the nodal variables for the axisymmetric shell element that correspond to the polynomial orders p ξ and p η . The element approximation functions and the nodal variables are both hierarchical and therefore the element matrices and the vectors corresponding to orders p ξ and p η are a subset of those corresponding to the polynomial orders ( p ξ + 1) and ( p η + 1). The formulation ensures C 0 continuity across the interelement boundaries. The geometry of the element is described in the usual way by the Cartesian coordinates of the nodes located at the middle surface ( η = 0) of the element and the nodal vectors representing nodal values of the lamina thicknesses. Weak formulation of the Fourier heat conduction equation for globally orthotropic material is constructed in the cylindrical coordinate system rz. The element properties are derived using the weak formulation (or the associated quadratic functional) and the hierarchical element approximation. This formulation is extended for generally orthotropic material behavior where the material directions are not necessarily parallel to the global axes. Further extension of the formulation for laminated composites is accomplished by incorporating the material properties of each layer through numerical integration of the element matrix for each layer. The element matrices and the equivalent heat vectors (due to convection, distributed heat flux and internal heat generation) are all hierarchical. The formulation permits any desired order temperature distribution in the ξ and η directions without remodeling. There is no restriction on the number of layers and the lay up pattern of the layers. Each layer can be generally orthotropic. The material directions and the layer thicknesses may vary from point to point within each layer. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation. For all examples, h-model results using isoparametric elements are also presented. Comparisons are made with analytical solutions wherever possible.