Three-dimensional curved shell element based on completely hierarchical p-approximation for heat conduction in laminated composites

Abstract

This paper presents a finite element formulation for the laminated three-dimensional curved shell element for heat conduction where the temperature approximation for the element can be of arbitrary polynomial order p ξ , p n , and p ξ in the ξ, n, and ξ directions. This is accomplished by first constructing element approximation functions and the corresponding nodal variable operators for each of the three directions ξ, n, and ξ using Lagrange interpolating polynomials and then taking their products (sometimes also referred to as tensor products). This procedure yields the desired approximation functions as well as the nodal variables that correspond to the polynomial orders p ξ , p n, and p ξ . The element approximation functions as well as the nodal variables are hierarchical and therefore the element matrices and the equivalent nodal heat vectors due to distributed heat flux, convection and internal heat generation are hierarchical also, i.e., the element properties corresponding to the polynomial orders p ξ , p n and p ξ are a subset of those corresponding to the polynomial orders ( p ξ + 1), ( p h +1), and ( p ζ + 1). The element formulation retains continuity or smoothness of temperature across the inter-element boundaries, i.e., C 0 continuity is ensured. Since true derivatives of the temperature are retained as nodal variables in the thickness direction of the shell, the element formulation permits step change in the thickness at a mating boundary resulting in continuous temperature distribution along such a mating face. The curved shell geometry is constructed in the usual manner by the coordinates of the element nodes lying on the middle surface of the element (ζ = 0) and the lamina thicknesses at the nodes. The temperature approximation for the element is described by the hierarchical approximation functions and the nodal variables both obtained through the use of tensor product. The element properties, i.e., element matrices and the equivalent nodal heat vectors are derived using weak formulation (or the quadratic functional) of three-dimensional Fourier heat conduction equation and the hierarchical element temperature approximation. The properties of the composite laminate are incorporated in the element formulation through numerically integrating the element matrices for each lamina. The formulation has no restriction on the number of laminas or the layup pattern of the laminas. Each lamina can be generally orthotropic and the material conductivities and the lamina thicknesses may vary from point to point within each lamina. The element formulation is equally effective for very thin as well as very thick laminated curved shells. In fact, in most of the three-dimensional applications the element can be used to replace the hierarchical three-dimensional solid element without any loss of accuracy but significant gain in modeling convenience. Numerical examples are presented to establish the accuracy of the formulation and to demonstrate its efficiency, modeling convenience, and fast rate of convergence. The numerical results obtained from the present formulation are compared with analytical solutions whenever possible. The h-approximation results obtained using three dimensional solids and thick shells are also presented for comparison purposes.