Axisymmetric shell elements for heat conduction with p-approximation in the thickness direction

Abstract

This paper presents a finite element formulation for the axisymmetric shell elements for heat conduction where the element temperature approximation in the direction of the shell thickness can be of an arbitrary polynomial order p. This is made possible by introducing additional nodal variables in the element temperature approximation corresponding to the complete Lagrange interpolating polynomials in the direction of the shell thickness. The elements presented have an important hierarchical property, i.e. the element properties corresponding to an approximation order p are a subset of the properties corresponding to an approximation order p + 1. The element formulation retains continuity or smoothness of temperature across the interelement boundaries, i.e. C0 continuity is enforced automatically.The axisymmetric shell geometry is constructed in the usual way by using the coordinates of the nodes lying on the middle surface of an element (η = 0) and the node point normals to the middle surface. The element temperature approximation consists of hierarchical approximation functions, nodal temperatures and the derivatives of the nodal temperatures corresponding to the complete polynomials in the direction of the shell thickness. The weak formulation (or the quadratic functional) of the Fourier heat conduction equation is constructed. The element properties are derived using weak formulation (or the quadratic functional) and the hierarchical element approximation. The element matrices and the equivalent heat vectors (resulting from convective boundaries, distributed heat flux and internal heat generation) are all hierarchical. The element formulation permits a temperature distribution of any desired order in the shell thickness direction.Numerical examples demonstrate the accuracy, efficiency and overall superiority of the present formulation over existing axisymmetric shell elements. The results are also compared with analytical solutions and for all four examples, h-approximation results are presented for comparison.