P-Version axisymmetric solid element for heat conduction

Abstract

A nine-node two-dimensional axisymmetric solid element is presented for linear steady state axisymmetric heat conduction where the temperature field in the ξ and η directions of the element can be of arbitrary polynomial orders p ξ and p η . This is accomplished by first constructing the one-dimensional hierarchical approximation functions and the corresponding nodal variable operators in the ξ and η directions using Lagrange interpolating polynomials and then taking the product (sometimes also called tensor product) of these hierarchical one-dimensional approximation functions and the corresponding nodal variable operators. The resulting approximation functions and the nodal variables are hierarchical, i.e. the element approximation functions and the nodal variables corresponding to the polynomial orders p ξ and p η are a subset of those corresponding to the polynomial orders p ξ + 1 and p η + 1 and as a consequence the element matrices and the equivalent nodal heat vectors are hierarchical also. The formulation ensures C 0 continuity, i.e. the continuity of the temperature across the interelement boundaries is guaranteed. The weak formulation of the Fourier heat conduction equation in the cylindrical coordinate system r, z is constructed and the element properties are derived using the weak formulation (or the associated quadratic functional) and the hierarchical temperature approximation for the element. The element formulation permits isotropic as well as generally orthotropic material properties. The formulation allows distributed heat flux, convective boundaries and internal heat generation. Numerical examples are presented to demonstrate the accuracy, simplicity of modeling, faster convergence rate and overall superiority of the present formulation over existing isoparametric axisymmetric solid elements. For each example the results obtained from the present formulation are compared with available analytical solutions and h-models employing isoparametric elements. Convergence graphs showing the rates at which the errors are decreasing are presented for both h and p-models. The results obtained from the present formulation exhibit excellent accuracy, faster convergence rate and require extremely simple models.