Higher Order Finite Elements for the Accurate Prediction of Temperature Gradients in Heat Conduction Problems

Abstract

When the Finite Element Method (FEM) is used to solve heat conduction problems in solids, the domain is typically discretized using elements that only include the nodal values of the temperature as Degrees of Freedom (DOFs). If the values of the spatial temperature gradients are needed, they are typically computed by differentiating the functional representation for the temperature inside the elements. Unfortunately, this differentiation process usually leads to less accurate results for the temperature gradients as compared to the temperature values. For elliptic problems, like steady state heat conduction, with Neumann Boundary Conditions (BCs), recent research related to Adini’s element suggests that higher order elements that include spatial derivatives of the primary field variable as nodal DOFs are promising for obtaining accurate values for those quantities as well as providing a higher order of convergence than conventional elements. In this paper, steady state and transient heat conduction problems which involve Dirichlet BCs or both Dirichlet and Neumann BCs are studied and a new auxiliary BC is proposed to increase the accuracy of the FE solution when Dirichlet BCs are present. Examples are used to illustrate that Adini’s elements converge faster and are more computationally economical than the conventional Lagrange linear elements and Serendipity quadratic elements when auxiliary BCs are used.