A hierarchical flux-based finite element method is developed for both a one and two dimensional thermal structural analyses. Derivation of the finite element equations is presented. The resulting finite element matrices associated with the flux based formulation are evaluated in a closed form. The hierarchical finite elements include additional degrees of freedom in the approximation of the element variable distributions by the use of nodeless variables. The nodeless variables offer increased solution accuracy without the need for defining actual nodes and rediscretizing the finite element model. Thermal and structural responses are obtained from a conventional linear finite element method and exact solutions. Results show that the hierarchical flux-based method can provide improved thermal and structural solution accuracy with fewer elements when compared to results for the conventional linear element method. Polesky, Sandra P. Langley Research Center APPLICATIONS OF MATHEMATICS; FINITE ELEMENT METHOD; STRUCTURAL ANALYSIS; THERMAL ANALYSIS; TWO DIMENSIONAL MODELS; ALGORITHMS; JACOBI MATRIX METHOD; MATHEMATICAL MODELS; TEMPERATURE GRADIENTS; TRANSIENT RESPONSE...