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Based on flux-based formulation, a nodeless variable element method is developed to analyze two-dimensional steady-state and transient heat transfer problems. The nodeless variable element employs quadratic interpolation functions to provide higher solution accuracy without necessity to actually generate additional nodes. The flux-based formulation is applied to reduce the complexity in deriving the finite element equations as compared to the conventional finite element method. The solution accuracy is further improved by implementing an adaptive meshing technique to generate finite element mesh that can adapt and move along corresponding to the solution behavior. The technique generates small elements in the regions of steep solution gradients to provide accurate solution, and meanwhile it generates larger elements in the other regions where the solution gradients are slight to reduce the computational time and the computer memory. The effectiveness of the combined procedure is demonstrated by heat transfer problems that have exact solutions. These problems are: (a) a steady-state heat conduction analysis in a square plate subjected to a highly localized surface heating, and (b) a transient heat conduction analysis in a long plate subjected to a moving heat source.

A combined characteristic-based split algorithm and an adaptive meshing technique for analyzing two-dimensional viscous incompressible flow are presented. The method uses the three-node triangular element with equal-order interpolation functions for all variables of the velocity components and pressure. The main advantage of the combined method is that it improves the solution accuracy by coupling an error estimation procedure to an adaptive meshing technique that generates small elements in regions with a large change in solution gradients, and at the same time, larger elements in the other regions. The performance of the combined procedure is evaluated by analyzing one test case of the flow past a cylinder, for their transient and steady-state flow behaviors.

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