Time-dependent shape functions for modeling highly transient geothermal systems

This paper presents new time‐dependent finite element shape functions suitable for modeling high‐gradient transient conductive heat flow in geothermal systems. The shape functions are made adaptive by enhancing the approximation functions with time‐dependent variables, which may vary according to the transient process without adding extra degrees of freedom or applying mesh adaptation. Two different approaches are presented. First, an iterative method is proposed, in which an exponential approximation function, which is optimized continually during the transient process, is incorporated in the shape function. Second, an analytical method is suggested, in which an analytical solution of a simplified process is incorporated in the shape function, enabling an explicit update of the shape functions in each time step. A methodology for modeling the variation of temperature in one and two dimensions is introduced. The ability of the method to capture high‐gradient temperature profiles using relatively large elements is illustrated with numerical examples of cases in which equally large standard finite elements fail. Copyright © 2008 John Wiley & Sons, Ltd.