Comparison of approximation schemes for three-dimensional thermal stress analysis and applications to cracks with fluid flowing

Abstract In this paper, we first present an approximate scheme for thermal problem with heat sources varying with time (time-varying). It is different from one that the authors used previously. Both schemes use series of stepwise constant heat sources to approximate the continuously varying heat source. But the constant sources of the two schemes are different. The previous scheme divides the time period from the initial time to the solution time, at which the solutions are sought, into some time-steps and assumes that each of the series of constant sources is applied at the start time of the time-steps and all the constant sources last to the solution time. The new scheme assumes that each of the constant sources starts at the start time of a time-step but lasts only for one step of time. When solutions for temperature, displacement and stress are sought at multiple time instants or the boundary conditions are obtained step by step from other problems, it is easier and computing efficient to use the new scheme. If the solution is required at just one time instant, then computations of the two schemes are basically the same. The schemes are compared for the thermal effects caused by time-varying temperature on surface of a spherical cavity in an infinite body and the results of the two schemes agree very well. Comparison of the two schemes is also done for the temperature change in rock due to fluid flowing through a fracture inside the rock.