An adaptive‐recursive staggering strategy for simulating multifield coupled processes in microheterogeneous solids

Abstract In this work an adaptive‐recursive staggering strategy is developed for the solution of partial differential equations arising from descriptions of multifield coupled processes in microheterogeneous solids. In order to illustrate the solution strategy, a multifield model problem is studied which describes the diffusion of a detrimental dilute solute into a solid material. The coupled equations to be solved are (1) a phenomenological diffusion‐reaction equation, (2) a phenomenological damage evolution law, (3) a balance of energy, and (4) a balance of momentum. The finite element method is used for the spatial discretization, and finite differences for the temporal discretization. In order to accurately capture the microstructure of the material, use of very fine finite element meshes is inescapable. Therefore, in order to reduce computation time, one would like to take as large time steps as possible, provided that the associated numerical accuracy can be maintained. Classical staggering approaches solve each field equation in an uncoupled manner, by allowing only the primary field variable to be active, and momentarily freezing all others. After the solution of each field equation, the primary field variable is updated, and the next field equation is treated in a similar manner. In standard approaches, after this process has been applied, only once, to all of the field equations, the time step is immediately incremented. This non‐recursive process is highly sensitive to the order in which the staggered field equations are solved. Furthermore, since the staggering error accumulates with each passing time step, the process may require very small time steps for sufficient accuracy. In the approach developed here, in order to reduce the error within a time step, the staggering methodology is formulated as a recursive fixed‐point iteration, whereby the system is repeatedly re‐solved until fixed‐point type convergence is achieved. A sufficient condition for the convergence of such a fixed‐point scheme is that the spectral radius of the coupled operator, which depends on the time step size, must be less than unity. This observation is used to adaptively maximize the time step sizes, while simultaneously controlling the coupled operator's spectral radius, in order to deliver solutions below an error tolerance within a prespecified number of desired iterations. This recursive staggering error control allows substantial reduction of computational effort by the adaptive use of large time steps. Three‐dimensional numerical examples are given to illustrate the approach. Copyright © 2001 John Wiley & Sons, Ltd.