Abstract :
In this work an adaptive-recursive staggering strategy is developed for the solution of partial di9erential
equations arising from descriptions of multi3eld coupled processes in microheterogeneous solids.
In order to illustrate the solution strategy, a multi3eld model problem is studied which describes the
di9usion of a detrimental dilute solute into a solid material. The coupled equations to be solved are
(1) a phenomenological di9usion-reaction equation, (2) a phenomenological damage evolution law, (3)
a balance of energy, and (4) a balance of momentum. The 3nite element method is used for the spatial
discretization, and 3nite di9erences for the temporal discretization. In order to accurately capture
the microstructure of the material, use of very 3ne 3nite 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 3eld equation in an uncoupled manner, by allowing only the primary 3eld variable to be active,
and momentarily freezing all others. After the solution of each 3eld equation, the primary 3eld variable
is updated, and the next 3eld equation is treated in a similar manner. In standard approaches,
after this process has been applied, only once, to all of the 3eld equations, the time step is immediately
incremented. This non-recursive process is highly sensitive to the order in which the staggered
3eld equations are solved. Furthermore, since the staggering error accumulates with each passing time
step, the process may require very small time steps for su?cient accuracy. In the approach developed
here, in order to reduce the error within a time step, the staggering methodology is formulated
as a recursive 3xed-point iteration, whereby the system is repeatedly re-solved until 3xed-point type
convergence is achieved. A su?cient condition for the convergence of such a 3xed-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 prespeci3ed number of desired iterations. This recursive staggering error control
allows substantial reduction of computational e9ort by the adaptive use of large time steps. Threedimensional
numerical examples are given to illustrate the approach.
