Control strategies for timestep selection in finite element simulation of incompressible flows and coupled reaction-convection-diffusion processes

Abstract

SUMMARYWe propose two timestep selection algorithms, based on feedback control theory, for ÿnite element simulation of steady state and transient 2D viscous ow and coupled reaction-convection-di usion processes. To illustrate performance of the schemes in practice, we solve Rayleigh-Benard-Marangoni ows, ow across a backward-facing step, unsteady ow around a circular cylinder and chemical reaction systems. Numerical experiments conÿrm that the feedback controllers produce in some cases a very smooth stepsize variation, suggesting that robust control algorithms are possible. These experiments also show that parameter selection can improve timesteps when co-ordinated with the convergence control of non-linear iterations. Further, computational cost of the selection procedures is negligible, since they involve only storing a few extra vectors, computation of norms and evaluation of kinetic energy.

Stabilized edge‐based finite element simulation of free‐surface flows

“…A nodal-block diagonal preconditioner is employed for the flow and a simple diagonal preconditioner for the marker. Moreover, for both the fluid flow and the marking function advection we use an adaptive time stepping procedure based on a proportional-integral-derivative (PID) controller (see [39] for further details). Most of the computational effort spent in this solution procedure is due to the matrix-vector products within the GMRES driver for both flow and marker.…”

On decoupled time step/subcycling and iteration strategies for multiphysics problems

“…The efficiency of the control was demonstrated by Valli et al [6,7] in numerical simulations of Rayleigh-Benard-Marangoni problems, flow over a backward-facing step and unsteady flow past a cylinder. Further, the computational overhead of the selection procedure is insignificant compared with solver operations, since time-step selection involves only storing a few extra vectors and computation of associated norms.…”

“…Boundary conditions and initial conditions for temperature and velocity complete the mathematical statement of the problem. Note that we can decouple at the continuous operator level or at the algebraic level [6,7]. The decoupled flow subproblem is obtained by iteratively 'lagging' temperature for the Boussinesq term in Equation (1).…”

Using fuzzy controllers as adaptive procedures in the finite element method

“…The basic ideas of the PID control can be generalized and be used in computational mechanics problems as demonstrated by Valli et al [9] and references therein. They presented a feedback PID control to choose the size of time steps.…”