The solution of large viscoelastic flow problems using parallel iterative techniques
Author(s)Caola, Anthony E
Massachusetts Institute of Technology. Dept. of Chemical Engineering.
Robert A. Brown and Robert C. Armstrong.
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Efficient parallel computation of complex viscoelastic flows is essential to bring modem computer power to bear on fluid calculations where complicated constitutive descriptions are required. We have developed an efficient, parallel time integration scheme for calculating viscoelastic flows. The method is a synthesis of an operator splitting and time integration method that decouples the calculation of the polymer portion of the stress by solution of a hyperbolic constitutive equation from the temporal updating of the velocity and pressure fields by solution of a generalized Stokes problem. Finite element discretizations using the DEVSS-G 1 method are used; here a direct interpolation of the components of the velocity gradient is introduced. The key to the parallelization is the incorporation of iterative matrix solution for each portion of the split calculation. The linear system that results from the generalized Stokes problem is asymmetric, indefinite, and block singular. At each timestep, this system is solved using an algorithm which combines a parallel preconditioner with a Krylov subspace method. The parallel preconditioner, called the Block Complement and Additive Levels Method (BCALM) preconditioner, is based on treating pressure unknowns separately from the variables velocities and gradients. A pressure preconditioner is constructed from a factor of the Schur complement of the pressures using a Jacobi iteration. The viscous operator is treated using the additive Schwarz method. The number of Krylov iterations required per implicit step with increasing numbers of processors is stabilized using a two-level additive Schwarz method. The coarse grid is generated using addition as a restriction operator and insertion for extension. The resulting iterative method is demonstrated to have high parallel efficiency, subject to effective domain decomposition. Robustness and good performance result from using geometric partitioning of the unknowns by the nested bisection routines supplied in CHAC02. The software is implemented using MPI and the PETSc toolkit, so as to be readily portable to a variety of parallel computers. This solver is developed in the context of the natural convection problem, where in the limit of high Grashof number, the linear system that arises during solution for steady state using Newton's method is stiff, assymmetric and indefinite. With this model problem, the preconditioner's robustness is tested and its efficiency is measured. The solver's generalizability for viscous flow problems is also discussed, as the only assumption about problem physics is that the continuity equation exists. For a given velocity field, the discretized equations from a typical differential constitutive model, e.g. the Oldroyd-B model, yields an asymmetric, nonsingular system of linear equations when nonlinearities are updated explicitly. Krylov iterations preconditioned with inexact solves in a two-level additive Schwarz framework are used to solve this system at each time step. Added efficiency is introduced by using a discontinuous Galerkin spatial discretization, which allows element-by-element condensation of the linear systems and higher parallel efficiency. The effectiveness of the algorithm is demonstrated by the calculation of transient, two dimensional flow of an Oldroyd-B fluid past an isolated cylinder confined symmetrically between parallel plates and by calculation of the stability of the steady-state motion to small-amplitude, three-dimensional disturbances The resulting method is far superior to the use of serial, frontal methods.
Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2000.Includes bibliographical references (p. 306-325).
DepartmentMassachusetts Institute of Technology. Dept. of Chemical Engineering.
Massachusetts Institute of Technology