Parallel edge-based solution of viscoplastic flows with the SUPG/PSPG formulation

Abstract

A parallel edge-based solution of three dimensional viscoplastic flows governed by the steady Navier-Stokes equations is presented. The governing partial differential equations are discretized using the SUPG/PSPG stabilized finite element method on unstructured grids. The highly nonlinear algebraic system arising from the convective and material effects is solved by an inexact Newton-Krylov method. The locally linear Newton equations are solved by GMRES with nodal block diagonal preconditioner. Matrix-vector products within GMRES are computed edge-by-edge (EDE), diminishing flop counts and memory requirements. A comparison between EDE and element-by-element data structures is presented. The parallel computations were based in a message passing interface standard. Performance tests were carried out in representative three dimensional problems, the sudden expansion for power-law fluids and the flow of Bingham fluids in a lid-driven cavity. Results have shown that edge based schemes requires less CPU time and memory than elementbased solutions.

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

“…The generalized trapezoidal rule is employed in the time discretization. The nonlinearities due to the convective term on the Navier-Stokes equation are treated by an Inexact Newton-GMRES algorithm as described in Elias et al [38]. In this solution algorithm, at the beginning of the nonlinear iterations in each time step, the algorithm computes large linear tolerances, producing fast nonlinear steps.…”

An Iterative Method for the Stokes-Type Problem with Variable Viscosity

“…The latter condition, however, does not hold exactly for the numerical solution, especially if a regularized model is used. For the driven cavity problem, some numerical results, including the prediction of the rigid zones, can be found in [10,11,22,26,38]. In [22,11], a regularized model was used, while [10,26,38] Tables 5.3 and 5.4 show that for small ε the number of outer nonlinear iterations increases.…”

Stabilized edge‐based finite element computation of gravity currents in lock‐exchange configurations

“…More recently, Ribeiro et al [30] presented an edge-based implementation for stabilized semi-discrete and space-time finite element formulations for shallow water equations, Catabriga and Coutinho [31] for the implicit SUPG solution of the Euler equations, Soto et al [32] for incompressible flow problems with fractional step methods and Kraft et al [33] for a segregated symmetric stabilized solution of incompressible flow with heat transfer and the parallel simulation of viscoplastic and free surface flows [34,35]. It has been shown by Ribeiro and Coutinho [36] that, for unstructured grids composed by tetrahedra, edge-based data structures decrease the number of floating point operations and indirect addressings in matrix-vector products needed in the Krylov space solvers and diminish the storage area to hold Jacobians compared with element and pointwise data structures, particularly for problems involving many degrees of freedom.…”

“…It has been shown by Ribeiro and Coutinho [36] that, for unstructured grids composed by tetrahedra, edge-based data structures decrease the number of floating point operations and indirect addressings in matrix-vector products needed in the Krylov space solvers and diminish the storage area to hold Jacobians compared with element and pointwise data structures, particularly for problems involving many degrees of freedom. The construction of edge operations are completely algebraic [30,31,[33][34][35], based on the concept of disassembling element operators, regardless of the particular underlying finite element formulation, thus providing a fast platform for simulation of complex problems such as those found in the present work.…”