Parallel solution of large-scale free surface viscoelastic flows via sparse approximate inverse preconditioning

Abstract

a b s t r a c tThough computational techniques for two-dimensional viscoelastic free surface flows are well developed, three-dimensional flows continue to present significant computational challenges. Fully coupled free surface flow models lead to nonlinear systems whose steady states can be found via Newton's method. Each Newton iteration requires the solution of a large, sparse linear system, for which memory and computational demands suggest the application of an iterative method, rather than the sparse direct methods widely used for two dimensional simulations. The Jacobian matrix of this system is often ill-conditioned, resulting in unacceptably slow convergence of the linear solver; hence preconditioning is essential. We propose a variant sparse approximate inverse preconditioner for the Jacobian matrix that allows for the solution of problems involving more than a million degrees of freedom in challenging parameter regimes. Construction of this preconditioner requires the solution of small least squares problems that can be simply parallelized on a distributed memory machine. The performance and scalability of this preconditioner with the GMRES solver are investigated for two-and three-dimensional free surface flows on both structured and unstructured meshes in the presence and absence of viscoelasticity. The results suggest that this preconditioner is an extremely promising candidate for solving large-scale steady viscoelastic flows with free surfaces.

An efficient parallel and fully implicit algorithm for the simulation of transient free-surface flows of multimode viscoelastic liquids

“…Finally, the performance of the solution of the constitutive equation is perfect irrespective of the number of processors, while the use of an overlapping scheme of submeshes enhances the performance of the algorithm because it reduces the multiple exchanges of data. In all cases, both memory requirements and execution times where significant smaller compared with those of the conventional monolithic scheme [15]. These features constitute the proposed algorithm suitable for the simulation of transient viscoelastic flows.…”

“…Apparently, the time integration of more than two million unknowns with this fully implicit algorithm can be done in less than fifty seconds, if fourteen processors of this hardware system are used. For comparison purposes, it is useful to claim that the solution of a single system of 3656 equations with the conventional monolithic approximation [15] requires 15 s on eight processors. Moreover, the efficiency of our algorithm with regard to two processors is about 60%, which is higher than those of the flow and the mesh subproblems (∼50%), and becomes higher as the number of viscoelastic stress modes increases.…”

“…The most efficient way for speeding-up numerical simulations and especially time-consuming calculations, is by parallelizing the solution algorithm, along with the distribution of the data on separate workstations interconnected with a high-speed network (Castillo et al [15], Dellis et al [16]). The very first attempt for parallelizing a viscoelastic code was made by the CESAME group (Aggarwal and Keunings [17]).…”

“…Their solution algorithm was based on a semi-implicit Taylor-Galerkin/pressure-correction, time-marching scheme. Finally, Castillo et al [15] developed a parallel, finite element algorithm for solving three-dimensional viscoelastic free-surface flows. It was a fully coupled approximation for steady-state calculations.…”

A fast and efficient iterative scheme for viscoelastic flow simulations with the DEVSS finite element method

“…More recently, Kim et al [9] applied an adaptive incomplete LU (AILU) preconditioned Bi-CGSTAB iterative method for both coupled and decoupled viscoelastic flows formulated with DEVSS-G/SUPG and Cai and Westphal [10] applied nonlinear nested iteration, an adaptive mesh refinement with an inexact Newton iteration, to their least-square finite element method for viscoelastic fluids. Another notable advance in the application of iterative solvers to viscoelastic flow simulations was recently made by Castillo et al [11]. They constructed a preconditioner for viscoelastic free surface flows using the sparse approximate inverse method (specifically a modification of the SPAI scheme of Grote and Huckle [13] for their Jacobian matrix in Newton's method) and reported that both the CPU time and the memory usage increases almost linearly with the problem size.…”