Performance Of Multigrid Method Research Articles

Efficient solvers for coupled models in respiratory mechanics.

We present efficient preconditioners for one of the most physiologically relevant pulmonary models currently available. Our underlying motivation is to enable the efficient simulation of such a lung model on high-performance computing platforms in order to assess mechanical ventilation strategies and contributing to design more protective patient-specific ventilation treatments. The system of linear equations to be solved using the proposed preconditioners is essentially the monolithic system arising in fluid-structure interaction (FSI) extended by additional algebraic constraints. The introduction of these constraints leads to a saddle point problem that cannot be solved with usual FSI preconditioners available in the literature. The key ingredient in this work is to use the idea of the semi-implicit method for pressure-linked equations (SIMPLE) for getting rid of the saddle point structure, resulting in a standard FSI problem that can be treated with available techniques. The numerical examples show that the resulting preconditioners approach the optimal performance of multigrid methods, even though the lung model is a complex multiphysics problem. Moreover, the preconditioners are robust enough to deal with physiologically relevant simulations involving complex real-world patient-specific lung geometries. The same approach is applicable to other challenging biomedical applications where coupling between flow and tissue deformations is modeled with additional algebraic constraints. Copyright © 2016 John Wiley & Sons, Ltd.

A positivity-preserving, implicit defect-correction multigrid method for turbulent combustion

A novel, robust multigrid method for the simulation of turbulent and chemically reacting flows is developed. A survey of previous attempts at implementing multigrid for the problems at hand indicated extensive use of artificial stabilization to overcome numerical instability arising from non-linearity of turbulence and chemistry model source-terms, small-scale physics of combustion, and loss of positivity. These issues are addressed in the current work. The highly stiff Reynolds-averaged Navier-Stokes (RANS) equations, coupled with turbulence and finite-rate chemical kinetics models, are integrated in time using the unconditionally positive-convergent (UPC) implicit method. The scheme is successfully extended in this work for use with chemical kinetics models, in a fully-coupled multigrid (FC-MG) framework.To tackle the degraded performance of multigrid methods for chemically reacting flows, two major modifications are introduced with respect to the basic, Full Approximation Storage (FAS) approach. First, a novel prolongation operator that is based on logarithmic variables is proposed to prevent loss of positivity due to coarse-grid corrections. Together with the extended UPC implicit scheme, the positivity-preserving prolongation operator guarantees unconditional positivity of turbulence quantities and species mass fractions throughout the multigrid cycle. Second, to improve the coarse-grid-correction obtained in localized regions of high chemical activity, a modified defect correction procedure is devised, and successfully applied for the first time to simulate turbulent, combusting flows.The proposed modifications to the standard multigrid algorithm create a well-rounded and robust numerical method that provides accelerated convergence, while unconditionally preserving the positivity of model equation variables. Numerical simulations of various flows involving premixed combustion demonstrate that the proposed MG method increases the efficiency by a factor of up to eight times with respect to an equivalent single-grid method, and by two times with respect to an artificially-stabilized MG method.

A dimension split method for the incompressible Navier–Stokes equations in three dimensions

SUMMARYIn this paper, we describe a new method for the three‐dimensional steady incompressible Navier–Stokes equations, which is called the dimension split method (DSM). The basic idea of DSM is that the three‐dimensional space is split up into a cluster of two‐dimensional manifolds and then the three‐dimensional solution is approximated by the solutions on these two‐dimensional manifolds. Through introducing some technologies, such as SUPG stabilization, multigrid method, and such, we firstly make DSM feasible in the computation of real flow. Because of split property of DSM, all computation is carried out on these two‐dimensional manifolds, namely, a series of two‐dimensional problems only need to be solved in the computation of three‐dimensional problem, which greatly reduces the difficulty and the computational cost in the mesh generation. Moreover, these two‐dimensional problems can be computed simultaneously and a coarse‐grained parallel algorithm would be constructed, whereas the two‐dimensional manifold is considered as the computation unit. In the last, we explore the behavior and the accuracy of the proposed method in two numerical examples. Firstly, error estimates, performance of multigrid method, and parallel algorithm are well‐demonstrated by the known analytical solution case. Secondly, the computations of three‐dimensional lid‐driven cavity flows with different Reynolds numbers are compared with other numerical simulations. Results show that the present implementation is able to exhibit good stability and accuracy properties for real flows. Copyright © 2013 John Wiley & Sons, Ltd.

Convergence analysis of multigrid methods with residual scaling techniques

In this paper, multigrid methods with residual scaling techniques for symmetric positive definite linear systems are considered. The idea of perturbed two-grid methods proposed in [7] is used to estimate the convergence factor of multigrid methods with residual scaled by positive constant scaling factors. We will show that if the convergence factors of the two-grid methods are uniformly bounded by σ ( σ < 0.5 ), then the convergence factors of the W-cycle multigrid methods are uniformly bounded by σ / ( 1 − σ ) , whether the residuals are scaled at some or all levels. This result extends Notay’s Theorem 3.1 in [7] to more general cases. The result also confirms the viewpoint that the W-cycle multigrid method will converge sufficiently well as long as the convergence factor of the two-grid method is small enough. In the case where the convergence factor of the two-grid method is not small enough, by appropriate choice of the cycle index γ , we can guarantee that the convergence factor of the multigrid methods with residual scaling techniques still has a uniform bound less than σ / ( 1 − σ ) . Numerical experiments are provided to show that the performance of multigrid methods can be improved by scaling the residual with a constant factor. The convergence rates of the two-grid methods and the multigrid methods show that the W-cycle multigrid methods perform better if the convergence rate of the two-grid method becomes smaller. These numerical experiments support the proposed theoretical results in this paper.

Adaptive non‐nested multigrid methods

This paper presents some linear adaptive non‐nested multigrid methods which are applied to linear elastic problems discretized with triangular and tetrahedral finite elements using unstructured and Delaunay mesh generators. The Zienkiewicz‐Zhu error estimator and a h‐refinement procedure are used to obtain the non‐nested meshes used by the multigrid methods. We solve problems with a specified percentage error in the energy norm using the optimal performance of multigrid methods.