This paper presents results of an ongoing research program directed towards developing fast and efficient finite element solutionalgorithms for the simulation of large-scale flow problems. Two main steps were taken t...
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This paper presents results of an ongoing research program directed towards developing fast and efficient finite element solutionalgorithms for the simulation of large-scale flow problems. Two main steps were taken towards achieving this goal. The first step was to employ segregatedsolution schemes as opposed to the fully coupled solution approach traditionally used in many finite element solutionalgorithms. The second step was to replace the direct Gaussian elimination linear equation solvers used in the first step with iterative solvers of the conjugate gradient and conjugate residual type. The three segregated solution algorithms developed in step one are first presented and their integrity and relative performance demonstrated by way of a few examples. Next, the four types of iterative solvers (i.e. two options for solving the symmetric pressure type equations and two options for solving the non-symmetric advection-diffusion type equations resulting from the segregatedalgorithms) together with the two preconditioning strategies employed in our study are presented. Finally, using examples of practical relevance the paper documents the large gains which result in computational efficiency, over fully coupled solutionalgorithms, as each of the above two main steps are introduced. It is shown that these gains become increasingly more dramatic as the complexity and size of the problem is increased.
In this paper, we present a SIMPLE based algorithm in the context of the discontinuous Galerkin method for unsteady incompressible flows. Time discretization is done fully implicit using backward differentiation formu...
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In this paper, we present a SIMPLE based algorithm in the context of the discontinuous Galerkin method for unsteady incompressible flows. Time discretization is done fully implicit using backward differentiation formulae (BDF) of varying order from 1 to 4. We show that the original equation for the pressure correction can be modified by using an equivalent operator stemming from the symmetric interior penalty (SIP) method leading to a reduced stencil *** assess the accuracy as well as the stability and the performance of the scheme, three different test cases are carried out: the Taylor vortex flow, the Orr-Sommerfeld stability problem for plane Poiseuille flow and the flow past a square cylinder. (1) Simulating the Taylor vortex flow, we verify the temporal accuracy for the different BDF schemes. Using the mixed-order formulation, a spatial convergence study yields convergence rates of k + 1 and k in the L-2-norm for velocity and pressure, respectively. For the equal-order formulation, we obtain approximately the same convergence rates, while the absolute error is smaller. (2) The stability of our method is examined by simulating the Orr-Sommerfeld stability problem. Using the mixed-order formulation and adjusting the penalty parameter of the symmetric interior penalty method for the discretization of the viscous part, we can demonstrate the long-term stability of the algorithm. Using pressure stabilization the equal-order formulation is stable without changing the penalty parameter. (3) Finally, the results for the flow past a square cylinder show excellent agreement with numerical reference solutions as well as experiments. Copyright (c) 2015 John Wiley & Sons, Ltd.
In this paper we present how the well-known SIMPLE algorithm can be extended to solve the steady incompressible Navier-Stokes equations discretized by the discontinuous Galerkin method. The convective part is discreti...
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In this paper we present how the well-known SIMPLE algorithm can be extended to solve the steady incompressible Navier-Stokes equations discretized by the discontinuous Galerkin method. The convective part is discretized by the local Lax-Friedrichs fluxes and the viscous part by the symmetric interior penalty method. Within the SIMPLE algorithm, the equations are solved in an iterative process. The discretized equations are linearized and an equation for the pressure is derived on the discrete level. The equations obtained for each velocity component and the pressure are decoupled and therefore can be solved sequentially, leading to an efficient solution procedure. The extension of the proposed scheme to the unsteady case is straightforward, where fully implicit time schemes can be used. Various test cases are carried out: the Poiseuille flow, the channel flow with constant transpiration, the Kovasznay flow, the flow into a corner and the backward-facing step flow. Using a mixed-order formulation, i.e. order k for the velocity and order k - 1 for the pressure, the scheme is numerically stable for all test cases. Convergence rates of k + 1 and k in the L-2-norm are observed for velocity and pressure, respectively. A study of the convergence behavior of the SIMPLE algorithm shows that no under-relaxation for the pressure is needed, which is in strong contrast to the application of the SIMPLE algorithm in the context of the finite volume method or the continuous finite element method. We conclude that the proposed scheme is efficient to solve the steady incompressible Navier-Stokes equations in the context of the discontinuous Galerkin method comprising hp-accuracy. (c) 2012 Elsevier Inc. All rights reserved.
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