SLAC – a semi-Lagrangian artificial compressibility solver for steady-state incompressible flows

Purpose The purpose of this paper is the development of a new density-based (DB) semi-Lagrangian method to speed up the conventional pressure-based (PB) semi-Lagrangian methods. Design/methodology/approach The semi-Lagrangian-based solvers are typically PB, i.e. semi-Lagrangian pressure-based (SLPB) solvers, where a Poisson equation is solved for obtaining the pressure field and ensuring a divergence-free flow field. As an elliptic-type equation, the Poisson equation often relies on an iterative solution, so it can create a challenge of parallel computing and a bottleneck of computing speed. This study proposes a new DB semi-Lagrangian method, i.e. the semi-Lagrangian artificial compressibility (SLAC), which replaces the Poisson equation by a hyperbolic continuity equation with an added artificial compressibility (AC) term, so a time-marching solution is possible. Without the Poisson equation, the proposed SLAC solver is faster, particularly for the cases with more computational cells, and better suited for parallel computing. Findings The study compares the accuracy and the computing speeds of both SLPB and SLAC solvers for the lid-driven cavity flow and the step-flow problems. It shows that the proposed SLAC solver is able to achieve the same results as the SLPB, whereas with a 3.03 times speed up before using the OpenMP parallelization and a 3.35 times speed up for the large grid number case (512 × 512) after the parallelization. The speed up can be improved further for larger cases because of increasing the condition number of the coefficient matrixes of the Poisson equation. Originality/value This paper proposes a method of avoiding solving the Poisson equation, a typical computing bottleneck for semi-Lagrangian-based fluid solvers by converting the conventional PB solver (SLPB) to the DB solver (SLAC) through the addition of the AC term. The method simplifies and facilitates the parallelization process of semi-Lagrangian-based fluid solvers for modern HPC infrastructures, such as OpenMP and GPU computing.

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Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow

Journal of Fluid Mechanics ◽

10.1017/s002211206900084x ◽

1969 ◽

Vol 37 (4) ◽

pp. 727-750 ◽

Cited By ~ 183

Author(s):

Gareth P. Williams

Keyword(s):

Poisson Equation ◽

Incompressible Flows ◽

Stokes Equations ◽

Three Dimensional ◽

Navier Stokes ◽

Wave Flow ◽

Trigonometric Interpolation ◽

Time Step ◽

Navier Stokes Equations ◽

The Poisson Equation

A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow.Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to’ advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 × 14 × 14 grid upward to 96 sec for a 60 × 34 × 34 grid.As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.

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Modified iterative method with red-black ordering for image composition using poisson equation

Bulletin of Electrical Engineering and Informatics ◽

10.11591/eei.v10i4.3076 ◽

2021 ◽

Vol 10 (4) ◽

pp. 2016-2027

Author(s):

Zakariah Aris ◽

Nordin Saad ◽

A’qilah Ahmad Dahalan ◽

Andang Sunarto ◽

Azali Saudi

Keyword(s):

Poisson Equation ◽

Linear Equations ◽

Numerical Differentiation ◽

Laplacian Operator ◽

Source Image ◽

Relaxation Methods ◽

Image Composition ◽

The Poisson Equation ◽

Speed Up ◽

Two Parameter

Image composition involves the process of embedding a selected region of the source image to the target image to produce a new desirable image seamlessly. This paper presents an image composition procedure based on numerical differentiation using the laplacian operator to obtain the solution of the poisson equation. The proposed method employs the red-black strategy to speed up the computation by using two acceleration parameters. The method is known as modified two-parameter over-relaxation (MTOR) and is an extension of the existing relaxation methods. The MTOR was extensively studied in solving various linear equations, but its usefulness in image processing was never explored. Several examples were tested to examine the effectiveness of the proposed method in solving the poisson equation for image composition. The results showed that the proposed MTOR performed faster than the existing methods.

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Numerical solutions to the Poisson equation in media surrounding multiple arbitrarily shaped bodies

10.32469/10355/12185 ◽

2010 ◽

Author(s):

Zebadiah M. Smith

Keyword(s):

Poisson Equation ◽

Numerical Solutions ◽

The Poisson Equation

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Solvability of one nonlocal Dirichlet problem for the Poisson equation

Novi Sad Journal of Mathematics ◽

10.30755/nsjom.08942 ◽

2019 ◽

Vol 50 (1) ◽

pp. 67-88

Author(s):

Batirkhan Turmetov ◽

Valery Karachik

Keyword(s):

Dirichlet Problem ◽

Poisson Equation ◽

The Poisson Equation

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Lattice-BGK Methods for Simulating Incompressible Fluid Flows

International Journal of Modern Physics C ◽

10.1142/s0129183197000680 ◽

1997 ◽

Vol 08 (04) ◽

pp. 793-803 ◽

Cited By ~ 10

Author(s):

Yu Chen ◽

Hirotada Ohashi

Keyword(s):

Fluid Flow ◽

Incompressible Fluid ◽

Poisson Equation ◽

General Model ◽

Incompressible Flows ◽

Fluid Flows ◽

Incompressible Limit ◽

Numerical Example ◽

Incompressible Fluid Flows

The lattice-Bhatnagar-Gross-Krook (BGK) method has been used to simulate fluid flow in the nearly incompressible limit. But for the completely incompressible flows, two special approaches should be applied to the general model, for the steady and unsteady cases, respectively. Introduced by Zou et al.,1 the method for steady incompressible flows will be described briefly in this paper. For the unsteady case, we will show, using a simple numerical example, the need to solve a Poisson equation for pressure.

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Parallel finite-element method using domain decomposition and Parareal for transient motor starting analysis

COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering ◽

10.1108/compel-12-2018-0516 ◽

2019 ◽

Vol 38 (5) ◽

pp. 1507-1520 ◽

Cited By ~ 1

Author(s):

Yasuhito Takahashi ◽

Koji Fujiwara ◽

Takeshi Iwashita ◽

Hiroshi Nakashima

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Parallel Computing ◽

Domain Decomposition ◽

Parallel Computation ◽

Domain Decomposition Method ◽

Space Time ◽

Content Type ◽

Parallel Performance ◽

Element Method

Purpose This paper aims to propose a parallel-in-space-time finite-element method (FEM) for transient motor starting analyses. Although the domain decomposition method (DDM) is suitable for solving large-scale problems and the parallel-in-time (PinT) integration method such as Parareal and time domain parallel FEM (TDPFEM) is effective for problems with a large number of time steps, their parallel performances get saturated as the number of processes increases. To overcome the difficulty, the hybrid approach in which both the DDM and PinT integration methods are used is investigated in a highly parallel computing environment. Design/methodology/approach First, the parallel performances of the DDM, Parareal and TDPFEM were compared because the scalability of these methods in highly parallel computation has not been deeply discussed. Then, the combination of the DDM and Parareal was investigated as a parallel-in-space-time FEM. The effectiveness of the developed method was demonstrated in transient starting analyses of induction motors. Findings The combination of Parareal with the DDM can improve the parallel performance in the case where the parallel performance of the DDM, TDPFEM or Parareal is saturated in highly parallel computation. In the case where the number of unknowns is large and the number of available processes is limited, the use of DDM is the most effective from the standpoint of computational cost. Originality/value This paper newly develops the parallel-in-space-time FEM and demonstrates its effectiveness in nonlinear magnetoquasistatic field analyses of electric machines. This finding is significantly important because a new direction of parallel computing techniques and great potential for its further development are clarified.

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