A Parallel Domain Decomposition Method for 3D Unsteady Incompressible Flows at High Reynolds Number

Chen, Rongliang; Wu, Yuqi; Yan, Zhengzheng; Zhao, Yubo; Cai, Xiao‐Chuan

doi:10.1007/s10915-013-9732-x

Cited by 18 publications

(9 citation statements)

References 25 publications

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“…Domain decomposition method is regarded as the basic mathematical background for many parallel applications [100][101][102]. A domain decomposition algorithm for time fractional reaction-diffusion equation with implicit finite difference method was proposed [103].…”

Section: Parallel Computingmentioning

confidence: 99%

Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

Gong

Bao

Tang

et al. 2015

Mathematical Problems in Engineering

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We present a survey of fractional differential equations and in particular of the computational cost for their numerical solutions from the view of computer science. The computational complexities of time fractional, space fractional, and space-time fractional equations areO(N2M),O(NM2), andO(NM(M+N)) compared withO(MN) for the classical partial differential equations with finite difference methods, whereM,Nare the number of space grid points and time steps. The potential solutions for this challenge include, but are not limited to, parallel computing, memory access optimization (fractional precomputing operator), short memory principle, fast Fourier transform (FFT) based solutions, alternating direction implicit method, multigrid method, and preconditioner technology. The relationships of these solutions for both space fractional derivative and time fractional derivative are discussed. The authors pointed out that the technologies of parallel computing should be regarded as a basic method to overcome this challenge, and some attention should be paid to the fractional killer applications, high performance iteration methods, high order schemes, and Monte Carlo methods. Since the computation of fractional equations with high dimension and variable order is even heavier, the researchers from the area of mathematics and computer science have opportunity to invent cornerstones in the area of fractional calculus.

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Section: Parallel Computingmentioning

confidence: 99%

Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

Gong

Bao

Tang

et al. 2015

Mathematical Problems in Engineering

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“…We also observe that the fully implicit method with both preconditioners offer superlinear speedup with up to 2,048 processors. We believe that the reason for the superlinear speedup is due to the increasingly better cache performance in the sparse LU factorization as the subdomain problem becomes smaller [15,21]. Since the RAS preconditioner does not change the accuracy of the Newton-Krylov solver, we further experiment by freezing the preconditioner at each time step.…”

Section: Npmentioning

confidence: 99%

A Fully Implicit Method for Lattice Boltzmann Equations

Huang

Yang²,

Cai³

2015

SIAM J. Sci. Comput.

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Existing approaches for solving the lattice Boltzmann equations with finite difference methods are explicit and semi-implicit; both have certain stability constraints on the time step size. In this work, a fully implicit second-order finite difference scheme is developed. We focus on a parallel, highly scalable, Newton-Krylov-RAS algorithm for the solution of a large sparse nonlinear system of equations arising at each time step. Here, RAS is a restricted additive Schwarz preconditioner based on a first-order spatial discretization. We show numerically that by using the fully implicit method the time step size is no longer constrained by the CFL condition, and the Newton-Krylov-RAS algorithm is scalable on a supercomputer with more than ten thousand processors. Moreover, to calculate the steady state solution we investigate an adaptive time stepping strategy. The total compute time required by the implicit method with adaptive time stepping is much smaller than that of an explicit method for several test cases.

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“…In this work, we study an efficient and highly parallel finite element method based on domain decomposition method for the Navier–Stokes equations 5,16,19‐21 . With the method, a simulation of a full 3D patient‐specific hepatic flow on a mesh with around 10 million elements can be accomplished in a few hours.…”

Section: Introductionmentioning

confidence: 99%

A highly parallel simulation of patient‐specific hepatic flows

Lin

Chen

Gao

et al. 2021

Numer Methods Biomed Eng

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Computational hemodynamics is being developed as an alternative approach for assisting clinical diagnosis and treatment planning for liver diseases. The technology is non‐invasive, but the computational time could be high when the full geometry of the blood vessels is taken into account. Existing approaches use either one‐dimensional model of the artery or simplified three‐dimensional tubular geometry in order to reduce the computational time, but the accuracy is sometime compromised, for example, when simulating blood flows in arteries with plaque. In this work, we study a highly parallel method for the transient incompressible Navier–Stokes equations for the simulation of the blood flows in the full three‐dimensional patient‐specific hepatic artery, portal vein and hepatic vein. As applications, we also simulate the flow in a patient with hepatectomy and calculate the S (PPG). One of the advantages of simulating blood flows in all hepatic vessels is that it provides a direct estimate of the PPG, which is a gold standard value to assess the portal hypertension. Moreover, the robustness and scalability of the algorithm are also investigated. A 83% parallel efficiency is achieved for solving a problem with 7 million elements on a supercomputer with more than 1000 processor cores.

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A Parallel Domain Decomposition Method for 3D Unsteady Incompressible Flows at High Reynolds Number

Cited by 18 publications

References 25 publications

Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

A Fully Implicit Method for Lattice Boltzmann Equations

A highly parallel simulation of patient‐specific hepatic flows

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