An implicit free element method for simulation of compressible flow

Liu, Hua-Yu; Gao, Xiao‐Wei; Xu, Bing‐Bing

doi:10.1016/j.compfluid.2019.104276

Cited by 12 publications

(4 citation statements)

References 32 publications

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“…The repeated index i and j represent the summation over d components. The typical engineering problem represented by eqn ( 12) is the energy equation in fluid mechanics [2], [19].…”

Section: Finite Line Methods For Solving Convection-diffusion Equationsmentioning

confidence: 99%

“…To overcome these drawbacks in stability, accuracy and suitability of irregular geometries, Gao et al [15] proposed a new type of collocation method, called the free element method (FrEM), which absorbs the advantages of FEM in stability, FDM in easy use and MFM in suitability for complicated geometries. FrEM has been successfully used to solve heat conduction [16], piezoelectric [17], solid mechanics [18] and fluid mechanics [19] problems. However, when using high-order free elements in FrEM, a lot of element nodes are required.…”

Section: Introductionmentioning

confidence: 99%

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Finite Line Method for Solving Convection–diffusion Equations

Gao

LIU

DING

2022

WIT Transactions on Engineering Sciences

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In this paper, a creative collocation-type numerical method, the Finite Line Method (FLM), is proposed for solving general convection-diffusion equations. The method is based on the use of a finite number of lines crossing each collocation point, and the Lagrange polynomial interpolation formulation to construct the shape functions over each line. The directional derivative technique is proposed to derive the first-order partial derivatives of any physical variables with respect to the global coordinates for the high-dimensional problems from the lines' ones and the high-order derivatives are evaluated from a recurrence formulation. The derived spatial partial derivatives are directly substituted into the governing partial differential equations and related boundary conditions of the convection-diffusion equations to set up the system of equations. The finite number of lines crossing each collocation point is called the line set. To evaluate the convection and diffusion terms accurately, two different line sets are used for these two terms, which are called the convection line set and central line set, respectively. The former is formed according to the velocity direction and is used for performing the upwind scheme in the computation of the convection term, and the latter is formed by the crossed lines including the collocation point at the center. A numerical example will be given to verify the correctness and stability of the proposed method.

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“…The repeated index i and j represent the summation over d components. The typical engineering problem represented by eqn ( 12) is the energy equation in fluid mechanics [2], [19].…”

Section: Finite Line Methods For Solving Convection-diffusion Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Finite Line Method for Solving Convection–diffusion Equations

Gao

LIU

DING

2022

WIT Transactions on Engineering Sciences

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“…In the former works, 33,34 authors introduced midpoints to construct a virtual element to discretize NS equations. This study inherits the notation of using the symbol “'” on the sub/super‐script to represent the variables at midpoints.…”

Section: Momentum Equationmentioning

confidence: 99%

“…The method has been developed to solve thermal and mechanic problems 31,32 . By employing midpoints between collocation points and field points, FECM can also be applied to solve Navier‐Stokes equations robustly 33,34 . Because the elements in FECM are generated locally and independently at each collocation point, the complexity of handing the connectivity between elements required in the conventional FEM can be avoided.…”

Section: Introductionmentioning

confidence: 99%

A free element scheme for simulating two‐ and three‐dimensional incompressible fluid flow

Liu

Gao

2020

Numerical Methods in Fluids

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In this work, Free Element method (FECM) is extended to solve incompressible Navier‐Stokes equations. The momentum equations are discretized by FECM, which permits overlapped elements. Then, the velocities at midpoints are interpolated by improved Momentum Interpolation Method to avoid oscillation caused by decoupling of velocity and pressure. At last, the pressure correction equation is solved to make sure the results satisfying continuity equation. The interpolation of midpoint's velocity is proved to be independent of the under‐relaxation factor and time step size. The deferred correction is employed for utilization of higher‐order discretization of convective terms. Taylor‐Green vortex, flow over a cylinder, lid‐driven flow, cooling tunnel, and flow over airfoil are simulated. The results computed by the proposed method are compared with exact solutions and benchmark solutions, which indicates that the new method has the second order accuracy in space. What is more, the proposed method works well on random node distributions as well, which means that the method is much robust.

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Cross-Line Method for Solving Heat Conduction Problems

Gao

2022

Lecture Notes in Mechanical Engineering

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An implicit free element method for simulation of compressible flow

Cited by 12 publications

References 32 publications

Finite Line Method for Solving Convection–diffusion Equations

Finite Line Method for Solving Convection–diffusion Equations

A free element scheme for simulating two‐ and three‐dimensional incompressible fluid flow

Cross-Line Method for Solving Heat Conduction Problems

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