A projection technique for incompressible flow in the meshless finite volume particle method

Keck, Rainer; Hietel, Dietmar

doi:10.1007/s10444-004-1831-7

Cited by 24 publications

(24 citation statements)

References 17 publications

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“…Note that (2)(3)(4) are general enough for different classical methods and for both strong and weak formulations. The unsymmetric RBF collocation method was first proposed by Kansa [13,14].…”

Section: Introductionmentioning

confidence: 99%

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On convergent numerical algorithms for unsymmetric collocation

2008

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In this paper, we are interested in some convergent formulations for the unsymmetric collocation method or the so-called Kansa's method. We review some newly developed theories on solvability and convergence. The rates of convergence of these variations of Kansa's method are examined and verified in arbitraryprecision computations. Numerical examples confirm with the theories that the modified Kansa's method converges faster than the interpolant to the solution; that is, exponential convergence for the multiquadric and Gaussian radial basis functions (RBFs). Some numerical algorithms are proposed for efficiency and accuracy in practical applications of Kansa's method. In double-precision, even for very large RBF shape parameters, we show that the modified Kansa's method, through a subspace selection using a greedy algorithm, can produce acceptable approximate solutions. A benchmark algorithm is used to verify the optimality of the selection process.

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Section: Introductionmentioning

confidence: 99%

“…Many successful applications of recently developed mesh-free methods can be found in different Mathematics, Physics and Engineering journals; for example, see [1][2][3][4][5][6][7][8][9][10]. In this paper, we are interested in the radial basis function (RBF) method for solving partial differential equations (PDE) in strong form.…”

Section: Introductionmentioning

confidence: 99%

On convergent numerical algorithms for unsymmetric collocation

2008

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“…This method was also the subject of concern in [3][4][5][6][7][8][9][10]. The extension of the projection technique for incompressible flows to the FVPM is done in [4,5].…”

Section: Introductionmentioning

confidence: 99%

“…The extension of the projection technique for incompressible flows to the FVPM is done in [4,5]. Isotropic and anisotropic adaptive strategies are investigated in [7].…”

Section: Introductionmentioning

confidence: 99%

A finite‐volume particle method for conservation laws on moving domains

Teleaga

Struckmeier

2008

Numerical Methods in Fluids

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SUMMARYThe paper deals with the finite-volume particle method (FVPM), a relatively new method for solving hyperbolic systems of conservation laws. A general formulation of the method for bounded and moving domains is presented. Furthermore, an approximation property of the reconstruction formula is proved. Then, based on a two-dimensional test problem posed on a moving domain, a special Ansatz for the movement of the particles is proposed. The obtained numerical results indicate that this method is well suited for such problems, and thus a first step to apply the FVPM to real industrial problems involving free boundaries or fluid-structure interaction is taken. Finally, we perform a numerical convergence study for a shock tube problem and a simple linear advection equation.

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“…Nevertheless, the SPH method with artificial viscosity cannot achieve the high-order accuracy, and other MMs with the upwind algorithm for nonlinear conservation laws do not enforce the conservation law locally, which renders them quite inefficient when strong shock waves exist. In 1998, the finite-volume particle method (FVPM) [6][7][8] for solving hyperbolic systems of conservation laws was developed by Hietel and his co-workers. The basic idea in the FVPM is to incorporate elements of the finite volume method (FVM) into a meshless method, which is realized by a keystone employing a Shepard function instead of the test functions in the standard FVM.…”

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confidence: 99%

Meshless local discontinuous Petrov-Galerkin method with application to blasting problems

Hu¹,

Gao²

2008

Trans. Tianjin Univ.

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Abstract：A meshless local discontinuous Petrov-Galerkin (MLDPG) method based on the local symmetric weak form (LSWF) is presented with the application to blasting problems. The derivation is similar to that of mesh-based Runge-Kutta Discontinuous Galerkin (RKDG) method. The solutions are reproduced in a set of overlapped spherical sub-domains, and the test functions are employed from a partition of unity of the local basis functions. There is no need of any traditional nonoverlapping mesh either for local approximation purpose or for Galerkin integration purpose in the presented method. The resulting MLDPG method is a meshless, stable, high-order accurate and highly parallelizable scheme which inherits both the advantages of RKDG and meshless method (MM), and it can handle the problems with extremely complicated physics and geometries easily.Three numerical examples of the one-dimensional Sod shock-tube problem, the blast-wave problem and the Woodward-Colella interacting shock wave problem are given. All the numerical results are in good agreement with the closed solutions. The higher-order MLDPG schemes can reproduce more accurate solution than the lower-order schemes. Keywords：meshless method; discontinuous Galerkin method; meshless local discontinuous PetrovGalerkin (MLDPG) method; finite-volume particle method; convection-dominated flow Conventional computational fluid dynamics (CFD) methods need a priori definition of the connectivity of nodes, i.e., they rely on a mesh, which leads to complications for certain classes of problems. The generation of good quality non-overlapping meshes presents fateful difficulties in the analysis of engineering systems (especially in the multi-dimensional problems). In recent years, more and more attention has been paid to the socalled meshless method(MM), as the method can solve partial differential equations only based on a scatter set of nodes without the need for an additional mesh, which eliminates the disadvantages of mesh-based method essentially. Nevertheless, MM is mostly applied to the solid mechanics problems, only very few work is reported using MM for convection-dominated flows.In the numerical simulation of blast phenomenon, the existence of the convection term makes that the solution will develop strong discontinuity as well as shock waves. In order to handle this kind of weak solutions numerically, one needs to introduce a special treatment to stabilize the numerical approximation. Some of the meshless schemes have been incorporated into the stabilization algorithms in the mesh-based method to obtain a good accuracy for convection-dominated problems, e.g., the smoothed particle hydrodynamics (SPH) [1,2] employs artificial viscosity in the formulations to improve the stability in the simulation of shock phenomenon, and a few of sophisticated MMs introduce the upwind algorithm for the calculation of convection-dominated problems including the finite particle method (FPM) [3] , reproducing kernel particle method (RKPM) [4] and meshless local Petrov-Galerkin (MLP...

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A projection technique for incompressible flow in the meshless finite volume particle method

Cited by 24 publications

References 17 publications

On convergent numerical algorithms for unsymmetric collocation

On convergent numerical algorithms for unsymmetric collocation

A finite‐volume particle method for conservation laws on moving domains

Meshless local discontinuous Petrov-Galerkin method with application to blasting problems

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