Abstract

The paper deals with the convergence problem of the SPH (Smoothed Particle Hydrodynamics) meshfree method for the solution of fluid dynamics tasks. In the introductory part, fundamental aspects of mesh- free methods, their definition, computational approaches and classification are discussed. In the following part, the methods of local integral representation, where SPH belongs are analyzed and specifically the method RKPM (Reproducing Kernel Particle Method) is described. In the contribution, also the influence of boundary conditions on the SPH approximation consistence is analyzed, which has a direct impact on the convergence of the method. A classical boundary condition in the form of virtual particles does not ensure a sufficient order of consistence near the boundary of the definition domain of the task. This problem is solved by using ghost particles as a boundary condition, which was implemented into the SPH code as part of this work. Further, several numerical aspects linked with the SPH method are described. In the concluding part, results are presented of the application of the SPH method with ghost particles to the 2D shock tube example. Also results of tests of several parameters and modifications of the SPH code are shown.

Highlights

  • Fluid dynamics problems are solved mostly by traditional numerical methods, such as FDM, FVM or FEM

  • This problem is solved by using ghost particles as a boundary condition, which was implemented into the SPH code as part of this work

  • Boundary conditions: In section 3.3, we have shown that the SPH method in conjunction with virtual particles does not satisfy the C1 consistency condition at the near boundary area

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Summary

Introduction

Fluid dynamics problems are solved mostly by traditional numerical methods, such as FDM, FVM or FEM. The common feature of these methods is the use of a lagrangian mesh or eulerian grid in the domain discretization process. For some types of problems these methods are suited poorly. The complications arising while solving these problems result from the use of a mesh or a grid. The idea of meshfree methods evolves naturally, first being used in 1977 by Lucy L.B. It was the SPH method applied to astrophysical problems of modeling the movement of stars and space objects. The main idea of meshfree methods lies in modeling of the domain through field nodes without any information about relations between these nodes. Function approximation is performed with the help of field nodes in support domains

Domain representation
Function approximation
Formation of system equations
Classification
SPH method
Formulation
Consistency
Boundary treatment
Virtual particles
Ghost particles
RKPM method
Shock tube 2D problem
Description of the shock tube 2D problem
Initial settings
Parameter settings
Simulation results
Conclusion
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