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MESHFREE METHOD EXAMPLES SERIES
Xu, Gas-Kinetic Schemes for Unsteady Compressible Flow Simulations, Lecture Series ( van Karman Institute for Fluid Dynamics, 1998). Poberezhnyi, “ The one-dimensional Riemann problem on an elliptic curve,” Math. Part II-Application to hyperbolic conservation laws on unstructured meshes,” J. Dumbser, “ Multidimensional Riemann problem with self-similar internal structure. Levveqye, “ Wave propagation algorithms for multidimensional hyperbolic systems,” J. Speares, “ Restoration of the contact surface in the HLL-Riemann solver,” Shock Waves 4(1), 25– 34 (1994). Liou, “ A sequel to AUSM, Part II: AUSM+-up for all speeds,” Comput. Rho et al., “ Cures for the shock instability: Development of a shock-stable Roe scheme,” Comput. Oñate et al., “ A finite point method for compressible flow,” Int. Balakrishnan, “ An upwind finite difference scheme for meshless solvers,” J. Morinishi, “ An implicit gridless type solver for the Navier-Stokes equations,” Comput. Ball, “ A free-Lagrange method for unsteady compressible flow: Simulation of a confined cylindrical blast wave,” Shock Waves 5, 311– 325 (1996). Turkel, “ Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes,” AIAA Paper No. Batina, “ A gridless Euler/Navier-Stokes solution algorithm for complex-aircraft applications,” AIAA Paper No. Monaghan, “ The collapse of a rotating non-axisymmetric isothermal cloud,” R. Lucy, “ A numerical approach to the testing of the fission hypothesis,” Astron. Jensen, “ Finite difference techniques for variable grids,” Comput. Huang et al., “ Smoothed particle hydrodynamics (SPH) for complex fluid flows: Recent developments in methodology and applications,” Phys. Jameson, “ Meshless scheme based on alignment constraints,” AIAA J. Kao, “ A general finite difference method for arbitrary meshes,” Comput. Zhang et al., “ A maximum-principle-preserving third order finite volume SWENO scheme on unstructured triangular meshes,” Adv. Shephard, “ A modified quadtree approach to finite element mesh generation,” IEEE Comput. Several representative examples, such as shock tube problems, implosion problem, couette flow, lid-driven cavity flow, flow in a channel with a backward-facing step, supersonic flow around a ramp segment, and flow around staggered NACA0012 biplane configuration, are simulated to validate the proposed meshfree-DGKS. In addition, the DGKS can simultaneously calculate inviscid and viscous fluxes when simulating viscous flow problems, which gives an improved algorithm consistency. More importantly, the fluxes at the mid-point are reconstructed with the DGKS using the local solution of the Boltzmann equation, which can describe its physical properties well, thus easily and stably capturing the shock wave. This breaks through the limitations of the grid topology and is suitable to handle complex geometries. The meshfree-DGKS maintains the advantages of the meshless method as it is implemented at arbitrarily distributed nodes. The corresponding particle velocity components and distribution functions are integrated based on moment relations to obtain the flux. The fluxes at the mid-points are calculated using the DGKS based on the discrete particle velocity model. Then, the concept of numerical flux is introduced, which enables computing both compressible and incompressible problems. To simulate compressible problems with discontinuities, the virtual mid-points between adjacent nodes, which are regarded as Riemann discontinuities, are established. In this approach, the governing equations are discretized using the meshfree method based on the least squares-based finite difference approach. A meshfree method based on the discrete gas-kinetic scheme (DGKS) (called the meshfree-DGKS) for simulation of incompressible/compressible flows is proposed in this work.
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