# A particle-gridless hybrid methods for incompressible flows

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• A particle-gridless hybrid methods for incompressible flows

• Introduction

The lecture is mainly based on the paper of H. Y. Yoon, S. Koshizuka and Y. Oka, A Particle-gridless Hybrid Method for Incompressible Flows, Int. Journal for Numerical Methods in Fluids, 30, 1999, 407-424The paper presents a method and a numerical scheme for the analysis of incompressible flows. This scheme is very often used in fluids animation applications.

• Introduction

The animation of fluid phenomena is widely used in the film industryComputer simulation has been required to analyze many thermal-hydraulic problemsThe physics of fluid dynamics has been described by the Navier-Stokes equations since about 1821Recently, numerical methods that do not use any grid structure were developedA ball splashes into a tank water (N. Foster and R. Fedkiw)

• Fluid motionApplying Newtons second law on a fixed particle is incorrectHowever, the fluid flow gives an indication of the average motion of particles per unit volumeAs an example, consider the temperature change by time as a volume is tracked in a velocity field

• Fluid motion: Derivative Operator for Vector FieldsAssume that we are tracking a particle with coordinates {x(t), y(t), z(t)} The time varying temperature field is given as T: 4 The temperature of the particle is determined as T(x(t), T(y(t), T(z(t), t)Then the total derivative with respect to t is found to be

using u = [u,v,w]T as the velocity vector of the particle.The substantial operator for a flow u and in a dimensionless formulation is given as

where is the spatial gradient operator.Newtons second law of motion in terms of the substantial derivative per unit volume is found to be

where F is the sum forces acting on the volume.

• Fluid motion: Eulers EquationsEuler (1707-1783) was the first to formulate an equation of the motion of fluids using Newtons second equation, and the pressure gradient as an internal force, augmented with an equation for mass conservation under the assumption of incompressible fluids (the density is constant)

The mass conservation equation is the consequence of Gauss` divergence theorem for a closed region of a fluid V as shown in Figure

• Fluid motion: The Navier-Stokes EquationsNavier (1785-1836) was the first to derive the equations including friction (from a pure theoretical consideration)

Stokes (1819-1903) later derived the same equations, where he made the physical of clear as the magnitude of the fluids viscosity.

• Moving-particle Semi-implicit (MPS) Numerical Scheme

In the MPS method a particle interacts with others in its vicinity covered with a weight function w(r), where r is the distance between two particles. Many weighted functions have been proposed in the literature. One of the best is the Gaussian.

In this study, the following function is employed

re defines the radius of the interaction area

• Moving-particle Semi-implicit (MPS) Numerical Scheme

The particle number density at a co-ordinate ri is defined by

A gradient vector between two particles i and j possessing scalar quantities i and j at coordinates ri and rj is defined by(i -j ) (ri - rj )/| ri - rj |2.The gradient vector at the particle i is given as the weighted average of these gradient vectors:

where d is the number of space dimensions and n0 is the particle number density.Laplacian is an operator representing diffusion. In the MPS method, diffusion is modeled by distribution of a quantity from a particle to its neighboring particles:

where is a parameter. When the space is two dimensional and the mentioned above weight function w(r) is employed, = (31/140)re2.

• Moving-particle Semi-implicit (MPS) Numerical Scheme

The continuity equation for incompressible fluid is follows:

In the MPS method, material derivative (Lagrangian time derivative) = 0 is used for the incompressibility model where the particle number density keeps a constant value of n0.When the calculated particle number density n* is not n0 , it is implicitly corrected to n0.However, the particle number density dose not change in Eulerian coordinates.By using these difficulties can be eliminated

• Moving-particle Semi-implicit (MPS) Numerical Scheme

The velocity divergence between two particles i and j is defined by (uj-ui)(rj-ri)/| rj-ri |2.The velocity divergence at the particle i is given by the following equation

From equations and pressure is calculated implicitly:

where u*i is the temporal velocity obtained from the explicit calculation and u**i is the new-time velocity.

_1247524810.unknown

• Moving-particle Semi-implicit (MPS) Numerical Scheme

The right side of Equation (*****) is the velocity divergence, which is calculated using Equation (***)The left side of Equation (*****) is calculated using the Laplacian model shown in equation (**)The one have simultaneous equations expressed by a linear matrix. This can be solved various liar solvers.After the pressure calculation, the new-time velocity u**i is calculated from Equation (****)

• Moving-particle Semi-implicit (MPS) Numerical Scheme

In the problems where inlet and outlet boundaries exist, it is difficult to trace a particle in Lagrangian coordinates.The Lagrangian and Eulerian calculations are combined using a cubic interpolation in area coordinates. A solution of a convection equation is substituted by f(t+t,r) = f(t, r- t u ). First, a particle located at rni moves to a new position r**i using the velocity u**i obtained from the Lagrangian calculation by a particle method.Then, the new-time properties at rn+1i are calculated by f(t+t, rn+1i) = f(t, r**i - t uai ). Depending on the velocity uai, an arbitrary Lagrangian-Eulerian calculation is possible between the fully Eulerian (uai = u**i) and the fully Lagrangian (uai = 0)

• Moving-particle Semi-implicit (MPS) Numerical Scheme

A convection schemeConsidering the flow direction of each computing point (particle), a one dimensional local grid is generated as shown in Figure.For the three local grid points on the one dimensional local grid, the physical properties are interpolated from those of neighboring particles.Any higher-order difference scheme can be applied since a one-dimentional grid has been obtained along the flow direction.Maximum and minimum limits are calculated at each time step and the solution of a higher-order calculation is bounded by them.

• Moving-particle Semi-implicit (MPS) Numerical Scheme

The overall algorithm of the particle-gridless hybrid method

• Moving-particle Semi-implicit (MPS) Numerical Scheme

For more details applying MPS method in Computer graphics, see, Simon Premoze, Tolga Tasdizen, James Bigler, Aaron Lefohn, Ross T. Whitaker, Particle-Based Simulation of Fluids, Proceedings of Eurographics 2003,

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