1. A particle-gridless hybrid methods for incompressible flows

2. 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,
The paper presents a method and a numerical scheme for the analysis of incompressible flows. This scheme is very often used in fluid’s animation applications.

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

4. Fluid motion
Applying Newton’s second law on a fixed particle is incorrect
However, the fluid flow gives an indication of the average motion of particles per unit volume
As an example, consider the temperature change by time as a volume is tracked in a velocity field

5. Fluid motion: Derivative Operator for Vector Fields
Assume 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.
Newton’s 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.

6. Fluid motion: Euler’s Equations
Euler ( ) was the first to formulate an equation of the motion of fluids
using Newton’s 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

7. Fluid motion: The Navier-Stokes Equations
Navier ( ) was the first to derive the equations including friction (from a pure theoretical consideration)
Stokes ( ) later derived the same equations, where he made the physical of  clear as the magnitude of the fluid’s viscosity.

8. 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

9. 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.

10. 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

11. 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.

12. 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 (****)

13. 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)

14. Moving-particle Semi-implicit (MPS) Numerical Scheme
A convection scheme
Considering 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.

15. Moving-particle Semi-implicit (MPS) Numerical Scheme
The overall algorithm of the particle-gridless hybrid method

16. 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,