Transcript Deformation Transfer - 法政大学 [HOSEI UNIVERSITY]

A particle-gridless hybrid methods
for incompressible flows
Introduction
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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, 407-424
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.
Introduction
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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 NavierStokes 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)
Fluid motion
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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
Fluid motion: Derivative Operator for Vector Fields
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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
dT T T dx T dy T dz T T
T
T






u
v
w,
dt
t x dt y dt z dt t x
y
z
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
d

  u  ,
dt t
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  d u  F
[ x , y , z ]T
dt



 u   u  F,
 t


where F is the sum forces acting on the volume.
Fluid motion: Euler’s Equations
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Euler (1707-1783) 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)


 u   u  p,
 t

u  0


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 Equations

Navier (1785-1836) was the first to derive the equations including friction (from a
pure theoretical consideration)


 u   u  p   2 u  F external ,
 t

u  0


Stokes (1819-1903) later derived the same equations, where he made the physical of
 clear as the magnitude of the fluid’s viscosity.
Moving-particle Semi-implicit (MPS) Numerical Scheme
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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
 (2r / re ) 2  2 (0  r  0.5re )

w(r )  (2r / re  2) 2
(0.5re  r  re )
0
(re  r )
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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
n
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j
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:
  

d


  w( rj ri ).
i
i

n
0
 
j i
j
i
2
 ri  rj
(r j  ri ) w( r j  ri ),



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:
 2
i

2d
n 0
 (
j i
j

 i ) w( rj  ri ) ,

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
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The continuity equation for incompressible fluid is follows:
D
   (  u)  0.
Dt
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In the MPS method, material derivative (Lagrangian time derivative) DDt = 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  (  u)  0 these difficulties can be eliminated
Moving-particle Semi-implicit (MPS) Numerical
Scheme
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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
 (u  u )  (r  r )

j
i
j
i

w( r j  ri ).


2


j i
r

r
i
j


D


2
external
   (  u)  0
  u   u  p   u  F
Dt
 t

**
*
ui  ui
1
  P n1 i ,

t

d
u i  0
n

From equations
implicitly:
and
 2 P n 1
i


t
  u* ,
i

where u*i is the temporal velocity obtained from
the explicit calculation and u**i is the new-time velocity.
pressure is calculated
Moving-particle Semi-implicit (MPS) Numerical
Scheme
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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
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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 LagrangianEulerian calculation is possible between the fully Eulerian (uai
= u**i) and the fully Lagrangian (uai = 0)
Moving-particle Semi-implicit (MPS) Numerical
Scheme
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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 higherorder calculation is bounded by them.
Moving-particle Semi-implicit (MPS) Numerical
Scheme
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The overall algorithm of the particlegridless hybrid method
Moving-particle Semi-implicit (MPS) Numerical
Scheme
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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,