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US 2010/0250213 A1Published application
Semi-lagrangian cip fluid solver without dimensional splitting
AssigneeSeoul National University
InventorsDoyub KIM, Oh-Young SONG, Hyeong-Seok KO
Life of this patent
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Abstract
A new constrained interpolation profile method, which is stable and accurate but requires less amount of computation, is provided. CIP is a high-order fluid advection solver that can reproduce rich details of fluids. It has third-order accuracy but its computation is performed over a compact stencil. A novel modification of the original CIP method that fixes all of the above problems without increasing the computational load or reducing the accuracy is provided. The proposed method brings significant improvements in both accuracy and speed.
Claims19
A method for simulating fluid using semi-Lagrangian CIP fluid solver without dimensional splitting, the method comprising steps for:
- modeling multiphase materials with grid of nodes for dealing with the multiphase behaviors including the dynamics of the borders between liquids and gas;
- suppressing numerical dissipation for getting rid of loss of mass of material of one phase from numerical dissipations due to the coarseness of the modeling of the materials in terms of grid of nodes;
- and
- suppressing numerical diffusion for reducing dampening of the fluid motion of materials in liquid phase,
- wherein the step of modeling multiphase materials comprises steps of:
- describing liquid and gas with a set of nonlinear partial differential equations that describe the flow of the fluids;
- representing the liquid-gas interface as an implicit surface;
- and
- determining properties of the materials, from the information about the liquid-gas interface, including the surface curvature and the surface tension,
- wherein the set of nonlinear partial differential equations comprises multiphase incompressible Navier-Stokes equations, wherein the step of representing the liquid-gas interface comprises a level set equation, φ,
- wherein the flow of fluid is described by the incompressible Navier-Stokes equations:
- ∇
- ·
- u
- =
- 0
- ;
- and
-
-
- ∂
- u
- ∂
- t
- =
- -
- u
- ·
- ∇
- u
- +
- f
- ρ
- +
- v
- ρ
-
- ∇
- 2
-
- u
- -
- ∇
- ρ
- ρ
- ,
- wherein u denotes the velocity field of the fluid, p pressure, ρ the density, v the kinetic viscosity, and f represents the external forces per volume,
- wherein the liquid evolves dynamically in space and time according to the underlying fluid velocity field, u, wherein the level set function changes according to the dynamical evolution of liquid and is updated by the level set equation,
- ∂
- φ
- ∂
- t
- +
- u
- ·
- ∇
- φ
- =
- 0
- ,
- further comprising the step of solving the incompressible Navier-Stokes equations and the level set equation at each time step,
- wherein the step of solving the incompressible Navier-Stokes equations and the level set equation comprises steps of:
- advecting the level set according to the level set equation;
- updating the velocity by solving the Navier-Stokes equations;
- and
- simulating droplets and bubbles,
- wherein the level set function φ and the fluid velocity field u are updated,
- wherein the step of updating the velocity comprises steps of:
- calculating the advection component u·∇u using the semi-Lagrangian method;
- applying the forces f/ρ;
- adding the effect of the viscous term v/ρ∇2u by employing implicit central differencing;
- and
- projecting the velocity field so that the condition ∇·u=0 is met,
- wherein the step of calculating the advection component comprises steps of:
- applying an unsplit semi-Lagrangian constrained interpolation profile (USCIP) method of using the function values at the grid points and the spatial derivatives at those points for constructing the profile inside the grid cell;
- and
- solving the level set equation to advect the level set values, and
- wherein, with the CIP method, the profile corresponding to the interval [xi,xi+1] in a two-dimensional case is represented by a polynomial,
- Φ
-
- (
- x
- ,
- y
- )
- =
- ∑
- 0
- ≤
- i
- +
- j
- ≤
- 3
-
- C
- ij
-
- x
- i
-
- y
- j
- +
- C
- 31
-
- x
- 3
-
- y
- +
- C
- 13
-
- xy
- 3
- .
- (
- 7
- )
- ,
- where C00, . . . , C31 are coefficients of the polynomial.
120 more elements
A method for simulating fluid using semi-Lagrangian CIP fluid solver without dimensional splitting, the method comprising steps for:
- modeling multiphase materials with grid of nodes for dealing with the multiphase behaviors including the dynamics of the borders between liquids and gas;
- suppressing numerical dissipation for getting rid of loss of mass of material of one phase from numerical dissipations due to the coarseness of the modeling of the materials in terms of grid of nodes;
- and
- suppressing numerical diffusion for reducing dampening of the fluid motion of materials in liquid phase,
- wherein the step of modeling multiphase materials comprises steps of
- describing liquid and gas with a set of nonlinear partial differential equations that describe the flow of the fluids;
- representing the liquid-gas interface as an implicit surface;
- and
- determining properties of the materials, from the information about the liquid-gas interface, including the surface curvature and the surface tension,
- wherein the set of nonlinear partial differential equations comprises multiphase incompressible Navier-Stokes equations, wherein the step of representing the liquid-gas interface comprises a level set equation, φ,
- wherein the flow of fluid is described by the incompressible Navier-Stokes equations:
- ∇
- ·
- u
- =
- 0
- ;
- and
- ∂
- u
- ∂
- t
- =
- -
- u
- ·
- ∇
- u
- +
- f
- ρ
- +
- v
- ρ
-
- ∇
- 2
-
- u
- -
- ∇
- ρ
- ρ
- ,
- wherein u denotes the velocity field of the fluid, p pressure, ρ the density, v the kinetic viscosity, and f represents the external forces per volume,
- wherein the liquid evolves dynamically in space and time according to the underlying fluid velocity field, u, wherein the level set function changes according to the dynamical evolution of liquid and is updated by the level set equation,
- ∂
- φ
- ∂
- t
- +
- u
- ·
- ∇
- φ
- =
- 0
- ,
- further comprising the step of solving the incompressible Navier-Stokes equations and the level set equation at each time step,
- wherein the step of solving the incompressible Navier-Stokes equations and the level set equation comprises steps of:
- advecting the level set according to the level set equation;
- updating the velocity by solving the Navier-Stokes equations;
- and
- simulating droplets and bubbles,
- wherein the level set function φ and the fluid velocity field u are updated,
- wherein the step of updating the velocity comprises steps of:
- calculating the advection component u·∇u using the semi-Lagrangian method;
- applying the forces f/ρ;
- adding the effect of the viscous term v/ρ∇2u by employing implicit central differencing;
- and
- projecting the velocity field so that the condition ∇·u=0 is met,
- wherein the step of calculating the advection component comprises steps of:
- applying an unsplit semi-Lagrangian constrained interpolation profile (USCIP) method of using the function values at the grid points and the spatial derivatives at those points for constructing the profile inside the grid cell;
- and
- solving the level set equation to advect the level set values, and
- wherein, with the CIP method, the profile corresponding to the interval [xi,xi+1] in a three-dimensional case is represented by a polynomial,
- Φ
-
- (
- x
- ,
- y
- ,
- z
- )
- =
- ∑
- 0
- ≤
- i
- +
- j
- +
- k
- ≤
- 3
-
- C
- ijk
-
- x
- i
-
- y
- j
-
- z
- k
- +
- C
- 310
-
- x
- 3
-
- y
- +
- C
- 301
-
- xz
- 3
- +
- C
- 130
-
- xy
- 3
- +
- C
- 031
-
- y
- 3
-
- z
- +
- C
- 103
-
- z
- 3
-
- x
- +
- C
- 013
-
- z
- 3
-
- y
- +
- C
- 211
-
- x
- 2
-
- yx
- +
- C
- 121
-
- xy
- 2
-
- z
- +
- C
- 112
-
- xyz
- 2
- +
- C
- 311
-
- x
- 3
-
- yx
- +
- C
- 131
-
- xy
- 3
-
- z
- +
- C
- 113
-
- xyz
- 3
- .
194 more elements
Dependencies are inferred from the claim text.