3D realtime fluid animation by Navier-Stokes equations

Claim 1

A method for generating a three-dimensional(x, y, z) animation model on a computer that simulates a fluid flow and displaying the simulated fluid flow on a three dimensional graphics display, comprising the steps of: (a) forming a two dimensional (x and y) representation grid that models a liquid fluid flow on a computer; (b) calculating pressure values, p, along the two dimensional representation grid, wherein the pressure values, p, are proportional to a third dimensional representation z, by solving differential equations that includes: p n+1 i;j =(Δ 1 x u n+1 i;j +Δ 1 y v n+1 i;j )/ε, where i, j are respective x, y mesh coordinates, n+1 represents next state, and "ε" is a constant less than 1 and greater than 0; and ##EQU20## and where f means either u, v or p, u is velocity in the direction of the x axis represented as meter per seconds, v is velocity in the direction of the y axis represented as meter per seconds, p is the pressure of fluid represented as newton per square meter, and terms l and m are arbitrary mesh coordinates in two-dimensions; and (c) displaying the two dimensional(x, y) and three dimensional(z) representations on a graphics display, wherein an animated three dimensional model of the liquid fluid flow is formed.

Claim 2

A method for generating a three-dimensional(x, y, z) animation model on a computer that simulates a fluid flow and displaying the simulated fluid flow on a three dimensional graphics display, comprising the steps of: (a) forming a two dimensional (x and y) representation grid that models a liquid fluid flow on a computer; (b) calculating pressure values, p, along the two dimensional representation grid, wherein the pressure values, p, are proportional to a third dimensional representation z, by solving differential equations that includes: p n+1 i;j =(Δ 1 x u n+1 i;j +Δ 1 y v n+1 i;j )/ε, where i, j are respective x, y mesh coordinates, n+1 represents next state, and "ε" is a constant less than 1 and greater than 0; and ##EQU20## and where f means either u, v or p, u is velocity in the direction of the x axis represented as meter per seconds, v is velocity in the direction of the y axis represented as meter per seconds, p is the pressure of fluid represented as newton per square meter, and terms l and m are arbitrary mesh coordinates in two-dimensions; and (c) displaying the two dimensional(x, y) and three dimensional(z) representations on a graphics display, wherein an animated three dimensional model of the liquid fluid flow is formed.

Claim 6

A method for generating a three-dimensional(x, y, z) fluid flow animation model from a two dimension(x,y) representation of a fluid flow field on a computer and displaying the animation model of the three-dimensional(x, y, z) fluid flow on a graphics display, comprising the steps of: (a) providing a fluid flow within a border to form a fluid flow field, the fluid flow field having two dimensional(x and y) coordinates, wherein interior surfaces of the border are external boundaries to the fluid flow field, and where internal obstructions within the fluid flow field are interior boundaries to the fluid flow field; (b) calculating velocity and pressure values at external boundaries of the fluid flow field from the two dimensional(x,y) coordinates on a computer-processor; (c) calculating velocity and pressure values at internal obstruction boundaries of the fluid flow field from the two dimensional(x,y) coordinates on the computer-processor, wherein the velocity and the pressure values of the external obstruction boundaries and the velocity and the pressure values of the internal obstruction boundaries are calculated by solving the equations: p.sup.n+1.sub.i;j =(Δ.sup.1.sub.x u.sup.n+1.sub.i;j +Δ.sup.1.sub.y v.sup.n+1.sub.i;j)/ε, where i, j are respective x, y mesh coordinates, n+1 represents next state, and ε is a constant less than 1 and greater than 0; and ##EQU24## where f means either u, v or p, u is velocity in the direction of the x axis represented as meter per seconds, v is velocity in the direction of the y axis represented as meter per seconds, p is the pressure of fluid represented as newton per square meter, and terms l and m are arbitrary mesh coordinates in two-dimensions; (d) repeat steps (a) and (b) when external boundaries are updated; (e) repeat steps (a) and (c) when internal boundaries are updated; and (f) displaying the external boundaries and the internal boundaries of the fluid flow field on a graphics display, wherein the calculated pressure and velocity values represent third dimension coordinates, z.

Claim 7

A method for generating a three-dimensional(x, y, z) fluid flow animation model from a two dimension(x,y) representation of a fluid flow field on a computer and displaying the animation model of the three-dimensional(x, y, z) fluid flow on a graphics display, comprising the steps of: (a) providing a fluid flow within a border to form a fluid flow field, the fluid flow field having two dimensional(x and y) coordinates, wherein interior surfaces of the border are external boundaries to the fluid flow field, and where internal obstructions within the fluid flow field are interior boundaries to the fluid flow field; (b) calculating velocity and pressure values at external boundaries of the fluid flow field from the two dimensional(x,y) coordinates on a computer-processor; (c) calculating velocity and pressure values at internal obstruction boundaries of the fluid flow field from the two dimensional(x,y) coordinates on the computer-processor, wherein the velocity and the pressure values of the external obstruction boundaries and the velocity and the pressure values of the internal obstruction boundaries are calculated by solving the equations: p.sup.n+1.sub.i;j =(Δ.sup.1.sub.x u.sup.n+1.sub.i;j +Δ.sup.1.sub.y v.sup.n+1.sub.i;j)/ε, where i, j are respective x, y mesh coordinates, n+1 represents next state, and ε is a constant less than 1 and greater than 0; and ##EQU24## where f means either u, v or p, u is velocity in the direction of the x axis represented as meter per seconds, v is velocity in the direction of the y axis represented as meter per seconds, p is the pressure of fluid represented as newton per square meter, and terms l and m are arbitrary mesh coordinates in two-dimensions; (d) repeat steps (a) and (b) when external boundaries are updated; (e) repeat steps (a) and (c) when internal boundaries are updated; and (f) displaying the external boundaries and the internal boundaries of the fluid flow field on a graphics display, wherein the calculated pressure and velocity values represent third dimension coordinates, z.

Claim 11

The method for producing a three-dimensional (x, y, z) fluid flow animation model from a two dimension(x,y) representation of a fluid flow field on a computer and displaying the animation model of the three-dimensional(x, y, z) fluid flow on a graphics display, comprising the steps of: (a) providing a fluid flow within a border to form a fluid flow field, the fluid flow field having two dimensional(x and y) coordinates, wherein interior surfaces of the border are external boundaries to the fluid flow field, and where internal obstructions within the fluid flow field are interior boundaries to the fluid flow field; (b) calculating velocity and pressure values at external boundaries of the fluid flow field from the two dimensional(x,y) coordinates on a computer-processor; (c) calculating velocity and pressure values at internal obstruction boundaries of the fluid flow field from the two dimensional(x,y) coordinates on the computer-processor, wherein the velocity and the pressure values of the external obstruction boundaries and the velocity and the pressure values of the internal obstruction boundaries are calculated by solving the equations: ##EQU25## where i, j are respective x, y mesh coordinates, n represents current state, n+1 represents the next state, ε is a constant less than 1 and greater than 0, Δt represents the time between current state and next state, and Re is ##EQU26## where L and V are a characteristic length and velocity, respectively, and ν is the kinematic viscosity; and ##EQU27## and where f means either u, v or p, and terms l and m are arbitrary mesh coordinates in two-dimensions: (d) repeat steps (a) and (b) when external boundaries are updated; (e) repeat steps (a) and (c) when internal boundaries are updated; and (f) displaying the external boundaries and the internal boundaries of the fluid flow field on a graphics display, wherein the calculated pressure and velocity values represent third dimension coordinates, z.

Claim 12

The method for producing a three-dimensional (x, y, z) fluid flow animation model from a two dimension(x,y) representation of a fluid flow field on a computer and displaying the animation model of the three-dimensional(x, y, z) fluid flow on a graphics display, comprising the steps of: (a) providing a fluid flow within a border to form a fluid flow field, the fluid flow field having two dimensional(x and y) coordinates, wherein interior surfaces of the border are external boundaries to the fluid flow field, and where internal obstructions within the fluid flow field are interior boundaries to the fluid flow field; (b) calculating velocity and pressure values at external boundaries of the fluid flow field from the two dimensional(x,y) coordinates on a computer-processor; (c) calculating velocity and pressure values at internal obstruction boundaries of the fluid flow field from the two dimensional(x,y) coordinates on the computer-processor, wherein the velocity and the pressure values of the external obstruction boundaries and the velocity and the pressure values of the internal obstruction boundaries are calculated by solving the equations: ##EQU25## where i, j are respective x, y mesh coordinates, n represents current state, n+1 represents the next state, ε is a constant less than 1 and greater than 0, Δt represents the time between current state and next state, and Re is ##EQU26## where L and V are a characteristic length and velocity, respectively, and ν is the kinematic viscosity; and ##EQU27## and where f means either u, v or p, and terms l and m are arbitrary mesh coordinates in two-dimensions: (d) repeat steps (a) and (b) when external boundaries are updated; (e) repeat steps (a) and (c) when internal boundaries are updated; and (f) displaying the external boundaries and the internal boundaries of the fluid flow field on a graphics display, wherein the calculated pressure and velocity values represent third dimension coordinates, z.

Claim 16

A system of modelling a two-dimensional(x, y) liquid fluid flow for a three-dimensional(x, y, z) computer animation display, comprising: (a) a fluid flow to be modelled, the fluid flow having external surface boundaries and internal obstruction boundaries; (b) means for calculating pressure values and velocity values on a computer-processor at both the external surface boundaries and at the internal surface boundaries by the equations: p n+1 i;j =(Δ 1 x u n+1 i;j +Δ 1 y v n+1 i;j )/ε, where i,j are respective x,y mesh coordinates, n+1 represents next state and ε is a constant less than 1 and greater than 0; and ##EQU28## where f means either u, v or p, and terms l and m are arbitrary mesh coordinates in two-dimensions; and solving the equations: ##EQU29## where i, j are respective x, y mesh coordinates, n represents current state, n+1 represents the next state, ε is a constant less than 1 and greater than 0, Δt represents the time between current state and next state, and Re is ##EQU30## where L and V are a characteristic length and velocity, respectively, and ν is the kinematic viscosity; and ##EQU31## and where f means either u, v or p, u is velocity in the direction of the x axis represented as meter per seconds, v is velocity in the direction of the y axis represented as meter per seconds, p is the pressure of fluid represented as newton per square meter, and terms l and m arc arbitrary mesh coordinates in two-dimensions; and a 3D graphics display for displaying three dimensional representations of the liquid fluid flow with the external and internal boundary conditions for animation.

Claim 17

A system of modelling a two-dimensional(x, y) liquid fluid flow for a three-dimensional(x, y, z) computer animation display, comprising: (a) a fluid flow to be modelled, the fluid flow having external surface boundaries and internal obstruction boundaries; (b) means for calculating pressure values and velocity values on a computer-processor at both the external surface boundaries and at the internal surface boundaries by the equations: p n+1 i;j =(Δ 1 x u n+1 i;j +Δ 1 y v n+1 i;j )/ε, where i,j are respective x,y mesh coordinates, n+1 represents next state and ε is a constant less than 1 and greater than 0; and ##EQU28## where f means either u, v or p, and terms l and m are arbitrary mesh coordinates in two-dimensions; and solving the equations: ##EQU29## where i, j are respective x, y mesh coordinates, n represents current state, n+1 represents the next state, ε is a constant less than 1 and greater than 0, Δt represents the time between current state and next state, and Re is ##EQU30## where L and V are a characteristic length and velocity, respectively, and ν is the kinematic viscosity; and ##EQU31## and where f means either u, v or p, u is velocity in the direction of the x axis represented as meter per seconds, v is velocity in the direction of the y axis represented as meter per seconds, p is the pressure of fluid represented as newton per square meter, and terms l and m arc arbitrary mesh coordinates in two-dimensions; and a 3D graphics display for displaying three dimensional representations of the liquid fluid flow with the external and internal boundary conditions for animation.