Upwind algorithm for solving lubrication equations
AssigneeSeiko Epson Corporation
InventorsJie Zhang, Jiun-Der Yu, Shinri Sakai
Life of this patent
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Abstract
An embodiment of the present invention may be a system or method for simulating a physical process. The physical process being simulated may be in a droplet. The process being simulated may be the drying of a droplet on a substrate. Simulating the physical process may include using a finite difference scheme to approximate a differential of a function. The function may be dependent on a plurality of variables. The location in space at which one or more of the variables is evaluated may depend on the sign of one or more of the variables and upon which portion of the finite difference equation is being evaluated.
Claims8
A non-transitory computer-readable medium encoded with instructions for a processor to perform a method for simulating a final pattern of a droplet of a fluid, the instructions comprising:
- instructions for using lubrication equations to represent behavior of the droplet on a substrate, wherein the lubrication equations include:
- a simulation space that is representative of an environment in which the droplet is being simulated;
- a velocity function that is representative of motion of fluid inside the droplet as a function of space and time;
- a thickness function that is representative of a height of the droplet above the substrate as a function of space and time;
- a concentration function that is representative of a concentration of solute in the droplet as a function of space and time;
- a solute amount function that is a product of the concentration function and the thickness function;
- and
- a first function that is a product of a solute amount and the velocity function;
- instructions for using the lubrication equations to represent using a first equation to represent a relationship between a temporal derivative of the solute amount and a spatial derivative of the first function;
- instructions for approximating the spatial derivate of the first function at a first node in the simulation space by calculating a difference between the first function evaluated at a second point in the simulation space and the first function evaluated at a third point in the simulation space;
- instructions for evaluating the first function at the second point in the simulation space includes:
- instructions for determining a sign of the velocity function at the second point in space;
- instructions for evaluating the first function at the second point in the simulation space, if an averaged radial velocity at the second point in the simulation space is positive, as a product of:
- the velocity function at the second point in the simulation space and
- a solute amount at the first point in the simulation space;
- instructions for evaluating the first function at the second point in the simulation space, if an averaged radial velocity at the second point in the simulation space is negative as, a product of:
- the velocity function at the second point in the simulation space and
- a solute amount at a fourth point in the simulation space;
- instructions for evaluating the first function at the third point in the simulation space, if an averaged radial velocity at the third point in the simulation space is positive as, a product of:
- the velocity function at the third point in the simulation space and
- a solute amount at the fifth point in the simulation space;
- instructions for evaluating the first function at the third point in the simulation space, if an averaged radial velocity at the third point in the simulation space is negative as, a product of:
- the velocity function at the third point in the simulation space and
- a solute amount at a first point in the simulation space;
- instructions for deriving the final pattern of the droplet from results from solving the lubrication equations;
- and
- instructions for storing the final pattern.
Dependencies are inferred from the claim text.