Source decay parameter system and method for automatic grid generation
A system and method for automatically generating a computation mesh for use with an analytical tool, the computation mesh having a plurality of ξ-grid lines and η-grid lines intersecting at grid points positioned with respect to an inner boundary and an outer boundary. The method includes receiving from a user information corresponding to a shape to be analyzed using the analytical tool and solving one or more mesh equation for a plurality of point locations, the one or more mesh equations depending on a source decay factor that is inversely proportional to the number of η-grid lines.
1. A computer-implemented method for automatically generating a computation mesh for use with an analytical tool, the computation mesh having a plurality of ξ-grid lines and η-grid lines intersecting at grid points positioned with respect to an inner boundary and an outer boundary, the method comprising:
receiving from a user information corresponding to a shape to be analyzed using the analytical tool;
solving, with a computer, one or more mesh equation for a plurality of point locations, the one or more mesh equations depending on a source decay factor that is inversely proportional to the number of η-grid lines;
generating the computation mesh as a function of the plurality of point locations; and
outputting one or more indicia representing the computational mesh.
2. A method according to claim 1 , wherein the source decay factor equals a source decay parameter divided by the number of η-grid lines.
3. A method according to claim 2 , wherein said source decay parameter has a value that is from about 8 to about 16.
4. A method according to claim 2 , wherein said source decay parameter has a value that is from 8 to 16.
5. A method according to claim 2 , wherein said source decay parameter has a value of about 14.
6. A method according to claim 2 , wherein said source decay parameter has a value of 14.
7. A method according to claim 2 , wherein said solving one or more mesh equations is iterated, each iteration having a source decay parameter that is constant across the iterations.
8. A method according to claim 1 , wherein said solving one or more mesh equations includes determining a plurality of point locations where one or more source terms of the one or more mesh equations are related to a source decay parameter as:
P (ξ,η)≡ p (ξ) e −σ(η−η 1 )/M +r (ξ) e −σ(η M −η)/M
Q (ξ,η)≡ q (ξ) e −σ(η−η 1 )/M +s (ξ) e −σ(η M −η)/M
where σ is a source decay parameter, M is the number of grid points along the ξ-grid lines, P (ξ,η)and Q (ξ,η) are source terms, p(ξ) is a source, q(ξ) is a source, r(ξ) is a source, and s(ξ) is a source.
9. A method according to claim 1 , wherein said solving one or more mesh equations includes determining a plurality of point locations where one or more source terms of the one or more mesh equations are related to a source decay parameter as:
P (ξ,η)≡ p (ξ) e −k(η−η 1 ) +r (ξ) e −k(η M −η)
Q (ξ,η)≡ q (ξ) e −k(η−η 1 ) +s (ξ) e −k(η M −η)
where k is a source decay factor that is inversely proportional to the number of η-grid lines, P (ξ,η)and Q (ξ,η) are source terms, p(ξ) is a source, q(ξ) is a source, r(ξ) is a source, and s(ξ) is a source.
10. A method according to claim 9 , wherein k equals a source decay parameter divided by the number of η-grid lines.
11. A system for generating a computation mesh for use with an analytical tool, the computation mesh having a plurality of ξ-grid lines and η-grid lines that intersect at grid points, the grid points positioned with respect to an inner boundary and an outer boundary, the system comprising:
a means for receiving a shape to be analyzed using the analytical tool;
a means for solving one or more mesh equations for a plurality of point locations including a storage device having stored therein one or more mesh equations that depend on a source decay factor that is inversely proportional to the number of η-grid lines;
a means for generating the computation mesh as a function of the plurality of point locations; and
a means for outputting one or more indicia representing the computation mesh.
12. A system according to claim 11 , further comprising a means for iterating the one or more mesh equations.
13. A system according to claim 11 , wherein the source decay factor equals a source decay parameter divided by the number of η-grid lines.
14. A system according to claim 13 , wherein said source decay parameter is 14.
15. A machine-readable medium containing machine readable instructions for performing a method of generating a computation mesh for use with an analytical tool, the computation mesh having a plurality of ξ-grid lines and η-grid lines that intersect at grid points positioned with respect to an inner boundary and an outer boundary, the machine-readable instructions comprising:
a set of instructions for receiving from a user information corresponding to a shape to be analyzed using the analytical tool;
a set of instructions for solving one or more mesh equation for a plurality of point locations, the one or more mesh equations depending on a source decay factor that is inversely proportional to the number of η-grid lines;
a set of instructions for generating the computation mesh as a function of the plurality of point locations; and
a set of instructions for outputting one or more indicia representing the computational mesh.
16. A machine-readable-medium according to claim 15 , wherein the source decay factor equals a source decay parameter divided by the number of η-grid lines, the source decay parameter having a value that is from about 8 to about 16.
17. A machine-readable-medium according to claim 15 , wherein the source decay factor equals a source decay parameter divided by the number of η-grid lines, the source decay parameter having a value of about 14.
18. A machine-readable medium according to claim 15 , wherein said set of instructions for solving one or more mesh equations includes a set of instructions for determining a plurality of point locations where one or more source terms of the one or more mesh equations are related to a source decay parameter as:
P (ξ,η)≡ p (ξ) e −σ(η−η 1 )/M +r (ξ) e −σ(η M −η)/M
Q (ξ,η)≡ q (ξ) e −σ(η−η 1 )/M +s (ξ) e −σ(η M −η)/M
where σ a is a source decay parameter, M is the number of grid points along the ξ-grid lines, P (ξ,η)and Q (ξ,η) are source terms, p(ξ) is a source, q(ξ) is a source, r(ξ) is a source, and s(ξ) is a source.
19. A machine-readable medium according to claim 15 , wherein said set of instructions for solving one or more mesh equations includes a set of instructions for determining a plurality of point locations where one or more source terms of the one or more mesh equations are related to a source decay parameter as:
P (ξ,η)≡ p (ξ) e −k(η−η 1 ) +r (ξ) e −k(η M −η)
Q (ξ,η)≡ q (ξ) e −k(η−η 1 ) +s (ξ) e −k(η M −η)
where k is a source decay factor that is inversely proportional to the number of η-grid lines, P (ξ,η)and Q (ξ,η) are source terms, p(ξ) is a source, q(ξ) is a source, r(ξ) is a source, and s(ξ) is a source.