First-point distance parameter system and method for automatic grid generation

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 mesh points positioned with respect to an inner boundary and an outer boundary, the method comprising:

receiving using the computer from a user information corresponding to a shape to be analyzed using the analytical tool, a ξ-grid line mesh parameter value corresponding to a desired number of ξ-grid lines for the computation mesh, and an η-grid line mesh parameter value corresponding to a desired number of η-grid lines for the computation mesh;

solving using the computer one or more mesh equations for a plurality of point locations without the need for additional information from the user, said solving including:

determining if the η-grid line mesh parameter value exceeds a threshold resolution parameter, wherein the one or more mesh equations depend on an outer boundary distance parameter;

determining the outer boundary distance parameter as a function of an inner boundary distance parameter and the natural log of the η-grid line mesh parameter value, if the η-grid line mesh parameter value exceeds the threshold resolution parameter; and

determining the outer boundary distance parameter as a function of an inner boundary distance parameter and the square root of the η-grid line mesh parameter value, if the η-grid line mesh parameter value does not exceed the threshold resolution parameter;

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 inner boundary distance parameter is defined as a length of a ξ-grid line divided by the η-grid line mesh parameter value.

3. A method according to claim 2 , wherein the length of a ξ-grid line is a length selected from the group consisting of a straight line distance between the inner boundary of the computation mesh and the outer boundary of the computation mesh, an actual length of the ξ-grid line between the inner boundary and the outer boundary of a computation mesh, and a distance between a point formed by an intersection of the ξ-grid line and the inner boundary of the computation mesh to a point formed by an intersection of the ξ-grid line and the outer boundary of the computation mesh.

4. A method according to claim 1 , wherein the inner boundary distance parameter is defined as:

D

GBL

=

∑

i

⁢

D

⁡

(

ξ

i

)

LOC

N

where D GBL is the inner boundary distance parameter, D(ξ i ) Loc is a local average mesh size, and N is the number of ξ-grid lines.

5. A method according to claim 1 , wherein the one or more mesh equations depend on an outer boundary distance parameter that is a function of an inner boundary distance parameter and the natural log of the η-grid line mesh parameter value.

6. A method according to claim 5 , wherein the outer boundary distance parameter is determined by the equation:

D M =ln( M )* D 1

wherein D M is the outer boundary distance parameter, D 1 is the inner boundary distance parameter, and ln(M) is the natural log of the η-grid line mesh parameter value.

7. A method according to claim 1 , wherein the one or more mesh equations depend on an outer boundary distance parameter that is a function of an inner boundary distance parameter and the square root of the η-grid line mesh parameter value.

8. A method according to claim 7 , wherein the outer boundary distance parameter is determined by the equation:

D M =√{square root over (M)}*D 1

wherein D M is the outer boundary distance parameter, D 1 is the inner boundary distance parameter, and √{square root over (M)} is the square root of the η-grid line mesh parameter value.

9. A method according to claim 1 , wherein the one or more mesh equations includes a source decay factor that is inversely proportional to the η-grid line mesh parameter value.

10. A method according to claim 1 , wherein the one or more mesh equations includes a Jacobian scaling parameter that is not equal to two.

11. A method according to claim 1 , wherein the one or more mesh equations includes a source decay factor that is inversely proportional to the η-grid line mesh parameter value, and the one or more mesh equations include a Jacobian scaling parameter that is not equal to two.

12. 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 mesh points positioned with respect to an inner boundary and an outer boundary, the method comprising:

providing a shape to be analyzed using the analytical tool;

providing to the computer an η-grid line mesh parameter value corresponding to a desired number of η-grid lines for the computation mesh;

if the η-grid line mesh parameter value exceeds a threshold resolution parameter, solving using the computer one or more mesh equations for a plurality of point locations, the one or more mesh equations depending on an outer boundary distance parameter determined by a function of an inner boundary distance parameter and the natural log of the η-grid line mesh parameter value;

if the η-grid line mesh parameter value does not exceed the threshold resolution parameter, solving using the computer the one or more mesh equations for a plurality of point locations, the one or more mesh equations depending on an outer boundary distance parameter determined by a function of an inner boundary distance parameter and the square root of the η-grid line mesh parameter value;

generating the computation mesh as a function of the plurality of point locations; and

outputting one or more indicia representing the computational mesh.

13. A method according to claim 12 , wherein the one or more mesh equations depend on an outer boundary distance parameter that is a function of an inner boundary distance parameter and the natural log of the η-grid line mesh parameter value.

14. A method according to claim 13 , wherein the outer boundary distance parameter is determined by the equation:

D M =ln( M )* D 1

wherein D M is the outer boundary distance parameter, D 1 is the inner boundary distance parameter, and ln(M) is the natural log of the η-grid line mesh parameter value.

15. A method according to claim 12 , wherein the one or more mesh equations depend on an outer boundary distance parameter that is a function of an inner boundary distance parameter and the square root of the η-grid line mesh parameter value.

16. A method according to claim 15 , wherein the outer boundary distance parameter is determined by the equation:

D M =√{square root over (M)}*D 1

wherein D M is the outer boundary distance parameter, D 1 , is the inner boundary distance parameter, and √{square root over (M)} is the square root of the η-grid line mesh parameter value.

17. A method according to claim 12 , wherein the one or more mesh equations includes a source decay factor that is inversely proportional to the η-grid line mesh parameter value.

18. A method according to claim 12 , wherein the one or more mesh equations includes a Jacobian scaling parameter that is not equal to two.

19. A method according to claim 12 , wherein the one or more mesh equations includes a source decay factor that is inversely proportional to the η-grid line mesh parameter value, and the one or more mesh equations include a Jacobian scaling parameter that is not equal to two.

20. A system for generating a computation for use with an analytical tool, the computation mesh including a plurality of ξ-grid lines and η-grid lines intersecting at grid points, the grid points positioned with respect to an inner boundary and an outer boundary, the method 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 an outer boundary distance parameter determined according to an η-grid line mesh parameter value that corresponds to a desired number of η-grid lines for the computation mesh, wherein if the η-grid line mesh parameter value exceeds a threshold resolution parameter, the outer boundary distance parameter is determined by a function of an inner boundary distance parameter and the natural log of the η-grid line mesh parameter value, if the η-grid line mesh parameter value does not exceed the threshold resolution parameter, the outer boundary distance parameter is determined by a function of an inner boundary distance parameter and the square root of the η-grid line mesh parameter value;

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.

21. The system of claim 20 , further comprising a means for receiving from a user information corresponding to a shape to be analyzed using the analytical tool, a ξ-grid line mesh parameter value corresponding to a desired number of ξ-grid lines for the computation mesh, and the η-grid line mesh parameter value.

22. The system of claim 20 , wherein the means for outputting includes a display device.

23. 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 mesh points positioned with respect to an inner boundary and an outer boundary, the method comprising:

receiving using the computer from a user information corresponding to a shape to be analyzed using the analytical tool, a ξ-grid line mesh parameter value corresponding to a desired number of ξ-grid lines for the computation mesh, and an η-grid line mesh parameter value corresponding to a desired number of η-grid lines for the computation mesh;

solving using the computer one or more mesh equations for a plurality of point locations, the one or more mesh'equations including

an outer boundary distance parameter that is a function of an inner boundary distance parameter and one of a natural log of the η-grid line mesh parameter value and a square root of the η-grid line mesh parameter value;

a source decay factor that is inversely proportional to the η-grid line mesh parameter value; and

a Jacobian scaling parameter that is not equal to two;

generating the computation mesh as a function of the plurality of point locations; and

outputting one or more indicia representing the computational mesh.

24. A method according to claim 23 , wherein the inner boundary distance parameter is defined as a length of a ξ-grid line divided by the η-grid line mesh parameter value.

25. A method according to claim 23 , wherein the inner boundary distance parameter is defined as:

D

GBL

=

∑

i

⁢

D

⁡

(

ξ

i

)

LOC

N

where D GBL is the inner boundary distance parameter, D(ξ i ) LOC is a local average mesh size, and N is the number of ξ-grid lines.

26. A method according to claim 23 , wherein the outer boundary distance parameter is determined by the equation:

D M =ln( M )* D 1

wherein D M is the outer boundary distance parameter, D 1 is the inner boundary distance parameter, and ln(M) is the natural log of the η-grid line mesh parameter value.

27. A method according to claim 23 , wherein the outer boundary distance parameter is determined by the equation:

D M =√{square root over (M)}*D 1

wherein D M is the outer boundary distance parameter, D 1 is the inner boundary distance parameter, and √{square root over (M)} is the square root of the η-grid line mesh parameter value.

28. A method according to claim 23 , wherein the source decay factor is a function of 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.

29. A method according to claim 28 , wherein the source decay parameter has a value of about 14.

30. A method according to claim 23 , wherein the Jacobian scaling parameter has a value of about 1.25.

31. A hardware machine readable medium containing machine readable instructions for performing a method for automatically generating a computation mesh for use with an analytical tool, the computation mesh having a plurality of ξ-lines and η-lines intersecting at mesh points positioned with respect to an inner boundary and an outer boundary, the instructions comprising:

a set of instructions for receiving from a user information corresponding to a shape to be analyzed using the analytical tool, a ξ-grid line mesh parameter value corresponding to a desired number of ξ-grid lines for the computation mesh, and an η-grid line mesh parameter value corresponding to a desired number of η-grid lines for the computation mesh;

a set of instructions for solving one or more mesh equation for a plurality of point locations without the need for additional information from the user, said solving including:

determining if the η-grid line mesh parameter value exceeds a threshold resolution parameter, wherein the one or more mesh equations depend on an outer boundary distance parameter;

determining the outer boundary distance parameter as a function of an inner boundary distance parameter and the natural log of the η-grid line mesh parameter value, if the η-grid line mesh parameter value exceeds the threshold resolution parameter; and

determining the outer boundary distance parameter as a function of an inner boundary distance parameter and the square root of the η-grid line mesh parameter value, if the η-grid line mesh parameter value does not exceed the threshold resolution parameter;

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.

32. A machine readable medium according to claim 31 , further comprising:

a set of instructions for determining if the η-grid line mesh parameter value exceeds a threshold resolution parameter, wherein the mesh equation depends on an outer boundary distance parameter;

a set of instructions for determining the outer boundary distance parameter as a function of an inner boundary distance parameter and the natural log of the η-grid line mesh parameter value, if the η-grid line mesh parameter value exceeds the threshold resolution parameter; and

a set of instructions for determining the outer boundary distance parameter as a function of an inner boundary distance parameter and the square root of the η-grid line mesh parameter value, if the η-grid line mesh parameter value does not exceed the threshold resolution parameter.

33. A hardware 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 intersecting at mesh points positioned with respect to an inner boundary and an outer boundary, the instructions comprising:

a set of instructions for receiving from a user information corresponding to a shape to be analyzed using the analytical tool, a ξ-grid line mesh parameter value corresponding to a desired number of ξ-grid lines for the computation mesh, and an η-grid line mesh parameter value corresponding to a desired number of η-grid lines for the computation mesh;

a set of instructions for solving one or more mesh equations for a plurality of point locations, the one or more mesh equations including

an outer boundary distance parameter that is a function of an inner boundary distance parameter and one of a natural log of the η-grid line mesh parameter value and a square root of the n-grid line mesh parameter value;

a source decay factor that is inversely proportional to the η-grid line mesh parameter value; and

a Jacobian scaling parameter that is not equal to two;

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.

34. A machine readable medium according to claim 33 , wherein the inner boundary distance parameter is defined as a length of a ξ-grid line divided by the η-grid line mesh parameter value.

35. A machine readable medium according to claim 33 , wherein the inner boundary distance parameter is defined as:

D

GBL

=

∑

i

⁢

D

⁡

(

ξ

i

)

LOC

N

where D GBL is the inner boundary distance parameter, D(ξ i ) LOC is a local average mesh size, and N is the number of ξ-grid lines.

36. A machine readable medium according to claim 33 , wherein the outer boundary distance parameter is determined by the equation:

D M =ln( M )* D 1

wherein D M is the outer boundary distance parameter, D 1 is the inner boundary distance parameter, and ln(M) is the natural log of the η-grid line mesh parameter value.

37. A machine readable medium according to claim 33 , wherein the outer boundary distance parameter is determined by the equation:

D M =√{square root over (M)}*D 1

wherein D M is the outer boundary distance parameter, D 1 is the inner boundary distance parameter, and √{square root over (M)} is the square root of the η-grid line mesh parameter value.

38. 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 mesh points positioned with respect to an inner boundary and an outer boundary, the method comprising:

receiving using the computer from a user information corresponding to a shape to be analyzed using the analytical tool, a ξ-grid line mesh parameter value corresponding to a desired number of ξ-grid lines for the computation mesh, and an η-grid line mesh parameter value corresponding to a desired number of η-grid lines for the computation mesh;

solving using the computer one or more mesh equations for a plurality of point locations without the need for additional information from the user, wherein the one or more mesh equations depend on an outer boundary distance parameter that is a function of an inner boundary distance parameter and the natural log of the η-grid line mesh parameter value;

generating the computation mesh as a function of the plurality of point locations; and

outputting one or more indicia representing the computational mesh.

39. A method according to claim 38 , wherein the outer boundary distance parameter is determined by the equation:

D M =ln( M )* D 1

wherein D M is the outer boundary distance parameter, D 1 is the inner boundary distance parameter, and ln(M) is the natural log of the η-grid line mesh parameter value.