Jacobian scaling 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 Jacobian scaling parameter that is not equal to 2.
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 equations for a plurality of point locations, the one or more mesh equations depending on a Jacobian scaling parameter that is from about 0.5 to less than 2;
generating the computation mesh as a function of the plurality of point locations; and
outputting one or more indicia representing the computation mesh, wherein said solving one or more mesh equations includes relating a plurality of sources to the inner and outer boundaries according to:
J
2
p
(
ξ
)
=
(
J
J
1
)
λ
[
R
1
y
η
J
1
-
R
2
x
η
J
1
]
η
1
J
2
q
(
ξ
)
=
(
J
J
1
)
λ
[
-
R
1
y
ξ
J
1
+
R
2
x
ξ
J
1
]
η
1
J
2
r
(
ξ
)
=
(
J
J
M
)
λ
[
R
3
y
η
J
M
-
R
4
x
η
J
M
]
η
M
J
2
s
(
ξ
)
=
(
J
J
M
)
λ
[
-
R
3
y
ξ
J
M
+
R
4
x
ξ
J
M
]
η
M
where J identifies a Jacobian defining a cell area of a grid cell located within the interior of a grid of the computation mesh; J 1 identifies a Jacobian evaluated at the inner boundary; J M identifies a Jacobian evaluated at the outer boundary; λ identifies the Jacobian scaling parameter; p(ξ), q(ξ), r(ξ) and s(ξ) are sources; and R 1 , R 2 , R 3 and R 4 are curvature terms.
2. A method according to claim 1 , wherein the Jacobian scaling parameter has a value that is from 0.5 to less than 2.
3. A method according to claim 1 , wherein the Jacobian scaling parameter has a value that is equal to about 1.25.
4. A method according to claim 1 , wherein the Jacobian scaling parameter has a value that is equal to 1.25.
5. A method according to claim 1 , wherein the curvature terms are
defined according to:
R 1 =−└g 22 x ξξ −2 g 12 x ξη +g 11 x ηη ┘ η 1
R 2 =−└g 22 y ξξ −2 g 12 y ξη +g 11 y ηη ┘ η 1
R 3 =−└g 22 x ξξ −2 g 12 x ξη +g 11 x ηη ┘ η M
R 4 =−└g 22 y ξξ −2 g 12 y ξη −g 11 y ηη ┘ η M .
6. A method according to claim 1 , wherein said receiving step includes receiving 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.
7. A method according to claim 6 , wherein said generating step occurs without the need for additional information from the user.
8. A system for generating a computation mesh for use with an analytical tool, the computation mesh including 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, the computational means including a storage device having stored therein one or more mesh equations that depend on a Jacobian scaling parameter that has a value that is from about 0.5 to less than 2;
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,
wherein the one or more mesh equations include a relation of a plurality of sources to the inner and outer boundaries according to:
J
2
p
(
ξ
)
=
(
J
J
1
)
λ
[
R
1
y
η
J
1
-
R
2
x
η
J
1
]
η
1
J
2
q
(
ξ
)
=
(
J
J
1
)
λ
[
-
R
1
y
ξ
J
1
+
R
2
x
ξ
J
1
]
η
1
J
2
r
(
ξ
)
=
(
J
J
M
)
λ
[
R
3
y
η
J
M
-
R
4
x
η
J
M
]
η
M
J
2
s
(
ξ
)
=
(
J
J
M
)
λ
[
-
R
3
y
ξ
J
M
+
R
4
x
ξ
J
M
]
η
M
where J identifies a Jacobian defining a cell area of a grid cell located within the interior of a grid of the computation mesh; J 1 identifies a Jacobian evaluated at the inner boundary; J M identifies a Jacobian evaluated at the outer boundary; λ identifies the Jacobian scaling parameter; p(ξ), q(ξ), r(ξ) and s(ξ) are sources; and R 1 , R 2 , R 3 and R 4 are curvature terms.
9. A system according to claim 8 , wherein the Jacobian scaling parameter has a value that is equal to about 1.25.
10. A system according to claim 8 , wherein the curvature terms are
defined according to:
R 1 =−└g 22 x ξξ −2 g 12 x ξη +g 11 x ηη ┘ η 1
R 2 =−└g 22 y ξξ −2 g 12 y ξη +g 11 y ηη ┘ η 1
R 3 =−└g 22 x ξξ −2 g 12 x ξη +g 11 x ηη ┘ η M
R 4 =−└g 22 y ξξ −2 g 12 y ξη −g 11 y ηη ┘ η M .
11. 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 Jacobian scaling parameter that has a value that is from about 0.5 to less than 2;
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,
wherein the one or more mesh equations include a relation of a plurality of sources to the inner and outer boundaries according to:
J
2
p
(
ξ
)
=
(
J
J
1
)
λ
[
R
1
y
η
J
1
-
R
2
x
η
J
1
]
η
1
J
2
q
(
ξ
)
=
(
J
J
1
)
λ
[
-
R
1
y
ξ
J
1
+
R
2
x
ξ
J
1
]
η
1
J
2
r
(
ξ
)
=
(
J
J
M
)
λ
[
R
3
y
η
J
M
-
R
4
x
η
J
M
]
η
M
J
2
s
(
ξ
)
=
(
J
J
M
)
λ
[
-
R
3
y
ξ
J
M
+
R
4
x
ξ
J
M
]
η
M
where J identifies a Jacobian defining a cell area of a grid cell located within the interior of a grid of the computation mesh; J 1 identifies a Jacobian evaluated at the inner boundary; J M identifies a Jacobian evaluated at the outer boundary; λ identifies the Jacobian scaling parameter; p(ξ), q(ξ), r(ξ) and s(ξ) are sources; and R 1 , R 2 , R 3 and R 4 are curvature terms.
12. A machine-readable medium according to claim 11 , wherein the Jacobian scaling parameter has a value that is equal to about 1.25.
13. A machine-readable medium according to claim 11 , wherein the Jacobian scaling parameter has a value that is equal to 1.25.
14. A machine-readable medium according to claim 11 , wherein the curvature terms are
defined according to:
R 1 =−└g 22 x ξξ −2 g 12 x ξη +g 11 x ηη ┘ η 1
R 2 =−└g 22 y ξξ −2 g 12 y ξη +g 11 y ηη ┘ η 1
R 3 =−└g 22 x ξξ −2 g 12 x ξη +g 11 x ηη ┘ η M
R 4 =−└g 22 y ξξ −2 g 12 y ξη −g 11 y ηη ┘ η M .
15. A machine-readable medium according to claim 11 , wherein said set of instructions for receiving information includes a set of instructions for receiving 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.
16. A machine-readable medium according to claim 11 , wherein said set of instructions for generating a computation mesh occurs without the need for additional information from the user.