Methods and systems for reducing finite element simulation time for acoustic response analysis
Methods and systems for reducing finite element simulation time for acoustic response analysis are disclosed. In one embodiment, a method includes analytically creating a finite element model, the finite element model including a plurality of subdivisions. A plurality of cross-correlations between respective pairings of the subdivisions is then specified. A portion of the cross-correlations are then eliminated to provide a reduced set of cross-correlations between respective pairings of the subdivisions. The elimination includes determining a spatial distance value between at least two subdivisions, and discarding at least one of the cross-correlations for which the spatial distance value is greater than a specified threshold value. The finite element simulation is then performed using the reduced set of cross-correlations.
1. A computer implemented method of performing a finite element simulation of a phenomenon, comprising:
analytically creating a finite element model of the phenomenon, the finite element model including a plurality of subdivisions;
specifying a plurality of cross-correlations between respective pairings of the subdivisions; each cross-correlation being based on a combination of a pair of variable values, each variable value occurring at a respective one of the subdivisions;
eliminating a portion of the cross-correlations to provide a reduced set of cross-correlations between respective pairings of the subdivisions, wherein the eliminating includes: determining a spatial distance value between at least two subdivisions, and discarding at least one of the cross-correlations for which the spatial distance value is greater than a specified threshold value, the specified threshold value being a spatial distance between a respective pairing of subdivisions involved in the cross-correlation calculation, wherein discarding at least one of the cross-correlations includes discarding at least one of the cross-correlations based on a specified value of epsilon e, wherein e is defined as follows.
2. The method of claim 1 , wherein analytically creating a finite element model of the phenomenon includes analytically creating a finite element model of a structural response to an acoustic excitation.
3. The method of claim 1 , wherein analytically creating a finite element model of the phenomenon includes defining at least one of structural preload and thermal preload.
4. The method of claim 1 , wherein specifying a plurality of cross-correlations between respective pairings of the subdivisions comprises implementing a method of modal superposition including integrating a cross-correlation function.
5. The method of claim 1 , wherein specifying a plurality of cross-correlations between respective pairings of the subdivisions includes defining the cross-correlations as follows:
C
(
ξ
,
η
,
ω
)
=
G
(
ω
)
exp
{
-
ξ
λ
ξ
(
ω
)
-
η
λ
η
(
ω
)
-
ⅈ
ω
ξ
U
ξ
}
where
ξ=( r − r ′)| x =Spatial distance in x direction
η=( r − r ′)| y =Spatial distance in y direction
λ ξ , λ η =Correlation scale along ξ and η
U ξ =Convective velocity along ξ.
6. The method of claim 1 , wherein discarding at least one of the cross-correlations for which the spatial distance value is greater than a specified error threshold value associated with a result of the finite element simulation includes discarding at least one of the cross-correlations for which the spatial distance value is greater than a specified value of epsilon ε, wherein ε<<1.
7. The method of claim 1 , wherein discarding at least one of the cross-correlations includes providing a logarithmic relationship as follows:
−|ξ|<ln ελ ξ (ω); and
recasting the logarithmic relationship as follows:
|ξ|>|ln ε| λ ξ
where
λ ξ =min{λ ξ (ω)}.
8. A computer-readable media containing instructions that, when executed by a computer, perform a finite element simulation of a phenomenon, comprising:
a first portion configured to analytically create a finite element model of the phenomenon, the finite element model including a plurality of subdivisions;
a second portion configured to specify a plurality of cross-correlations between respective pairings of the subdivisions, each cross-correlation being based on a combination of a pair of variable values, each variable value occurring at a respective one of the subdivisions;
a third portion configured to eliminate a portion of the cross-correlations to provide a reduced set of cross-correlations between respective pairings of the subdivisions, wherein the eliminating includes:
determining a spatial distance value between at least two subdivisions, and
discarding at least one of the cross-correlations for which the spatial distance value is greater than a specified threshold value, the specified threshold value being a spatial distance between a respective pairing of subdivisions involved in the cross-correlation calculation wherein the third portion is further configured to discard at least one of the cross-correlations based on a specified value of epsilon ε, wherein ε is defined as follows:
exp
{
-
ξ
λ
ξ
(
ω
)
}
<
ɛ
where
ξ=( r − r ′)| x =Spatial distance in a first direction,
η=( r − r ′)| y =Spatial distance in a second direction,
λ ξ =Correlation scale along ξ and η, and
ω=2*pi*f, where f is the frequency of the acoustic excitation in Hz; and
a fourth portion configured to perform the finite element simulation using the reduced set of cross-correlations.
9. The computer-readable media of claim 8 , wherein the first portion is further configured to analytically create a finite element model of a structural response to an acoustic excitation.
10. The computer-readable media of claim 8 , wherein the first portion is further configured to define at least one of a structural preload and a thermal preload.
11. The computer-readable media of claim 8 , wherein the second portion is further configured to implement a method of modal superposition that includes integrating a cross-correlation function.
12. The computer-readable media of claim 8 , wherein the second portion is further configured to define the cross-correlations as follows:
C
(
ξ
,
η
,
ω
)
=
G
(
ω
)
exp
{
-
ξ
λ
ξ
(
ω
)
-
η
λ
η
(
ω
)
-
ⅈ
ω
ξ
U
ξ
}
where
ξ=( r − r ′)| x =Spatial distance in x direction
η=( r − r ′)| y =Spatial distance in y direction
λ ξ , λ η =Correlation scale along ξ and η
U ξ =Convective velocity along ξ.
13. The computer-readable media of claim 8 , wherein the third portion is further configured to discard at least one of the cross-correlations for which the spatial distance value is greater than a specified value of epsilon ε, wherein ε<<1.
14. A system configured to perform a finite element simulation of a phenomenon, comprising:
a memory component configured to store information; a processor operatively coupled to the memory component and configured to perform a plurality of computations, wherein the processor is enabled to perform finite element calculations including:
analytically creating a finite element model of the phenomenon, the finite element model including a plurality of subdivisions;
specifying a plurality of cross-correlations between respective pairings of the subdivisions, each cross-correlation being based on a combination of a pair of variable values, each variable value occurring at a respective one of the subdivisions;
eliminating a portion of the cross-correlations to provide a reduced set of cross-correlations between respective pairings of the subdivisions, wherein the eliminating includes: determining a spatial distance value between at least two subdivisions, and discarding at least one of the cross-correlations for which the spatial distance value is greater than a specified threshold value, the specified threshold value being a spatial distance between a respective pairing of subdivisions involved in the cross-correlation calculation wherein discarding at least one of the cross-correlations includes discarding at least one of the cross-correlations based on a specified value of epsilon e, wherein e is defined as follows:
exp
{
-
ξ
λ
ξ
(
ω
)
}
<
ɛ
;
and
performing the finite element simulation using the reduced set of cross-correlations.
15. The system of claim 14 , wherein analytically creating a finite element model of the phenomenon includes defining at least one of structural preload and thermal preload.
16. The system of claim 14 , wherein specifying a plurality of cross-correlations between respective pairings of the subdivisions includes defining the cross-correlations as follows:
C
(
ξ
,
η
,
ω
)
=
G
(
ω
)
exp
{
-
ξ
λ
ξ
(
ω
)
-
η
λ
η
(
ω
)
-
ⅈ
ωξ
U
ξ
}
where
ξ=( r − r ′)| x =Spatial distance in x direction
η=( r − r ′)| y =Spatial distance in y direction
λ ξ , λ η =Correlation scale along ξ and η
U ξ =Convective velocity along ξ.
17. The system of claim 14 , wherein discarding at least one of the cross-correlations for which the spatial distance value is greater than a specified error threshold value associated with a result of the finite element simulation includes discarding at least one of the cross-correlations for which the spatial distance value is greater than a specified value of epsilon ε, wherein ε<<1.