Fully-integrated hexahedral elements configured for reducing shear locking in finite element method
View Patent ↗Improved 8-node hexahedral elements configured for reducing shear locking in finite element method are disclosed. According to one aspect, aspect-ratio based scale factors are introduced to modify partial derivatives of the isoparametric shape function of the hexahedral element with respect to isoparametric dimensions, respectively. The modified derivatives are used for calculating the Jacobian matrix thereby the rate-of-strain. The scale factor is configured such that no changes for a perfect cubic solid element (i.e., element having aspect ratio of 1 (one) in all three spatial dimensions), while significant changes for element having poor aspect ratio. In other words, elements with poor aspect ratio are mapped to a perfect cubic element using the aspect-ratio based scale factors. According to anther aspect, off-diagonal components in the local Jacobian matrix are directly modified by cancelling terms related to spurious shear deformation modes. This measure completely alleviates the shear locking effect even for perfectly shaped elements.
1. A method of reducing transverse shear locking effects of a fully-integrated 8-node hexahedral element in finite element analysis comprising:
defining, in a computer system, a finite element analysis model having at least one fully-integrated 8-node hexahedral element, said finite element analysis model representing an engineering product;
calculating a set of aspect-ratio based scale factors for each of the at least one fully-integrated 8-node hexahedral element, wherein the aspect-ratio based scale factors are defined according to formula:
λ
i
j
=
min
(
1
,
L
j
L
i
)
i
,
j
=
1
,
2
,
3
where L 1 , L 2 and L 3 are respective lengths in three isoparametric dimensions of said each of the at least one fully-integrated 8-node hexahedral element, and
λ i j represents one of the aspect-ratio based scale factors;
creating an isoparametric Jacobian matrix of said each of the at least one fully-integrated 8-node hexahedral element by modifying, the isoparametric Jacobian matrix's off-diagonal terms with partial derivatives of an isoparametric shape functios (N I ) that is derived from aspect-ratio based scale factors such that said each of the at least one fully-integrated 8-node hexahedral element's transverse shear locking effects are reduced; and
conducting an engineering simulation using finite element analysis in the computer system with the finite element analysis model having the at least one fully-integrated 8-node hexahedral element and the modified isoparametric Jacobian matrix, the engineering simulation's results are used for assisting a user in making design decisions for improving the engineering product.
2. The method of claim 1 , wherein said each of the at least one fully-integrated 8-node hexahedral element has three local spatial dimensions.
3. The method of claim 1 , wherein said partial derivatives of the isoparametirc shape function are modified from
∂
N
I
∂
ξ
j
(
ξ
1
,
ξ
2
,
ξ
3
)
to
∂
N
I
∂
ξ
j
(
λ
1
j
ξ
1
,
λ
2
j
ξ
2
,
λ
3
j
ξ
3
)
,
j
=
1
,
2
,
3
,
where ξ 1 , ξ 2 , and ξ 3 are isoparametric coordinates in the three isoparametric dimensions, respectively.
4. The method of claim 1 , wherein said each of the at least one fully-integrated 8-node hexahedral element comprises 8 integration points.
5. A system for reducing transverse shear locking effects of a fully-integrated 8-node hexahedral element in finite element analysis comprising:
a main memory for storing computer readable code for a finite element analysis application module;
at least one processor coupled to the main memory, said at least one processor executing the computer readable code in the main memory to cause the finite element analysis application module to perform operations by a method of:
defining a finite element analysis model having at least one fully-integrated 8-node hexahedral element, said finite element analysis model representing an engineering product;
calculating a set of aspect-ratio based scale factors for each of the at least one fully-integrated 8-node hexahedral element, wherein the aspect-ratio based scale factors are defined according to formula:
λ
i
j
=
min
(
1
,
L
j
L
i
)
i
,
j
=
1
,
2
,
3
where L 1 , L 2 and L 3 are respective lengths in three isoparametric dimensions of said each of the at least one fully-integrated 8-node hexahedral element, and
λ i j represents one of the aspect-ratio based scale factors;
creating an isoparametric Jacobian matrix of said each of the at least one fully-integrated 8-node hexahedral element by modifying, the isoparametric Jacobian matrix's off-diagonal terms with partial derivatives of an isoparametric shape function (N I ) that is derived from aspect-ratio based scale factors such that said each of the at least one fully-integrated 8-node hexahedral element's transverse shear locking effects are reduced; and
conducting an engineering simulation using finite element analysis with the finite element analysis model having the at least one fully-integrated 8-node hexahedral element and the modified isoparametric Jacobian matrix, the engineering simulation's results are used for assisting a user in making design decisions for improving the engineering product.
6. The system of claim 5 wherein said partial derivatives of the isoparametric shape function are modified from
∂
N
I
∂
ξ
j
(
ξ
1
,
ξ
2
,
ξ
3
)
to
∂
N
I
∂
ξ
j
(
λ
1
j
ξ
1
,
λ
2
j
ξ
2
,
λ
3
j
ξ
3
)
,
j
=
1
,
2
,
3
,
where ξ 1 , ξ 2 , and ξ 3 are isoparametric coordinates in the three isoparametric dimensions, respectively.
7. A computer readable medium containing instructions for controlling a computer system for reducing transverse shear locking effects of a fully-integrated 8-node hexahedral element in finite element analysis by a method comprising:
defining, in a computer system, a finite element analysis model having at least one fully-integrated 8-node hexahedral element, said finite element analysis model representing an engineering product;
calculating a set of aspect-ratio based scale factors for each of the at least one fully-integrated 8-node hexahedral element, wherein the aspect-ratio based scale factors is defined according to formula:
λ
i
j
=
min
(
1
,
L
j
L
i
)
i
,
j
=
1
,
2
,
3
where L 1 , L 2 and L 3 are respective lengths in three isoparametric dimensions of said each of the at least one fully-integrated 8-node hexahedral element, and
λ i j represents one of the set of aspect-ratio based scale factors;
creating an isoparametric Jacobian matrix of said each of the at least one fully-integrated 8-node hexahedral element by modifying, the isoparametric Jacobian matrix's off-diagonal terms with partial derivatives of an isoparametric shape function (N I ) that is derived from aspect-ratio based scale factors such that said each of the at least one fully-integrated 8-node hexahedral element's transverse shear locking effects are reduced; and
conducting an engineering simulation using finite element analysis in the computer system with the finite element analysis model having the at least one fully-integrated 8-node hexahedral element and the modified isoparametric Jacobian matrix, the engineering simulation's results are used for assisting a user in making design decisions in improvement of the engineering product.
8. The computer readable medium of claim 7 wherein said partial deprvatives of the isoparametric shape function are modified from
∂
N
I
∂
ξ
j
(
ξ
1
,
ξ
2
,
ξ
3
)
to
∂
N
I
∂
ξ
j
(
λ
1
j
ξ
1
,
λ
2
j
ξ
2
,
λ
3
j
ξ
3
)
,
j
=
1
,
2
,
3
,
where ξ 1 , ξ 2 , and ξ 3 are isoparametric coordinates in the three isoparametric dimensions, respectively.