incremental volumetric remapping method - analysis and error evaluation
DESCRIPTION
Incremental Volumetric Remapping Method: Analysis and Error Evaluation A.J. Baptista 1, J.L. Alves 2, M.C. Oliveira 1, D.M. Rodrigues 1, L.F. Menezes 1 1 CEMUC, University of Coimbra, Pólo II, Rua Luís Reis Santos, Pinhal de Marrocos, 3030-788 Coimbra, Portugal 2 Department of Mechanical Engineering, University of Minho, Campus de Azurém, 4800-058 Guimarães, Portugal Abstract. In this paper the error associated with the remapping problem is analyzed. A range of numerical results that assess the performance of three different remapping strategies, applied to FE meshes that typically are used in sheet metal forming simulation, are evaluated. One of the selected strategies is the previously presented Incremental Volumetric Remapping method (IVR), which was implemented in the in-house code DD3TRIM. The IVR method fundaments consists on the premise that state variables in all points associated to a Gauss volume of a given element are equal to the state variable quantities placed in the correspondent Gauss point. Hence, given a typical remapping procedure between a donor and a target mesh, the variables to be associated to a target Gauss volume (and point) are determined by a weighted average. The weight function is the Gauss volume percentage of each donor element that is located inside the target Gauss volume. The calculus of the intersecting volumes between the donor and target Gauss volumes is attained incrementally, for each target Gauss volume, by means of a discrete approach. The other two remapping strategies selected are based in the interpolation/extrapolation of variables by using the finite element shape functions or moving least square interpolants. The performance of the three different remapping strategies is address with two tests. The first remapping test was taken from a literature work. The test consists in remapping successively a rotating symmetrical mesh, throughout N increments, in an angular span of 90º. The second remapping error evaluation test consists of remapping an irregular element shape target mesh from a given regular element shape donor mesh and proceed with the inverse operation. In this second test the computation effort is also measured. The results showed that the error level associated to IVR can be very low and with a stable evolution along the number of remapping procedures when compared with the other two methods. Besides, the method proved to be very robust even in critical remapping situations such as poor geometrical definition of the mesh domain boundaries. Keywords: Remapping, Mesh Transfer Operator, Numerical Simulation, Deep-Drawing, Error Evaluation, Incremental Volumetric Remapping.TRANSCRIPT
Incremental Volumetric Remapping Method:
Analysis and Error Evaluation
Centro de Engenharia Mecânica da Universidade de Coimbra
A.J. Baptista*, J.L. Alves**, M.C. Oliveira*, D.M. Rodrigues*, L.F. Menezes*
* Department of Mechanical Engineering, University of Coimbra, PORTUGAL
** Department of Mechanical Engineering, University of Minho, PORTUGAL
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
Donor mesh Target mesh
Transfer operator
• Nodal Variables
(forces, displacements, etc.)
• State Variables
(tensions, densities, etc.)
Φ
Remapping types
• Remapping basis
Donor mesh Target mesh
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
Original meshes Extrapolation Interpolation I Interpolation II
2N
i ig i ig
ig
I w x x x
1
1
, ,ng
i ig i ig
ig
N
• Finite element shape functions inversion
• Moving least squares interpolants
1
, ,n
j i j i
i
N
1
, ,n
ig j ig j
j
N
• Common remapping strategies
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
Direct transfer of state variables using
a weighted average funtion
Incremental Volumetric Remapping Method
Φ(v)
• Weighted average remapping method
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
Gauss Volume
Gauss Point
“constant variables”
i) Divide donor elements in Gauss Volumes
• Incremental volumetric remapping method
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
ii) Divide each target element to remapp in Gauss Volumes
• Incremental volumetric remapping method
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
DIFICULTY:
Calculus of the intersecting volumes
iii) Intersect each target Gauss Volume with the donor Gauss Volumes
• Incremental volumetric remapping method
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
iv) Divide each target Gauss Volume in small parts and obtain their centroids
NL
Small volume part
• Incremental volumetric remapping method
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
NL
Small volume part
3
1
1
NLi
jNGj
iii tot
V
V
Weighted average
Φ(v)
v) Find the donor Gauss Volume that contains the centroid of each small volume part
• Incremental volumetric remapping method
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
T x
• Simetrical mesh relative to the perpendicular planes YOZ and XOZ
• N angular increments between [0°, 90°]
• N consecutive remapping operations
• Variable comparison, between the initial and N states, in the same Gauss points positions
2 2
2220 1 cos 2 ,x y
T r r ra
x
Test characteristics
• Test 1 – Remapping of rotated circular meshes
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
Test ilustration: 3 rotation increments (α = 90°/3):
1st Remapping
Increment 1
1I
30
Initial state
• Test 1 – Remapping of rotated circular meshes
Increment 1
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
Test ilustration: 3 rotation increments (α = 90°/3):
1st Remapping
Increment 1
1I
30
Initial state
• Test 1 – Remapping of rotated circular meshes
Increment 1
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
2nd Remapeamento
Increment 1
1I
30
Test ilustration: 3 rotation increments (α = 90°/3): Increment 2
• Test 1 – Remapping of rotated circular meshes
Increment 2
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
2nd Remapeamento
Increment 1
1I
30
Test ilustration: 3 rotation increments (α = 90°/3): Increment 2
• Test 1 – Remapping of rotated circular meshes
Increment 2
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
3rd Remapping
Increment 2
1
I30
Test ilustration: 3 rotation increments (α = 90°/3): Increment 3
• Test 1 – Remapping of rotated circular meshes
Increment 3
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
3rd Remapping
Increment 2
1
I30
Test ilustration: 3 rotation increments (α = 90°/3): Increment 3
• Test 1 – Remapping of rotated circular meshes
Increment 3
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
Error evolution with the number of rotation increments (N)
Normalized RMS error Normalized maximum error
Method III – Incremental volumetric remapping (IVR)
Method II – Moving least squares interpolants
Method I – Extrapolation/Interpolation
• Test 1 – Remapping of rotated circular meshes
0.00
0.03
0.06
0.09
0.12
0.15
0.18
0 1 2 3 4 5 6 7 8 9
Número de incrementos de rotação
Err
o R
MS
[%
]
Método I Método II Método III
Number of rotation increments
Method I Method II Method III
RM
S e
rro
r [%
]
Err
o m
áx
imo
[%
]
115.7
219.7
0
4
8
12
16
20
0 1 2 3 4 5 6 7 8 9
Número de incrementos de rotação
Err
o m
áx
imo
RM
S [
%]
Método I Método II Método III
0.00
0.03
0.06
0.09
0.12
0.15
0.18
0 1 2 3 4 5 6 7 8 9
Número de incrementos de rotação
Err
o R
MS
[%
]
Método I Método II Método III
Err
o m
áx
imo
[%]
Method I Method II Method III
Number of rotation increments
Max
imu
m e
rro
r [%
]
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
• Test 2 – Remapping between two meshes of different discretizations
1st Remapping
2nd Remapping
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
• Test 2 – Remapping between two meshes of different discretizations
1st Remapping
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
• Test 2 – Remapping between two meshes of different discretizations
1st Remapping
2nd Remapping
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
• Test 2 – Remapping between two meshes of different discretizations
1st Remapping
2nd Remapping
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
• Test 2 – Remapping between two meshes of different discretizations
1st Remapping
2nd Remapping
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
• Test 2 – Remapping between two meshes of different discretizations
RMS error and CPU effort evolutions for each studied method
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0 1 2 3 4 5 6 7 8 9 10
Variação do parâmetro nl (método III)
Err
o R
MS
[%
]
0
200
400
600
800
1000
1200
1400
1600
1800
Tem
po
de
CP
U [
s]
Erro RMS - Método I Erro RMS - Método II
Erro RMS - Método III Tempo de CPU - Método I
Tempo de CPU - Método II Tempo de CPU - Método III
RMS Error – Method I RMS Error – Method III
RMS Error – Method III CPU Time – Method I
CPU Time – Method II CPU Time – Method III
RM
S e
rro
r [%
]
CP
U T
ime
[s]
Parameter nl (Method III)
Incremental Volumetric Remapping Method: Analysis and Error Evaluation CEMUC
• The error level associated to IVR method can be very low and with a
stable evolution when increasing the number of remapping operations,
compared with the other two studied methods
• IVR method achieves good relations between accuracy and the
CPU effort
• The Extrapolation-interpolation method requires low CPU effort,
although it achieved the worst results in terms of the error level
• Moving least squares interpolants lead to slightly better results
of error level relatively to the extrapolation-interpolation method
• The algorithms included in IVR have proven their reliability and
robustness even in critical remapping situations, such as poor
geometrical definition of the mesh domain boundaries
• Conclusions
Incremental Volumetric Remapping Method:
Analysis and Error Evaluation
Centro de Engenharia Mecânica da Universidade de Coimbra
A.J. Baptista*, J.L. Alves**, M.C. Oliveira*, D.M. Rodrigues*, L.F. Menezes*
* Department of Mechanical Engineering, University of Coimbra, PORTUGAL
** Department of Mechanical Engineering, University of Minho, PORTUGAL