This document analyzes and evaluates the error of an incremental volumetric remapping method. Two tests are performed: 1) remapping of rotated circular meshes, and 2) remapping between meshes with different discretizations. The incremental volumetric remapping method achieves very low and stable error even with increasing remapping operations, with better accuracy-computation efficiency tradeoff compared to extrapolation-interpolation and moving least squares interpolation methods. The incremental volumetric remapping method is concluded to be reliable and robust for critical remapping situations.

1. 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

2. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
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

3. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
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

4. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
Direct transfer of state variables using
a weighted average funtion
Incremental Volumetric Remapping Method
Φ(v)
• Weighted average remapping method

5. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
Gauss Volume
Gauss Point
“constant variables”
i) Divide donor elements in Gauss Volumes
• Incremental volumetric remapping method

6. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
ii) Divide each target element to remapp in Gauss Volumes
• Incremental volumetric remapping method

7. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
DIFICULTY:
Calculus of the intersecting volumes
iii) Intersect each target Gauss Volume with the donor Gauss Volumes
• Incremental volumetric remapping method

8. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
iv) Divide each target Gauss Volume in small parts and obtain their centroids
NL
Small volume part
• Incremental volumetric remapping method

9. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
NL
Small volume part
3
1
1
NL
i
jNG
j
ii
i 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

10. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
 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
22
20 1 cos 2 ,
x y
T r r r
a


  x
Test characteristics
• Test 1 – Remapping of rotated circular meshes

11. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
Test ilustration: 3 rotation increments (α = 90°/3):
1st Remapping
Increment 1
1
I 30
Initial state
• Test 1 – Remapping of rotated circular meshes
Increment 1

12. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
Test ilustration: 3 rotation increments (α = 90°/3):
1st Remapping
Increment 1
1
I 30
Initial state
• Test 1 – Remapping of rotated circular meshes
Increment 1

13. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
2nd Remapeamento
Increment 1
1
I
30

Test ilustration: 3 rotation increments (α = 90°/3): Increment 2
• Test 1 – Remapping of rotated circular meshes
Increment 2

14. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
2nd Remapeamento
Increment 1
1
I
30

Test ilustration: 3 rotation increments (α = 90°/3): Increment 2
• Test 1 – Remapping of rotated circular meshes
Increment 2

15. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
3rd Remapping
Increment 2
1I30
Test ilustration: 3 rotation increments (α = 90°/3): Increment 3
• Test 1 – Remapping of rotated circular meshes
Increment 3

16. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
3rd Remapping
Increment 2
1I30
Test ilustration: 3 rotation increments (α = 90°/3): Increment 3
• Test 1 – Remapping of rotated circular meshes
Increment 3

17. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
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
ErroRMS[%]
Método I Método II Método III
Number of rotation increments
Method I Method II Method III
RMSerror[%]
Erromáximo[%]
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
ErromáximoRMS[%]
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
ErroRMS[%]
Método I Método II Método III
Erromáximo
[%]
Method I Method II Method III
Number of rotation increments
Maximumerror[%]

18. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
• Test 2 – Remapping between two meshes of different discretizations
1st Remapping
2nd Remapping

19. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
• Test 2 – Remapping between two meshes of different discretizations
1st Remapping

20. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
• Test 2 – Remapping between two meshes of different discretizations
1st Remapping
2nd Remapping

21. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
• Test 2 – Remapping between two meshes of different discretizations
1st Remapping
2nd Remapping

22. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
• Test 2 – Remapping between two meshes of different discretizations
1st Remapping
2nd Remapping

23. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
• 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)
ErroRMS[%]
0
200
400
600
800
1000
1200
1400
1600
1800
TempodeCPU[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
RMSerror[%]
CPUTime[s]
Parameter nl (Method III)

24. Incremental Volumetric Remapping Method:
Analysis and Error EvaluationCEMUC
• 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

25. 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