This document discusses the application of the incremental volumetric remapping method for simulating multi-step deep drawing processes. The method divides elements into Gaussian volumes to calculate intersecting volumes between source and target meshes, then performs a weighted average to remap state variables. It provides more accurate remapping than standard interpolation methods, with the level of accuracy controlled by a parameter. The method is self-consistent, conserves properties like yield criteria, and reduces errors and smoothing effects compared to other approaches. The authors conclude the incremental volumetric remapping method is effective for simulating multi-step forming simulations.

1. Application of the Incremental Volumetric
Remapping Method in the Simulation of
Multi-Step Deep Drawing Processes
A.J. Baptista*, J.L. Alves**, M.C. Oliveira*, D.M. Rodrigues*, L.F. Menezes*
* Department of Mechanical Engineering, University of Coimbra,
Polo II, 3030 Coimbra, PORTUGAL
** Department of Mechanical Engineering, University of Minho,
Campus de Azurém,4080-058,Guimarães, PORTUGAL
CENTRO DE ENGENHARIA MECÂNICA DA UNIVERSIDADE DE COIMBRA
THE 6th INTERNATIONAL CONFERENCE AND WORKSHOP ON
NUMERICAL SIMULATION OF 3D SHEET FORMING PROCESSES
August 15-19, 2005, Detroit, Michigan, USA

2. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
 OUTLOOK
I. Introduction
II. Remapping algorithms
III. Numerical example
IV. Results
V. Conclusions

3. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
I. Introduction
II. Remapping algorithms
III. Numerical example
IV. Results
V. Conclusions
 OUTLOOK
CEMUC

4. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
 INTRODUCTION
 The Remapping operation
CEMUC
Donor mesh Target mesh
 Generic definition of a remapping procedure (2D example)
 Remapping in the Nodes 
Nodal Variables:
Force, displacement, etc.
 Remapping in the Gauss Points 
State Variables:
Stress, density, etc.

5. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
 INTRODUCTION
 Remapping characterization
CEMUC
APPLICATION FIELDS
• Solid Mechanics
• Fluid Dynamics
• Combustion
• Multidisciplinary subjects
• Adaptive mesh operations
• Multigrid methods
• Texture mapping
• Trimming operations
REMAPPING
NECESSITY
• Keep the equilibrium state
• Minimize the transfer error
• Overall accuracy of the simulations
OPERATION REQUIREMENTS

6. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
 INTRODUCTION
 Remapping methodologies and features
CEMUC
 Remapping methodologies (X. Jiao and M.T. Heath 2004)
 Pointwise interpolation and extrapolation
 Area / Volume averaging (Rezoning techniques)
 Mortar elements (Project the data interface of subdomains)
 Common refinement (Intersection of two overlay meshes)
 Specialized methods
 Methods desirable features (M. M. Rashid 2002)
 Self-consistency (Identity operator for the degenerate case)
 Locality (Avoid wrong domain/interfaces contributions)
 Freedom from excessive smoothing
 Freedom from spurious local extremes
 Potential to incorporate constrains (Equilibrium, yield criteria, etc.)

7. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
I. Introduction
II. Remapping algorithms
III. Numerical example
IV. Results
V. Conclusions
 OUTLOOK

8. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
 REMAPPING ALGORITHMS
CEMUC
 Standard method
 Standard extrapolation-interpolation remapping
Donor mesh Target mesh
Original meshes Extrapolation Interpolation I Interpolation II
INCREMENTAL VOLUMETRIC REMAPPING – IVR
Using the finite element shape functions: i i
i
  
 Volume averaging method
 Incremental / discrete intersecting volumes calculation
DD3TRIM

9. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
 Remapping basis
Donor mesh Target mesh
(State variable ) (Unload)
Transfer Operator
 REMAPPING ALGORITHMS
 Incremental Volumetric Remapping

10. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
 Step 1 – Divide all donor elements in 8 Gauss Volumes
Gauss Volume
Gauss Point
 REMAPPING ALGORITHMS
 Incremental Volumetric Remapping
 2D case view: Quadrilateral meshes overlay
Homogeneous
properties
Step 1

11. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
(Step 1)
CEMUC
 Step 2 – For each target element to treat: division in 8 Gauss Volumes to remap
 REMAPPING ALGORITHMS
 Incremental Volumetric Remapping
 2D case view: Quadrilateral meshes overlay
Step 2

12. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
 Step 3 – Intersect each target Gauss volume with the donor Gauss volumes
 REMAPPING ALGORITHMS
 Incremental Volumetric Remapping
 2D case view: Quadrilateral meshes overlay
Step 3(Step 2)

13. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
 Step 4 – For each target Gauss volume: Gauss volume division
 REMAPPING ALGORITHMS
 Incremental Volumetric Remapping
 2D case view: Quadrilateral meshes
Step 4
NL
Gauss
volume part
(Step 3)

14. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
 Step 5 – For each target Gauss Volume part centroid:
Find the donor Gauss volume that encloses it
 REMAPPING ALGORITHMS
 Incremental Volumetric Remapping
3
1
1
NL
i
jNG
j
ii
i tot
V
V
 




 Remap state variable calculus
Weighted average as function of
the intersection Gauss volumes

15. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
I. Introduction
II. Remapping algorithms
III. Numerical example
IV. Results
V. Conclusions
 OUTLOOK

16. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
 NUMISHEET Benchmark#3: Channel Draw/Cylindrical Cup 2-Stage Test (DP600)
 NUMISHEET Benchmark#3
Stage 1: Channel Draw
 NUMERICAL EXAMPLE
 Cyclic bending and unbending
 Three layers in thickness direction
 More elements in the longitudinal direction
Good in thickness gradients prediction
Deep-Drawing simulations: DD3IMP (Static Implicit)

17. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
 NUMISHEET Benchmark#3: Channel Draw/Cylindrical Cup 2-Stage Test (DP600)
 NUMISHEET Benchmark#3
Stage 2: Cylindrical Cup
 NUMERICAL EXAMPLE
Intermediate State: Trimming
Specimen A
 Plane-strain conditions
Homogenize the number of elements
in the two principal directions
+ Remeshing
359 45
4 64

18. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
 Remapping operation: Comparison of methods
 Remapping operation tests
 NUMERICAL EXAMPLE
 Standard extrapolation-interpolation method
 Incremental Volumetric Remapping
 State variable analysed: sxx stress
Original
Mesh
Remeshed
Mesh
Remap
Remap 2
Test methodology
Stage 2
1

19. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
I. Introduction
II. Remapping algorithms
III. Numerical example
IV. Results
V. Conclusions
 OUTLOOK

20. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
 Remapping operation: Error comparison of the methods (IVR - NL5)
 Remapping operation tests
 Results
Initial state Extrapolation-Interpolation IVR ( NL = 5 )
0
500
1000
1500
2000
2500
3000
3500
4000
5 25 45 65 85 105 125 145 165
Error [MPa]
Numberofnodes
0
500
1000
1500
2000
2500
3000
3500
4000
5 25 45 65 85 105 125 145 165
Error [MPa]
Numberofnodes
X = 54.7 X = 29.8
– 45%

21. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
0
500
1000
1500
2000
2500
3000
3500
4000
5 25 45 65 85 105 125 145 165
Error [MPa]
Numberofnodes
CEMUC
 Remapping operation: Error comparison of the methods (IVR – NL10)
 Remapping operation tests
 Results
Initial state Extrapolation-Interpolation IVR ( NL = 10 )
X = 17.5
0
500
1000
1500
2000
2500
3000
3500
4000
5 25 45 65 85 105 125 145 165
Error [MPa]
Numberofnodes
X = 54.7
– 68%

22. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
0
500
1000
1500
2000
2500
3000
3500
4000
5 25 45 65 85 105 125 145 165
Error [MPa]
Numberofnodes
0
500
1000
1500
2000
2500
3000
3500
4000
5 25 45 65 85 105 125 145 165
Error [MPa]
Numberofnodes
CEMUC
 Remapping operation: Error comparison of the methods (IVR – NL15)
 Remapping operation tests
 Results
Initial state Extrapolation-Interpolation IVR ( NL = 15 )
X = 54.7 X = 12.4
– 77%

23. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
I. Introduction
II. Remapping algorithms
III. Numerical example
IV. Results
V. Conclusions
 OUTLOOK

24. THE 6th INTERNATIONAL NUMISHEET CONFERENCE
“Application of the Incremental Volumetric Remapping Method
in the Simulation of Multi-Step Deep Drawing Processes”
CEMUC
 Remarks and main conclusions
 The developed IVR method of DD3TRIM prove to be a very effective
and straightforward way to remap a given mesh.
 The method is both self-consistent and makes use of constrains
such as yield criteria conservation.
 The mean error, the smoothing effects and gradients distortion, can be greatly
reduced when compared with the extrapolation- interpolation base methods.
 The discrete approximation used for calculating the intersecting volumes is a
reliable option face the complex geometrical methods.
 An adjustable parameter (NL) allows the accuracy control of the remapping
operation and also, the flexibility face the meshes dimensions and asymmetries.
 Conclusions

25. Application of the Incremental Volumetric
Remapping Method in the Simulation of
Multi-Step Deep Drawing Processes
A.J. Baptista*, J.L. Alves**, M.C. Oliveira*, D.M. Rodrigues*, L.F. Menezes*
* Department of Mechanical Engineering, University of Coimbra,
Polo II, 3030 Coimbra, PORTUGAL
** Department of Mechanical Engineering, University of Minho,
Campus de Azurém,4080-058,Guimarães, PORTUGAL
CENTRO DE ENGENHARIA MECÂNICA DA UNIVERSIDADE DE COIMBRA
THE 6th INTERNATIONAL CONFERENCE AND WORKSHOP ON
NUMERICAL SIMULATION OF 3D SHEET FORMING PROCESSES
August 15-19, 2005, Detroit, Michigan, USA