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# Incremental Volumetric Remapping Method - Analysis and Error Evaluation

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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.

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### Incremental Volumetric Remapping Method - Analysis and Error Evaluation

1. 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. 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. 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. 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. 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. 6. Incremental Volumetric Remapping Method: Analysis and Error EvaluationCEMUC ii) Divide each target element to remapp in Gauss Volumes • Incremental volumetric remapping method
7. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 18. Incremental Volumetric Remapping Method: Analysis and Error EvaluationCEMUC • Test 2 – Remapping between two meshes of different discretizations 1st Remapping 2nd Remapping
19. 19. Incremental Volumetric Remapping Method: Analysis and Error EvaluationCEMUC • Test 2 – Remapping between two meshes of different discretizations 1st Remapping
20. 20. Incremental Volumetric Remapping Method: Analysis and Error EvaluationCEMUC • Test 2 – Remapping between two meshes of different discretizations 1st Remapping 2nd Remapping
21. 21. Incremental Volumetric Remapping Method: Analysis and Error EvaluationCEMUC • Test 2 – Remapping between two meshes of different discretizations 1st Remapping 2nd Remapping
22. 22. Incremental Volumetric Remapping Method: Analysis and Error EvaluationCEMUC • Test 2 – Remapping between two meshes of different discretizations 1st Remapping 2nd Remapping
23. 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. 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. 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