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