Download - Incremental Volumetric Remapping Method - Analysis and Error Evaluation

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


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


Direct transfer of state variables using

a weighted average funtion

Incremental Volumetric Remapping Method

Φ(v)

• Weighted average remapping method


Gauss Volume

Gauss Point

“constant variables”

i) Divide donor elements in Gauss Volumes

• Incremental volumetric remapping method


ii) Divide each target element to remapp in Gauss Volumes



DIFICULTY:

Calculus of the intersecting volumes

iii) Intersect each target Gauss Volume with the donor Gauss Volumes



iv) Divide each target Gauss Volume in small parts and obtain their centroids

NL

Small volume part



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



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


Test ilustration: 3 rotation increments (α = 90°/3):

1st Remapping

Increment 1

1I

30

Initial state


Increment 1


2nd Remapeamento

Increment 1

1I

30

Test ilustration: 3 rotation increments (α = 90°/3): Increment 2


Increment 2


3rd Remapping

Increment 2

1

I30

Test ilustration: 3 rotation increments (α = 90°/3): Increment 3


Increment 3


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


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


Err

o m

áx

imo

RM

S [

%]


0.00

0.03

0.06

0.09

0.12

0.15

0.18

0 1 2 3 4 5 6 7 8 9


Err

o R

MS

[%

]


Err

o m

áx

imo

[%]

Method I Method II Method III

Number of rotation increments

Max

imu

m e

rro

r [%

]


• Test 2 – Remapping between two meshes of different discretizations

1st Remapping

2nd Remapping



1st Remapping



1st Remapping

2nd Remapping



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)


• 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

Download - Incremental Volumetric Remapping Method - Analysis and Error Evaluation

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