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Author manuscript, published in "ECCM 2010, IV European Comparative study of the use of C 1 continuous finite Conference on Computational Mechanics, Paris : France (2010)" hal-00657382, version 1 - 6 Jan 2012 elements and splines

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SLIDE 1

Comparative study of the use of C 1−continuous finite elements and splines for contact problems with large slidings

  • B. Magnaina, A. Bataillyb, N. Chevaugeaonc, M. Legrandb, C. Pierreb

(a) Institut PRISME (b) Structural dynamics and (c) G´ eM, UMR CNRS 6183 ENSI Bourges vibration laboratory Pˆ

  • le calculs et structures

McGill University ´ Ecole Centrale de Nantes

Institut de Recherche en Génie Civil et Mécanique

G Ge

eM

M

hal-00657382, version 1 - 6 Jan 2012

Author manuscript, published in "ECCM 2010, IV European Conference on Computational Mechanics, Paris : France (2010)"

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Problem description

Problem description

Contact problems are highly nonlinear and may lead to complex and inefficient simulations In particular, high sensitivity of contact simulations when large slidings occur on curved contact interfaces discontinuity of the orientation of the normal to the contact surface (facetization) non smooth approximation of the gap function mesh refinement may not be a solution when large models are involved

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 2 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Practical manifestation

Practical manifestation

Rotor/stator interaction study may lead to the simulation of permanent contact between a blade and the surrounding casing rubbing modal interaction

C ∀α, g = 0

  • x
  • y

α = π

2

Contact detection results from discretization errors

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 3 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Practical manifestation

Practical manifestation

Beam: Q4 (20×5) Ring: Q4 (120×8)

  • x
  • y

α = π

2

1215 1029

π 10 2π 10 3π 10 4π 10 π 2

α (rad) 0.1 0.2 0.3 contact effort (N)

π 10 2π 10 3π 10 4π 10 π 2

α (rad)

  • 4

4 8 12 displacement (µm)

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 4 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Practical manifestation

Layout

1

Introduction Problem description Practical manifestation

2

C 1−continuous methods:Hermite and B-splines Description Gap inaccuracy Contact normal orientation approximation Contact detection

3

Large slidings cases validation case:cube and rings blade tip/casing

4

Conclusion

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 5 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Description

C 1−continuous methods

Objective: better representation of a curved contact surface Means: mortar elements superposition of linear elements and smoothing methods (B-splineS) integration of smoothing methods within the element (Hermite element) Focus of our study: compare B-spline and Hermite methods that are compatible with our solution algorithm.

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 6 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Description

B-splines

Description: third degree polynomial basis interpolation splines: control points chosen in agreement with mesh Assets: Independent on the element type of the mesh can be easily extended to 3D cases Requirements: specific detection procedure with Newton-Raphson method higher computation time

n n + 1

  • nn+1
  • nn
  • ns

spline/contact line

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 7 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Description

Hermite finite element

Description : 24 dof isoparametric element with cubic edge dof include coordinates of tangent vector to the edges Assets : continuity of normal vector orientation from an element to another automatic update of the contact surface with mesh deformation Requirements : specific mesh construction higher computation time

1 2 4 3 2 1 4 η ξ Y X

u3 v3 ∂yv3 ∂xv3 ∂xu3 ∂yu3

ui = {ui,∂ui ∂x ,∂ui ∂y ,vi,∂vi ∂x ,∂vi ∂y } i = (1,2,3,4).

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 8 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Description

C 1−continuous methods: validation

The pertinence of each strategy is assessed by:

1

checking geometrical errors

◮ gap inaccuracy ◮ contact normal orientation approximation 2

validating contact detection

3

simulating large sliding contact cases All the simulations presented are quasi-static simulations without friction using bi-potential method for contact management in total Lagrangian framework.

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 9 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Gap inaccuracy

Gap inaccuracy

5 10 15 20 25 30

  • 1

1 gap inaccuracy g (%) ξ

(a) Linear discretization

2 4 6 8 10 12 14 16 18

  • 1.5
  • 1

1 gap inaccuracy g (%) ξ

(b) Spline discretization

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

  • 1

1 gap inaccuracy g (%) ξ

(c) Hermite discretization

Gap inaccuracy for η = 2 ; η = 4 ; η = 8 ; η = 14 and η = 20 , η being the number of elements over the 1

2 ring.

η = 8 η = 4 ξ = −1 ξ = −1 ξ = 1 ξ = 1

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Contact normal orientation approximation

Contact normal orientation approximation

0.2 0.4 0.6 0.8

  • 0.8
  • 0.6
  • 0.4
  • 0.2
  • 1

1 contact orientation error θ (rad) ξ

(d) Linear discretization

0.2 0.4 0.6

  • 0.3
  • 0.2
  • 0.1
  • 1

1 0.1 0.3 0.5 contact orientation error θ (rad) ξ

(e) Spline discretization

  • 1

1 contact orientation error θ (rad)

  • 0.036
  • 0.027
  • 0.018
  • 0.009

0.009 0.018 0.027 0.036 ξ

(f) Hermite discretization

Contact orientation error for η = 2 ; η = 4 ; η = 8 ; η = 14 and η = 20 .

η = 8 η = 4 ξ = −1 ξ = −1 ξ = 1 ξ = 1

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Contact normal orientation approximation

Contact normal orientation approximation

Both the addition of B-spline over linear elements and the use of Hermite elements allow for significant improvements of the gap inaccuracy and the contact normal orientation error. The choice of the mesh parameters for our study is driven by the following conditions: maximum gap error must be gmax < 1% of the outer radius of the ring convergence must be observed for static simulations (sensitive bending of structures modeled with Q4 elements) Mesh parameters: 10×8

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Contact detection

Contact detection

Assessment of contact detection: beam in contact over a 1

2 ring, flexible or rigid.

  • x
  • y
  • F

X Y Z

Load applied on the tip

  • f the beam.

maximum displacement is equal for the three discretization methods no penetration detected Good contact detection

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Contact detection

Contact detection

Assessment of contact detection: beam in contact over a 1

2 ring, flexible or rigid.

  • x
  • y
  • F

0.143 0.285

  • x
  • y

(loading)

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 14 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

validation case: cube and rings

Large slidings: cube and rings

Contact simulation with large slidings is first tested in the following situation:

δ O

  • x
  • y

O1 O2 c δu node 1 node 2

Forced displacements on the superior edge of the cube Two distinct contact areas Symmetric contact case

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

validation case: cube and rings

Large slidings: cube and rings

Contact simulation with large slidings is first tested in the following situation:

X Y Z

(loading)

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

validation case: cube and rings

Large slidings: cube and rings

increment displacement (mm) 20 40 60 80 100

  • 5
  • 10
  • 15
  • 20
  • 25

node 2, uy increment displacement (mm) 20 40 60 80 100

  • 8
  • 4

4 8 node 2, ux increment contact effort (N) 20 40 60 80 100 1 0.2 0.6 1.4 1.8 node 1, F increment displacement (mm) 20 40 60 80 100 1 2 3 node 1, ux

linear finite elements , B-splines on linear finite elements , Hermite elements and refined mesh with linear finite elements

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 17 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

validation case: cube and rings

Large slidings: cube and rings

increment displacement (mm) 20 40 60 80 100

  • 5
  • 10
  • 15
  • 20
  • 25

node 2, uy increment displacement (mm) 20 40 60 80 100

  • 8
  • 4

4 8 node 2, ux increment contact effort (N) 20 40 60 80 100 1 0.2 0.6 1.4 1.8 node 1, F increment displacement (mm) 20 40 60 80 100 1 2 3 node 1, ux

linear finite elements , B-splines on linear finite elements , Hermite elements and refined mesh with linear finite elements

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 18 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

blade tip/casing

Large slidings: blade/casing

As mentionned in the introduction, the contact cases are:

  • x
  • x
  • y
  • y

α = π

2

C ∀α, g = 0 case 1 case 2

case 1: the 1

2 ring is clamped on its extremities

case 2: the 1

2 ring is clamped on its extremities and for R = Re

Mesh of the ring is 120×8 for linear elements and 10×8 for Hermite elements.

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 19 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

blade tip/casing

Large slidings: blade/casing, results

case 1:

casing, uy 20 40 60 80 100 increment displacement (×10−5 m)

  • 0.4

0.4 0.8 1.2 casing, uy 20 40 60 80 100 increment

  • 2
  • 1

1 displacement (×10−5 m)

case 2:

node 1, ux 20 40 60 80 100 increment displacement (×10−5 m) 0.1 0.2 0.3 casing, uy 20 40 60 80 100 increment

  • 4
  • 2

4 displacement (×10−7 m)

linear finite elements , B-splines on linear finite elements , Hermite elements

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 20 / 24

hal-00657382, version 1 - 6 Jan 2012

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

blade tip/casing

Large slidings: blade/casing, results

case 1:

blade, F 20 40 60 80 100 increment 0.1 0.2 0.3 contact effort (N)

case 2:

blade, F 20 40 60 80 100 increment 0.2 0.4 0.6 contact effort (N)

linear finite elements , B-splines on linear finite elements , Hermite elements

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 21 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

blade tip/casing

Large slidings: blade/casing, mesh size influence

Previous results show the smoothing obtained with the use of a B-spline over linear finite elements Facetization and discretization errors phenomenon obviously depends on mesh size Smoothing methods’ sensitivity to mesh size (?) mesh 120×8 ⇒ mesh 60×4

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

blade tip/casing

Large slidings: blade/casing, mesh size influence

case 1: (similar results are obtained for case 2)

node 1, ux 20 40 60 80 100 increment displacement (×10−5 m) 1 2 3 casing, uy 20 40 60 80 100 increment displacement (×10−5 m)

  • 4
  • 2

4 blade, F 20 40 60 80 100 increment 0.2 0.4 0.6 contact effort (N)

linear finite elements , B-splines on linear finite elements , Hermite elements

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 23 / 24

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Introduction C1−continuous methods:Hermite and B-splines Large slidings cases Conclusion

Conclusion

B-splines and Hermite elements allow for an important reduction of geometrical errors Both methods ensure contact normal orientation continuity It has been shown that when large slidings occur, discontinuous displacements and efforts appear due to discretization errors Using a smoothing method not only allow for the suppression of these discontinuities but may also allow for mesh size reduction In 2D, Hermite elements allow for a significant diminution of the number of dof Future work: Dynamic simulations must be carried out to highlight the consequences of displacements and efforts discontinuities 3D simulations on industrial models .

May 19th 2010 ECCM 2010: C1−continuous finite elements and splines for contact problems with large slidings 24 / 24

hal-00657382, version 1 - 6 Jan 2012