isogeometric analysis application to car crash simulation · isogeometric analysis application to...

Isogeometric Analysis Applicationto Car Crash Simulation

S. Bouabdallah2, C. Adam2,3, M. Zarroug2, H. Maitournam3

1De Vinci Engineering Lab, Ecole Superieure d’Ingenieurs Leonard de Vinci2Direction Scientifique et Technologies Futures, PSA Peugeot Citroen

3Laboratoire de Mecanique des Solides, Ecole Polytechnique

Forum Teratec 2014 - July 2, 2014

Introduction Timoshenko beams Reissner-Mindlin plates Reissner-Mindlin shells Non-linear solid shells Conclusions

Outline

1 IntroductionIndustrial contextNumerical locking

2 Timoshenko beamsPure bending problemMembrane and transverse shear locking1D B-Spline reduced quadrature rules

3 Reissner-Mindlin platesPure bending problem2D B-Spline reduced quadrature rules

4 Reissner-Mindlin shellsShell modelShell obstacle course

5 Non-linear solid shellsBenchmark problems

6 Conclusions

S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 2014 - IGA Application to Car Crash Simulation 1 / 26


Industrial context

Good predictive capability expected.

Complete mesh size: ' 3 million elements.

=⇒ Explicit simulations performed on the whole model.

Stable time step: ' 1 µs.

10 hours to simulate ' 130ms on 96 CPUs.

=⇒ Excessive computational cost.

≥ 90% of the geometry meshed with linear Reissner-Mindlin shellelements.

Figure: Model size evolution



Industrial context

Geometry has proved to be very significant.Need to refine some components.

=⇒ Stable time step greatly penalized.Accurate and smooth geometry preferred for contact problems.Large time used for preparing and meshing parts.

21

32

14

4

14

6 6 84 5

1

Figure: Estimation of relative time costs for model generation and analysisprocess



Industrial context

=⇒ IsoGeometric Analysis for automobile crashworthiness.

Parametrized CAD geometry for optimisation process.

Analysis and Design share the same model.

Higher regularity between elements within a patch.

Figure: Numerical simulation process in isogeometric analysis.



Numerical locking

Fully integrated low order elements, based on Timoshenko/Mindlinhypothesis, present poor performances with thin beams, plates andshells.Transverse shear and membrane locking appear.Classical Lagrange-based reduced and selective integration rules donot remove these pathologies.

=⇒ Need to define B-Spline/NURBS-based integration schemes.Framework:

Multi-patch problems with uniform regularity within each patch.Quadratic and cubic polynomials, regularity r from 0 to p − 1.

C0

Cr

Figure: Multi-patch problems.S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 2014 - IGA Application to Car Crash Simulation 5 / 26







6 Conclusions



Pure bending problem

Thick curved beam of length L and radius R clamped at one end.

Moment of inertia for a rectangular section: I = bh3

12 .

Distributed loading moment proportional to thebending stiffness:

m(s) = EI( π

2L

)2

sin( π

2Ls).

=⇒ Exact solution known, depends only of L and R:

un(s) =2Lπ

1− ( 2LπR )2

(cos

( s

R

)− cos

( π2L

s))

.

Exact membrane and shear strains are zero.

Severe test for membrane and shear locking.



Membrane and transverse shear locking

One element with B-Splines of degree p from 1 to 5.

Shear locking for slenderness ratio L/h� 1 and membrane lockingfor L/h� R/L.

Both shear and membrane locking appear for all degrees p.

100

101

102

103

104

105

106

0

20

40

60

80

100

L/h

||θbex

−θ bh || L2

/ ||θ

bex|| L2

(%)

Rotation relative error with R/L = 103

LinearQuad.CubicQuarticQuintic

SHEARLOCKING

MEMBRANELOCKING

R/L

(a) Linear scale

100

101

102

103

104

105

106

10−4

10−3

10−2

10−1

100

101

102

L/h

||θbex

−θ bh || L2

/ ||θ

bex|| L2

(%)

Rotation relative error with R/L = 103

LinearQuad.CubicQuarticQuintic

SHEARLOCKING

R/L

MEMBRANELOCKING

(b) Logarithmic scale

Figure: One element curved thick beam: rotation relative error in L2 norm.



Transverse shear locking

Different effects on convergence according to the couple (p,r).Degree 1: no pre-asymptotic convergence.Degree 2: slow pre-asymptotic convergence when C 0 and nopre-asymptotic convergence with few elements when C 1.Spurious constraints strengthened by the high regularity.

=⇒ Excessive bending stiffeness of the structure.

100

101

102

10−6

10−4

10−2

100

Number of elements ∝ 1 / l

||θbex

−θ bh || L2

/ ||θ

bex|| L2

(%)

Rotation relative error with L/h = 103

Linear

Quad. C0

Quad C1

Cubic C0

Cubic C1

Cubic C2

1

1

3

2

1

4

1

5

(a) Pre-asymptotic convergence withL/h = 103

100

101

102

103

10−8

10−6

10−4

10−2

100

Number of elements ∝ 1 / l

||θbex

−θ bh || L2

/ ||θ

bex|| L2

(%)

Rotation relative error with L/h = 102

Linear

Quad. C0

Quad C1

Cubic C0

Cubic C1

Cubic C2

4

1

31

4

1

3

1

1

5

21 1 2

(b) Pre-asymptotic and asymptoticconvergence with L/h = 102

Figure: Rotation relative error in L2 norm for a straight Timoshenko beam.S. Bouabdallah, C. Adam, M. Zarroug, H. Maitournam Teratec 2014 - IGA Application to Car Crash Simulation 9 / 26


Mathematical framework

Based on the work in FEA of [Prathap,93].

Field-consistency paradigm in thick structural elements.

Transverse shear strain in straight beam:

γs(s) = w,s(s)− θ(s).

In the limit of an infinitely thin beam, the shear strain energy mustvanish.

=⇒ Different physical and spurious constraints are obtained according tothe couple (p,r).

Mathematical induction is performed adding elements one-by-one.

New unknowns and conditions are numbered.

An over-constrained linear system leads to numerical lockingwhereas an under-constrained system produces zero energy modes.



1D B-Spline reduced quadrature rules

Optimal selective reduced integration schemes adapted to highregularity basis functions.

The higher the regularity is, the fewer Gauss points are needed.

Shear and membrane locking are completely removed.

Improved accuracy of curved beam B-Spline elements.

0 1

Order 3 C2

Order 3 C1

Order 3 C0

Order 2 C1

Order 2 C0

Order 1 C0

ξ = s / L

Reduced integration for transverse shear: Gauss points

100

101

102

10−6

10−4

10−2

100

Number of elements

||θex

− θ

h || L2 / |

|θex

|| L2Quadratic B−spline

C0 full

C0 selec.

C1 full

C1 selec.

1

3

21

1

4

100

101

102

10−6

10−4

10−2

100

Number of elements

||θex

− θ

h || L2 / |

|θex

|| L2

Cubic B−spline

C0 full

C0 selec.

C1 full

C1 selec.

C2 full

C2 selec.

1

1

3

1

4

5

Figure: Rotation relative error in L2 norm with full and selective reducedintegrations.








6 Conclusions




Same behaviours as with beams.

High regularity wanted for more accurate results with fewer controlpoints.

Need to unlock B-Splines especially when C p−1 regularity.

Possibility to extend integration rules from beams to plates andshells.

0 5 10 15

0

20

40

60

80

100

Number of elements in each direction

wc /

wcre

f (%

)

Vertical displacement of point C: concentrated load

Reference

Linear C0 L/h = 103

Linear C0 L/h = 105

Quad. C0 L/h = 103

Quad. C0 L/h = 105

Quad. C1 L/h = 103

Quad. C1 L/h = 105

Figure: Central deflection of a simply supported plate (L/h = 103, 105).




Same behaviours as with beams.

High regularity wanted for more accurate results with fewer controlpoints.

Need to unlock B-Splines especially when C p−1 regularity.

Possibility to extend integration rules from beams to plates andshells.

0 5 10 15

0

20

40

60

80

100


wc /

wcre

f (%

)

Vertical displacement of point C: uniform pressure

Reference

Linear C0 L/h = 103

Linear C0 L/h = 105

Quad. C0 L/h = 103

Quad. C0 L/h = 105

Quad. C1 L/h = 103

Quad. C1 L/h = 105





Uni-dimensional B-Spline-based reduced quadrature rules extendedto bi-dimensional rules by tensor product.

The high regularity lowers the required number of Gauss points.

Hourglass modes are suppressed using additional quadrature pointsin boundary elements, where the control points accumulate.

0 0.25 0.5 0.75 1 0

0.25

0.5

0.75

1

B−spline selective integration: Gauss points

ξ

η

(a) Quadratic C 1

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1


ξ

η

(b) Cubic C 1

0 0.25 0.5 0.75 1 0

0.25

0.5

0.75

1


ξ

η

(c) Cubic C 2

Figure: B-Spline selective integration quadrature rules and central deflection.




Resulting under-integrated elements are free from transverse shearlocking.Numerical solutions independent of the thickness.Improved accuracy of plate isogeometric elements.No spurious zero energy modes.

0 5 10 15

0

20

40

60

80

100


wc /

wcre

f (%

)


Reference

Linear C0 L/h = 103

Linear C0 L/h = 105

Quad. C0 L/h = 103

Quad. C0 L/h = 105

Quad. C1 L/h = 103

Quad. C1 L/h = 105

(a) Full integration

0 5 10 150

20

40

60

80

100


wc /

wcre

f (%

)


Reference

Linear C0 L/h = 103

Linear C0 L/h = 105

Quad. C0 L/h = 103

Quad. C0 L/h = 105

Quad. C1 L/h = 103

Quad. C1 L/h = 105

(b) Reduced integration





Resulting under-integrated elements are free from transverse shearlocking.Numerical solutions independent of the thickness.Improved accuracy of plate isogeometric elements.No spurious zero energy modes.

0 5 10 15

0

20

40

60

80

100


wc /

wcre

f (%

)


Reference

Linear C0 L/h = 103

Linear C0 L/h = 105

Quad. C0 L/h = 103

Quad. C0 L/h = 105

Quad. C1 L/h = 103

Quad. C1 L/h = 105

(c) Full integration

0 5 10 150

20

40

60

80

100


wc /

wcre

f (%

)


Reference

Linear C0 L/h = 103

Linear C0 L/h = 105

Quad. C0 L/h = 103

Quad. C0 L/h = 105

Quad. C1 L/h = 103

Quad. C1 L/h = 105

(d) Reduced integration









6 Conclusions



Shell model

Reissner-Mindlin shell model obtained froma degenerated three-dimensional model.

First-order kinematic description throughthe thickness with transverse shear.

Exact geometry of the shell defined by

x(ξ) =nm∑A=1

RA(ξ, η)XA +h

2ζn(ξ, η).

Interpolated displacements described by

u(ξ) =nm∑A=1

RA

(UA +

h

2ζθA ∧ n

).



Characterization of the normal

Several possibilities:1 Exact normal:

n =x,ξ ∧ x,η

||x,ξ ∧ x,η||2.

2 Collocated normal:

n = nA ∀A ∈ [|1, nm|].a. nA defined by the orthogonal projection of the associated control

point XA onto the mid-surface of the shell,b. nA defined by a uniform distribution of the shell normals in the

parametric space,c. nA defined at the Greville abscissae in the parametric space.

Exact normal formulation is incompatible with reduced integration.

Collocation at Greville abscissae results in good compromise betweenaccuracy and computation time.



Shell obstacle course problems

Evaluate time efficiency and accuracy of the proposedunder-integrated elements.

Linear elasticity benchmark problems.

Bending dominated problems: transverse shear and membranelocking.

(a) Scordelis-Lo roof (b) Pinched cylinder (c) Pinched hemisphere

Figure: The shell obstacle course: problem descriptions and data.



Accuracy

0 5 10 15 20 25 30 35

0

20

40

60

80

100

Number of elements per side

wB /

wBre

f (%

)

Vertical displacement of point B : full integration

Reference

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(a) Roof: full

0 5 10 15 20 25 30 35

0

20

40

60

80

100


wC

/ w

Cref (

%)

Vertical displacement of point C: full integration

Reference

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(b) Cylinder: full

0 5 10 15 20 25 30 35

0

20

40

60

80

100


u A /

u Aref (

%)

Horizontal displacement of point A: full integration

Reference

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(c) Hemisphere: full

0 5 10 15 20 25 30 35

0

20

40

60

80

100


wB /

wBre

f (%

)

Vertical displacement of point B : reduced integration

Reference

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(d) Roof: reduced

0 5 10 15 20 25 30 35

0

20

40

60

80

100


wC

/ w

Cref (

%)

Vertical displacement of point C: reduced integration

Reference

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(e) Cylinder: reduced

0 5 10 15 20 25 30 35

0

20

40

60

80

100


u A /

u Aref (

%)

Horizontal displacement of point A: reduced integration

Reference

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(f) Hemisphere: reduced

Figure: Shell obstacle course: displacement convergence



Time efficiency

10−2

10−1

100

101

0

20

40

60

80

100

Computational time (s)

wB /

wBre

f (%

)

Vertical displacement of point B : full integration

Reference

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

CubicC2

(a) Roof: full

10−2

10−1

100

101

0

20

40

60

80

100


wC

/ w

Cref (

%)

Vertical displacement of point C: full integratione

Reference

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(b) Cylinder: full

10−2

10−1

100

101

0

20

40

60

80

100


u A /

u Aref (

%)

Horizontal displacement of point A: full integration

Reference

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(c) Hemisphere: full

10−2

10−1

100

101

0

20

40

60

80

100


wB /

wBre

f (%

)

Vertical displacement of point B : reduced integration

Reference

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(d) Roof: reduced

10−2

10−1

100

101

0

20

40

60

80

100


wC

/ w

Cref (

%)

Vertical displacement of point C: reduced integration

Reference

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(e) Cylinder: reduced

10−2

10−1

100

101

0

20

40

60

80

100


u A /

u Aref (

%)

Horizontal displacement of point A: reduced integration

Reference

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(f) Hemisphere: reduced

Figure: Shell obstacle course: displacement convergence



Total number of Gauss points

Lagrange-based quadrature:

nbendingGP =

(e(⌈

2p+12

⌉))2and nshear

GP =(e(⌈

2p+12

⌉− 1

))2.

B-Spline-based quadrature:

nbendingGP = nshear

GP =(e(⌈

2p+12

⌉− 1− r

)+ 2r

)2.

Computational effort depends on the difference p − r whenperforming B-Spline-based reduced integration.Speed up factor greater than 5 (resp. 6) is reached with theunder-integrated quadratic (resp. cubic) C p−1 elements.

0 5 10 15 20 25 30 350

5

10

15

20

Computation time: full integration


Com

puta

tion

time

(s)

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(a) Full integration

0 5 10 15 20 25 30 350

5

10

15

20

Computation time: selective integration


Com

puta

tion

time

(s)

Linear C0

Quad. C0

Quad. C1

Cubic C0

Cubic C1

Cubic C2

(b) Selective integration

Figure: Computation time for the pinched cylinder problem.








6 Conclusions



Benchmark problems

Solid-shell elements with one quadratic element in the thickness.

Geometric non-linear elasticity benchmark problems.

Bi-variate B-Spline-based quadrature rules applied.

(a) Pinched hemisphere (b) Stretched cylinder

Figure: Geometric non-linear analysis of shells: problem descriptions and data.



Accuracy and time efficiency

Bi-quadratic C 1 shape functions in the in-plane directions.

Improved accuracy and time efficiency.

Speed up factor of approximately 3.

0 2 4 6 8 100

20

40

60

80

100

Displacement (m)

Load

P in

% o

f Pm

ax (

N)

Load − displacement curve

12x12 full12x12 reducedShell reference S4R

(a) Pinched hemisphere

0 1 2 3 4 50

20

40

60

80

100

Displacement (m)

Load

P in

% o

f Pm

ax (

N)

Load − displacement curve

12x12 full12x12 reducedShell reference S4R

(b) Stretched cylinder

Figure: Displacement convergence with Total Lagrangian Formulation



Conclusions

Severe numerical locking appear in quadratic and cubic C p−1

elements.

=⇒ Reduced integration to improve the performances of IG elements.

Classical reduced/selective quadrature rules remove numericallocking in C 0 elements only.

Mathematical framework to define B-Spline-based reduced 1Dquadrature rules.

Extension to 2D quadrature rules by tensor product.

=⇒ Efficient thick shell elements obtained as in FEA.

Improved accuracy and time efficiency.

Computational cost depends on the difference p − r .


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