Current Status of Isogeometric Analysis in LS-DYNA

12th LS-DYNA Forum 24 - 25 September 2013, Filderstadt, Germany

Current Status of Isogeometric Analysisin LS -DYNA

Stefan Hartmann

Developer Forum , September 24th, 2013, Filderstadt, Germany

Development in cooperation with:D.J. Benson: Professor of Applied Mechanics, University of California, San Diego, USA


Outline� Isogeometric Analysis

- motivation / definition

� NURBS-based finite elements- NURBS / *ELEMENT_SHELL_NURBS_PATCH / elements - connectivity

� Todo-List- mass-scaling / merging / trimmed NURBS / NURBS-solids / LSPP / …

� Current status- MPP / examples / limits / LSPP / …

� Contact with NURBS- smoothness / example

� Shell formulations- continuity / rotation free shells / blended shells


Isogeometric Analysis – motivation & definition

� ISOPARAMETRIC (FE-Analysis)use same approximation for geometry and deformation

GEOMETRY �� DEFORMATION

� ISOGEOMETRIC (CAD - FEA)same description of the geometry in the design (CAD) and the analysis (FEA)

CAD �� FEA

� common geometry descriptions in CAD- NURBS (Non-Uniform Rational B-splines) � most commonly used- T-splines � enhancement of NURBS- subdivision surfaces � mainly used in animation industry- and others

� reduce effort of geometry conversion from CAD into a suitable mesh for FEA (?)

� reduce discretization errors � adaptivity

� increase continuity of geometry representation � contact / rot. free shells / …


� NURBS = Non Uniform Rational Basis Spline

NURBS-based finite elements

Control-Points

Control-Net

Elements

� A typical NURBS-Patch and the elements- elements are defined through the knot-vectors- shape functions for each control-point (�DOFs)

� B-spline basis functions- recursive- positive everywhere- dependent on knot-vector and polynomial order- normally C(P-1)-continuity- „partition of unity“ (like Lagrange polynomials)- refinement (h/p and k) without changing theinitial geometry � adaptivity

- control points are normally not a part of thephysical geometry (non-interpolatory basis functions)

� NURBS- B-spline basis functions + control net with weights- all mentioned properties for B-splines apply for NURBS


xy

z

*ELEMENT_SHELL_NURBS_PATCH$---+--EID----+--PID----+--NPR----+---PR----+--NPS----+---PS----+----7----+----8

11 12 4 2 5 2$---+--WFL----+-FORM----+--INT----+-NISR----+-NISS----+IMASS----+----7----+----8

0 0 1 2 2 0$rk-+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8

0.0 0.0 0.0 1.0 2.0 2.0 2.0$sk-+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8

0.0 0.0 0.0 1.0 2.0 3.0 3.0 3.0$net+---N1----+---N2----+---N3----+---N4----+---N5----+---N6----+---N7----+---N8

1 2 3 45 6 7 89 10 11 12

13 14 15 1617 18 19 20

Control Points

Control Net


r

s

20

3

4

1

2

5 6

9

7

8

1011

12

13

14

15

16

17

1819

Elements


NURBS-based finite elements� A typical NURBS-Patch and the definition of elements

- elements are defined through the knot-vectors (interval between different values)- shape functions for each control-point

r

s

0 1 20

1

2

3

M1

M2

M3

M4

M5

N1 N2 N3N4

NURBS-Patch(physical space)

NURBS-Patch(parameter space)

Control-Points

Control-Netrknot=[0,0,0,1,2,2,2]

skno

t=[0

,0,0

,1,2

,3,3

,3]

Element

polynomial order:- quadratic in r-direction (pr=2)- quadratic in s-direction (ps=2)

r

s

xy

z


Nurbs-Element (automatic)

Interpolation Node (automatic)

Interpolation Element (automatic)

Control Points (*NODE)

Control Net

(*ELEMENT_SHELL_NURBS_PATCH)

Connectivity of

Nurbs-Element (automatic)

Nurbs-Surface (physical space)

3937

38

16

15

14

13

12

11

1718 19

36

35

34

33

32

31

26

25

24

23

22

21

2728 29

6

5

3

1

78

9

4

2r

s

Interpolation Elements

(postprocessing/contact/BC)

NISR=2 / NISS=2

Nurbs-Surface (parameter space)

r7

B-spline basis functions (r-direction)

B-s

pli

ne

ba

sis

fun

ctio

ns

(s-d

ire

ctio

n)

r-knot: [0,0,0,1,2,3,4,5,6,7,7,7]

s-k

no

t: [

0,0

,0,1

,2,2

,2]

s

2


LS-DYNA®

KEYWORDUSER’S MANUAL

VOLUME I

Livermore SoftwareTechnology Corporation

(LSTC)

from:


Shell formulations

NURBS(bi-quadratic)

LAGRANGE(bi-quadratic)C1-lines

� FORM=0:- shear deformable shell theorywith rotational DOFs(6DOFs/control point)

� FORM=1: (requires C1)- rotation free shell theorywithout rotational DOFs(3DOFs/control point)

� FORM=4/-4:- hybrid shell formulationcombination of FORM=0 andFORM=1 (especially necessaryat patch boundaries wherecontinuity drops to C0)

� continuity at element boundary- Lagrange: C0

- NURBS: Cp-1

support of shape functions(control points) do overlap

vs.


� G1 continuity along patch boundaries - at patch boundaries: C0-lines (no transmission of bending moments with rot-free formulation)- add rotational DOFs at patch boundary (hybrid/blended shell: FORM=4/-4)

One single patch(rot-free)

Two patches(rot-free, coupled)

Two patches(rot-free, kink)

Kink! Kink!

Shell formulations – multi-patch coupling (rotation free formulation)


� keep angle at discontinuities (internal C0-lines) - no transmission of bending moments with rot-free formulation � kinks- add rotational DOFS at interior control points (FORM=4)

rot-free (kink)

90 -angle

90 -angle

Kink!

coupling orformulation with rotational DOFs

piano hinges

Shell formulations – multi-patch coupling (rotation free formulation)


� all (penalty based) contacts available- use interpolation shell elements (lin. Quads) - loss of smooth (contact) surface representation� may lead to problems in tangential sliding

interpolation shells (rough)

interpolation shells (fine)

NURBS-surface (smooth)

� NURBS contact for (SMP only )*CONTACT_AUTOMATIC_1WAY_S2S - use NURBS representation on master side - smooth contact behavior (i.e. tangential sliding)

Contact with NURBS

*CONTROL_CONTACT$Optional 6 -2 -3 -4 -5 -6 -7 -8$---SHLEDG----PSTIFF----ITHCNT----TDCNOF-----ftall---------6----shltrw----igactc

1


� comparison *Contact_Automatic_One_Way_Surface_To_Surface- IGACTC=0: master surface defined by interpolation elements (faceted)- IGACTC=1: master surface defined by NURBS surface (smooth)- _SMOOTH: smooth curve-fitted surface to represent master surface

Contact with NURBS – sliding example

� Example: Tube-In-Tube (both elastic bodies – properties of steel)- frictionless sliding- outer tube is fixed- inner tube is free to move- inner tube with initial rotationalvelocity (death: t=0.01)

- termination time: t=0.1

t

v (rz)

xz

y0.01


� discretization- inner tube: 8x12 bi-quadratic NURBS element- outer tube: 4x16 bi-quadratic NURBS element- 2x2 bi-linear interpolation shells per NURBS element


� Version A- inner tube is master- outer tube is slave

NURBS

interpolationelements

x

z

y P

P

free to rotate

_SMOOTH

IGACTC=0

IGACTC=1


� discretization- inner tube: 8x12 bi-quadratic NURBS element- outer tube: 4x16 bi-quadratic NURBS element- 2x2 bi-linear interpolation shells per NURBS element


� Version B- inner tube is slave- outer tube is master

NURBS

interpolationelements

x

z

y P

P

free to rotate

_SMOOTH

IGACTC=0

IGACTC=1


� computational time


Version A Version B

CPU time #of cycles CPU time #of cycles

IGACTC=0 3 min 32 sec 346535 3 min 18 sec 346484 SMP (single) ncpu=1

IGACTC=1 9 min 27 sec 346537 7 min 55 sec 346536 SMP (single) ncpu=1

_SMOOTH 8 min 43 sec 346484 6 min 10 sec(loss of contact ataround t=0.03)

349561 MPP (single)-np 1

� NURBS contact (IGACTC=1)- insensitive to Slave-Master-Pair definition- more expensive than standard contact (IGACTC=0) but comparable with _SMOOTH option- NOTE: comparison is not representative (SMP vs. MPP), _SMOOTH option and Version B

looses contact around t=0.03 (30% of computation time)


Current status - summary

� Keyword: *ELEMENT_SHELL_NURBS_PATCH- definition of NURBS-surfaces- shell formulations with/without rotational DOFs and hybrid shell- NURBS elements run in MPP with good scaling- NURBS contact(*CONTACT_AUTOMATIC_ONE_WAY_SURFACE_TO_SURFACE, IGACTC=1, SMP only)

� Pre- and Postprocessing- work in progress for LS-PrePost … current status (lspp4.1beta)� visualization of 2D-NURBS-Patches � import IGES-format and construct *ELEMENT_SHELL_NURBS_PATCH� modification of 2D-NURBS geometry� … much more to come!

� Postprocessing and boundary conditions (i.e. contact) mainly with- interpolation nodes (automatically created)- interpolation elements (automatically created)

� Analysis capabilities- implicit and explicit time integration- eigenvalue analysis- other capabilities (e.g. geometric stiffness for buckling) implemented but not yet tested

� LS-DYNA material library available (including umats)


Example – Vibration of a square plate

L

L

1,0E-14

1,0E-12

1,0E-10

1,0E-08

1,0E-06

1,0E-04

1,0E-02

1,0E+00

1 10 100 1000 10000

fully integrated shell (ELFORM=16)

NURBS P=2 (rotation free)



� eigenvalue analysis- simply supported square plate- linear elastic material- consistent mass matrix- rotation free shell formulation

� accuracy of first eigenvalue

Number of Nodes

Err

or=

(Cal

cula

ted-

Exa

ct)/

Exa

ct


� nonlinear dynamic buckling- isotropic elastic-plastic material- linear plastic hardening- NURBS (P=2,3,4)- Reissner-Mindlin formulation

Example – dynamic buckling

Experimental results from Kyriakides S. Kyriakides, Personal communication, 2008

� good agreement of - buckling load & buckling pattern- Reissner-Mindlin shell formulations (P=2,3,4)


D.J. Benson

� mesh:- 640 quartic (P=4) elements- 1156 control points- 3 integration points through thickness

� standard benchmark for automobile crashworthiness� quarter symmetry to reduce cost� perturbation to initiate buckling mode� J2 plasticity with linear isotropic hardening

Example - buckling


� spotweld modeling with*CONTACT_TIED_SHELL_EDGE_TO_SURFACE_OFFSET

Example – column crush


Example - forming


FEA:

FEA:

FEA:

� regular NURBS-patch

� trimmed NURBS

� 4 regular NURBS-patches

Control Points

Control Net

trimmingcurve

- reparametrization- C0-lines at patchboundaries

- standard in CAD

Current status - limits


Summary

� higher order accurate isogeometric analysis may be cost competitive

� code optimization necessary to make it faster

� NURBS contact works well

� NURBS-based elements run stable and scale good in MPP

� further implementation- post-processing directly with NURBS (partially available already)- mass scaling- NURBS contact implementation for MPP- merging of non-matching NURBS patches- trimmed NURBS (research currently done at UCSD)- make pre- and post-processing more user-friendly and faster- introduce 3D NURBS elements- … much more

ToDo-List� perform a lot more studies in different fields � experience� motivate customers (and researchers) to “play” with these elements

� model creation with standard NURBS is currently the bottleneck!

Current Status of Isogeometric Analysis in LS-DYNA

Documents

Transcript of Current Status of Isogeometric Analysis in LS-DYNA