master hh crash worthiness p 2

June 2007 Crashworthiness, 1 Fabian Duddeck

Hamburg University of Applied SciencesComputational Mechanics for Car Body Design I

Crash Simulation of Car Bodieswith FEM – Part 2

Fabian [email protected]


First Numerical Models for Crashworthiness 1960s

Lumped Mass Models First numerical methods for crash analysis were developed in the 1960s.


Lumped Mass Models

• The frontal side member undergoes progressive folding, i.e. a phase of elastic deformations, followed by plastic reactions and finally, after reaching a stability limit, buckling.

• The first peak for buckling is the highest; the integral over the curve gives the energy.

• Normally the crush test finishes with a dramatic column buckling mode (bending).

• This sequential failure behaviour can be represented by a series of elasto-plastic collapse elements.

• One of the simplest models for thisis obtained via a single DOF system with nonlinear force resistance elements FSi defined via elasto-plastic collapse concepts.

• The FSi are activated sequentially.

Forc

e

Displacement

I : 1st Elasto-plasticII : 1st BucklingIII : 2nd Elasto-plasticVI : 2nd Buckling

Kim (2001)PhD thesis

A typical force-displacement curve of a frontal side member crushed by a moving mass


Lumped Mass Models

• Each nonlinear force resistance element FSi is defined for one special characteristic of the force-displace-ment curve, i.e. each element works only in its displacement domain.

• The differential equation is now:

• If the function has a positive slope, it represents an elastic behaviour for a structure. If the function has a negative slope, then it represents the collapse behaviour. Zero slope is interpreted as a perfect plastic behaviour.

• Two different slopes characterize elasto-plastic behaviour with hardening or softening

• Higher order approximations avoid difficulties in identifying model parameters

Forc

e

Displacement

Sub-division of the force-displacement curve to define elasto-plastic collapse elements

Force-displacement curve for one elasto-plastic collapse elements


Forc

e

Displacement

0),,,,()( 11 =+ ++ iiiiiii xddSSFStxM &&


Lumped Mass Models

A three DOF system with six resistance force elements


Free body diagram for the 3 DOF system with 6 resistance force elements

[ ]000)0(,0)0(;0)(

vvvxxxfxCxM==

=++&

&&&

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

00000

;00

0000

211

11

3

2

1

CCCCC

CM

MM

M

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+−−++−

++=

653

542

321

)(FSFSFSFSFSFS

FSFSFSxf


Origin of FEM

The Finite Element Method (FEM) was developed by R. Clough, J. Argyris, and O. Zienkiewicz in the early 1960s.

Olgierd ZienkiewiczJohn Argyris Ray Clough


History of Finite Element Codes

• In 1986 Hallquist developed at the Lawrence Livermore National Laboratory the finite element program DYNA3D, which is based on an explicit integration method combined with a vectorization for Cray super computers.

• Belytschko and Marchertas adapted the code WHAMS from the nuclear industry for crash simulation.

• These two codes have formed the fundamentals of current commercial software available for crash analysis.

• Thus all concepts have similar principles.

Finite element model for frontal impact analysis, Benson 1986.

• DYNA-3D

• LS-DYNA3D

• PAM-CRASH

• RADIOSS

• ABAQUS EXPLICIT

• …


First Crash Analyses with Finite Elements, 1983

Shell structure for static and dynamic analysis of frontal side member (1983).

1983

• Local static and dynamic crimpling• Global elastoplastic buckling• Member collapse, with local plastic

hinge formation and large cross-section distortions.

• Multiple contact and impact between structure and supports and different structural parts.

• Localized member and spot welds or rivet failure.

• Complex elastoplastic and viscoplastic material behaviour.

• Peak and extensive post-peak behaviour of structural components and structures loaded statically and dynamically.

• Rate dependent constitutive equations.



• Non-linear, accurate, thin-shell and -plate finite elements that permit large displacements, rotations, curvatures and distortions to be simulated accurately, economically and free from numerical instabilities.

• Supplementary finite elements (such as truss, beam and eccentric stiffener elements) and nodal tie options (such as master-slave and Lagrange constraint options) that permit easy simulation of secondary parts of the structure (such as reinforcements, connections, rigid bodies, and the application of complex loading mechanisms).

• Element elimination• Barrier model (volume elements)



Beam structure for crashworthiness (rollover) of a bus (1983)

1983 Stelzmann et al. (2006)


First Full Vehicle Models, 1986

• First finite element computations for crash were performed by Belytschko et al. (1975);

• First full car crash analysis were realized by Benson and Hallquist(1986) and Du Bois et al. (1986);

• The Figure on the right-hand side shows the finite element model developed by Bretz et al. for BMW also in 1986.

• It consists of 2,800 finite shell elements with 2,604 nodes, where the smallest element was of a size of 18 mm.

• The rear and the engine were assumed to be rigid;

• An explicit FE-code and special contact algorithms were applied;

BMW, Bretz et al., 1986

FEM-computation of a frontal impact


Rising Model Size, 1990

Full car model with 25,000 elements.

Audi 1990

BMW 2005

• Maximal time step size is determined by the size of the smallest element;

• Appropriate modelling of local physical phenomena requires very small elements (characteristic length of less than 1 mm);

• This is up to now not realizable, the required time step is then too small and the computation time too large.

• In current models (2006), the characteristic length of an element is about 5mm and the time step is circa 10-6 s.

• FE models with up to 3 million elements have been realized.

Full car model with 1,200,000 elements.


History of FE Simulations for Crashworthiness

Simulation for global analysis

First productive results

First feasibility studies

2001

1996

1991

1986

CAE-driven design process

Reduction of the number of prototypes

2006

Simulation for local analysis

BMW, 2006


Examples of current crash simulations, BMW.

Current Crash Analysis

• In early stages of crash simulations, every load case required a separate FE model where the critical parts were refined.

• Currently the OEMs intend to realize one common model for all crash load cases (and NVH=noise, vibration and harshness). Nevertheless, several features are load case specific and the models are hence still different.

• A current FE model is developed in about 8 weeks;

• They are developed in a modular form such that a single module can be exchanged easily.

• Each part has to be validated.


Full Vehicle Models

Audi


Finite Element Method - Principle

• The principle idea is to divide the medium into small, finite elements.

• The quantities of the partial differential equation are then approximated by finite sums of known shape functions Nk(x) and unknown coefficients ak:

• The coefficients correspond to nodal values either at the corners of the elements or at so-called Gaussian points in the interior where the integration is performed.

• For multi-dimensional problems, each component is approximated by shape functions, e.g.:

Nau =≈= ∑k

kk xNaxu )()(

Partition of the domain Ω with the boundary Γ

Sample of a shape functionyxiyxNayxu

kkiki ,;),(),( =≈ ∑

Ω

Γ


Γ+Ω−Ω=

Γ+Ω−Ω=

∫∫∫

∫∫∫

ΓΩΩ

ΓΩΩ

ddd0

ddd0

tuDεεbu

tuσεbu

TTT

TTT

δδδ

δδδ


• The other quantities (strain and stress) are approximated in a similar manner (here 2D):

Strain:

Stress:

BaSNaSuε

ε

===

⎭⎬⎫

⎩⎨⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∂∂∂∂∂∂

∂∂=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=y

x

xy

y

x

uu

xyy

x

///00/

γεε

Dεσ =⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=

xy

yy

xx

σσσ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

=2/)1(00

0101

1 2

νν

ν

νED

• The potential energy (interior deformation energy) is given by

• The total virtual work is :

b is the vector of volume forces in the interior;

t is the vector of tractions at the boundary.

VUV

T d21∫= Dεε



• After inserting the approximation into the weak form of the differential equation, an algebraic system of equations is obtained, which can be solved for the remaining unknown coefficients a.

• K is the stiffness matrix

• f is the known force vector

Kaf

tNa

DBaBabNa

+=

Γ−

Ω+Ω−=

∫

∫∫

Γ

ΩΩ

0

d

dd0

TT

TTTT

• The method is hence converting a problem with an infinite number of unknown quantities to a problem with a finite number of degree of problems. A continuous model is approximated by a discrete model.

• The stiffness matrix is symmetric and has a band structure:

Γ−Ω−= ∫∫ΓΩ

dd tNbNf TT Stiffness matrixof 100 shellelements

∫Ω

Ω= dDBBTK


Governing Equations and Weak Form

Lagrangian meshes• Nodes and elements move with the

material;• Boundaries and interfaces remain

coincident with element edges;• Quadrature points move with the

material Constitutive equations are

always evaluated at the same material points, which is advanta-geous for history dependent materials;

Nonlinearities• Geometric and material

nonlinearities are considered;• Large deformations;• Material nonlinearities;• No large distortions of the

elements.

Lagrangian coordinates• Updated Lagrangian formulation:

– Derivatives are taken with respect to the spatial (Eulerian) coordinates;

– Weak form involves integrals over the deformed configuration.

• Total Lagrangian formulation:– Derivatives are taken with

respect to the material (Lagrangian) coordinates;

– Weak form involves integrals over the initial configuration.


Updated Lagrangian Formulation

• Conservation of mass

• Conservation of linear momentum

• Conservation of angular momentum

)()()()()( 000 XXJXXJX ρρρ ==

matrix Jacobidensity),( time

coord. Lagrangian Eulerian,,

====

J(X)tXt

Xx

ρ

dtdvb

xi

ij

ij ρρσ

=+∂

∂

jiij σσ =

• Conservation of energy

Constitutive equation

• Rate of deformation

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂+

∂∂

=i

j

j

iij x

vxvD

21

velocityforces volume tensor,stress

===

i

iijv

bσ

( )etc.,,σDσ DtSσ=∇

rateJaumann stress,Cauchy =∇σ

sxqDw

i

iijij ρσρ +

∂∂

−=int&

fluxheat entropy

energy internalint

===

iqs

w


FE Discretization (Updated Lagrangian formulation)

• Partition of the domain in elements:

• Nodal coordinates in the current configuration:

• Nodal coordinates in the undeformed configuration:

• The motion is approximated by

are the shape functions

• Nodal displacements

Partition of the domain Ω with the boundary Γ

Ω

Γ

Ωe

ee

Ω=Ω U

)3(3,2,1;..1)2(2,1;..1;DinI

DinIx

Ne

Ne

iI

====

iIX

)()(),( txNtx iIIiI XX ≅

)(XIN

iIiIiI Xtxtu −= )()(

)()(),(),( XXX IiIiii NtuXtxtu =−=• Displacement field

• Velocity and acceleration fields

)()(),()()(),(XXXX

IiIi

IiIi

NtutuNtutu

&&&&

&&

==


FE Discretization (Updated Lagrangian formulation)

• Velocity gradient

• Rate of deformation

• Test functions

• Weak form of the momentum equation

jIiIjiij NuuL ,, && ==

( ) ( )iIjIjIiIjiijij NuNuLLD ,,21

21

&& +=+=

)()( XX IiIi Nuu δδ =

{ }uiiii

i

uCuuUUu

Γ=∈=∈

on 0),(|;)(

00

0

δδδδ

XX

0dd

dd,

=Ω+Γ−

Ω−Ω

∫∑ ∫

∫∫

ΩΓ

ΩΩ

iIi

iI

iIjijI

uNtN

bNN

tj

&&ρ

ρσ

• Internal nodal forces

• External nodal forces

• Inertia forces

• Mass matrix

• Semi-discrete momentum equation

(not discretized with respect to time)

∫Ω

Ω= d,intern

jijIiI Nf σ

∑ ∫∫ΓΩ

Γ−Ω=i

iIiIiI

tj

tNbNf ddextern ρ

∫Ω

Ω= dJIijijIJ NNM ρδ

iJJIiI uNNf &&∫Ω

Ω= dinertia ρ

externinterniIiIjJijIJ ffuM =+&&


Standard Initial Value Problem

( )( )( ) iIQjIijjiijQkl

QklijQij

iIiIjJijIJ

uNLLLD

etcDSffuM

&

&&

)(;21)( where

points quadrature allfor .,)(

,

externintern

XX

XX

=+=

=

=+∇σ

• Geometrical linear relation• Cauchy stress• Rate dependent constitutive

equations• Damping term not included

• Isoparametric approach

• Linear shape functions (volume, shell, bar, and beam)

• Rigid bodies, kinematic joints


Single Degree of Freedom System (1 DoF)

m

)(extern tf

uukuf

)()(intern

=)(tuc &

)(tum &&

)(tum

c)(uk

externintern )( fufucum =++ &&&

)()()()( tFtkutuctum =++ &&&Linear elastic systems :

)(extern tf


externintern )( fufuCuM =++ &&&

)(1 tu )(1 tf1m

1c)( 11 uk

)(2 tu )(2 tf2m

2c)( 22 uk(load) forces External

on vectorAccelerati ectorVelocity v

nt vectorDisplaceme forcesnonlinear Internal )(

matrix Damping matrix Mass

extern

intern

fuuu

ufCM

&&

&

Multi - Degree of Freedom System (MDoF)


Implicit Integration (1 DoF)

Direct methods can be divided into implicit and explicit methods. In the former, the equation of motion is integrated at the end of the time step (t+Δt) :

All quantities are unknown, a direct solution is therefore not possible. From the physical point of view, this formulation is more correct than the explicit integration scheme, where equilibrium is not necessarily fulfilled. By applying a forward difference scheme, the following relations are obtained:

t

)(tu t

)(tu& t

)(tu&&

nt 1+nt1−nt

1+nu&&nu&&

nu& 1+nu&

nu 1+nu

ntΔ111 +++ =+ nnn fkuum &&

tuuu nn

n Δ−

= ++

&&&& 1

1

tuuu nn

n Δ−

= ++

11&

( )ktm

uutmfu nnnn +Δ

−Δ+= −+

+ 21

21

/2/

1

This scheme works independently of the time step value; it is unconditionally stable. Disadvantage is that the stiffness matrix has to be inverted. Additionally, the contact cannot be easily controlled. Hence implicit methods are not used for crash simulations.


The size of the time step has to be adapted to the highest eigen form which is related to the size of the smallest element L and the wave velocity c of the material.

Explicit Integration (1 DoF)

t

)(tu t

)(tu& t

)(tu&&

nt 1+nt1−nt

1−nu&&nu&&

2/1−nu& 2/1+nu&

nu 1+nu

ntΔ

2/1+Δ nt

).()()( tftkutum =+&&m

)(),(),( tututu &&&

)(tfk

)(1nnn

nnn

kufmukufum

−=−=

−&&

&&

2/12/11

2/12/1 ,+++

−+

Δ+=Δ+=

nnnn

nnnn

utuuutuu&

&&&&

mcEL

cLtt

k/;E//

2crit

==

===Δ≤Δ

ωρρω

? QUESTION:What can be done, if the time step is getting smaller than the critical time step during a crash computation?

Ordinary differentialequation (1 DOF)

Explicit methods are only conditionally stable!


Critical Time Step

Critical Time Step• The critical time step corresponds

to the condition that the computation should not advance faster than a physical phenomenon (e.g. choc wave) can transverse an element;

• The critical element size is for 1-D elements the length of the element and for 2-D and 3-D elements the length of the shortest edge of the element;

• The critical time step depends of the density and the Young’s modulus.

Number of Time Steps• Although unconditionally stable,

implicit methods require many time steps in order to trace the physical phenomenon (e.g. contact) studied. For crash simulation, this exceeds the feasible amount. Explicit analysis requires a small time step. This leads to many steps which demand, due to their simplicity, few CPU time.

Equations to be solved• Implicit analysis requires matrix

inversion. The solution of non-linear sets has to be done by iterative solution strategies.

• Explicit methods require no iteration and no matrix inversion.

Static problems• Implicit methods can solve static

problems directly, while explicit methods can solve quasi-static problems if defined carefully.


Mass Scaling, Subcycling

• When a model contains a few very small or stiff elements, the efficiency is very low since the time step is set by these elements;

• This can be approved by adding additional mass (mass scaling): the masses of stiffer elements are set such that the time step is not affected by these elements;

• Mass scaling should be used where high frequency (inertia) effects are not important;

• Selective mass scaling (LS-DYNA) does not affect the rigid body behaviour and leads therefore to better scaling properties;

• In some cases, the Young modulus might be changed to avoid small time step development during crash simulation.

• Alternatively, a subcycling approach can be used: the model is split into subdomains and each is integrated with its own stable time step. Crucial is hereby the treatment of the interfaces between the subdomains;


Elements


Volume Elements

• Solid finite elements are used for discretization of bulk materials.

• The standard element is defined by 8 corner nodes while other types (degenerated elements) can be obtained by repeating nodes, i.e.

• Normally, linear shape functionsare chosen for the interpolation of the coordinates.

• To insure automatic satisfaction of convergence and completeness criteria, an isoparametric formulation is used, i.e. for geometry and for the displacements the same shape functions are used.

Definition of solid elements (8-nodes, 4-nodes, and 6-nodes)

4-node element: n1, n2 , n3, n3 , n4, n4 , n4, n4



• These elements are first order C0-solids, thus the displacement field is smooth inside the element and continuous across the element borders. The strains and stresses are in general discontinuous.


Volume Elements

• The Galerkin version of the weak form is based on choosing as test and as trial functions the same functions. The unknown displacements are approximated by predefined shape functions and unknown coefficients:

• Inserted in the main equation in the weak form, the following expression is obtained:

.)(~)(),( ∑=J

iJJi tuNtu xx

....d

dd,

cicbb

u

ii

iiijji

++Ω=

Ω+Ω

∫

∫∫

Ω

ΩΩ

η

ρηση &&

).1)(1)(1(81),,( ζζξξηηζηξ IIIIN +++=

.)(~),,(),,(

),(~),,(),,(

),(~),,(),,(

8

1

8

1

8

1

∑

∑

∑

=

=

=

=

=

=

III

III

III

txNx

tuNu

tuNu

ζξηζξη

ζξηζξη

ζξηζξη

&&

• The shape functions for the 8 node solid element are:

• A formulation is called isoparametricif the same interpolation is used for the displacements as for the geometry:


Eight Node Solid Elements


Volume Elements

• First order elements are computa-tionally efficient. They provide good results when localization problems arise (plasticity, shocks, etc.).

• In explicit calculations, where the time step is governed by stability conditions resulting in small elements, the C0-element is particular attractive.

• The elements are integrated numerically by a Gaussian quadrature rule either using a selective or a reduced integration scheme.

• An unmodified full eight point integration of the isoparametric solid formulation results in an exact evaluation of the nodal quantities.

σ, ε

u

Scheme of C0-continuous elements (1D)

Drawbacks• The approximation in constrained

media (e.g. incompressible or nearly incompressible materials and deviatoric plasticity) may be poor independent of the size of the mesh.

• To overcome this (and locking) a selective or reduced integration method is applied.


Shear Locking

• Full integration is often too expensive; due to shear locking, the element reacts to stiff.

• Uniform reduced integration is less expensive, but spurious zero energy modes have to be treated (hourglass control).

• Therefore, different integration rules for deviatoric and volumetric properties in solid elements (selective reduced integration) are necessary.

• Fully integrated elements may be disadvantageous in cases of incompressible media or for bending when locking is occurring. For this, selective or reduced integration is proposed, cf. the literature on finite element methods by for example Zienkiewicz, Bathe, Belytschko.

Shear locking

• Fully integrated C0-elements show a tendency to lock in bending caused by shear resulting from the kinematic constraints.

• One drawback of the reduced integration is the rank deficiency of the element. This results in zero energy modes called hourglassmodes, which are non-physical.


Hourglass Modes (2-D)

• Full numerical integration is often numerically too expensive; so-called under-integrated elements are therefore often used;

• These elements can lead to hourglass effects, i.e. deformation modes where the computed internal energy is not affected (zero-energy modes);

• Artificial oscillations are static deformations are the consequences.

Hourglass modes of a 4-nodes shell element (there is an additional,

the y-rotation).

x-displacement y-displacement

z-displacement x-rotation

Integration point

Without (left) and with (right)hourglass control


Hourglass Modes (3-D)

• One-point reduced integration for both deviatoric and volumetric part;

• Zero-energy modes due to rank deficiency of stiffness matrix;

• Special hourglass control algorithms avoid these unrealistic deformations.

• Use hourglass control, i.e. calculate hourglass strain/stress corresponding to the hourglass modes.

• Hourglass control is often orthogonal to rigid body motion, i.e. no hourglass energy will be generated by rigid body displacement or rotation

• CPU efficient since one point integration

Hourglass modes of a hexahedral volume element.


Numerical Integration

• The integrals in the FE discretization are integrated using a Gaussian quadrature rule;

• Exact integration of the highest degree monomials for the C0-elements is achieved using an 8-point integration rule for the solid elements with the volume element

dx = J(ξ) dξwhere J(ξ) is the Jacobian of the transformation from the x-domain to the ξ-domain;

∑

∫ ∫ ∫∫

=

− − −Ω

=

=Ω

int

1

1

1

1

1

1

1

)()(

)()(d)(

n

iiii Jf

dJffe

ωξξ

ξξξx

• The sum is formed at the discrete Gauss points with the weights wi.

• In the case of the 8 integration points, nint = 8, ξi = (1/sqrt 3) ξawhere the ξi are the natural coordinates of the element.

• A reduced one point integrationrule for the solid element, one order less than the 8 point rule, is given by:

).0()0(8

)0()0(d)(

Jf

wJffi

i

e

=

=Ω ∑∫Ω

x


Shell Elements

• Thin shell elements can be employed to discretize structures made of plates and shells.

• For elasticity problems the number of integration points across the element thickness is not important while for non-linear stress distributions due to plastifications, a numerical integration is required.

• Normally, 3 integration points are sufficient although for the sake of precision 5 points are commonly used. Restricting to one integration point would degenerate the shell element to a membrane element where only normal stresses are accounted for.

• Most of the elements have 5 degrees of freedom (DOF), i.e. two rotations and three displacements.

Definition of shell elements

• All elements have C0-continuity. • The element formulation is based on

the shell theory of Mindlin, which means that plane sections remain plane but not necessarily perpendicular to the mid-surface.


Belytschko-Lin-Tsai Element (BT)

• This element is a bi-linear four node quadrilateral isoparametric element formulation with uniform reduced integration for bending and shear and hourglass control.

• The shape functions for the reference element are in natural coordinates :

• The coupling between curvature and translations is neglected, whereas all nodes have six degrees of freedom per node; at each node, six equations have to be solved.

• The BT-element is simple, cheap and robust but lacks for accuracy for coarser and warped elements.

).1)(1(41),( ξξηηηξ IIIN ++=

The four hourglass modes of the BT element.


Hughes-Tezduyar Element (HT)

• This element is a four node bi-linear element with full 2x2 quadrature integration and a special interpolation rule for the transverse shear field.

• The element exhibits correct rank, thus no hourglass controlling is necessary.

• The bending performance is improved with respect to other element types.

• The HT-element is three to four times more expensive than the BT-element.

• The integration of a quantity fbecomes then

∑

∫ ∫∫

=

− −Ω

=

=Ω

int

1

1

1

1

1

.),(),(

dd),(),(d

n

llllll

e

WJf

Jffe

ηξηξ

ηξηξηξ


Belytschko-Wong-Chiang Element (BW)

• In order to improve the BT-element for warped meshes and to pass the quadratic Kirchhoff patch test in the thin plate limit, a particular projection scheme was added.

• Additional terms added to the strain-displacement relation account for the coupling between curvature and translation, which improves the performance for warped elements.

• A nodal projection (anti-drilling projection) is used to calculate the transverse shear strain field, resulting in improved behavior at the thin shell limit (Kirchhoff theory).

Ted BelytschkoThomas J.R. Hughes

• The BW-element is under-integrated, thus a hourglass control is required.

• To overcome this, a fully integrated BW-element was also proposed. It requires about three times more CPU time but remains stable even if largely deformed elements occur.


Kinematic Joints


Mesh Quality

To assure correct performance of shell elements, the following geometrical checks are normally included in the initial phase of the FE computation:

1. Check for bad aspect ratio:The aspect ratio AR :

is ideally equal to 1, which means that the ideal element is a square. It is recommended that AR should not be higher than 4.

..max:4123

3412 ARLLLLAR ≤

++

=

12L

23L41L

34L

2. Angle check: (4- and 3-lateral elements)The inner angles of a quadrilateral and a C0-trilateral shell element are checked to assure that the element is not highly distorted. Elements with ideal shapes are squares and equilateral triangles. Elements with highly distorted angles may lead to bad results, time step deterioration and divergence of the algorithm. The check verifies:

For quadrilateral elements, the angle should lie between 40° and 140° and for trilateral elements it should remain between 30° and 100°.

.maxmin θθθ ≤≤


Mesh Quality

3. Check for bad warping angleFor quadrilateral shell elements, the warping angle is measured as the difference of the angle between the normal at a node and the corres-ponding diagonal (as sketched in the figure below) and 90°:

A warped element can lead to bad results in in certain cases to divergence of the algorithm. The maximal warping angle is 10°. An example for a warped element is:

2νϑ

4. Check for initial penetration and perforation:The mesh should be also checked for initial perforations and penetrations. The latter can cause severe stability problems and artificial deformation.


Contact Algorithm

•The contact algorithm is crucial for the numerical effectiveness of the crash simulation; it often takes up to 1/3 of the total computation time;

• In general, a penalty approach is used for the contact;

•On the surfaces contact thicknesses t are defined (for shell elements in the range, but not necessarily identical of the physical thickness);

•The contact algorithm controls if the nodes of a second shell element are penetrating the contact surface;

•A penalty force F is generated, which is proportional to the distance d by which the second shell has penetrated the first element.

•Perforation occurs if the velocity of the approaching element is too high.

• At the start of a computation, initial penetrations must be avoided because they generate high contact forces, which lead to unrealistic deformations of the structure.

• These initial penetrations are often generated by automatic transfer of CAD data to FEM models.

• Kc represents the virtual spring, which is determined by the mass of the contact partners, the duration of the contact and the distance d


Contact: Master Segments and Slave Nodes

• For reasons of computational efficiency, distinctions are often made in the numerical treatment of contacts of a deformable structures with: rigid walls, other deformable structures, and with itself.

• In crash analysis, the CPU time may remarkably be determined by the performance of the contact algorithms.

• In general the algorithm is divided into a part in which the contact partners are searched and another part in which the interface problem is solved, i.e. the partners are checked for penetration. Then either penalty forces are applied to drive the partners from one another or a Lagrange multiplier approach is selected. One-sided contact

Self contact


Contact: Master Segments and Slave Nodes

• The algorithm defines slave and master nodes, which are nodes of the FE mesh belonging to the contact segments and interfaces.

• In one-sided contact definitions, slave nodes are belonging to a slave interface and master nodes to a master interface. For each slave node it is tested if it penetrates a master segment.

• For two-sided contact, this distinction is not made, both interfaces are treated equally. Thus, first the nodes of one interface are tested against the segments of the other, then vice versa.

Two-sided contact where both sides are treated symmetrically.

Two-sided contact


Contact and Interface Search

• For shell elements a contact thickness is defined. When the slave node is within a distance less than this value it is penetrating the master segment and a penalty force is applied to push it back.

• If the slave node has penetrated even the mid-surface of the master segment, this penalty force would even augment the penetration because it is normally defined outwards of the middle-surface.

• FE crash models consist today of about 1.5 million elements. A contact search for all these elements is by far not feasible. Thus the search problem should be restricted by specifying foreseeable contact regions and omitting other regions of the structure from contact treatment.

Ridge

penetration

perforation

contactthickness

t

slave

masterslave node

Penalty forces

perforation

penetration


Search Algorithms

In some codes, there are two search algorithms:

1. Node-to-node proximity search:Here, the algorithm searches for each slave node for the actual closest master node. Before, all nodes are sorted with respect to their distance according to a given search direction (e.g. the direction of the largest extension of the FE model) to facilitate the search. Alternatively, boxes can be defined in which the search is restricted. Normally the search is limited to a specific search radius around the slave node.

2. Node-to-segment correspondence search:Once the master node closest to the slave node within its contact sphere has been located, the corresponding segment has to be identified. Then the penetration check is performed. To prevent that in the case of a ridge, the algorithm fails, the contact surface is extended in direction of the mid-surface line by t/2. In case of a valley, the occurring ambiguity is less critical.


Meshing

•One of the major uncertainties in FE modelling is the proper choice of the mesh density, that is the level of spatial discretization, needed to achieve a homogeneous solution accuracy over the analyzed domain;

• In regions where high gradients of the solution variables occur or where local effects (fracturing, localizations, plastifications, etc.)are generated, a finer mesh than in other regions is required;

• In standard crash analysis, the mesh is created initially (currently with a minimal element size of ca. 5 mm) and left unchanged over the simulation.

•Minimal element size is restricted by minimal time step and the related maximal feasible CPU time.

• Therefore, the mesh is only refined in zones previously known as critical areas;

• Fracturing, failing cannot be modelled;

• Subcycling is currently developed to enable finer meshes;

• Element splitting is still not realized.


Mesh Size


Mesh Size

• Check of the convergence by refining the mesh;

• The results should be independent on the choice of mesh


Model Check

The following topics should be checked:

• Number of elements:Does a refinement of the mesh change the results of the computations?

• Element type:Does a change of the type of element affect the results? Do I have shear locking or hourglass modes?

• Number of integration points:How many integration points should I select for the numerical integration (at least 5)?

• Hourglass coefficient:What is the effect of the hourglass control, i.e. the hourglass coefficient?

Number of elements

Source: ESI, PAMcrash

Thin walled box beam / axial loading


Model Check

• Element size has the most important influence on the results;

• The actual mesh size used in crash models seems to be too large to get a mesh independent behaviour;

• All element types converge to slightly the same result if the mesh is fine enough;

• Results converge to same value with increasing number of elements for a given mesh formulation;

• The recommended element type (B.T./Stiffness HC using plastic modulus) shows the best convergence behaviour;

• For coarse meshes, 3 integration points are sufficient even if 5 integration points give a more accurate results. But if the mesh density becomes higher, 5 or even 7 integration points are strongly recommended.

• In current crash models, 5 integration points should be used.


Adaptive Meshing

1. r-adaptivity: Relocation of existing nodes to refine near areas where high precision is needed; no increase of total number of nodes and elements;

2. p-adaptivity:Adaptation of degree of polynomial interpolation p to local requirements of precision; no change of mesh topology;

3. h-adaptivity:Adaptation of grid spacing haccording to local precision requirements by subdivision of existing elements into smaller elements, and vice versa; increase of number of nodes and elements;

4. h-p-adaptivity:Combination of h- and p-adaptivity;


Adaptive Meshing

Advantages and disadvantages of adaptive FEM strategies:

• r-adaptivity is not effective; adaptation is restricted;

• p-adaptivity is disadvantageous because for explicit FE modelling the complex polynomial interpolation leads to difficulties in forming the lumped mass matrices;

• Regions with high discontinuous like plastic hinges would require unpractical high polynomial degrees.

• The generation of new nodes and elements has to be considered in the contact algorithm. After each model adaptation it is necessary to re-initialize the contact tables when adapted elements form part of contact interfaces

The main problem in realizing an adaptivity strategy for crash analysis lies in the parallelization of the computations;

Criteria for model adaptation:• Detection of regions with large

gradients of membrane stresses or energy;

• Detection of regions with large plastic deformations (membrane and bending);

• Detection of regions with important levels of hourglass energy;

• Detection of regions with large changes of the inter-elemental angles, which is also an indicator of local buckling effects.


Adaptive Meshing

• In early simulations, the computations were performed on a SMP platform, i.e. a shared memory parallelization was implemented.

• Currently, the crash simulations are performed on a cluster of processors with a DMP strategy (distributed memory parallelization).

• There, the fission of elements into smaller is not evident and therefore not realized.

• The computation time is highly depending on the amount of required communications between the processors; adaptation is rendering this infeasible.

• Nevertheless, for stamping simulation, which is also done by explicit finite element codes, adaptivity is used commonly.

Right: Example of an adaptive stamping simulation; Left:: Example of an adaptive crash simulation of a box beam, ESI, Theory Manual for PAMcrash.


Parallel Computing

• Since the beginning of the crash simulations, the numerical performance was one of the most important aspects for industrial applicability of the developments.

• In the mid of the 1980s vector computation was massively used in the automotive industry.

• The early FE models of up to 5,000 elements were then easily solved. Nevertheless the contemporary models had already sizes of about 60 to 90,000 elements, which required on a CRAY C90 about 20 to 25 hours.

• New element formulations, new material models for composites and foams were developed and the models became more sophisticated.

Model partition for parallel computing

First node Second node

Third node Fourth node

Total FE model


Parallel Computing

• Natural limit of further developments of scalar or vectorial computers.

• Implementation of parallelized algorithms was the next step.

• First experiences were made for metal forming in the mid 1990s.

• Simulations with shared memory (SMP) and with distributed memory (DMP) were realized.

• One of the challenges is the organization of contact algorithms.

• 4 main sources of overhead that can degrade ideal parallel performance: • non-optimal algorithm overhead• system software overhead• computational load imbalance• communication overhead Different partitions of a FEM model


Robustness of Crash Simulations / Repeatability

• Small changes in model parameter may result in large differences in simulation results;

• Repeated identical computations (identical input decks, identical machines, etc.) may lead to significant scatter in simulation results;

• There is no 100% correspondence between simulation model and finally built vehicles;

• Scatter in manufacturing, delivered materials, physics;

• Additional changes originating from the product evolution process.

• Etc.

Input scatter

Instabilities in crash simulations1. Numerical instabilities2. Physical instabilities

Scatter of the results

• Round-off errors on parallel machines;

• Small changes of input parameter;• Amplification of the input scatter by

modelling and by FE-Code;• Physics of the real vehicle.


Round-off Errors on Parallel Machines

• Small round-off errors are due to the random order of the operations to merge the results from the different processors;

• By augmenting the numerical effort, the round-off errors can be reduced: the order of the operations is then pre-defined (option PIPE in PAMcrash).

• The round-off errors can lead to large differences in the results (cf. example at the right-hand side);

• Therefore, instable physical designs can be identified in some cases;

• Bad modelling and incorrect code performance may also result in round-off errors.

dbcadcba +++≠+++


Instabilities in Crash Models

• Round-off errors may lead to different contact performances or buckling initiation;

• Most of the crash departments of the car companies have chosen to accept the round-off error as an indicator to real physical instability;

• (“Do not kill the butterfly”).

Stochastic Simulations (Monte-Carlo)Robustness Analysis


Developments in Computer Architectures

Source: IBM, ESI


Scalability

Source: IBM, ESI

Number of CPU

CPU times [h]


Mesh Independent Spotweld Elements

System of beam elements (LS-DYNA)

Elasto-plastic spotweld element (hybrid Trefftz Finite

Element).Early mesh-independent realization of spotwelds

Elasto-plastic spotweld (Trefftz-FE) with Hencky plasticity


Crack Propagation

Simulation of a cross member with failure option.Comparison of simulation to experimental results

BMW, 1998


First Rollover Simulations (MBS)

• The rollover was first modelled by two-dimensional multi-body systems (MBS). Robbins simulated about 2 seconds of a vehicle with one dummy to study the hurling out of the dummy. The kinematics were sufficiently modelled while no structural deformations could be assessed.


Hybrid Finite Element - Multi-body Simulation, 1990

• 3D-analysis were performed where a multi-body simulation was used for the phases without ground contact and a finite element simulation for the contact periods.

• They simulated 3 seconds of a rollover due to a test with 56 km/h. • For such long simulation time, the fact that explicit time schemes are

conditionally stable becomes remarkably important. • It has to be assured that no artificial numerical energy is accumulated.


First Rollover Simulations, 1990Hybrid Finite Element - Multi-body Simulation

• In 1990 when the study was published, a finite element simulation would have required 200 or 400 hours of computation time.

• The computation would have become instable.• The total vehicle was modelled by rigid bodies and only some parts were

switched to deformable finite elements when needed. This may occur several times during one single rollover.


Rollover Simulation, 2004


End.

Exploded representation of the finite element model of the BMW X5. The model is relatively small, it consists of ca. 250,000 finite elements in 300 parts. Today (i.e. 2005), the models have up to 1.5 Mio elements. Dummies, seats, interior parts have to be added.

master hh crash worthiness p 2

Documents

gaussian quadrature

uniform reduced

full car model

lumped mass

updated lagrangian

finite element

full vehicle

minimal element