master hh crash worthiness p 2
TRANSCRIPT
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]
June 2007 Crashworthiness, 2 Fabian Duddeck
First Numerical Models for Crashworthiness 1960s
Lumped Mass Models First numerical methods for crash analysis were developed in the 1960s.
June 2007 Crashworthiness, 3 Fabian Duddeck
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
June 2007 Crashworthiness, 4 Fabian Duddeck
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
Kim (2001)PhD thesis
Forc
e
Displacement
0),,,,()( 11 =+ ++ iiiiiii xddSSFStxM &&
June 2007 Crashworthiness, 5 Fabian Duddeck
Lumped Mass Models
A three DOF system with six resistance force elements
Kim (2001)PhD thesis
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
June 2007 Crashworthiness, 6 Fabian Duddeck
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
June 2007 Crashworthiness, 7 Fabian Duddeck
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
• …
June 2007 Crashworthiness, 8 Fabian Duddeck
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.
June 2007 Crashworthiness, 9 Fabian Duddeck
First Crash Analyses with Finite Elements, 1983
• 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)
June 2007 Crashworthiness, 10 Fabian Duddeck
First Crash Analyses with Finite Elements, 1983
Beam structure for crashworthiness (rollover) of a bus (1983)
1983 Stelzmann et al. (2006)
June 2007 Crashworthiness, 11 Fabian Duddeck
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
June 2007 Crashworthiness, 12 Fabian Duddeck
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.
June 2007 Crashworthiness, 13 Fabian Duddeck
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
June 2007 Crashworthiness, 14 Fabian Duddeck
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.
June 2007 Crashworthiness, 15 Fabian Duddeck
Full Vehicle Models
Audi
June 2007 Crashworthiness, 16 Fabian Duddeck
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
June 2007 Crashworthiness, 17 Fabian Duddeck
Γ+Ω−Ω=
Γ+Ω−Ω=
∫∫∫
∫∫∫
ΓΩΩ
ΓΩΩ
ddd0
ddd0
tuDεεbu
tuσεbu
TTT
TTT
δδδ
δδδ
Finite Element Method - Principle
• 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
June 2007 Crashworthiness, 18 Fabian Duddeck
Finite Element Method - Principle
• 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
June 2007 Crashworthiness, 19 Fabian Duddeck
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.
June 2007 Crashworthiness, 20 Fabian Duddeck
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
June 2007 Crashworthiness, 21 Fabian Duddeck
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
&&&&
&&
==
June 2007 Crashworthiness, 22 Fabian Duddeck
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 =+&&
June 2007 Crashworthiness, 23 Fabian Duddeck
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
June 2007 Crashworthiness, 24 Fabian Duddeck
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
June 2007 Crashworthiness, 25 Fabian Duddeck
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)
June 2007 Crashworthiness, 26 Fabian Duddeck
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.
June 2007 Crashworthiness, 27 Fabian Duddeck
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!
June 2007 Crashworthiness, 28 Fabian Duddeck
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.
June 2007 Crashworthiness, 29 Fabian Duddeck
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;
June 2007 Crashworthiness, 30 Fabian Duddeck
Elements
June 2007 Crashworthiness, 31 Fabian Duddeck
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
5-node element: n1, n2 , n3, n4 , n5, n5 , n5, n5
6-node element: n1, n2 , n3, n4 , n5, n5 , n6, n6
• 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.
June 2007 Crashworthiness, 32 Fabian Duddeck
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:
June 2007 Crashworthiness, 33 Fabian Duddeck
Eight Node Solid Elements
June 2007 Crashworthiness, 34 Fabian Duddeck
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.
June 2007 Crashworthiness, 35 Fabian Duddeck
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.
June 2007 Crashworthiness, 36 Fabian Duddeck
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
June 2007 Crashworthiness, 37 Fabian Duddeck
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.
June 2007 Crashworthiness, 38 Fabian Duddeck
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
June 2007 Crashworthiness, 39 Fabian Duddeck
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.
June 2007 Crashworthiness, 40 Fabian Duddeck
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.
June 2007 Crashworthiness, 41 Fabian Duddeck
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
June 2007 Crashworthiness, 42 Fabian Duddeck
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.
June 2007 Crashworthiness, 43 Fabian Duddeck
Kinematic Joints
June 2007 Crashworthiness, 44 Fabian Duddeck
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 ≤≤
June 2007 Crashworthiness, 45 Fabian Duddeck
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.
June 2007 Crashworthiness, 46 Fabian Duddeck
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
June 2007 Crashworthiness, 47 Fabian Duddeck
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
June 2007 Crashworthiness, 48 Fabian Duddeck
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
June 2007 Crashworthiness, 49 Fabian Duddeck
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
June 2007 Crashworthiness, 50 Fabian Duddeck
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.
June 2007 Crashworthiness, 51 Fabian Duddeck
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.
June 2007 Crashworthiness, 52 Fabian Duddeck
Mesh Size
June 2007 Crashworthiness, 53 Fabian Duddeck
Mesh Size
• Check of the convergence by refining the mesh;
• The results should be independent on the choice of mesh
June 2007 Crashworthiness, 54 Fabian Duddeck
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
June 2007 Crashworthiness, 55 Fabian Duddeck
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.
June 2007 Crashworthiness, 56 Fabian Duddeck
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;
June 2007 Crashworthiness, 57 Fabian Duddeck
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.
June 2007 Crashworthiness, 58 Fabian Duddeck
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.
June 2007 Crashworthiness, 59 Fabian Duddeck
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
June 2007 Crashworthiness, 60 Fabian Duddeck
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
June 2007 Crashworthiness, 61 Fabian Duddeck
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.
June 2007 Crashworthiness, 62 Fabian Duddeck
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 +++≠+++
June 2007 Crashworthiness, 63 Fabian Duddeck
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
June 2007 Crashworthiness, 64 Fabian Duddeck
Developments in Computer Architectures
Source: IBM, ESI
June 2007 Crashworthiness, 65 Fabian Duddeck
Scalability
Source: IBM, ESI
Number of CPU
CPU times [h]
June 2007 Crashworthiness, 66 Fabian Duddeck
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
June 2007 Crashworthiness, 67 Fabian Duddeck
Crack Propagation
Simulation of a cross member with failure option.Comparison of simulation to experimental results
BMW, 1998
June 2007 Crashworthiness, 68 Fabian Duddeck
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.
June 2007 Crashworthiness, 69 Fabian Duddeck
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.
June 2007 Crashworthiness, 70 Fabian Duddeck
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.
June 2007 Crashworthiness, 71 Fabian Duddeck
Rollover Simulation, 2004
June 2007 Crashworthiness, 72 Fabian Duddeck
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.