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Solution of impact tasks,assessment of result reliability
M. Okrouhlík
Institute of Thermomechanics
Scope of the lecture
• What is a ‘good agreement’ in dynamical transient analysis in solid continuum mechanics
• Vehicle for comparison• Continuum model and its limits• Experimental and FE analysis• Assessment of agreement quality• Synergy of FE and experimental analyses• Conclusions
A good agreement of experimental and FE results
0 0.5 1 1.5 2 2.5 3 3.5
x 10-4
-4
-2
0
2x 10
-4 LOC1 - exp vs. FE 3D, NM
G1R90L, mesh1, medium striker for all figures
urmas data march 2007
experimentFE analysis, filtered
0 0.5 1 1.5 2 2.5 3 3.5
x 10-4
-4
-2
0
2x 10
-4 LOC2 - exp vs. FE 3D, NM
axi
al s
tra
in
time
experimentFE analysis, filtered
0 0.5 1 1.5 2 2.5 3 3.5
x 10-4
-4
-2
0
2x 10
-4 LOC3 - exp vs. FE 3D, NM
time
dt = 1e-7; sr = 1/dt; nf = sr/2 = 5000000; rel cut off = 1/50 = 0.02abs cut off = nf*rel cut off = 100000f order = 2; [b,a] = butter(f order,rel cut off)
experimentFE analysis, filtered
Experimental and FE treatment are limited by cut-off frequencies, which are generally different – in this case the filter applied on FE results has the same frequency limit as raw experimental data.
Impact loading of a tube with four spiral slots
Dimensions and points of interest
3D elements and node numbering for mesh1
Nodes in unfold location
Nodes in one layer
Location
Strain gauges
Strain gauges
Geometry, material and elements
G1R90L_NM ... four slots with 15 deg. indentation, axial length = 17.28 mm, slots and mesh rotate 90 deg
Geometry and elements first part of the tube = 800 mm 800 layers, 144 elem in one layer
second part, ie. axial length of slots = 17.28 mm 18 layers, 120 elem in one layerthird part of the tube = 800 mm 800 layers, 144 elem in one layertotal lenght of tube = 1617.28 mm 1618 total number of layersouter to inner diameters D/d = 22/16 mmnumber of spiral slots = 4
Element propertiestrilinear bricks elements, 8 nodes, Gauss order quadrature NG = 3consistent mass matrix for NM (Newmark)diagonal mass matrix for CD (central differences)number of elements 232 560number of nodes 310 576number of degrees of freedom 931 728max. front width 606
Material propertiesYoung modulus = 2.05e11 PaPoisson's ratio = 0.24density = 7800 kg/m^3
Loading and FE technologyLoading
Vertical arrangement of the loading is assumed.In experiment the upper face of the tube is loaded by a vertically falling striker, which has been released from a certain height.Lower face of the tube is fixed. In FE model only axial displacements of the lower face are constrained.The striker is made of the same material as the tube, has the same outer and inner diameters.For computational purposes the upper side of the tube is loaded by uniform pressure,whose time dependence is given by a rectangular pulse.The loading pressure corresponds to the height from which the striker was released.
The time of the pulse corresponds to the length of the striker.In this case the loading pressure is 88.5198 Mpa, which corresponds to the height of h = 1m.Velocity of the striker just before the impact is sqrt(2*g*h) = 4.42944 m/s.Material particle velocity immediately after the impact is v = 0.5*sqrt(2*g*h) = 2.2147 m/s.The time length of the pulse is 15.6 microsec,
which corresponds to a medium length striker, Ls = 40 mm.Pressure is evaluated from p = E*v/c0, with c0 = sqrt(E/ro).Input energy from the striker is mgh = 0.548086 J. m = 05587 kg.
FE technologyNewmark with no algorithmic damping was used with consistent mass matrixCentral differences with diagonal mass matrixtimestep = 0.1 microsec = 1e-7 stotal number of time steps = 3500 (Newmark); 5000 (central differences)this corresponds to total time = 350 microsec (Newmark); 500 microsec (central differences)
Wavefront timetable
FE strain distribution in location 1
Raw comparison – no tricks
0 1 2 3 4 5 6
x 10-5
-4
-3
-2
-1
0
1
2x 10
-4 LOC0 - exp vs. FE axisym, CD
axia
l str
ain
mesh1, medium striker for all figures
urmas data march 2007
0 0.5 1 1.5 2 2.5 3 3.5
x 10-4
-4
-3
-2
-1
0
1
2x 10
-4 LOC1 - exp vs. FE 3D, NM
0 0.5 1 1.5 2 2.5 3 3.5
x 10-4
-4
-3
-2
-1
0
1
2x 10
-4 LOC2 - exp vs. FE 3D, NM
axia
l str
ain
time
0 0.5 1 1.5 2 2.5 3 3.5
x 10-4
-4
-3
-2
-1
0
1
2x 10
-4 LOC3 - exp vs. FE 3D, NM
time
experiment
FE analysis
experiment
FE analysis
experiment
FE analysis
experiment
FE analysis
Are FE or experimental results closer to reality? Where is the truth?
Where is the truth? Are experimental or FE results closer to reality?
• When trying to reveal the ‘true’ behavior of a mechanical system we are using the experiment
• When trying to predict the ‘true’ behavior of a mechanical system we are accepting a certain model of it and then solve it analytically and/or numerically
• Physical laws (models) as– Newton’s – Energy conservation– Theory of relativity
cannot be proved (in mathematical sense)
• Often we say that it is the experiment which ultimately confirms the model
• But experiments, as well as the numerical treatment of models describing the nature, have observational thresholds. Sometimes, the computational threshold of computational analysis are narrower than those of experiment
• Let’s ponder about limits of applicability of continuum model as well as about limits of modern computational approaches when applied to approximate solution of continuum mechanics
Continuum mechanics
• Deals with response of solid and/or fluid medium to external influence
• By response we mean description of motion, displacement, force, strain and stress expressed as functions of time and space
• By external influence we mean loading, constraints, etc. Expressed as functions of time and space
Solid continuum mechanics
• Macroscopic model disregarding the corpuscular structure of matter
• Continuous distribution of matter is assumed – continuity hypothesis
• All considered material properties within the observed infinitesimal element are identical with those of a specimen of finite size
• Quantities describing the continuum behaviour are expressed as piecewice continuous functions of time and space
Governing equations
• Cauchy equations of motion
• Kinematic relations
• Constitutive relations
itt
it
jtji
tt xfx
j
kt
i
kt
i
jt
j
it
ij x
u
x
u
x
u
x
u00002
1angeGreen_Lagr
i
jt
j
it
ij x
u
x
u002
1eng
engengklijklij C Lagrange_Green
klijklij DS
For transient problems, we are interested in, the inertia forces are to be taken into account
In linear continuum mechanicsthe equations are simplified to
klijkliji
j
j
iij
ii
j
ji Cx
u
x
u
t
uf
x
,2
1,
2
2
• Inertia forces are considered
• loading is
• localized in space
• of short duration with a short rise time
The high frequency components are of utmost importance
Wave equation 2D - plane stress, Lamé equations – expressed in displacements only
,1
2
2
22
2
2
22
2
xy
v
y
uG
yx
v
x
uE
t
u
.1
2
2
22
2
2
22
2
yx
u
x
vG
yx
u
y
vE
t
v
tcxfu 3
tcyhv 3 tcyFu 2
tcxHv 2
23 1
Ec
G
c 2 S … shearP … primary
Longitudinaldilatationalirrotationalextension
Transversalshearrotationaldistortionequivolumetrical
Nondispersive and uncoupled solutions in unbound region
Velocities for 2D plane strain and 3D
211
121
EGc
211
E 12
EG
P … primary
Within the scope of linear theory of elasticity,P and S waves are uncoupled.
Typical values for steel in m/s
For E = 2.1e11 Pa, = 7800 kgm^-3, = 0.3
0.3for Rayleigh wave,R29849274.0
stress planefor waveP5439))1(/(
shear wave,S3218/
3D strain, planefor waveP6020/)2(
barslender wave,D15189/
2R
23
2
1
0
cc
Ec
Gc
Gc
Ec
Analytical solutions for bounded regions are difficult and exist
for bodies with simple initial and boundary conditions only
One way to solve the problem is to apply the Fourier transform in space and the Laplace transform in time on equations of motion. The subsequent inverse procedure leads to infinite series of improper integrals.
To indicate amount of effort to be exerted, the formulas derived by F. Valeš are shown.
A typical expression for a stress component
in a transiently loaded thin elastic strip is shown
Relations between integrand quantities and integral variable
Roots of these so called dispersive relations have to be evaluated numerically before the integration itself.
Roots of dispersive relations
After all, the analytical solution ends up
with a numerical evaluation
• Only finite number of terms can be summed up• Integration itself has to be provided numerically
• This led to development of approximate numerical solutions
Today, approximate methods of solution prevail Discretization Finite difference method Transfer matrix method Matrix method Displacement formulation Force formulation Finite element method Displacement formulation Force formulation Hybrid finite element method Mixed finite element method
Numerical methods in FEA of continuum Equilibrium problems
QqqK )( solution of algebraic equations Steady-state vibration problems
0qMK 2Ω generalized eigenvalue problem Propagation problems
VV
d, Tintextint σBFFFqM
step by step integration in time In linear cases we have
)(tPKqqCqM
Unbound frequency response of continuum
For fast transient problems as shock and impact the high frequency components of solutions are of utmost importance. In continuum, there is no upper limit of the frequency range of the response. In this respect continuum is able to deal with infinitely high frequencies. This is a sort of singularity deeply embedded in the continuum model.
As soon as we apply any of discrete methods for the approximate treatment of transient tasks in continuum mechanics, the value of upper cut-off frequency is to be known in order to ‘safely’ describe the frequencies of interest.
Dispersive propertiesof 1D and 2D constant strain elements
When looking for the upper frequency limit of a discrete approach to continuum problems,
we could proceed as follows
• Characteristic element size• Wavelength to be registered• How many elements into the
wavelength• Wavelength to period relation• Wave velocity in steel• Frequency to period relation• The limit frequency• For 1 mm element we get
s
s5
m/s5000ccT
Tf /1)5/( scf
MHz 1 Hz101001.05
5000 6
f
10-10
10-8
10-6
10-4
10-2
10-2
100
102
104
106
108
size in [m]
freq
uenc
y in
[Mhz
]
characteristic sizes and corresponding frequencies
atom sizeaustenite steel grain size1 mm finite element1 MHz level1 GHz levelFE analysis range from 0.1mm to 100mmmaximum exp. sampling limit 100 MHz - 14 bits
Where is the continuum limit?
Limits of continuum, FE analysis and experiment
Validity limits of a model
• Model is a purposefully simplified concept of a studied phenomenon invented with the intention to predict – what would happen if ..
• Accepted assumptions (simplifications) specify the validity limits of the model
• Model is neither true nor false
• Regardless of being simple or complicated, it is good, if it is approved by an experiment
Using a model outside its limits is a blunder
• Using a model outside of its validity limits leads to erroneous results and conclusions
• This is not, however, the fault of the model, but pure consequence of a poor judgment of the model’s user
• Model gives no warning. Lot of checks might be satisfied and still …
Blunders are easy to commit
• Point force is– prohibited in continuum,– frequently used in FE analysis
• Employing smaller and smaller elements, leads to singularity, since we are coming closer and closer to ‘continuum’ revealing thus unacceptable behavior of the point force in continuum
• An example follow
Classical Lamb’s problem
ra d ia l
B
A C
1 m
1 m
a xia l
p 0
T im p
t
L or Q
ElementsL, Q, full int.Consistent massaxisymmetricMeshCoarse 20x20Medium 40x40Fine 80x80Newmark
Loading a point force equiv. pressure
Example of a transient problem
Primary wave
Lamb, presure loading, rectangular pulse, velocity distribution, FEA
P waves
S waves
R waves
Again, where is the first nonzero P-wave appearance?
Pollution-free energy production
How to avoid blunders
By knowing and understanding– the assumptions– instrumental limitations
By providing– validity checks
• within the model itself• comparing with other methods
FE self-check considerations
Numerický experiment může mez rozlišení simulovat a napomoci tak experimentu.
Dá se ukázat, jak zjištěná rychlost šíření rozruchu závisí na hodnotě meze pozorování
Experimental
threshold
V pozorovaném místě známe
- vzdálenost od aplikace budící síly,
- čas, kdy dorazí „nenulový“ signál
můžeme tedy vypočítat rychlost šíření rozruchu.
Numerical analysis should be robust
It should inform us about its limits
Should be independent of – mass matrix formulation– the method of integration– meshsize– element type
It is not always so, very often our results are method dependent. See the next example.
Validity self-assessments, NM vs. CD
C o m p u t a t i o n a l i n f i n i t e s p e e d o f w a v e p r o p a g a t i o n c a n b e e x p l a i n e d b y a n a l y z i n g t h e t i m e m a r c h i n g a l g o r i t h m s f o r tPKqqM
E x p l i c i t ( c e n t r a l d i f f e r e n c e s ) I m p l i c i t ( N e w m a r k ) E q u i l i b r i u m i s c o n s i d e r e d a t t i m e t tt a n d l e a d s t o r e p e a t e d s o l u t i o n s o f
ttttPMq~
2
1 tttt PqK ˆˆ
b o u n d edn o t is np ro p ag atio o f sp eed n alco mp u tatiod iag o n alb en ev ercan
d iag o n al is ifo n ly d iag o n al,isfu ll;b an d ed ;g en erally 1
K
MMMKMK 11
ˆ,,
ttttt tt
MqqMKPP
22
22~ MKK2
1
t
ˆ
ttttttt ccc qqqMPP 321 ˆ
Instrumental limitation
Floating point representation of real numbers threshold
Memory size limitation
Meshsize and time step limitations
machine epsilon
FFT frequency analysis
Now, let’s concentrate on the frequency analysis frequency analysis of the loading pulse and of axial and radial displacements obtained in the outer corner node of location C by means of NM and CD operators for the mesh1. The normalized power spectra are plotted in the range from 0 to Nyquist frequency together with the power spectrum of the loading pulse.
timestep [s] sampling rate [MHz] Nyquist frequency [MHz]mesh1 1e-7 10 5mesh2 1e-7/2 20 10 mesh3 1e-7/4 40 20 mesh4 1e-7/8 80 40
Assessment by frequency analysis
0 1 2 3 4
x 10-5
0
0.5
1
1.5
2
2.5x 10
4 imput pulse
time0 1 2 3 4
x 10-5
0
1
2
3
x 10-5 mesh1, disp, corner node
time
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 106
0
1
2
3
4
5
6x 10
-10
frequency from 0 to Nyquist
transfer function, input vs. rad. disp
radial NMaxial NMradial CDaxial CD
NMCDbreathing frequencyzig-zag L frequencyzig-zag S frequencyFE limit frequency consFE limit frequency diag
Validity self-assessments Mesh- and timestep refinement
0 0.5 1 1.5 2
x 106
0
0.5
1
1.5
2
2.5
3
3.5
x 10-10
tra
nsf
er
fun
ctio
n, r
ad
. dis
p, m
ed
ium
str
ike
r axisym, mesh1, corner node
0 0.5 1 1.5 2
x 106
0
0.2
0.4
0.6
0.8
1x 10
-5 axisym, mesh2, corner node
0 0.5 1 1.5 2
x 106
0
0.2
0.4
0.6
0.8
1x 10
-5 axisym, mesh3, corner node
frequency range from 0 to 2 MHz
tra
nsf
er
fun
ctio
n, r
ad
. dis
p, m
ed
ium
str
ike
r
0 0.5 1 1.5 2
x 106
0
0.2
0.4
0.6
0.8
1x 10
-5 axisym, mesh4, corner node
frequency range from 0 to 2 MHz
NMCDbreathing frequencyzig-zag L frequencyzig-zag S frequency
NMCDbreathing frequencyzig-zag L frequencyzig-zag S frequency
NMCDbreathing frequencyzig-zag L frequencyzig-zag S frequency
NMCDbreathing frequencyzig-zag L frequencyzig-zag S frequency
Synergy of experiment and FE analysis
FE analysis needs input data for computational models from experiment
Experimental analysis could benefit from FE in ‘proper’ settings of observational thresholds
Four pulses, their mean values, power spectra and input energies
Mean value, c0 value, norm and square root of the input energy for different pulses
Terms and statistical tools for
quality assessment of close solutions
• Standard deviation (směrodatná odchylka) of a sample vector (also called series of measurements, signal, list of samples, variables) having n elements is a scalar quantity defined by
• Variance (rozptyl) is the square of the standard deviation.
2
1
1
2
1
1
n
ii xx
ny
n
iixn
x1
1
Covariance
• is the measure of how much two variables vary together (as distinct from variance, which measures how much a single variable varies).
• The algorithm for covariance is
[n,p] = size(X);X = X – ones(n,1)*mean(X);C = X’*X/(n-1);
Correlation
• The correlation, also called correlation coefficient, indicates the strength and direction of a linear relationship between two (or more) variables. The correlation refers to the departure of two (or more) variables from independence.
yx
n
iii
xy ssn
yyxxr
)1(
))((1
Respective means
and
Corresponding standard deviations
Geometric interpretation of correlation
• The correlation coefficient can also be viewed as the cosine of the angle between the two vectors of samples drawn from the two random variables.
yx
yxuncenteredcos
Validity assessments, Axisymmetric and 3D elements
0 500 1000 1500-1
-0.5
0
0.5
1
1.5
2
2.5x 10
-6
steps
rad
ial d
isp
lac
em
en
ts, l
oc
ati
on
1, l
ay
er
1, n
od
e 1
[m]
Axisym vs. 3D elements, medium striker, CD
axi3D
0 500 1000 15000
1
2
3
4
5
6x 10
-5
steps
ax
ial d
isp
lac
em
en
ts, l
oc
ati
on
1, l
ay
er
1, n
od
e 1
[m
]
Axisym vs. 3D elements, medium striker, CD
axi3D
Quantitative measure of agreement quality
The agreement of solutions obtained by a different element types is excellent – for a given loading and the employed time and space discretizations, there is almost no ‘measurable’ difference.
One way to measure the difference between two solutions, having the form of a vector in n-dimensional space (n is the number of time steps in this case), is to compute the angle between them. In our case the quality of agreement could be quantified by
320209978474978.0cos
yx
yx
This value is called the uncentered correlation coefficient.
Assessment of solutions obtained by means of NM and CD operators
and for different time and space discretizations
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10-5
-2
-1
0
1
2
x 10-6 four meshes, original data, axisym, NM
rad
ial d
isp
lace
me
nts
, co
rne
r n
od
e, l
ocC
time
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
x 10-5
-2
-1
0
1
2
x 10-6 four meshes, original data, axisym, CD
time
rad
ial d
isp
lace
me
nts
, co
rne
r n
od
e, l
ocC
m1m2m3m4
m1m2m3m4
Measure of sameness by correlation
2 3 40
0.005
0.01
0.015
0.02
0.025
m2 to m4, <j> index
relative directional error of m<i> to m<j> data, NM
2 3 40
0.005
0.01
0.015
0.02
0.025
m2 to m4, <j> index
relative directional error of m<i> to m<j> data, CD
m1, <i> indexm2, <i> indexm3, <i> index
m1, <i> indexm2, <i> indexm3, <i> index
Measure of sameness by variance and covariance
1 2 3 41.46
1.48
1.5
1.52
1.54
1.56
1.58
1.6x 10
-12 variance and covariance for m1 to m4
meshes m1 to m4
variance for NM
variance for CDcovariance NM. vs. CD
To reveal the true nature of reality good-agreement solutions have to analyzed.
How?• Avoid crude mistakes and omissions • Avoid blunders – poor judgment• The ‘sameness’ of close solutions should be assessed by statistical
tools• Employed models should be robust, which means they should be
able give warning of their misuse. This is a pretty demanding task. Presently, the robustness can only be achieved indirectly by means of a posteriori checks by– Comparing results obtained by different variant of the employed model
(coarse and fine meshes, different types of elements, coarse and fine timesteps, different integration methods, etc)
– Analyzing the frequency contents of signals by Fourier analysis
• Results obtained by models should be checked by other models and/or by experiments
• Since the experiment is just another tool for revealing the true nature of reality, its results have to viewed by the prism of its validity limits
Conclusions
When trying to ascertain the reliability of modelling approaches and the extent of their validity one has to realize that the models as a rule do not have self-correction features. That’s why we have to let the models to check themselves, be checked by independent models and let the systematic doubt be our everyday companion.
Leftovers