Download - O metod ě konečných prvků Lect_ 1 5
O metodě konečných prvkůLect_15.ppt
M. Okrouhlík
Ústav termomechaniky, AV ČR, PrahaPlzeň, 2010
FFT and FEM
Obsah
• Terminologie
• Fourierova řada
• CFT, DFT a FFT
• Pár příkladů
• Mudrování o platnosti MKP výsledků
TerminologieFourierova řada, CFT, DFT a FFT
• Fourier series
• Continuous Fourier Transform
• Discrete Fourier Transform
• Fast Fourier Transform
Fourierova řada, 20 členů T = Tmax = n * Timp, n = 2,4,8,16
Obdélníkový puls – CFTtři obdélníkové pulsy různé délky – různá normalizace
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 105
0
1
2
3
4
5x 10
-5 C() = 2*abs(sin(T/2))/
Am
plit
ud
e s
pe
ctru
m
timp = 1e-5
timp = 2e-5timp = 5e-5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 105
0
0.2
0.4
0.6
0.8
1C()/T = 2*abs(sin(T/2))/(T)
frequency in Hz
Am
plit
ud
e s
pe
ctru
m
f3 3 fft c3.m, f ig1
timp = 1e-5
timp = 2e-5timp = 5e-5
FFT je způsob výpočtu DFT
Základní vztahy_1
Sampling rate,vzorovací frekvence
0 5 10 15 20 25 30 35 40-20
-15
-10
-5
0
5
10
15
20
Continuous and sampled forms of a signal
0 5 10 15 20 25 30 35 40-20
-15
-10
-5
0
5
10
15
20Coarse sampling spoils the signal
timecontinuous and sampled fig1
Základní vztahy_2
Pro diskusi o významu termínu power spectrum jsou následující důležité vztahymean(y) = sum(y)/NArea = tmax*sum(y)/(N–1) = tmax*mean(y)*N/(N-1) ~ tmax*mean(y)
první členy řad Y a pYY(1) = mean(y) * NpY(1) = mean(y) * mean(y) * N
final
initial
dd0i0
t
t
t ttPtetPc
Je to vztah mezi číselnými hodnotami
Základní vztahy_3
Obdélníkový pulsFFT (Fast Fourier Transform) a MPS (Matlab Power Spectrum)
0 1 2 3 4 5 6 70
1
2
3
4sampled signal
time [s]
0 10 20 30 40 50 60-10
0
10
20FFT spectrum - real part
counter
Nyquist
0 10 20 30 40 50 60-10
0
10
20FFT spectrum - imaginary part
counter
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
1
2
3
4
5FFT power spectrum
frequency [Hz]
Nyquist
kkk c1 f ig1
Význam členů y(1) a Y(1)
Normalization of FFT data
Základní vztahy_4
Základní vztahy_5
Nyquistova frekvence závisí na tom, zda počet vzorků je sudý či lichý.
Je-li počet vzorků velký, rozdíl nestojí za řeč.
function [f, p_x, n_f] = power_spect_c2(t,x) % compute fft power spectrum for x signal in time domain % input % t ... time ..... t(i) ... i = 1 : N % time range ..... t = 0 : dt : tmax % x ... signal ... x(i) ... i = 1 : N % output % f frequency f(i) ..... i = 1 : NF % p_x complete fft power spectrum p_x(i) ... i = 1 : N % n_f ... Nyquist frequency % number of sampled points N = length(t); % dt .. time increment, assumed uniform dt = t(2) - t(1); % sampling frequency = sampling rate s_r = 1/dt; % Nyquist frequency n_f = 0.5*s_r; % is N odd or even? use the reminder function rm = rem(N,2); % calculate increment of frequencies and frequencies if rm == 0,, df = s_r/N; f = (0:N/2)*df; % N is even else df = s_r/(N-1); f = (0:(N-1)/2)*df; % N is odd end NF = length(f);
% Fourier spectrum f_x = fft(x); % complete power spectrum p_x = f_x.*conj(f_x)/N; % the first half of the power spectrum p_x = p_x(1:NF);
% end of power_spect_c2
Obdélníkový puls – CFTtři obdélníkové pulsy různé délky – různá normalizace
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 105
0
1
2
3
4
5x 10
-5 C() = 2*abs(sin(T/2))/
Am
plit
ud
e s
pe
ctru
m
timp = 1e-5
timp = 2e-5timp = 5e-5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 105
0
0.2
0.4
0.6
0.8
1C()/T = 2*abs(sin(T/2))/(T)
frequency in Hz
Am
plit
ud
e s
pe
ctru
m
f3 3 fft c3.m, f ig1
timp = 1e-5
timp = 2e-5timp = 5e-5
CFT vs. FFTthree different rectangular pulses, the same sampling and the same window
0 0.2 0.4 0.6 0.8 1
x 10-4
0
0.2
0.4
0.6
0.8
1
time
do
ma
in
time [s]
timp = 0.1
0 0.2 0.4 0.6 0.8 1
x 10-4
0
0.2
0.4
0.6
0.8
1
timp = 0.2
time [s]0 0.2 0.4 0.6 0.8 1
x 10-4
0
0.2
0.4
0.6
0.8
1
timp = 0.5
time [s]
0 0.5 1 1.5 2
x 105
0
1
2
3
4
5
ma
tlab
po
we
r sp
ect
rum
0 0.5 1 1.5 2
x 105
0
5
10
15
20
0 0.5 1 1.5 2
x 105
0
50
100
0 0.5 1 1.5 2
x 105
0
10
20
30
mp
s/m
ea
n/ti
mp
frequency [Hz]F3 3 fft c3.m, f ig2 ... the same w indow and sampling rate, three different pulses
0 0.5 1 1.5 2
x 105
0
10
20
30
frequency [Hz]0 0.5 1 1.5 2
x 105
0
10
20
30
frequency [Hz]
1/dt1/timp
CFT vs. FFTthe same rectangular pulse, the same sampling and three different windows
0 0.2 0.4 0.6 0.8 1 1.2
x 10-4
0
0.2
0.4
0.6
0.8
1
tmax = 2*timp
time [s]
rect
angu
lar
puls
e
0 0.2 0.4 0.6 0.8 1 1.2
x 10-4
0
0.2
0.4
0.6
0.8
1
tmax = 4*timp
time [s]0 0.2 0.4 0.6 0.8 1 1.2
x 10-4
0
0.2
0.4
0.6
0.8
1
tmax = 8*timp
time [s]
0 1 2 3 4 5
x 105
0
20
40
60
80
pow
er s
pect
rum
frequency [Hz]
Nyquist = 5000000 [Hz]
0 1 2 3 4 5
x 105
0
10
20
30
40
frequency [Hz]
Nyquist = 5000000 [Hz]
0 1 2 3 4 5
x 105
0
5
10
15
20
frequency [Hz]
Nyquist = 5000000 [Hz]
0 1 2 3 4 5
x 105
0
0.1
0.2
0.3
0.4
0.5
norm
aliz
ed p
ower
spe
ctru
m
frequency [Hz]0 1 2 3 4 5
x 105
0
0.1
0.2
0.3
0.4
0.5
frequency [Hz]0 1 2 3 4 5
x 105
0
0.1
0.2
0.3
0.4
0.5
frequency [Hz]
mean1 = 0.5 mean3 = 0.125mean2 = 0.25
c02 = mean22 * N2c01 = mean12 * N1 c03 = mean12 * N3
df1
df2
Fyzikální význam Denoting the discretized time evolution of reference pulse T
21 nPPPP we can express its
mean value
n
in 1mean
1iPP and the approximate the area under the pulse
final
initial
dt
t
ttP by TmeanP .
The physical meaning of the last expression is the mean impulse. The area under the
pulse can also be expressed by means of the first term of the continuous Fourier’s transform,
CFT [Doyle, 1997]. This is the term corresponding to zero-th frequency and has the form
final
initial
dd0i0
t
t
t ttPtetPc .
Pokud je tedy vstupní veličinou síla,
pak power spectrum (výkonové)
by se mělo nazývat
momentum (hybnostní) spectrum
Momentum - hybnost
Four different input pulses and their average
The correlation between 0c and meanP values is shown in the first column of subfigures in
Fig. 12. They are plotted as functions of the pulse counter. Both of them are readily available at
the time of experiment and bear information related to the total momentum since the impulse of
the loading force is equal to the rate of momentum, i.e. vmtP dd . Observing these two
quantities one can easily judge the magnitude of the total impulse of the loading force.
The second couple of quantities that could be correlated (see the subfigures in the right-hand side column of Fig. 12) are the Euclidean norm of the time history of the loading
pulse
n
i 1
2iP and the square root of the total input energy. This follows from the fact
that the energy is proportional to the quadratic form of displacements and velocities, i.e. qMqKqq TT and could also be expressed as proportional to the quadratic form of
applied forces TT QMQQKQ 1T1T , where qq , are global arrays of displacements and velocities ‘measured’ at time finalt , KM, are mass and stiffness
matrices, T is the time interval during which the mean global loading vector, defined by T
mean 000000 PQ , is applied. So observing the norm value one can estimate
amount of energy being delivered by the pulse to the mechanical system.
Tube with four spiral slots, coarse mesh, dimensions
Four different meshes
Comparison of FE and EXP data for axial strains in different locationsC:\spiral_slot_2006_backup_from_2007_10_24\1_u_2007_december_changed_data\u_2007_data_from_urmas_march_2007\ToMila\plot_urmas_data_march_2007_c1, f6
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
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
axia
l str
ain
time
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
experiment
FE analysis
experiment
FE analysis
experiment
FE analysis
Comparison of filtered FE and EXP data for axial strains in different locations
C:\spiral_slot_2006_backup_from_2007_10_24\1_u_2007_december_changed_data\u_2007_data_from_urmas_march_2007\ToMila\plot_urmas_data_march_2007_c1, f5
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 - filtered to experimental frequency cutoff
G1R90L, mesh1, medium striker for all figuresurmas data march 2007
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
axia
l str
ain
time
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 = 0.02
abs cut off = nf*rel cut off = 100000
f order = 2; [b,a] = butter(f order,rel cut off)
experiment
FE analysis, filtered
experiment
FE analysis, filtered
experiment
FE analysis, filtered
FE data were filtered by a filter having the same upper frequency as the experiment sampling frequency
And this leads to a question.
That is, up to which frequency limit is the FE approach trustworthy?
We know that FE method is a model of continuum. Continuum – also a model – being based on the continuity hypothesis, disregards the corpuscular structure of matter. It is assumed that matter within the observed specimen is distributed continuously and its properties do not depend on the specimen size.
Quantities describing the continuum behavior are expressed as piecewise continuous functions of time and space. It is known, see [2], that such a conceived continuum has no upper frequency limit.
To find a ‘meaningful’ frequency limit of FE model, which is of discrete – not continuous – nature, one might pursue the following heuristic reasoning.
Rule of thump1mm corresponds to 1 MHz
Easy to rememberImagine a uniform finite element mesh with a characteristic element size, say s . Trying to safely ‘grasp’ a harmonic component (having the wavelength ) by this element size we require that at least five-element length fits the wavelength. This leads to s5 . What is the frequency of this harmonics? Taking a typical wave speed value in steel of about m/s5000c and realizing that cT and Tf /1 , we get the sought-after ‘frequency limit’ in the form )5/( scf . For a one-millimeter element we
get MHz 1 Hz101001.05
5000 6
f . Let’s call it the five-element frequency, denoting
it elem5f in the text.
Bathe recommends 10 here
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 have
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
All considered material properties within the observed infinitesimal element are identical with those of a specimen of finite size
10 MHz
10 kHz
Back to our example
Observing the original (or raw) FE signal in Fig. 4 we may notice its three significant characteristics. First, the negative peak representing the input rectangular pulse, as it was changed on its way from the loading face of the tube to the measurement location; second, the slow frequency variation of the tail of the signal and finally the high frequency components superimposed on the signal everywhere. To estimate the low frequencies, appearing in the signal, let’s consider the lowest frequency of the unsupported infinitely long thin shell of the radius r . In [21] there is
derived the formula)21)(1(
1
2
1
E
rf , which when applied to our case gives
the value of 93 kHz. Due to the corresponding mode of vibration, let’s call this frequency the lowest breathing frequency.
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
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
axia
l str
ain
time
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
experiment
FE analysis
experiment
FE analysis
experiment
FE analysis
The first eigenfrequency of an infinitely long thin-walled tubeBreathing frequency
.
The tube has the thickness h, which is substantially smaller than its radius r. The material constants are Young’s modulus E, density and the Poisson’s ratio . Assuming that the symmetry, radial and circumferential axes are denoted x, y and z respectively, we can write
zz
xy
yy
xx
zz
xy
yy
xx
E
10
02/)21(00
01
01
)21)(1(
Let’s assume that in this case only radial displacement yu is non-zero.
Assume the free body diagram of the mass element whose elementary mass is zrhm ddd . The radial displacement evokes stresses zz acting on both sides of the
element having the surface area hdz. The corresponding elementary forces are zhP zz dd . The equation of motion for the radial direction with the resultant of P-forces is
0)2/dsin(d2d Pum y .
Furthermore
.2/d)2/dsin(;;)21)(1(
)1(
r
uE yzzzzzz
Assuming the harmonic motion in the form t
y Uu ie and its second time derivative,
substituting this and the previous relations into the equation of motion, and rearranging we get
)21)(1(
11
E
r.
Zig-zag frequencyThe faster frequency appearing in time distributions of displacements and strains is called the zig-zag frequency in the text.
For the zig-zag frequency estimation let’s pursue the following reasoning. The case we are dealing with is three-dimensional even if its axial dimension is predominant and the thickness of the tube is rather small comparing to its axial length. Also the applied loading is rather mild – meaning that the time length of the pulse is relatively long with respect to time needed for a wave to pass through the overall length of the tube. Still, in reality there is a fully 3D wave motion pattern appearing within the tube cross section that is dutifully detected by the FE modeling we are employing. According to Huygens’ Principle each point on the surface being hit by a wave is a source of two kinds of waves – the longitudinal and transversal (shear) waves, respectively. [3].
transversal,shear,S-wave
primary,longitudinal,P-wave
Von Schmidt wavefront
2D wavefronts, Huygen’s principleMaterial points having been hitby primary wave becomesources of both types of wavesi.e. P and S
dp007
At the beginning of the loading process the frequencies of evoked waves can be crudely estimated the following way. Each type of wave, being emanated from the outer surface, propagates through the tube thickness, is reflected from the inner surface, and hits the outer surface after the time interval
LL
TS
2,
2
c
st
c
st ,
where the tube thickness is denoted by s . The process is repeated. The corresponding estimates of frequencies of S- and L-waves hitting the outer surface are
LL
SS
1,
1
tf
tf .
Considering the given geometry and material properties the numerical values for these frequencies are
MHz93.0,MHz54.0 LS ff .
In the text we will call them zig-zag frequencies with attributes L (for longitudinal waves) and with S (for shear waves) respectively.
Transfer function for mesh1. NM vs.CD. Limit frequencies.C:\miok_2007\spiral_slot_2006_backup_from_2007_10_24\axisym\mesh_refinement_medium_striker_nm\ check_data_10.m, f5
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
radial NMaxial NMradial CDaxial CD
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 106
0
2
4
6x 10
-10
frequency from 0 to Nyquist
transfer function, input vs. rad. disp
NMCDbreathing frequencyzig-zag L frequencyzig-zag S frequencyFE limit frequency consFE limit frequency diag5elem frequency
Transfer functions for different meshes from 0 to NyquistC:\miok_2007\spiral_slot_2006_backup_from_2007_10_24\axisym\mesh_refinement_medium_striker_nm\ check_data_10.m, f13
0 1 2 3 4 5
x 106
0
1
2
3
4
5
6x 10
-10
tra
nsf
er
fun
ctio
n
axisym, corner, rad. disp, mesh1
0 2 4 6 8 10
x 106
0
0.5
1
1.5
2
2.5
3x 10
-5 axisym, corner, rad. disp, mesh2
0 0.5 1 1.5 2
x 107
0
0.5
1
1.5
x 10-5 axisym, corner, rad. disp, mesh3
frequency from 0 to Nyquist [Hz]
tra
nsf
er
fun
ctio
n
0 1 2 3 4
x 107
0
0.2
0.4
0.6
0.8
1
1.2
x 10-5 axisym, corner, rad. disp, mesh4
frequency from 0 to Nyquist [Hz]
NMCDbreathing frequencyzig-zag L frequencyzig-zag S frequencyFE limit frequency consFE limit frequency diag5elem limit frequency
NMCDbreathing frequencyzig-zag L frequencyzig-zag S frequencyFE limit frequency consFE limit frequency diag5elem limit frequency
NMCDbreathing frequencyzig-zag L frequencyzig-zag S frequencyFE limit frequency consFE limit frequency diag5elem limit frequency
NMCDbreathing frequencyzig-zag L frequencyzig-zag S frequencyFE limit frequency consFE limit frequency diag5elem limit frequency
Transfer functions for different meshes from 0 to 2 MHzC:\miok_2007\spiral_slot_2006_backup_from_2007_10_24\axisym\mesh_refinement_medium_striker_nm\ check_data_10.m, f14
0 0.5 1 1.5 2
x 106
0
1
2
3
4x 10
-10
tra
nsf
er
fun
ctio
n
axisym, corner, rad. disp, mesh1
0 0.5 1 1.5 2
x 106
0
0.2
0.4
0.6
0.8
1x 10
-5 axisym, corner, rad. disp, mesh2
0 0.5 1 1.5 2
x 106
0
0.2
0.4
0.6
0.8
1x 10
-5 axisym, corner, rad. disp, mesh3
frequency range from 0 to 2 MHz
tra
nsf
er
fun
ctio
n
0 0.5 1 1.5 2
x 106
0
0.2
0.4
0.6
0.8
1x 10
-5 axisym, corner, rad. disp, mesh1
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
Observing Fig. 7 one should notice the subsequent ‘convergence’ of CD and NM peaks within the zig-zag frequency interval. The natural explanation is that with the finer meshsize, and with the correspondingly smaller timestep, both methods operate in ‘good’ frequency intervals where their spatial and temporal discretization errors are minimized.
The presented transfer spectra for four studied meshes show a distinct indication of the breathing and zig-zag frequencies, the ‘convergence’ of CD and NM responses, subsequent disappearance of ‘false’ CD responses and that the ‘dubious’ CD frequency peaks do not have their counterparts in NM
responses.
What remains to be compared is the ‘raw’ FE signal with that the frequencies higher than the five-element frequency were filtered out. The results for the ‘raw’ and filtered FE signals, for the mesh1 and the NM operator with consistent mass matrix, are presented in Fig. 8.
FE raw signal compared to that in which the frequencies higher than five-element ones were filtered out.
C:\spiral_slot_2006_backup_from_2007_10_24\1_u_2007_december_changed_data\u_2007_data_from_urmas_march_2007\ToMila\plot_urmas_data_march_2007_c1.m, f10
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
x 10-4
-4
-3
-2
-1
0
1x 10
-4 LOC1 - FE 3D, mesh1, NM raw signal vs. filtered to FE 5elem cutoff
time [s]
axi
al s
tra
in
FE analysis, filtered to 5elem cutoffFE analysis, raw data
In future these results might be confirmed by a new and more sophisticated experiment having a lower observational threshold, a higher sampling rate and also a higher frequency amplifier cut-off .
Conclusions
The FE analysis is a robust tool giving reliable results with a satisfactory engineering precision in standard tasks of continuum mechanics.
Nevertheless, employing the FE method in cases on borders of their applicability is tricky and obtained results have to be treated with utmost care, since they might be profoundly influenced by intricacies of finite element technology.
It should be emphasized, however, that testing the methods in the vicinity of borders of their applicability we do not want to discredit them, on the contrary, the more precise knowledge of their imperfections makes us – users – more confident in them.