O metod ě konečných prvků Lect_ 1 5

1. O metodě konečných prvkůLect_15.ppt FFT and FEM M. Okrouhlík Ústav termomechaniky, AV ČR, Praha Plzeň, 2010

2. Obsah • Terminologie • Fourierova řada • CFT, DFT a FFT • Pár příkladů • Mudrování o platnosti MKP výsledků

3. TerminologieFourierova řada, CFT, DFT a FFT • Fourier series • Continuous Fourier Transform • Discrete Fourier Transform • Fast Fourier Transform

5. Fourierova řada, 20 členůT = Tmax = n * Timp, n = 2,4,8,16

7. Obdélníkový puls – CFTtři obdélníkové pulsy různé délky – různá normalizace

8. FFT je způsob výpočtu DFT

9. Základní vztahy_1 Sampling rate, vzorovací frekvence

11. Základní vztahy_2 Je to vztah mezi číselnými hodnotami Pro diskusi o významu termínu power spectrum jsou následující důležité vztahy mean(y) = sum(y)/N Area = tmax*sum(y)/(N–1) = tmax*mean(y)*N/(N-1) ~ tmax*mean(y) první členy řad Y a pY Y(1) = mean(y) * N pY(1) = mean(y) * mean(y) * N

12. Základní vztahy_3

13. Obdélníkový pulsFFT(Fast Fourier Transform) a MPS(Matlab Power Spectrum)

14. Význam členů y(1) a Y(1)

15. Normalization of FFT data

16. Základní vztahy_4

17. Základní vztahy_5

18. Nyquistova frekvence závisí na tom, zda počet vzorků je sudý či lichý. Je-li počet vzorků velký, rozdíl nestojí za řeč.

19. 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

20. Obdélníkový puls – CFTtři obdélníkové pulsy různé délky – různá normalizace

21. CFT vs. FFTthree different rectangular pulses, the same sampling and the same window

22. CFT vs. FFTthe same rectangular pulse, the same sampling and three different windows

23. Fyzikální význam Momentum - hybnost Pokud je tedy vstupní veličinou síla, pak power spectrum (výkonové) by se mělo nazývat momentum (hybnostní) spectrum

24. Four different input pulses and their average

27. Tube with four spiral slots, coarse mesh, dimensions Four different meshes

28. 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

29. Comparison of filtered 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, f5 FE data were filtered by a filter having the same upper frequency as the experiment sampling frequency

30. 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.

31. Rule of thump1mm corresponds to 1 MHz Easy to remember Bathe recommends 10 here

32. 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

33. Limits of continuum, FE analysis and experiment 10 MHz 10 kHz All considered material properties within the observed infinitesimal element are identical with those of a specimen of finite size

34. Back to our example

35. The first eigenfrequency of an infinitely long thin-walled tubeBreathing frequency .

37. Zig-zag frequency

39. 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

40. 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

41. 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

43. 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 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 .

44. 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.