SPECTRAL FINITE ELEMENT METHOD

Transcription

1 SPECTRAL FINITE ELEMENT METHOD Originally proposed by Patera in 1984 for problems in fluid dynamics Adopted for problems of propagation of acoustic and seismic waves Snapshot of the propagation of seismic waves in the Earth during the December 26, 2004 Sumatra Island earthquake (S. Tsuboi, N. Takeuchi) Recently used for problems of propagation of elastic wave in structural elements Damage modelling techniques can be adopted easily from FEM (Krawczuk, Ostachowicz)

2 SPECTRAL SHELL FINITE ELEMENT

3 SPECTRAL SHELL FINITE ELEMENT (1/3) Element nodes at Gauss-Lobatto-Legendre (GLL) points:

4 SPECTRAL SHELL FINITE ELEMENT (2/3) Lagrange interpolation polynomials used as shape functions Shape functions are orthogonal in a discrete sense: The form of the displacement field: The form of the strain field according to the FSDT

5 SPECTRAL SHELL FINITE ELEMENT (3/3) Characteristic mass and stiffness matrices of the element

6 2D Spectral Finite Element Shape Functions No. 30 No. 36 No. 15 No. 35

7 Integration of the wave equation MU ( t) KU( t) F( t) Orthogonality of the shape functions Diagonal mass matrix! Integration in time domain: central difference scheme

8 Idea of Spectral Finite Element Method Idea: Write the equation of motion in a weak form Use the spectral element approximation Use the Gauss-Lobatto-Legendre integration rule Form the stiffness and mass matrices Assembly the element equations Use the central difference method for time integration

9 Kudela P., Ostachowicz W.: A multilayer delaminated composite beam and plate elements: reflections of Lamb waves at delamination. 3rd ECCOMAS Thematic Conference Smart Materials and Structures. 9th-11th July 2007, Gdansk, Poland Higher order theory u(x,z)= u0(x)+ (x) z w(x,z)=w0(x)+ (x) z K I = K II + K III Delaminated composite beam

10 Delaminated composite plate Mindlin s plate theory homogenization 36 nodes 5 degrees of freedom per node 36 node plate spectral finite element Kudela P., Ostachowicz W.: A multilayer delaminated composite beam and plate elements: reflections of Lamb waves at delamination. 3rd ECCOMAS Thematic Conference Smart Materials and Structures. 9th-11th July 2007, Gdansk, Poland Kudela P., Ostachowicz W.: A multilayer delaminated composite beam and plate elements: reflections of Lamb waves at delamination, Mechanics of Advanced Materials and Structures, Vol. 16, No. 3, 2009, pp

11 Mesh generation (1) No. of nodes > The small size of the element = small integration time step! Delaminated region Nonuniform mesh of automatically generated quad elements

12 Mesh generation (2) Delaminated region Mesh with quadratic approximation of the geometry of delaminated region

13 Beam - longitudinal waves

14 Beam - transverse waves

15 Numerical simulation Composite plate [0] 1, vol=20% Glass-epoxy Graphite-epoxy

16 Numerical simulation Composite plate [+45/-45] 5, vol=20% Glass-epoxy Graphite-epoxy

17 Numerical simulation Graphite/epoxy composite plate [+45/-45] 1, vol=30%

18 Mindlin-Reisner theory Materials Epoxy-glass E m =3.43, E f =66.5 [GPa] Epoxy-graphite E m =3.43, E f =275.6 [GPa] Analysed cases Wave propagation in composite plates One-layered plate Multilayered plate Investigated parameters which influence on wave propagation Orientation angle of reinforcing fibres [ ] Volume fraction of reinforcing fibres vol [%]

19 Composite plate with cracks (1) 40 mm length (2) 20 mm length Damage scenario 1 Damage scenario composite plate [+45/-45] 5, vol=50% plate dimensions: 1m x 1m x 1 cm

20 Interaction with damage Graphite-epoxy composite plate [+45/-45] 5, vol=50% Damage scenario 1 Damage scenario 2 Signal 33 khz 3 cycles Hanning Signal 48 khz 3 cycles Hanning

21 Group velocity surface as a function of propagation angle 50% fibres volume fraction Material Young s modulus Epoxy Glass fibres Glass/epoxy composite plate [0/90] 5, A0 wave Graphite fibres E=3.43 GPa E=66.5 GPa E=275.6 GPa Graphite/epoxy composite plate [0/90] 5, A0 wave

22 Glass/epoxy composite plate [0] 1, A0 wave anisotropic isotropic (epoxy) Material Epoxy Glass fibres Graphite fibres Young s modulus E=3.43 GPa E=66.5 GPa E=275.6 GPa Group velocity surface as a function of propagation angle for different volumes of reinforcement (0,10,20,30,40,50%)

23 Group velocity surfaces in multilayered plate composite plate [+45/-45] 5, vol=30% Glass-epoxy Units [m/s] Graphite-epoxy

24 Group velocity surfaces composite plate [0] 1, vol=0,20,40,60,80,100% Glass-epoxy Graphite-epoxy Units [m/s]

25 Graphite/epoxy composite plate [0] 1, A0 wave anisotropic isotropic (epoxy) Material Epoxy Glass fibres Graphite fibres Young s modulus E=3.43 GPa E=66.5 GPa E=275.6 GPa Group velocity surface as a function of propagation angle for different volumes of reinforcement (0,10,20,30,40,50%)

26 Glass/epoxy composite panel [0/90] 5, S & P waves P-wave (longitudinal) S-wave (shear) Material Epoxy Glass fibres Graphite fibres Young s modulus E=3.43 GPa E=66.5 GPa E=275.6 GPa Group velocity surface as a function of propagation angle 50% fibres volume fraction

27 Graphite/epoxy composite panel [0/90] 5, S & P waves P-wave (longitudinal) S-wave (shear) Material Epoxy Glass fibres Graphite fibres Young s modulus E=3.43 GPa E=66.5 GPa E=275.6 GPa Group velocity surface as a function of propagation angle 50% fibres volume fraction

28 Glass/epoxy composite panel [0] 1, P wave anisotropic isotropic (epoxy) Material Epoxy Glass fibres Graphite fibres Young s modulus E=3.43 GPa E=66.5 GPa E=275.6 GPa Group velocity surface as a function of propagation angle for different volumes of reinforcement (0,10,20,30,40,50%)

29 Glass/epoxy composite panel [0] 1, S wave anisotropic isotropic (epoxy) Material Epoxy Glass fibres Graphite fibres Young s modulus E=3.43 GPa E=66.5 GPa E=275.6 GPa Group velocity surface as a function of propagation angle for different volumes of reinforcement (0,10,20,30,40,50%)

30 Traveling wave in delaminated plate A0 mode Graphite-epoxy composite plate 1m x 1m x 1cm [+45/-45] 5 vol=50% The region of delamination - ellipse of the size a=16mm and b=8mm

31 Traveling wave in cracked plate Graphite-epoxy composite plate [+45/-45] 5, vol=20% 1 cm long crack through thickness of the plate

32 DAMAGE DETECTION ALGORITHMS (2/6) Propagation of out-of-plane waves in a pipe section with no stiffener concept of a clock-like PZT sensor array

33 DAMAGE DETECTION ALGORITHMS (3/6) Propagation of out-of-plane waves in a pipe section with no stiffener concept of a clock-like PZT sensor array

34 DAMAGE DETECTION ALGORITHMS (4/6) Propagation of out-of-plane waves in a pipe section with no stiffener concept of a clock-like PZT sensor array damage influence map

35 Signal processing

36 Experimental signals

37 Experiment: results Clock diameter 4.4 cm Clock diameter 8 cm

38 Damage detection algorithm

39 Damage detection algorithm Damage influence maps based on damage state signals only crack #1 crack #2 and crack #3

40 Damage detection algorithm Damage influence maps based on damage state signals only crack #1 crack #2 and crack #3

41 Damage detection algorithm Damage influence maps based on differential signals crack #1 crack #2 and crack #3

42 Motivation (3D Models)

43 Lamb dispersion curves (aluminium) Symmetric modes (red) Antisymmetric modes (blue)

44 Lamb waves approximation Lamb waves can be modelled assuming an appropriate displacement field, which approximate accurately mode shapes of particle displacement through the thickness of the solid Symmetric modes (red) Antisymmetric modes (blue) S0 A1 poor approximation Midlin theory of plates A0

45 Spectral plate element S0 SH0 A0 Group velocities: theory vs Mindlin s model (thin black lines)

46 MU ( t) KU( t) F( t) Orthogonality of the shape functions Diagonal mass matrix! True for Mindlin s theory and symmetric laminates and 3D models Integration in time domain: central difference scheme Integration of the wave equation

47 Mindlin s plate theory 5 degrees of freedom per node Transverse displacement In-plane displacements 2 independent rotations homogenization 36 nodes Spectral plate element

48 3D Spectral Element 216 nodes 3 mechanical degrees of freedom per node - displacements u, v, w one electrical degree of freedom Typical size of the task for the plate of dimensions 200x200x1mm and excitation frequency 100 khz: 1 million degrees of freedom!

49 Numerical example (1) Aluminium plate of the thickness 1 mm with excitation point 1 and response points 1-5

50 Numerical example (2) Response of the plate at the centre of the plate (point 1) for excitation frequency 100 khz and 200 khz

51 Numerical example (3) Response of the plate at point 3 of the plate for excitation frequency 100 khz and 200 khz

52 Numerical example (4) Displacement distribution along the thickness of the plate - Lamb wave theory Displacement distribution along the thickness of the plate for 2D and 3D models

53 Lamb wave-based damage detection in composite structures: potentials and limitations

54 Wave modes in composite laminates Group velocity surface of [+45/ 45/0/90/+45/ 45] carbon/epoxy laminate

55 Wave mode conversion, reflection

56 Mode conversion at delamination Note: A0 mode is not only reflected at delamination but also converted into faster S0 mode

57 Influence of environmental conditions on propagating waves Temperature, initial stresses, moisture, icing Free thermal expansion of a structure + wave velocity dependence on temperature Lu, Y. and J.E. Michaels A methodology for structural health monitoring with diffuse ultrasonic waves in the presence of temperature variations, Ultrasonics, 43: Konstantinidis, G., P.D. Wilcox and B.W. Drinkwater An Investigation Into the Temperature Stability of a Guided Wave Structural Health Monitoring System Using Permanently Attached Sensors, IEEE Sens. J., 7(5): Stiffness reduction of the structure due to temperature field = velocity changes Kudela, P., W. Ostachowicz and A. Żak Influence of temperature fields on wave propagation in composite plates, in Key Engineering Materials, L. Garibaldi, C. Surace, K. Holford and W.M. Ostachowicz, TTP, pp

58 Temperature effect compensation Free thermal expansion of a structure + wave velocity dependence on temperature Optimal Baseline Subtraction Collect a set of baseline signals recorded over full range of temperature conditions Use best match baseline for subtraction Optimal Stretch Method Collect a set of baseline signals at a known temperature Predict effect of temperature variation (stretch frequency axis of signal spectrum)

59 Temperature field effect Stiffness reduction of the structure due to temperature field = velocity changes metallic alloys composite laminates Young s modulus as function of temperature

60 Temperature field effect glass-epoxy laminate [+45/-45] 5 changes in the Young s modulus up to 40% Temperature field Note: propagating wave slows down travelling through heated up region without reflection!

61 Wave attenuation in composites Attenuation reasons: Geometrical wave attenuation (energy conservation) Material damping Leakage (due to surroundings, ie. steel rod in rock) Note: damping in composite laminates depends on stack sequence so that wave attenuation depends on angle of propagation

62 Wave attenuation in composites ~85% amplitude reduction after 400 mm!!! Relative damping of propagating waves measured experimentally Note: In composites wave packet amplitude drastically decrease with travelled distance due to material damping

63 CONTACT EFFECT AT DELAMINATION Delaminated composite beam Higher order theory u(x,z)= u0(x)+j(x) z w(x,z)=w0(x)+y(x) z K I = K II + K III

64 Solution of equation of motion External forces Ma t t K u P t t F t Internal forces Orthogonality of the shape functions CONTACT EFFECT AT DELAMINATION Diagonal mass matrix! Integration in time domain: implicit Newmark algorithm u Generalized nodal displacement increment found iteratively

65 CONTACT EFFECT AT DELAMINATION Contact modelling technique Discrete frictionless contact model for contacting element nodes. Modified implicit Newmark algorithm. Compatibility conditions at delaminated condition of no penetration between the nodes in contact in the transverse direction conservation of momentum in the same direction When on the time step Δt the penetration for any pair of contacting nodes is detected the algorithm adjusts the current time step and calculates a new time step for which the penetrations vanish and only one pair of contacting nodes stays in contact. Once compatibility conditions are applied calculations continue with the original time step

66 L=1000 mm b=20 mm h=12 mm L 1 =226.6 mm h 1 =6 or h 1 =3 mm a=31.25 mm Property Glass fibres Epoxy resin Young s modulus [GPa] Shear modulus [GPa] Poisson ratio Density [kg/m 3 ] Loss coefficient the relative volume fraction of the reinforcing glass fibres 0.5 ply stacking sequence [±45 ] 6S Numerical examples

67 Low frequency example

68 non-contact model Low frequency example contact model Transverse displacements in the middle of delamination at the upper and lower node of interface (delamination location h 1 /h=0.25, excitation 3kHz in transverse direction)

69 High frequency example

70 non-contact model High frequency example contact model Transverse displacements in the middle of delamination at the upper and lower node of interface (delamination location h 1 /h=0.25, excitation 30kHz in transverse direction)

71 Optimal sensor network for damage localization The aim of this work is development of an algorithm for damage localisation in composite plates based on wave propagation signals registered by sensors

72 Laptop Device for generation and registration of elastic waves The network of piezoelectric transducers The test specimen - an aluminium alloy plate dimensions: 1m x 1m x 0.001m

73 Results True locations of the holes Stronger responses than from the holes

74 Results After removing a part of the response, holes locations are indicated The difference between found and true location of the hole was 3.7 mm for the hole closer to the middle of the specimen, and 4.3 mm for the second one.

75 Experience and space for improvements (1) Clock-like sensor array Damage influence map

76 Dead zones due to edge reflections Experience and space for improvements (2) Contribution to the damage influence map from edge reflections Kudela P., Ostachowicz W., Zak A.: Damage detection in composite plates with embedded PZT transducers. Mechanical Systems and Signal Processing, Vol. 22(2008) pp

77 Experience and space for improvements (3) Dead zones due to high attenuation in composites

78 Sensor configurations E( x, y) 10log 10( E D / ER) D Damage R - Reference pitch-catch and pulse-echo pulse-echo

79 Multi triangular grid method (1) Triangular grids of piezoelements (black box active, empty box - inactive) Damage influence maps for triangular grids given above

80 Multi triangular grid method (2) In the multi triangular method particular damage influence maps are combined into one map by summation crack sum of maps for triangular grids smoothed map by mean average filter

81 Transducers configuration 12 transducers Aluminium (3 alloy 3Transducers 2 mm specimen 3 ) in vertices coordinates 1000of 700 a eqilateral 1 mmtriangle m

82 magnet (2g, Ø 6 mm) Localisation result - scenario 1 Sensing configuration

83 Localisation result - scenario 1 Regions with highest ampification due to edges reflections magnet position Error in position estimation: 48 mm

84 Localisation result - scenario 1 Setting the threshold level to 0.96 confirms that the strongest ampification comes from the magnet region

85 magnet (2g, Ø 6 mm) Localisation result - scenario 2

86 Localisation result - scenario 2 magnet position Error in position estimation: 20 mm

87 Localisation result - scenario 2 Setting the threshold level to 0.97 confirms that the strongest ampification comes from the magnet region

88 Direct path monitoring result magnet (2g, Ø 6 mm) Wave exciting transducer no. Wave registering transducers no. Damage indication value M(P) for magnet inside the triangle in (0.48 m, 0.33 m) Damage indication value M(P) for magnet outside the triangle in (0.28 m, 0.36 m) Damage indication value M(P) for magnet on triangle side in (0.56 m, 0.33 m) 7 1,2,3, ,10,11,

89 Piezoelectric systems a) b) c) d) Four maps with denoted failures for the frequency of excitation: 180 khz. The Figures present four systems of PZT s: a) strip b) circle c) cross d) square stars denote failures

90 Damage detection in helicopter panel

91 Experimental results: damage influence maps crack of the length 10 mm crack depth 0.5 mm through-thickness crack

92 Distributed vs concentrated sensor networks E( x, y) 10log 10( E D / ER) D Damage R - Reference pitch-catch and pulse-echo pulse-echo

93 Experimental results (Rig at IFFM) Waveform Generator: 1, 2 or 4 independent channels sampling frequency to 400 MHz sine waves and square. Waves up to: 16MHz, 12 bit. Piezo Linear Amplifier Model EPA-104 Provides 40 watts peak power from DC to 250kHz. DC Offset Voltage Control.

94 Tests in the IFFM laboratory

95 Tests in the IFFM laboratory

96 Tests in the IFFM laboratory

97 Equipment of the IFFM laboratory

98 Equipment of the IFFM laboratory