Two Dimensional FEM Two Dimensional FEM Simulation of Ultrasonic Wave Simulation of Ultrasonic Wave Propagation in Isotropic Solid Propagation in Isotropic Solid
Media Using COMSOLMedia Using COMSOL®®
BikashBikash GhoseGhose1 *1 *, Krishnan Balasubramaniam, Krishnan Balasubramaniam22 ##
C V KrishnamurthyC V Krishnamurthy33, A , A SubhanandaSubhananda RaoRao11
11 High Energy Materials Research Laboratory, High Energy Materials Research Laboratory, SutarwadiSutarwadi, , PunePune -- 212122 Center for Non Destructive Evaluation, IIT Madras, Chennai Center for Non Destructive Evaluation, IIT Madras, Chennai -- 3636
33 Department of Physics, IIT Madras, Chennai Department of Physics, IIT Madras, Chennai --3636
EE--mail: * mail: * [email protected]@hemrl.drdo.in , # , # [email protected]@iitm.ac.in
Presented at the COMSOL Conference 2010 India
MotivationMotivation�� Application of Ultrasonic wave propagationApplication of Ultrasonic wave propagation
•• Under Water AcousticsUnder Water Acoustics
•• Medical DiagnosticMedical Diagnostic
•• Structural Health Monitoring (SHM)Structural Health Monitoring (SHM)
•• NonNon--Destructive Evaluation (NDE)Destructive Evaluation (NDE)
•• Material Characterization Material Characterization
�� NonNon--Destructive EvaluationDestructive Evaluation•• Bulk wave Bulk wave
�� Longitudinal wave (Pulse echo technique, Through Longitudinal wave (Pulse echo technique, Through transmission etc)transmission etc)
�� Transverse WaveTransverse Wave
•• Surface WaveSurface Wave
•• Guided Wave / Lamb Wave etc.Guided Wave / Lamb Wave etc.
�� Guided WaveGuided Wave•• Many different modes of ultrasonic vibration Many different modes of ultrasonic vibration (Symmetric, Anti symmetric)(Symmetric, Anti symmetric)
�� Numerical Simulation of Wave propagationNumerical Simulation of Wave propagation•• Fluid (Only Longitudinal wave)Fluid (Only Longitudinal wave)
•• Solid (Longitudinal, Transverse, Surface, Lamb Solid (Longitudinal, Transverse, Surface, Lamb etc) etc)
�� Use of COMSOL for simulation of wave Use of COMSOL for simulation of wave propagation in solids propagation in solids
Transient Analysis for Wave Transient Analysis for Wave PropagationPropagation
�� Transient Analysis always pose difficult problemsTransient Analysis always pose difficult problems
�� Length of element (Length of element (∆∆x)x)�� Time steps (Time steps (∆∆t)t)�� Type of Elements (Triangular, Quad)Type of Elements (Triangular, Quad)
�� Integration scheme for integration over timeIntegration scheme for integration over time
�� IssuesIssues
•• InstabilityInstability
•• Numerical DispersionNumerical Dispersion
•• ConvergenceConvergence
Numerical Simulation of Ultrasonic Numerical Simulation of Ultrasonic Wave Propagation using COMSOLWave Propagation using COMSOL®®
Initial Displacement Displacement at diff positions with time
Solution at 7.0×10-5 s Solution at 1.7×10-4 s
Thumb Rule ?•Length of Element (∆x) = λ/12•Time Step = ∆x/Cph
Materials & Ultrasonic Wave Materials & Ultrasonic Wave Properties Properties
�� YoungYoung’’s Modulus (E) = 2 s Modulus (E) = 2 ×× 10101111 PaPa
�� PoissonPoisson’’s ratio (s ratio (νν) = 0.33) = 0.33
�� Density (Density (ρρ) = 7850 kg/m3 ) = 7850 kg/m3
�� Ultrasonic velocity (CUltrasonic velocity (CLL) (longitudinal ) (longitudinal
wave) = 5850 wave) = 5850 m/sm/s
�� Frequency of incident ultrasonic wave (Frequency of incident ultrasonic wave (ff) )
= 20 kHz = 20= 20 kHz = 20××101033 ss--11
�� Wavelength of the longitudinal ultrasonic Wavelength of the longitudinal ultrasonic
wave (wave (λλLL) = C) = C
LL//ff = 0.2925 m= 0.2925 m
�� Source Length : 0.04 m (40 mm) Source Length : 0.04 m (40 mm)
(located at the middle)(located at the middle)
�� Source Excitation : On a line and points Source Excitation : On a line and points
on lineon line
�� Simulation Domain : 5.0 m (L) x 2.5 m Simulation Domain : 5.0 m (L) x 2.5 m
(H)(H)
0 0.5 1 1.5
x 10-4
-10
-8
-6
-4
-2
0
2
4
6
8x 10
-7
Time (s)
Dis
plac
emen
t (m
)0 2 4 6 8 10 12 14 16 18 20
x 104
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
-7 Single-Sided Amplitude Spectrum of y(t)(Hanning windowed signal)
Frequency (Hz)|Y
(f)|
Meshing and Application ModesMeshing and Application Modes
�� Triangular ElementsTriangular Elements
�� Elements Automatically Elements Automatically
GeneratedGenerated
�� Approximation : Plane StainApproximation : Plane Stain
�� Element Type : Lagrange Element Type : Lagrange
QuadraticQuadratic
�� Analysis : TransientAnalysis : Transient
�� Solver : Time DependentSolver : Time Dependent
Effect of Length of Element (Effect of Length of Element (∆∆x)x)
The time steps chosen initially for simulation for different ∆xmax
is 2.5×10-6 s
ssm
m
C
x
C
xt
Lphcritical
6maxmax 101.3/5850
0182.0 −×==∆=∆=∆
As per CFL Criteria
-5.00E-08
-4.00E-08
-3.00E-08
-2.00E-08
-1.00E-08
0.00E+00
1.00E-08
2.00E-08
3.00E-08
4.00E-08
5.00E-08
0 0.0001 0.0002 0.0003 0.0004 0.0005
Time (s)
Dis
plac
emen
t (m
)
Ratio = 2Ratio = 3Ratio = 4Ratio = 5Ratio = 8Ratio = 9Ratio = 12Ratio = 16
-8.00E-08
-6.00E-08
-4.00E-08
-2.00E-08
0.00E+00
2.00E-08
4.00E-08
6.00E-08
8.00E-08
1.00E-07
0 0.0001 0.0002 0.0003 0.0004 0.0005
Time (s)
Dis
plac
emen
ts (
m)
Ratio = 2Ratio = 3Ratio = 4Ratio= 5Ratio = 8Ratio = 9Ratio = 12Ratio = 16
Onward Propagating Wave at various Onward Propagating Wave at various Distances for Different Distances for Different λλλλλλλλLL//∆∆∆∆∆∆∆∆xxmaxmax at at t = 4t = 4××1010--44 ss
-6.00E-08
-4.00E-08
-2.00E-08
0.00E+00
2.00E-08
4.00E-08
6.00E-08
0 0.0001 0.0002 0.0003 0.0004 0.0005
Time (t)
Dis
plac
emen
t (m
)
Ratio = 2Ratio = 3Ratio = 4Ratio = 5Ratio = 8Ratio = 9Ratio = 12Ratio = 16
-4.00E-08
-3.00E-08
-2.00E-08
-1.00E-08
0.00E+00
1.00E-08
2.00E-08
3.00E-08
4.00E-08
0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003 0.00035 0.0004 0.00045
Time (t)
Dis
plac
emen
t
Ratio = 2Ratio = 3Ratio = 4Ratio = 5Ratio = 8Ratio = 9Ratio = 12Ratio = 16
At 0.3 m At 0.5 m
At 0.8 m At 1.0 m
• Oscillations remains after passing by of signal• Convergence of the solution for the ratio (λ
L/∆x
max) ≥ 8
• Time steps are taken from solver or exactly what has been given does not make any difference to the propagating signal
Onward Propagating Wave for Different Onward Propagating Wave for Different λλλλλλλλLL//∆∆∆∆∆∆∆∆xxmaxmax at at t = 4t = 4××1010--44 ss
-2.50E-08
-2.00E-08
-1.50E-08
-1.00E-08
-5.00E-09
0.00E+00
5.00E-09
1.00E-08
1.50E-08
2.00E-08
2.50E-08
0 0.5 1 1.5 2 2.5 3
Distance from Source center (m)
Dis
plac
emen
t (m
)Ratio = 2Ratio = 3ratio = 4ratio = 5Ratio = 8Ratio = 9Ratio = 12
Line profile (Displacement Vs Distance from source) at 0.0004s for different wavelength to element length ratio
Effect of Time Steps (Effect of Time Steps ( ∆∆∆∆∆∆∆∆t)t)�� As per CFL criteria the time steps should be less than As per CFL criteria the time steps should be less than ∆∆x/Cx/C
LL
for example, for the maximum element length of for example, for the maximum element length of λλLL/10/10 the the
time steps should be time steps should be ≤≤ ((λλLL/10/10××CC
LL)) that means ifthat means if
�� Signal so far obtained is not as expectedSignal so far obtained is not as expected
�� Major change in the shape of signalMajor change in the shape of signal
�� Time steps further decreased and solution was checked for Time steps further decreased and solution was checked for
convergenceconvergence
ft
CC
xt
x
L
L
L
L
×≤∆⇒
×=∆≤∆⇒
=∆
101
10
10
max
max
λ
λ
Case: I Case: I λλLL//∆∆xxmaxmax= 5= 5st 6105 −×=∆ st 6100.2 −×=∆
st 6100.1 −×=∆ st 6105.0 −×=∆ st 6102.0 −×=∆
st 61010 −×=∆
Case: II Case: II λλLL//∆∆xxmaxmax= 8= 8st 61026.6 −×=∆ st 6105 −×=∆ st 6100.2 −×=∆
st 6100.1 −×=∆ st 6105.0 −×=∆ st 6102.0 −×=∆
Case: III Case: III λλLL//∆∆xxmaxmax= 12= 12st 6105 −×=∆
st 6100.1 −×=∆ st 6105.0 −×=∆
st 6105.2 −×=∆
Effect of Excitation on Line and Points on Effect of Excitation on Line and Points on Line for Line for λλLL//∆∆xxmaxmax= 8 and = 8 and ∆∆t = 2.5x10t = 2.5x10--66 ss
(Line source) (Points on Line source)At 1.0 m At 1.0 m
At 0.5 m At 0.5 m
Effect of Excitation on Line and Points on Effect of Excitation on Line and Points on Line for Line for λλLL//∆∆xxmaxmax= 8 and = 8 and ∆∆t = 2.5x10t = 2.5x10--66 ss
(Line source) (Points on Line source)
Line profile at t = 4x10-4 s
Time and Frequency Domain Signal for Time and Frequency Domain Signal for Forward Propagating WaveForward Propagating Wave
-1.20E-07
-8.00E-08
-4.00E-08
0.00E+00
4.00E-08
8.00E-08
0.00E+00 2.00E-04 4.00E-04 6.00E-04
Time (s)
Dis
plac
emen
t (m
)
1 2 3 4 5 6 7 8 9
x 104
0
1
2
3
4
5
6
7
8
9
10x 10
-9 Single-Sided Amplitude Spectrum
Frequency (Hz)
|Y(f
)|
-1.20E-07
-8.00E-08
-4.00E-08
0.00E+00
4.00E-08
8.00E-08
0.00E+00 2.00E-04 4.00E-04 6.00E-04
Time (s)
Dis
plac
emen
t (m
)
1 2 3 4 5 6 7 8 9
x 104
2
4
6
8
10
12
14x 10
-9 Single-Sided Amplitude Spectrum
Frequency (Hz)
|Y(f
)|-1.20E-07
-8.00E-08
-4.00E-08
0.00E+00
4.00E-08
8.00E-08
0.00E+00 2.00E-04 4.00E-04 6.00E-04
Time (s)
Dis
plac
emen
t (m
)
1 2 3 4 5 6 7 8 9
x 104
2
4
6
8
10
12
x 10-9 Single-Sided Amplitude Spectrum
Frequency (Hz)
|Y(f
)|
12max
=∆x
Lλ8
max
=∆x
Lλ
All signals are for All signals are for ∆∆t =0.5x10t =0.5x10--66 ss
5max
=∆x
Lλ
Time and Frequency Domain Signal for Time and Frequency Domain Signal for Back Wall Reflected WaveBack Wall Reflected Wave
-6.00E-08
-4.00E-08
-2.00E-08
0.00E+00
2.00E-08
4.00E-08
0.00E+00 4.00E-04 8.00E-04 1.20E-03
Time (s)
Dis
plac
emen
t (m
)
0 1 2 3 4 5 6 7 8 9
x 104
1
2
3
4
5
6
7x 10
-9 Single-Sided Amplitude Spectrum
Frequency (Hz)
|Y(f
)|
-6.00E-08
-4.00E-08
-2.00E-08
0.00E+00
2.00E-08
4.00E-08
0.00E+00 4.00E-04 8.00E-04 1.20E-03
Time (s)D
ispl
acem
ent (
m)
1 2 3 4 5 6 7 8 9
x 104
1
2
3
4
5
6
7
8x 10
-9 Single-Sided Amplitude Spectrum
Frequency (Hz)
|Y(f
)|-6.00E-08
-4.00E-08
-2.00E-08
0.00E+00
2.00E-08
4.00E-08
0.00E+00 4.00E-04 8.00E-04 1.20E-03
Time (s)
Dis
plac
emen
t (m
)
1 2 3 4 5 6 7 8 9
x 104
1
2
3
4
5
6
7
8x 10-9 Single-Sided Amplitude Spectrum
Frequency (Hz)
|Y(f
)|
All signals are for All signals are for ∆∆t =0.5x10t =0.5x10--66 ss
12max
=∆x
Lλ8
max
=∆x
Lλ5
max
=∆x
Lλ
Simulation Results for Simulation Results for λλLL//∆∆xxmaxmax= 8 = 8 ∆∆t = 0.5 x 10t = 0.5 x 10--6 6 s s at different timeat different time
At t = 1x10At t = 1x10--44 ss
Simulation Results for Simulation Results for λλLL//∆∆xxmaxmax= 8 = 8 ∆∆t = 0.5 x 10t = 0.5 x 10--6 6 s s at different timeat different time
At t = 4x10At t = 4x10--44 ss
Simulation Results for Simulation Results for λλLL//∆∆xxmaxmax= 8 = 8 ∆∆t = 0.5 x 10t = 0.5 x 10--6 6 s s at different timeat different time
At t = 5x10At t = 5x10--44 ss
Simulation Results for Simulation Results for λλLL//∆∆xxmaxmax= 8 = 8 ∆∆t = 0.5 x 10t = 0.5 x 10--6 6 s s at different timeat different time
At t = 8x10At t = 8x10--44 ss
ConclusionsConclusions�� Wave Propagation can well be modeled using Wave Propagation can well be modeled using
COMSOL with excitation on line segment and / or COMSOL with excitation on line segment and / or
on points on line (Difference only in the on points on line (Difference only in the
maximum displacements)maximum displacements)
�� λλLL//∆∆xxmaxmax ≥≥ 88 irrespective of any time steps less irrespective of any time steps less
than calculated by CFL criteria ( For triangular than calculated by CFL criteria ( For triangular
free meshing)free meshing)
�� Time step Time step ∆∆tt should be about T/100 even if should be about T/100 even if λλLL//∆∆x x
≈≈ 1616
�� No substantial difference in frequency content of No substantial difference in frequency content of
forward as well as back wall reflected ultrasonic forward as well as back wall reflected ultrasonic
signal observedsignal observed
EE--mail: mail: [email protected]@hemrl.drdo.in , , [email protected]@iitm.ac.in
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