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: * ghose.bikash@hemrl.drdo.inghose.bikash@hemrl.drdo.in , # , # balas@iitm.ac.inbalas@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: ghose.bikash@hemrl.drdo.inghose.bikash@hemrl.drdo.in , , balas@iitm.ac.inbalas@iitm.ac.in