Two Dimensional FEM Simulation of Ultrasonic Wave Propagation in Isotropic Solid Media using COMSOL

Transcription

1 Excerpt from the Proceeding of the COMSO Conference 0 India Two Dimenional FEM Simulation of Ultraonic Wave Propagation in Iotropic Solid Media uing COMSO Bikah Ghoe *, Krihnan Balaubramaniam *, C V Krihnamurthy 3, A Subhananda Rao High Energy Material Reearch aboratory, Sutarwadi, Pune, India PIN Center for Non-Detructive Evaluation, Dept of Mechanical Engg, IIT, Madra, Chennai Department of Phyic, IIT Madra, Chennai 3 * Author for correpondence: ghoe.bikah@hemrl.drdo.in, bala@iitm.ac.in Abtract: Ultraonic wave propagation in olid can be modeled uing Finite Element Method (FEM) that help in undertanding of the interaction of wave with material. The FEM ue variou parameter which need to be optimized properly to obtain the olution cloer to the exact one. A two dimenional FEM model i deigned in COMSO and ha been imulated for ultraonic wave propagation in an iotropic olid media to optimize the FEM parameter for getting cloer olution. The change in the hape of incident wave, amplitude and frequency with repect to the change in length of element and time tep ha been tudied. It wa oberved that free mehing provided in COMSO with triangular element i capable of imulating the wave propagation for the ratio of wavelength to the element length of more than 8. The time tep required for the imulation to obtain proper reult hould be better than that i merely atified by CF criteria. Keyword: Ultraonic, CF Criteria, Element length, time tep, FEM. Introduction Ultraonic Teting (UT) i one of the important Non-Detructive Evaluation (NDE) technique widely ued for characteriation of material a well a detection and characteriation of flaw preent in the material ued in variou indutrie. Undertanding of ultraonic wave propagation and it capability for flaw detection i an important apect for proper evaluation of wave characteritic for characteriation of material or detection of flaw. The ultraonic wave propagation can be imulated uing Finite Element Method (FEM). The imulation parameter ued in FEM play very important role for the correctne of the obtained data. Thi paper preent the Finite Element Method (FEM) imulation of ultraonic wave propagation in iotropic olid media uing COMSO. The ultraonic wave characteritic like hape of wave, frequency content, and wave amplitude of the onward and reflected propagated wave in material with repect to important imulation parameter like time tep, element length ha been tudied for getting proper olution.. Two Dimenional FEM Model of ultraonic wave propagation A two-dimenional finite element model for propagation of ultraonic wave in an iotropic olid media ha been tudied with COMSO.. Material and ultraonic wave propertie The material choen wa teel with following propertie Young Modulu (E) Pa Poion ratio (ν) 0.33 Denity (ρ) 7850 kg/m 3 Ultraonic velocity (C ) (longitudinal wave) 5850 m/ Wavelength of the longitudinal ultraonic wave (λ ) C / m. Incident ultraonic wave A 0 khz frequency ignal ha been applied to the iotropic material. The ignal i choen a the 3 cycle of coine function and operated with a hanning window. The applied diturbance a the input ignal i applied to a line of length 0.0 m which mean the tranducer ha been modeled a the diturbance on a line. The initial diturbance i given a the diplacement in the ve y direction only for modeling the normal incidence of ultraonic wave. The input ignal i provided from a data file and the linear interpolation i conidered for the value of diplacement at the undefined time in the data file. The time domain repreentation of input diplacement pule ued in the model i hown in the figure. The frequency content of the input ignal wa evaluated a the Fourier tranform of the time domain ignal and i a hown in figure.

2 Diplacement (m) 8 x Time () x - Figure : Input diplacement pule a ource excitation x -7 Single-Sided Amplitude Spectrum of y(t)(hanning windowed ignal) x Figure : Input ignal in frequency domain.3 Model Geometry A the tudy wa mainly focued on the bulk longitudinal wave propagation, a imple rectangular geometry ha been choen and the dimenion of the geometry wa choen uch that no ide wall reflection reache to the point of obervation within the pecified time. The initial diturbance wa applied on a line of length 0.0 m located at the centre of the top horizontal line. For tudy of point ource a the model for the tranducer, nine number of equally paced point of initial diturbance are located within the line length of 0.0 m which wa earlier conidered a the line ource. A all the obervation were made for the longitudinal wave only the ignal wa oberved on the perpendicular line that pae through the middle of the ource.. Mehing The meh wa generated automatically with the triangular element. Free mehing provided in COMSO wa ued for the generation of meh. The element were choen a uniform through out the ub-domain. The meh ha been refined at the region below the excitation ource. Minimum meh quality that ha been enured for each of the model that ha been olved i 0.7 with an element area ratio of about Application Mode For modeling of the two dimenional geometry, the cae of plane train wa conidered becaue of the dimenion of the material conidered to be large enough in the third dimenion a compared to the x and y direction. Time dependent analyi, agrange quadratic type of element and time dependent olver wa ued for the olution. The duration of time pan for the olution wa choen uch that back-wall reflected ignal come back to the front wall or from where excitation tarted. All the imulation were carried out in the -bit XP environment. 3. Simulation Reult for Effect of Maximum ength of Element and Time Step 3. Effect of Maximum ength of Element The length of element in the meh ued for olution of any FEM model play a crucial role for correctne of the obtained olution. The length of the element need to be maller for evaluation of proper olution wherea a the length of element decreae the cot of computation increae. So it i alway neceary to evaluate the optimum element length for correct olution along with the leer computational effort. In the current tudy, the maximum element ize (x max ) wa varied from λ / to λ / and the model wa imulated for the ultraonic wave propagation. λ i the wavelength of the longitudinal ultraonic wave propagated in the direction of incidence. Following table how the different value of λ for which the x max imulation wa carried out along with the correponding maximum element ize for each cae. The time tep choen initially for imulation for different x max i.5 -. The reaon behind Ratio of Wavelength to x max (λ /x max ) Maximum element ize (x max ) (m)

3 the choen time tep i a follow. A per CF criteria, the critical time tep to be ued for imulation on FEM model for a time dependent olver i, where C ph i the x C ph ultraonic phae velocity. A only longitudinal ultraonic wave wa conidered for thi tudy, the C ph value wa taken a the longitudinal ultraonic velocity in the teel material which i about 5850 m/. In thi regard, for different x max value different t hould be evaluated which i to be ued for olution. The critical time t would be mallet for the mallet value of x max. If the mallet x max wa choen for evaluating t then the t mut be ufficiently maller for other cae. Hence t wa evaluated for the leat value of x max. The leat value of x max i 0.08 m and correpondingly the critical value of t wa calculated to be 0.08m tcritical 3. C C 5850m / ph The value of t ued in the imulation wa.5 - which i le than the critical time tep for the cae of λ /x max. The ultraonic ignal were plotted at different perpendicular ditance from the middle of line ource for different value of λ at t x max -. A cro ectional plot (line profile for the plot of diplacement V ditance or poition) wa alo plotted at t -. The time t - i choen for the above aid plot a by that time the ignal jut reache to the back-wall. The above plot were ued to evaluate the minimum value of λ required for the x max olution to converge irrepective of time tep if the olution remain conitent for the variou element length. The ignal at a variou perpendicular ditance from the middle of the line ource wa plotted for different λ /x max a hown below. The following plot (figure 3) how the forward propagating ignal at a time of - plotted along the perpendicular line paing through the centre of the line ource. Figure 3: Onward propagating wave ignal at variou ditance for different wavelength to element length ratio Diplacement (m).00e E-07.00E-07.00E E E E E-07 (a) At 0. m Diplacement (m).00e E-08.00E-08.00E E E-08 Time () Ratio Ratio3 Ratio Ratio5 Ratio8 Ratio 9 Ratio Ratio E-08 (b) At 0.3 m Diplacement 3.00E-08.00E-08.00E E E-08 Time () Ratio Ratio 3 Ratio Ratio 5 Ratio 8 Ratio 9 Ratio Ratio E-08 (c) At.0 m Diplacement (m).50e-08.00e-08.50e-08.00e E E E E E-08 Time (t) Ratio Ratio 3 Ratio Ratio 5 Ratio 8 Ratio 9 Ratio Ratio E-09 Ditance from Source center (m) Figure : ine profile (Diplacement V Ditance from ource) at for different wavelength to element length ratio Ratio Ratio 3 ratio ratio 5 Ratio 8 Ratio 9 Ratio

4 The plot indicate that the olution converge for λ /x max 8 or more for contant time tep which i ufficient for λ /x max a per CF criteria. In each of the above cae, the time tep wa taken a t.5 -. The t taken i ufficient a per CF criteria i.e., t (x/c ). x max for condition of λ /x max i 0.08 which account for the t a x/c 0.08m/5850 m/ The plot indicate that ocillation till perit after complete paing out of the ignal which may be becaue of the leer x. Thi plot alo indicate the converging of the olution for the ratio (λ /x max ) 8. It ha been oberved that whether the time tep are taken from olver or exactly what ha been given doe not make any difference to the propagating ignal. 3. Effect of time tep All of the above olution have been obtained for the input ignal of 0 khz. The ampling of the ignal wa done at MHz and the time tep ued wa.5 - for all of the above olution. A per CF criteria the time tep hould be le than x/c. A per the criteria, for example, for the maximum element length of λ / the time tep hould be (λ /C ) that mean if λ λ t C C t f In the above cae λ varied from to x max wherea in all the cae time tep wa taken a t.5 which 3 0 f 0 0 wa coniderably leer than required a per the criteria. A already been oberved, irrepective of the better time tep the olution converge only for λ 8 i.e. there i no more change in the x max olution beyond λ 8. x max Unfortunately, although there i no change in the ignal for any further reduction of element length but the ignal o far obtained i not a per expected one. The ultraonic wave form hould not change after propagated through an iotropic media. The amplitude wa expected to change conitently but at leat not the hape of the input ignal. In thi regard, it may be thought that probably time tep ued o far wa not ufficient for a correct olution although the CF criteria wa atified well for all of the above cae. In thi regard, the time tep ued were further reduced for three value of λ to obtain the x max olution and checked for the convergence of olution a per the expected reult. Cae I: λ 8 x max For the cae of λ 8, the olution i x max obtained for the following time tep (i) A per CF criteria x 0.03m t. C 5850m / (ii) t 5 (iii) t (iv) t.5 (v) t.0 (vi) t 0.5 (vii) t 0. The following plot (figure 5) how the ignal at a perpendicular ditance of.0 m from the middle of the line ource for different time tep. For all the following cae the ratio λ i taken x max a 8. Figure 5: Time domain ignal at different time tep for λ 8 at.0 m x max (a) Original applied ignal at ource

5 (b) t. (Critical time tep a per CF Criteria) (c) t 5 (d) t.5 (e) t.0 (f) t.0 (g) t 0.5

6 (h) t 0. λ Cae II: 5 x max (i) A per CF criteria x 0.0m t C 5850m / (ii) t 5 (iii) t.5 (iv) t (v) t (vi) t 0.5 (vii) t 0. The following plot (figure ) how the ignal at a perpendicular ditance of.0 m from the middle of the line ource for different time tep. For all the following cae the ratio λ i taken x max a 5. Figure : Time domain ignal at different time tep for λ 5 at.0 m x max (b) t (Critical time tep a per CF Criteria) (c) t 5 (d) t.5 (a) Original applied ignal at ource

7 (h) t 0. Cae III: λ (i) A per CF criteria x.0m t.7 5 C 5850m / 0 (e) t.0 (ii) t.5 (iii) t (iv) t 0.5 The plot in figure 7 how the ignal at a perpendicular ditance of.0 m from the middle of the line ource for different time tep. For all the following cae the ratio λ i taken a. x max Figure 7: Time domain ignal at different time tep for λ at.0 m x max (f) t.0 (a) t 5 (g) t 0.5 (b) t.5

8 (c) t.0 (b) λ 8 t 0.5 x max (d) t 0.5 To compare the plot for convergence of the olution in term of time tep of 0.5 -, the total ignal i hown below for three different λ value. x max Figure 8: Comparion between converged olution at λ 8, and 5 at t 0.5 x max (a) λ t 0.5 x max (c) λ 5 t 0.5 The ignal again indicate that the optimum value of the ratio λ. More ripple in the 8 ignal are een for higher x max in pite of very le t value.. Effect of line and point ource excitation In all of the earlier cae the ource wa modeled a the line ource which get excitation a a whole. It wa alo poible to model the ource a per Huygen principle i.e., conidering the ource a many econdary ource of excitation. Few of the imulation were carried out by conidering point ource a the detector. In that cae, within total 0.0 m length of ource, nine number of equiditant ource excitation point were placed. The imulation wa done for line a well a point ource for λ 8 t.5. The following plot in figure 9 how the ultraonic ignal for both cae (i) for line ource (ii) for point ource oberved at different perpendicular ditance from middle of ource.

9 The line profile (diplacement V ditance from ource) wa alo plotted at a time intant of -. Figure 9: Comparion between ource with excitation on line and excitation on point λ t.5 8 (line ource) At.0 m λ t.5 8 (point ource) At.0 m At 0.5 m At 0.5 m ine profile at t - ine profile at t - 5. Frequency content of the forward propagated ignal The frequency content of the onward propagating wave ha been evaluated by Fourier tranform of the onward propagating wave. The figure how the onward propagating wave in

10 time and frequency domain for three cae of λ, 8 and 5 for t 0.5 x max (converged final olution for each cae) Figure : Time and Frequency domain ignal for onward propagating wave λ t 0.5 Time domain Diplacement (m) 8.00E-08.00E-0.00E-0.00E E E-07 Frequency domain 9 8 Time () Single-Sided Amplitude Spectrum x -9 Frequency domain x -9 Single-Sided Amplitude Spectrum x λ t Time domain Diplacement (m) 8.00E-08.00E-0.00E-0.00E E x λ 8 t 0.5 Time domain -.0E-07 Frequency domain x -9 8 Time () Single-Sided Amplitude Spectrum 8.00E-08 Diplacement (m).00e-0.00e-0.00e e E-07 Time () x. Frequency content of the back-wall reflected ignal The figure how the back-wall reflected wave in time and frequency domain for three cae of λ, 8 and 5 for t 0.5 x max (converged final olution for each cae)

11 Figure : Time and Frequency domain ignal for reflected wave λ t 0.5 Time domain Diplacement (m).00e-08.00e E-0.0E E E-08 Frequency domain 5 3 Time () Single-Sided Amplitude Spectrum x -9 7 Frequency domain 8 x -9 Single-Sided Amplitude Spectrum x λ t Time domain Diplacement (m).00e-08.00e E-0.0E E E-08 Time () Frequency domain x x -9 8 Single-Sided Amplitude Spectrum λ t Time domain Diplacement (m).00e-08.00e E-0.0E E E-08 Time () x 7. Dicuion All reult hown in figure 3 were obtained for the time tep of.5 - which i well below the value calculated uing CF criteria. The time domain ignal hown in figure 3 (a) indicate that for all the cae of λ /x max, the ignal follow the ame time profile path (no change in time cale) till the major ignal pae through the point of obervation. For all the cae, after the major ignal pae through, coniderable amount of the reidual diplacement are till hown to be exit. At that region, the

12 curve for λ /x max, 3 & doe not follow to curve for λ /x max 5, 8, 9, &. There i minor variation een for the cae of 5 with other. For the main ignal, although there i no change in term of time cale but it i een that the amplitude i le in cae of higher value of λ /x max. When the ame ignal i oberved at a further ditance it i een from figure 3(b) that there i change in the hape of wave form that wa propagated from the earlier point. From figure 3(a) to 3(c), it i oberved that the olution for the major ignal alway follow for all λ /x max wherea at the region of reidual diplacement, the olution getting converged only from λ /x ma 5 onward. But in pite of very poor value of λ /x max, 3 &, the olution evolved correctly in term of time for all λ /x max. Again for all plot under figure 3, the ignal i not of the hape a expected. For a nondiperive media like teel, the ultraonic ignal mut not change it hape during propagation. Thi indicate that in pite of following the CF criteria alo the ignal could not really converge for the amplitude detail. Figure how that for λ /x max, 3 & there i coniderable error in the reidual ignal (which remain after the actual ignal pa through the point of obervation). Figure 5 indicate that the time tep ued earlier for obtaining the olution wa not ufficient although it followed the CF criteria. Becaue of change in time tep, there i a change in both frequency information a well a intenity information of the propagating ignal. At the time tep of only, the propagated ignal reume the hape of original ignal. Further decreae in the time tep below doe not make any further improvement of the ignal which indicate the poible convergence of the olution. Similarly the olution converge for the cae of λ /x max 8 only for time tep of Even for λ /x max 5 alo the olution converge with the time tep of Thi clearly indicate that time tep i a very important parameter for getting correct olution for wave propagation. For very poor value of λ /x max like or 3 can at leat give ome information regarding the ignal with a better value of t but any good value of x i of no ue if a proper value of t i not ued for obtaining the olution. At leat in three cae it i een that the required time tep i about (/0*f) which i independent of the value of x. The value of x ha to be taken independently at about λ/8. From figure 9 it i een that at leat for the normal incidence of ultraonic beam no difference in term of hape of the ignal i oberved between the olution obtained for two way of excitation viz. excitation on a line and excitation on point to reemble the ource. The only difference oberved i in term of the intenity. Intenity oberved in cae of point ource a the excitation i leer than that oberved in cae of line ource. The obervation i obviou a power input in the cae of line ource will alway be more than that from the point ource a the excitation. Figure how that there i no ubtantial change in the frequency domain ignal for the onward propagating wave for λ /x max 8, & 5 in the final converged olution. Similarly no difference i oberved in the frequency domain reflected ignal. However a little difference in intenity i oberved in both onward and reflected wave. Intenity i higher for λ /x max 5 and leat for λ /x max. 8. Concluion Ultraonic wave propagation can well be modeled with COMSO with excitation on a line egment a tranducer or excitation on point on a line egment. The triangular element free mehing can be ued for the imulation of ultraonic wave propagation for a value of λ /x max 8 or more. The time tep hould be ued near to /0*f. Smaller value of λ /x max uch a 5 can alo be ued with expected maller variation of amplitude information with ue of proper time tep. No ubtantial difference in the frequency content of the onward a well a back-reflected converged olution i oberved.