Ultrasonics 42 (2004) 781–786

www.elsevier.com/locate/ultras

Numerical time-domain simulation of diffusive ultrasound in concrete

Frank Schubert *, Bernd Koehler

Fraunhofer-Institute for Nondestructive Testing, Branch Lab, EADQ, Kruegerstrasse 22, D-01326 Dresden, Germany

Abstract

Certain aspects of diffusive ultrasound fields in concrete are still unknown and thus, systematic parameter studies using numerical

time-domain simulations of the ultrasonic propagation process could lead to further insights into theoretical and experimental

questions. In the present paper, the elastodynamic finite integration technique (EFIT) is used to simulate a diffusive reverberation

measurement at a concrete specimen taking aggregates, pores, and viscoelastic damping explicitly into account. The numerical

results for dissipation and diffusivity are compared with theoretical models. Moreover, the influence of air-filled pores in the cement

matrix is demonstrated.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Diffusive ultrasound; Concrete; Time-domain simulation; Elastodynamic finite integration technique

1. Introduction

For elastic wave propagation in the ultrasonic regime,

concrete represents a strongly heterogeneous medium

consisting of statistically varying aggregates and pores

embedded in a cement matrix. Additionally, structural

components like reinforcing bars and tendon ducts affectthe wave propagation process significantly. In nonde-

structive testing, the interpretation of the received

ultrasonic signals may become extremely difficult due to

multiple scattering, strong attenuation, and mode con-

versions. Even the use of sophisticated averaging and

reconstruction techniques like synthetic aperture meth-

ods only allows the detection of defects being signifi-

cantly larger than the maximum aggregate size. Thus,localised damage zones developing at length scales

smaller than the size of the majority of aggregates can-

not be detected with typically used frequencies of less

than 100 kHz.

Increasing the frequency up to a few hundred kHz

changes the character of the wave propagation process

in concrete dramatically. While wave propagation in the

low-frequency region below 100 kHz is dominated bythe coherent field, ultrasound becomes diffusive at

higher frequencies which means dominance of the

*Corresponding author. Fax: +49-351-2648219.

E-mail address: schubert@eadq.izfp.fraunhofer.de (F. Schubert).

0041-624X/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.ultras.2004.01.040

incoherent scattered field. This is shown in Fig. 1 where

the elastic wave field caused by a 200 kHz ultrasonic

pulse was detected at a 9 · 9 cm2 outer surface of a

concrete specimen using scanning Laser vibrometer [1].

Yet before the first parts of the primary wave field reach

the lower surface of the sample, the acoustic energy

propagation is almost diffusive with coherent wavefronts no longer visible.

In this regime ultrasonic spectral energy density

Eðz; tÞ evolves in accordance with diffusion equations. In

a specimen with lateral dimensions being small com-

pared to the longitudinal dimension, z, a one-dimen-

sional diffusion equation with dissipation can be applied

[2,3]. The solution of this equation––after taking the

natural logarithm––is given by

lnEðz; tÞ þ 0:5 ln t ¼ C0 �z2

4Dt� rt; ð1Þ

where D is ultrasonic diffusivity, r the dissipation or

absorptivity, and C0 a factor connected with the initial

energy deposition at z ¼ 0 and time t ¼ 0.

Measuring Eðz; tÞ at a certain sensor location, z 6¼ 0,

for different time windows and frequency bands andperforming a least-squares fit to the data allows the

determination of the three parameters in Eq. (1) which

completely characterise the diffusion process. The dissi-

pation, r, describes the exponential decay at late times

while diffusivity, D, is related to the arrival time of the

maximum energy density.

Fig. 1. Formation of a diffusive ultrasound field in a concrete specimen caused by 200 kHz pulse excitation. The first picture shows aggregates and

pores at an outer surface of the sample (9 · 9 cm2). The following three pictures represent time snapshots of the ultrasonic wave field obtained by

scanning laser detection at the same surface. The ultrasonic transducer is located at the top surface of the specimen.

782 F. Schubert, B. Koehler / Ultrasonics 42 (2004) 781–786

2. Numerical time-domain simulation of a diffusion

process

For the numerical investigations we simulated a realexperiment similar to that performed by Anugonda

et al. [2]. They used several cylindrical concrete speci-

mens with diameters of 5 cm and a length of 46 cm. The

maximum aggregate size was 12.7 mm. The beams were

excited at one end by a contact transducer and the sig-

nals were detected at the other end by a point-like pin

transducer. Their experiment was done in narrow-band

technique, i.e. they used several narrow-band pulseswith varying center frequencies from 100 to 900 kHz

covering the whole frequency band of interest.

For the numerical calculations we used the elasto-

dynamic finite integration technique (EFIT) which is

particularly suitable for simulation of ultrasonic waves

in strongly heterogeneous media [4–6]. For reasons of

computational effort, we used a two-dimensional model

of the beam according to a plane strain propagationprocess. In contrast to the experiment described above, a

broadband excitation with frequencies from zero to

about 1.5 MHz was used. To avoid problems connected

with the generation of Rayleigh waves at the source

position, we used a transducer aperture with the same

width than the lateral beam dimension. The point-likesensor was located at the other end of the beam at a

distance of 45 cm from the source.

The concrete was modelled by using gravel and sand

aggregates with an area fraction of about 66% embed-

ded in a homogeneous cement matrix and according to a

standardised grading curve with a maximum aggregate

size of 12 mm. The acoustic parameters of the constit-

uents of the model are summarised in Table 1. Toincorporate dissipative loss we used the Kelvin–Voigt

model of viscoelasticity introducing two additional

parameters into the governing equations, i.e. volume

and shear viscosity, gV and gS. For reasons of simplicity

and interpretation of the results, we used the same val-

ues for both, aggregates and cement matrix, being aware

that in fact dissipation is likely dominated by the vis-

coelasticity of the cement paste alone.Air-filled pores were modelled as voids with stress-

free boundary conditions. The porosity was 0.7% with

Table 1

Material parameters as used for the numerical simulations

Material cL (m/s) cT (m/s) q (kg/m3) gV (Pa s) gS (Pa s)

Cement 3950 2250 2050 65 17

Aggregates 4300± 215 2475±124 2610± 130 65± 3.25 17± 0.85

Air-filled pores were modelled as voids with stress-free boundary conditions. The values for cL, cT, and q are typical values for concrete well known

from field measurements. They fluctuate from one grain to another by 5% (maximum) around the given mean value. The values for gV and gS are only

roughly estimated. They were chosen in such a way that the signal damping in the simulation was close to typical experimental observations as given

in [2].

F. Schubert, B. Koehler / Ultrasonics 42 (2004) 781–786 783

pore sizes uniformly distributed in the range between 0.3

and 2 mm.

Fig. 2 shows the numerical concrete model consisting

of about one million cells (grid spacing Dx � 150 lm)

and various time snapshots of the wave propagation

process after pulse excitation at the bottom end of the

beam. The pictures are taken at equidistant time inter-

vals of 22.9 ls and represent the absolute value of theparticle velocity vector. The coherent wave front is

clearly visible. Due to scattering, the higher frequencies

are strongly attenuated during propagation and thus,

the wave contains only the lower frequencies when it

arrives at the sensor position at z ¼ 45 cm. Due to mode

conversions at the lateral stress-free boundaries, trans-

verse head waves followed by secondary ultrasonic

Fig. 2. Two-dimensional EFIT simulation of diffusive ultrasound in a concret

of aggregates and pores embedded in a cement matrix (porosity 0.7%). The m

of the elastic wave field calculated in equidistant time steps of Dt ¼ 22:9 ls.

waves are generated and a significant energy transfer

from the primary wave front to the first secondary wave

is evident. One can already see from one of the last

snapshots that the low-frequency parts of the diffusive

field following the coherent wave fronts arrive earlier at

the top end of the beam than the high-frequency parts

which are still localised in the lower beam region.

Fig. 3 shows the calculated time-domain signal at thereceiver position. The first arrival of the primary

coherent wave front is at t � 107 ls. The first secondarywave (with larger amplitude) arrives at t � 126 ls. Al-

though these coherent phenomena are interesting, they

are not taken into account for further consideration. In

fact, the diffusive field behind the coherent wave fronts is

used for evaluation.

e beam. The first picture shows the numerical concrete model consisting

odel contains 333 · 2997 grid cells. The following pictures are snapshots

They represent the absolute value of the particle velocity vector.

Fig. 3. Calculated time-domain signal after broadband pulse excitation

using the model from Fig. 2. The signal gives the normal particle

velocity, vz, at the receiver position at z ¼ 45 cm. The arrival of the

primary coherent wave front is at t � 107 ls. The first secondary echo

arrives at t � 126 ls.

Fig. 5. Results of the numerical experiment for dissipation as a func-

tion of frequency (obtained at a single material configuration in each

case). The theoretical mean value according to the Kelvin–Voigt model

784 F. Schubert, B. Koehler / Ultrasonics 42 (2004) 781–786

3. Results for dissipation and diffusivity

The calculated data was analysed using standard

time–frequency techniques [2,3]. We used 40 time win-

dows of length Dt ¼ 37:5 ls and 10 frequency windows

of width Df ¼ 100 kHz. The latter were centred at 100,200, 300, . . ., 1000 kHz. The results are shown in Fig. 4

together with the corresponding least-squares curve fits

to Eq. (1), i.e. the solution of the 1-D diffusion equation.

It can be seen that the late time curve fits as well as the

early time fits for the high frequencies are quite good

while the early time fits for the lower frequencies are

rather bad. This is most likely caused by the fact that in

the low-frequency regime wave propagation is stilldominated by the coherent field and thus, the diffusion

equation is not well suited to describe energy propaga-

tion. This assumption is confirmed by the fact that for

the low-frequency data in Fig. 4 the positions of the

Fig. 4. Results of the numerical experiment (as described in Figs. 2 and

3) for the logarithm of energy density as a function of time and fre-

quency (from 100 to 1000 kHz in steps of 100 kHz each). The solid

lines are least-squares curve fits to Eq. (1), i.e. the solution of the 1-D

diffusion equation.

maxima of lnE coincide with the arrival times of the

primary and secondary coherent wave fronts, respec-

tively.

The diffusivity, D, and the dissipation, r, can directly

be determined from the curve fits shown in Fig. 4. Theresults for various numerical experiments with different

parameters are given in Figs. 5 and 6. According to the

comments above the diffusivity results for the low fre-

quencies are expected to have larger errors than the

high-frequency results.

The dissipation is significantly increasing with

increasing frequency (Fig. 5). This rise is stronger than

linear which is in accordance with the Kelvin–Voigtmodel of viscoelasticity (solid line). Following this

model, the theoretical dissipation for the longitudinal

wave is given by r ¼ 2p2f 2ðgV þ ð4=3ÞgSÞ=ðqc3LÞ, wherewe used the mean values from Table 1. It is important to

point out that for concrete as a heterogeneous medium,

this curve should be interpreted as the ensemble average

over numerous different scatterer arrangements. In

Fig. 6. Results of the numerical experiment for diffusivity as a function

of frequency and porosity (obtained at a single material configuration

in each case).

is given by the solid line. The porosity in this case was 0.7%.

F. Schubert, B. Koehler / Ultrasonics 42 (2004) 781–786 785

contrast to that, the numerical experiment was carried

out by using one single sensor position and one certain

arrangement of aggregates and pores. Thus, a quanti-

tative difference between the theoretical curve and the

simulation data as seen in Fig. 5 can be expected. Apartfrom that, the quadratic frequency dependence of r is

reproduced well by the numerical data.

In addition to the viscous model, the calculation was

repeated by setting both viscoelastic parameters to zero.

The resulting absorptivities are not vanishing totally but

have only very small values (Fig. 5, lower curve). This

also demonstrates the consistency of the evaluation

method.To investigate the dependency of the diffusion coef-

ficient on frequency, we used three different concrete

models with varying porosity (0%, 0.7%, and 1.4%). As

one can see from Fig. 6, diffusivity is significantly low-

ered by increasing porosity. This is consistent with dif-

fusion theory since a larger number of pores means

stronger scattering and thus, a further delayed energy

propagation. It is striking that the differences betweenthe nonporous model and the model with 0.7% porosity

are significantly larger than the differences between 0.7%

and 1.4% porosity.

The general fact that small changes in porosity have a

dramatic effect on the diffusivity seems to be surprising

at first view. However, one has to keep in mind that the

largest pores (2 mm diameter) are not negligible com-

pared to the smallest wavelengths of 4–8 mm (for fre-quencies between 500 and 1000 kHz at a wave speed of

cL � 4000 m/s). Moreover, it is well known from pre-

ceding investigations that although size and concentra-

tion of air-filled pores are clearly smaller than that of the

aggregates, porosity plays a significant––if not domi-

nant––role for scattering of ultrasonic waves in concrete

[7]. This is caused by the large impedance mismatch

between air and matrix. Apart from that it should alsobe taken into account that scattering at a 2-D plane

strain concrete model is quantitatively different from

scattering at a 3-D model, as discussed in [7]. As a

consequence, the numerical results obtained so far

should only be compared qualitatively with 3-D exper-

iments.

In order to predict diffusivity in concrete, Turner and

Anugonda proposed a statistical scattering theory usinga two-phase approach of cement matrix and aggregates

with perfect bonding between the two phases [2,8]. They

found a good agreement with experimental data. So far,

we did not find a satisfactory agreement between our

numerical 2-D results and the 3-D Anugonda–Turner

model. The diffusion coefficients of the upper curve in

Fig. 6 should be comparable to the Anugonda–Turner

model due to the lacking porosity but the values aresignificantly larger then predicted by the model. This is

probably caused by the fact that scattering at aggregates

and pores in the numerical 2-D model is larger than in

3-D if using the same volume/area fractions for aggre-

gates and pores [7].

Moreover, the strong rise of diffusivity at low fre-

quencies as predicted by the model [2], cannot be found

in our numerical results. This is maybe caused by thedomination of the coherent field in this regime but the

exact reason for this discrepancy should be investigated

further. A 2-D version of the Anugonda–Turner model

or alternatively a 3-D simulation would help to answer

these questions. Independent from that it is worth not-

ing that preceding experiments and simulations clearly

revealed that porosity in concrete cannot be neglected

compared to the aggregates [7]. This is also confirmed bythe results shown in Fig. 6. As a consequence, appro-

priate enhancements of the Anugonda–Turner model

need to be taken into consideration.

4. Conclusions

The results of the numerical experiments are plau-sible and consistent with the assumptions of a diffusive

energy transport. While absorptivities obtained at the

discrete model are in a good agreement with the

Kelvin–Voigt model, the diffusion coefficients do not fit

well to the Anugonda–Turner model so far. However,

the numerical results reveal a significant impact of

porosity on the diffusion coefficient. Due to the fact

that microstructural damage (e.g. by micro cracks) isexpected to have similar effects on wave scattering than

porosity, the use of high-frequency diffusive ultrasound

could lead to new nondestructive methods for cha-

racterising material properties and damage in concrete

structures.

In the future, an ensemble averaging of the numerical

data by using various models with different scatterer

arrangements should lead to even more stable and reli-able results. Moreover, a direct calculation of the energy

density from the elastic field components will possibly

lead to further insights into still unsolved aspects of

diffusivity in concrete.

Acknowledgements

The authors would like to thank an anonymous re-

viewer for useful comments to improve the presentation

of the manuscript.

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