Ultrasonics 42 (2004) 221–229 www.elsevier.com/locate/ultras Numerical time-domain modeling of linear and nonlinear ultrasonic wave propagation using finite integration techniques––theory and applications Frank Schubert * Fraunhofer-Institute for Nondestructive Testing, Branch Lab EADQ, Kruegerstrasse 22, D-01326 Dresden, Germany Abstract This survey paper summarizes the basic principles of the Elastodynamic Finite Integration Technique (EFIT) for linear and nonlinear elastic wave propagation in homogeneous and heterogeneous media and presents various examples of application in the field of ultrasonic nondestructive testing. These examples involve modeling of pulse-echo, acoustic emission, and Lamb wave problems in various structures as well as nonlinear wave propagation simulations based on third-order elasticity. 2004 Elsevier B.V. All rights reserved. Keywords: Time-domain modeling; Ultrasonic wave propagation; Elastodynamic finite integration technique; Nondestructive testing 1. Introduction Due to the rapid development of new materials and structures, the requirements for powerful and reliable nondestructive testing methods are increasing significantly. In ultrasonic testing, an adequate understanding of the interaction of elastic waves with external and internal boundaries and defects of the structure is essential. Especially in solids and heterogeneous media, the interpretation of the detected signals becomes difficult due to multiple scattering, strong attenuation, mode conversion, and nonlinear effects. Thus, there is an increasing demand for powerful, flexible, and easily manageable modeling techniques. temporal grid. The starting point of EFIT for inhomogeneous anisotropic media is the integral form of the linear governing equations, the Cauchy equation of motion, Z Z Z qðrÞ v_ ðr; tÞ dV v ¼ tS¼ov n Tðr; tÞ dS þ Z Z Z fðr; tÞ dV ; v and the equation of deformation rate, Z Z Z _ tÞ dV s : Tðr; v ¼ tS¼ov symfn vðr; tÞg dS þ 2. Elastodynamic finite integration technique The elastodynamic finite integration technique (EFIT) is a numerical time-domain scheme to model elastic wave propagation in isotropic and anisotropic, homogeneous and heterogeneous as well as dissipative and non-dissipative elastic media [1–3]. EFIT uses a velocity-stress formalism on a staggered spatial and * Fax: +49-351-26482-19. E-mail address: [email protected] (F. Schubert). 0041-624X/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultras.2004.01.013 ð1Þ Z Z Z hðr; tÞ dV : ð2Þ v Here, the dot and colon indicate single and double contraction. q denotes the mass density at rest tensor, v the particle velocity vector, n the outward normal unit vector of the closed surface S ¼ oV of the volume V , T the Cauchy stress tensor, f the vector of volume force density, h the source of deformation rate, sym fn vðr; tÞg the symmetric part of the dyad n vðr; tÞ, and s the fourth rank compliance tensor. For inhomogeneous isotropic media, only a scalar mass density is required, and hence qðrÞ ¼ qðrÞI with the unit tensor of rank two, I. With the two independent 222 F. Schubert / Ultrasonics 42 (2004) 221–229 Lame’s constants, kðrÞ and lðrÞ, the compliance tensor for isotropic media is given by sðrÞ ¼ kðrÞ 1 ðII1324 þ II1342 Þ; II þ lðrÞ½6kðrÞ þ 4lðrÞ 4lðrÞ ð3Þ where we applied the so-called upper indicial notation. According to the equations above, EFIT performs an integration over certain control volumes, V , and over the surfaces of these cells, S, assuming constant v and T within V and on each of the surfaces S of V . Obviously this method requires staggered grids and leads to a very stable and efficient numerical code allowing an easy and flexible treatment of various boundary conditions. In Fig. 1, the spatial staggered grid of the 3-D EFIT code in Cartesian, cylindrical, and spherical coordinates is shown as an example. The usual EFIT procedure uses the mid-point rule for integration, resulting in a numerical scheme of second order accuracy in space and time. However, this code can easily be extended to higher order accuracy if necessary. For homogeneous media, EFIT is similar to finite difference time-domain (FDTD) methods if the latter also use the velocity-stress formalism on a staggered grid. However, the EFIT treatment of heterogeneous media is different and allows sophisticated methods for discretizing the material parameter distribution in space (e.g. treatment of partly filled material cells). One of the main features of EFIT is its relative simplicity and its great flexibility and effectiveness. Even millions of grid cells can be tackled on an ordinary PC within reasonable computing time. 3. Examples of application In order to give an impression of the flexibility and capability of EFIT, various examples of application in the field of nondestructive testing are presented in the following sections. 3.1. Ultrasonic waves in concrete Ultrasonic nondestructive testing of strongly heterogeneous materials is a difficult task due to multiple scattering, mode conversion, and attenuation. Concrete for example is of high practical interest in civil engineering and there is a great demand for efficient and reliable testing methods. In order to optimise these methods or to develop new ones, it is necessary to study the wave propagation process in concrete systematically. Fig. 2 shows the numerical EFIT simulation of a scattering experiment in a cross-sectional model of a concrete specimen containing two cylindrical drill holes. The maximum size of the gravel and sand aggregates was 16 mm. The porosity due to air-filled pores was 1% with pore sizes up to 2 mm. The pores were approximated as voids with stress-free boundary conditions. The width of the transducer aperture was 6 cm. A center frequency of 200 kHz was used for the input pulse. The time-snapshots of the elastic wave field in Fig. 2 represent the absolute value of particle velocity using a linear gray-scale. The simulations revealed that porosity significantly affects the wave propagation process. The coherent wave front is attenuated very much and a diffusive energy transport can be observed behind it. Fig. 1. 3-D discretization of different specimens (upper row) and corresponding EFIT control volumes (lower row) in Cartesian, cylindrical, and spherical coordinates. The velocity vector components vi , i ¼ 1, 2, 3 (denoted by arrows) and the stress tensor components Tij , i; j ¼ 1, 2, 3 (denoted by dots (i ¼ j) and cruxes (i 6¼ j)) are located at different positions inside the grid cell (Ôstaggered grid’). F. Schubert / Ultrasonics 42 (2004) 221–229 223 Fig. 2. EFIT time snapshots of wave propagation and scattering in a 2-D concrete model with gravel and sand aggregates and 1% porosity. The snapshots are taken at t ¼ 34:4, 68.8, and 103.2 ls after 200 kHz pulse excitation. They represent the absolute value of the particle velocity vector using a linear grey scale. This diffusive ultrasound in concrete is discussed in more detail in [4]. Due to the strong attenuation of the coherent wave, special averaging and reconstruction techniques like SAFT (Synthetic Aperture Focusing Technique) are used in practice in order to increase the signal-to-noise ratio [5]. The EFIT simulations can efficiently be used to produce synthetic data and to optimize the imaging techniques by taking into account the strong backscattering at aggregates and pores. They can further be used for planning as well as for interpretation and evaluation of experimental data. Another important testing method for concrete inspection is the impact-echo method where low-frequency stress waves are introduced into the structure by mechanical impact and the response is detected by a point-like transducer and evaluated in the time- and frequency domain. Numerical simulations of impactecho testing are described in [6]. investigate the influence of pre-stressing and reinforcement on inversion algorithms for source location and source mechanism. As can be seen from the wave front snapshots in Fig. 3, the P-wave is diffracted around the tendon and additionally a strong creeping shear wave is generated (whispering gallery mode). Only a small portion of the waves is transmitted through the duct. One can summarize that the tendon duct has a significant effect on wave propagation and thus can influence the accuracy of source localization algorithms. By calculating the timedomain signals of velocity or displacement at different sensor positions, the numerical simulations can systematically be used to optimize the inversion algorithms. The important advantage of a numerical method like EFIT compared to experimental measurements is, that reproduceable AE sources at fixed locations and with clearly defined fracture mechanisms can be implemented. More details of the simulation of acoustic emission events in concrete can be found in [7]. 3.2. Acoustic emission modeling For modeling of acoustic emission (AE) analysis, the geometry of an existing pre-stressed concrete beam including steel reinforcement and a polyethylene tendon duct filled with mortar and steel strands was chosen (Fig. 3). The specimen was used for fatigue tests and was assigned for AE measurements as well. The goal of the simulations was to reach a better understanding of the wave propagation process in the specimen and to 3.3. Lamb wave propagation in plates and cylindrical shells EFIT can also be used to investigate Lamb wave propagation in plates and their interaction with various kinds of defects. As an example, Fig. 4 shows the interaction of a transient pulse (center frequency 500 kHz, asymmetric A0 mode) with a flat-bottom hole in a 1.5 mm thick aluminum plate. The propagation 224 F. Schubert / Ultrasonics 42 (2004) 221–229 Fig. 3. 2-D acoustic emission simulation in a pre-stressed concrete beam including steel reinforcement and a polyethylene tendon duct filled with mortar and steel strands. The point source was located immediately at the tendon/concrete interface. The snapshots represent the absolute value of the particle velocity vector using a linear grey scale. direction of the incident pulse is from left to right. The wave front pictures represent the normal component of the particle velocity vector, detected at the outer surface of the 3-D plate model (top view in Fig. 4). Calculations as described above can directly be compared with experimental wave propagation measurements obtained by scanning Laser vibrometer [8]. By that, the influence of various kinds of defects on Lamb wave propagation can be studied systematically, leading to enhanced and optimized data processing and imaging algorithms that can be used in plate testing and structural health monitoring. Similar wave modes as described above for plates can also be found in cylindrical shells. For simulation of wave propagation in such environments, an EFIT discretization based on cylindrical coordinates can be used. Details F. Schubert / Ultrasonics 42 (2004) 221–229 225 Fig. 4. Lamb wave propagation and scattering at a circular flat-bottom hole in a 150 · 150 · 1.5 mm3 aluminum plate, calculated by the 3-D EFIT code. The incident plane wave (500 kHz, asymmetric A0 mode) propagates from left to right. The time snapshots represent the out-of-plane particle velocity using a linear colour scale (here reproduced as grey scale). of this code and further examples of application are described in [2,9]. To give a demonstration, Fig. 5 shows structural wave propagation in a steel tube with length ¼ 1.35 m, diameter ¼ 10.7 cm, and wall thickness ¼ 9 mm, as calculated by the 3-D cylindrical EFIT code. The center frequency of the input pulse was 60 kHz. 3.4. Acoustic signatures of electrical contacts The residual lifetime of electrical contacts or other devices can be determined by so called acoustic signature analysis. Thereby, a combined actuator/sensor system is coupled to the contact. The acoustic signature of the complete system (actuator/sensor module + contact element) is evaluated as a whole by using time-frequency representations (spectrograms), obtained either by windowed FFT or continuous wavelet transforma- tion, as acoustic fingerprints. During the lifetime of a contact, its fingerprint changes due to contact erosion. These changes can be recorded and used for lifetime predictions. In order to develop efficient and reliable data analysis techniques, the wave propagation in the complete system was simulated by using the 3-D EFIT code. The discretized model of the electrical contact and the sensor/actuator module is shown in Fig. 6 together with exemplary wave front snapshots in a certain cross-section of the 3-D model. The sensor/actuator module can be identified in the top right picture of Fig. 6 by the longish rectangular element coupled beneath the oblate rectangular contact plate. The latter in turn is located at the top surface of the contact element. For more details about the method of lifetime determination by acoustic signature analysis, the reader is referred to [10]. 226 F. Schubert / Ultrasonics 42 (2004) 221–229 3.5. NDT in the railway industry Fig. 5. Structural wave propagation (60 kHz, asymmetric A0 mode) caused by point-source excitation in a steel tube of length ¼ 1.35 m, diameter ¼ 10.7 cm, and wall thickness ¼ 9 mm, calculated by the 3-D EFIT code in cylindrical coordinates. The snapshots represent the absolute value of the particle velocity vector using a linear grey scale. In the railway industry, nondestructive testing of rails and wheels is often performed by ultrasonic methods. For that, an adequate understanding of the wave propagation inside the complex shaped structures is necessary. For this purpose, EFIT algorithms in two and three dimensions can be efficiently used as was already demonstrated for wave propagation in railway rails [3,11]. In Fig. 7, wave propagation in the wheel of a German ICE-2 train is presented. The pictures represent a crosssectional view of the upper half of the wheel. The model is bounded at the bottom by a hollow shaft in the wheelset, where ultrasonic sensors can be placed. The top part of the wheel in Fig. 7 is the contact surface to the rail. Possible cracks in this surface produce acoustic emission events each time when the crack gets in contact with the rail surface. These signals can be averaged during ride of the train and used for detection of the flaws. Due to the fact that the complex shaped wheel represents a dispersive wave guide, wave propagation from Fig. 6. 3-D EFIT model of the electrical contact and snapshots of wave propagation caused by broadband excitation of the piezoelectric actuator stack. The dimensions of the contact are 24.5 · 14 · 11 mm3 . The maximum frequency of the signal is 1 MHz. The whole model consists of more than 1.2 million grid cells. The snapshots represent the absolute value of the particle velocity vector using a linear grey scale. F. Schubert / Ultrasonics 42 (2004) 221–229 227 Fig. 7. EFIT simulation of ultrasonic wave propagation in the wheel of a German ICE-2 train caused by crack-induced acoustic emission at the contact surface. The pictures represent the absolute value of particle velocity using a logarithmic colour scale (here reproduced as grey scale). the contact surface to the hollow shaft was calculated by the EFIT code. The corresponding wave front snapshots are shown in Fig. 7. It turned out that the lower frequencies propagate significantly faster through the wheel than the high-frequency parts of the signal. This fact can be used for adapted data analysis routines. 4. Nonlinear wave propagation The EFIT procedure as described in Section 1 can be extended to nonlinear elastic wave propagation. For that, nonlinearity is typically introduced by using the general nonlinear expression for the strain tensor and a nonlinear version of the constitutive equation. There are two main approaches to elastic nonlinearity (used for different kinds of media) which seem to have the most practical relevance for NDT applications, that is the so called Ôfive-constant-theory’, also known as third order elasticity [11], as well as the implementation of hysteretic stress-strain relations based on nonlinear mesoscopic elasticity [12]. The latter is based on the PMspace model and first attempts to use EFIT-like numerical methods to model resonant bar measurements in media with localized damage are presented in [13]. In the present paper we focus on the Ôclassical’ nonlinearity and discuss some basic numerical problems to overcome. The central result of third-order elasticity for a plane longitudinal wave in a nonlinear five-constant medium states that the wave speed depends on powers of the linear term of strain, ou=ox, where u is displacement [11]. For the corresponding EFIT code this means that in addition to the linear field components for velocity and stress, also the displacements must be considered explicitly. This can be done in a straightforward way since displacement can simply be obtained by numerical integration of velocity. In addition to the adapted spatial discretization, an appropriate temporal discretization with linear interpolations between displacements at certain time steps is also necessary [11]. 228 F. Schubert / Ultrasonics 42 (2004) 221–229 quency-dependent absorption into the constitutive equations which produces an attenuation of the highest frequencies and thus, stops the distortion process at a certain level. We implemented a viscoelastic (linear) loss mechanism based on the Kelvin–Voigt model [1]. FoloS lowing this approach, an additional term g : ot is added to the right hand side of the stress-strain relation. In this connection, S is the second rank strain tensor and g the Fig. 8. Wave profile distortion of finite amplitude sine waves in Massilon Sandstone modeled by the NL-EFIT code (at the top). For a better presentation, all amplitudes were normalised to one. Due to the profile distortion, higher harmonics are generated in the spectrum (bottom picture). The base frequency of the initial sine wave was 10 kHz in this case. In Fig. 8 (at the top), the wave profile distortion of a finite amplitude sine wave in a Massilon Sandstone model was calculated by the nonlinear EFIT code. In the figure, the time-domain signal of the stress component, Txx , is shown. The calculations were performed using three different amplitudes of the initial sine wave. They are all normalized to one in order to better compare the wave profiles. From the picture at the bottom, it can be seen that the wave profile distortion is connected with the generation of higher harmonics in the spectrum showing frequency peaks at multiple values of the base frequency of 10 kHz. This causes some problems for numerical methods like EFIT because the maximum frequency in the signal becomes higher and higher if the traveling distance of the wave increases. Therefore, for a fixed grid spacing, Dx, numerical dispersion occurs at a certain traveling distance and the calculation is no longer accurate due to the inadequate discretization of the shortest wavelengths. This problem can be solved by adaptive mesh refinements as well as by additionally introducing fre- fourth rank viscosity tensor. Due to the time derivatives of strain, the time derivatives of the velocity components have to be considered additionally within the EFIT framework. This can also be done very easily because they appear in the equation of motion (1) anyway. Concerning numerical stability, in 1-D the discrete time step Dt has to fulfil the criterion Dt 6 Dx=cmax , where Dx is the spatial discretization and cmax the maximum wave speed in the model. In contrast to the linear case, where cmax is simply given by the linear wave speed of the longitudinal wave, in a nonlinear lossy medium the maximum wave speed depends on ou=ox and on the larger effective stiffness caused by viscoelasticity [1]. The latter in turn depends on the viscosity parameters and the maximum frequency of the signal. To give a concluding example, the results of various EFIT pulse transmission simulations, analysing the variation of wave speed with applied external stress, are shown in Fig. 9 for a Portland sandstone model. The amplitudes of the ultrasonic pulses traveling through the nonlinear medium were very small so that no wave profile distortion appeared. The calculations were performed for various amounts of external stress and the longitudinal wave speed was determined from travel time and distance. In Fig. 9, the numerical results (black dots) are compared with analytical solutions of third-order elasticity (black curve) as well as with experimental measurements of Winkler and Liu [14] (gray dots). Fig. 9. Longitudinal wave speed as a function of external hydrostatic stress in Portland Sandstone as modeled by the nonlinear EFIT code ( : NL-EFIT, ––: analytical, : experiment by Winkler and Liu [14]). F. Schubert / Ultrasonics 42 (2004) 221–229 The agreement between numerical and analytical results is excellent demonstrating the accuracy of the nonlinear code. For stresses larger than 35 MPa, the experimental results are significantly smaller than numerical and analytical results. This is obviously caused by the fact that the five-constant-theory is an accurate approach for moderate stresses but for larger stresses, higher orders of nonlinearity have to be considered as well. Moreover, possibly nonlinear mesoscopic elasticity would be a better theoretical approach in this regime [12,13]. 5. Conclusions As it has been demonstrated in this survey paper, the Elastodynamic Finite Integration Technique (EFIT) represents a powerful numerical time-domain method to calculate elastic wave propagation and scattering in various media and structures. More technical details can be found in the literature given below. The EFIT results can be used for planning of test setups, for optimization of inverse techniques, as well as for interpretation and evaluation of experimental data. The EFIT scheme is easy to implement and very flexible concerning the realization of various boundary conditions. Each calculation presented in this paper was performed on an ordinary PC (Pentium IV, 2 GHz, 1 GB RAM). 229 References [1] R. Marklein, R. Baermann, K.-J. Langenberg, in: Rev. Progr. Quant. Nondestr. Eval. (QNDE), vol. 14, Plenum Press, New York, 1995, p. 251. [2] F. Schubert, A. Peiffer, B. Koehler, T. Sanderson, J. Acoust. Soc. Am. 104 (5) (1998) 2604. [3] F. Schubert, O. Sacharova, B. Koehler, in: Rev. Progr. Quant. Nondestr. Eval. (QNDE), vol. 18B, Plenum Press, New York, 1999, p. 2129. [4] F. Schubert, B. Koehler, in: Proceedings of Ultrasonics International 2003, June 30–July 3, 2003, Granada, Spain, Ref UI319. [5] B. Koehler, G. Hentges, W. Mueller, NDT&E Int. 31 (1998) 281. [6] F. Schubert, H. Wiggenhauser, R. Lausch, in: Proceedings of Ultrasonics International 2003, June 30–July 3, 2003, Granada, Spain, Ref UI361. [7] F. Schubert, B. Schechinger, NDTnet 7 (9) (2002) (http:// www.ndt.net/article/v07n09/07/07.htm). [8] B. Koehler, M. Kehlenbach, R. Bilgram, in: Acoustical Imaging, vol. 27, Kluwer Academic/Plenum Publishers, Dordrecht & New York, 2004, p. 315, in print. [9] P.P. Delsanto, F. Schubert, Z. Prevorovsky, I. Genesio, C. Chiroiu, unpublished manuscript (2001), available from the authors. [10] F. Berger, O. Duehr, K.-J. Froehlich, F. Schubert, in: Proceedings of 21st International Conference on Electrical Contacts (ICEC), September 9–12, 2002, Zuerich, Switzerland. [11] F. Schubert, B. Koehler, presented at: EUROMECH 419 colloquium ÔElastic Waves in NDT’, Prague, Czech Republic, October 2–5, 2000, available from the authors). [12] K.R. McCall, R.A. Guyer, Nonlinear Proces. Geophys. 3 (1996) 89. [13] K. Van Den Abeele, F. Schubert, V. Aleshin, F. Windels, J. Carmeliet, in: Proceedings of Ultrasonics International 2003, June 30–July 3, 2003, Granada, Spain, Ref UI325. [14] K.W. Winkler, X. Liu, J. Acoust. Soc. Am. 100 (3) (1996) 1392.