
Computer Simulation of Ultrasonic NonDestructive Testing of Concrete Using the Elastodynamic Finite Integration Technique (EFIT)
R. Marklein, K. Mayer, P. Ampha, K.J. Langenberg
University of Kassel, Department of Electrical Engineering/Computer Science
Electromagnetic Theory, Wilhelmshöher Allee 71, D34121 Kassel, Germany
marklein@unikassel.de, www.unikassel.de/fb16/tet/marklein
kmayer@unikassel.de, amphasuk@unikassel.de
langenberg@unikassel.de, www.tet.etechnik.unikassel.de
Abstract
This paper presents the computer simulation of ultrasonic nondestructive testing of concrete. The numerical modeling tool uses a geometry generator which generates a surface or volume material model on a discrete grid system in 2D or 3D and models the very heterogeneous character of the concrete material which consists in general of cement, a random distribution of several aggregates, and a certain water/air concentration. The propagation of ultrasonic waves in such a random material exhibits a complex mixture of multiple mode conversions and multiple scattering effects as well as absorption effects. In order to model these effects we apply the Elastodynamic Finite Integration Technique (EFIT). For the numerical simulations in 3D we developed a MessagePassing Interface (MPI) version of our EFIT code. Results are presented by means of timedomain data like snapshots of the ultrasonic wavefield. Appropriate and successful modeling with EFIT can also be utilized by an interactive design procedure for concrete test specimens as well as the synthetic data can serve as a test bed for imaging algorithms like the Fourier TransformSynthetic Aperture Focusing Technique (FTSAFT).
Introduction
For the quantitative numerical modeling of ultrasonic waves in concrete we apply the Elastodynamic Finite Integration Technique (EFIT) [3, 12]. Fig. 1a shows a crosssection of a typical concrete sample. The concrete material can be modeled by a random material model. Fig. 1b shows such a model using a random distribution of pixels/voxels defined on a computational grid and Fig 1c) displays a more realistic model using a random distribution of ellipsoids/spheroids representing a random distribution of aggregates (see [9, 11, 12] for first EFIT modeling results for concrete).
Fig 1: a) crosssection of a concrete sample; b) random mode using a pixel/voxel distribution on the used grid system; c) random model using a random distribution of ellipsoids/spheroids with different sizes, orientations, and fillings.

To generate such geometry models we developed a 2D/3D geometry generator which generates a random distribution of ellipsoids/spheroids defined on a 2D/3D surface/volume material grid. The ellipsoids/spheroids can vary randomly in size and orientation and can be filled with different aggregate materials. The random distribution of aggregates is computed according to a selected grading curve.
The mathematical description of the propagation of ultrasonic waves in such a random material is given by the governing equations of linear elastodynamics, the NewtonCauchy equation of motion and the deformation rate equation, which read for linear, inhomogeneous, anisotropic, instantaneously and locally reacting materials in integral form [2, 12]
 (1)

 (2)

v [m/s]
 Particle velocity vector
 [N/m]
 Cauchy's stress dyad

f [N/m^{3}]  Volume force density vector  [1/s]  Injected deformation rate dyad

r_{e0 }[kg/m^{3}]  Mass density at rest  [m^{2}/N]  Compliance tetrad

For isotropic media the compliance tensor can be written as a function of Lamé constants, l and m, in the following form [2]
 (3)

 (4)

After introducing an appropriate staggered grid complex in space and time (Fig. 2) we discretize Eq. (1) and (2) and obtain by introducing welldefined algebraic vectors and matrices the following explicit marchingonintime algorithm of leapfrog type [12]:
 (5)

 (6)

 (7)

 (8)

Fig 2: a) spatial staggered grid complex; b) temporal staggered grid complex; c) discrete Cartesian field components with 1 = x, 2 = y, and 3 = z.

Fig. 2 makes clear that we discretize a 3D concrete test specimen in small volume cells called voxels. If the region of interest, the part of the concrete sample which is under evaluation, is very large, then the number of volume cells as well as the number of unknowns are very large. This means that a computer system with sufficient main memory is required. To handle large 3D grid systems we recently installed a Beowulf compute cluster with 16 nodes connected via 1Gbps Ethernet. Each node consists of a single processor board with a Pentium IV, 3GHz, 800MHz system bus, and 1GBRAM; in total 16GBRAM can be used for simulations. The compute cluster represents a distributed memory computer. In order to run our EFIT code on such a distributed memory computer system we developed an MPI version of our EFIT code, where MPI stands for MessagePassing Interface [4, 5, 17]. On our computer systems which run Windows XP or Linux we use the freely available MPI implementation named MPICH [18]. In many reallife NDTCE situations the numerical modeling of the ultrasonic wave propagation and scattering must be restricted to a finite subdomain of the concrete sample. For these cases it is necessary to implement an absorbing boundary or an absorbing layer at the boundaries of the computational grid which is always of finite size. We have implemented the open boundary condition by Higdon [6] as an absorbing boundary and the perfectly matching layer (PML) [1] as an absorbing layer. For the numerical modeling of nondestructive testing procedures different measurement techniques are implemented, for instance, common techniques like the pulseecho and pitchcatch technique as well as LLT or TOFD [14, 17].
Computer Simulation of Ultrasonic Waves in Concrete Applying EFIT
2D and 3D Modeling Results Published in the Literature
2D and 3D EFIT modeling results of the numerical modeling of ultrasonic wave propagation and scattering can be found in [914] and references therein. Further examples are given in [7, 19, 20] and references therein.
Optimization of an Ultrasonic Phased Array System for LowFrequencies
An ultrasonic phased array system using commercial probes with a center frequency of 200 kHz has been built at the Federal Institute for Materials Research and Testing (BAM). We have implemented a special transducer model for the phased array system with adjustable amplitude and phase tapering and performed several EFIT simulations for different material configurations. Results can be found in [16].
2D EFIT Modeling  Validation of Imaging Algorithms  FTSAFT
For the validation of imaging algorithms applied in ultrasonic NDTCE carefully designed and welldefined concrete samples are needed. The fabrication of such concrete samples needs a lot of resources. These efforts can be minimized by using synthetic data generated by computer simulations for a first validation step. In collaboration with the Federal Institute for Materials Research and Testing (BAM) we performed a parameter study to design an optimal concrete test sample for this task. The EFIT simulation comprises a pulseecho experiment for 90probe positions and a 2D grid of 4000x1400 cells which corresponds to a sample size of 1000x350mm. The computations were performed on a Sun parallel computer system at the University of Ulm, Ulm, Germany. The computational time was ca. 380CPUhours. By varying the material properties we are able to differentiate between the influence of multiple scattering effects and effects which are inherently given by the Synthetic Aperture Focusing Technique (SAFT) und FTSAFT (Fourier TransformSAFT) (see [15] for FTSAFT applied in electromagnetic NDTCE and references therein).
Fig. 3 shows a result out of 8 simulations for a concrete sample with a grading curve A16 and an air concentration of 2% (see Fig. 3a). A selected snapshot of the ultrasonic wavefield is displayed in Fig. 3b. At the displayed time point the pressure wave is reflected at the backwall. The white spaces represent the backwallbreaking notches. Fig. 3c displays a Bscan obtained from 90 pulseecho simulations. The backwall can be recognized at t @ 150ms and the echoes from the notches are visible at t@ 130ms. In the reconstruction in Fig. 3d) the upper flat surfaces of the notches are displayed at a depth of 300mm which are detectable from a width of 20mm. This result motivated the reduction of the sample depth from 350mm to 250mm.
Fig 3: a) 2D geometry of a planned concrete sample with different backwallbreaking notches; b) 2D wavefield snapshot; c) Bscan image; d) FTSAFT reconstruction.

Several simulations were performed with different material configuration and reconstructions were computed with FTSAFT. The FTSAFT reconstruction of the synthetic data for the modified sample depth is shown in Fig. 4 and, in contrast, Fig.5 displays the FTSAFT reconstruction of the experimental data (see also [21]). Due to the smaller depth the S/N ratio increases and a better resolution is obtained. A reconstruction of the experimental data obtained for the test sample BAM NB 3 displays an even better resolution. This is due to the fact that the scattering amplitude of a cylindrical air inclusion (2D) is higher than that of a spherical air inclusion (3D). 3D EFIT simulations are impossible at the moment because of the huge memory requirements and computational time. Nevertheless, we developed a MessagePassing Interface (MPI) version of our EFIT code to perform such reallife 3D simulations. First results are discussed in the following section.
Fig 4: FTSAFT reconstruction using synthetic data of a modified test sample.

Fig 5: FTSAFT reconstruction using experimental data (150 kHz) of a realized test sample BAMNB3 (x=0 is not adjusted).

3D Modeling Applying the Standard and the MPI Version of EFIT
Fig.6 shows a 3D concrete model generated with the developed 3D geometry generator. In a first example we consider a relatively small 3D grid of 100x100x100 cells. We used a grading curve A16 and three different aggregates. We simulated a plane wave excitation with the standard and the MPI version of 3D EFIT. In the simulation with the standard version we applied a plane wave boundary condition at the vertical boundaries and in the simulation with the MPI version a PML boundary at all boundaries except the boundary at x = 0. Fig.7 displays the results for the standard version and Fig. 8 gives the results of the MPI version where we used a Cartesian MPI topology of 2x2x2=8 computational nodes. The results differ in some details because of the different boundary conditions. For example the absorbing effects of the PML boundary condition are visible clearly in the x slice and y slice given in Fig. 8; the backwalls in the 3D view in Fig. 8 are also white. The white crosses in Fig. 8 indicate the positions of the MPI communication boundaries. These results obtained on our recently installed compute cluster are very promising and we are now ready to simulate reallife 3D concrete structures.
Fig 6: 3D material index distribution as an output of the 3D geometry generator defining the geometry of a 3D concrete sample for the 3D EFIT modeling.

Fig 7: Computer simulation of ultrasonic wave propagation in concrete using the standard version of the 3D EFIT code.

Fig 8: Computer simulation of the ultrasonic wave propagation in concrete using the MPI version of the 3D EFIT code. The white crosses indicate the MPI communication boundaries. Here a Cartesian topology of 2 x 2 x 2 = 8 computational nodes is used.

Conclusions
We have presented computer simulations of the ultrasonic wave propagation and scattering in concrete using the Elastodynamic Finite Integration Technique (EFIT). The EFIT scheme is very robust, stable and flexible concerning the handling of complex materials and boundary conditions like the absorbing layer (PML). The results show that EFIT is a very successful tool for the simulation and optimization of reallife ultrasonic NDTCE situations, even in 3D, when using the MPI version.
Acknowledgements
The presented work is part of a project within the framework of a German Research Council funded research group (FOR 384, www.for384.unistuttgart.de) which is also supported by several industrial companies. We would like to thank Dr. M. Krause, D. Streicher, and F. Mielentz of the Federal Institute for Materials Research and Testing (BAM), Berlin, Germany for their fruitful collaboration.
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