Table of Contents ECNDT '98 Session: Simulation Copenhagen 26 - 29 May 1998

NDT Related Quantitative Modeling of Coupled Piezo electric and Ultrasonic Wave Phenomena R. Marklein Electromagnetic Theory, Department of Electrical Engineering, University of Kassel, D-34109 Kassel, Germany E-mail: marklein@tet.e-technik.uni-kassel.de Dedicated to Professor Dr. H. Wustenberg on the Occasion of his 60th Birthday This work received the NDT Innovation Prize for outstanding basic research

Abstract

This paper presents an overview of the development of numerical tools for the computational modeling of acoustic (A), elasto dynamic (E), and coupled piezo electric (P) and ultrasonic wave phenomena in the time domain related to nondestructive testing with ultrasound. F or this purpose, a numerical method, the Finite Integration T echnique (FIT), is directly applied to the set of governing equations of the underlying wave propagation in integral form. Space and time discretization of the governing equations on a staggered grid system results in a set of discrete matrix equations of three explicit time domain modeling codes acronymed PFIT, EFIT, and AFIT. Each of the discrete matrix equations has a one-to-one correspondent in the original set of equations. Analytical properties of the original set of equations are conserved. Mathematical properties like consistency and convergence of the codes are proofed and stability conditions are derived. The sparse matrices are highly suitable for high-performance vector computers, workstations, or even for personal computers and state-of-the-art laptops. To handle complex geometries, an IGES interface to commercial CAD systems is implemented, and to solve arbitrary initial-boundary-value problems several kinds of boundary conditions can be defined. The numerical codes PFIT, EFIT, and AFIT have widespread applications in quantitative computational NDT, for example starting from piezoelectric transducer mo deling and optimization to ultrasonic wave propagation, difraction, and scattering in inhomogeneous anisotropic dissipative and even statistically heterogeneous materials. Some of the analytical and experimental validations and applications to real life NDT problems are given in this paper illustrating the potential of these numerical modeling tools.

1 Introduction

The numerical time domain modeling of transient waves of all types in various disciplines of science and engineering is utilized for increasingly sophisticated applications to communication, remote sensing, medical imaging, and nondestructive testing. In recent years ultrasonic nondestructive evaluation has been more and more characterized by quantitative methods, which is particularly true for modern difraction tomographic imaging algorithms. These developments and their appropriate applications to increasingly complex NDT problems - complex relating to materials as well as to geometries - like inhomogeneous anisotropic austenitic welds and fier-reinforced composites, require a deep and fundamental physical understanding of ultrasonic wave propagation, difraction and scattering. Consequently, numerical codes to model these phenomena should be as close to real life as possible with only marginal or even no mathematical approximations. Only for canonical problems, where the geometries comply with coordinate systems in which the governing equations are separable, analytical solutions can be constructed, but, for the general case it is necessary to turn to direct numerical methods. This means that a direct discretization technique has to be applied to the governing equations of the underlying physical phenomena. Well known numerical methods of that kind comprise the
• Finite Diference Method (FDM) [e. g. Alterman , 1968; Bamberger et al. , 1980; Chinet al., 1984; Temple, 1988; Bondet al., 1988; Harker, 1988; Harker et al., 1990, 1991],
• Finite Element Method (FEM) [e.g. Ludwig, 1986; Ludwig & Lord, 1988; Ludwig et al., 1989; Lord et al., 1990; Lerch, 1990, 1991; Stam, 1990; Stam & Hoop, 1990; You et al., 1991; Roberts, 1991], and
• Finite-Diference Time-Domain (FD-TD) method [e.g. Madariaga, 1976; Aki & Richards, 1980; Virieux, 1984, 1986; Stanke & Chang, 1989].

In this paper, the well-established Finite Integration Technique (FIT) [Weiland, 1977; Fellinger, 1991a; Fellinger et al., 1995; Wolter, 1995b; Marklein, 1998; Bihn, 1998] which is a generalization of the Finite Volume Method (FVM) [e.g. Versteeg & Malalaskera, 1995], is applied to the governing equations in integral form to deriven umerical codes to simulate typical ultrasonic NDT situations as shown in Figure 1 including the piezo electricelements and external electrical circuit elements [ Marklein, 1998].

Figure 1: Typical Ultrasonic NDT situation: Nondestructive Testing with ultrasound of a block of steel with a backwall breaking crack

Figure 2: Numerical ultrasonic modeling tools: PFIT, EFIT, and AFIT based on FIT

Figure 2 relates the three explicit time domain modeling tools based on FIT [Marklein, 1998]:

• PFIT - Piezoelectric Finite Integration Technique,
• EFIT - Elasto dynamic Finite Integration Technique, and
• AFIT - Acoustic Finite Integration Technique.

PFIT is particularly developed for the modeling of the excitation and reception mode of a piezoelectric transducer including an external electrical load. Furthermore, PFIT includes the features of EFIT and AFIT. EFIT and AFIT have been designed to model the elasto dynamic wavefields in solids and acoustic wavefields in fluids and gases. Besides that, AFIT can be also used to model scalar approximations of elastic wavefields in solids.

Concluding Remarks

In this paper an overview on the n umerical modeling tools PFIT, EFIT, and AFIT has been giv en starting from the application of the Finite Integration Technique (FIT) to the governing equations of elasto dynamics and acoustics, the derivation of the algorithms EFIT and AFIT, the validation of the codes with analytical and numerical references, and the application to ultrasonic NDT real life situations. In the last section the recently dev eloped PFIT algorithm has been presented, which allows the computer simulation of a typical ultrasonic NDT problem including the piezo electric element and an external electrical load. As an example, the validation of the PFIT-code with experimental data has been given. All presented results are documenting the power of the established n umerical tools and give trust for further industrial applications:
• to help to understand ultrasonic wave propagation in complex materials and complex geometries,
• to optimize well-accepted NDT tec hniques, or
• to dev elop new and more efcient NDT techniques.

From an umerical point of view, some of the open problems, which are presently subject of the continuing work, are for example

• 3-D modeling of real life ultrasonic NDT situations,
• CAD/IGES in terface for inhomogeneous anisotropic materials (e. g., austenitic V-butt weld)
• irregular and triangular (Voronoigrid, Delaunay tessellation), and
• angled piezo electric excitation.

Acknowledgements

It is an honor for the author to dedicate this overview on AFIT, EFIT, and PFIT to Professor Dr. H. Wüstenberg on the occasion of his 60th birthday.
The author would like to express his gratitude to Professor Dr. K. J. Langenberg, who has not only stimulated the development of EFIT, but his continuous interest was extremely motivating to make the codes AFIT, EFIT, and PFIT what they are to day: powerful modeling tools in quantitative ultrasonic NDT.

Numerous former and present scientists of Professor Langenberg's group at the University of Kassel have contributed to the evaluation of AFIT, EFIT, and PFIT as well as their application to ultrasonic imaging:
Dr. P. Fellinger, S. Klaholz, Ch. Hofmann, T. Kaczorowski, Dr. K. Mayer, R. B Ärmann, J. Kostk a, R. Hannemann, and P. Zanger. Their help in preparing the material for various examples is gratefully acknowledged. Mrs. R. Brylla has carefully read the paper.

Finally, the author would like to appreciate the supp ort and the trustful collaboration with the following scientists:
O. Glitza, Measurement Techniques, University of Kassel, Kassel; Dr. V. Schmitz, F. Walte, W. Müller, and Dr. M. Spies, S. Schuhmacher, Fraunhofer-Institut für zerstörungsfreie Prüfverfahren (IzfP), Saarbrücken; Dr. C. Schurig and Dr. B. Köhler, Fraunhofer-Einrichtung für Akustische Diagnostik und Qualitätssicherung (EADQ) Dresden; Dr. M. Krause and Dr. H. Wiggenhauser, Bundesanstalt für Materialforschung und -prüfung (BAM), Berlin; Professor Dr. R. Ludwig, Worcester Polytechnic Institute, Worcester, USA; 38 Professor Dr. T. Weiland, H. Wolter, and M. Bihn, Theorieelektromagnetischer Felder, Darmstadt University of Technology, Darmstadt; Dr. A. Hecht, BASF, Ludwigshafen; H. Willems, Pipetronix, Stutensee; W. Biesle and W.-B. Klemmt, Daimler-Benz Aerospace Airbus, Bremen; Dr. T. Schmeidl, Siemens-KWU, Erlangen.