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The Role of Model Based Inversion

by R. Bruce Thompson*

The objective of NDT is to indirectly measure material properties or characterize anomalies. Model based inversion is a technique for achieving this goal. R. Bruce Thompson provides the historical setting for the need for crack sizing and proceeds to present a tutorial on model based inversion. Here he illustrates the organic development of model based inversion, starting with empirical relationships that the practitioner is familiar with and ending with the use of advanced models. Jeremy S. Knopp Guest Technical Editor

 

BACKGROUND: NEED FOR CRACK SIZING

In any NDT problem, the most important question is, "Given a set of experimental data, what is the condition of the material?" The answer is crucial to the disposition of the part: that is, whether to accept it for future use of to remove it from service. In every test, there is an accept/reject criterion implicit in the procedure. For example, in an ultrasonic test, the procedure might classify as a flaw (rejectable discontinuity) any indication that produced pulse/echo signals greater than 80% full screen height when the instrumentation was calibrated in a specified way. Such accept/reject criteria are ultimately related to the desire to remove from service discontinuities above a given size, and to the assumption that the signal depends on discontinuity size in a prescribed way.

In every test, there is an accept/reject criterion implicit in the procedure.

The determination of the flaw threshold (the size above which a discontinuity becomes rejectable) is based on the desire to maintain structural integrity, with the specifics depending on the design philosophy used. For example, many structures designed in the 1960s were based on a safe life design (Grandt, 2004). In that approach, the expected lives of structures were based on a mean life, determined by component fatigue tests, coupled with a safety factor (often a factor of 4). It was assumed that the initial condition of the fatigue test specimens was representative of that of the material placed into service. Failure was described in terms of the development of a macroscopic flaw. Accordingly, any discontinuity found in a test was grounds for rejection of the part since no credit was given for crack propagation beyond the initiation stage. Safe life design poses two serious problems. Nominally identical parts will generally have a wide distribution of fatigue lives, a consequence of the statistical nature of fatigue crack initiation and growth. If all are to be removed from service at the same time, and if that time is chosen such that the probability of failure during that time is small (10-3 is sometimes used by the US Air Force), then many of the parts removed from service could have been used for a greater period of time. It has been estimated in some cases that 75% of potentially useful life is lost in this strategy for some problems. Moreover, from the NDT perspective, there is another dilemma. As NDT techniques became more sensitive, smaller and smaller discontinuities can be detected. Hence, more and more components would be removed from service, even though the expected lifetime would not change. Within this scenario, there is no need for accurate sizing, since the existence of a discontinuity is grounds for rejection.

In the late 1960s, the development of fracture mechanics provided a major step forward in the understanding of the failure of metal parts under static and cyclic loads. Given the knowledge of stress levels, flaw size and some material parameters such as fracture toughness and crack growth parameters, it became possible to predict how fast cracks might grow under conditions of fatigue and at what size they would lead to catastrophic failure. This capability paved the way for a new damage tolerant design philosophy (Grandt, 2004). The better understanding of how cracks grew and failed allowed structures to be designed for use under conditions in which it was recognized that undetected cracks might be present and growing. For single load path structures, stress levels and allowable service life (before an inspection) are set such that a presumed pre-existent crack (of a size just below the detection limit of NDT) would not grow to failure. That detection limit is generally related to an estimate of the largest flaw that might be missed by NDT, since this would be the flaw that would lead to the shortest service life. As is shown schematically in Figure 1a, "for increased safety, the allowed service life is usually obtained by dividing the total crack growth period by a factor of 2" (Grandt, 2004). As illustrated in Figure 1b, the part could be used for additional periods of service if testing revealed that flaws larger than the detection limit had not yet developed, a strategy referred to as "retirement for cause" by the US Air Force. Related damage tolerant strategies include redundant load paths. These damage tolerant design and maintenance strategies are extremely attractive, since they allow many components to be used far beyond the life that would be determined based on safe life design while still providing a mechanism to identify those parts in which the fatigue process has led to significant cracks that would compromise the integrity of the structure.

Figure 1 - Schematic representation of the slow crack growth approach to damage tolerant design and maintenance: (a) determination of initial allowable life; (b) maintenance strategy designed to keep pre-existing discontinuities from growing to a size that causes failure of the structure (after Grandt, 2004).

The advent of damage tolerant design created a demand for the ability to quantitatively determine the sizes of discontinuities. Knowledge that a discontinuity was present was not enough. If testing procedures could be used to identify (for rejection) all discontinuities greater than a given size, then it would be known that the component could be used safely for some additional interval of time - perhaps well beyond that allowed in a safe life design.

MODEL BASED INVERSION

Definition

This incentive to be able to estimate the size of a discontinuity, rather than just establishing its existence, is the motivation for data inversion. By inversion, we refer to the process whereby one starts with measured data and then estimates the size (or other characteristics) of a discontinuity. Put in other words, one seeks to quantify the discontinuity or other damage that caused the signal. Inversion is simply a formal word for this process. Two ways to carry out this process are through the development of empirical relationships and through model based inversion. The former can be accomplished in a wide variety of ways, ranging from the knowledge that an inspector has gained over years of experience to more formal procedures based on artificial intelligence techniques such as neural networks. These have, as a common feature, the requirement that a large set of experimental experience, including NDT observations and independent determination of the sizes of the discontinuities by some metallurgical procedures, be available for training. Though very useful in a number of situations, these empirical approaches suffer from two problems. Gathering the necessary experimental information can be costly in terms of time and direct expense, and the empirical observations based on a particular problem (for example, the sizing of cracks in welds) may have little applicability to other problems (such as sizing second layer cracks in fastener holes).

Figure 2 - Essential elements of model based inversion: (a) forward problem; (b) inverse problem.

Model based inversion seeks to reduce the need for empiricism by taking advantage of insights from physics based models of the testing process (Thompson, 2005). Figure 2 illustrates the basic idea. Figure 2a conceptually illustrates what is sometimes called the forward problem. Given the specification of the component (including both its geometry and the properties of the materials from which it is fabricated), the discontinuity and the measurement instrumentation, the NDT signal(s) produced by the discontinuity is predicted. As illustrated in Figure 2b, the inverse problem involves working backwards. The inputs are the knowledge of the measurement system, component and NDT signal(s), while the outputs are the characteristics of the discontinuity (size, shape and orientation). At an intuitive level, it is clear that solutions to the forward problem can greatly assist in solving the inverse problem. One could imagine asking the simple question, "if the discontinuity were of a particular size, shape and orientation, what NDT signal would be predicted?" If the answer does not match the experimental observation (assuming that the accuracy of the physics based model used to predict that observation has been demonstrated via validation experiments), then one must pose the question again for a different candidate discontinuity. A process of iteration could be used to find a discontinuity whose predicted signal comes "closest" to the data. This sort of thinking has been practiced in elementary form for some time, as will be discussed in the next subsection. This will be followed by a discussion of current research directions that go far beyond current practice, enabled by advances in experimental instrumentation and in digital computers. In the context of developing these advanced techniques, a number of challenges must be considered that are indicated in Figure 2b.

Early Model Based Inversion Procedures

A common assumption underlying most NDT techniques is that some aspect of the signal becomes larger as the discontinuity size increases. This is the reason that thresholds can be used to determine whether discontinuities producing a given signal are candidates for further evaluation or rejection. Simple physics based models, some of which have been available for decades, are often used to quantify the relationship between discontinuity size and response for idealized discontinuities. Those idealized relationships are used to guide the interpretation of experimental data, with a correction for the anticipated differences in the responses of natural discontinuities from those of the idealized discontinuities often being introduced. Since these relationships were generally developed at a time in which NDT instrumentation was based on analog electronics, relatively simple features of the discontinuity signals were used in the interpretation. A few examples of these early, model based inversion strategies follow.

In the ultrasonic detection of discontinuities, accept/reject decisions are often made by comparing the amplitude of the signal to a threshold. Physics based models have long been used to guide this process. For example, it is well known that, for planar reflectors oriented perpendicularly to the ultrasonic propagation direction and smaller in size than the beam, the discontinuity response is proportional to the area of the reflector and inversely proportional to the square of the distance from the transducer to the reflector (as long as the reflector is in the far field of the beam). This behavior is graphically captured in distance/gain/size (DGS) diagrams, as are used in a number of industries (Krautkrämer and Krautkrämer, 1990). DGS diagrams, sometimes extended to take into account the effects of attenuation, are used in the interpretation of ultrasonic data obtained in the testing of forgings, weldments and other structures. Included in some test procedures are the use of DGS overlays on the oscilloscope display, to assist in the interpretation of test data. Of course, what the DGS approach determines is an equivalent reflector size, only a substitute for the true discontinuity size, which is generally greater. Hence, in setting rejection thresholds, the size of the equivalent reflector that is to be rejected is generally less than the size of the naturally occurring discontinuities of concern. This correction is intended to take into account the effects of the morphology of naturally occurring discontinuities (tilt, roughness, branching and so forth) on the pulse/echo response.

Because of these discontinuity morphology effects, the amplitudes of reflected signals are not always good indicators of the discontinuity size, and other features must be used. For example, in weld testing, it is not possible to illuminate a crack normal to its surface. In such cases, the majority of the energy of the reflected beam might not be directed back toward the transducer in a pulse/echo measurement, a problem that is most pronounced when the discontinuity is many wavelengths in extent. One widely used sizing technique to overcome this problem is based on the relative arrival times of signals diffracted from the tips of the cracks, which spread in all directions (Halmshaw, 1991). Such signals are not explicitly considered in the most elementary physics based models, such as those upon which the DGS approach is built, but are described by higher order models, for example, those based on the geometrical theory of diffraction (Achenbach et al., 1982). Crack sizing can be achieved by measuring the relative times of signals diffracted from opposite tips of an internal crack - known as the time of flight diffraction technique - or by comparing the time of the tip diffracted signal to the corner crack signal in a surface breaking crack - the relative arrival time technique (R/D Tech, 2004). These arrival times are related to size through simple formulae, derived from the geometry of the measurement configuration.

Similar examples can be given in other measurement modalities. For example, the need to monitor subcritical crack growth in the large offshore structures used to bring oil and gas to the surface in the North Sea greatly enhanced the development of the alternating current potential drop and field measurement techniques (Dover and Rudlin, 1995; Dover et al., 1986; Topp and Dover, 1991). Currents are injected into a sample and either the voltage drop between two electrodes straddling the crack (which has generally been detected by other means) or the fields around the crack are measured. For example, this technique has been used to measure crack profiles in threaded members of offshore structures, as reported in Dover et al. (1986). Models, based on the theory of electromagnetism, are an integral part of those procedures, being used to relate the potential drop and field distribution signals to the depth of a crack. In order to make the problem tractable, some simplifying assumptions were made. The field was assumed to be uniform in the vicinity of the crack and an "unfolding model" was used to describe the fields in the ferromagnetic steel. Practical tests have shown accuracy of 10% in crack sizing (Dover and Rudlin, 1995).

A unifying feature of all of these approaches is that a relatively simple algorithm, based on a physics based model that relates the experimental observations to the geometry of the discontinuity, is used to size the discontinuity. Generally, the model has been developed for an idealized situation. Good results can be obtained when the discontinuities and measurement geometry are in accord with the assumptions of the model or when the particular features that are the basis for the sizing are not strongly influenced by deviations from the idealized assumptions. However, the approaches can break down when the experimental situation deviates from this situation.

Emerging Model Based Inversion Procedures

Each of the examples cited above have been in practice for well over a decade, some considerably longer, and they have served the NDT community well. In the ensuing period, two changes have occurred that are opening the way for a new generation of more sophisticated, model based inversion techniques. Physics based models and simulation tools are considerably more mature in terms of the complexity of experimental situations that can be handled and the speed with which the necessary computations can be done (Amos et al., 2004; Thompson, 2001; Thompson, 2005). These advances have been made possible by increased understanding of the underlying physics, its description in simple cases by analytical formulae, the development of computational models for more general cases that take advantage of these analytical results, and the advances in the capabilities of modern computational systems that make these computational models widely accessible with reasonable computational times. In parallel, instrumentation has progressed from being primarily analog to primarily digital, providing permanent access to a much more complete set of experimental data for analysis. Consequences include the ability to implement more sophisticated calibration procedures that increase the fidelity of the experimental data and the availability of a richer data set as input to model based inversion procedures. It is now possible to implement a number of model based inversion procedures that could not be previously envisioned, and the articles in this technical focus issue provide examples.

There are a number of different strategies that are utilized in model based inversion, but all are based, in one way or the other, on the common idea of directly comparing the predictions of a physics based model to the experimental observations and varying the parameters in the description of the discontinuity until a "best fit" to the data is obtained. If one has sufficient independent information to know the general nature of the discontinuity (for example, that it is very likely that it is a corner crack growing out of a fastener hole), then the parametric description of the discontinuity may only have a few parameters that need to be determined. For example, suppose that the measured data, exp(f,x), are the ultrasonic or eddy current responses as a function of frequency, f, and probe position with respect to the discontinuity. Also suppose that a physics based model can predict these experimental observations with the discontinuity dimensions a and c as parameters, pred(f,x; a,c). Then one approach is to vary a and c until the fit is best, that is, to minimize the residual R, defined to be

In Equation 1, the subscripts denote the discrete frequencies and positions at which data are available. Figure 3 provides an illustration. Given the experimental data, a and c are adjusted until R is minimized, with a = 2 and c = 3 being the answers in this simple illustration. An early example of this approach to the eddy current sizing of an irregular crack was reported by Bowler (1995).

Figure 3 - Schematic illustration of inversion by the least squares method.

If initial information reducing the discontinuity description to a few parameters is not available, the interpretation may need to be more complex. For example, in approaches based on Bayesian statistics (Meeker and Escobar, 1998), one might assume that one has prior knowledge of a distribution of possible discontinuities (types and sizes) that might be present. The comparison of the experimental data to the physics based model predictions of the responses of those possible discontinuities allows one to reduce the probability of the presence of those that are inconsistent with the data. Thus, the output is a distribution of discontinuities that are consistent with the data rather than the deterministic estimate of a discontinuity size, assuming that such a discontinuity was indeed the cause of the signal. If the NDT technique is effective, this distribution will be considerably narrower than the distribution that was considered possible before the evaluation was performed.

The papers in this technical focus issue are a snapshot of current efforts in what is a very broad field of investigation, ranging from the development of simply implemented approximate approaches (such as were described in the previous section) to a careful examination of a number of fundamental issues, including the nature of the underlying mathematics. A detailed discussion of that work is beyond the scope of this article. However, to provide a flavor for the interested reader, a few examples with which the author is familiar are given. DeFacio (1982) has provided formal comments on the effects of noise and the ill-posed nature of the problem. Rose (2002) considered the relationship between applied imaging and exact inverse scattering theory for the case of ultrasonic NDT. A set of papers appearing in a special issue of the journal Inverse Problems dedicated to electromagnetic and ultrasonic NDT (Lesselier and Bowler, 2002) described a number of current research efforts. A search of the literature would reveal far more work whose discussion is beyond the scope of this article.

ROLE OF MODEL BASED INVERSION IN IMPROVING STRUCTURAL INTEGRITY

In the final analysis, the effectiveness of an NDT technique is measured by the probability that flaws that would produce failure during the intended lifetime of a part are removed from service. The exact way in which this is quantified depends on the application and on the organization defining and implementing the structural integrity program. For example, the US Air Force uses as input to their lifing strategies, which were schematically illustrated in Figure 1, the size of the "inspectable flaw." This is often defined as the minimum flaw size deemed reliably detected (probability of detection estimated to be 90% with a 95% confidence level). When a simple algorithm is used, (for example, one based on comparing the amplitude of an ultrasonic signal or an eddy current impedance change to a threshold), adjusting the threshold such that "all" undesirable flaws are rejected opens the possibility that many smaller discontinuities that would not be of immediate concern would also be rejected. As illustrated in Figure 4a, this is a consequence of the distribution of signal amplitudes produced by naturally occurring discontinuities of the same size. As is illustrated by the distributions shown for three discontinuity sizes, all naturally occurring discontinuities of the same size, a, do not produce the same signal amplitude. Small discontinuities, a₁, with characteristics favorable for testing, may produce signal amplitudes as large as those of significantly larger size, a₃, with less favorable characteristics (for example, tightness, tilt or morphology in the case of cracks). Given the testing threshold, the probability of detection is the area under the signal distribution to the right of the threshold, as indicated by the shaded region under the distributions. Because of the broad nature of these signal distributions, there will be considerable overlap, leading to a fairly slowly varying probability of detection curve, as shown in Figure 4c.

A more sophisticated, model based inversion would allow an estimate of size that is more accurate than is possible with the simple algorithm based on the implicit assumption that the size is proportional to amplitude. Put in other words, the distribution of discontinuity sizes estimated by a successful inversion algorithm could be much narrower than the distribution of signal amplitudes, as shown in Figure 4b. Comparing this estimated size to a size rejection threshold would lead to a much sharper probability of detection curve, as shown in Figure 4c.

Figure 4 - Schematic illustration of the role of model based inversion in sharpening the probability of detection curve: (a) broad distribution of signal amplitudes for discontinuities of sizes a1 < a2 < a3 = aNDT; (b) sharper distribution of estimated sizes for discontinuities resulting from model based inversion; (c) improved probability of detection curve that results. Shaded area in parts (a) and (b) indicates probability of detection for a particular threshold (accept/reject criteria) selection.

In the example given, the threshold was selected such that the 90% probability of detection curves of the two approaches coincided. This would have the consequence that, using the inversion strategy, fewer discontinuities larger than a_NDT would be accepted (greater safety), while avoiding rejecting parts with smaller discontinuities that would not be of concern in the intended usage (greater economics or readiness). Of course, the inversion algorithm might allow one to lower aNDT, such that smaller discontinuities could be rejected with confidence, thereby allowing stress levels or service intervals to increase.

References

Achenbach, J.D., A.K. Gautesan and H. McMaken, Ray Methods for Waves in Elastic Solids: With Applications to Scattering by Cracks, Boston, Pitman, 1982.

Amos, J., J. Gray, A. Lhemery and R.B. Thompson, "Future Applications of NDE Simulators," Review of Progress in Quantitative Nondestructive Evaluation, Vol. 23B, D.O. Thompson and D.E. Chimenti, eds., Melville, New York, AIP, 2004, pp. 1620-1632.

Bowler, J., "Eddy Current Inversion Using Gradient Methods," Nondestructive Testing of Materials, R. Collins et al., eds., Amsterdam, IOS Press, 1995, pp. 31-40.

DeFacio, B., "Rigorous Results on Inverse Source and Inverse Scattering Theory," Review of Progress in Quantitative Nondestructive Evaluation, Vol. 1, D.O. Thompson and D.E. Chimenti, eds., New York, Plenum Press, 1982, pp. 219-226.

Dover, W.D. and J.R. Rudlin, "Reliability of Crack Detection and Sizing for ACPD and ACFM," Nondestructive Testing of Materials, R. Collins et al., eds., Amsterdam, IOS Press, 1995, pp. 87-102.

Dover, W.D., R. Collins and D.H. Michael, "The Use of AC Field Measurements for Crack Detection and Sizing in Air and Underwater," Philosophical Transactions of the Royal Society of London, Vol. A320, 1986, pp. 271-283.

Grandt, A.F., Fundamentals of Structural Integrity: Damage Tolerant Design and Nondestructive Evaluation, Hoboken, New Jersey, Wiley, 2004.

Halmshaw, R., Non-destructive Testing, second edition, London, Edward Arnold, 1991.

Krautkrämer, Josef and Herbert Krautkrämer, Ultrasonic Testing of Materials, fourth edition, Berlin, Springer-Verlag, 1990.

Lesselier, D. and J. Bowler, eds., Special Issue on Electromagnetic and Ultrasonic NDE, Inverse Problems, 2002.

Meeker, W.Q. and L.A. Escobar, "Introduction to the Use of Bayesian Methods for Reliability Data," Statistical Methods for Reliability Data, Hobocken, New Jersey, Wiley, 1998, pp. 343-368.

R/D Tech, Introduction to Phased Array Ultrasonic Technology Applications, Quebec, R/D Tech, 2004.

Rose, J.H., "Time Reversal, Focusing and Exact Inverse Scattering," Imaging of Complex Media with Acoustic and Seismic Waves, Topics in Applied Physics, Vol. 84, M. Fink et al., eds., Berlin, Springer-Verlag, 2002, pp. 97-105.

Thompson, R.B., "Using Physical Models of the Testing Process in the Determination of Probability of Detection," Materials Evaluation, Vol. 59, 2001, pp. 861-865.

Thompson, R.B., "Simulation Models: Critical Tools in NDE Engineering," Materials Evaluation, Vol. 63, 2005, pp. 300-308.

Topp, D.A. and W.D. Dover, "Review of ACPD/ACFM Crack Measurement System," Review of Progress in Quantitative Nondestructive Evaluation, Vol. 10A, D.O. Thompson and D.E. Chimenti, eds., New York, Plenum Press, 1991, pp. 301-308.

 

* Center for Nondestructive Evaluation, Iowa State University, 1915 Scholl Road, Ames, IA 50011-3042; e-mail thompsonrb@cnde.iastate.edu.

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