تجزیه و تحلیل حساسیت خستگی با استفاده از روش متغیر مختلط

The sensitivity of the computed cycles-to-failure and other lifing estimates to the various input parameters is a valuable, yet largely unexploited, aspect of a fatigue lifing analysis. Two complex variable sensitivity methods, complex Taylor series expansion (CTSE) and Fourier differentiation (FD), are adapted and applied to fatigue analysis through the development of a complex variable fatigue analysis software (CVGROW). The software computes the cycles-to-failure and the sensitivities of the computed cycles-to-failure with respect to parameters of interest such as the initial crack size, material properties, geometry, and loading. The complex variable methods are shown to have advantages over traditional numerical differentiation in that more accurate and stable first and second order derivatives are obtained using CTSE and more accurate and stable higher order derivatives are obtained using FD.

The fatigue of structural components due to repeated loading cycles is a detrimental and dangerous problem. Structural failure due to fatigue can lead to costly and time consuming repairs, early retirement or catastrophic failure of structural and mechanical systems. Therefore, the calculation of the estimated “life” of a component or a structure is a critical element in a fracture control plan. As a result, there are a number of in-house and commercial computer codes such as AFGROW [1], NASGRO [2], and DARWIN [3] for fatigue crack growth evaluation. These codes compute an estimate of the cycles-to-failure of a cracked geometry under load by integrating the crack growth rate equation until failure occurs, with failure typically defined as net-section-yield or unstable fracture. Other ancillary outputs such as crack size and the stress intensity factor as a function of loading cycles, the residual strength, and the critical crack size are also provided. The inputs that are traditionally considered are the initial crack size, the applied loading, the geometry, and the material properties, and, in some cases, inspection schedules and probability-of-detection curves. The sensitivity of the computed cycles-to-failure to the various input parameters is a valuable, yet largely unexploited, aspect of a fatigue analysis. The fatigue analysis often contains: (a) a significant uncertainty or variability in the input parameters; (b) simplifications in the modeling such as a simplified geometry and loading; and (c) numerical approximations, e.g., when computing the geometry correction factor using handbook solutions or curve fits to numerical results such as weight functions. Sensitivity analysis can convey the significance of the various inputs on the computed cycles-to-failure. For example, the sensitivities express the expected change in the computed cycles-to-failure given a small change in the initial crack size, the part geometry, the loading cycles (gust, maneuver, ground-air-ground), the material properties (crack growth rate, fracture toughness), etc. If a sensitivity is relatively large, then a more thorough data collection effort or thorough analysis may be warranted. As a result, methods that determine the sensitivity of the cycles-to-failure to the inputs are valuable additions to a fatigue analysis. Both deterministic and probabilistic approaches can be applied for sensitivity analysis. Deterministic studies to determine the partial derivative of the cycles-to-failure with respect to each input of interest can be obtained using numerical finite differencing. McGinty discussed the sensitivities of damage tolerance analysis (DTA) elements by examining the deterministic ratio of the percent change of output to the percent change of input, i.e., the nondimensionalized derivative of the governing equation [4]. The following studies were developed: sensitivity of the stress intensity factor to the stress intensity geometry correction factor (β), the applied stress, and the crack size; sensitivity of the crack growth rate to β, Paris coefficient, Paris exponent, and crack size; sensitivity of the fatigue life to β, Paris coefficient, Paris exponent, and initial and final crack sizes. The results indicated that the geometry correction factor (β) and the applied stress are the important factors with respect to fatigue life. Fawaz and Harter studied the impact of various parameters on DTA estimates using deterministic parameter studies [5]. The expected variation in the parameters was estimated using engineering judgment. The cycles-to-failure was used as a metric of importance. Five different cracking scenarios were studied of a transport aircraft fuselage crack under remote tension: a single through crack at a hole, a double through crack at a hole, a single corner crack at a hole, a double corner crack at a hole, and a double oblique through crack at a hole. The material properties and their variation were chosen as representative of 2024-T3 aluminum sheet. Eight parameters were deemed important and, per the analysis results, divided into first and second order effects. The first order parameters were the initial flaw size, the geometry correction factor, the load interaction model, the crack growth rate data, and the stress intensity factor. The secondary parameters were the yield stress, fracture toughness, and threshold stress intensity factor. The distinction between first and second order effects was based on whether the life-cycle costs could be reduced via a more appropriate inspection schedule or if flight safety was affected. The parameters within each category were not comparatively ranked. Millwater and Wieland considered the probabilistic sensitivities of a T-38 wing (corner crack growing from a fastener hole) with respect to initial crack size and aspect ratio, hole diameter, crack growth rate, hole edge distance, geometry correction factor, fracture toughness, retardation and loading spectrum [6]. The conclusion from their study was that the probability-of-failure was sensitive to the fastener hole diameter, β, the crack growth rate at higher ΔK’s, and the stress spectrum scale factor. Of these variables, only the geometry correction factor was sensitive to variation in both its mean and standard deviation. Probabilistic approaches, while more comprehensive than deterministic approaches, are more arduous since significantly more data are required to determine the probability distributions of the inputs and the analysis is more time consuming to execute. Also, the results typically identify, after the fact, unimportant variables for which the data collection effort was not warranted. Therefore, fast yet accurate deterministic sensitivity methods have an important role to play. Numerical finite differencing is a straightforward commonly-used method to evaluate the derivatives of implicit functions. The method is simple in concept: change a parameter, rerun the analysis and determine the ratio of change in cycles-to-failure to the change in the input parameter. However, an estimate of the derivative is only accurate when the step size is small. On the other hand, when subtracting near-equal numbers, machine round off can also introduce error. As a result, the method is sensitive to the step size, which cannot be too large or too small. This becomes even more of a problem when calculating higher order derivatives since more subtraction operations are required. In summary, finite differencing, while easy to do, is prone to numerical issues that may be difficult to discern. It is typical that a laborious trial-and-error effort is required to locate a step size that results in a satisfactory derivative. Alternatives to finite differencing methods are complex variable sensitivity methods, in particular, complex Taylor series expansion (CTSE) and Fourier differentiation (FD). CTSE was originally described by Lyness and Moler [7] and [8] and was brought to the attention of the engineering community by Squire and Trapp [9]. In CTSE, the first derivative is calculated by perturbing the parameter of interest along the imaginary axes. For example, the initial crack size is given an imaginary perturbation, e.g., a0 + ih, where i denotes an imaginary number and h the step size. As derived below, the sensitivity is then estimated by evaluating the imaginary component of the cycles-to-failure and dividing by the step size. Consequently, CTSE involves no difference operations, thus, allowing for the step size to be made arbitrarily small. Hence the issue of choosing an accurate step size is avoided, making CTSE an easy to implement and highly accurate method for the numerical calculation of first derivatives. In order to calculate the higher order derivatives using CTSE, additional sample points along the imaginary axis are needed, then, in this case, differencing operations are necessary and the step size must be chosen carefully. Fourier differentiation (FD) is analogous complex variable sensitivity method for determining higher order derivatives using sampling in the complex plane. This technique was first described by Lyness et al. [7] and [8] FD requires the evaluation of sample points along a circular contour around the initial point in the complex plane and a fast Fourier transform (FFT) of the evaluated sample points is used to calculate the derivatives. The use of the FFT as a method to calculate derivatives was described by Lyness et al. [10] and by Henrici [11] and more recently by Bagley [12] to compute of sensitivities of implicit functions. The advantage of FD is that it can compute higher order derivatives more accurately than CTSE or finite differencing. The number of derivatives that can be obtained from the FFT is related to the number of sample points chosen. CTSE has been applied in several engineering fields but as yet is not widely known nor applied. In fluid dynamics, CTSE has been used to find sensitivities for the solution of the Navier–Stokes equation [13]. Furthermore, researchers have been able to apply CTSE techniques to finite element methods in the field of aerodynamics and aero-structural analysis [14] and [15]. CTSE has also been applied in the study of heat transfer [16], dynamic system optimization [17], pseudospectral [18] and eigenvalue sensitivity methods [19]. CTSE has been compared with automatic differentiation and shown that it is equivalent to the forward mode with comparable accuracy and much simpler implementation [20]. CTSE has been implemented within a solid mechanics finite element program to compute shape sensitivities with good accuracy and it was shown that the accuracy of the derivatives were a function of the finite element mesh [21]. Each of these applications uses the same mathematical CTSE approach, thus, substantiating the generality of the method. A review of the literature has not found instances of CTSE being applied to fatigue problems. Standard finite difference techniques have been used in fatigue problems [22] and [23], as well as probabilistic sensitivity methods [6] and [24]. However, by adopting complex variable numerical differentiation techniques, it is possible to obtain more accurate sensitivity estimates. It is worth noting that several alternatives to numerical differentiation exist including the equation-based methods of direct differentiation and adjoint differentiation, as well as the code-based method of automatic differentiation. These methods calculate derivates via explicit differentiation and can offer high accuracy and are highly efficient for systems with a large number of input or output variables. However, these methods require either the lengthy derivation of auxiliary equations or heavy modification of existing code and as such have a high cost-of-entry. In contrast, complex variable sensitivity methods are straightforward in concept and offer excellent accuracy; with the caveat that the fatigue crack growth evaluation must be computed using complex variables. For this reason, this paper compares complex variable sensitivity and finite differencing methods. The goal of this paper is to explore the use of complex variable methods as an alternative to the finite differencing method in the determination of the sensitivities of cycles-to-failure with respect to input quantities such as loading amplitude, crack growth parameters, initial crack size, and fracture toughness, for typical fatigue analyses.

Complex variable numerical differentiation techniques provide a straightforward and accurate method to calculate sensitivities useful for fatigue analysis. CTSE is analogous to finite differencing however the parameter of interest is perturbed along the imaginary axis rather than the real axis. The advantage this provides is that the first order derivative can be obtained without any differencing operations; thus, allowing the step size to be as small as desired and avoiding the troublesome step size issue that is problematic for finite differencing. FD requires real and imaginary perturbations to the parameter of interest such that samples are obtained in a circle within the complex plane. The complex cycles-to-failure results are then post processed through an FFT routine to obtain the derivatives. A complex variable fatigue analysis software was written that is emblematic of the solutions required for a typical fatigue analysis. The complex variable version required only minor alterations of a traditional real valued fatigue analysis code. The software is complex in that any input (initial crack size, material property, loading) can be a complex variable; as a result, the cycles-to-failure becomes complex. For CTSE, the real portion of the cycles-to-failure variable holds the traditional cycles-to-failure results and the imaginary portion holds the derivative times the imaginary step size. All three methods, CD, CTSE and FD were shown to give accurate first and second order derivatives in general; however, CTSE outperforms the traditional CD method for the calculation of first and second order sensitivities in that the results are more stable, particularly for larger crack growth strides (Δc). In terms of computational cost, CTSE requires half the number of sample points required by CD, albeit each complex sample point requires takes about three times longer to evaluate. This effectively makes the computational cost of CTSE approximately 1.5 times that of CD. FD clearly outperforms CD and CTSE for the calculation of derivatives above 2nd order. The calculation of high order derivatives through FD requires an increase in the number of sample points required, however this increase also leads to an increase in the stability and accuracy of the low order derivatives. By utilizing more sample points in FD, the stability of the low order derivatives approaches the stability of the low order derivatives as calculated by CTSE. In addition, the complex variable sensitivity methods can also be used to compute partial derivatives of any order.