Equivalent Normal Distribution Research Articles

Melt pool geometry and morphology variability for the Inconel 718 alloy in a laser powder bed fusion additive manufacturing process

Expanding on prior process mapping work by the authors, multiple melt pool cross-sections are measured at multiple process parameter combinations for the Inconel 718 alloy in a Laser Powder Bed Fusion (L-PBF) process. Collection of such data enables the study of the variability of melt pool geometry (e.g. width, depth, and cross-sectional area) across process space. Furthermore, the statistical distribution of the measured melt pool geometries is compared to that of an equivalent normal distribution and intriguing outliers are observed. The cross-sectional morphology of the melt pools are associated with defects such as keyholing porosity and balling and the variability of the defects is quantified. The final product of this work is a robust description of L-PBF In718 melt pool behavior, based on ex-situ observations, which can be linked to in-situ observations of melt pool morphology in future work.

A Modification to HL-RF Method for Computation of Structural Reliability Index in Problems with Skew-distributed Variables

The Hasofer-Lind and Rackwitz-Fiessler (HL-RF) method in reliability analysis is a popular iterative method for obtaining the reliability index. However, in the cases of limit state functions with skew-distributed variables, HL-RF method may give inappropriate answers. This paper represents a modification to HL-RF method in order to improve its performance in such problems. Based on this modification, non-normal distributions are replaced with equivalent skew-normal distributions instead of equivalent normal distributions. By this modification, asymmetric non-normal distributions are not replaced with symmetric distributions anymore. It is demonstrated that this consideration of skewness of non-normal distributions improves the behavior of HL-RF method and makes the proposed method more reliable. This improvement is shown through illustrative examples.

Reliability Analysis of Alumina-Based Ceramic Cutting Tool's Wear Life by Saddle Point Approximation

Under continuous cutting conditions, the reliability of alumina-based ceramic cutting tools wear life with abrasion wear was discussed extensively. An ultimate state equation of alumina-based ceramic cutting tools wear life was proposed according to stress-strength interference theory. Based on the premise that the probability distribution of random parameters has been known, the fracture toughness KIC and the hardness H which obey Weibull distribution were transformed into normal distribution by means of the equivalent normal distribution method, and then the reliability of alumina-based ceramic cutting tools wear life was analyzed by saddle point approximation method. The probability density function and cumulative distribution function of the ultimate state equation were accurately and quickly obtained by way of saddle point approximation, as a result, the saddle point approximation method was proved accurate with high computing speed in comparison with the Monte-Carlo method. Therefore, the application of the saddle point approximation method developed the reliability analysis theory of alumina-based ceramic cutting tools wear life.

Uncertainty in patient set-up margin analysis in radiation therapy

We investigated the uncertainty in patient set-up margin analysis with a small dataset consisting of a limited number of clinical cases over a short time period, and propose a method for determining the optimum set-up margin. Patient set-up errors from 555 registration images of 15 patients with prostate cancer were tested for normality using a quantile-quantile (Q-Q) plot and a Kolmogorov–Smirnov test with the hypothesis that the data were not normally distributed. The ranges of set-up errors include the set-up errors within the 95% interval of the entire patient data histogram, and their equivalent normal distributions were compared. The patient set-up error was not normally distributed. When the patient set-up error distribution was assumed to have a normal distribution, an underestimate of the actual set-up error occurred in some patients but an overestimate occurred in others. When using a limited dataset for patient set-up errors, which consists of only a small number of the cases over a short period of time in a clinical practice, the 2.5% and 97.5% intervals of the actual patient data histogram from the percentile method should be used for estimating the set-up margin. Since set-up error data is usually not normally distributed, these intervals should provide a more accurate estimate of set-up margin. In this way, the uncertainty in patient set-up margin analysis in radiation therapy can be reduced.

An SLP filter algorithm for probabilistic analytical target cascading

Decision-making under uncertainty is particularly challenging in the case of multidisciplinary, multilevel system optimization problems. Subsystem interactions cause strong couplings, which may be amplified by uncertainty. Thus, effective coordination strategies can be particularly beneficial. Analytical target cascading (ATC) is a deterministic optimization method for multilevel hierarchical system design that has been extended to probabilistic formulations. Solving the probabilistic optimization problem requires propagation of uncertainty, namely, evaluating or estimating the output distributions, a task that is computationally expensive for highly nonlinear functions. This article presents the use of sequential linear programming (SLP) for probabilistic ATC. By linearizing and solving a problem successively, the strategy takes advantage of the simplicity and ease of uncertainty propagation for a linear system under the assumption that inputs are normally distributed or can be transformed into equivalent normal distributions. A suspension strategy, developed for a deterministic SLP coordination strategy for ATC, is applied to reduce computational cost by suspending the analyses of subsystems that do not need considerable redesign. The accuracy and effectiveness of the proposed coordination strategy is demonstrated with several numerical examples.

THE AUTHORS' ANSWER TO DISCUSSION BY YAN-GANG ZHAO AND HIDEKI IDOTA

The authors thank Yan-Gang Zhao and Hideki Idota for their discussion. The answers are as follows;(1) We corrected the description of the definition of normalization according to their indication. And also we corrected the description related to the correlation coefficient ρ0,12 and the equation ƒ(χ1, χ2)(2) The cause of other indications seems to be the normalization. Joint probability density function for non-normal random variables was derived by bivariate normal distribution function. In the process of derivation, we used the normalization at any point of non-normal random variables. By this normalization the equivalent normal distribution can be gotten according to the point. Then the mean and standard deviation of the equivalent normal distribution are different at each point. But in the process of derivation, normal variate by normalization is transformed to standard normal variate, whose mean and standard deviation is zero and one respectively, even though the points are different. Finally the random variables included in the derived equation are standard normal variates and the original distribution function. So we think the derivation is correct.

Transformation of a nonnormal distribution of quantitative traits for breeding purposes

Some continuous quantitative traits such as yield are not always normally distributed. This article proposes an underlying normal distribution—called the equivalent normal distribution—to help analyze and interpret the distribution of such data. The mean and standard deviation of this distribution are estimated through the regression of the theoretical (assuming normality) selection intensity on the corresponding observed standardized selection differential at each selection rate that may be applied (as many selection rates as individuals in the distribution). This simple and pragmatic transformation does not require scale transformations, which sometimes obscure biological interpretation. A possible biological interpretation of the Equivalent Normal Distribution is presented; it considers the difference between the potential and the eventually observed performance. This method allows comparisons between distributions of apparently different shapes. Simulations furthermore indicated that Equivalent statistics, compared with observed statistics, allowed an improved prediction of genetic values and estimation of realised heritability.

Efficient Reliability Evaluation Using Spreadsheet

A practical method using spreadsheets is proposed for calculating the Hasofer-Lind second moment reliability index 13. Two example problems involving correlated normals and correlated nonnormals, respectively, are solved and compared with established mathematical approach. The proposed method is based on the per­ spective of an ellipsoid that is tangent to the failure surface in the original space of the random variables; concepts of transformed space or reduced variates are not required. Correlation is accounted for by setting up the quadratic form in the spreadsheet. Nonnormals are dealt with using established relationships between non­ normal distribution and its equivalent normal distribution. Iterative searching is performed automatically by invoking a spreadsheet's optimization tool. An advantage of the spreadsheet method is that partial derivatives are not part of the input; the method may therefore be especially attractive for cases with complicated or nonexplicit performance functions.

Geomagnetic storms, the Dst ring-current myth and lognormal distributions

The definition of geomagnetic storms dates back to the turn of the century when researchers recognized the unique shape of the H-component field change upon averaging storms recorded at low latitude observatories. A generally accepted modeling of the storm field sources as a magnetospheric ring current was settled about 30 years ago at the start of space exploration and the discovery of the Van Allen belt of particles encircling the Earth. The Dst global ‘ring-current’ index of geomagnetic disturbances, formulated in that period, is still taken to be the definitive representation for geomagnetic storms. Dst indices, or data from many world observatories processed in a fashion paralleling the index, are used widely by researchers relying on the assumption of such a magnetospheric current-ring depiction. Recent in situ measurements by satellites passing through the ring-current region and computations with disturbed magnetosphere models show that the Dst storm is not solely a main-phase, growth to disintegration, of a massive current encircling the Earth. Although a ring current certainly exists during a storm, there are many other field contributions at the middle-and low-latitude observatories that are summed to show the ‘storm’ characteristic behavior in Dst at these observatories. One characteristic of the storm field form at middle and low latitudes is that Dst exhibits a lognormal distribution shape when plotted as the hourly value amplitude in each time range. Such distributions, common in nature, arise when there are many contributors to a measurement or when the measurement is a result of a connected series of statistical processes. The amplitude-time displays of Dst are thought to occur because the many time-series processes that are added to form Dst all have their own characteristic distribution in time. By transforming the Dst time display into the equivalent normal distribution, it is shown that a storm recovery can be predicted with remarkable accuracy from measurements made during the Dst growth phase. In the lognormal formulation, the mean, standard deviation and field count within standard deviation limits become definitive Dst storm parameters.

Use of Stress-Strength Model in Determination of Safety Factor for Pressure Vessel Design

In pressure vessel design, the values of safety factor are still determined on the basis of engineering experience. Thus, they cannot properly reflect the influence of the consequences of failure and the variabilities in stress and strength. As a result, designsare often excessively conservative, while on the other hand, the possibility of failure still exists. Two approaches for determining the value of the safety factor, which are based on reliability analysis, are presented in this paper. As a result of a comparison, one approach based on a stress-strength model is found to be appropriate for pressure vessel design practice. By transforming the interference parts of the distributions of stress and strength into equivalent normal distributions, the approach allows stress and strength to have arbitrary distributions. Three examples, one in which a vessel is subjected to internal pressure, one in which a tall vessel is subjected to combined loads, and one in which a vessel is subjected to external pressure, are given in the paper. From threexamples, the principles for determining target reliability and the factors affecting the safety factor are discussed. It is concluded that by using the approach presented in this paper for pressure vessel design, different consequences of failure as well as variabilities in stress and strength can be taken into account. The approach yields a value for the safety factor that leads to a design which will be safer and yet more economical.