A transformation technique to estimate the process capability index for non-normal processes

Estimating the process capability index (PCI) for non-normal processes has been discussed by many researches. There are two basic approaches to estimating the PCI for non-normal processes. The first commonly used approach is to transform the non-normal data into normal data using transformation techniques and then use a conventional normal method to estimate the PCI for transformed data. This is a straightforward approach and is easy to deploy. The alternate approach is to use non-normal percentiles to calculate the PCI. The latter approach is not easy to implement and a deviation in estimating the distribution of the process may affect the efficacy of the estimated PCI. The aim of this paper is to estimate the PCI for non-normal processes using a transformation technique called root transformation. The efficacy of the proposed technique is assessed by conducting a simulation study using gamma, Weibull, and beta distributions. The root transformation technique is used to estimate the PCI for each set of simulated data. These results are then compared with the PCI obtained using exact percentiles and the Box-Cox method. Finally, a case study based on real-world data is presented.

A transformation-based multivariate chart to monitor process dispersion

Multivariate monitoring techniques such as multivariate control charts are used to control the processes that contain more than one correlated characteristic. Although the majority of previous researches are focused on controlling only the mean vector of multivariate processes, little work has been performed to monitor the covariance matrix. In this research, a new method is presented to detect possible shifts in the covariance matrix of multivariate processes. The basis of the proposed method is to eliminate the correlation structure between the quality characteristics by transformation technique and then use an S chart for each variable. The performance of the proposed method is then compared to the ones from other existing methods and a real case is presented.

Introducing the Logarithmic finite element method: a geometrically exact planar Bernoulli beam element

We propose a novel finite element formulation that significantly reduces the number of degrees of freedom necessary to obtain reasonably accurate approximations of the low-frequency component of the deformation in boundary-value problems. In contrast to the standard Ritz–Galerkin approach, the shape functions are defined on a Lie algebra—the logarithmic space—of the deformation function. We construct a deformation function based on an interpolation of transformations at the nodes of the finite element. In the case of the geometrically exact planar Bernoulli beam element presented in this work, these transformation functions at the nodes are given as rotations. However, due to an intrinsic coupling between rotational and translational components of the deformation function, the formulation provides for a good approximation of the deflection of the beam, as well as of the resultant forces and moments. As both the translational and the rotational components of the deformation function are defined on the logarithmic space, we propose to refer to the novel approach as the “Logarithmic finite element method”, or “LogFE” method.

An approximate method for the solution of an influence function foil rolling model

Fleck and Johnson (Int. J. Mech. Sci. 29 (1987) 507) and Fleck et al. (Proc. Inst. Mech. Eng. 206 (1992) 119) have developed foil rolling models which allow for large deformations in the roll profile, including the possibility that the rolls flatten completely. However, these models require computationally expensive iterative solution techniques. A new approach to the approximate solution of the Fleck et al. (1992) Influence Function Model has been developed using both analytic and approximation techniques. The numerical difficulties arising from solving an integral equation in the flattened region have been reduced by applying an Inverse Hilbert Transform to get an analytic expression for the pressure. The method described in this paper is applicable to cases where there is or there is not a flat region.

A Petrov-Galerkin method for a singularly perturbed ordinary differential equation with non-smooth data

In this paper, a singularly perturbed ordinary differential equation with non-smooth data is considered. The numerical method is generated by means of a Petrov-Galerkin finite element method with the piecewise-exponential test function and the piecewise-linear trial function. At the discontinuous point of the coefficient, a special technique is used. The method is shown to be first-order accurate and singular perturbation parameter uniform convergence. Finally, numerical results are presented, which are in agreement with theoretical results.