Analysis of Linear Elastic Fracture Mechanics for Cracked Functionally Graded Composite Plate by Enhanced Local Enriched Consecutive-interpolation Elements

This paper reports the application of consecutive-interpolation procedure into four-node quadrilateral elements for analysis of two-dimensional cracked solids made of functionally graded composite plate. Compared to standard finite element method, the recent developed consecutive-interpolation has been shown to possess many desirable features, such as higher accuracy and smooth nodal gradients it still satisfies the Kronecker-delta property and keeps the total number of degrees of freedom unchanged. The discontinuity in displacement fields along the crack faces and stress singularity around the crack tips are mathematically modeled using enrichment functions. The Heaviside function is employed to describe displacement jump, while four branch functions being developed from asymptotic stress fields are taken as basis functions to capture singularities. The interesting characteristic of functionall graded composite plate is the spatial variation of material properties which are intentionally designed to be served for particular purposes. Such variation has to be taken into account during the computation of Stress Intensity Factors (SIFs). Performance of the proposed approach is demonstrated and verified through various numerical examples, in which SIFs are compared with reference solutions derived from other methods available in literatures.

Riccati PDEs That Imply Curvature-Flatness

In this paper the following three goals are addressed. The first goal is to study some strong partial differential equations (PDEs) that imply curvature-flatness, in the cases of both symmetric and non-symmetric connection. Although the curvature-flatness idea is classic for symmetric connection, our main theorems about flatness solutions are completely new, leaving for a while the point of view of differential geometry and entering that of PDEs. The second goal is to introduce and study some strong partial differential relations associated to curvature-flatness. The third goal is to introduce and analyze some vector spaces of exotic objects that change the meaning of a generalized Kronecker delta projection operator, in order to discover new PDEs implying curvature-flatness. Significant examples clarify some ideas.

An efficient meshless radial point collocation method for nonlinear p-Laplacian equation

This paper considered the spectral meshless radial point interpolation (SMRPI) method to unravel for the nonlinear p-Laplacian equation with mixed Dirichlet and Neumann boundary conditions. Through this assessment, which includes meshless methods and collocation techniques based on radial point interpolation, we construct the shape functions, with the Kronecker delta function property, as basis functions in the framework of spectral collocation methods. Studies in this regard require one to evaluate the high-order derivatives without any kind of integration locally over the small quadrature domains. Finally, some examples are given to illustrate the low computing costs and high enough accuracy and efficiency of this method to solve a p-Laplacian equation and it would be of great help to fulfill the implementation related to the element-free Galerkin (EFG) method. Both the SMRPI and the EFG methods have been compared by similar numerical examples to show their application in strongly nonlinear problems.

A Technique for Seamless Forecast Construction and Validation from Weather to Monthly Time Scales

Seamless prediction means bridging discrete short-term weather forecasts valid at a specific time and time-averaged forecasts at longer periods. Subseasonal predictions span this time range and must contend with this transition. Seamless forecasts and seamless validation methods go hand-in-hand. Time-averaged forecasts often feature a verification window that widens in time with growing forecast leads. Ideally, a smooth transition across daily to monthly time scales would provide true seamlessness—a generalized approach is presented here to accomplish this. We discuss prior attempts to achieve this transition with individual weighting functions before presenting the two-parameter Hill equation as a general weighting function to blend discrete and time-averaged forecasts, achieving seamlessness. The Hill equation can be tuned to specify the lead time at which the discrete forecast loses dominance to time-averaged forecasts, as well as the swiftness of the transition with lead time. For this application, discrete forecasts are defined at any lead time using a Kronecker delta weighting, and any time-averaged weighting approach can be used at longer leads. Time-averaged weighting functions whose averaging window widens with lead time are used. Example applications are shown for deterministic and ensemble forecasts and validation and a variety of validation metrics, along with sensitivities to parameter choices and a discussion of caveats. This technique aims to counterbalance the natural increase in uncertainty with forecast lead. It is not meant to construct forecasts with the highest skill, but to construct forecasts with the highest utility across time scales from weather to subseasonal in a single seamless product.

A computational homogenization analysis of materials using the stabilized mesh-free method based on the radial basis functions

This study presents a novel application of mesh-free method using the smoothed-radial basis functions for the computational homogenization analysis of materials. The displacement field corresponding to the scattered nodes within the representative volume element (RVE) is split into two parts including mean term and fluctuation term, and then the fluctuation one is approximated using the integrated radial basis function (iRBF) method. Due to the use of the stabilized conforming nodal integration (SCNI) technique, the strain rate is smoothed at discreted nodes; therefore, all constrains in resulting problems are enforced at nodes directly. Taking advantage of the shape function satisfies Kronecker-delta property, the periodic boundary conditions well-known as the most appropriate procedure for RVE are similarly imposed as in the finite element method. Several numerical examples are investigated to observe the computational aspect of iRBF procedure. The good agreement of the results in comparison with those reported in other studies demonstrates the accuracy and reliability of proposed approach. Keywords: homogenization analysis; mesh-free method; radial point interpolation method; SCNI scheme.

Discrete Probability Theory

The chapter presents an overview of various interpretations of probability. It introduces a ‘model probability,’ which assumes that all microscopic states that are essentially alike have the same probability in equilibrium. A justification for this fundamental assumption is provided. The basic definitions used in discrete probability theory are introduced, along with examples of their application. One such example, which illustrates how a random variable is derived from other random variables, demonstrates the use of the Kronecker delta function. The chapter further derives the binomial and multinomial distributions, which will be important in the following chapter on the configurational entropy, along with the useful approximation developed by Stirling and its variations. The Gaussian distribution is presented in detail, as it will be very important throughout the book.

Image Preservation using Wavelet Based On Kronecker Mask, Birge - Massart And Parity Strategy

Image carries more information about the ideas than text. Growth of social media, images has become the universal language because it is more interactive. Images are used in different fields like medical, multimedia, industries etc. When using the images, we need to find effective storage and transmission methods to reduce storage size and transmission time. Lossless and lossy are two ways to compress the image to reduce the storage and transmission time. The proposed method implements the concept of lossless image compression using the method of Kronecker delta notation, wavelet based on Birge-Massart strategy and parity strategy. This paper presents that enhancing the image by applying the Kronecker delta notation as the mask and applying the wavelet based on Birge-Massart strategy, finally applying the parity threshold to compress the image. The proposed method is compared the compression ratio (CR) with the existing lossless compression methods such as Birge – Massart without the enhanced method and Unimodal method. This proposed algorithm is very simple and more efficient to reduce the storage capacity and maintain the quality of an image than the existing lossless compression techniques. The experimental result shows that the Birge-Massart strategy combined with Kronecker mask and parity threshold produces the best CR than the simple Birge- Massart(without enhancement and threshold) strategy. This efficient method is proved by without loss of information of an original image with low MSE, high PSNR and high CR. An experimental result shows that the proposed algorithm achieved maximum CR of 146% on medical images and maximum CR of 19.2% on standard images than the existing methods.