Implicit implementation of the nonlocal operator method: an open source code

In this paper, we present an open-source code for the first-order and higher-order nonlocal operator method (NOM) including a detailed description of the implementation. The NOM is based on so-called support, dual-support, nonlocal operators, and an operate energy functional ensuring stability. The nonlocal operator is a generalization of the conventional differential operators. Combined with the method of weighed residuals and variational principles, NOM establishes the residual and tangent stiffness matrix of operate energy functional through some simple matrix without the need of shape functions as in other classical computational methods such as FEM. NOM only requires the definition of the energy drastically simplifying its implementation. The implementation in this paper is focused on linear elastic solids for sake of conciseness through the NOM can handle more complex nonlinear problems. The NOM can be very flexible and efficient to solve partial differential equations (PDEs), it’s also quite easy for readers to use the NOM and extend it to solve other complicated physical phenomena described by one or a set of PDEs. Finally, we present some classical benchmark problems including the classical cantilever beam and plate-with-a-hole problem, and we also make an extension of this method to solve complicated problems including phase-field fracture modeling and gradient elasticity material.

Modeling of electric mass transfer process in controlled electrochemical resistance

The aim of the study is to mathematically simulate the concentration field of an electrolyte in a controlled electrochemical resistance. All processes occurring in the object are considered isothermal, the electric fields in all fragments of the electrochemical resistance are potential, plane-parallel. All physical and chemical parameters of the object’s materials are constant values. The kinetics of all electrode processes in the object of study is controlled by the diffusion stage in the electrolyte. As a research method, the operator method of Laplace and Fourier – expansion is used to obtain an analytical description of the concentration field of the electrolyte. In the course of the study, a mathematical model of the concentration field of the electrolyte with a uniform distribution of the current density was obtained. An analytical expression is obtained that allows to construct the surface of the change in the concentration of the electrolyte in the controlled electrochemical resistance. The concentration profiles of the electrolyte at different temperatures and the time dependences of the change in the concentration of the electrolyte are calculated.

OPERATOR METHOD IN THE PROBLEM OF A PLANE ELECTROMAGNETIC WAVE DIFFRACTION BY AN ANNULAR SLOT IN THE PLANE OR BY A RING

Purpose: The problem of a plane electromagnetic wave diffraction by an annular slot in the perfectly conducting zero thickness plane is considered. As a dual problem, the problem of diffraction by a perfectly conducting zero thickness ring is also considered. The paper aims at developing the operator method for the axially symmetric structures placed in free space. Design/methodology/approach: The problem is considered in the spectral domain. The scattered field is expressed in terms of unknown Fourier amplitudes (spectral functions). The annular slot is given as a unity of two simple discontinuities, namely of a disk and a circular hole in the plane, which interact with each other. The Fourier amplitude of the scattered field is sought as a sum of two amplitudes, the Fourier amplitude of the field of currents on the disk and Fourier amplitude of the field of currents on the perfectly conducting plane with circular hole. The operator equations are written for these amplitudes, which take into account the electromagnetic coupling of the disk and the hole in the plane. The equations use the reflection operators of a single isolated disk and a single hole in the plane. They are supposed to be known and can be obtained for example by the method of moments.The reflection operators can have singularities. After transformations, the equations are obtained, which are equivalent to the Fredholm integral equations of second kind and they can be solved numerically. Findings: The operator equations relative to the Fourier amplitudes of the field scattered by the discussed structure are obtained. The far zone scattered field for an annular slot and a ring for different values of parameters are studied. Conclusions: The rigorous solution of the problem of the electromagnetic wave diffraction by an annular slot in the plane and by a circular ring is obtained. The problem is reduced to the Fredholm integral equations of second kind. The far field distribution for different parameters is studied. The developed approach is an effective instrument for a number of problems of antenna technique to be solved. Key words: circular hole; disk; annular slot; ring; operator method; diffraction

VIKOR method for MCDM based on bipolar fuzzy soft β-covering based bipolar fuzzy rough set model and its application to site selection of solar power plant

Bipolarity indicates the positive and negative aspects of a particular problem. The concept behind the bipolarity is that a huge range of human decision analysis is involved in bipolar subjective thoughts. The VIKOR (Vlse Kriterijumska Optimizacija Kompromisno Resenje) which means multicriteria optimization and compromise solution, has already become a quite popular multi-criteria decision making tool for its computational simplicity and solution accuracy. In this article, we propose a hybrid model for multi-criteria decision-making (MCDM) based on bipolar fuzzy soft β-covering based bipolar fuzzy rough sets using VIKOR technique. It consists of a suitable redesign of the VIKOR approach so that it can use information with bipolar configurations. This method focuses on selecting and ranking from a set of feasible alternatives, and determines compromise solution for a problem with conflicting criteria to help the decision maker in reaching a final course of action. It determines the compromise ranking list based on the particular measure of closeness to the ideal solution. For illustration, the proposed technique is applied to a decision-making problems, namely, the selection of site for renewable energy project (solar power plant). A comparison of this method with another aggregation operator method and with the existing decision making algorithm Fuzzy VIKOR is also presented.

Development and Validation of a Model to Predict Rebleeding Within Three Days After Endoscopic Hemostasis for Peptic Ulcer Bleeding

Background: Peptic ulcer bleeding remains a typical medical emergency with significant morbidity and mortality. Peptic ulcer rebleeding often occurs within three days after emergency endoscopic hemostasis. Our study aims to develop a nomogram to predict rebleeding within three days after emergency endoscopic hemostasis for peptic ulcer bleedingMethods: We retrospectively reviewed the data of 386 patients with bleeding ulcers who underwent emergency endoscopic hemostasis between March 2014 and October 2018. The least absolute shrinkage and selection operator method were used to identified predictors. The model was displayed as a nomogram. Internal validation was carried out using bootstrapping. The model was evaluated using the calibration plot, decision-curve analyses and clinical impact curve. Results: Overall, 386 patients meeting the inclusion criteria were enrolled, with 48 patients developed rebleeding within three days after initial endoscopic hemostasis. Predictors contained in the nomogram included albumin, prothrombin time, shock, haematemesis/melena and Forrest classification. The model showed good discrimination and good calibration with a C-index of 0.854 (C-index: 0.830 via bootstrapping validation). Decision-curve analyses and clinical impact curve also demonstrated that it was clinically valuable.Conclusion: This study presents a nomogram that incorporates clinical, laboratory, and endoscopic features, effectively predicting rebleeding within three days after emergency endoscopic hemostasis and identifying high-risk rebleeding patients with peptic ulcer bleeding.Trial registration: This clinical trial has been registered in the ClinicalTrials.gov (ID: NCT04895904) approved by the International Committee of Medical Journal Editors (ICMJE).

Time-ordering in Heisenberg’s equation of motion as related to spontaneous radiation

Despite many years of research into Raman phenomena, the problem of how to include both spontaneous and stimulated Raman scattering into a unified set of partial differential equations persists. The issue is solved by formulating the quantum dynamics in the Heisenberg picture with a rigorous accounting for both time- and normal-ordering of the operators. It is shown how this can be done in a simple, straightforward way. Firstly, the technique is applied to a two-level Raman system, and comparison of analytical and numerical results verifies the approach. A connection to a fully time-dependent Langevin operator method is made for the spontaneous initiation of stimulated Raman scattering. Secondly, the technique is demonstrated for the much-studied two-level atom both in vacuum and in a lossy dielectric medium. It is shown to be fully consistent with accepted theories: using the rotating wave approximation, the Einstein A coefficient for the rate of spontaneous emission from a two-level atom can be derived in a manner parallel to the Weisskopf–Wigner approximation. The Lamb frequency shift is also calculated. It is shown throughout that field operators corresponding to spontaneous radiative terms do not commute with atomic/molecular operators. The approach may prove useful in many areas, including modeling the propagation of next-generation high-energy, high-intensity ultrafast laser pulses as well as spontaneous radiative processes in lossy media.

Reconstructed Interpolating Differential Operator Method with Arbitrary Order of Accuracy for the Hyperbolic Equation

We propose a family of multi-moment methods with arbitrary orders of accuracy for the hyperbolic equation via the reconstructed interpolating differential operator (RDO) approach. Reconstruction up to arbitrary order can be achieved on a single cell from properly allocated model variables including spatial derivatives of varying orders. Then we calculate the temporal derivatives of coefficients of the reconstructed polynomial and transform them into the temporal derivatives of the model variables. Unlike the conventional multi-moment methods which evolve different types of moments by deriving different equations, RDO can update all derivatives uniformly via a simple linear transform more efficiently. Based on difference in introducing interaction from adjacent cells, the central RDO and the upwind RDO are proposed. Both schemes enjoy high-order accuracy which is verified by Fourier analysis and numerical experiments.