3rd and 11th orders of accuracy of ‘linear’ and ‘quadratic’ elements for Poisson equation with irregular interfaces on Cartesian meshes

Purpose The purpose of this paper is as follows: to significantly reduce the computation time (by a factor of 1,000 and more) compared to known numerical techniques for real-world problems with complex interfaces; and to simplify the solution by using trivial unfitted Cartesian meshes (no need in complicated mesh generators for complex geometry). Design/methodology/approach This study extends the recently developed optimal local truncation error method (OLTEM) for the Poisson equation with constant coefficients to a much more general case of discontinuous coefficients that can be applied to domains with different material properties (e.g. different inclusions, multi-material structural components, etc.). This study develops OLTEM using compact 9-point and 25-point stencils that are similar to those for linear and quadratic finite elements. In contrast to finite elements and other known numerical techniques for interface problems with conformed and unfitted meshes, OLTEM with 9-point and 25-point stencils and unfitted Cartesian meshes provides the 3-rd and 11-th order of accuracy for irregular interfaces, respectively; i.e. a huge increase in accuracy by eight orders for the new 'quadratic' elements compared to known techniques at similar computational costs. There are no unknowns on interfaces between different materials; the structure of the global discrete system is the same for homogeneous and heterogeneous materials (the difference in the values of the stencil coefficients). The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of OLTEM at a given stencil width. The numerical results with irregular interfaces show that at the same number of degrees of freedom, OLTEM with the 9-points stencils is even more accurate than the 4-th order finite elements; OLTEM with the 25-points stencils is much more accurate than the 7-th order finite elements with much wider stencils and conformed meshes. Findings The significant increase in accuracy for OLTEM by one order for 'linear' elements and by 8 orders for 'quadratic' elements compared to that for known techniques. This will lead to a huge reduction in the computation time for the problems with complex irregular interfaces. The use of trivial unfitted Cartesian meshes significantly simplifies the solution and reduces the time for the data preparation (no need in complicated mesh generators for complex geometry). Originality/value It has been never seen in the literature such a huge increase in accuracy for the proposed technique compared to existing methods. Due to a high accuracy, the proposed technique will allow the direct solution of multiscale problems without the scale separation.

A Reliable Treatment for Nonlinear Differential Equations

In this paper, we use the concept of homotopy, Laplace transform, and He’s polynomials, to propose the auxiliary Laplace homotopy parameter method (ALHPM). We construct a homotopy equation consisting on two auxiliary parameters for solving nonlinear differential equations, which switch nonlinear terms with He’s polynomials. The existence of two auxiliary parameters in the homotopy equation allows us to guarantee the convergence of the obtained series. Compared with numerical techniques, the method solves nonlinear problems without any discretization and is capable to reduce computational work. We use the method for different types of singular Emden–Fowler equations. The solutions, constructed in the form of a convergent series, are in excellent agreement with the existing solutions.

A review of benchmark experiments for the validation of peridynamics models

Peridynamics (PD), a non-local generalization of classical continuum mechanics (CCM) allowing for discontinuous displacement fields, provides an attractive framework for the modeling and simulation of fracture mechanics applications. However, PD introduces new model parameters, such as the so-called horizon parameter. The length scale of the horizon is a priori unknown and need to be identified. Moreover, the treatment of the boundary conditions is also problematic due to the non-local nature of PD models. It has thus become crucial to calibrate the new PD parameters and assess the model adequacy based on experimental observations. The objective of the present paper is to review and catalog available experimental set-ups that have been used to date for the calibration and validation of peridynamics. We have identified and analyzed a total of 39 publications that compare PD-based simulation results with experimental data. In particular, we have systematically reported, whenever possible, either the relative error or the R-squared coefficient. The best correlations were obtained in the case of experiments involving aluminum and steel materials. Experiments based on imaging techniques were also considered. However, images provide large amounts of information and their comparison with simulations is in that case far from trivial. A total of 6 publications have been identified and summarized, that introduce numerical techniques for extracting additional attributes from peridynamics simulations in order to facilitate the comparison against image-based data.

SELF-SIMILARITY TECHNIQUES FOR CHAOTIC ATTRACTORS WITH MANY SCROLLS USING STEP SERIES SWITCHING

Highly applied in machining, image compressing, network traffic prediction, biological dynamics, nerve dendrite pattern and so on, self-similarity dynamic represents a part of fractal processes where an object is reproduced exactly or approximately exact to a part of itself. These reproduction processes are also very important and captivating in chaos theory. They occur naturally in our environment in the form of growth spirals, romanesco broccoli, trees and so on. Seeking alternative ways to reproduce self-similarity dynamics has called the attention of many authors working in chaos theory since the range of applications is quite wide. In this paper, three combined notions, namely the step series switching process, the Julia’s technique and the fractal-fractional dynamic are used to create various forms of self-similarity dynamics in chaotic systems of attractors, initially with two, five and seven scrolls. In each case, the solvability of the model is addressed via numerical techniques and related graphical simulations are provided. It appears that the initial systems are able to trigger a self-similarity process that generates the exact or approximately exact copy of itself or part of itself. Moreover, the dynamics of the copies are impacted by some model’s parameters involved in the process. Using mathematical concepts to re-create features that usually occur in a natural way proves to be a prowess as related applications are many for engineers.

Legendre-Galerkin Method, Dynamical System, and Optimal Control for Solving Covid-19 Model

The advancement of numerical demonstrating of irresistible illnesses is a key examination territory in different fields including the nature and the study of disease transmission. One point of these models is to comprehend the elements of conduct in irresistible infections. For the new strain of Covid (Coronavirus), there is no immunization to secure individuals and to forestall its spread up until now. All things being equal, control procedures related to medical services, for example, social separating, isolation, travel limitations, can be adjusted to control the pandemic of Coronavirus. This article reveals insights into the dynamical practices of nonlinear Coronavirus models dependent on strategy: the Legendre-Galerkin strategy. We summon a novel sign stream chart that is utilized to depict the Coronavirus model. Based on the Legendre-Galerkin method, the covid-19 model. Mathematica, as one of the world's leading computational software, was employed for the implementation of solutions. The proposed numerical techniques provide are excellent. Through our numerical investigations, it is uncovered that social removing between possibly tainted people who are conveying the infection and solid people can diminish or intrude on the spread of the infection. The mathematical reenactment results are insensible concurrence with the investigation forecasts. The free balance and dependability focus for the Coronavirus model is researched. Likewise, the presence of a consistently steady arrangement is demonstrate.

Modelling of Planetary Gravitational Microlensing Events

<p>This thesis describes and develops procedures for the generation of theoretical lightcurves that can be used to model gravitational microlensing events that involve multiple lenses. Of particular interest are the cases involving a single lens star with one or more orbiting planets, as this has proven to be an effective way of detecting extrasolar planets. Although there is an analytical expression for microlensing lightcurves produced by single lensing body, the generation of model lightcurves for more than one lensing body requires the use of numerical techniques. The method developed here, known as the semi-analytic method, involves the analytical rearrangement of the relatively simple ‘lens equation’ to produce a high-order complex lens polynomial. Root-finding algorithms are then used to obtain the roots of this ‘lens polynomial’ in order to locate the positions of the images and calculate their magnifications. By running example microlensing events through the root-finding algorithms, both the speed and accuracy of the Laguerre and Jenkins-Traub algorithms were investigated. It was discovered that, in order to correctly identify the image positions, a method involving solutions of several ‘lens polynomials’ corresponding to different coordinate origins needed to be invoked. Multipole and polygon approximations were also developed to include finite source and limb darkening effects. The semi-analytical method and the appropriate numerical techniques were incorporated into a C++ modelling code at VUW (Victoria University of Wellington) known as mlens2. The effectiveness of the semi-analytic method was demonstrated using mlens2 to generate theoretical lightcurves for the microlensing events MOA-2009-BLG-319 and OGLE-2006-BLG-109. By comparing these theoretical lightcurves with the observed photometric data and the published models, it was demonstrated that the semi-analytic method described in this thesis is a robust and efficient method for discovering extrasolar planets.</p>

Modelling of Planetary Gravitational Microlensing Events

<p>This thesis describes and develops procedures for the generation of theoretical lightcurves that can be used to model gravitational microlensing events that involve multiple lenses. Of particular interest are the cases involving a single lens star with one or more orbiting planets, as this has proven to be an effective way of detecting extrasolar planets. Although there is an analytical expression for microlensing lightcurves produced by single lensing body, the generation of model lightcurves for more than one lensing body requires the use of numerical techniques. The method developed here, known as the semi-analytic method, involves the analytical rearrangement of the relatively simple ‘lens equation’ to produce a high-order complex lens polynomial. Root-finding algorithms are then used to obtain the roots of this ‘lens polynomial’ in order to locate the positions of the images and calculate their magnifications. By running example microlensing events through the root-finding algorithms, both the speed and accuracy of the Laguerre and Jenkins-Traub algorithms were investigated. It was discovered that, in order to correctly identify the image positions, a method involving solutions of several ‘lens polynomials’ corresponding to different coordinate origins needed to be invoked. Multipole and polygon approximations were also developed to include finite source and limb darkening effects. The semi-analytical method and the appropriate numerical techniques were incorporated into a C++ modelling code at VUW (Victoria University of Wellington) known as mlens2. The effectiveness of the semi-analytic method was demonstrated using mlens2 to generate theoretical lightcurves for the microlensing events MOA-2009-BLG-319 and OGLE-2006-BLG-109. By comparing these theoretical lightcurves with the observed photometric data and the published models, it was demonstrated that the semi-analytic method described in this thesis is a robust and efficient method for discovering extrasolar planets.</p>