One-dimensional Problem Research Articles

Peridynamics method for problems solve of solids destruction

Abstract. The article investigates the method of peridynamics, which is an alternative approach to solving destruction problems based on integral equations. It is assumed that particles in a continuum interact with each other at a finite distance, as in molecular dynamics. Damage is part of the theory at the level of two-particle interactions, so damage finding and destruction occurs when solving the equation of motion. During this work, bondbased and state-based peridynamics models of destruction used in the Sandia Laboratory were described and implemented within the framework of the MoDyS molecular dynamics software package. In the bond-based model, the defining relationship is the bond stiffness function, which corrects the force of particle-particle interaction and imposes a restriction on the use of the Poisson’s ratio. The state-based model generalizes the bond-based approach and may be applied to materials with any Poisson’s ratio. The relationship of both models is ascertained. Calculation convergence is demonstrated on the example of a one-dimensional elasticity problem. The possibility of using the implemented models for fracture problems is also shown.

Comparison of Corrected and Uncorrected Enthalpy Methods for Solving Conduction-Driven Solid/Liquid Phase Change Problems

The numerical study of solid/liquid phase change problems represents a large and ongoing field of research with many applications. These simulations should run as fast and accurately as possible. Therefore, proceeding from previous work and findings from the literature, this study investigates enthalpy methods for solving solid/liquid phase change problems. The relationship between temperature and enthalpy is strongly non-linear and requires special treatment; iteratively corrected methods, as well as approaches that do not correct the temperature/enthalpy relationship at all or only once per time step, were considered for a one-dimensional test problem. Based on the results of this study, two solvers can be recommended, the so-called optimum approach and a simple explicit method; both provide accurate results. The explicit method is easy to program, but the optimum approach allows larger time steps and is, therefore, faster. The influence of several parameters was investigated. The mesh resolution strongly influenced the accuracy and the computational speed, and the time step size barely influenced the accuracy but did affect the computational speed. An artificial melting temperature range influenced the accuracy but had hardly any influence on the simulation speed. Higher-order time discretization schemes were not superior compared to the first-order implicit optimum approach.

Temperature profile and thermal piston component of photoacoustic response calculated by the fractional dual-phase-lag heat conduction theory

We present the temperature distribution predictions for photothermal systems by considering an extension of dual-phase lag. It is an extension of the GCE-II and GCE-III models with a fractional dual-phase lag from kinetic relaxation time. Solving the one-dimensional problem considering a planar and periodic excitation, we obtained the temperature distribution and the Photoacoustic (PA) signal for the transmission setup. We also analyze the effects of fractional order derivatives and kinetic relaxation time. It is shown that the derived models have promising results that could be used to explain the experimentally observed behavior of PA signals measured on thin films with an inhomogeneous internal structure.

The dynamical problem on acting distributed load on the elastic layer

The wave field of an elastic half-layer is constructed, when a dynamic normal load distributed over a rectangular area acts on upper face at the initial moment of time. The lower face of the half-layer is rigidly fixed to the foundation, and the side border is in the conditions of a smooth contact. The method of decomposing the system of motion equations into a system of equations and an independently solvable equation is used, this approach was proposed by Popov~G.~Ya. Laplace and Fourier integral transformations are applied directly to the motion equations and boundary conditions, which reduces the problem to a vector one-dimensional boundary value problem, which is solved by the matrix differential calculus method. The output displacements are obtained using inverse integral transformations. The case of steady oscillations was considered and the amplitude of vertical displacement occurring in the layer was analyzed depending on the shape of the distributed load section, the material of the layer medium and the values of the natural frequency of the layer oscillations.

Numerical Analysis of Fourier Finite Volume Element Method for Dirichlet Boundary Optimal Control Problems Governed by Elliptic PDEs on Complex Connected Domains

In this research, we investigate an optimal control problem governed by elliptic PDEs with Dirichlet boundary conditions on complex connected domains, which can be utilized to model the cooling process of concrete dam pouring. A new convergence result for two-dimensional Dirichlet boundary control is proven with the Fourier finite volume element method. The Lagrange multiplier approach is employed to find the optimality systems of the Dirichlet boundary optimal control problem. The discrete optimal control problem is then obtained by applying the Fourier finite volume element method based on Galerkin variational formulation for optimality systems, that is, using Fourier expansion in the azimuthal direction and the finite volume element method in the radial direction, respectively. In this way, the original two-dimensional problem is reduced to a sequence of one-dimensional problems, with the Dirichlet boundary acting as an interval endpoint at which a quadratic interpolation scheme can be implemented. The convergence order of state, adjoint state, and Dirichlet boundary control are therefore proved. The effectiveness of the method is demonstrated numerically, and numerical data is provided to support the theoretical analysis.

Locally order-preserving mapping for WENO methods

In our previous studies (Li and Zhong, 2021a; Li and Zhong, 2021b), the commonly reported issue that most of the existing mapped WENO schemes suffer from either losing high resolutions or generating spurious oscillations in long-run simulations of hyperbolic problems has been successfully addressed, by devising the improved mapped WENO schemes, namely MOP-WENO-X, where “X” stands for the version of the existing mapped WENO scheme. However, all the MOP-WENO-X schemes bring about the serious deficiency that their resolutions in the region with smoothly varying high-frequency waves are dramatically decreased compared to their associated WENO-X schemes. The purpose of this paper is to overcome this drawback. We first present the locally order-preserving (LOP) mapping. Then, by using a new proposed posteriori adaptive technique, we apply this LOP property to obtain the new mappings from those of the WENO-X schemes. The essential idea of the posteriori adaptive technique is to identify the global stencil in which the existing mappings fail to preserve the LOP property, and then replace the mapped weights with the weights of the classic WENO-JS scheme to recover the LOP property. We build the resultant mapped WENO schemes and denote them as LOP-WENO-X. The numerical experiments demonstrate that the resolutions in the region with high-frequency but smooth waves of the LOP-WENO-X schemes are similar to or even better than those of their associated WENO-X schemes and naturally much higher than the MOP-WENO-X schemes. Furthermore, the LOP-WENO-X schemes gain all the great advantages of the MOP-WENO-X schemes, such as attaining high resolutions and in the meantime preventing spurious oscillations near discontinuities when solving the one-dimensional linear advection problems with long output times, and significantly reducing the post-shock oscillations in the simulations of the two-dimensional problems with shock waves.

On Adaptive Grid Approximations in the Weight Norm

The purpose of this paper is to develop an algorithm construction of adaptive variation-grid methods for classes of one-dimensional boundary value problems of the second order. Classes of non-degenerate problems are considered, as well as classes of problems with weak and strong degeneracy. The results obtained are suitable for learning computer systems designed to solve problems of the aforementioned classes. To achieve the set goals, the corresponding approximation theorems are established with degeneration. Ways of adaptive choice approximation space at a variation-grid method in a one-dimensional boundary value problem are considered. The locality of the approximation is substantially used. The considerations are reduced to an iterative process, while building an adaptive grid. Numerical examples illustrating the effectiveness of the proposed approach are given.

On the application of Jacobian-free Riemann solvers for relativistic radiation magnetohydrodynamics under M1 closure

Radiative transfer plays a major role in high-energy astrophysics. In multiple scenarios and in a broad range of energy scales, the coupling between matter and radiation is essential to understand the interplay between theory, observations and numerical simulations. In this paper, we present a novel scheme for solving the equations of radiation relativistic magnetohydrodynamics within the parallel code Lóstrego. These equations, which are formulated taking successive moments of the Boltzmann radiative transfer equation, are solved under the grey-body approximation and the M1 closure using an IMEX time integration scheme. The main novelty of our scheme is that we introduce for the first time in the context of radiation magnetohydrodynamics a family of Jacobian-free Riemann solvers based on internal approximations to the Polynomial Viscosity Matrix, which were demonstrated to be robust and accurate for non-radiative applications. The robustness and the limitations of the new algorithms are tested by solving a collection of one-dimensional and multi-dimensional test problems, both in the free-streaming and in the diffusion radiation transport limits. Due to its stable performance, the applicability of the scheme presented in this paper to real astrophysical scenarios in high-energy astrophysics is promising. In future simulations, we expect to be able to explore the dynamical relevance of photon-matter interactions in the context of relativistic jets and accretion discs, from microquasars and AGN to gamma-ray bursts.

Variable stiffness optimization algorithm for vibration optimization of variable-stiffness composite plates

A novel variable stiffness optimization (VSO) algorithm is proposed for vibration optimization of variable-stiffness composite (VSC) plates uisng linear variation fiber path function. Firstly, an independent 2D-sampling optimization method (2DSOind) is employed to generate a good initial point. Then, optimization is conducted in multiple phases, within each phase, the problem is simplified to an approximately one-dimensional linear search problem. By dividing the fiber angles into two groups, VSO algorithm designs the laminate between two fiber angle groups sequentially and iteratively, making the stiffness of VSC plate stiffened part by part. Lastly, the VSC plate is redesigned accounting for the curvature constraint. The sequential permutation search (SPS) algorithm is coupled into VSO to optimize one group of fiber angles whilst the angles of another group are fixed during the iteration. Under a variety of boundary conditions, the free vibration frequencies of VSC plates are optimized using VSO. In comparison to conventional layerwise optimization (LO) and genetic algorithm (GA), the present results demonstrate much higher efficiency and similar robustness. Moreover, the maximum fundamental frequency of VSC plates can be improved by about 10% in comparison to straight fiber laminates.

A novel safety measure with random and fuzzy variables and its solution by combining Kriging with truncated candidate region

For the safety assessment of models with random and fuzzy variables, this work proposes a novel measure of substandard credibility of reliability (SCR), which is defined by the credibility of reliability less than the minimum allowed reliability. In order to improve the computational efficiency of SCR, the truncated candidate region (TCR) based adaptive Kriging model (AK) combined with secant method (SM) is developed. In the proposed method, the solution of SCR is first divided into a double-loop process. In the inner loop, the fuzzy variables are converted into interval variables under a specific membership degree, and then the Kriging model is built and updated with TCR to search the adding points, which can accurately predict the reliability bounds. While in the outer loop, a numerical iterative method of SM is used to solve a one-dimensional root-finding problem to estimate SCR. Compared with traditional method, the proposed method of AK-TCR-SM introduces TCR into the Kriging model to reduce the size of the candidate sample pool, and in the iterative process, only a few new samples are added to sample pool, which significantly improves the computational efficiency. The advantages of the proposed AK-TCR-SM method are demonstrated by several examples.

Thermodynamically Consistent Modified Lord–Shulman Generalized Thermoelasticity With Strain-Rate

Abstract The analysis of thermoelastic wave propagation in continuum solids at micro/nano-seconds is especially significant for ultra-fast heating technologies, where strain relaxation effects will increase significantly. In most cases, it is commonly accompanied by a relatively small strain-rate; however, this is questionable in the environment of transient thermal wave propagation under the ultra-fast heating case. The present work is dedicated to constitutive modeling of a novel generalized thermoelasticity model by introducing an additional strain-rate term associated with a relaxation time parameter in the Lord–Shulman (LS) thermoelasticity with the aid of an extended thermodynamics framework. As an application, the newly developed model is applied to a one-dimensional half-space problem which is traction free at one end; a time-dependent thermal shock is imposed at the same end to analyze transient responses of thermodynamic field variables (temperature, displacement, strain, and stress). The inclusion of strain-rate in the LS model eliminates the probable propagating jump discontinuities of the strain and stress fields at the wavefront. The current work is expected to be useful in the mathematical modeling and numerical simulation of thermoelastic processes under an ultra-fast heating environment.

Spatially One-Dimensional Boundary Value Problems of Coupled Thermoelasticity: Generalized Functions Method

Spatially One-Dimensional Boundary Value Problems of Coupled Thermoelasticity: Generalized Functions Method

A Study of One-dimensional Weak Shock Propagation Under the Action of Axial and Azimuthal Magnetic Field: An Analytical Approach

The present paper presents an analytical study of the one-dimensional weak shock wave problem in a perfect gas under the action of a generalized magnetic field subjected to weak shock jump conditions (R-H conditions). The magnetic field is considered axial and azimuthal in cylindrically symmetric configuration. By considering a straightforward analytical approach, an explicit solution exhibiting time-space dependency for gas-dynamical flow parameters and total energy (generated during the propagation of the weak shock from the center of the explosion) has been obtained under the significant influence of generalized magnetic fields (axial and azimuthal) and the results are analyzed graphically. From the outcome, it is worth noticing that for an increasing value of Mach number under the generalized magnetic field, the decay process of physical parameters (density, pressure, and magnetic pressure) is a bit slower, whereas the velocity profile and total energy increase rapidly with respect to time. Moreover, for increasing values of Shock-Cowling number the total energy grows rapidly with respect to time.

Optimal allocation of irrigation water in a single-reservoir and a single-pumping-station system under deficit irrigation conditions

Abstract Aiming at the optimal allocation of irrigation water in a multi-water source project in a water resource shortage area, this study developed a water resource joint scheduling optimization model for the reservoir and the pumping station under deficit irrigation conditions. In the model, the maximum annual yield of the irrigation area was the objective function; the water supply, water spill of the reservoir and replenishment water of the pump station at each stage were the decision variables; and the total annual water supply of the system, the reservoir operation criteria, the water rights of the pumping station, and the water demand of the crop during the entire growth period were the constraint conditions. According to the characteristics of the model, a large system decomposition aggregation dynamic programming (DADP) method is proposed to transform the N + 1 dimensional dynamic programming problem into a N + 1 one-dimensional dynamic programming problem for solution. In addition, this study also uses the real-coded genetic algorithm (RGA) and DADP to compare the algorithms, and discusses the performance of the two algorithms from the optimization of the algorithm and the applicability of the algorithm.

On one problem of controlling the heating of a rod system under uncertainty

The problem of control of a parabolic system, which describes the heating of a given number of rods, is considered. The density functions of the internal heat sources of the rods are not exactly known, and only the segment of their change is given. Control are point heat sources that are located at the ends of the rods. The goal of the choice of control is to ensure that at a fixed time the modulus of the linear function determined using the average temperatures of the rods does not exceed the given value for any admissible functions of the density of internal heat sources. A technique has been developed for reducing this problem to a one-dimensional control problem under uncertainty. Necessary and sufficient termination conditions are found.

Optimal Transmission Probabilities of Information and Artificial Noise in Covert Communications

This letter studies a wireless covert communication system, where a jammer generating intermittent artificial noise (AN) is employed to confuse the warden. The transmission probabilities of information and AN are jointly optimized to maximize the communication covertness under a throughput requirement. We first analyse the detection performance of the warden and the throughput of the system, based on which we formulate the optimization problem. Then, we simplify this two-dimensional nonconvex problem to a one-dimensional problem and obtain the closed-form optimal solutions in special cases. The results illustrate that the optimal transmission probabilities of information and AN are 0.5 and 1, respectively, when the throughput requirement is less than a threshold. However, 0.5 and 1 are not optimal when the throughput requirement exceeds the aforementioned threshold, which has not been reported yet.

A Market Segmentation Assessment Weighted Scoring for Using WSM method An Study for Different Market

Market segmentation evaluation and selection is one of the important marketing issues for all companies. This study is for selecting the market segment and can be used as a model for future studies. Segmentation change factors are common in five main areas: a segmentation factor, Competitive, Financial and economic factors, Technological and socio-political factors are taken as evaluation parameters. WSM methodology for market segmentation evaluation and selection. This article will summarize presents a new approach. In MCDM methods, WSM alone cannot be used for multidimensional problems. Multidimensional means all dimensions have different units. Besides using WSM, There are other methods that can solve a one-dimensional problem. However, they were originally developed with the aim of solving multidimensional problems, and it is for solving one-dimensional problems a special method makes the WSM method. Finally result manufacturing process technology required is in 1st rank and contribution margins are last rank.

Ionization dynamics and gauge invariance

Photoionization is one of the most fundamental processes in laser-matter interaction. It plays a crucial role also from the practical point of view since the electron-density dynamics in a gaseous medium is a process that affects the field propagation. Photoionization has been addressed in different regimes: linear, multiphoton, and tunneling. The ideal tool that allows for the description of ionization, in all aforementioned regimes, is the numerical evaluation of the time-dependent Schr\"odinger equation. The determination of the electron density needs the computation of the time-dependent ionization probability which unfortunately is an ambiguous quantity due to the gauge dependence of the latter. In this paper, we show how to overcome this difficulty by properly defining the time-dependent ionization probability in the context of the resolvent operator method. We show in particular that the velocity gauge allows for a definition of adiabatic states that is suitable to define an ionization threshold at all times during the interaction to compute ionization probability. Applications to linear, multiphoton, and tunneling regimes are presented for the one-dimensional problem. The extension to the nondipole case is discussed and we show that time-dependent ionization probability cannot be defined unambiguously due to the introduction of the magnetic-field component. We also discuss the case of gauge invariance in a subspace of the eigenbasis defined by the Hamiltonian.

Extension of an exponential discretization scheme to multidimensional convection–diffusion problems

ENATE (Enhanced Numerical Approximation of a Transport Equation) is a high-order exponential scheme for convection–diffusion problems, such as those that govern the transport of fluid properties in a flow field. The scheme was intended to be employed in fluid-related transport equations, although it can be used for any inhomogeneous second-order ordinary differential equation. The value of a variable ϕ at a generic point is related to those of adjacent nodes via an algebraic equation. Thus, a three-point stencil is associated to each node. The coefficients of this equation contain integrals of some fluid and flow parameters. One important property is that the scheme allows to obtain a machine-accurate solution of an inhomogeneous transport equation if these integrals can be obtained analytically. As the scheme is essentially one-dimensional, getting the machine-accurate solution of multidimensional problems is not guaranteed even in cases where ENATE integrals are analytic along each coordinate. In this regard the paper presents a simple way of getting solutions in multidimensional problems while still using the one-dimensional formulation. Moreover, if the problem is such that the solution is machine-accurate in the one-dimensional problem along coordinate lines, it will also be for the multidimensional domain. Two different methods of evaluating those terms that come out of the discretization will be explained and compared in various cases.

An interface sharpening technique for the simulation of underwater explosions

In this study, a simple and effective interface sharpening technique is proposed to simulate underwater explosions. The five-equation model is used as the numerical model for underwater explosions. The multi-dimensional problem can be split into multiple one-dimensional problems using the Strang splitting. The MUSCL–Hancock Method (MHM) is employed to solve the governing equations in one-dimensional space, and the HLLC (Harten–Lax–van Leer contact) scheme is adopted to solve the Riemann problem approximately. The numerical results show that the five-equation model cannot accurately simulate underwater explosions due to the diffusion problem. To overcome this problem, a new interface sharpening model based on the five-equation model is obtained by adding the anti-diffusion source term to the volume-fraction transport equation. A fractional step method is employed to solve equations of the interface sharpening model in two separate steps. The model is validated by conducting numerical cases. On this basis, the one-dimensional and two-dimensional underwater explosion simulations are conducted. The numerical results indicate that the interface sharpening model based on the five-equation model can accurately simulate underwater explosions.