Initial Value Method Research Articles

Buckling analysis of single‐walled carbon nanotubes using the initial value method and approximate transfer matrix based on nonlocal elasticity theory

AbstractIn this study, a novel method is presented to analyze the buckling behavior of single‐walled carbon nanotubes (SWCNTs) using the initial value method (IVM) in conjunction with the approximate transfer matrix, within the framework of nonlocal elasticity theory. The study aims to accurately approximate critical buckling load parameters under various boundary conditions, without encountering high computational requirements. IVM enables the computation of displacements and stress resultants along the entire beam from given initial conditions. The approximate transfer matrix is employed to analyze system states at different points through successive integration of solutions, generating the principal matrix needed for IVM and ensuring systematic and precise results that optimize the accuracy of the analysis. A convergence study confirms the effectiveness and precision of the proposed method, revealing a decrease in the critical buckling load parameters as the nonlocal parameter increases, applicable across all boundary conditions studied (simply supported, clamped–clamped, clamped–simply supported, and clamped‐free). These results underscore the need to incorporate nonlocal effects for more accurate nanostructure mechanics predictions. The integration of IVM and the approximate transfer matrix provides a computationally efficient alternative to traditional numerical and semi‐analytical methods, aiding researchers and engineers working with SWCNTs and other nanomaterials.

Nonlocal buckling analysis of double-walled carbon nanotubes using initial value method and matricant approach

In this study, the buckling behavior of double-walled carbon nanotubes (DWCNTs) is investigated using the initial value method (IVM) combined with the approximate transfer matrix (ATM) approach within the framework of nonlocal elasticity theory. The aim is to provide a method that best estimates the critical buckling loads with improved accuracy and computational efficiency. To validate the method, the critical buckling loads obtained without considering nonlocal effects were compared with results from the literature, showing a high degree of agreement. Various boundary conditions, including simply supported (S-S), clamped-simply supported (C-S), clamped-free (C-F), and clamped-clamped (C-C), have been considered for investigation. The effects of nonlocal parameters, aspect ratios, and boundary conditions on the critical buckling loads are analyzed. The findings emphasize the importance of accounting for these parameters in mechanical analyses to ensure accurate predictions. The proposed framework provides an efficient alternative to existing numerical methods and offers valuable benchmarks for future research on the mechanical behavior of nanostructures.

Fast simulation strategy for capacitively-coupled plasmas based on fluid model

Fluid simulations are widely used in optimizing the reactor geometry and improving the performance of capacitively coupled plasma (CCP) sources in industry, so high computation speed is very important. In this work, a fast method for CCP fluid simulation based on the framework of Multi-physics Analysis of Plasma Sources (MAPS) is developed, which includes a multi-time-step explicit upwind scheme to solve electron fluid equations, a semi-implicit scheme and an iterative method with in-phase initial value to solve Poisson's equation, an explicit upwind scheme with limited artificial diffusion to solve heavy particle fluid equations, and an acceleration method based on fluid equation modification to reduce the periods required to reach equilibrium. In order to prove the validity and efficiency of the newly developed method, benchmarking against COMSOL and comparison with experimental data have been performed in argon discharges on the Gaseous Electronics Conference (GEC) reactor. Besides, the performance of each acceleration method is tested, and the results indicated that the multi-time-step explicit Euler scheme can effectively decline the computational burden in the bulk plasma and reduce the time cost on the electron fluid equations by half. The in-phase initial value method can greatly decrease the iteration times required to solve linear equations and lower the computational time of Poisson's equation by 77 %. The acceleration method based on equation modification can reduce the periods required to reach equilibrium by two-thirds.

Approximate frequency analysis of isotropic shear beams using initial value method

Approximate frequency analysis of isotropic shear beams using initial value method

Wind farm power optimization using system identification

The wake effect reduces the total power production of wind farms. This paper presents a method for wind farm power optimization through wake effect reduction. The proposed method optimizes the yaw angle offsets and de-rating settings of all turbines to maximize total power generation. The optimization approach is gradient-based, with gradients at each iteration obtained through system identification using field test data, eliminating the need for physical models. In system identification, test signal design, model estimation and model validation problems are solved in a systematic manner; in the gradient-based optimization, in order to achieve fast convergence, methods for initial value and initial step-size determination, variable step-size iteration and iteration termination are developed. The method is verified using the FLORIS wind farm model developed by National Renewable Energy Laboratory (NREL), USA. The studied wind farm consists of 80 wind turbines configured similarly to the Horns Rev I offshore wind farm in Denmark. The result of the developed optimization method is highly consistent with those obtained using FLORIS's built-in optimization tool.

A generalized midpoint-based boundary value method for unstable partial differential equations

We present a class of Boundary Value Methods (BVMs) that can be applied to semi-stable and unstable Partial Differential Equation (PDE) models. Step-by-step (initial value) methods, such as Runge–Kutta and linear multistep methods have numerical stability regions which do not intersect with certain regions of the complex plane that are significant to the time-integration of unstable PDEs. BVMs, which need extra numerical conditions at the final time, are global methods and are, in some sense, free of such barriers. We discuss BVMs based on generalized midpoint methods, combined with appropriate numerical initial and final conditions. The stability regions of these methods intersect with a significant part of the complex plane. In several numerical experiments we will illustrate the usefulness of such methods when applied to PDE models, such as a dispersive wave equation, a space-fractional PDE and the backward heat equation.

Stability of Segmented Circular Arch under Constant Pressure

The N-segment circular arch with rigid members and semirigid joints under external pressure is studied. The problem is formulated in coupled difference equations. An efficient initial value method is presented to solve the stability problem. The product of buckling pressure and N tends to be that of the continuous elastic arch as N→∞. Critical buckling is mostly antisymmetric but may be symmetric at larger opening angles. Optimum shapes (for maximum critical pressure) are determined for both segmented and continuous circular arches.

A class of linearization-based collocation methods for initial value and boundary value engineering problems

Nonlinear initial value and boundary value problems are generally classified as two different categories of problems and the methods utilized to solve them are also different. This paper proposed a uniform linearization-collocation framework through which the initial value and boundary value problems can be solved in a similar way. An optional pre-conditioning procedure is provided to guarantee the computational accuracy. Rigorous analysis about the convergence rate is given. Different from the commonly seen convergence rate analysis based on one-dimensional system and mean-value theorem, this analysis is based on multidimensional system and polynomial approximation. A perturbed orbit propagation problem, a radiation heat transfer problem and an orbit optimization problem are solved to verify the validity and efficiency of the proposed methods. The proposed methods not only outperform the highly optimized MATLAB built-in integrator ODE45 in orbit propagation, but also outperform the bvp4c and bvp5c in solving the boundary value problem. Most importantly, the proposed methods are capable of solving some problems where the bvp4c and bvp5c are invalid. The quasi-linearization-collocation method proved to be a highly accurate, efficient and stable solver for ordinary differential equations with various types of constraints.

Median robust nonlinear weighted total least squares estimator of nonlinear EIV models: three algorithms

To improve robust estimation performance of fully correlated nonlinear error-in-variables model, three median robust estimate methods are discussed on base of Nonlinear Weighted Total Least Squares (NWTLS), namely, median parameter method, median parameter initial value method and median variance method. After the discussion of the median robust estimation theory and the derivation of the calculation formula of the three median robust estimation methods, three iterative algorithms are developed. Through two nonlinear regression examples, four schemes including no artificial outliers, different location of outliers, different degrees of correlation, and different ratios of outliers are adopted, which proves the feasibility and effectiveness of the proposed algorithm. The experimental results show that from the statistical point of view, although the three median methods have different antioutlier effects for different nonlinear models, they are more effective than RNWTLS and NWTLS; the robust estimation results of three median methods are related to the functional model.

A Fuzzy Multi-Criteria Evaluation Model for the Coordination of Industrial Agglomeration and Regional Economic Growth

The coordinated development between industrial agglomeration and regional economic growth is of great significance. Based on the theory, this article constructs a fuzzy multi-criteria evaluation model for the coordination of industrial agglomeration and regional economic growth. The model reveals the impact of industrial agglomeration on economic growth by establishing a regression model. The output value of the secondary industry and the tertiary industry are classified, and the initial value method is used for dimensionless processing. The experimental results show that using the panel unit root test, panel cointegration test, panel regression analysis, and gray correlation analysis to conduct empirical analysis and research, the capital agglomeration and labor agglomeration in regional industries both promote economic growth, and the correlation degree of sustainable agglomeration components reached 89.7%, which significantly and indirectly played a role in promoting the coordinated development of economic growth in regional economic growth.

A second order convergent initial value method for singularly perturbed system of differential-difference equations of convection diffusion type

In this article, a system of second order singularly perturbed delay differential equations of convection diffusion type problem is considered. An asymptotic expansion approximation of the solution is constructed. Further the asymptotic expansion approximation is numerically approximated using the Runge Kutta methods and hybrid finite difference methods. The error estimate is obtained and it is of almost second order. Numerical examples are given to illustrate the present method.

Initial Value Methods for the Numerical Simulation of Fuzzy Two-Point Boundary Value Problems Using General Linear Method

This article considers the numerical simulation of fuzzy two-point boundary value problems (FBVP) using general linear method (GLM). The author derived the method, which is a combination of a Runge-Kutta type method and multi-step method. It is originally designed to solve initial value problems. It requires fewer function evaluations than the traditional Runge-Kutta methods making it computationally more efficient in achieving the required accuracy. The author will utilize the combination of the GLM with initial value methods to solve the linear fuzzy BVP's and a shooting-like method for the nonlinear cases. Numerical testing and simulation of several examples, considered by other authors, will be presented to show the efficiency of the proposed method.

Sinc-Galerkin method for solving system of singular perturbed reaction-diffusion problems

In this paper, the system of singularly Perturbed Reaction-Diffusion p roblems w hich a re commonly used in physics and chemistry branches of science, were investigated. Sinc-Galerkin Method was used to obtaining the solution of problems. Because of there is no article about Sinc-Galerkin Method related to singularly perturbed Reaction-Diffusion problems in liter- ature, the efficiency of the method was shown via this problem. There are important results that occurred after our research and application. Sinc Galerkin Method which was used in this paper as the main solution method gave better results according to parameter robust method and asymptotical initial value method. The figures and the tables show this competence and low errors.

Imaginary-time mean-field method for collective tunneling

Background: Quantum tunneling in many-body systems is the subject of many experimental and theoretical studies in fields ranging from cold atoms to nuclear physics. However, theoretical description of quantum tunneling with strongly interacting particles, such as nucleons in atomic nuclei, remains a major challenge in quantum physics. Purpose: An initial-value approach to tunneling accounting for the degrees of freedom of each interacting particle is highly desirable. Methods: Inspired by existing methods to describe instantons with periodic solutions in imaginary time, we investigate the possibility to use an initial value approach to describe tunneling at the mean-field level. Real-time and imaginary-time Hartree dynamics are compared to the exact solution in the case of two particles in a two-well potential. Results: Whereas real-time evolutions exhibit a spurious self-trapping effect preventing tunneling in strongly interacting systems, the imaginary-time-dependent mean-field method predicts tunneling rates in excellent agreement with the exact solution. Conclusions: Being an initial-value method, it could be more suitable than approaches requiring periodic solutions to describe realistic systems such as heavy-ion fusion.

Scaling invariance theory and numerical transformation method: A unifying framework

In a transformation method, the numerical solution of a given boundary value problem is obtained by solving one or more related initial value problems. Therefore, a transformation method, like a shooting method, is an initial value method. The peculiar difference between a transformation and a shooting method is that the former is conceived and formulated within scaling invariance theory. The main aim of this paper is to propose a unifying framework for numerical transformation methods. The non-iterative method is an extension of the Töpfer’s non-iterative algorithm developed as a simple way to solve the celebrated Blasius problem. As many boundary value problems cannot be solved non-iteratively because they lack the required scaling invariance an iterative extension of the method has been developed. This iterative method provides a simple numerical test for the existence and uniqueness of solutions, as shown by this author in the case of free boundary problems [Appl. Anal., 66 (1997) pp. 89-100] and proved herewith for a wide class of boundary value problems defined on a semi-infinite interval.

The initial-boundary value problem for a class of third order pseudoparabolic equations

In this paper, a priori estimate for a linear third pseudoparabolic operator with bound is established, and applying the above result, the existence and uniqueness theorem of solutions for a class of nonlinear pseudoparabolic equations is obtained with the help of the homeomorphism method and the initial value method. Furthermore, an existence and uniqueness theorem of the semilinear equation is obtained as a corollary.

Solving Systems of Singularly Perturbed Convection Diffusion Problems via Initial Value Method

In this paper, an initial value method for solving a weakly coupled system of two second-order singularly perturbed Convection–diffusion problems exhibiting a boundary layer at one end is proposed. In this approach, the approximate solution for the given problem is obtained by solving, a coupled system of initial value problem (namely, the reduced system), and two decoupled initial value problems (namely, the layer correction problems), which are easily deduced from the given system of equations. Both the reduced system and the layer correction problems are independent of perturbation parameter, ε. These problems are then solved analytically and/or numerically, and those solutions are combined to give an approximate solution to the problem. Further, error estimates are derived and examples are provided to illustrate the method.

Approximate analytical solution of singularly perturbed boundary value problems in MAPLE

In this paper, based on a new initial value method, an approximate analytical solution of singularly perturbed boundary value problems is obtained using a new and simple Maple program spivp. The algorithm is designed for practicing engineers or applied mathematicians who need a practical tool for solving these problems. Some examples are solved to illustrate the implementation of the program and the accuracy of the algorithm. Analytical and numerical results are compared with results in literature. The results confirm that the present method is accurate and offers a simple and easy practical tool for obtaining approximate analytical and numerical solutions of singularly perturbed boundary value problems.

Magnetoelastic buckling properties of type-II superconducting thin strip subjected to perpendicular magnetic field and parallel distributed uniform mechanical load

The buckling analyses of type-II superconducting strip under applied perpendicular magnetic field and/or distributed uniform mechanical load are investigated in this paper. Based on the Bean critical state model, the electromagnetic body force is firstly given. Then, based on the classical plate theory and two-point initial value method, the critical buckling states of the superconducting strip with different boundary conditions are analyzed. Numerical results show the effects of both the thickness and boundary conditions of superconducting strip on the corresponding critical buckling loads. The present work should be helpful to the research and application of superconducting thin strips.

Solving Linear Second-Order Singularly Perturbed Differential Difference Equations via Initial Value Method

In this paper, an initial value method for solving a class of linear second-order singularly perturbed differential difference equation containing mixed shifts is proposed. In doing so, first, the given problem is modified in to an equivalent singularly perturbed problem by approximating the term containing the delay and advance parameters using Taylor series expansion. From the modified problem, two explicit initial value problems which are independent of the perturbation parameter are produced; namely, the reduced problem and the boundary layer correction problem. These problems are then solved analytically and/or numerically, and those solutions are combined to give an approximate solution to the original problem. An error estimate for this method is derived using maximum norm. Several test problems are considered to illustrate the theoretical results. It is observed that the present method approximates the exact solution very well.