Nonlinear Relaxation Method Research Articles

Numerical Study to Predict Temperature and Moisture Profiles in Unventilated Grain Silos at Prolonged Time Periods

The experimental acquisition of data for monitoring cereal grain storage in silos for long periods is not practical in most cases; in such situations, mathematical modeling is an excellent alternative to predict temperature and mass transfer behavior. The model was developed in two dimensions using the equations of heat, mass, and momentum transport. The governing equations were solved using discretization of the spatial coordinates by orthogonal collocation with Jacobi polynomials and the resulting algebraic system was solved by employing nonlinear relaxation method that uses less computational resources. For the ambient boundary conditions, it was used as a more accurate equation to represent the diurnal–nocturnal temperature dynamic. This study analyzes the effects of moisture migration and temperature interactions between the interstitial air and the sorghum cereal grain bed defined as a porous medium in a cylindrical cavity, including the variations affected by meteorological conditions over 1 year. The mass–thermal gradients were predicted, and the effects of environmental temperature were analyzed for flow patterns, isotherms and moisture distribution changes, indicating how the environmental fluctuations influence the heat and moisture of the ambient spots in the grain storage. The maximum temperature inside the silo was 42 °C, considering the respiration heat of the grain, and the maximum temperature of the grain without the climatic boundary conditions was reported to be approximately 35 °C in previous studies. The proposed model can be easily extrapolated to be used for other cereal grains or to incorporate additional effects such as solar radiation, shadow sunlight path, wind velocity, equations to predict more precise ambient temperatures, etc. as boundary conditions.

A simple and efficient technique to accelerate the computation of a nonlocal dielectric model for electrostatics of biomolecule

<p style='text-indent:20px;'>The nonlocal modified Poisson-Boltzmann equation (NMPBE) is one important variant of a commonly-used dielectric continuum model, Poisson-Boltzmann equation (PBE). In this paper, we use a nonlinear block relaxation method to develop a new nonlinear solver for the nonlinear equation of <inline-formula><tex-math id="M1">\begin{document} $\Phi $ \end{document}</tex-math></inline-formula> and thus a new NMPBE solver, which is then programmed as a software package in <inline-formula><tex-math id="M2">\begin{document} $\texttt{C}\backslash\texttt{C++}$ \end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document} $\texttt{Fortran}$ \end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document} $\texttt{Python}$ \end{document}</tex-math></inline-formula> for computing the electrostatics of a protein in a symmetric 1:1 ionic solvent. Numerical tests validate the new package and show that the new solver can improve the efficiency by at least <inline-formula><tex-math id="M5">\begin{document}$ 40\%$ \end{document}</tex-math></inline-formula> than the finite element NMPBE solver without compromising solution accuracy.

The Ozone Mapping and Profiler Suite (OMPS) Limb Profiler (LP) Version 1 aerosol extinction retrieval algorithm: theoretical basis

Abstract. The theoretical basis of the Ozone Mapping and Profiler Suite (OMPS) Limb Profiler (LP) Version 1 aerosol extinction retrieval algorithm is presented. The algorithm uses an assumed bimodal lognormal aerosol size distribution to retrieve aerosol extinction profiles at 675 nm from OMPS LP radiance measurements. A first-guess aerosol extinction profile is updated by iteration using the Chahine nonlinear relaxation method, based on comparisons between the measured radiance profile at 675 nm and the radiance profile calculated by the Gauss–Seidel limb-scattering (GSLS) radiative transfer model for a spherical-shell atmosphere. This algorithm is discussed in the context of previous limb-scattering aerosol extinction retrieval algorithms, and the most significant error sources are enumerated. The retrieval algorithm is limited primarily by uncertainty about the aerosol phase function. Horizontal variations in aerosol extinction, which violate the spherical-shell atmosphere assumed in the version 1 algorithm, may also limit the quality of the retrieved aerosol extinction profiles significantly.

Analysis of second order and unconditionally stable BDF2‐AB2 method for the Navier‐Stokes equations with nonlinear time relaxation

In this study, we first consider a second order time stepping finite element BDF2‐AB2 method for the Navier‐Stokes equations (NSE). We prove that the method is unconditionally stable and accurate. Second, we consider a nonlinear time relaxation model which consists of adding a term “” to the Navier‐Stokes Equations with the algorithm depends on BDF2‐AB2 method. We prove that this method is unconditionally stable, too. We applied the BDF2‐AB2 method to several numeral experiments including flow around the cylinder. We have also applied BDF2‐AB2 method with nonlinear time relaxation to some problems. It is observed that when the equilibrium errors are high, applying BDF2‐AB2 with nonlinear time relaxation method to the problem yields lower equilibrium errors.

Mechanically driven domain wall movement in magnetoelastic nanomagnets

Magnetic domain walls are fundamental objects arising in ferromagnetic materials, largely investigated both through micromagnetic simulations and experiments. While current- and field-based techniques for inducing domain wall propagation have been widely studied for fundamental understanding and application-oriented purposes, the possibility to manipulate domain walls using mechanical stress in magnetoelastic materials has only recently drawn interest. Here, a complete analytical model describing stress-induced transverse domain wall movement in ferromagnetic nanostripe with variable cross-section is presented. This approach yields a nonlinear integro-differential equation describing the magnetization field. Its numerical implementation, based on the nonlinear relaxation method, demonstrates the possibility to precisely control the position of a domain wall through mechanical action.

Geodesic shape finding of membrane structure with geodesic string by the dynamic relaxation method

The explicit nonlinear dynamic relaxation method (DRM) is applied to the nonlinear geodesic shape finding analysis by introducing fictional tensioned \'strings\' along the desired seams with a three or four-node membrane element. A number of results from the numerical example for the nonlinear geodesic shape finding and patterning analysis are obtained by the proposed method to demonstrate the accuracy and efficiency of the developed method. Therefore, the proposed geodesic shape finding algorithm may improve the applicability of a four-node membrane element to membrane structural engineering and design analysis simultaneously for the shape finding, stress, and patterning analysis.

Asynchronous parallel nonlinear multisplitting relaxation methods for large sparse nonlinear complementarity problems

In accordance with the principle of sufficiently using delayed information, and making use of the nonlinear multisplitting and the nonlinear relaxation techniques, we present in this paper a class of asynchronous parallel nonlinear multisplitting successive overrelaxation (SOR) methods for solving large sparse nonlinear complementarity problems on high-speed MIMD multiprocessor systems. These new methods particularly include the so-called asynchronous parallel nonlinear multisplitting SOR-Newton method, asynchronous parallel nonlinear multisplitting SOR-chord method and asynchronous parallel nonlinear multisplitting SOR-Steffensen method. Under suitable conditions we establish the local convergence theory of this class of new methods. Numerical imitations show that our new methods are feasible and efficient for solving the nonlinear complementarity problems on the MIMD multiprocessor systems.

A class of parallel nonlinear multisplitting relaxation methods for the large sparse nonlinear complementarity problems

By making use of the nonlinear multisplitting and the nonlinear relaxation techniques, we present, in this paper, a class of parallel nonlinear multisplitting successive overrelaxation methods for solving the large sparse nonlinear complementarity problems on the modern high-speed multiprocessors. These new methods particularly include the so-called nonlinear multisplitting SOR-Newton method. Under suitable conditions, we establish the local convergence theories of the new methods, and investigate their asymptotic convergence rates. A lot of numerical results show that our new methods are feasible and efficient for parallel solving the nonlinear complementarity problems.

New comparison theorem for the nonlinear multisplitting relaxation method for the nonlinear complementarity problems

A new comparison theorem on the monotone convergence rates of the parallel nonlinear multisplitting accelerated overrelaxation (AOR) method for solving the large scale nonlinear complementarity problem is established. Thus, the monotone convergence theory of this class of method is completed.

The monotone convergence of a class of parallel nonlinear relaxation methods for nonlinear complementarity problems

We set up a class of parallel nonlinear multisplitting AOR methods by directly multisplitting the nonlinear mapping involved in the nonlinear complementarity problems. The different choices of the relaxation parameters can yield all the known and a lot of new relaxation methods, as well as a lot of new relaxed parallel nonlinear multisplitting methods for solving the nonlinear complementarity problems. The two-sided approximation properties and the influences on the convergence rates from the relaxation parameters about our new methods are shown, and sufficient conditions guaranteeing the methods to converge globally are discussed. Finally, a lot of numerical results show that our new methods are feasible and efficient.

Improved comparison theorem for the nonlinear multisplitting relaxation method

A new comparison theorem is given for the nonlinear multisplitting relaxation method [1], and an important modification is proposed for it, too.

Parallel nonlinear multisplitting relaxation methods

By further generalizing Frommer’s results in the sense of nonlinear multisplitting, we build a class of nonlinear multisplitting AOR-type methods, which covers many rather practical nonlinear multisplitting relaxation methods such as multisplitting AOR-Newton method, multisplitting AOR-chord method and multisplitting AOR-Steffensen method, etc., Furthermore, a general convergence theorem for the nonlinear multisplitting AOR-type methods and the local convergence for the multisplitting AOR-Newton method are discussed in detail. A lot of numerical tests show that our new methods are feasible and satisfactory.

Efficient Parallel Nonlinear Multigrid Relaxation Algorithms for Low-Level Vision Applications

Multigrid techniques have been shown to significantly improve the convergence rate of the nonlinear relaxation algorithms used in computer vision for the extraction of low-level image features. It is also well known that the computations involved with relaxation algorithms are regular and local, and lead naturally to massive data parallelism. However, standard data parallelism does not exploit the large computing resources of the now available massively parallel 2D processor arrays when coarse image resolutions (i.e., small image grids) have to be processed, like in multigrid methods. In this research note, we present an algorithmic framework which enables us making a full use of the large potential of data parallelism for the implementation of nonlinear multigrid relaxation methods. The approach combines two different levels of parallelism: parallel updating of the image sites and concurrent explorations of the configuration space of the problem. The efficiency of the method is demonstrated on two different low-level vision applications: restoration of noisy images and optical flow computation.

Numerical solution of the nonlinear Poisson–Boltzmann equation: Developing more robust and efficient methods

AbstractWe present a robust and efficient numerical method for solution of the nonlinear Poisson‐Boltzmann equation arising in molecular biophysics. The equation is discretized with the box method, and solution of the discrete equations is accomplished with a global inexact‐Newton method, combined with linear multilevel techniques we have described in an article appearing previously in this journal. A detailed analysis of the resulting method is presented, with comparisons to other methods that have been proposed in the literature, including the classical nonlinear multigrid method, the nonlinear conjugate gradient method, and nonlinear relaxation methods such as successive overrelaxation. Both theoretical and numerical evidence suggests that this method will converge in the case of molecules for which many of the existing methods will not. In addition, for problems which the other methods are able to solve, numerical experiments show that the new method is substantially more efficient, and the superiority of this method grows with the problem size. The method is easy to implement once a linear multilevel solver is available and can also easily be used in conjunction with linear methods other than multigrid. © 1995 by John Wiley &amp; Sons, Inc.

Modeling and simulation of solid-state transducers: the thermal and electrical energy domain

Abstract This paper deals with a basic strategy towards the (efficient) numerical modeling of semiconductor micro-structures in which electric, thermal and thermoelectric effects are important. In particular, we discuss a very efficient method of O(n) for solving the nonlinear semiconductor thermoelectric model equations, where n is the total number of elements in the grid. The method is based on the multigrid method in combination with the mixed finite element discretization method and a nonlinear Vanka-type relaxation method. For a simple, but representative test problem we demonstrate the feasibility of this approach by showing that (nearly) mesh-size independent convergence rates can be obtained.

A CLASS OF ASYNCHRONOUS PARALLEL NONLINEAR MULTISPLITTING RELAXATION METHODS

In this paper, we establish a class of asynchronous parallel nonlinear multisplitting relaxation methods for solving system of nonlinear equations. With special choices of the relaxed parameters in the new methods, not only can the convergence properties of them be improved, but also many applicable and efficient asynchronous parallel nonlinear multisplitting iteration methods such as the Jacobi, Gauss-Seidel, SOR as well as the asynchronous parallel nonlinear multisplitting AOR-Newton, -Chord and -Steffensen programs, etc., can be obtained. Under proper conditions, we build convergence theories about these asynchronous methods, and estimate their asymptotic convergence rates in detailed manner.

AN EXPERT SYSTEM FOR INTERPRETING MESOSCALE FEATURES IN OCEANOGRAPHIC SATELLITE IMAGES

Thermal infrared images of the ocean obtained from satellite sensors are widely used for the study of ocean dynamics. The derivation of mesoscale ocean information from satellite data depends to a large extent on the correct interpretation of infrared oceanographic images. The difficulty of the image analysis and understanding problem for oceanographic images is due in large part to the lack of precise mathematical descriptions of the ocean features, coupled with the time varying nature of these features and the complication that the view of the ocean surface is typically obscured by clouds, sometimes almost completely. Towards this objective, the present paper describes a hybrid technique that utilizes a nonlinear probabilistic relaxation method and an expert system for the oceanographic image interpretation problem. This paper highlights the advantages of using the contextual information in the feature labeling algorithm. The need for an expert system and its feedback in automatic interpretation of oceanic features is discussed. The paper presents some important results of the series of experiments conducted at the Remote Sensing Branch, of the Naval Oceanographic and Atmospheric Research Laboratory, on the National Oceanic and Atmospheric Administration Advanced Very High Resolution Radiometer (AVHRR) imagery data. The results clearly indicate the drastic improvement in labeling due to the oceanographic expert system.

Feature labelling in infrared oceanographic images

Thermal infrared images of the ocean obtained from satellite sensors are widely used for the study of ocean dynamics. The derivation of mesoscale ocean information from satellite data depends to a large extent on the correct interpretation of infrared oceanographie images. The difficulty of the image analysis and understanding problem for oceanographic images is due, in large part, to the lack of precise mathematical descriptions of the ocean features, coupled with the time varying nature of these features and the complication that the view of the ocean surface is typically obscured by clouds, sometimes almost completely. Towards this objective a technique that utilizes a non-linear probabilistic relaxation method for the oceanographie feature labelling problem is described. A unified mathematical framework that helps in solving the problem is presented, and the advantages of using the contextual information in the feature labelling algorithm is highlighted. The feature labelling technique makes use of a new, efficient edge detection algorithm based on cluster shade texture measures. This algorithm is found to be more suitable for labelling the mesoscale features present in the oceanographie satellite images. Some important results of a series of experiments conducted at NOARL's Remote Sensing Branch on NOAA AVHRR imagery data are presented. A motivation for using this technique to build an automatic image interpretation system is also given.

Nonlinear relaxation methods for solving symmetric linear complementarity problems

A family of iterative algorithms is presented for the solution of the symmetric linear complementarity problem, $$Mx - q \geqslant 0, x \geqslant 0, x^T (Mx - q) = 0,$$ whereM is a givenn×n symmetric real nonnegative matrix andq is a given positiven×1 vector. The algorithms are derived from a nonlinear relaxation method first proposed by Gold and Scofield for solving linear systems that arise from the discretization of certain linear integral equations. It is shown that the original algorithm has been used in several different fields of application like deconvolution, atmospheric sciences, computer graphics and image processing, and image reconstruction from projections. Convergence proofs are given.

Quasi-Simultaneous Solution Method: A New Highly Efficient Strategy for Numerical MOST Simulations

Most numerical MOS-simulators are based on Gummel's nonlinear relaxation method, for which severe convergence problems arise above threshold. A new hierarchical approach for the numerical simulation of MOS-transistors (MOST's) is presented, which overcomes these shortcomings resulting in an improvement of convergence speed of more than one order of magnitude.