Search results

The purpose of this paper is to find out some new rational non-traveling wave solutions and to study localized structures for (2+1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation.

Along with some special transformations, the Lie group method and the rational function method are applied to the (2+1)-dimensional AKNS equation.

Some new non-traveling wave solutions are obtained, including generalized rational solutions with two arbitrary functions of time variable.

As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.

With the help of the Lie group method, special transformations and the rational function method, new non-traveling wave solutions are derived for the AKNS equation by Maple software. These results are much useful for investigating some new localized structures and the interaction of waves in high-dimensional models, and enrich dynamical features of solutions for the higher dimensional systems.

The initial stiffness method has been extensively adopted for elasto‐plastic finite element analysis. The main problem associated with the initial stiffness method, however, is its slow convergence, even when it is used in conjunction with acceleration techniques. The Newton‐Raphson method has a rapid convergence rate, but its implementation resorts to non‐symmetric linear solvers, and hence the memory requirement may be high. The purpose of this paper is to develop more advanced solution techniques which may overcome the above problems associated with the initial stiffness method and the Newton‐Raphson method.

In this work, the accelerated symmetric stiffness matrix methods, which cover the accelerated initial stiffness methods as special cases, are proposed for non‐associated plasticity. Within the computational framework for the accelerated symmetric stiffness matrix techniques, some symmetric stiffness matrix candidates are investigated and evaluated.

Numerical results indicate that for the accelerated symmetric stiffness methods, the elasto‐plastic constitutive matrix, which is constructed by mapping the yield surface of the equivalent material to the plastic potential surface, appears to be appealing. Even when combined with the Krylov iterative solver using a loose convergence criterion, they may still provide good nonlinear convergence rates.

Compared to the work by Sloan et al., the novelty of this study is that a symmetric stiffness matrix is proposed to be used in conjunction with acceleration schemes and it is shown to be more appealing; it is assembled from the elasto‐plastic constitutive matrix by mapping the yield surface of the equivalent material to the plastic potential surface. The advantage of combining the proposed accelerated symmetric stiffness techniques with the Krylov subspace iterative methods for large‐scale applications is also emphasized.

Extrapolation is a process used to accelerate the convergence of a sequence of approximations to the true value. Different stepsizes are used to obtain approximate solutions, which are combined to increase the order of the approximation by eliminating leading error terms. The smoothing technique is also applied to suppress order reduction and to dampen the oscillatory component in the numerical solution when solving stiff problems. The extrapolation and smoothing technique can be applied in either active, passive or the combination of both active and passive modes. In this paper, the authors investigate the best strategy of implementing extrapolation and smoothing technique and use this strategy to solve stiff ordinary differential equations. Based on the experiment, the authors suggest using passive smoothing in order to reduce the computation time.

The two-step smoothing is a composition of four steps of the symmetric method with different weights. It is used as the final two steps when combined with many steps of the symmetric method. The aim is to preserve symmetry and provide damping for stiff problem and to be more robust than the one-step smoothing. The two-step smoothing is L-stable. The new method is then applied with extrapolation process in passive and active modes to investigate the most efficient and accurate method of implementation.

In this paper, the authors constructed the two-step smoothing to be more robust than the one-step smoothing. The two-step smoothing is constructed to achieve as high order as possible and able to restore the classical order of particular method compared to the one-step active smoothing that is only able to achieve order-1 condition. The two-step smoothing for ITR is also superior in solving stiff case since it has the super-convergent order-4 behavior. In our experiments with extrapolation, it is proven that the two-step smoothing is more accurate and more efficient than the one-step smoothing, namely 1ASAX. It is also observed that the method with smoothing is comparable if not superior to the existing base method in certain cases. Based on the experiment, the authors would suggest using passive smoothing if the aim is to reduce computation time. It is of interest to conduct more experiment to validate the accuracy and efficiency of the smoothing formula with and without extrapolation.

The implementation of extrapolation on two-step symmetric Runge–Kutta method has not been tested on variety of other test problems yet. The two-step symmetrization is an extension of the one-step symmetrization and has not been constructed by other researchers yet. The method is constructed such that it preserves the asymptotic error expansion in even powers of stepsize, and when used with extrapolation the order might increase by 2 at a time. The method is also L-stable and eliminates the order reduction phenomenon when solving stiff ODEs. It is also of interest to observe other ways of implementing extrapolation using other sequences or with interpolation.

This paper aims to deal with two-dimensional magneto-hydrodynamic (MHD) Falkner–Skan boundary layer flow of an incompressible viscous electrically conducting fluid over a permeable wall in the presence of a magnetic field.

Using the Lie group approach, the Lie algebra of infinitesimal generators of equivalence transformations is constructed for the equation under consideration. Using these suitable similarity transformations, the governing partial differential equations are reduced to linear and nonlinear ordinary differential equations (ODEs). Further, Haar wavelet approach is applied to the reduced ODE under the subalgebra 4.1 for constructing numerical solutions of the flow problem.

A new type of solutions was obtained of the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis.

To find a solution for the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis is a new approach for fluid problems.

Detailed results of numerical calculations of transient, 2D incompressible flow around and in the wake of a square prism at Re = 100, 200 and 500 are presented. An implicit finite‐difference operator‐splitting method, a version of the known SIMPLEC‐like method on a staggered grid, is described. Appropriate theoretical results are presented. The method has second‐order accuracy in space, conserving mass, momentum and kinetic energy. A new modification of the multigrid method is employed to solve the elliptic pressure problem. Calculations are performed on a sequence of spatial grids with up to 401 × 321 grid points, at sequentially halved time steps to ensure grid‐independent results. Three types of flow are shown to exist at Re = 500: a steady‐state unstable flow and two which are transient, fully periodic and asymmetric about the centre line but mirror symmetric to each other. Discrete frequency spectra of drag and lift coefficients are presented.

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.

In this paper, an interpolation method called the quasi‐higher order element method (QHOEM) is proposed for solving elastoplastic problems using symmetric Galerkin boundary element method (SGBEM). At the initial stage, it uses higher order elements to interpolate the coordinates and the field variables. Then, for the numerical integration involved, it further uses interpolation to decompose the higher order elements into lower order elements so that the existing analytical integration formulas can be applied. By adopting this procedure, the proposed method yields good adaptability and reduces the computational cost. Its accuracy and efficiency are demonstrated by analyzing three practical examples.

The objective of this research is to describe and apply a method with which to measure and evaluate mutual trust in dyadic business relationships with both one‐to‐one and multiple informants, as well as symmetric and asymmetric interactions.

Application of the perceptual bi‐directionality method, i.e. the PBD‐method, is introduced in this context to measure and evaluate the relationship properties of mutual trust in the context of one‐to‐one and multiple informants, as well as the symmetric and asymmetric interactions, in dyadic business relationships.

The overall impression of the mutual trust in the studied relationships is that they are well balanced. It may erroneous, since there may be balance deficiencies between the various trust dimensions. The stepwise evaluation of a case example reveals that there are substantial differences between the various trust dimensions measured in the dyadic business relationship.

It is argued that the measurement and evaluation of unidirectional trust issues is not enough in itself to understand the relationship properties of trust between two actors in dyadic business relationships, but that a bi‐directional approach should be applied in terms of the relationship properties of mutual trust. Furthermore, the approach of one‐to‐one key informants might also be insufficient. Instead multiple informants should be used in the measurement and evaluation of the relationship properties of mutual trust in dyadic business relationships and, symmetric, as well as asymmetric, interactions of mutual trust in dyadic contexts should be considered.

The PBD‐method is a formalised and structured method of how to measure and evaluate the mutuality of relationship properties in dyadic contexts. The outcome of the present research is also a set of generic mutuality measures that can be useful in different management and research applications.

The PBD‐method may be used to measure and evaluate multiple key informants' in asymmetric interactions of mutual relationship properties in dyadic contexts, instead of only matching one‐to‐one key informants in symmetric interactions.

The purpose of this paper is to propose a computational model for thin layers, for problems of linear time-dependent heat conduction. The thin layer is replaced by a zero-thickness interface. The advantage of the new model is that it saves the need to construct and use a fine mesh inside the layer and in regions adjacent to it, and thus leads to a reduction in the computational effort associated with implicit or explicit finite element schemes.

Special asymptotic models have been proposed for linear heat transfer and linear elasticity, to handle thin layers. In these models the thin layer is replaced by an interface with zero thickness, and specific jump conditions are imposed on this interface in order to represent the special effect of the layer. One such asymptotic interface model is the first-order Bövik-Benveniste model. In a paper by Sussmann et al., this model was incorporated in a FE formulation for linear steady-state heat conduction problems, and was shown to yield an accurate and efficient computational scheme. Here, this work is extended to the time-dependent case.

As shown here, and demonstrated by numerical examples, the new model offers a cost-effective way of handling thin layers in linear time-dependent heat conduction problems. The hybrid asymptotic-FE scheme can be used with either implicit or explicit time stepping. Since the formulation can easily be symmetrized by one of several techniques, the lack of self-adjointness of the original formulation does not hinder an accurate and efficient solution.

Most of the literature on asymptotic models for thin layers, replacing the layer by an interface, is analytic in nature. The proposed model is presented in a computational context, fitting naturally into a finite element framework, with both implicit and explicit time stepping, while saving the need for expensive mesh construction inside the layer and in its vicinity.

This paper aims to propose a semi-analytical benchmarking framework for enthalpy-based methods used in problems involving phase change with latent heat. The benchmark is based on a class of semi-analytical solutions of spatially symmetric Stefan problems in an arbitrary spatial dimension. Via a public repository this study provides a finite element numerical code based on the FEniCS computational platform, which can be used to test and compare any method of choice with the (semi-)analytical solutions. As a particular demonstration, this paper uses the benchmark to test several standard temperature-based implementations of the enthalpy method and assesses their accuracy and stability with respect to the discretization parameters.

The class of spatially symmetric semi-analytical self-similar solutions to the Stefan problem is found for an arbitrary spatial dimension, connecting some of the known results in a unified manner, while providing the solutions’ existence and uniqueness. For two chosen standard semi-implicit temperature-based enthalpy methods, the numerical error assessment of the implementations is carried out in the finite element formulation of the problem. This paper compares the numerical approximations to the semi-analytical solutions and analyzes the influence of discretization parameters, as well as their interdependence. This study also compares accuracy of these methods with other traditional approach based on time-explicit treatment of the effective heat capacity with and without iterative correction.

This study shows that the quantitative comparison between the semi-analytical and numerical solutions of the symmetric Stefan problems can serve as a robust tool for identifying the optimal values of discretization parameters, both in terms of accuracy and stability. Moreover, this study concludes that, from the performance point of view, both of the semi-implicit implementations studied are equivalent, for optimal choice of discretization parameters, they outperform the effective heat capacity method with iterative correction in terms of accuracy, but, by contrast, they lose stability for subcritical thickness of the mushy region.

The proposed benchmark provides a versatile, accessible test bed for computational methods approximating multidimensional phase change problems. The supplemented numerical code can be directly used to test any method of choice against the semi-analytical solutions.

While the solutions of the symmetric Stefan problems for individual spatial dimensions can be found scattered across the literature, the unifying perspective on their derivation presented here has, to the best of the authors’ knowledge, been missing. The unified formulation in a general dimension can be used for the systematic construction of well-posed, reliable and genuinely multidimensional benchmark experiments.