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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.

The purpose of this paper is to survey and express the advantages and disadvantages of the existing approaches for solving grey linear programming in decision-making problems.

After presenting the concepts of grey systems and grey numbers, this paper surveys existing approaches for solving grey linear programming problems and applications. Also, methods and approaches for solving grey linear programming are classified, and its advantages and disadvantages are expressed.

The progress of grey programming has been expressed from past to present. The main methods for solving the grey linear programming problem can be categorized as Best-Worst model, Confidence degree, Whitening parameters, Prediction model, Positioned solution, Genetic algorithm, Covered solution, Multi-objective, Simplex and dual theory methods. This survey investigates the developments of various solving grey programming methods and its applications.

Different methods for solving grey linear programming problems are presented, where each of them has disadvantages and advantages in providing results of grey linear programming problems. This study attempted to review papers published during 35 years (1985–2020) about grey linear programming solving and applications. The review also helps clarify the important advantages, disadvantages and distinctions between different approaches and algorithms such as weakness of solving linear programming with grey numbers in constraints, inappropriate results with the lower bound is greater than upper bound, out of feasible region solutions and so on.

The purpose of this paper is to solve non‐linear parametric thermal models defined in degenerated geometries, such as plate and shell geometries.

The work presented in this paper is based in a combination of the proper generalized decomposition (PGD) that proceeds to a separated representation of the involved fields and advanced non‐linear solvers. A particular emphasis is put on the asymptotic numerical method.

The authors demonstrate that this approach is valid for computing the solution of challenging thermal models and parametric models.

This is the first time that PGD is combined with advanced non‐linear solvers in the context of non‐linear transient parametric thermal models.

Introduction Operations research, i.e. the application of scientific methodology to operational problems in the search for improved understanding and control, can be said to have started with the application of mathematical tools to military problems of supply bombing and strategy, during the Second World War. Post‐war these tools were applied to business problems, particularly production scheduling, inventory control and physical distribution because of the acute shortages of goods and the numerical aspects of these problems.

A simple numerical algorithm is developed for the implementation of the harmonic balance method to analyse periodic responses of a general dynamic system having geometrical non‐linearities of the quadratic and cubic types. The resulting non‐linear algebraic equations which are not explicitly determined are solved by non‐linear equation routines available in most mathematical libraries. Various non‐linear responses, such as the combinational resonances of a hinged‐clamped beam, the non‐linear effect on degenerate vibration modes of a square plate and the non‐linear oscillation of thin rings, are presented to demonstrate the versatility of the algorithm.

The paper gives the description of boundary element method (BEM) with subdomains for the solution of convection—diffusion equations with variable coefficients and Burgers’ equations. At first, the whole domain is discretized into K subdomains, in which linearization of equations by representing convective velocity by the sum of constant and variable parts is carried out. Then using fundamental solutions for convection—diffusion linear equations for each subdomain the boundary integral equation (in which the part of the convective term with the constant convective velocity is not included into the pseudo‐body force) is formulated. Only part of the convective term with the variable velocity, which is, as a rule, more than one order less than convective velocity constant part contribution, is left as the pseudo‐source. On the one hand, this does not disturb the numerical BEM—algorithm stability and, on the other hand, this leads to significant improvement in the accuracy of solution. The global matrix, similar to the case of finite element method, has block band structure whereas its width depends only on the numeration order of nodes and subdomains. It is noted, that in comparison with the direct boundary element method the number of global matrix non‐zero elements is not proportional to the square of the number of nodes, but only to the total number of nodal points. This allows us to use the BEM for the solution of problems with very fine space discretization. The proposed BEM with subdomains technique has been used for the numerical solution of one‐dimensional linear steady‐state convective—diffusion problem with variable coefficients and one‐dimensional non‐linear Burgers’ equation for which exact analytical solutions are available. It made it possible to find out the BEM correctness according to both time and space. High precision of the numerical method is noted. The good point of the BEM is the high iteration convergence, which is disturbed neither by high Reynolds numbers nor by the presence of negative velocity zones.

The purpose of this paper is to report the implementation of an alternative time integration procedure for the dynamic non-linear analysis of structures.

The time integration algorithm discussed in this work corresponds to a spectral decomposition technique implemented in the time domain. As in the case of the modal decomposition in space, the numerical efficiency of the resulting integration scheme depends on the possibility of uncoupling the equations of motion. This is achieved by solving an eigenvalue problem in the time domain that only depends on the approximation basis being implemented. Complete sets of orthogonal Legendre polynomials are used to define the time approximation basis required by the model.

A classical example with known analytical solution is presented to validate the model, in linear and non-linear analysis. The efficiency of the numerical technique is assessed. Comparisons are made with the classical Newmark method applied to the solution of both linear and non-linear dynamics. The mixed time integration technique presents some interesting features making very attractive its application to the analysis of non-linear dynamic systems. It corresponds in essence to a modal decomposition technique implemented in the time domain. As in the case of the modal decomposition in space, the numerical efficiency of the resulting integration scheme depends on the possibility of uncoupling the equations of motion.

One of the main advantages of this technique is the possibility of considering relatively large time step increments which enhances the computational efficiency of the numerical procedure. Due to its characteristics, this method is well suited to parallel processing, one of the features that have to be conveniently explored in the near future.

This paper concerns the study of non‐linear effects in optical fibres with a core made of a Kerr type medium. The aim is to propose an algorithm to find spatial solitons, i.e. solutions with a harmonic behaviour in time and along the fibre but with a field distribution in the cross‐section corresponding to a self‐trapped propagation of the electromagnetic field.

The field is supposed to be harmonic in time and along the direction of invariance of the fibre but inhomogeneous in the cross‐section. This modifies the refractive index profile of the fibre (a step‐index one in this study). A scalar model of the fibre, together with the finite element method (that is well suited to deal with inhomogeneous media), is used and a new iterative algorithm is proposed to obtain the non‐linear solutions. An adaptive meshing is necessary to guarantee the accuracy of the model.

The new algorithm converges to self‐coherent solutions that are different from those obtained via a fixed power algorithm. The equivalents both of a fundamental mode and of a second order mode are studied.

The approach acknowledges the findings of the previously known spatial solitons (with a slight modification of the algorithm) together with a new family of solutions. It opens a new field of investigation to understand this whole family of non‐linear solutions as it shows that only a small part of them was known up to now.

This paper attempts to establish the general formula for computing the inverse of grey matrix, and the results are applied to solve grey linear programming. The inverse of a grey matrix and grey linear programming plays an important role in establishing a grey computational system.

Starting from the fact that missing information often appears in complex systems, and therefore that true values of elements are uncertain when the authors construct a matrix, as well as calculate its inverse. However, the authors can get their ranges, which are called the number-covered sets, by using grey computational rules. How to get the matrix-covered set of inverse grey matrix became a typical approach. In this paper, grey linear programming was explained in detail, for the point of grey meaning and the methodology to calculate the inverse grey matrix can successfully solve grey linear programming.

The results show that the ranges of grey value of inverse grey matrix and grey linear programming can be obtained by using the computational rules.

Because the matrix and the linear programming have been widely used in many fields such as system controlling, economic analysis and social management, and the missing information is a general phenomenon for complex systems, grey matrix and grey linear programming may have great potential application in real world. The methodology realizes the feasibility to control the complex system under uncertain situations.

The paper successfully obtained the ranges of uncertain inverse matrix and linear programming by using grey system theory, when the elements of matrix and the coefficients of linear programming are intervals and the results enrich the contents of grey mathematics.

The purpose of this paper is to present a comparison of nonlinear and linear simulations of aircraft dynamics to determine the divergence of the linear solution from the nonlinear solution.

The general equations of motion of a transport aircraft are presented both in nonlinear and linear form. The nonlinear equations are solved by using the Runge Kutta method. Linear equations are solved numerically by using the Runge Kutta method and they are also solved exactly by using the Laplace transformation method. All of these solutions are obtained by using the body axis system. The results of the simulations are plotted for different control deflections.

Solution of linear equations by both methods gave the same results as expected. There are important differences in amplitude and frequency of oscillations which are obtained by using nonlinear and linear equations. These differences increase with growing input control deflection. Therefore, it is appropriate to prefer nonlinear approach to obtain more satisfactory results.

Accurate determination of the aerodynamic derivatives is important for the accuracy of the nonlinear solutions.

Many classical approaches use stability axis system for the solution of linear equations. However, in this paper transfer functions of the aircraft are redefined in the body axis system, because stability axes change with angle of attack and some of the stability derivatives need to be re‐evaluated for each angle of attack. Moreover, in addition to classical text book, linear equations are also solved by using the 4th order Runge Kutta medhod.