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The purpose of this paper is to propose a stable high-order absorbing boundary condition (ABC) based on new continued fraction for scalar wave propagation in 2D and 3D unbounded layers.

The ABC is obtained based on continued fraction (CF) expansion of the frequency-domain dynamic stiffness coefficient (DtN kernel) on the artificial boundary of a truncated infinite domain. The CF which has been used to the thin layer method in [69] will be applied to the DtN method to develop a time-domain high-order ABC for the transient scalar wave propagation in 2D. Furthermore, a new stable composite-CF is proposed in this study for 3D unbounded layers by nesting the above CF for 2D layer and another CF.

The ABS has been transformed from frequency to time domain by using the auxiliary variable technique. The high-order time-domain ABC can couple seamlessly with the finite element method. The instability of the ABC-FEM coupled system is discussed and cured.

This manuscript establishes a stable high-order time-domain ABC for the scalar wave equation in 2D and 3D unbounded layers, which is based on the new continued fraction. The high-order time-domain ABC can couple seamlessly with the finite element method. The instability of the coupled system is discussed and cured.

Infinite elements provide one of the most attractive alternatives for dealing with differential equations in unbounded domains. The region where loads, sources, inhomogeneities and anisotropics exist is modelled by finite elements and the far, uniform region is represented by infinite elements. We propose a new infinite element which can represent any type of decay towards infinity. The element is so simple that explicit expressions can be obtained for the element matrix in many cases, yet large improvements in the accuracy of the solution are obtained as compared with the truncated mesh. Explicit expressions are in fact given for the Laplace equation and 1/rn decay. The element is conforming with linear triangles and bilinear quadrilaterals in two dimensions. The element can be used with any standard finite‐element program without having to modify the shape function library or the numerical quadrature library of the program. The structure or bandwidth of the stiffness matrix of the finite portion of the mesh is not modified when the infinite elements are used. An example problem is solved and the solution found to be better than several other methods in common usage. The proposed method is thus highly recommended.

This paper is concerned with infinite elements for dynamic problems, that is, those which change in time. It is a sequel to our earlier paper on static problems. The paper is in a number of sections. The first is an introduction. In the second the state‐of‐the‐art review of infinite elements is updated. In the third, ‘added mass’ type effects are considered. In the fourth, time dependent problems of the diffusion type, which only involve the first time derivative are considered. Wave problems are considered in the fifth and the necessary radiation conditions for such problems are summarized. Section six deals with dynamic problems of a repetitive nature, that is periodic or harmonic problems. In section seven completely transient problems are dealt with and some fundamental difficulties are noted. Conclusions are drawn in section eight.

This paper aims the application of a novel synergy between a neural network (NN) and the finite element method (FEM) in the solution of electromagnetic problem involving hysteretic material in unbounded domain.

The hysteretic nature of the material is taken into account by an original NN able to perform the modelling of any kind of quasi-static loop (saturated and non-saturated, symmetric or asymmetric). An appositely developed iterative FEM procedure is presented for the solution of this kind of problems in unbounded domains.

By starting from a small set of measured loops, the NN manages the values of the magnetic field, H, and the flux density, B, as inputs while the differential permeability is the output. In particular, the proposed NN is capable to perform the modelling of saturated and non-saturated, symmetric or asymmetric hysteresis loops.

The development of an efficient method for the solution of a complicated electromagnetic problem in unbounded domain by using an iterative approach and NNs, which can be implemented also in existing FEM code.

The paper shows that the combination of FEM, iterative procedure and NNs allows us to produce effective solutions of electromagnetic problems in unbounded domains involving also nonlinear hysteretic magnetic materials with an acceptable computational cost.

Much of the literature on theories of decision making under risk has emphasized differences between theories. One enduring theme has been the attempt to develop a distinction between “normative” and “descriptive” theories of choice. Bernoulli (1738) introduced log utility because expected value theory was alleged to have descriptively incorrect predictions for behavior in St. Petersburg games. Much later, Kahneman and Tversky (1979) introduced prospect theory because of the alleged descriptive failure of expected utility (EU) theory (von Neumann & Morgenstern, 1947).

A simple and efficient method for the finite‐element solution of three‐dimensional unbounded region field problems is presented in this paper. The proposed technique consists of a global mapping of the original unbounded region onto a bounded domain by applying a standard inversion transformation to the spatial coordinates. Same numerical values of the potential function are assigned to the transformed points. The functional associated to the field problem, which incorporates the boundary conditions, has the same structure in the transformed domain as that in the original one. This allows the implementation of the standard finite‐element method in the bounded transformed domain. The finite‐element solution is obtained on the basis of a complete discretization of the bounded, transformed domain by standard finite elements, with no approximate assumption made for the behaviour of the field at infinity, other than that introduced by the finite‐element idealization. This leads to improved accuracy of the numerical results, compared to those obtained in the original region, for the same number of nodes. Application to three test problems illustrates the high efficiency of the proposed method in terms of both accuracy and computational effort. The technique presented is particularly recommended for exterior‐field problems in the presence of material inhomogeneities and anisotropies.

A technique combining finite elements and boundary elements is promising for unbounded field problems. A hypothetical boundary is assumed in the unbounded domain, and the usual finite element method is applied to the inner region, while the boundary element method is applied to the outer infinite region. On the coupling boundary, therefore, both potential and flux must be compatible. In the finite element method, the flux is defined as the derivative of the potential for which a trial function is defined. In the boundary element method, on the other hand, the same polynomial function is chosen for the potential and the flux. Thus, the compatibility cannot be satisfied unless a special device is considered. In the present paper, several compatibility conditions are discussed concerning the total flux or energy flow continuity across the coupling boundary. Some numerical examples of Poisson and Helmholtz problems are demonstrated.

The finite‐element method can be used for an approximate solution of axisymmetric exterior‐field problems by truncating the unbounded domain, or by applying various techniques of coupling a finite region of interest with the remaining far region, which is properly modelled. In this paper, we propose the solution of axisymmetric exterior‐field problems by using the standard finite‐element method in a bounded, transformed domain obtained by conformal mapping from the original, unbounded one. The transformed functionals have very simple expressions and the exact transforms of the original boundary conditions are used in the transformed domain. Consequently no approximation is introduced in the proposed method and improvements in the accuracy of the solution are obtained as compared with several other methods in common usage, especially with the truncated mesh technique. A few example problems are solved and the presented method is found to be simple and computationally highly efficient. It is particularly recommended for problems with material inhomogeneities and anisotropies within large regions.

This paper is concerned with static problems, i.e. those which do not change with time. Dynamic problems will be considered in a sequel. The historical development of infinite elements is described. The two main developments, decay function infinite elements and mapped infinite elements, are described in detail. Results obtained using various infinite elements are given, followed by a discussion of possibilities and likely developments.

Survey of period infinite element developments The first infinite elements for periodic wave problems, as stated in Part 1, were developed by Bettess and Zienkiewicz, the earliest publication being in 1975. These applications were of ‘decay function’ type elements and were used in surface waves on water problems. This was soon followed by an application by Saini et al., to dam‐reservoir interaction, where the waves are pressure waves in the water in the reservoir. In this case both the solid displacements and the fluid pressures are complex valued. In 1980 to 1983 Medina and co‐workers and Chow and Smith successfully used quite different methods to develop infinite elements for elastic waves. Zienkiewicz et al. published the details of the first mapped wave infinite element formulation, which they went on to program, and to use to generate results for surface wave problems. In 1982 Aggarwal et al. used infinite elements in fluid‐structure interaction problems, in this case plates vibrating in an unbounded fluid. In 1983 Corzani used infinite elements for electric wave problems. This period also saw the first infinite element applications in acoustics, by Astley and Eversman, and their development of the ‘wave envelope’ concept. Kagawa applied periodic infinite wave elements to Helmholtz equation in electromagnetic applications. Pos used infinite elements to model wave diffraction by breakwaters and gave comparisons with laboratory photogrammetric measurements of waves. Good agreement was obtained. Huang also used infinite elements for surface wave diffraction problems. Davies and Rahman used infinite elements to model wave guide behaviour. Moriya developed a new type of infinite element for Helmholtz problem. In 1986 Yamabuchi et al. developed another infinite element for unbounded Helmholtz problems. Rajapalakse et al. produced an infinite element for elastodynamics, in which some of the integrations are carried out analytically, and which is said to model correctly both body and Rayleigh waves. Imai et al. gave further applications of infinite elements to wave diffraction, fluid‐structure interaction and wave force calculations for breakwaters, offshore platforms and a floating rectangular caisson. Pantic et al. used infinite elements in wave guide computations. In 1986 Cao et al. applied infinite elements to dynamic interaction of soil and pile. The infinite element is said to be ‘semi‐analytical’. Goransson and Davidsson used a mapped wave infinite element in some three dimensional acoustic problems, in 1987. They incorporated the infinite elements into the ASKA code. A novel application of wave infinite elements to photolithography simulation for semiconductor device fabrication was given by Matsuzawa et al. They obtained ‘reasonably good’ agreement with observed photoresist profiles. Häggblad and Nordgren used infinite elements in a dynamic analysis of non‐linear soil‐structure interaction, with plastic soil elements. In 1989 Lau and Ji published a new type of 3‐D infinite element for wave diffraction problems. They gave good results for problems of waves diffracted by a cylinder and various three dimensional structures.