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The spectral boundary element method for solving two‐dimensional transient heat conduction problems is developed. This method is combined with the fast Fourier transform (FFT) to convert the solution between the time and frequency domains. The fundamental solutions in the frequency domain, required for the present method, are discussed. The resulting line integrations in the frequency domain are discretized using constant boundary elements and used in a Fortran boundary element program. Three examples are used to illustrate the accuracy and effectiveness of the method in both the frequency and time domains. First, the frequency domain solution procedure is verified using the steady‐state example of a semi‐infinite half space with a heat flux applied to a patch of the surface. This spectral boundary element method is then applied to the problem of a circular hole in an infinite solid subjected to a time‐varying heat flux, and solutions in both the frequency and time domains are presented. Finally, the method is used to solve the circular hole problem with a convection boundary condition. The accurary of these results leads to the conclusion that the spectral boundary element method in conjunction with the FFT is a viable option for transient problems. In addition, this spectral approach naturally produces frequence domain information which is itself of interest.

This paper presents the generalized theory of the most important energy principles in structural analysis. All derive from two basic complementary theorems denoted as the principles of virtual displacements and virtual forces. Both exact and approximate methods are discussed and particular attention is paid to the derivation of upper and lower limits. The theory is not restricted to linearly elastic bodies but includes ab initio the effect of non‐linear stress‐strain laws and thermal strains. Finally the basic principles are illustrated on a number of simple examples in preparation for the more complex ones to appear in Parts II and III.

A hybrid formulation is proposed that incorporates finite element substructuring and Galerkin boundary elements in the numerical solution of Poisson's or Laplace's equation with open boundaries. Substructuring the problem can dramatically decreases the size of matrix to be solved. It is shown that the boundary integration that results from application of Green's first theorem to the weighted residual statement can be used to advantage by imposing potential and flux continuity through the contour which separates the interior and exterior regions. In fact, the boundary integration is of exactly the same form as that found in Galerkin boundary elements.

This paper aims to present a simple, yet accurate and efficient, formulation for computing the vertical soil stresses due to arbitrarily distributed surface pressures or loads over an arbitrarily shaped area.

By leveraging on the strength of Green’s theorem, the present approach is based on the formulation of the classical Boussinesq solution as a boundary-type problem over an arbitrarily shaped simply- or multiply-connected loaded region. The accuracy of the developed formulation was exemplified through a number of illustrative examples, which included both simply- and multiply-connected loaded areas.

The results of the test examples presented in this work indicated a high degree of accuracy and flexibility of the developed approach despite its simplicity.

The main contribution of the present work is the introduction of an efficient meshless approach and an algorithm that can be implemented in few lines of code on any programing platform, as either a stand-alone program or a computational module in larger engineering software packages.

The computation of 3‐D eddy current problems is an important topic both in theoretical and engineering areas. Based on Green's theorem, Kost theoretically developed a BEM formulation for general 3‐D nonlinear eddy current problems. The parallel implementation of this formulation in the linear case has been made on a massively parallel supercomputer, where a constant triangular element was used.

IT is natural in reviewing the developments of Sections 3 and 4 to inquire if it is possible to enlarge upon the conception of complementary work and strain energy in a similar way as accomplished for work and strain energy by the introduction of virtual displacements.

To propose a novel 3D hybrid approach, based on a discrete formulation of Maxwell equations (the cell method – CM), suitable for solving eddy current problems in unbounded domains.

Field equations for magnetodynamics are expressed directly in algebraic form thanks to the CM. The eddy current problem inside bulk conductors is formulated in terms of discrete modified vector potential, whereas magnetic scalar potential is used in order to model the free space. The CM is coupled to the boundary element method by using a surface boundary operator, which maps the surface magnetic fluxes to the surface magnetic scalar potentials. This leads to a unique set of linear equations to be solved in terms of discrete potentials. The eddy currents in bulk conductors are then obtained from discrete potentials.

It is shown that formulation of hybrid approaches can be simplified by expressing field equations directly in algebraic form without need of weighted residual techniques. An original strategy, based on Green's formula for the magnetic scalar potential, is proposed in order to couple conducting parts to the exterior domain.

Conducting bodies with multiply connected parts cannot be modelled by the proposed approach, since it is based on the magnetic scalar potential. The resulting global matrix is partially dense and non‐symmetric; therefore, standard iterative solvers such as GMRES have to be used.

The proposed approach can be suitably used for analyzing eddy current problems involving models with high degree of complexity, large air domains and moving parts. These are typical of induction heating processes.

This paper proposes a new 3D hybrid approach, based on a discrete formulation of Maxwell equations. A novel coupling strategy relying on integral electromagnetic variables, i.e. magnetic fluxes and magnetic scalar potentials, is devised in order to solve uniquely for eddy currents inside conducting bodies.

The purpose of this paper is to analyse the dynamics of a thermal field generated in a tubular bus with rated current by using two models of electrical resistivity of copper.

The boundary-initial problem of the modified heat equation was formulated for the tubular bus. Analytical solutions were obtained by means of Green’s functions as the kernels of the integral operator inverse to the corresponding differential operator. The results were presented graphically and verified using the finite element method. The calculations were made by considering the example of the Storm Power Components tubular bus (USA).

Analytical field models were used to determine time- and space-variable heating curves, time constants and steady-state current ratings.

This paper is related to the structure of a hollow cylinder. Other bus sections can be taken into account by using the coordinate systems of different curvilinear orthogonal symmetry.

Using the analytical method, the influence of the variable (temperature dependent) electrical resistivity on some important parameters and characteristics of the tubular bus was investigated. The system was considered as an element with distributed parameters.

The penalty boundary method (PBM) is a new method for performing finite element analysis using a regular overlapping mesh that does not have to coincide with the geometric boundaries. The PBM uses CAD solid geometry directly to generate element matrix equations and apply boundary conditions, removing the need for a separate representation of the geometry. The preliminary results show that the PBM can significantly reduce the time and manual intervention required to prepare finite element models and perform analyses. This paper presents the PBM approach for representing the problem domain on an overlapping mesh that results in a more traditional method for applying natural boundary conditions.

A finite element formulation for flexure of isotropic plates based on a recent refined theory is developed. The refined theory incorporates effects of transverse shear, transverse normal stress and transverse normal strain. The Galerkin finite element method was used to develop the finite element equations for both plate bending and inplane problems. The performance of the proposed finite element model was evaluated by solving problems of uniformly loaded thick plates with different support conditions. The results of the present formulation are compared with Mindlin/Reissner and elasticity solutions.