By Nicola Bianchi

Define OF ELECTROMAGNETIC FIELDSVector AnalysisElectromagnetic FieldsFundamental Equations SummaryReferencesBASIC rules OF FINITE aspect METHODSIntroductionField issues of Boundary ConditionsClassical strategy for the sector challenge SolutionThe Classical Residual technique (Galerkin's Method)The Classical Variational strategy (Rayleigh-Ritz's Method)The Finite point MethodReferencesAPPLICATIONS OF THE FINITE Read more...

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provides the instruments precious for the research and layout of any kind of electric computing device by way of integrating mathematical/numerical concepts with analytical and layout methodologies. This book Read more...

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Basic Principles of Finite Element Methods 29 Before illustrating the different procedures, let us introduce the inner product between two functions φ and ϕ. 10) where the symbol ~ indicates the complex conjugate. 10) could be used to verify the properties of the differential operator L. 1) directly. 1). 15) equal to zero (or at least very low) in the whole analysis domain. Fixing some weight functions wi, the residual method forces the integral of the residuals, weighed by wi to be zero over the domain volume τD.

1) is given by the Poisson, Laplace, or Helmholtz equation, in which φ is a scalar or a vector field. As an example, in the case of an electrostatic problem, φ indicates the scalar electric potential V, and its distribution is described by the Poisson equation. The forcing function is the distribution of the free charge density f = ρ. 2) in which a nonhomogeneous medium is considered, so that the electric permittivity ε could be nonconstant. 3) Boundary Conditions The field problem admits a solution not only if the differential equation that describes its distribution is known in all the points of the domain D, but also if the unknown function φ is given on the boundary Γ of the domain D itself.

Guarnieri and G. Malesani, Elettromagnetismo Stazionario e Quasi-Stazionario, Edizioni Progetto, Padova, 1999 (in Italian). 11. M. Jin, Electromagnetic Analysis and Design in Magnetic Resonance Imaging, CRC Press, Boca Raton, FL, 1998. 2 Basic Principles of Finite Element Methods Some basic concepts of finite element method are dealt with in this chapter and in the next. In this chapter, the differences of the finite element method compared to the classical methods of field problem analysis are highlighted.

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