Perturbation Expansion Approach Research Articles

Spectral-finite-element approach to two-dimensional electromagnetic induction in a spherical earth

We present a spectral-finite-element approach to the 2-D forward problem for electromagnetic induction in a spherical earth. It represents an alternative to a variety of numerical methods for 2-D global electromagnetic modelling introduced recently (e.g. the perturbation expansion approach, the finite difference scheme). It may be used to estimate the effect of a possible axisymmetric structure of electrical conductivity of the mantle on surface observations, or it may serve as a tool for testing methods and codes for 3-D global electromagnetic modelling. The ultimate goal of these electromagnetic studies is to learn about the Earth's 3-D electrical structure. Since the spectral-finite-element approach comes from the variational formulation, we formulate the 2-D electromagnetic induction problem in a variational sense. The boundary data used in this formulation consist of the horizontal components of the total magnetic intensity measured on the Earth's surface. In this the variational approach differs from other methods, which usually use spherical harmonic coefficients of external magnetic sources as input data. We verify the assumptions of the Lax-Milgram theorem and show that the variational solution exists and is unique. The spectral-finite-element approach then means that the problem is parametrized by spherical harmonics in the angular direction, whereas finite elements span the radial direction. The solution is searched for by the Galerkin method, which leads to the solving of a system of linear algebraic equations. The method and code have been tested for Everett & Schultz's (1995) model of two eccentrically nested spheres, and good agreement has been obtained.

High order nonlinear susceptibilities of composite medium

We propose an approximate general method for calculating the effective high-order nonlinear susceptibilities of a weakly nonlinear dielectric composite system. General expressions for effective high-order nonlinear susceptibilities are derived. These results are strictly valid if the electric field in each component is uniform. The approximation is applied to two different cases: Low Density Limit, Maxwell-Garnett Approximation, our results are comparable with those of other methods, such as T-matrix approach, perturbation expansion approach.

“Missing moment” and perturbative methods for polynomial iterated function systems

An iterated function system (IFS) over a compact metric space X is defined by a set of contractive maps w i : X → X, i = 1,…, N, with associated nonzero probabilities p i > 0, ⩞p i = 1 . The “parallel” action of the maps defines a unique compact invariant attractor set A ⊂ X which supports an invariant measure μ and which is balanced with respect to the p i . For linear w i on X ⊂ R , the invariance of μ yields a relation between the moments g n = ∫ χ n dμ which permits their recursive computation from the initial value g 0 = 1. For nonlinear w i , however, the moment relations are incomplete and do not permit a recursive computation. This paper describes two methods of obtaining accurate estimates of the moments when the IFS maps w i are polynomials: (i) application of the necessary Hausdorff conditions on the g i to obtain convergent upper and lower bounds and (ii) a perturbation expansion approach. The methods are applied to some model problems.