General overview

The FEM begins by casting the differential problem into an equivalent integral problem, which is more easily solvable with numeric techniques. In order to do this it employs

a classical method for solving boundary-value problems: either theRayleigh-Ritz varia- tional method or theGalerkin’s method of weighted residuals.

For example, suppose you want to solve the boundary-value problem defined by the inhomogeneous partial differential equation:

Lφ=f (2.7)

defined on a domain Ω and completed by some boundary conditions on the boundary Γ that encloses the domain. In Equation 2.7, L is a differential operator, f is the forcing function (typical for example of scattering or excitation problems where sources are present) and φ is the unknown solution. In the Rayleigh-Ritz method the solution of Equation 2.7 is sought by minimising the functional associated with the differential equation. In the Galerkin’s method (GM), which leads to the same solution provided that L is self-adjoint [148], the solution is obtained by minimising an appropriately defined residual of the differential equation.

The GM works in the following way: assume that ˜φis an approximate solution of (2.7) which can be expanded as

˜ φ= N X j=1 cjvj ={c}T{v} (2.8) where the vj are chosen expansion functions defined over the entire domain, cj are the constant coefficients to be determined and {·} is a column vector. The substitution of

˜

φforφin (2.7) would result in a non-zero residual:

r =Lφ˜−f 6= 0. (2.9) The solution of (2.7) can therefore be approximated by the search of ˜φwhich minimises

r in Ω, in some mathematical sense. The problem at this point is that r is a contin- uous function of the entire infinite space (r = r(r),r ∈ Ω); it is therefore a common approach to minimise instead a weighted integral ofr, calculated over the entire domain Ω. Weighted residual methods define a number of those weighted residuals Ri, upon which the following condition is enforced:

Ri =

Z

Ω

wir dΩ = 0 i= 1,2,3, . . . (2.10) wherewi are chosen weighting functions. In the particular case of the Galerkin method, the weighting functions are selected to be exactly the functions used in the expansion of the approximate solution, i.e. w_i =v_i fori= 1,2,3, . . . , N. Therefore Equation 2.10 becomes: Ri= N X j=1 cj Z Ω viLvjdΩ− Z Ω vif dΩ = 0 i= 1,2,3, . . . , N. (2.11)

This can be written as the matrix equation

[S]{c}={b} (2.12) where{c}is the vector of the unknown coefficients, [S] is anN×N matrix with elements

S_ij =

Z

Ω

v_iLv_jdΩ (2.13)

and the column vector{b}is composed by

bi =

Z

Ω

vif dΩ. (2.14)

So far we have simply transformed the original differential problem (2.7) with an infinite number of degrees of freedom (DOFs) into a linear system of integral equations with a finite number of DOFs. Nothing so far has been said about the expansion orinterpolating

functions vj. Generally the selection of a complete set of vj, defined over the entire domain and in such a way they can satisfy the boundary conditions, can be extremely difficult, if not impossible. To alleviate this difficulty, in the FEM, the domain Ω is divided into smaller subdomains orfinite elements which, in the case of two-dimensional problems usually assume the shape of triangles and for three-dimensional problems that of tetrahedrons. The interpolating functions are then defined only locally, over each subdomain (local shape functions), in such a way that they are identically zero outside it. Since the subdomains are small, the local functions do not need to vary in a complicated manner to approximate φ(r) accurately: such trial functions are usually much simpler than the globally-defined functions used in other methods.

Without entering into the details of the method (for which the reader is pointed to the book by Jin [148]) here we will just add a few considerations:

• In the simpler FEM formulations, based on nodal elements, the local shape func- tions are chosen as simple polynomials. For example in 2D these take the form of

Ni(x, y) =ax+by+cfor a linear case orNi(x, y) =ax2+by2+cxy+dx+ey+f for the quadratic case. For each triangle a number of 3 and 6 nodal points – or DOFs – respectively is therefore required. Increasing the order of the polynomial improves the overall accuracy at the expense of the computational time;

• By employing locally defined functionsN_i the integral in (2.13) can be expressed as a sum of many integrals, each of which operates only on a single subdomain Ωe:

Sij = X e Z Ωe NiLNjdΩe. (2.15) Each integral will now comprise the product of local shape functions and/or their derivatives, depending on the specific form of the differential operator L. By a

careful choice of theN_i, each integral can be solved analytically, transformed into a sum of rational functions and stored for a subsequent quick evaluation. The elements of [S] can therefore be easily obtained by the combination or assembly

of the local integrals. Before the system of equations is ready to be solved the boundary conditions can be also simply imposed, directly on the elements of the matrix;

• The initial differential problem has been thus transformed into the linear algebraic system (2.12), where the size of [S] is DOF×DOF. With DOF typically on the order of 100,000 or more, it is easy to understand why the FEM requires the largest amount of memory amongst the various numerical methods, as was observed in the comparison in Ref. [147];

• In current computers however, memory is generally not the limiting factor. More- over, from the choice of employing local functions which are zero outside the sub- domain where they are defined, all the elements Sij for which i and j belong to different subdomains are identically zero. The matrix [S] is therefore sparse – or mainly composed by zero elements – and numerical procedures optimised for solving Equation 2.12 for sparse matrices can be exploited, reducing considerably the time and memory requirements;

• Recent FEM formulations can employ more complicated shape functions, either based onvector oredge elements or on a combination of (hybrid) edge and nodal elements [149]. The latter is the formulation that will be employed throughout this thesis. Although the specific form of the local integrals is more complicated, the general FEM procedure highlighted above remains unaltered. These vector elements provide at least two advantages over conventional nodal implementations: they allow the elimination of spurious solutions (see [148] for more details) and automatically enforce the field continuity at the interface between media with different permittivity (see the next Section).