After stating the fundamental laws of electromagnetism, the eigenvalue problem for the time-harmonic fields in a resonator has been derived. A closed form solution is available only when trivial shapes are considered, e.g. parallelepipeds or cylinders. In the second part of the chapter we have defined the most common quantities of interest used in RF cavity design and we have proposed a coupled model for simulation of the Lorentz detuning effect.
Our main interest is the simulation of devices whose performance is highly sensitive to the domain shape. As such we propose the application of Isogeometric methods for the numerical treatment. In the following chapter we focus on the discretisation of Maxwell’s eigensystem (2.28) and of linear elasticity problem (2.52) using IGA, giving particular attention to the construction of suitable approximation spaces for the electromagnetic fields.
3 Isogeometric Analysis
We now enter the world of discrete, the solution we apply is quite neat: from computer designs
we can take the B-Splines,
and the diagram will be complete.
The first step in the numerical treatment of a PDE on a domain of interest Ω is, typically, the discretisation of the domain itself. One classical example is the meshing process of classical FEM which subdivides the geometry into triangles (quadrilaterals) in the 2D case or tetrahedra (hexahedra) in the 3D case. Whenever the geometry is non trivial, this immediately introduces an approximation. The use of curved boundary meshes and higher order elements [71] alleviates this problem, but does not solve it, since even common curves like conical sections (circles, ellipses, etc. . . ) cannot be exactly represented by a polynomial map.
In many practical applications the domain Ω is generated through a CAD software, thus a set of basis functions and a parametrisation of it (or at least of its boundary representation) are already available. IGA was introduced in 2005 by Hughes et al. [58] with the idea of using the same classes of basis functions commonly used for geometry description in CAD software, for the representation of the solution of scalar (or vector) fields. Such basis functions are the so called B-Splines and NURBS basis functions [12, 83].
This distinctive feature allows for the exact representation of geometries defined via CAD, independently of the level of refinement of the computational grid. Moreover, the commonly employed piece-wise polynomial spaces are embedded into NURBS spaces, thus IGA can be seen as a generalisation of standard FEMs. It is worth mentioning the fact that IGA shares the same Galerkin approach as FEM: a weak formulation of the PDE is constructed and cast in a suitable spaceV , and a set of basis functions is selected in order to construct a sequence of proper subspaces Vh → V as h → 0. IGA expands the set of basis functions from polynomials to the superset of rational polynomials. In many cases a pre-existent code can be easily modified to work in an IGA setting by changing the basis function construction routines only. Given the higher regularity of the basis functions employed, the method has been shown to present several advantages over FEM in addition to the better handling of CAD geometries, like a faster convergence with respect to the number of degrees of freedom [62] and the possibility to treat higher order differential operators [4].
As mentioned in the Introduction, these properties have made IGA appealing for a wide variety of applications (see chapter 1).
In the simulation of RF cavities it is especially interesting to be able to accurately represent the geometry throughout the analysis without the mesh approximation, since the use of NURBS guarantees exact representation of the elliptical arcs that defines the half-cells (see Fig. 2.7). Furthermore, IGA allows for the domain deformations to be handled in an accurate and straightforward way by changing only a relatively small number of control points.
As a first step in this chapter we state the variational formulation of the cavity eigenproblem. The weak formulation requires us to define Sobolev spaces of both scalar and vector valued functions and important relations that hold between them. The concepts of IGA are then introduced with focus on the application to RF cavity simulation. In particular we define B-Splines and Non-Uniform Rational B-Splines basis functions and show how they are used to parametrise geometries. In section 3.4 those functions and mapping are used to discretise Maxwell’s equations and the linear elasticity problem (2.52). Finally a brief explanation on how to treat complicated domains using a multipatch approach is given.
3.1 Weak Formulation
To numerically solve Maxwell’s eigenvalue problem (2.28) the Galerkin Ansatz is used. For this purpose the equations must be cast into weak formulation and suitable choices for the spaces must be made.
A preliminary step before the discretisation is the construction of its weak form and the definition of the proper spaces in which to cast it.
3.1.1 Vector Functions with Well Defined Curl or Divergence
For the definition of the Sobolev spaces we follow [71, chapter 3] and we denote byLp(Ω) the classical Lebesgue spaces endowed with the norm k·kLp(Ω), and byLp(Ω) their vector valued counterparts. The
Hilbert spacesHk(Ω) denote the functions in Lp(Ω) such that their k-th order derivatives also belong to Lp(Ω), and Hk(Ω) are their vector valued counterparts. When a fixed value is prescribed on a section of the boundary Γ ⊆ ∂ Ω, we will use the common notation Hk
0,Γ(Ω) and Hk0,Γ(Ω), where the 0 subscript
denotes homogeneous boundary conditions.
To correctly deal with Maxwell’s equations we need to introduce the spaces of L2(Ω) functions with
divergence or curl in L2(Ω). Under the hypothesis mentioned above, the space of functions with
divergence in L2(Ω) is denoted by H (div; Ω) and defined by:
H (div; Ω) = v ∈ L2(Ω) : ∇ · v ∈ L2(Ω)
(3.1) with the associated norm
kvkH(div;Ω)=kvk2L2(Ω)+ k∇ · vk2L2(Ω) 1/2
. (3.2)
To solve problems in which the normal component of a vector field is specified on ∂ Ω, it is also useful to consider the subspace ofH (div; Ω):
H0(div; Ω) = {v ∈ H (div; Ω) : v · n |∂ Ω= 0} , (3.3)
corresponding to the kernel of the trace operator
γn(v) = v |∂ Ω·n. (3.4)
We denote the space of three-dimensional vector functions with curl inL2(Ω) as H (curl; Ω) = v ∈ L2(Ω) : ∇ × v ∈ L2(Ω)
endowed with the norm
kvkH(curl;Ω)=kvk2L2(Ω)+ k∇ × vk2L2(Ω) 1/2
. (3.6)
The spaceH (curl; Ω) is of great importance in the study of Maxwell’s equations since it corresponds to
the space of finite-energy solutions [71].
Let us now assume, for the sake of simplicity, that our domain Ω is simply connected and that its boundary ∂ Ω is split into two disjoint parts, ∂ Ω = ΓD∪ ΓN with ΓD6= 0. We denote by H0,ΓD(curl; Ω) the
space of functions inH (curl; Ω) with vanishing tangential trace on ΓD:
γt(v) = v|ΓD× n = 0. (3.7)
A more in depth discussion about the trace properties of functions inH (curl; Ω) can be found in [71].
3.1.2 Weak Formulation of the Eigenvalue Problem
The standard variational formulation of (2.28) is [11]: Find ω ∈ R and E ∈ H0(curl; Ω) with E 6= 0 such that
(∇ × E, ∇ × v) = ω2"0µ0(E, v) ∀v ∈ H0(curl; Ω) (3.8a)
(E, ∇φ) = 0 ∀φ ∈ H10(Ω) , (3.8b)
where (·, ·) identifies the usual L2 scalar product in Ω, i.e.
(u, v) = Z
Ω
u · v dr. (3.9)
In practice, however, the following formulation is used [11, 98]: Find ω ∈ R and E ∈ H0(curl; Ω) with E 6= 0 such that
(∇ × E, ∇ × v) = ω2"0µ0(E, v) ∀v ∈ H0(curl; Ω) . (3.10)
In general, there exists two groups of solution of (3.10): static fields (with ω = 0 and ∇ × E = 0) and resonant fields (with ω 6= 0). The first case corresponds to the infinite dimensional eigenspace given by the gradient of functions in H1
0(Ω), while the other eigenvalues form a diverging sequence with
eigenfunctions inH0(curl; Ω) ∩ H div0; Ω, where we have introduced the space H div0; Ω := v ∈ L2
(Ω) s.t. ∇ · v = 0 (3.11)
of integrable functions with zero divergence [11, 71].
Given weak formulation (3.10), it is possible to construct a sequence of finite dimensional subsets Vh⊂ H0(curl; Ω) in terms of a set of basis functions:
Vh= span {vi}Ni=1dof, dim(Vh) = Ndof. (3.12)
These basis functions are used both for the testing and for expressing the solution field:
E ≈ Eh∈ Vh, Eh= Ndof
X
j=1
ejvj. (3.13)
The discretised weak formulation becomes: Find ωh∈ R and Eh∈ Vh withEh6= 0 such that Ndof X j=1 ej ∇ × vj, ∇ × vi = ω2h"0µ0 Ndof X j=1 ej vj, vi ∀i = 1, . . . , Ndof. (3.14)
By introducing the stiffness and the mass matrixK ∈ RNdof×Ndof andM ∈ RNdof×Ndofwith elements
ki j= ∇ × vj, ∇ × vi, mi j = vj, vi
, (3.15)
we obtain to the generalised eigenvalue problem
Ke = λMe. (3.16)