Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results Eigenvalues of the curl operator: variational formulation and numerical approximation Alberto Valli Dipartimento di Matematica, Universit` a di Trento, Italy A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results Joint paper with: Ana Alonso Rodr´ ıguez Dipartimento di Matematica, Universit` a di Trento, Italy Jessika Cama˜ no Departamento de Matem´ atica y F´ ısica Aplicadas, Universidad Cat´ olica de la Sant´ ısima Concepci´ on, Chile Rodolfo Rodr´ ıguez Departamento de Ingenier´ ıa Matem´ atica, Universidad de Concepci´ on, Chile Pablo Venegas Departamento de Matem´ atica, Universidad del B´ ıo B´ ıo, Chile A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results Outline Introduction 1 The mathematical problem 2 The variational problem 3 Spectral analysis 4 Finite element approximation 5 Numerical results 6 A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results Introduction A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results Physical framework By the Lorentz law the density of the magnetic force is given by F = J × B , where J is the current density and B is the magnetic induction. Linear isotropic media: B = µ H (the scalar function µ being the magnetic permeability). Eddy current or static approximation: J = curl H . If curl H = λ H ( λ a scalar function) the magnetic force vanishes: F = curl H × µ H = λ H × µ H = 0 . A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results Physical framework (cont’d) Fields satisfying curl H = λ H are called force-free fields. If λ is a constant are called linear force-free fields. [Clearly, the most interesting case is for λ not identically vanishing.] [In fluid dynamics, force-free fields are called Beltrami fields, and a Beltrami field u that is divergence-free and tangential to the boundary is a steady solution of the Euler equations for incompressible inviscid flows (with pressure given by p = −| u | 2 / 2).] A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results Physical framework (cont’d) A couple of interesting physical remarks: a field which is divergence-free and tangential to the boundary (e.g., the magnetic field), and which minimizes the magnetic energy with fixed helicity is a linear force-free field [Woltjer (1958)]; linear force-free fields are time-asymptotic configurations (they remain force-free as time changes) [Jette (1970)]. [Helicity of a vector field v in a domain Ω, i.e., � � H ( v ) = 1 x − y v ( x ) × v ( y ) · | x − y | 3 d x d y , 4 π Ω Ω is a “measure of the extent to which the field lines wrap and coil around one another” (Cantarella et al. (2000:a); other physical remarks can be found there and in Cantarella et al. (2000:b)).] A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results The mathematical problem A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results Mathematical framework Let us focus now on the mathematical aspects of this problem. A linear force-free field is an eigenfunction of the curl operator: curl u = λ u . It is thus interesting to see when it is possible to define self-adjoint realizations of the curl operator. The starting point is clearly the Green’s formula (here and in the sequel we write Γ = ∂ Ω) � � ( v · curl w − curl v · w ) = v × n · w , Ω Γ � and the analysis is driven by the need of obtaining Γ v × n · w = 0. A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results The spectral problem: Ω simply-connected � It is clear that Γ v × n · w = 0 when v × n = 0 on Γ. However, this boundary condition is too strong for the spectral problem. Weaker boundary conditions can be devised (see, e.g., Kress (1972, 1986); Picard (1976, 1998); Yoshida and Giga (1990)). When Ω is a simply-connected domain, it is sufficient to assume that curl v · n = 0 on Γ. Since for an eigenfunction u of the curl operator the condition u · n = 0 implies curl u · n = 0, it is thus natural to consider the spectral problem curl u = λ u in Ω div u = 0 in Ω (1) u · n = 0 on Γ . The numerical approximation of (1) has been analyzed in Rodr´ ıguez and Venegas (2014). A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results The spectral problem: Ω not simply-connected The condition curl v · n = 0 is not enough if the physical domain is not simply-connected. A possible additional condition is the following (see, e.g., Kress (1972, 1986); Picard (1976, 1998); Yoshida and Giga (1990)): the curl operator is self-adjoint if curl v · n = 0 on Γ (which is equivalent to curl v ⊥∇ H 1 (Ω)) and curl v ⊥ K T , where K T is the space of the so-called harmonic Neumann fields h (those fields satisfying curl h = 0 in Ω, div h = 0 in Ω and h · n = 0 on Γ). This space K T is finite dimensional, its dimension being the first Betti number of Ω; it is trivial for a simply-connected domain Ω. ⊥ It can be proved that H 0 ( curl ; Ω) = ∇ H 1 (Ω) ⊕ K T . The numerical approximation of the eigenvalues and eigenfunctions of this problem has been studied in Lara et al. (2016). A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results The spectral problem: Ω not simply-connected (cont’d) It is worth noting that the condition curl v ⊥ K T is not essential, but only sufficient for the proof that the curl operator is self-adjoint. In this respect, an in-depth analysis has been recently presented in Hiptmair et al. (2012). The authors, by incorporating in problem (1) additional conditions related to the first homology group of Γ, devise suitable self-adjoint realizations of the curl operator. A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results Homological tools Let us show how this family of eigenvalue problems can be described. We first need to recall some geometrical results. Let g be the first Betti number Ω; then the first Betti number of Γ is equal to 2 g . From algebraic topology we know that: on Γ there are 2 g non-bounding cycles { γ j } g j =1 ∪ { γ ′ j } g j =1 , that are the generators of the first homology group of Γ { γ j } g j =1 are the generators of the first homology group of Ω ′ , with Ω ′ = B \ Ω, B being an open ball containing Ω (tangent vector on γ j denoted by t j ); j } g { γ ′ j =1 are the generators of the first homology group of Ω (tangent vector on γ ′ j denoted by t ′ j ); A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results Homological tools (cont’d) in Ω there exist g ‘cutting’ surfaces { Σ j } g j =1 , that are connected orientable Lipschitz surfaces satisfying Σ j ⊂ Ω and ∂ Σ j ⊂ Γ, such that every curl-free vector in Ω has a global potential in the ‘cut’ domain Ω 0 := Ω \ � g j =1 Σ j ; each surface Σ j satisfies ∂ Σ j = γ j , ‘cuts’ the corresponding cycle γ ′ j and does not intersect the other cycles γ ′ i for i � = j ; in Ω ′ there exist g ‘cutting’ surfaces { Σ ′ j } g j =1 , that are j ⊂ Ω ′ and connected orientable Lipschitz surfaces satisfying Σ ′ j ⊂ Γ, such that every curl-free vector in Ω ′ has a global ∂ Σ ′ potential in the ‘cut’ domain (Ω ′ ) 0 := Ω ′ \ � g j =1 Σ ′ j ; each surface Σ ′ j satisfies ∂ Σ ′ j = γ ′ j , ‘cuts’ the corresponding cycle γ j , and does not intersect the other cycles γ i for i � = j . A. Valli Eigenvalus of the curl
Introduction The mathematical problem The variational problem Spectral analysis Finite element approximation Numerical results Homological tools (cont’d) [Some misunderstanding appears when looking back at the literature on this topic; it is thus interesting to make clear that: the statement concerning the ‘cutting’ surfaces Σ j does not mean that the ‘cut’ domain Ω 0 is simply-connected nor that it is homologically trivial: an example in this sense is furnished by Ω = Q \ K , where Q is a cube and K is the trefoil knot.] A. Valli Eigenvalus of the curl
Recommend
More recommend
