Minimum variational principles for time-harmonic waves in a dissipative medium and associated variational principles of Hashin–Shtrikman type

Abstract

Minimization variational principles for linear elastodynamic, acoustic or electromagnetic time-harmonic waves in dissipative media were obtained by Milton et al. (Milton et al. 2009 Proc. R. Soc. A 465, 367-396 (doi:10.1098/rspa.2008.0195)), generalizing the quasistatic variational principles of Cherkaev and Gibiansky (Cherkaev & Gibiansky 1994 J. Math. Phys. 35, 127-145 (doi:10.1063/1.530782)). Here, a further generalization is made to allow for a much wider variety of boundary conditions, and in particular Dirichlet and Neumann boundary conditions. In addition minimization or maximization principles of the Hashin-Shtrikman type, incorporating 'polarization fields', are developed. The corresponding principles for static problems have found substantial use in bounding the effective static properties of composite materials. The new dynamical principles offer the prospect of developing bounds on the effective dynamic properties of such materials.

Exact relations for Green’s functions in linear PDE and boundary field equalities: a generalization of conservation laws

“…The preceeding three equations have been written in this form so Im L(x) ≥ 0 when Im ω ≥ 0, where complex frequencies have the physical meaning of the solution increasing exponentially in time. Under assumptions that the material moduli are lossy, or that the frequency ω is complex with positive imaginary part, one can easily manipulate them into equivalent forms similar to the Gibiansky-Cherkaev form in (10.11) with a positive semidefinite tensor entering the constitutive law [33,35]. Of course, the boundary fields ∂E and ∂J then need to be appropriately redefined.…”

“…Furthermore, some components of the fields E ∈ E are not necessarily derivatives of potentials but could also involve, say, divergence-free vector fields (and the corresponding components of the fields J ∈ J would then be gradients of potentials). Such mixed formulations are useful in quasistatic and wave equations in lossy media when one wants to reformulate the problem so that L(x) is real and positive definite [8,33,35,36].…”

Bounds on complex polarizabilities and a new perspective on scattering by a lossy inclusion

“…The imaginary part of the quadratic form associated with Y 0 * has a physical interpretation in terms of the power absorbed and scattered by the inclusion. In elastodynamics the power absorption by a body Ω, having a possibly complex density ρ 1 = ρ 1 + iρ 1 (with real and imaginary parts ρ 1 and ρ 1 ), is given by formula (2.5) in [53] and (taking into account our choice of e −iωt for the time dependence, rather than e iωt ) can be written as…”

“…We mention that minimization principles have been obtained by Milton, Seppecher and Bouchitté [51] and Milton and Willis [53] for the full time harmonic acoustic equations, Maxwell's equations, and elastodynamic equations, in bodies of finite extent containing inhomogeneous lossy media. This advance was made possible by the key realization that these equations can all be suitably manipulated into a form where it is easy to see that one can directly apply the transformation techniques of Cherkaev and Gibiansky [14] to obtain minimization variational principles.…”

A numerical minimization scheme for the complex Helmholtz equation

“…The classical methods of deriving a weak form for this equation (with Dirichlet boundary conditions, for example) result in the variational equation 2) physically to dissipated energy in the system, and is valid even for arbitrarily small coefficients of loss. While the framework presented in [9] results in nonstandard boundary conditions, Milton and Willis extend the principles to handle the classical Dirichlet and Neumann boundary conditions in [8].…”