Alban Gruget

Morgan Roger

Van Tam Nguyen

Caroline Lelandais-Perrault

Philippe Bénabès

Patrick Loumeau

Wide-band multipath A to D converter for Cognitive Radio applications

Keywords: Analog-digital conversion, ADC, Bandpass charge sampling, Integration sampler, Hybrid filter bank, Wideband receiver, Wide-band sensing

This article presents a digital-enhanced radio frequency receiver for fast wide-band spectrum sensing. It is based on charge sampling and hybrid filter bank techniques. The charge sampling method is employed to design analog bandpass filters. Using a hybrid filter bank for wide-band analog-to-digital conversion improves the speed and resolution of the conversion. We propose to use these techniques in combination of frequencydivision multiplexing with time-division multiplexing to design an integrated, completely software reconfigurable and reliable backend of radio frequency receiver for cognitive radio applications.

I. INTRODUCTION

The term of "Cognitive Radio" was coined by Joseph Mitola in 1999. It describes a radio where hardware parameters are defined by software. Furthermore such a radio link is aware of its surrounding and adapts intelligently. Future wireless communications systems should benefit from the cognitive radio approach to achieve high data rate services while managing interferences. Cognitive radio is indeed an emerging concept in wireless access, aimed at vastly improving the radio spectrum use by reusing unoccupied spectrum "holes" while minimizing the generated interference. The principle of cognitive radio is to detect transmission opportunities in spatial, temporal and spectral domains and reuse them to deploy new opportunistic services without causing any harmful degradation to the other systems in the band.

Spectrum sensing function (SSF) is arguably the most important task for the establishment of cognitive radio. SSF includes awareness of interferences and presence of primary users.

This work is part of a project studying cognitive radio systems to fulfill the needs for seamless mobile connectivity and optimum spectrum management. This paper focuses on the radio frequency (RF) receiver, especially on the sampling and quantization process for wide-band and reliable spectrum sensing. SSF can be realized in two steps:

• Wide sensing: coarse exploration of the whole band to discriminate used and available frequency bands; • Fine sensing: focusing on some bands selected trough wide sensing with a higher resolution to find more information about them, their occupation and availability.

The requirements for this receiver are wideband conversion, software reconfigurability, CMOS integrability.

This paper is organized as follows. Section II introduces the hybrid filter bank (HFB) architecture for wide bandpass analog-to-digital conversion (ADC). Section III presents the bandpass charge sampling (BCS) technique and its use in a HFB context. Section IV presents the studied system specifications. Simulation results are shown in Section V. The proposed RF receiver architecture is shown in Fig. 1. RF filters limit the reconfigurability of the receivers and current research works toward the suppression of these filters (SAW-less architecture). In this work, it is however decided that at the beginning of the project, the RF filter is still used in order to relax the requirement on both the low noise amplifier and the back-end stage. After the wideband RF filter, a wideband (or multi) low noise transconductance amplifier (LNTA) is used, which allows to implement bandpass charge sampling filters (see section III). The RF signal is then processed by a parallel M-path HFB by employing bandpass charge sampling filters as analysis filters (H i (s) in Fig. 1). On each path the subband signals are sampled at F s = F OUT / M, where F OUT =1/T is the effective sample rate of the system, and converted to digital signals with a n-bit quantizer. The resulting digital signals are then upsampled by M and filtered by the digital "synthesis" filters F i (z). The effective sample rate of the system is M times that of the subband quantizers. In this hybrid filter bank (HFB) architecture, the digital synthesis filters are computed to minimize reconstruction error (see section V).

II. HYBRID FILTER BANK ANALOG TO DIGITAL CONVERSION

Another interesting feature of HFB is the frequency focusing (FF) capability [START_REF] Lelandais-Perrault | Bandpass, and Versatile Hybrid Filter Bank A/D Conversion for Software Radio[END_REF]. The digital synthesis filters coefficients can be computed to get a constant resolution over the whole band, or rather achieve a higher resolution on a selected narrower band.

In most HFB architectures, analog filters are continuoustime. Continuous-time filters are well suited for very high speed applications and can offer high signal-to-noise ratio (SNR). However HFB architecture is highly sensitive to analog imperfections, and manufacturing errors of continuoustime filters that become more and more inaccurate in advanced CMOS technology. Furthermore, the architecture of a continuous-time filter becomes increasingly complex and expensive to integrate as the filter order or the quality factor becomes higher [START_REF] Asemani | Influences of oversampling and analog imperfections on hybrid filter bank A/D converters[END_REF]. That is why we propose to use bandpass charge sampling filters as analysis filters.

III. BANDPASS CHARGE SAMPLING

The bandpass charge sampling (BCS) filter [START_REF] Yuan | A charge sampling mixer with embedded filter function for wireless applications[END_REF] (also called bandpass integration sampler [START_REF] Mirzaei | Analysis of First-Order Anti-Aliasing Integration Sampler[END_REF]) provides the abovementioned requirements for the design of a digital enhanced receiver: BCS are simple (only switches and capacitors), CMOS integrable [START_REF] Latiri | A reconfigurable RF sampling receiver for multi-standard applications[END_REF], and robust to nonidealities such as clock jitter [START_REF] Mirzaei | Analysis of First-Order Anti-Aliasing Integration Sampler[END_REF]). The input waveform is integrated over a fixed time window, and the resulting integral is taken as a sample. The window can approach the sample period, and its repetition rate defines the sampling frequency. Fig. 2.a shows the bandpass charge sampling (BCS) filter architecture, and Fig. 2.b shows its clocks scheme.

The first element, the transconductance G m , represents the last stage of the LNTA. Its output i in is a differential analog current signal. During the integration window T i the switches are turned on alternatively at frequency F c . The resulting chopped current is integrated into the capacitor C s . At the end of the integration window the sampled voltage on the capacitor is read in V out through charge transfer during the time s Δ . Then the capacitor is reset during the time r Δ . Thus, the BCS combines a filtering and a sampling function. The filtering transfer function is a cardinal sine centered on frequency F c and with notches located at F c ±k.F i for all k in Z* [START_REF] Latiri | A reconfigurable RF sampling receiver for multi-standard applications[END_REF]. Fig. 3 shows this transfer function for F c =900MHz and two values of F i . This filtered signal is sampled at frequency F s .

T i is limited by T s and technology speeds since

s T r T i s Δ + + Δ =
. Thus with this architecture F i will always be higher than F s . Example of frequency response of a BCS filter.

IV. SYSTEM SPECIFICATIONS

The HFB architecture proposed in this paper includes BCS filters as analysis filters. Considering the properties of the BCS filters seen above, parameters F c and F i of each filter can be chosen to maximize the performance of the system, F s being fixed at 2*B/M (see section II).

The rest of the paper examines the variations of the performances of the system with respect to these parameters under some chosen specifications. These settings are:

• The frequency band of the input signal is [954-1762] MHz, corresponding to a bandwidth of 808 MHz. • The total band converted by the quantizers, which must be slightly wider than the input band to ensure acceptable performances for any HFB, is [900-1800] MHz.

• This band is divided into M=8 sub-bands, each being processed by a single quantizer. The sampling frequency is then F s =225 MHz. With these settings, there are 16 parameters to tune, two for each BCS filter. To reduce this number, we made the choice to fix the central frequency of each BCS filter on the medium frequency of its sub-band, which seems like a reasonable choice. The value of parameter F i was also chosen to be the same for all filters.

It seems likely that the results in section V can be partially generalized to any specification.

The performance of the architecture is measured by the Signal-to-Noise Ratio (SNR) defined for an input frequency f as:

∑ ∑ = + = M k k j j f F psde f f f 1 2 2 2 10 ) ( . ) ( Alias ) Trans( log 10 ) SNR( (1)
where Trans(f) is the transmittance introduced by analysis and synthesis filters, Alias j (f) is the j-th aliasing term produced by sub-sampling, psde is the power spectral density of quantification error (considered here as a white noise so psde is constant over the frequency band), and F k is the transfer function of synthesis filter k.

In the same way the overall performance is defined as the global SNR over the whole frequency band: This measure is related to the global resolution of the wideband ADC since 6dB of SNR correspond roughly to a resolution of 1 bit.

∫ ∑ ∫ ∑ ∫ = + = B M k k B j j B f f F psde f f f f

V. RESULTS

For a given set of analysis filters, the Least Mean Squares Global Approximation method [START_REF] Lelandais-Perrault | Bandpass, and Versatile Hybrid Filter Bank A/D Conversion for Software Radio[END_REF] computes the digital filter bank that minimizes the reconstruction error on the band (which takes into account the distortion, aliasing and quantification noise effects). This reconstruction error can be weighted to maximize the resolution over a selected narrower band, allowing for the frequency focusing feature.

Due to the specificity of BCS filters, the performance of the resulting HFB can be improved by introducing phase diversity between the paths of the HFB. Consequently small delays are added between the sampling instant of BCSs. We define δ as the delay between the sampling instant of two consecutive BCS (so if the first BCS samples at instant nT s , the i-th BCS samples at instant δ

).

( -+ i nTs

). This modifies the phase of the transfer function of each BCS filter and leads the architecture to be a combination of frequency-division multiplexing with time-divison multiplexing. To help the comparison with time-interleaved architectures, we introduce the normalized delay

1 . - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = M T d s δ (3)
T s /M being the delay introduced between paths in a timeinterleaved architecture.

Similarly, to ease the generalization of the results, a normalized width parameter is defined as

M B F T T r i i s = = * 2 (4)
Given the condition F i >F s , the value of r must be greater than 2.

The overall performance of the system is limited by the resolution of the quantizers and the number of coefficients of the FIR synthesis filters. Figures 4 (resp. 5) shows the variation of this performance with respect to d (resp. r) for various values of r (resp. d), when the quantizers have a resolution of n=14 bits (corresponding to a maximum SNR tot of 86dB) and the analysis filters have 128 coefficients. As can be seen on Fig. 4, the conditioning problem induces poor performances when d is near 0, but the resolution approaches the ideal value (n=14 bits) for values of d between 0.25 and 0.4. For those values, the performance does not vary much with respect to r, but is rather better for low values of r. However very low values could be difficult to implement in practice due to technological minimum for r Δ the capacitor reset time and for s Δ the sampled charge transfer time. Considering this, d=0.36 and r=2.3 could be reasonable and near optimal values for the parameters.

With these specific values, the global SNR is 77.4dB, which corresponds to about 13 bits of resolution. Figure 6 shows the SNR versus frequency in this case. It can be noted that the SNR is almost constant on the whole band, except near the borders of the band, where it decreases somewhat substantially. This could be improved easily by slightly increasing the sampling frequency of the quantizers and is the subject of further work. The overall performance of the ADC being near the maximum obtainable performance (86dB) given the resolution of the quantizers, the frequency focusing feature cannot improve it much on a narrower selected band. The interest of this feature can yet be illustrated when the number of coefficients of the FIR synthesis filters is reduced, which diminishes the overall performance of the ADC. To this aim, Fig. 7 compares the SNR versus frequency with and without frequency focusing, when the number of coefficients is 64 instead of 128.

VI. CONCLUSION

Combining bandpass charge sampling filters with hybrid filter banks offers a promising and potential solution for designing wideband A to D conversion in the context of a digital-enhanced radio frequency application. The proposed resulting architecture is a combination of frequency-division and time-division multiplexing.

Optimizing the BCS filters parameters, we showed that this architecture is very suitable for spectrum sensing feature: It can approach the ADCs resolution over a wide band, provided that the digital complexity is sufficient; or with lower digital complexity the frequency focusing feature can improve resolution on any shorter frequency range.

Fig. 1 .

 1 Fig. 1. Front end with Hybrid Filter Bank ADC.

Fig. 2 .

 2 Fig.2.a Bandpass Charge Sampling architecture.

Fig. 2 .

 2 Fig.2.b BCS clock scheme.

Fig. 3 .

 3 Fig.3.Example of frequency response of a BCS filter.

Fig. 4 .

 4 Fig.4. SNR tot versus normalized delay d for various values of r.

Fig. 5 .

 5 Fig.5. SNR tot versus normalized delay r for various values of d.

Fig. 6 .

 6 Fig.6. SNR versus frequency for near optimal values of the parameters (r= 2.3 and d=0.36) and 128-tap synthesis filters.

Fig. 7 .

 7 Fig.7. Illustration of the frequency focusing feature: SNR(f) of the system with 64-tap synthesis filters without and with frequency focusing over a 24MHz band around 1.6GHz.

ACKOWLEDGMENT

This work is part of TEROPP project, supported by ANR, the French Research National Agency. This project works on designing a full terminal for opportunistic radio behaviour, both at physical and MAC layers.


Graeme W Milton

Pierre Seppecher

Guy Bouchitté

Minimization variational principles for acoustics, elastodynamics, and electromagnetism in lossy inhomogeneous bodies at fixed frequency

Keywords: variational principles, acoustics, elastodynamics, electromagnetism

The classical energy minimization principles of Dirichlet and Thompson are extended as minimization principles to acoustics, elastodynamics and electromagnetism in lossy inhomogeneous bodies at fixed frequency. This is done by building upon ideas of Cherkaev and Gibiansky, who derived minimization variational principles for quasistatics. In the absence of free current the primary electromagnetic minimization variational principles have a minimum which is the time-averaged electrical power dissipated in the body. The variational principles provide constraints on the boundary values of the fields when the moduli are known. Conversely, when the boundary values of the fields have been measured, then they provide information about the values of the moduli within the body. This should have application to electromagnetic tomography. We also derive saddle point variational principles which correspond to variational principles of Gurtin, Willis, and Borcea.

Introduction

As our goal is to extend the classical minimization principles to acoustics, elastodynamics, and electromagnetism, let us begin by recalling the well-known classical results of electrostatics. Consider the equation of electrostatics

∇ • ε∇V = 0, (1.1)
for the real valued potential V (x) in a body Γ containing a locally isotropic material with scalar real positive dielectric constant ε(x). This equation can be rewritten in the equivalent form

D = εE, E = -∇V, ∇ • D = 0, (1.2)
where the first relation is known as the constitutive relation linking the electric displacement field D(x) and the electrical field E(x) where these two fields satisfy the differential constraints embodied in the second two relations. Two variational principles are well known. We have the Dirichlet variational principle that

W (V ) = inf V W (V ) where W (V ) = Γ ε|∇V | 2 /2, (1.3)
in which the infimum is taken over all trial potentials V (x) satisfying the boundary condition that V (x) = V (x) on the boundary ∂Γ. We also have dual Thompson variational principle that

W (D) = inf D ∇ • D = 0 W (D) where W (D) = Γ |D| 2 /(2ε), (1.4)
where the infimum is taken over all divergence free trial displacement fields D satisfying the boundary condition that D • n = D • n on ∂Γ. The quantity ε|∇V | 2 /2 = |D| 2 /(2ε) is the electrostatic energy density inside the body, and so these variational principles are also known as energy minimization principles. In the mathematically analogous DC electrical conductivity problem, where ε(x) and D(x) are replaced by the conductivity σ(x) and the current J(x), this quantity is half the electrical power dissipation and in that context the variational principles could more accurately be called power dissipation minimization principles. An appealing feature of these variational principles is that the minimum values W (V ) and W (D) can be expressed in terms of the values of V (x) and D • n at the boundary ∂Γ:

W (V ) = W (D) = - 1 2 ∂Γ (D • n)V (x).
(1.5) So these variational principles provide a constraint on the values of V 0 (x) = V (x) and q 0 (x) ≡ -D(x) • n at the boundary ∂Γ. The relationship between these boundary functions q 0 and V 0 is linear and the relationship q 0 = NV 0 defines the Dirichlet to Neumann map N, which is a real self-adjoint positive semi-definite operator. Therefore N is determined by knowledge of the quadratic form

w(V 0 ) = V 0 , NV 0 /2, (1.6)
where for any two scalar or vector, real or complex, valued functions p(x) and q(x) defined on the boundary ∂Γ, we denote p, q = ∂Γ p(x) • q(x).

(1.7)

Given trial potentials V (x, V 0 ), parameterized by the function V 0 , with V (x, V 0 ) = V 0 (x) on ∂Γ, the Dirichlet variational principle implies the inequality w(V 0 ) ≤ W (V ), (1.8) and provides an "upper bound" on the map N. By "upper bound" we mean an upper bound on the associated non-negative quadratic form w(V 0 ). To obtain a "lower bound" one uses the Thompson variational principle. Since w(V 0 ) = 0 when V 0 is constant the null-space of N consists of all constant fields and so N is determined by knowledge of the quadratic form w(q 0 ) = q 0 , N -1 q 0 /2, (1.9) for all fields q 0 (x) with zero average value on ∂Γ. The functionals w(V 0 ) and w(q 0 ) are Legendre transforms of each other,

w(V 0 ) = inf q [ V 0 , q -w(q)], (1.10)
where the infimum is over all fields q(x) with zero average value on ∂Γ, and so an upper bound on the functional w(q) provides a lower bound on w(V 0 ). Given trial displacement fields D(x, q 0 ) parameterized by the function q 0 , with -D(x) • n = q 0 (x) on ∂Γ, the Thompson variational principle implies the inequality w(q 0 ) ≤ W (D), (1.11) and provides a "lower bound" on the map N.

Thus the Dirichlet and Thompson variational principles allow one to bound the Dirichlet to Neuman map N. As shown by ?) they can also be used in an inverse fashion: if N and hence the functionals w(V 0 ) and w(q 0 ) are known one can obtain information about the permittivity distribution ε(x) from the inequalities (1.8) and (1.11) by appropriately chosing the trial potentials V (x, V 0 ) and trial displacement fields D(x, q 0 ). It has been proved that knowledge of N uniquely determines ε(x) when ε(x) is piecewise analytic (?), or more generally when ε(x) is smooth (?).

In this paper we extend these variational principles to acoustics, elastodynamics, and electromagnetism, in lossy inhomogeneous bodies at fixed frequency. By lossy we mean that the time-averaged dissipation of mechanical or electrical energy into heat is positive everywhere, which means that the imaginary part of certain tensors, such as the permittivity tensor, are positive definite. Like the Dirichlet and Thompson variational principles, the variational principles we obtain are minimization (not saddle point) variational principles. Other variational principles which extend the Dirichlet and Thompson variational principles to dynamics, not assuming time harmonicity, were derived for elastodynamics and electromagnetism by ?) and Willis (?, ?). These are minimizing variational principles in the Laplace domain but not in the frequency domain. We will recover the Gurtin-Willis variational principles in the time harmonic setting where they correspond to saddle point variational principles. We remark that ?) and Willis (?, ?) derived variational principles which are dynamic analogs of the Hashin-Shtrikman (?, ?, ?, ?) variational principles, involving trial polarizations fields. We do not consider such variational principles here.

Of course given a (inhomogeneous) linear differential operator A, mapping m-component complex vector fields to m-component complex vector fields, there is a trivial minimization variational principle associated with the equation Au = h for the m-component potential u, in which h is a m-component source term. When the equation Au = h with u = u 0 on ∂Γ has a unique solution for u, the infimum in

0 = inf u Γ (Au -h) • (Au -h),
(1.12)

where the overline denotes complex conjugation and the infimum is over all fields u with u = u 0 on ∂Γ, is clearly attained when u = u. What separates such variational principles from those of Dirichlet and Thompson is that, as explained above, the latter variational principles give us useful information on the boundary values of the fields. We desire variational principles with this property. Also, from a mathematical viewpoint, the existance of a solution cannot be deduced from (1.12), and in case there is a solution no interesting inequality results from (1.12).

In an important development ?) extended the Dirichlet and Thompson variational principles to the quasistatic electromagnetic equations (and also to the quasistatic elastodynamic equations) where the equations (1.1) and (1.2) still hold but ε(x), V (x), E(x) and D(x) are complex, and the imaginary part of ε(x) is positive. Since we build upon their work let us briefly review their ideas in this context.

By rewriting the constitutive law D = εE in terms of its real and imaginary parts (1.13) where the prime denotes the real part, while the double prime denotes the imaginary part, it is evident that since ε ′′ (x) > 0 one has the Cherkaev-Gibiansky saddle point variational principle

D ′′ D ′ = ε ′′ ε ′ ε ′ -ε ′′ E ′ E ′′ ,
Q(V ′ , V ′′ ) = inf V ′ sup V ′′ Q(V ′ , V ′′ ) = sup V ′′ inf V ′ Q(V ′ , V ′′ ), (1.14)
where (1.15) and the infimum and supremum are over trial potentials with V ′ (x) = V ′ (x) and V ′′ (x) = V ′′ (x) on the boundary ∂Γ. To obtain a minimization variational principle from this saddle point variational principle ?) make a partial Legendre transform of the saddle shaped quadratic form to convert it into a convex quadratic form. This is equivalent to rewriting the constitutive law in the form (1.16) where the matrix

Q(V ′ , V ′′ ) = GG ∇V ′ ∇V ′′ • ε ′′ ε ′ ε ′ -ε ′′ ∇V ′ ∇V ′′ ,
D ′′ E ′′ = E E ′ -D ′ ,
E = ε ′′ + (ε ′ ) 2 /(ε ′′ ) ε ′ /ε ′′ ε ′ /ε ′′ 1/ε ′′ (1.17)
is positive definite when ε ′′ (x) is positive. From this it is evident that one has the Cherkaev-Gibiansky minimization variational principle:

Y (V ′ , D ′ ) = inf V ′ inf D ′ ∇ • D ′ = 0 Y (V ′ , D ′ ), (1.18)
where

Y (V ′ , D ′ ) = Γ ∇V ′ D ′ • E ∇V ′ D ′ , (1.19)
and the infimums are over all trial real potentials with V ′ (x) = V ′ (x) on ∂Γ, and all trial real divergence free displacement fields with

D • n = D • n on ∂Γ.
As an aside we remark that, as shown in section 18 of ?), one can extend these ideas of Gibiansky and Cherkaev to obtain variational principles for problems where the constitutive law takes the form D = εE, where the matrix valued field ε(x) is not symmetric. For example, for conduction in a fixed magnetic field, the conductivity tensor σ(x) is real but not symmetric. Expressing σ(x) = σ s (x) + σ a (x), where σ s (x) = σ T s (x) and σ a (x) = -σ T a (x) are the symmetric and antisymmetric parts of the conductivity tensor, one considers current fields J(x) and electric fields E(x) which solve the conductivity equations

J = (σ s + σ a )E, E = -∇V, ∇ • J = 0, (1.20)
in conjunction with current fields J(x) and electric fields E(x) which solve the adjoint problem

J = (σ s -σ a ) E, E = -∇ V , ∇ • J = 0. (1.21)
By adding and subtracting the two constitutive laws one gets a new constitutive law

J s -J a = σ s σ a -σ a -σ s E s E a , (1.22) involving fields J s ≡ J + J, J a ≡ J -J, E s ≡ E + E, E a ≡ E -E, (1.23)
that satisfy the differential constraints

∇ • J s = ∇ • J a = 0, E s = -∇V s , E a = -∇V a , (1.24)
where V s ≡ V + V and V a ≡ V -V . Conversely, given any fields which solve (1.22) and (1.24), one can recover the associated fields which solve (1.20) and (1.21). The two problems are equivalent. Since the matrix entering the constitutive law (1.22) is real and symmetric there is clearly an associated saddle point variational principle when σ s (x) is positive definite for all x. ?) show how this variational principle provides useful asymptotic estimates for the mathematically analogous problem of convection enhanced diffusion. Also by rewriting the constitutive law (1.22) in the equivalent form

J s E a = σ s -σ a σ -1 s σ a σ a σ -1 s -σ -1 s σ a σ -1 s E s J a , (1.25)
which involves a matrix which is symmetric and positive definite when σ s (x) is positive definite for all x, one obtains an associated minimization variational principle (?). In this paper we do not treat such cases where the constitutive laws involve non-symmetric matrices: this will be covered elsewhere.

General Theory

Let d be the dimension of the space and let F (x) and G(x) be two complex valued fields of the form

F = F f , G = G g , (2.1)
where F(x) and G(x) are m × d dimensional matrix fields and f(x) and g(x) are m component vector fields, and which satisfy the differential constraints

F = ⊓u, h + ⊔G = 0, (2.2)
where u(x) is an m-component potential and h(x) is some m-component source term and

⊓ u ≡ ∇u u , ⊔G ≡ -∇ • G + g (2.
3) serve to define the operators ⊓ and ⊔. The differential constraints imply the key property that on any domain Γ

Γ G • F + h • u = ∂Γ (G • n) • u, (2.4)
as follows from integration by parts, where

G • F ≡ G • F + g • f, G • F ≡ Tr(G T F), (2.5)
and n is the outward unit normal to the surface ∂Γ. (Note that (2.5) is an inner product only when G and F are real, since it does not involve complex conjugation). Here, following standard notation, we define

{∇u} ij = ∂u i ∂x j , {∇ • G} i = d j=1 ∂G ij ∂x j . (2.6)
Now suppose the fields F and G are linked by the constitutive relation

G(x) = Z(x)F (x), (2.7)
where Z(x) = Z ′ (x) + iZ ′′ (x) is a complex symmetric linear map (which is represented by a complex symmetric m(d + 1) × m(d + 1) matrix when G and F are represented by m(d + 1) component vectors). The map Z can be expressed in block form as

Z = L K K T M , ( 2
G ij = L ijkℓ F kℓ + K ijk f k , g i = K kℓi F kℓ + M ik f k , (2.9)
in which sums of repeated indices are assumed. Since L and M are symmetric maps, we have

L ijkℓ = L kℓij , M ik = M ki .
(2.10)

The constitutive law and differential constraints imply that u satisfies

h + ⊔(Z ⊓ u) = 0, (2.11) or equivalently, ∇ • (L∇u + Ku) = h + K T ∇u + Mu.
(2.12)

Our analysis thus applies to equations of this form, with L and M having the symmetries (2.10). Let (u, F , G) be a solution of (2.2) and (2.7). Taking real and imaginary parts, the differential constraints imply

F ′ = ⊓u ′ , h ′ + ⊔G ′ = 0, F ′′ = ⊓u ′′ , h ′′ + ⊔G ′′ = 0, (2.13)
where the prime denotes the real part and the double prime the imaginary part, and the constitutive law can be written in the form

G ′′ G ′ = Z ′′ Z ′ Z ′ -Z ′′ F ′ F ′′ .
(2.14)

To make use of this representation let us further suppose that the imaginary part of Z satisfies (2.15) for some α > 0. Extending the ideas of ?) consider the real valued quadratic functional (2.16) subject to the boundary conditions that u ′ = u ′ and u ′′ = u ′′ on ∂Γ. Let s(x) be a real valued m component vector field with s = 0 on ∂Γ. Now Q(u ′ +s, u ′′ ) is a convex quadratic function of s (with a linear component) while Q(u ′ , u ′′ +s) is a concave quadratic function of s (with a linear component). Therefore on the space of fields satisfying the boundary conditions the pair (u ′ , u ′′ ) which are at the saddle point of the variational problem

Z ′′ (x) ≥ αI, ∀x ∈ Γ,
Q(u ′ , u ′′ ) = Γ ⊓u ′ ⊓u ′′ • Z ′′ Z ′ Z ′ -Z ′′ ⊓u ′ ⊓u ′′ + 2(h ′′ • u ′ + h ′ • u ′′ ) = Γ [(⊓u) • Z ⊓ u + 2h • u] ′′ ,
inf u ′ sup u ′′ Q(u ′ , u ′′ ), (2.17)
exists and is unique and corresponds to the stationary point of Q(u ′ , u ′′ ), and is also at the saddle point of the variational problem sup

u ′′ inf u ′ Q(u ′ , u ′′ ). (2.18)
Now let us prove we have the variational principle

Q(u ′ , u ′′ ) = inf u ′ sup u ′′ Q(u ′ , u ′′ ) = sup u ′′ inf u ′ Q(u ′ , u ′′ ), (2.19)
where the infimum and supremum are over fields with u ′ and u ′′ with any fixed values u ′ 0 and u ′′ 0 at the boundary ∂Γ and u ′ and u ′′ are the solutions with these prescribed boundary conditions.

Let (u, F , G) be a solution of (2.2) and (2.7). As s = 0 on ∂Γ, in a similar way as (2.4) was derived, we get

Γ G ′′ • (⊓s) + h ′′ • s = 0, Γ G ′ • (⊓s) + h ′ • s = 0.
(2.20)

Thus we see that

Q(u ′ + s, u ′′ ) = Q(u ′ , u ′′ ) + Γ 2G ′′ • (⊓s) + 2h ′′ • s + (⊓s) • Z ′′ ⊓ s = Q(u ′ , u ′′ ) + Γ (⊓s) • Z ′′ ⊓ s, (2.21)
and in the same way we get

Q(u ′ , u ′′ + s) = Q(u ′ , u ′′ ) - Γ (⊓s) • Z ′′ ⊓ s. (2.22)
So we see that the saddle point of (2.17) is at (u ′ , u ′′ ) = (u ′ , u ′′ ). Conversely, let (u ′ , u ′′ ) be a saddle point of (2.17) satisfying some boundary condition

u ′ = u ′ 0 and u ′′ = u ′′ 0 on ∂Γ. Consider inf s sup t Q(u ′ + s, u ′′ + t), (2.23)
where the infimum and supremum are taken over fields s and t satisfying s = t = 0 on ∂Γ. A necessary condition for the saddle point to be at s = t = 0 is that the first-order derivative of the functional Q vanishes: that is

Γ 2 ⊓s ⊓t • G ′′ G ′ + 2h ′′ • s + 2h ′ • t = 0, (2.24)
for any s and t satisfying s = t = 0 on ∂Γ, where we have introduced

G ′′ G ′ ≡ Z ′′ Z ′ Z ′ -Z ′′ • ⊓u ′ ⊓u ′′ .
(2.25)

Upon setting

G ′ = G ′ g ′ , G ′′ = G ′′ g ′′ , (2.26)
the condition (2.24) reduces to

Γ G ′′ • ∇s + (g ′′ + h ′′ ) • s + G ′ • ∇t + (g ′ + h ′ ) • t = 0. (2.27)
Integrating this by parts gives (2.28) and this will be satisfied for any s and t satisfying s = t = 0 on ∂Γ if and only if

Γ (-∇ • G ′′ + g ′′ + h ′′ ) • s + (-∇ • G ′ + g ′ + h ′ ) • t = 0,
h ′ + ⊔G ′ = 0, h ′′ + ⊔G ′′ = 0. (2.29)
Thus the Euler-Lagrange equations associated with the saddle point variational principle coincide with the original equations (2.2).

There is also a dual saddle point variational principle, analogous to the Thompson variational principle. By taking real and imaginary parts of the constitutive relation F = KG where K = Z -1 has negative definite imaginary part we obtain

F ′′ F ′ = K ′′ K ′ K ′ -K ′′ G ′ G ′′ .
(2.30)

The dual variational principle involves the quadratic form

R(G ′ , G ′′ ) = Γ G ′ G ′′ • K ′′ K ′ K ′ -K ′′ G ′ G ′′ = Γ (G • KG) ′′ , (2.31)
where G ′ and G ′′ are given by

G ′ = G ′ ∇ • G ′ -h ′ , G ′′ = G ′′ ∇ • G ′′ -h ′′ , (2.32)
and G ′ and G ′′ satisfy the boundary conditions

G ′ • n = G ′ • n and G ′′ • n = G ′′ • n on ∂Γ, (2.33)
where G ′ and G ′′ are the real and imaginary parts of G. Notice that the form of G given by (2.32) ensures that h + ⊔G = 0. inside Γ. By similar argument to the one establishing (2.19), we have the dual variational principle

R(G ′ , G ′′ ) = sup G ′ inf G ′′ R(G ′ , G ′′ ) = inf G ′′ sup G ′ R(G ′ , G ′′ ), (2.34)
where the supremum and infimum are over fields satisfying (2.33).

Again building upon the ideas of ?) let us rewrite the constitutive relation in the form

G ′′ F ′′ = L F ′ -G ′ , (2.35)
where straightforward algebra shows that

L = Z ′′ + Z ′ (Z ′′ ) -1 Z ′ Z ′ (Z ′′ ) -1 (Z ′′ ) -1 Z ′ (Z ′′ ) -1 . (2.36)
For all x ∈ Γ the matrix L(x) is positive definite since, using (2.14) and (2.35), we see that the associated quadratic form

F ′ -G ′ • L F ′ -G ′ = F ′ • G ′′ -F ′′ • G ′ = F ′ • Z ′′ F ′ + F ′′ • Z ′′ F ′′ (2.37)
is non-negative, and zero only when

F ′ = F ′′ = 0, i.e. when F ′ = G ′ = 0. Now consider the convex functional Y (u ′ , G ′ ) = Γ ⊓u ′ -G ′ • L ⊓u ′ -G ′ + 2h ′′ • u ′ , (2.38)
where G ′ is given by

G ′ = G ′ ∇ • G ′ -h ′ ,
(2.39) (to ensure that h ′ + ⊔G ′ = 0 inside Γ) and u ′ and G ′ are real and satisfy the boundary conditions that u ′ = u ′ 0 , and

G ′ • n = G ′ 0 • n on ∂Γ.
(2.40)

Let us prove we have the variational principle

Y (u ′ , G ′ ) = inf u ′ inf G ′ Y (u ′ , G ′ ), (2.41)
where the infimums are over all trial fields u ′ and G ′ satisfying the boundary conditions (2.40). Let (u, G) be a solution of (2.2) and (2.7) with

u ′ = u ′ 0 and G ′ • n = G ′ 0 • n on ∂Γ.
Suppose we are given any m component vector field s(x) with s = 0 on ∂Γ and also a m × d matrix valued field T(x) with T • n = 0 on ∂Γ. Then (2.20) and integration by parts implies that

Γ 2G ′′ • (⊓s) + 2h ′′ • s -2T • (⊓u ′′ ) = 0, where T = T ∇ • T . (2.42)
From this identity it follows that

Y (u ′ + s, G ′ + T) = Y (u ′ , G ′ ) + Γ ⊓s -T • L ⊓s -T , (2.43) which implies (u ′ , G ′
) is a mimimizer of the variational principle. Conversely, let (u ′ , G ′ ) be a mimimizer of the variational principle, satisfying some boundary conditions

u ′ = u ′ 0 and G ′ • n = G ′ 0 • n on ∂Γ. Consider inf s inf T Y (u ′ + s, G ′ + T), (2.44)
where the infimum is over all fields s and T satisfying s = 0 and T • n = 0 on ∂Γ. A necessary condition for the infimum to be at s = T = 0 is that the first-order derivative of the functional Y vanish: that is

Γ 2 ⊓s -T • G ′′ F ′′ + 2h ′′ • s = 0, (2.45)
for any s and T satisfying s = 0 and T • n = 0 on ∂Γ, where we have introduced

T ≡ T ∇ • T , G ′′ F ′′ ≡ L ⊓u ′ -G ′ . (2.46)
Upon setting

F ′′ = F ′′ f ′′ , G ′′ = G ′′ g ′′ , (2.47) the condition (2.45) reduces to Γ G ′′ • ∇s + (g ′′ + h ′′ ) • s -T • F ′′ -(∇ • T) • f ′′ = 0. (2.48)
Integrating this by parts gives (2.49) and this will be satisfied for any s and T satisfying s = 0 and T • n = 0 on ∂Γ if and only if

Γ (-∇ • G ′′ + g ′′ + h ′′ ) • s -T • (F ′′ -∇f ′′ ) = 0,
h ′′ + ⊔G ′′ = 0, F ′′ = ⊓f ′′ .
(2.50)

Upon identifying f ′′ with u ′′ we see that the Euler-Lagrange equations associated with the minimization variational principle coincide with the original equations (2.2).

The convex variational principle we have just stated is, as usual, associated with a dual variational principle:

Y (u ′′ , G ′′ ) = inf u ′′ inf G ′′ Y (u ′′ , G ′′ ), (2.51)
where

Y (u ′′ , G ′′ ) = Γ G ′′ ⊓u ′′ • L -1 G ′′ ⊓u ′′ -2h ′ • u ′′ , (2.52) in which G ′′ = G ′′ ∇ • G ′′ -h ′′ , (2.53)
and the infimum in (2.51) is over real fields u ′′ and G ′′ with u ′′ = u ′′ and

G ′′ • n = G ′′ • n on ∂Γ. Since L -1 = (Z ′′ ) -1 -(Z ′′ ) -1 Z ′ -Z ′ (Z ′′ ) -1 Z ′′ + Z ′ (Z ′′ ) -1 Z ′ ,
(2.54) the expression for Y (u ′′ , G ′′ ) can be written equivalently as

Y (u ′′ , G ′′ ) = Γ ⊓u ′′ -G ′′ • L ⊓u ′′ -G ′′ -2h ′ • u ′′ .
(2.55)

These two minimization variational principles are not the only ones one can derive. Indeed, there is a continuous two-parameter family of related variational principles. The differential constraints (2.2) and the constitutive relation (2.7) imply

F = ⊓ u, h + ⊔ G = 0, G(x) = Z(x) F (x),
(2.56)

where

u = e iτ u, F = e iτ F , G = e i(τ +θ) G, h = e i(τ +θ) h, Z(x) = e iθ Z(x), (2.57)
in which τ and θ are real parameters. So the fields with tildes on them satisfy the same equations as those without tildes, and the variational principles directly apply provided θ is chosen so that

Z ′′ (x) ≥ αI, ∀x ∈ Γ, (2.58)
for some α > 0. With the subsequent replacement (2.57) we obtain a new set of variational principles parameterized by τ and θ. Observe that even if (2.15) is not satisfied for any α > 0 we still may be able to find a range of values of θ for which (2.58) holds, and thus for which we can obtain variational principles. With the particular choice θ = 0 and e iτ = -i the variational principle (2.41) gets mapped to the dual variational principle (2.51).

All the previous minimization principles need the fundamental coercivity assumption (2.15). If Z ′′ (x) = 0 in some part Ψ of the domain Γ they cannot be applied directly. However we can still obtain variational principles by taking appropriate limits. If, in some region Ψ ⊂ Γ, Z ′′ (x) is very small in the sense that

δI ≥ Z ′′ (x), ∀x ∈ Ψ, (2.59)
for some very small δ > 0, then within this region

⊓u ′ -G ′ • L ⊓u ′ -G ′ ≈ [Z ′ ⊓ u ′ -G ′ ] • (Z ′′ ) -1 [Z ′ ⊓ u ′ -G ′ ].
(2.60)

If Ψ = Γ then the variational principle reduces to a variational principle similar to (1.12) which is not particularly interesting. If Ψ = Γ the variational principle will only be useful if one takes

G ′ ≈ Z ′ ⊓ u ′ in Ψ.
The variational principles still hold in the limit as δ → 0 provided one first takes

G ′ = Z ′ ⊓ u ′ , ∀x ∈ Ψ, (2.61)
corresponding to taking an exact solution of (2.11) in this region (but with (u ′ , G ′ ) not necessarily coinciding with the exact solution (u ′ , G ′ ) associated with the given boundary conditions). Then as δ → 0 the expression for Y reduces to

Y (u ′ , G ′ ) = Γ\Ψ ⊓u ′ -G ′ • L ⊓u ′ -G ′ - Γ 2h ′′ • u ′ . (2.62)
Outside Ψ we still require that Z ′′ (x) ≥ αI for some α > 0.

We now show how these variational principles may give useful information about the Dirichlet to Neumann map on the boundary ∂Γ. Let us assume h = 0 and introduce the m-component complex vector fields

q 0 = G(x) • n, u 0 (x) = u(x) for x ∈ ∂Γ.
(2.63)

The relation between q 0 and u 0 must be linear and we can write q 0 = Nu 0 which defines the Dirichlet to Neumann map N. If u

(1) 0 and u

(2) 0 denote two boundary conditions for u and G (j) , F (j) , and q (j) 0 , j = 1, 2, denote the associated fields then it follows from the key property and the symmetry of Z(x) that

Nu (1) 0 , u (2) 0 = ∂Γ q (1) 0 • u (2) 0 = Γ G (1) • F (2) = Γ F (1) • ZF (2) = Γ G (2) • F (1) = ∂Γ q (2) 0 • u (1) 0 = u (1) 0 , Nu (2) 0 ,
(2.64) which implies the map N is symmetric. It has positive semidefinite imaginary part since

u ′ 0 , N ′′ u ′ 0 + u ′′ 0 , N ′′ u ′′ 0 = ∂Γ q ′′ 0 • u ′ 0 -q ′ 0 • u ′′ 0 = Γ G ′′ • F ′ -G ′ • F ′′ = Γ F ′ • Z ′′ F ′ + F ′′ • Z ′′ F ′′ ≥ 0. (2.65)
Assuming N ′′ is invertible, we can write the relation q 0 = Nu 0 as

q ′′ 0 u ′′ 0 = N u ′ 0 -q ′ 0 , (2.66)
where

N = N ′′ + N ′ (N ′′ ) -1 N ′ N ′ (N ′′ ) -1 (N ′′ ) -1 N ′ (N ′′ ) -1 .
(2.67)

Thus the map N giving the boundary fields q ′′ 0 and u ′′ 0 in terms of q ′ 0 and u ′ 0 must have this special form. Now, using the key property, Y (u ′ , G ′ ) can be expressed (when h = 0) in terms of the boundary fields:

Y (u ′ , G ′ ) = Γ G ′′ • F ′ -G ′ • F ′′ = ∂Γ q ′′ 0 • u ′ 0 -q ′ 0 • u ′′ 0 = u ′ 0 -q ′ 0 , N u ′ 0 -q ′ 0 , (2.68)
and thus bounds on Y (u ′ , G ′ ), obtained from the variational principle, give upper bounds on the quadratic form associated with N . Bounds on Y (u ′′ , G ′′ ), obtained from the dual variational principle, give lower bounds on the quadratic form associated with N .

Application to Acoustics and Elastodynamics

Let us begin by studying the acoustic equation in d dimensions:

-∇ • ρ -1 ∇P = (1/κ)ω 2 P, (3.1)
where P (x) is the complex pressure, κ(x) is the bulk modulus, ρ(x) is the density tensor, and ω is the (fixed) frequency of oscillation. [As is well known this equation with d = 2 is also applicable in cylindrical bodies, for antiplane elastodynamics where the displacement field is directed parallel to the cylinder axis, and for electromagnetism when the fields are transverse electric (TE), where the electric field is directed parallel to the cylinder axis, or transverse magnetic (TM), where the magnetic field is directed parallel to the cylinder axis]. Normally one expects κ to have a negative imaginary part (due to bulk viscosity), and ρ to be ρI where ρ(x) is the positive mass density. However, viewed as the effective density tensor of a composite material at a given frequency ω, ρ can be anisotropic (?) and, at least in the context of elastodynamics, can be complex valued with a positive semidefinite imaginary part (?) with a real part that is not even necessarily positive [as established by ?), ?), and ?) in the context of antiplane elastodynamics, where 1/κ plays the role of density, and by ?) and ?) in the context of three-dimensional elastodynamics]. Such unusual effective densities are due to the fact that different parts of the microstructure can experience different accelerations, and can be out of phase with the overall applied force if this relative motion dissipates energy. Given the complex pressure field P (x) the associated complex velocity field of the fluid is

v = -i(ωρ) -1 ∇P. (3.2)
[The associated physical pressure and physical velocity are respectively (e -iωt P (x)) ′ and (e -iωt v(x)) ′ where the prime denotes the real part and t denotes the time.] Upon multiplying (3.1) by e iθ and comparing it with (2.12) we can make the identifications

u = P, L = -e iθ ρ -1 , K = 0, M = e iθ ω 2 /κ, h = 0, G = -ie iθ ωv. (3.3)
Thus the potential u is a scalar, corresponding to the case m = 1. We choose θ such that the imaginary part of

Z = -e iθ ρ -1 0 0 e iθ ω 2 /κ (3.4)
satisfies (2.15). When ρ is real and positive definite, as is typically the case, then we need to choose θ in the range 0 > θ > -π. If in addition κ is almost real and positive, with a small negative imaginary part, then we should choose θ appropriately small and negative.

Explicit expressions for the various variational principles in the acoustic case can of course be obtained by making the substitutions (3.3) in the relevant equations in the preceding section. For example, if ρ has a positive definite imaginary part (which is more applicable in the mathematically analogous TE or TM electromagnetic problems) and κ(x) has a negative imaginary part, then with θ = 0 we have the variational principle

Y (P ′ , v ′′ ) = inf P ′ inf v ′′ Y (P ′ , v ′′ ), (3.5)
where

Y (P ′ , v ′′ ) = Γ ∇P ′ -ωv ′′ • R ∇P ′ -ωv ′′ + ωP ′ -∇ • v ′′ • K ωP ′ -∇ • v ′′ , (3.6) in which R = r ′′ + r ′ (r ′′ ) -1 r ′ r ′ (r ′′ ) -1 (r ′′ ) -1 r ′ (r ′′ ) -1 , K = k ′′ + (k ′ ) 2 /k ′′ k ′ /k ′′ k ′ /k ′′ 1/k ′′ , (3.7)
and r = -ρ -1 , and k = 1/κ. In this variational principle the boundary values of P ′ and v ′′ •n are fixed, and the infimums are attained at fields P ′ and v ′′ associated with solutions of the acoustic equation satisfying the prescribed boundary conditions. That the Euler-Lagrange equations of (3.5) coincide with (3.1) and (3.2) follows from the analysis of (2.44)-(2.50), and can also be shown directly. When r ′′ is very small within all of Γ then we have

∇P ′ -ωv ′′ • R ∇P ′ -ωv ′′ ≈ [r ′ ∇P ′ -ωv ′′ ] • (r ′′ ) -1 [r ′ ∇P ′ -ωv ′′ ], (3.8)
and the variational principles will only be useful if we take trial fields with ωv ′′ ≈ r ′ ∇P ′ . The variational principles still hold in the limit as r ′′ → 0 provided we first choose v ′′ = r ′ ∇P ′ /ω. Then, when ρ is real, we have the variational principle

Y (P ′ ) = inf P ′ Y (P ′ ) (3.9)
where

Y (P ′ ) = Γ ωP ′ ∇ • ρ -1 ∇P ′ /ω • K ωP ′ ∇ • ρ -1 ∇P ′ /ω . (3.10)
In this variational principle the boundary values of P ′ and n • ρ -1 ∇P ′ are fixed, where the latter corresponds to fixing the boundary value of v ′′ • n. The infimum is attained at a field P ′ associated with solutions of the acoustic equation satisfying the prescribed boundary conditions. To show this, let P ′ be a minimizer and set P ′ = P ′ + S where S = 0 and n • ρ -1 ∇S = 0 on ∂Γ. A necessary condition for the minimum to be at S = 0 is that the first derivative of the functional Y vanish: that is

Γ U ωP ′′ • ωS ∇ • ρ -1 ∇S/ω = 0, (3.11)
where we have introduced

U ωP ′′ ≡ K ωP ′ ∇ • ρ -1 ∇P ′ /ω . (3.12)
Integrating by parts the constraint (3.11) gives (3.13) and this will be satisfied for any S with S = 0 and n • ρ -1 ∇S = 0 on ∂Γ if and only if

Γ S(ωU + ∇ • ρ -1 ∇P ′′ ) = 0,
U = -∇ • ρ -1 ∇P ′′ /ω. (3.14)
A straightforward calculation (similar to the equivalence of (2.7) and (2.35)) shows that (3.12) is equivalent to (3.1) with this value of U. Thus the acoustic equation is the Euler-Lagrange equation of (3.9), when P ′′ is defined by (3.12). This variational principle remains valid if we replace P by e iτ P where τ can be any constant.

In a similar way, if ρ has a positive definite imaginary part but κ is real (which is more applicable to the analogous TM electromagnetic problem, where the magnetic permeability plays the role of 1/κ) then we have the variational principle

Y (v ′′ ) = inf v ′′ Y (v ′′ ), (3.15)
where

Y (v ′′ ) = Γ ∇(κ∇ • v ′′ )/ω -ωv ′′ • R ∇(κ∇ • v ′′ )/ω -ωv ′′ , (3.16)
in which the boundary values of v ′′ • n and κ∇ • v ′′ are fixed, where the latter corresponds to fixing the boundary value of P ′ . One has to slightly modify the analysis to apply it to the elastodynamic equations,

-∇ • (C∇u) = b + ω 2 ρu, (3.17)
in which u(x) is the complex displacement field, b(x) is the complex body-force density ρ(x) is the complex density tensor, and C(x) is the complex elasticity tensor. Given the complex displacement field u(x), the complex stress τ (x) and complex momentum density p(x) are given by τ = C∇u, p = -iωρu.

(3.18)

[The associated physical displacement, body-force density, stress, and momentum density are respectively (e -iωt u) ′ , (e -iωt b) ′ , (e -iωt τ ) ′ and (e -iωt p) ′ .] Upon multiplying (3.17) by e iθ and comparing it with (2.12) we can make the identifications

L = -e iθ C, K = 0, M = e iθ ω 2 ρ, h = e iθ b, (3.19)
and (3.18) implies G = -e iθ τ , g = ie iθ ωp.

(3.20)

The problem is that C acting upon any antisymmetric matrix is zero, and so L and hence Z are singular, which is a problem for computing the inverses in the equation (2.36) for L. This problem is rectified in the standard way, by replacing ⊓ everywhere with ⊓ defined by satisfies (2.15) on the space of these fields, and we compute the inverse of Z ′′ on the space of these fields. When ρ is real and positive definite then we need to choose θ in the range π > θ > 0. If in addition C is almost real with a positive definite real part, and with a small negative definite imaginary part, then we should choose θ appropriately small and positive. With ⊓ replaced by ⊓ the analysis of equations (2.23)-(2.29) and (2.44)-(2.50) still holds, and we recover the elastodynamic equations from the Euler-Lagrange equations associated with the variational principles.

In the special case θ = 0 the functionals Q, R and Y reduce to (3.24) and

Q(u ′ , u ′′ ) = Γ (-ǫ • Cǫ + ω 2 u • ρu + 2b • u) ′′ , R(τ ′ , τ ′′ ) = Γ [-τ • C -1 τ + (∇ • τ + b) • (ω 2 ρ) -1 (∇ • τ + b)] ′′ , Y ((u ′ , τ ′ )) = Γ - ǫ ′ -τ ′ • C ǫ ′ -τ ′ + ωu ′ (b ′ + ∇ • τ ′ )/ω P ωu ′ (b ′ + ∇ • τ ′ )/ω + 2b ′′ • u ′ , (3.23) where ǫ = [∇u + (∇u) T ]/2,
C = C ′′ + C ′ (C ′′ ) -1 C ′ C ′ (C ′′ ) -1 (C ′′ ) -1 C ′ (C ′′ ) -1 , P = ρ ′′ + ρ ′ (ρ ′′ ) -1 ρ ′ ρ ′ (ρ ′′ ) -1 (ρ ′′ ) -1 ρ ′ (ρ ′′ ) -1 (3.25)
are respectively negative definite and positive definite matrices. It follows directly from the variational principles of ?) and ?), applied to the time harmonic case considered here, that the stationary points of the functionals Q and R correspond to solutions of the elastodynamics equations, in agreement with the results of the previous section. When ρ ′′ is very small we have

ωu ′ (b ′ + ∇ • τ ′ )/ω P ωu ′ (b ′ + ∇ • τ ′ )/ω ≈ [ω 2 ρ ′ u ′ +b ′ +∇•τ ′ ](ω 2 ρ ′′ ) -1 [ω 2 ρ ′ u ′ +b ′ +∇•τ ′ ]
(3.26) and the variational principles only will be useful if we take trial fields with ω 2 ρ ′ u ′ + b ′ + ∇ • τ ′ approximately zero. The variational principle involving the functional Y still holds in the limit as ρ ′′ → 0 provided we first choose

u ′ = -(ρ ′ ) -1 (b ′ + ∇ • τ ′ )/ω 2 .
(3.27)

Then, when ρ is real, we have the variational principle

Y (τ ′ ) = inf τ ′ Y (τ ′ ), (3.28)
where (3.29) in which u ′ and ǫ ′ are given by (3.27) and (3.24). In this variational principle the trial field τ ′ can be any real symmetric matrix valued field with prescribed values of τ ′ • n and ρ -1 ∇ • τ ′ at the boundary ∂Γ, where prescribing ρ -1 ∇ • τ ′ at ∂Γ corresponds to prescribing u ′ according to (3.27). The infimum is attained at a field τ ′ associated with solutions of the elastodynamic equation satisfying the prescribed boundary conditions.

Y (τ ′ ) = Γ - ǫ ′ -τ ′ • C ǫ ′ -τ ′ + 2b ′′ • u ′ ,

The saddle point variational principles for electromagnetism

A comparison of the continuum elastodynamic equation (3.17) with Maxwell's equations,

∇ × E = iωB, ∇ × H = -iωD + j, D = εE, B = µH, (4.1)
which can be rewritten (?) in the form

-∇ • (C∇E) = iωj + ω 2 εE, (4.2)
where now C ijkℓ = e ijm e kℓn {µ -1 } mn , (4.3)

in which E is the electric field, j is the free current ε the electric permittivity tensor, µ the magnetic permeability tensor, and e ijm = 1 (-1) if (i, j, m) is an even (odd) permutation of (1, 2, 3) and is zero otherwise, suggests that the preceeding analysis should also extend to three-dimensional electromagnetism. Upon multiplying (4.2) by e iθ and comparing it with (2.12) we can make the identifications

u = E, L = -e iθ C, K = 0, M = e iθ ω 2 ε, h = iωe iθ j. (4.4)
The problem is now that C acting upon any symmetric matrix is zero, and so L and hence Z are singular, which is a problem for computing the inverses in the equation (2.36) for L. This problem is rectified by replacing ⊓ everywhere with ⊓ defined by

⊓E ≡ [∇E -(∇E) T ]/2 E . (4.5)
Then the fields F and G have the form (2.1), with F and G being antisymmetric 3 × 3 matrices: thus F and G can be represented by 6 component vectors. We choose θ such that the imaginary part of

Z = -e iθ C 0 0 e iθ ω 2 ε (4.6)
satisfies (2.15) on the space of these fields, and we compute the inverse of Z ′′ on the space of these fields. Let us assume, for simplicity, that

ε ′′ (x) ≥ α 1 I, µ ′′ (x) ≥ α 2 I, ∀x ∈ Γ, (4.7)
for some α 1 > 0 and α 2 > 0. (By multiplying (4.2) by e iθ and redefining ε, µ and j if necessary.) Then we can take θ = 0 and (2.15) holds for some α > 0. From (4.1) and (4.5) we see that

F = ⊓E = F E , with F = iω 2   0 -B 3 B 2 B 3 0 -B 1 -B 2 B 1 0   , (4.8)
and the constitutive law (2.9) implies

G = G ω 2 D with G = -iω   0 -H 3 H 2 H 3 0 -H 1 -H 2 H 1 0   . (4.9)
Thus the key property (2.4) reduces to

Γ ω 2 (B • H + E • D) + iωj • E = ∂Γ -iω(H × n) • E = ∂Γ -iω(E × H) • n, (4.10)
and holds for all fields E, D, B,and H satisfying the differential constraints

∇ × E = iωB, ∇ × H = -iωD + j. (4.11) Now the first relation (2.20) will hold if ∂Γ (G ′′ • n) • s = ∂Γ -iω(H ′′ × n) • s = 0, (4.12)
and for this it suffices that the tangential component of s vanishes at the surface ∂Γ.

Similarly the second relation in (2.20) will hold if this condition is satisfied. Therefore the infimum and supremum in (2.17) can be extended to all u ′ and u ′′ with the same tangential components as, respectively, E ′ and E ′′ . Similarly in the minimization variational principle (2.41), or (2.51), we only need require that u ′ , or u ′′ , have the same tangential components as E ′ , or E ′′ . With ⊓ replaced by ⊓ the analysis of equations (2.23)-(2.29) and (2.44)-(2.50) still holds, and we recover Maxwell's equations from the Euler-Lagrange equations associated with the variational principles. If we introduce the tangential components

E (x) = E -n(n • E), H (x) = H -n(n • H) (4.13)
of the fields E(x) and H(x) at the boundary ∂Γ, then

q 0 (x) = G • n = -iω(H × n). (4.14)
When j = 0 we can write q 0 = NE , thus defining the map N which governs the electrodynamic response of the body at the frequency ω. The analysis of (2.63)-(2.68) shows that the map N is symmetric with positive semidefinite imaginary part, and can be bounded (when j = 0) using the minimization variational principles.

Equivalently we can find the variational principles directly from Maxwell's equations (4.1). The key property (basically Poynting's theorem) (4.10) is obtained by integrating over Γ the identity

∇ • (E × H) = H • (∇ × E) -E • (∇ × H) = iω(H • B + E • D) -j • E. (4.15)
Motivated by the form of (4.8) and (4.9) let us redefine the fields

F = F f = iωB E , G = G g = -iωH ω 2 D , (4.16)
which are subject to the differential constraints that F = ΩE, iωj + ℧G = 0, (4.17)

where the operators Ω and ℧ are defined by

ΩE = ∇ × E E , ℧G = ∇ × G + g (4.18)
so that the differential constraints (4.11) and (4.17) are equivalent. The key property (4.10) now takes the form

Γ G • F + iωj • E = ∂Γ (E × G) • n, (4.19)
where

G • F ≡ F • G + f • g. The constitutive law G = ZF holds with Z redefined as Z = -µ -1 0 0 ω 2 ε , (4.20)
and this can be reexpressed in the form (2.14), with

F ′ = ΩE ′ = -ωB ′′ E ′ , F ′′ = ΩE ′′ = ωB ′ E ′′ , G ′ = ωH ′′ ω 2 D ′ , G ′′ = -ωH ′ ω 2 D ′′ , (4.21
) where these fields satisfy the differential constraints

F ′ = ΩE ′ , -ωj ′′ + ℧G ′ = 0, F ′′ = ΩE ′′ , ωj ′ + ℧G ′′ = 0, (4.22)
implying, as a corollary of (4.19), that

Γ G ′ • Ωs -ωj ′′ • s = 0, Γ G ′′ • Ωs + ωj ′ • s = 0, (4.23)
for all real valued vector fields s such that the tangential component of s vanishes at the surface ∂Γ. Introducing the functional

Q(E ′ , E ′′ ) = Γ ΩE ′ ΩE ′′ • Z ′′ Z ′ Z ′ -Z ′′ ΩE ′ ΩE ′′ + 2ω(j ′ • E ′ -j ′′ • E ′′ ) = Γ [(ΩE) • ZΩE + 2iωj • E] ′′ , (4.24)
we see from (4.23) that

Q(E ′ + s, E ′′ ) = Q(E ′ , E ′′ ) + Γ (Ωs) • Z ′′ Ωs, Q(E ′ , E ′′ + s) = Q(E ′ , E ′′ ) - Γ (Ωs) • Z ′′ Ωs. (4.25)
Consequently the pair of fields (E ′ , E ′′ ) = (E ′ , E ′′ ) is at the saddle point of (2.17) and of (2.18), where the infimum and supremum are over all vector fields E ′ and E ′′ with the same tangential components at the boundary ∂Γ as, respectively, E ′ and E ′′ . Due to the diagonal structure (4.20) of the matrix Z(x) the formula (4.24) for Q(E ′ , E ′′ ) reduces to

Q(E ′ , E ′′ ) = Γ [(∇ × E) • m(∇ × E) + ω 2 E • εE + 2iωj • E] ′′ , (4.26)
where m = -µ -1 , like ε, has positive definite imaginary part. Hence we have the saddle point variational principles

Q(E ′ , E ′′ ) = inf E ′ sup E ′′ Q(E ′ , E ′′ ) = sup E ′′ inf E ′ Q(E ′ , E ′′ ), (4.27)
where

Q(E ′ , E ′′ ) = Γ G ′′ • F ′ + G ′ • F ′′ + 2ω(j ′ • E ′ -j ′′ • E ′′ ) = Γ ω 2 (H ′ • B ′′ + H ′′ • B ′ + E ′ • D ′′ + E ′′ • D ′ ) + 2ω(j ′ • E ′ -j ′′ • E ′ ).
(4.28)

When j = 0 we can use (4.15) to express Q(E ′ , E ′′ ) just in terms of the tangential components of the fields at the boundary:

Q(E ′ , E ′′ ) = ω ∂Γ (E ′′ × H ′′ ) • n -(E ′ × H ′ ) • n when j = 0. (4.29)
The result that the stationary point of the variational principle (4.27) corresponds to a solution of Maxwell's equations follows from a variational principle derived by ?). The Willis variational principle is expressed in terms of the electromagnetic potentials V (x) and A(x), which are associated with E and B through the relations

E = -∇V + iωA, B = ∇ × A. (4.30)
and, in the time harmonic case, involves (to within a proportionality constant) the functional

Q(V , A) = ω 2 ∂Γ {[-(∇×A)•m(∇×A)+(∇V -iωA)•ε(∇V -iωA)+2iV ∇•j/ω-2A•j]e -iωt } ′ ,
(4.31) where, as usual, the prime denotes the real part. His variational principle states that for any choice of t the functional Q(V , A) applied to trial fields V and A having any given fixed values at the boundary ∂Γ, has a stationary point at the potentials V and A solving Maxwell's equations, and having the prescribed boundary values. Setting E = -∇V + iωA one sees by integrating by parts that Q(V , A) is just a functional of E alone, and when t is chosen with e -iωt = -i one has the identity

Q(V , A) = Q(E ′ , E ′′ ). (4.32)
Furthermore the values of V and A on ∂Γ determine the tangential (but not normal) components of E ′ and E ′′ on ∂Γ.

There is also a dual variational principle. By taking real and imaginary parts of the constitutive relation F = KG where K = Z -1 has negative definite imaginary part we obtain (2.23). Let us introduce the functional (4.33) where G ′ and G ′′ are given by

R(H ′ , H ′′ ) = Γ G ′ G ′′ K ′′ K ′ K ′ -K ′′ G ′ G ′′ = Γ (G • KG) ′′ ,
G ′ = ω H ′′ j ′′ -∇ × H ′′ , G ′′ = ω -H ′ -j ′ + ∇ × H ′ , (4.34)
(to ensure that -ωj ′′ + ℧G ′ = 0 and ωj ′ + ℧G ′′ = 0 inside Γ) and H ′′ and H ′ have the same tangential components as, respectively, H ′′ and H ′ . We then have the saddle point variational principles

R(H ′ , H ′′ ) = sup H ′′ inf H ′ R(H ′ , H ′′ ) = inf H ′ sup H ′′ R(H ′ , H ′′ ), (4.35)
where the supremum and infimum are over fields with the required tangential components at the boundary. Due to the diagonal structure (4.20) of Z(x) the formula for R(H ′ , H ′′ ) reduces to (4.36) where e = -ε -1 , like µ, has positive definite imaginary part. From this expression one can see that R(H ′ , H ′′ ) is a special case of the functional associated with the ?) dual variational principle. We also have that

R(H ′ , H ′′ ) = Γ [(j -∇ × H) • e(j -∇ × H) + ω 2 HµH] ′′ ,
R(H ′ , H ′′ ) = Γ G ′′ • F ′ + G ′ • F ′′ = Γ ω 2 (H ′ • B ′′ + H ′′ • B ′ + E ′ • D ′′ + E ′′ • D ′ ), (4.37)
which when j = 0 can be expressed in terms of the tangential components of the fields at the boundary:

R(H ′ , H ′′ ) = ω ∂Γ (E ′′ × H ′′ ) • n -(E ′ × H ′ ) • n when j = 0. (4.38)
Consider a medium which is locally isotropic and such that ε(x) is purely imaginary (which may be a good approximation at low frequencies in conductive media) while µ(x) is purely real (and so (4.7) only holds in the limiting sense with α 2 = 0). We can write ε = iI/(ωρ), e = -iωρI, µ = µI, (4.39)

where the resistivity ρ(x) and the permeability µ(x) are purely real and positive. Then the expression (4.36) reduces to

R(H ′ , H ′′ ) = Γ ωρ(|j ′ -∇ × H ′ | 2 -|j ′′ -∇ × H ′′ | 2 ) + 2ω 2 µH ′ • H ′′ (4.40)
and then when j = 0 the saddle point variational principle (4.35) corresponds to the ?) variational principle. (Note that j has a different meaning in that paper.) Now Maxwell's equations still hold if we make the replacements

ε → -iε, µ → iµ, E → iE, B → iB, D → D, H → H, j → j, (4.41)
(which is a special case of the transformation (2.57)) and with these replacements applied to (4.39) the expression (4.36) reduces to

R(H ′ , H ′′ ) = Γ -2ωρ(j ′ -∇ × H ′ ) • (j ′′ -∇ × H ′′ ) + ω 2 µ(|H ′ | 2 -|H ′′ | 2 ), (4.42)
and when j = 0 the saddle point variational principle (4.35) then corresponds to the other Borcea variational principle. Multiplying ε and dividing µ by e iθ , where 0 > θ > -π/2 leads to a continuous family of saddle point variational principles which interpolate the two Borcea variational principles.

Minimization variational principles for electromagnetism

To obtain a minimization principle we rewrite the constitutive law in the form (2.35) where L is given by (2.36). Then we introduce the quadratic functional

Y (E ′ , H ′′ ) = Γ ΩE ′ -G ′ • L ΩE ′ -G ′ + 2ωj ′ • E ′ , (5.1)
where G ′ is given by

G ′ = ω H ′′ j ′′ -∇ × H ′′ , (5.2)
(to ensure that -ωj ′′ + ℧G ′ = 0 inside Γ) and E ′ and H ′′ have the same tangential components at the boundary ∂Γ as, respectively, E ′ and ωH ′′ . Given two vector fields s and t with zero tangential components at the boundary ∂Γ then (4.23) and the key property (4.19) imply

Γ G ′′ • (Ωs) + ωj ′ • s -T • (Ωu ′′ ) = 0, where T = ω t -∇ × t . (5.3)
As a result we deduce that

Y (E ′ + s, H ′′ + t) = Y (E ′ , H ′′ ) + Γ Ωs -T • L Ωs -T . (5.4)
Consequently, because L is positive definite under the assumption (4.7), we the minimization variational principle

Y (E ′ , H ′′ ) = inf E ′ inf H ′′ Y (E ′ , H ′′ ), (5.5)
where the infimums are over vector fields E ′ and H ′′ with the same tangential components at the boundary ∂Γ as, respectively, E ′ and H ′′ . Using (2.37) the minimum value of

Y (E ′ , H ′′ ) is Y (E ′ , H ′′ ) = Γ ω 2 (H ′ • B ′′ -H ′′ • B ′ + E ′ • D ′′ -E ′′ • D ′ ) + 2ωj ′ • E ′ .
(5.6)

Now the power dissipated into heat in the body Γ, averaged over a cycle of oscillation is

W (E, H) = Γ ω 2π 2π 0 (Ee -iωt ) ′ • ∂(De -iωt ) ′ ∂t + (He -iωt ) ′ • ∂(Be -iωt ) ′ ∂t + (Ee -iωt ) ′ • (je -iωt ) ′ dt = 1 2 Γ ω(H ′ • B ′′ -H ′′ • B ′ + E ′ • D ′′ -E ′′ • D ′ ) + E ′ • j ′ + E ′′ • j ′′ , (5.7)
which is proportional to (5.6) when j = 0. So the minimum value of Y has a physical interpretation when j = 0. Due to the diagonal structure (4.20) of Z the formula (5.1) reduces to (5.8) where E(x) and M(x) are the positive definite matrices

Y (E ′ , H ′′ ) = Γ ωE ′ ∇ × H ′′ -j ′′ • E ωE ′ ∇ × H ′′ -j ′′ + ∇ × E ′ -ωH ′′ • M ∇ × E ′ -ωH ′′ + 2ωj ′ • E ′ ,
E = ε ′′ + ε ′ (ε ′′ ) -1 ε ′ ε ′ (ε ′′ ) -1 (ε ′′ ) -1 ε ′ (ε ′′ ) -1 , M = m ′′ + m ′ (m ′′ ) -1 m ′ m ′ (m ′′ ) -1 (m ′′ ) -1 m ′ (m ′′ ) -1 .
(5.9)

in which m = -µ -1 . Also the constitutive law (2.35) reduces to

D ′′ E ′′ = E E ′ -D ′ , H ′ -B ′ = M B ′′ H ′′ .
(5.10) So given any arbitrary prescribed values for the tangential components E ′ and H ′′ at the boundary ∂Γ the minimum in the variational principle (5.5) will satisfy the Euler-Lagrange equation and therefore will correspond to a solution of Maxwell equations.

From the minimizing fields E ′ and H ′′ one can, according to (5.10), determine (5.11) and in particular one can determine the tangential components of these fields at the boundary.

E ′′ = (ε ′′ ) ε ′ E ′ + (ε ′′ ) -1 (∇ × H ′′ -j ′′ )/ω, H ′ = m ′ (m ′′ ) -1 H ′′ -[m ′′ + m ′ (m ′′ ) -1 m ′ ](∇ × E ′ )/ω,
It is often the case that µ is real, while ε has a positive definite imaginary part. In this event we redefine Z, G and j by multiplying them by e iθ . Again there is a continuous two-parameter family of related variational principles. We rewrite Maxwell's equations (4.1) in the form

∇ × E = iω B, ∇ × H = -iω D + j, D = ε E, B = µ H, (5.12)
where (5.13) in which the real constant t 0 (which may be viewed as a time interval) can be chosen freely, while θ must be chosen so that (5.14) for some α 1 > 0 and α 2 > 0. Then the fields with the tildes on them also satisfy Maxwell's equations and the variational principles directly apply to them. With the subsequent replacement (5.13) we obtain a new set of variational principles parameterized by θ and t 0 . When θ is non-zero the minimum value of the minimization principles no longer has a physical interpretation as the time averaged power dissipation in the body.

E = e -iωt 0 E, D = e i(θ-ωt 0 ) D, H = e i(θ-ωt 0 ) H, B = e -iωt 0 B, j = e i(θ-ωt 0 ) j, ε = e iθ ε, µ = e -iθ µ,
ε ′′ (x) ≥ α 1 I, µ ′′ (x) ≥ α 2 I, ∀x ∈ Γ,
In particular by choosing θ = 0 and t 0 with e -iωt 0 = -i we recover the dual variational principle that

Y (E ′′ , H ′ ) = Γ ωE ′′ j ′ -∇ × H ′ • E ωE ′′ j ′ -∇ × H ′ + ∇ × E ′′ ωH ′ • M ∇ × E ′′ ωH ′ + 2ωj ′′ • E ′′ , ( 5
W (E, H) ≡ [Y (E ′ , H ′′ ) + Y (E ′′ , H ′ )]/(4ω) (5.16) is minimized at E = E ′ + iE ′′ = E and H = H ′ + iH ′′ = H
, under the constraint that E and H have the same tangential values as, respectively E and H. By making the substitutions (5.13) in (5.6) it is easy to check that the minimum value of W (E, H) is the average power dissipation W (E, H) given by (5.7), even when j is non-zero. An appealing feature of the expression (4.39) is that it remains invariant under the replacement E → e -iωt 0 E, H → e -iωt 0 H for any real value of t 0 . However a disadvantage is that the boundary conditions are overprescribed: if one imposes boundary conditions on the tangential values of E and H that are not compatible with the moduli ε(x) and µ(x) then the fields which minimize W (E, H) will not satisfy Maxwell's equations. When µ ′′ , or equivalently m ′′ , is very small we have

∇ × E ′ -ωH ′′ • M ∇ × E ′ -ωH ′′ ≈ [m ′ ∇ × E ′ -ωH ′′ ] • (m ′′ ) -1 [m ′ ∇ × E ′ -ωH ′′ ] (5.17)
and the variational principle is only useful if we take trial fields with

H ′′ ≈ m ′ ∇ × E ′ /ω.
If we take H ′′ = m ′ ∇ × E ′ /ω we still obtain a useful variational principle in the limit when µ is real. From (5.5) and (5.8) we have

Y (E ′ ) = inf E ′ Y (E ′ ), (5.18)
where

Y (E ′ ) = Γ ωE ′ -∇ × µ -1 (∇ × E ′ )/ω -j ′′ • E ωE ′ -∇ × µ -1 (∇ × E ′ )/ω -j ′′ + 2ωj ′ • E ′ .
(5.19) In this variational principle the infimum is over vector fields E ′ with prescribed tangential components of E ′ and µ -1 ∇ × E ′ at the boundary ∂Γ. The infimum is attained at a field E ′ associated with the solution of Maxwell's equations satisfying the prescribed boundary conditions.

Application to Electromagnetic Tomography

Here we consider the tomography problem: how can one recover information about the functions ε(x) and µ(x) given measurements of the tangential components E (x) and H (x) of the fields E(x) and H(x) at the boundary ∂Γ? When it is known that the body Γ consists of small inclusions in a matrix this question has been answered by ?): see also the review by ?). For other geometries a different approach is needed and the variational principles provide this. First observe from (4.15) and (5.7) that the power dissipation W (E, H), only depends on the tangential values of these fields at the boundary

W (E, H) = - 1 2 ∂Γ (E ′ × H ′ ) • n + (E ′′ × H ′′ ) • n, (6.1)
and, as could be expected physically, is the time-averaged value of the flux of the Poynting vector into the body Γ. Thus when j = 0, and trial fields E ′ and H ′′ are chosen having the same tangential values at the boundary as, respectively E ′ and H ′′ , the inequality

W (E, H) ≤ Y (E ′ , H ′′ )/(2ω) (6.2)
provides for a given ε(x) and µ(x) a bound on the possible tangential components of the fields E ′′ and H ′ . Conversely if these tangential components have been measured the inequality provides one constraint on the possible values of ε(x) and µ(x). Additional inequalities on ε(x) and µ(x) may be obtained by chosing different trial fields, by working with other minimization variational principles in the family parameterized by t 0 and θ, and by conducting a series of physical experiments with various different sets of values of the tangential components of the fields E(x) and H(x) at the boundary ∂Γ. It is not necessary that the moduli ε(x) and µ(x) remain the same for each physical experiment provided we have some model for how they vary from experiment to experiment. For example, the frequency ω could be different for the different experiments if we know that the permittivity has the form ε(x) = ε 0 + iσ o /ω (or some other known parametric dependence on ω). The objective is then to constrain the possible values of the real functions ε 0 (x) and σ o (x). One criticism of this approach is that it only utilizes information (in the case θ = 0) about the time-averaged power dissipation in the body Γ. When t = θ = 0 the measured values of the functions E ′′ and H ′ around the boundary ∂Γ are not used except to calculate the single scalar quantity W (E, H). This seems like a tremendous waste of information. However some extra information from the measured fields can be incorporated if one has measurements of the boundary value fields E (j) (x) and H (j) (x) for n experiments indexed by the integer j = 1, 2, . . . , n, with the moduli ε(x) and µ(x) remaining the same for each experiment. Then one can infer the boundary values E and H associated with a linear combination of these fields:

E = n j=1 λ j E (j) , H = n j=1
λ j H (j) , (6.3)

where λ 1 , λ 2 , . . . λ n are a set of complex constants. Letting E and H denote the corresponding electric and magnetic fields inside Γ, it follows that W (E, H) can be expressed as the sum

W (E, H) = n j=1 n k=1 (λ ′ j λ ′ k + λ ′′ j λ ′′ k )W jk + (λ ′′ j λ ′ k -λ ′′ k λ ′ j )S jk = 1 2 n j=1 n k=1 (λ ′ j λ ′ k + λ ′′ j λ ′′ k )(W jk + W kj ) + (λ ′′ j λ ′ k -λ ′′ k λ ′ j )(S jk -S kj ) (6.4)
involving the real quantities (6.5) which can be extracted from the experimental measurements of the boundary fields. Thus W (E, H) depends not only on the n diagonal elements, W jj , which physically represent the time-averaged power dissipation in each experiment, but also on the n(n-1) elements W jk + W kj and S jk -S kj , with j > k, which have no such physical interpretation. It makes sense to choose trial fields which are also linear combinations: j) ,

W jk = - 1 2 ∂Γ (E ′ (j) × H ′ (k) ) • n + (E ′′ (j) × H ′′ (k) ) • n, S jk = - 1 2 ∂Γ (E ′ (j) × H ′′ (k) ) • n -(E ′′ (j) × H ′ (k) ) • n.
E ′ = n j=1 (λ j E (j) ) ′ = n j=1 λ ′ j E ′ (j) -λ ′′ j E ′′ (
H ′′ = n j=1
(λ j H (j) ) ′′ = n j=1 λ ′ j H ′′ (j) + λ ′′ j H ′ (j) . (6.6)

Then Y (E ′ , H ′′ )/(2ω) -W (E, H) must be a positive quadratic form in the 2n real variables λ ′ 1 , λ ′′ 1 , λ ′ 2 , λ ′′ 2 . . . λ ′ n , λ ′′ n , and the positive semi-definiteness of the associated 2n by 2n matrix will give constaints on the functions ε(x) and µ(x).

We can also use the variational principle (5.18) for tomography purposes. When µ is real and j = 0 the inequality W (E, H) ≤ Y (E ′ )/(2ω) (6.7) applies. Often one knows in advance that the material inside Γ is non-magnetic, so that µ = I inside Γ. Then from knowledge of W (E, H) (6.7) provides a constraint on the possible values of the electrical permittivity ε for every suitable choice of the trial field E ′ . On the other hand, suppose the medium inside Γ was two phase with an unknown phase boundary and with two different values of µ inside Γ. Then the variational principle (6.7) will be useless unless the choice of trial field is carefully correlated with the unknown phase boundary, so that the tangential component of µ -1 (∇ × E ′ ) is continuous across the phase boundary.

  F and G have the form (2.1), with F and G being symmetric d × d matrices: thus F and G can be represented by d(d + 3)/2 component vectors. We choose θ such that the imaginary part of Z = -e iθ C 0 0 e iθ ω 2 ρ (3.22)

  .15) is minimized at E ′′ = E ′′ and H ′ = H ′ , under the constraint that E ′′ and H ′ have the same tangential values as, respectively, E ′′ and H ′ . A corollary is that

Acknowledgements

The authors are grateful for support from the Université de Toulon et du Var, the University of Utah, and from the National Science Foundation through grant DMS-070978. The authors are grateful to Russell Richins for checking the manuscript, and to David Dobson for helpful discussions.

When j = 0 then we have the inequality (6.8) where W (E, H) is given by (5.16) and the trial fields E and H are required to have the same tangential values as, respectively, E and H. Given measurements of the tangential values of E and H, and trial fields E and H we expect that this inequality should provide a useful constraint on the possible values of ε(x) and µ(x).


Bernard Guy
email: guy@emse.fr

Banded skarns, an example of geochemical dissipative structure

Keywords: banded skarns, oscillatory precipitation, dissipative structure

published or not. The documents may come     Banded skarns, an example of geochemical dissipative structure

Introduction

Skarns are coarse-grained calc-silicate metasomatic rocks that develop from carbonates under metamorphic conditions, through the operation of pervasive aqueous hot (hydrothermal) solutions [START_REF] Einaudi | Skarn deposits[END_REF]. Some of these rocks may be banded, in the sense that the metasomatic minerals are deposited in the form of alternating or oscillatory strata of centimetre wave length ; and they develop at the expense of starting materials (marbles) that may be homogeneous [START_REF] Guy | Certaines alternances récurrentes rencontrées dans les skarns et les structures dissipatives au sens de Prigogine : un rapprochement[END_REF].

The purpose of this paper is to give an introduction to an understanding of these structures, through description and qualitative discussion of some natural examples, as a basis for further more quantitative studies. The temporal evolution of zoning patterns in skarns and the situation of banded skarns within this evolution will be stressed.

Ordinary zoning and evolution of zoning in skarns

The fluids responsible for the chemical transformation of (mainly) carbonate materials may be rich in silica, iron, manganese, and possibly aluminium and other elements. The conditions are metamorphic, in that temperatures are in the range of several hundreds of degrees Celsius, pressures are several hundreds of bars to 1 or 2 kilobars [START_REF] Einaudi | Skarn deposits[END_REF][START_REF] Einaudi | A special issue devoted to skarn deposits : introduction -terminology, classification and composition of skarn deposits[END_REF]. Because of their mode of formation, the minerals in skarns combine elements from the carbonates, such as calcium, with others added by the external fluid, such as silica, iron, aluminium. Thus these minerals are mainly calcsilicates such as garnet (grandite solid solution : Ca 3 (Al,Fe) 2 Si 3 O 12 ), clinopyroxene solid solution (Ca(Fe,Mg)Si 2 O 6 ), calcic amphibole (solid solution Ca 2 (Mg,Fe) 5 (OH) 2 Si 8 O 22 ), ilvaite (CaFe 2 2 Fe 3 OOHSi 2 O 7 ), epidote (Ca 2 (Al,Fe)Al 2 Si 3 O 12 ) and so on. One may also encounter other minerals, and particularly economic minerals (skarns may be ores of tungsten, tin, iron, copper and other metals). Skarns usually form decimetric to metric sized veins within marbles, or metric to decametric-sized lens-shape bodies along lithologic discontinuities, between marbles and other rocks (granites, schists...), that have guided the aqueous solutions.

One characteristic feature of ordinary skarns is the pattern of zoning some may have. For instance transformation of marble A may be made through different steps in space separated by sharp fronts defining zones B, C, D and so on. In Fig. 1, two zones (hedenbergitic clinopyroxene and granditic garnet) develop in a calcitic marble.

These features are best explained by hypothesizing that the different fronts of transformation induced by a pervading metasomatic fluid, simultaneously progress at different speeds, the first front progressing faster than the second, and so on. This type of phenomenon has been studied by [START_REF] Korzhinskii | Theory of Metasomatic Zoning[END_REF], who proposed a model based on local chemical equilibrium (no kinetics, no diffusion), and developed by several other workers including the present author (e.g. [START_REF] Guy | Contribution to the theory of infiltration metasomatic zoning, the formation of sharp fronts : a geometrical model[END_REF][START_REF] Guy | Contribution à l'étude des skarns de Costabonne (Pyrénées Orientales, France) et à la théorie de la zonation métasomatique[END_REF][START_REF] Guy | Mathematical revision of Korzhinskii's theory of infiltration metasomatic zoning[END_REF], Guy et al., 1984). In the example of Fig. 1, one part of the starting marble has thus been converted first to pyroxene and then to garnet. This collection of zones of synchronous growth is called a metasomatic column.

Another feature of ordinary skarns that is less appreciated is the hypothesized evolution with time of the zonation itself [START_REF] Guy | Contribution à l'étude des skarns de Costabonne (Pyrénées Orientales, France) et à la théorie de la zonation métasomatique[END_REF]. In a given skarn deposit, different types of zonings may form from the same starting material. In figure 2, another zoning of a skarn developing in calcitic marble has been portrayed : marble / ilvaite / andraditic garnet. A simple explanation for this variety of zonings is that they have been produced under different physical and/or chemical conditions i.e. by different activities of the mobile components in the fluid. When we compare zoning 1 (Fig. 1) and zoning 2 (Fig. 2) one (or several) parameter(s) must have changed and ilvaite has replaced hedenbergite.

The transformation of dolomites (calcium-magnesium carbonates) shows conspicuous evolution of zoning (e.g. [START_REF] Bucher-Nurminen | The formation of metasomatic reaction veins in dolomitic marble roof pendants in the Bergell intrusion (Province Sandrio, Northern Italy)[END_REF], Dubru, 1986[START_REF] Guy | Contribution à l'étude des skarns de Costabonne (Pyrénées Orientales, France) et à la théorie de la zonation métasomatique[END_REF], and the paragenetic studies indicate that one important factor governing these changes is the decrease of temperature during the history of skarn development. The evolution of zoning may be portrayed on diagrams such as figure 3, where different metasomatic columns are represented and substitute for one another along the vertical axis.

Banded skarns

Ordinary zoning generally shows a succession of zones A, B, C and so on ;

sometimes, rather infrequently, we observe that the skarns are banded (Fig. Other descriptions are reported in the literature : [START_REF] Trustedt | Die Erzlagerstatten von Pitkaranta an Ladoga See[END_REF] andJahns (1944) describe quite similar alternations in an identical context (skarn developing in marbles). In the second case these are thin alternations of magnetite and silicates, associated with fluorite. Other examples are given in table 1.

Such phenomena have been observed elsewhere [START_REF] Guy | Certaines alternances récurrentes rencontrées dans les skarns et les structures dissipatives au sens de Prigogine : un rapprochement[END_REF][START_REF] Guy | Contribution à l'étude des skarns de Costabonne (Pyrénées Orientales, France) et à la théorie de la zonation métasomatique[END_REF]

The genesis of banded skarns

The formation of banded skarns must be understood in the context of ordinary skarns. Their genesis mainly involves transport and chemical reaction.

As already stated by Knopf, the geometric connection between the banded veins and orbicules and the massive skarns of which they are an extension, is an indication of the similarity of formation of all the rocks ; according to Knopf, all result from the action of fluids emanating from the nearby granite and infiltrating through the calcareous wall-rock.

In this sense, a possible explanation for the banding would entail an oscillation of the composition of the metasomatic fluid or an oscillation of the fracturing of the system, as proposed by various workers (see table 1). These explanations will not be considered here since, 1) they are very unlikely, and 2) since other explanations may be put forward to understand the oscillating and nonoscillating patterning within a unified framework.

The first step is to understand the time of formation of banded structures.

An important feature noticed by [START_REF] Jahns | Ribbon rock", an unusual beryllium tactite[END_REF] strongly suggests that in the course of the temporal evolution of hydrothermal deposits, where some minerals (or "alteration minerals") may develop at the expense of previously formed minerals (or "primary minerals"), the banded skarns seem to develop at the intermediate time when primary and alteration minerals are "equivalent". This statement is based on the following reasoning.

So far as the minerals involved in an alternating structure are concerned, one most often observes that some (one type of strata) belong to the group of "primary mineral" of usual skarns, whereas others (second type of strata) belong to the group of "alteration minerals" of usual skarns. The textures of the rocks that display such oscillatory precipitation are original, and the two groups of minerals are on the same level ; they appear to be equivalent of competing with each other and none grows from alteration of the other. We can give the example of pyroxene (a mineral of the first group) and ilvaite (a mineral of the second group) that can be found in oscillatory patterns or also of andradite (first group) and of the assemblage magnetite, calcite and quartz (second group). The example of forsterite or diopside and magnetite that can form banded skarns in dolomites is also of the same type. The role of fluorite, together with silicates, in certain banded structures described in the literature, can be understood in the same scheme.

In keeping with these observations on the evolution of zoning in skarns, we have inferred that the physical parameters (pressure and temperature) and the chemical parameters (composition of the fluid at its source) could evolve during the history of a deposit (we are speaking here of a monotonous, non-oscillating evolution), and that the stability field of primary minerals could be overstepped, whereas the stability field of other alteration minerals (that can then develop at the expense of the first) could be reached. But if for any reason the parameters stay close to the equilibrium conditions of both types, primary and alteration minerals, while the fluid is still entering the system and transforming the original rock, we are led to a critical situation in the sense that one does not know in advance which of the two types of minerals will develop, since they are equivalent with respect to the zoning. In figure 3, this critical situation corresponds to the dotted line that stays along the border between phases B and D. In such a situation a loss of stability may occur, in the sense that the ordinary evolution that produces the nonrepetitive zoning (here A / B / C) may be upset ; small fluctuations of the parameters may cause precipitation of one phase for another (here B or D). In short, according to natural observations, this seems to be one case in which oscillations develop. In that case of oscillatory precipitation of the primary and alteration minerals the usual terminology becomes misleading since the two types of minerals precipitate contemporaneously.

Discussion

Let us go a little further and discuss a qualitative model trying to give a preliminary account for the banding of skarns. In the critical situations we have described in the preceding section, second order effects (effects of the supersaturation threshold necessary for nucleation, of crystal-growth kinetics, and of diffusion) may be important and even dominant. Different authors (cf table 1)

have compared the banding structures to that of Liesegang rings, stressing the role of diffusion ; in that respect they have already dismissed the action of an external template to explain the oscillation. Recently, several models have been put

forward to explain the onset of oscillations in geological systems and particularly in fluid-rock interaction ; these are reviewed by Ortoleva et al. (1987a). In our case, the problem of banded skarns is original in that the model must account at the same time for 1) a periodic precipitation of the two alternating phases or groups of phases, taken one by one, during the dissolution of the starting material, and 2) for an organized competition between the two new phases so that their precipitation is shifted by one wave-length. For clarity, this can be explained by a combination of two end-member processes. 2) The skew-synchronicity of the two precipitation processes may be accounted for by their coupling via aqueous species involved in the mass balance, the mass action laws, and the kinetics of crystal growth. As is well known, the rate of mineral growth or dissolution is limited by surface attachment. Since the two phases precipitate in approximately the same amounts and in oscillation together, the temporal laws for their precipitation must, as a first approximation, be the same. A coupling of the precipitations of the two phases may take place because of the instability of the precipitation in equal amounts of the two phases or groups of phases coupled : if there is any slight difference in surface area of the two phases, all other parameters being equal, including the number of nucleated grains, then the mineral having the larger surface area will grow faster than the other. The interstitial fluid will then be impoverished in the chemical components necessary to build the first phase until the system jumps to a more abundant development of the second mineral and so on. Because of the nucleation threshold discussed above, the precipitation of one mineral may be completely repressed in the domain where the other mineral precipitates. This may in brief account for the appearance of oscillatory precipitation. This discussion must be assessed by a more quantitative (numerical) approach.

In the model proposed here, the oscillations maintained provided the inlet fluid continues to bring in chemical components at the right speed to keep up with the growth of crystals. There is thus a critical value for the inlet velocity with respect to the kinetic rate and diffusion constants. If fluid velocity is too fast, the composition of the in-coming fluid is imposed everywhere and local disequilibrium allowing for the oscillations is no longer possible. We then return to the Korzhinskii's model (1970) for the propagation of fronts. It is also necessary that the composition of the inlet fluid is such that two minerals or groups of minerals may be stable together and so as a whole precipitate in the same time.

Other models may be envisaged to explain banding. Oscillations may appear with diffusion coefficients non-linearly dependant on concentration [START_REF] Hazewinkel | Pattern formation for a one's dimensional evolution equation based on Thom's river basin model[END_REF], in non-ideal system (Li [START_REF] Ru | Dissipative structures in non ideal reaction-diffusion systems[END_REF], and in the framework of hyperbolic systems of equations [START_REF] Bonnefille | Propagation des oscillations dans les systèmes hyperboliques de lois de conservation[END_REF]. In the last case, fronts and oscillations are treated only with an equilibrium formulation and without diffusion, whereas these features are more usually treated in terms of the kinetics of the chemical reactions and with diffusion. As a solution of the more general equation that could be written for such problems and involve diffusion, advection and chemical kinetics terms, several types of behaviours may be observed according to the importance of the different terms and the values of the parameters : fronts [START_REF] Ortoleva | Redox front propagation and banding modalities[END_REF], oscillations, combinations of both, and even chaotic behaviour. One will find in [START_REF] Nicolis | Self-organization in non-equilibrium systems[END_REF], or in [START_REF] Pacault | Mise au point : structures chimiques spatiales, ondes chimiques[END_REF], general features of the models that may account for the appearance of oscillatory chemical "waves". The instability effect of surface attachment evoked here has been termed autocatalysis by some workers : Slin'ko and Slin 'ko, 1978[START_REF] Chaix | Pétrologie et géochimie des marbres à brucite et des borates associés au gisement de tungstène de Costabonne (Pyrénées Orientales[END_REF][START_REF] Gruffat | Un modèle pour les précipitations alternantes de minéraux dans les roches métasomatiques : l'effet autocatalytique des surfaces[END_REF] There is some analogy with the model proposed by [START_REF] Haase | Oscillatory zoning in plagioclase feldspar[END_REF] to explain the oscillatory zoning of feldspar.

In conclusion, one must set the problem of the oscillatory precipitations in the overall context of where ordinary skarns are formed. Without pretending to offer a general explanation for banded skarns, it may be recalled that, in the qualitative model proposed, the critical condition responsible for the oscillations must be defined at the same time by 1) the magnitude of the transport parameters (fluid velocity, diffusion coefficient) compared to that of kinetics and 2) by the concentrations of the components in the fluid : 1) transport must not be too fast as compared to kinetics : this is in agreement with observation that oscillatory structures preferentially develop in places where advective transport seems to play a minor role, as at the borders of veins, around blocks that are completely surrounded by veins, or within thick skarns where pressure gradients must have been small ; 2) it is necessary that the concentrations of the components in the fluid lie close to the value where several phases may co-exist in chemical equilibrium, or rather, may compete for the same zone with respect to the dynamic development of the metasomatic column (this may occur only at particular times during the skarn evolution).

Banded skarns are a beautiful example of geochemical self-organization in the sense that the organization must not be due to oscillating boundary condition but to the interplay of competitive phenomena within the system. Speaking of these self-organized structures, the equivalent term of dissipative structures (Prigogine) has been used [START_REF] Guy | Certaines alternances récurrentes rencontrées dans les skarns et les structures dissipatives au sens de Prigogine : un rapprochement[END_REF], in the sense that their formation is manifested in a system that is out of equilibrium, with a dissipation of energy, and with feed-backs. Similar structures have been studied on a quantitative basis by Ortoleva et al. (1987 a and b). The ordinary zoning of skarn wherein the concentrations of components are organized into different domains separated by sharp fronts is already an example of self-organization [START_REF] Guy | Contribution à l'étude des skarns de Costabonne (Pyrénées Orientales, France) et à la théorie de la zonation métasomatique[END_REF].

Captions for figures

Figure 1 An example of skarn development within a marble, showing metasomatic zoning : marble (A) / pyroxene (B) / garnet (C). One will notice the constancy of the order of the metasomatic zones (marble, then pyroxene, then garnet) wherever they develop, in veins or veinlets, around marble remnants or in massive skarns.

Arrows indicate the supposed direction of the metasomatic fluid. Width of zone B may be centimetric to metric.

Zoning of this type is described in many skarn deposits (e.g. [START_REF] Soler | Pétrographie, thermochimie et métallogénie du gisement de scheelite de Salau (Pyrénées Ariégeoises[END_REF][START_REF] Burt | Mineralogy and geochemistry of Ca-Fe-Si skarn deposits[END_REF]).

Figure 2 Another example of metasomatic zoning in skarns developed on pure marble : marble (A) / ilvaite (D) / garnet (C). Width of zone D may be centimetric to metric.

Zoning of this type has been observed in the Empire Mine (New Mexico) (cf [START_REF] Burt | Skarn mineralization in prophyry and non prophyry environments[END_REF].

The choice of the letters A, B, C, D has been made consistant between the different figures.

Figure 3 The evolution of metasomatic zoning. A possible explanation for the variety of zoning patterns that develop on the same starting material (such as zonings 1 and 2 on figures 1 and 2) is to suppose that some parameters have changed during skarn evolution (time is along the vertical axis) and that zoning (1)

A / B / C is replaced by zoning (2) A / D / C (distance is along the horizontal axis), because mineral B is no longer stable and is replaced by mineral D.

As will be discussed in the following, we will suppose that when the We can give many examples where this situation may occur, although clear evidence of evolution of zoning in a skarn deposit is not common. (B,D) couples may be (pyroxene, ilvaite), (pyroxene, amphibole), (garnet, magnetite + quartz), (forsterite, magnetite), (diopside, magnetite) and so on.

We suppose in this picture that the changes in the physical and/or chemical conditions that take place along the vertical axis are slower than the time necessary to create a single zoning pattern developing along the horizonal axis ; In skarns we may for instance observe the disappearance of wollastonite from the sequence calcite / wollastonite / hedenbergite, with the later zoning calcite / hedenbergite. The oscillation between hedenbergite and wollastonite is then expected (it has been observed by numerical simulation, Ortoleva and Guy, pers. obs.). Photograph : ilvaite : pyroxene banded skarn (Rio Marina, Elba Island, Italy) seen in thin section. The ilvaite bands, in black, alternate with hedenbergitic pyroxene (with interstitial quartz and calcite). Notice the radial texture of ilvaite within a "stratified" distribution. This is also the case for hedenbergite, though less distinctly. On this sample, the ilvaite bands are 2 to 3 millimetres thick, the hedenbergite ones 4 to 6 millimetres. In other parts of the skarns, the bands may reach one centimetre. [START_REF] Guy | Certaines alternances récurrentes rencontrées dans les skarns et les structures dissipatives au sens de Prigogine : un rapprochement[END_REF][START_REF] Guy | Contribution à l'étude des skarns de Costabonne (Pyrénées Orientales, France) et à la théorie de la zonation métasomatique[END_REF].

Bernard GUY 20 TABLES [START_REF] Meinert | Skarn, manto and breccia pipe formation in sedimentary rocks of the Cananea Mining district, Sonora, Mexico[END_REF], in which case the template of replaced argillaceous layers is recognized; one will also find in [START_REF] Kwak | Geology and genesis of the laminar F-Sn-W(-Be-Zn) skarn at Moina, Tasmania, Australia[END_REF] and in [START_REF] Kwak | W-Sn skarn deposits and related metamorphic skarns and granitoids[END_REF] a review on what they call "wrigglite" (i.e. banded skarns).

Skarn minerals may also exhibit oscillatory zoning : garnets (e.g. [START_REF] Verkaeren | Les grenats biréfringents des skarns à magnétite de San Leone (Sardaigne S.W.)[END_REF][START_REF] Zimmernink | Geology and mineralogy of Felicidad and Santander, two Peruvian skarn-type deposits in Cretaceous (sedimentary) host rocks (with a case study of various garnet types)[END_REF], epidote (pers. observations of the author) and so on.

  photographs of banded skarns of these kinds.

  : in the Empire Mine (New Mexico) and at Rio Marina and Calamita (Elba Island), the skarns are made of centimetric alternations of hedenbergitic clinopyroxene and ilvaite (figure 4.4) giving beautifully banded rocks developed within homogeneous marbles, similar to that portrayed in figure 4.1. At San Leone (Sardinia), the clear bands of andradite and the dark bands of magnetite + quartz are particularly conspicuous (figure 4.5).

1)

  The supersaturation / nucleation / depletion cycle discussed byOrtoleva et al. (op. cit.) must be invoked in the periodic precipitation of one phase. In this phenomenon, the pervading reacting fluid dissolves the phases contained in the starting material causing an increase in the concentration of the chemical component contained therein. In addition to these components, the building of the new phases involves other components brought into the system by the inflowing fluid. A finite supersaturation is needed to begin nucleation of the new phases ; crystal growth of the phase locally impoverishes the medium in the chemical components including those taken from the starting material ; the possible nucleation of a new crystal is then possible only downstream where the concentration of the chemical components induced by dissolution of the initial minerals has reached the proper value for supersaturation, and this can explain the oscillatory precipitation of one phase.

  transformation operates in conditions close to the boundary between B and D (dotted line), the zoning may be oscillatory : A / B-D-B-D and so on / C or A / B-D-B-D and so on / B (or D) (oscillatory zoning of this sort is illustrated in figures 4.1 to 4.5).

  the horizontal arrow in the figure shows the direction of fluid movement during an interval of time when the conditions did not change much and were close to the boundary between two zoning systems. In this situation, we suppose that the spatial zoning is not merely that indicated by the arrow (A / D / B / C) but that there is an oscillation between B and D. The change in the zoning is sometimes defined by the disappearance of one zone, such as B in the A / B / C zoning pattern (this situation is different from that portrayed in the figure). If then the conditions are maintained in the vicinity of the triple point A B C, we may expect an oscillation between B and C phases. In that case the oscillatory zoning may involve minerals that may in other cases be involved in the same zoning pattern.

Figure 4

 4 Figure 4 Banded skarns

Figure 4. 1

 1 Figure 4.1 An example of banded skarn : skarn develops in an homogeneous marble. It is banded in the sense that the metasomatic minerals B and D are precipitated alternatively forming oscillatory structures B-D-B-D... The ribbons are parallel to the main transformation front, in the external part of the skarn or around remants of marbles.

Figure

  Figure 4.2 and 4.3 Two idealized cross-sections of banded skarns : in 4.2 mineral B dominates in the inner part of the skarn (as it is in the case in Rio Marina where B = hedenbergite dominates). In 4.3 another mineral or zone, C, develops at the expense of the "zone" B-D-B-D. This is the case in some other deposits.

Figure

  Figure 4.4

Figure

  Figure 4.5 Photograph : banded skarn in San Leone (Sardinia, Italy) : alternations of andraditic garnet (light) and magnetite + quartz (dark). The garnet bands are 2 to 5 mm thick, and the magnetite + quartz bands are 5 to 15 mm thick[START_REF] Guy | Certaines alternances récurrentes rencontrées dans les skarns et les structures dissipatives au sens de Prigogine : un rapprochement[END_REF]

TABLE 1

 1 Banded skarns in the literature. A few examples are given here, together with some comments. Additional examples include : Zimmernink, 1985 (Fe-sphalerite / calc-silicates, in Santander, Peru) ; Shabynin, 1977 (banded skarns and greisens) ; Alexandrov, 1982 (banded magnesian skarns) ; Guy, 1988 (forsterite / magnetite and diopside / magnetite at Traversella, Italy) ;

Acknowledgements

The matters exposed here have been discussed with several individuals whom the author thank here : G. Nicolis, A. McBirney, M. Fonteilles, J.J. Gruffat, P. Ortoleva.