Green s Dyadic for the Three-dimensional Linear Elasticity in Periodic Structures

Transcription

1 BULLETIN OF THE GREEK MATHEMATICAL SOCIETY Volume 5, Green s Dyadic for the Three-dimensional Linear Elasticity in Periodic Structures Constantinos Anestopoulos, Elias Argyropoulos, Drossos Gintides and Kiriakie Kiriaki Astract In this paper, we consider the Green s dyadic in three dimensional linear elasticity for periodic domains. Using the Floquet-Bloch oundary conditions for the elastic case, we give the exact form of the Green s dyadic in terms of sums of images. Applying also the integral representation of the Green s function for the Helmholtz equation with the heat kernel we present a convenient analytic form, which has slow convergence. The main result of this paper is to overcome this difficulty y implementing Ewald s method. This method uses interchanging of integrations and summation processes and specific representations of special functions which in elasticity also involve the dyadic character of the Green s dyadic. Finally, we discuss some details of the method concerning the choice of parameters descriing the dimension of the periodic cell or termination for the integration procedures. The potential use of the method in periodic oundary value prolems in elasticity is also enlightened. Keywords: Ewald s method, Green s dyadic, linear elasticity. Introduction In the last few years there is an increasing interest among scientists for wave propagation of acoustic, electromagnetic or elastic waves in periodic media. A lot of new technological applications are justifying this interests. For example, in electromagnetism very promising is the area of photonic crystals. A macroscopic periodic lattice of a dielectric can e considered as photonic crystal [0], []. It is well known that photonic crystals exhiit photonic andgaps, that is, regions of their frequency spectrum over which there are no waves propagating through the lattice. A great variety of technological applications incorporate such type of materials as for example filters, laser, microwave antennas e.t.c.. In elasticity a similar ehavior of periodic materials is estalished [0]. Some applications of periodic materials are elastic wave filters and sensors. 35

2 36Constantinos Anestopoulos, Elias Argyropoulos, Drossos Gintides and Kiriakie Kiriaki The mathematical formulation of these oundary value prolems in integral equations is one of the main methods for the theoretical investigation of well possedness of the prolems and also for numerical calculation of solutions. One of the most important prolems in this approach is the efficient calculation of periodic Green s functions. In periodic media Green s functions are produced via the method of summing images or eigenfunction ansions [6]. In any case, the resulting series present poor convergence. A lot of techniques are availale for accelerating slowly convergence series. One of the most effective methods is due to Ewald, []. This method accelerates the convergence of the series of periodic Green s function. It is ased on the integral representation of the free space Green s function via the heat kernel. This formula is applied in all free space Green s function entering in the infinite sum of images for the periodic Green s function. Using interchanging of integrations and summation and specific representations via special functions the resulting series exhiit rapid convergence. This procedure is usually descried in the literature in the context of two or three dimensional lattice, [7]. In [6] C. M. Linton also uses Ewald s method and reports the results of this method compared with other methods of this direction as Kummer s transformation or integral representations and lattice sum s. In [7] Ewald s method is revisited. It is extended for the Helmholtz equation in certain domains of R d with quite general oundary conditions. The goal of this work is except of the derivation of a rapidly converging periodic Green s function to set the asis for studying the oundary value prolems for periodic arrays of scatterers. This approach is used y S. Venakides et.al. in two-dimensional electromagnetic waves scattering modelling a Fary-Perot structure with mirrors which consist a photonic crystals, []. A oundary integral formulation is used to solve the prolem employing a periodic Green s function rapidly converging according Ewald s representation. In elasticity, a great variety of materials have periodic structure, for example composite materials, e.t.c., which makes necessary the use of periodic Green s dyadic in any integral formulation of the corresponding wave propagation and scattering prolems. We would like to mention that, the vector elastic waves are the superposition of longitudinal and transverse waves which interfere on the oundaries due to the mode conversion mechanism. This character of the waves makes the situation much more complicated. The computation of the very complicated Green s dyadic suffers from poor convergence. The purpose of this paper is to remedy this lack of fast convergence of the periodic Green s dyadic y employing Ewald s approach in elasticity. In section, we give the exact form of the Green s dyadic for periodic structures using the method of images in three-dimensional elasticity with the Floquet-Bloch oundary conditions. We use the heat kernel and a representation of the three-dimensional Green s function in acoustics. In section 3, we apply Ewald s technique for the elastic dyadic. We prove that this representation converges rapidly and consequently is appropriate for numerical computations. At the end we discuss some details of the method. It is

3 Green s Dyadic for the Three-dimensional Linear Elasticity in Periodic Structures 37 underlined that the choice of parameters descriing the dimension of the periodic cell does not alter the whole method and that the termination limits of the integration processes can e used to alance the decays of sums and integrals. The potential use of the method in periodic oundary value prolems in elasticity is also enlightened.. The Green s dyadic for periodic structures We consider the domain D in R 3 with D = 0, 0, r R 3r, r =,, 3, where r > 0, r =,, 3 and we assume that it consists the fundamental cell of the periodic structure. We also consider that the space is filled up y an isotropic and homogenous elastic medium with Lamé constants λ, µ where µ > 0, λ + µ > 0 and constant density ρ =. The propagation of elastic waves in this space is governed y the Navier time-reduce equation where the Lamé operator is given y the relation + ω ux = 0 = µ + λ + µ and u is the displacement field and ω > 0 is the angular frequency. It is well-known [4] that every solution of can e decomposed as a superposition of longitudinal and transverse components given y u p = k p u, u s = u u p. 3 In previous relations, the indices reflect the physical properties and assign the nomination of P-waves pressure and S-waves shear, k p = ω c p, k s = ω c s are the wave numers and c p = λ + µ, c s = µ are the phase-velocities of corresponding transverse and longitudinal waves. The fundamental solution Γx, y of Navier s equation in R 3, is given, [4], y Γx, y = [ e ık p xy ω x x x x + k sĩ e ık s xy ] 4 4π x y 4π x y with x, y R 3, x y and Ĩ the identity dyadic. From the periodicity of the structure in conformance with Bloch s theorem [], there are constants a j : a j j < π, j =,,..., r, such that the following Floquet- Bloch oundary conditions and ux,..., j,..., x 3 = e ıa j j ux,..., 0,..., x 3, 5 ux,..., j,..., x 3 = e ıa j j ux,..., 0,..., x 3, 6

4 38Constantinos Anestopoulos, Elias Argyropoulos, Drossos Gintides and Kiriakie Kiriaki hold, for j =,,..., r, r =,, 3. We are now in the position to prove the following theorem. Theorem. The Green s dyadic descried y equation with periodicity conditions 5 and 6 is given y the relation Gx, y = 4π ω m Z r e ıam { 4πt 7 0 k pt x y + m y + m x y + m tĩ] dt k st x y + m 0 4πt 7 y + m x y + m + 4kst Ĩ ] } dt, x, y R 3 7 where a = a,..., a r, 0,..., 0, a R 3 and m = m,..., m r r, 0,..., 0, m R 3, m r Z, r =,, 3, Ĩ the unit dyadic and t > 0. Proof. Let k a complex numer. If G n x, y; k is the Green s function of free space of partial deferential operator L = ki in R n, n, where I the identity operator, then [7] x y G n x, y; k = kt dt 8 4πt n with Rek < 0 and t > 0. The radial solution of equation 0 ux kux = δx, x R n 9 where δx the Dirac s function with pole at 0 R n, is G n x, y; k = k n n π n 4 x y K n k x y 0 where K m denotes the modified Bessel s function of second kind and m order. From 0, for n = 3, taking into account that K z = π z e z [5], we find that k xy G 3 x, y; k = e 4π x y, hence using 8 also for n = 3, we otain e k xy 4π x y = 0 4πt 3 kt x y dt

5 Green s Dyadic for the Three-dimensional Linear Elasticity in Periodic Structures 39 with Rek < 0, which is the asic relation of Ewald s method. For k = k α, α = p, s, especially for k = ık α gives e ıkα xy 4π x y = 0 4πt 3 k αt We use the relations x y x = t x x x y = x y dt, α = p, s, x y, x, y R 3. 3 x y x y, x, y R 3, 4 x y [x y x y t Ĩ ], 5 the fact that we can interchange the order of differentiation and integration in 3 and we conclude to x y x x = 4π k αt x y 0 4πt 7 y x y tĩ] dt. 6 Sustituting the last relation to 4, we otain { [ Γx, y = 4π ω 4πt 7 0 k pt x y y x y tĩ]] dt [ k st x y 0 4πt 7 y x y + 4kst Ĩ ]] } dt, x, y R 3. 7 We are now in the position to handle the prolem governed y and satisfying the Floquet-Bloch oundary conditions. The method of images gives that the Green s dyadic of this prolem is Gx, y = m Z r e ıam Γx + m, y 8 where a = a,..., a r, 0,..., 0 and m = m,..., m r r, 0,..., 0, m r Z, r =,, 3. Sustituting ressions given y 6 and 7 in 8 we derive formula 7. The series in 7 converges asolutely provided x y, with x, y D.

6 40Constantinos Anestopoulos, Elias Argyropoulos, Drossos Gintides and Kiriakie Kiriaki 3. The Ewald s representations Now, following Ewald s method, we derive a series representation of Gx, y, x y, which rapidly converges for all complex wave numers except a discrete countale set. Ewald s idea is to split G in two parts where G x, y = 4π ω Gx, y = G x, y + G x, y 9 { Ep 0 m Z r e ıam 4πt 7 k pt x y + m y + m x y + m tĩ] dt Es k st x y + m 0 4πt 7 y + m x y + m + 4kst Ĩ ] } dt, x, y R 3, 0 G x, y = 4π ω m Z r e ıam { E p 4πt 7 k pt x y + m y + m x y + m tĩ] dt k st x y + m E s 4πt 7 y + m x y + m + 4kst Ĩ ] } dt, x, y R 3, and the positive numers E p, E s are chosen appropriately, as we discuss in section 4, to accelerate the convergence of the series which involved in G and G. Firstly, we consider the full periodic case in R 3, which corresponds to the choice r = 3. We assume that Rek a < 0, a = p, s. In this case, we can interchange summation and integration in 0, so we otain { G x, y = 4π Ep ω 0 kpt [ 4πt 7 m Z 3 y + m x y + m tĩ]] dt x y + m ıa m

7 Green s Dyadic for the Three-dimensional Linear Elasticity in Periodic Structures 4 Es k [ st x y + m ıa m 0 4πt 7 m Z 3 y + m x y + m + 4kst Ĩ ]] } dt, x, y R 3. In, oth series are already rapidly convergent, for x y, since its general term decays in m like C m m, 3 4E where E = min{e s, E p }. Interchanging summation and integration in since G x, y converges uniformly, we otain { [ G x, y = 4π k pt ω E p 4πt 7 m Z 3 x y + m ıa m y + m x y + m tĩ]] dt k [ st x y + m ıa m E s 4πt 7 m Z 3 y + m x y + m + 4kst Ĩ ]] } dt, x, y R 3. 4 From Lemma of [7], we have that if s > 0 and z, γ C, then π e s z+nıγn = n Z s eıγz γ 4s n Z e π n s πγn s +πızn. 5 Using the last relation, whose proof is ased in Poisson Summation Formula, applied to the function fx = e Ax +Bx, A > 0, B C, we otain m Z 3 x y + m m Z 3 ıa m = 4πt 3 D [ 4π m + 4πa m where D = 3 is the volume of D and m = m, m, m3 3. Thus 4 takes the form G x, y = 4π ω D { E p [ k p t 4πt [ıa x y a t] [ıa x y a t] t + πıx y m ], 6

8 4Constantinos Anestopoulos, Elias Argyropoulos, Drossos Gintides and Kiriakie Kiriaki [ 4π m m Z 3 + 4πa m t + πıx y m ] y + m x y + m tĩ]] dt [ k s t E s 4πt [ıa x y a t] [ 4π m + 4πa m t + πıx y m ] m Z 3 y + m x y + m + 4kst Ĩ ]] } dt, x, y R 3. 7 Now, we interchange integration and summation in 7 ecause we have uniform convergence for this relation. If we set A p m = kp + a + 4π m + 4πa m 8 A s m = ks + a + 4π m + 4πa m, 9 with A p m > A s m, since k p < k s, we have G x, y = { ω D If A s m > 0, then m Z 3 [ [ ı a + π m ] x y x y + m x y + m 4 Ĩ m Z 3 +k sĩ E p A p ] mt dt t [ [ ı a + π m x y ] x y + m x y + m 4 E s E p E s A p mt t dt A s mt t dt ]} A s mtdt, x, y R ks < a + 4π m + 4πa m, 3 namely in the tail of the series, the last integral in the right term of 30 ecomes E s A s mtdt = ea s m E s A s m 3

9 Green s Dyadic for the Three-dimensional Linear Elasticity in Periodic Structures 43 and using the limit comparison test follows that the integral E s e A s m t t dt < E s t dt = E s 33 which means that the aove integrals converge. Then, using the same arguments we infer that the other two integrals of 30 exist. Therefore, the terms of four series in 30 decay in m like C m 4π m E, 34 where E = min{e s, E p }. From 3 follows that the only singularities of G x, y are a + 4π m + 4πa m which are the poles, in ks, of the left term of 3. Hence we should have ks a + 4π m + 4πa m which holds from 3. Now, let us discuss the case r < 3. For x = x, x, x 3 we denote or for 35 x = x + x, x = x, 0, 0, x = 0, x, x 3 36 x = x + x, x = x, x, 0, x = 0, 0, x 3 37 x = x r + x r, r =, 38 then, from the orthogonality of x r y r + m r and x r y r, we otain sustituting in, we have G x, y = 4π x y + m = x r y r + x r y r + m, 39 ω m Z r e ıam { k pt xr y r + x r y r + m E p 4πt 7 r y r + x r y r + m x r y r + x r y r + m tĩ] dt k st xr y r + x r y r + m E s 4πt 7 r y r + x r y r + m x r y r + x r y r + m +4kst Ĩ ] } dt, x, y R 3. 40

10 44Constantinos Anestopoulos, Elias Argyropoulos, Drossos Gintides and Kiriakie Kiriaki We set à m x, y = x r y r x r y r + m + x r y r + m x r y r, 4 using 6 in 40 and after some calculations { G x, y = 4π ω e ı a+π m x r y r à m x, y... r m Z r A E p 4πt 7r p t xr y r r y r x r y r tĩ] dt e ı a+π m x r y r à m x, y... r m Z r A E s 4πt 7r s t xr y r r y r x r y r + 4kst Ĩ ] } dt, x, y R 3. 4 The terms of the series in 4 decays in m like C m 4π m E a, a = p, s 43 respectively, for a fixed ka, a = p, s. In the tail of the series we can have any k a, a = p, s. There seems to e a prolem, for the convergence of the integrals, though with the first few terms of the series, where we may have ka > a +4π m +4πam, a = p, s. In 4 appear integrals in the form of [ ] k t c dt, 44 4πt n E where, c, E > 0 and n positive integer. Lemma in [7] say us that the only singularity of the function [ ] gλ = λt c dt, 45 4πt n E with Reλ > 0, λ C, is the ranch point at λ =. Thus, if r < d the only singularities of G x, y, as a function of λ = ka, a = p, s, are the ranch points at ka = a +4π m +4πam, a = p, s. These are also the singularities of Gx, y. For r = relation 4 ecomes { G x, y = 4π ω m Z e ı a+π m x y à m x, y

11 Green s Dyadic for the Three-dimensional Linear Elasticity in Periodic Structures 45 E E p 4πt 5 A p t x y y x y tĩ] dt e ı a+π m x y à m x, y m Z A E s 4πt 5 s t x y y x y + 4kst Ĩ ] } dt, x, y R 3, 46 with x, x as in 37. The integrals appear in the last relation are of the form [ ] λt c dt, 47 4πt q with q =, 3, 5. The integral for q = can e computed in terms of the special function erf cz defined y In fact holds [ ] λt c dt = 4πt E erfcz = erfz = e s ds. 48 π 4 λ z [ e c λ erfc E λ + +e c λ erfc E λ c E c E ], 49 this last relation can e proved sustituting t = s, differentiating with respect to E and finally taking into account that oth sides vanish when E. 4. Discussion From the previous analysis we conclude that: When r = 3, the dyadic G like in 9 with G and G which are given y 0 and 30, is ressed y a rapidly convergent series that is very convenient in numerical calculations, from the series which gives the Γ y relation 7 as certify 3 and 34. By these last two relations is demonstrated the role of E p and E s in the velocity of convergence of the series. Moreover, these relations show the different effect of E a, a = p, s in convergence of G from this of G. If we want to alance the decays of the terms of the series for G and G, as referred in [7], a reasonale choice is E a = [ r 4π r ], a = p, s, r =,, 3. 50

12 46Constantinos Anestopoulos, Elias Argyropoulos, Drossos Gintides and Kiriakie Kiriaki This choice is otained equating 3 and 34 or 3 and 43 and taking into account linear approximations for the onentials. Since especially in elasticity there are a lot of periodic materials, for example composite filters, the Ewald s method can e applied to the solution of oundary value prolems. In the case of Dirichlet or Neumann oundary conditions we can apply the Ewald s method, since the corresponding Green s dyadics Γ D and Γ N is a sum of free space Green s dyadic Γ and another dyadic Ũ of a similar form with appropriate constant coefficients. In the case r = everything holds as in the case for r = 3. Moreover one of the integrals which appear in the ression of G, as is given y 46, can e computed in terms of the special function, error function erfz, which is an entire function, in contradiction with acoustic case, where the terms of oth G and G can e computed in terms of the special function erfz. Acknowledgements The authors would like to ress their thanks to V. Papanicolaou, for ringing to their attention the new interest in Ewald s method, and the fruitful discussions in the details of the method. References [] Ashcroft and Merhin, Solid State Physics. Saunders College Pulishing, 976. [] P. P. Ewald, Die Berechnung optischer and elektrostatischer Gitterspotentiale. Ann.Phys., 64, 53-87, 9. [3] A. D. Klironomos and E. N. Economou, Elastic wave and gaps and single scattering. Solid State Commun., 055, 37-3, 998. [4] V. D. Kupradze, Three-dimensional prolems of the mathematical theory of elasticity and thermoelasticity. North-Holland Pulishing Company, Amsterdam, 979. [5] N. N. Leedev, Special functions and their applications. Dover Pulications,Inc., New York, 97. [6] C. M. Linton, The Green s function for the two-dimensional Helmholtz equation in periodic domains. J. Engin. Math., 33, , 998. [7] V. Papanicolaou, Ewald s method revisited: Rapidly convergent series representations of certain Green s-functions. J. Comp. Anal. Appl.,, 05-4, 999. [8] J. A. F. Santiago and L. C. Wroel, D modeling of shallow water acoustic wave propagation using suregions technique. Boundary Element Techniques-An International Conference. Queen Mary and Westfield College, 45-44, U.K.,999.

13 Green s Dyadic for the Three-dimensional Linear Elasticity in Periodic Structures 47 [9] J. A. F. Santiago and L. C. Wroel, A oundary element model for underwater acoustics in shallow waters. Comp. Model. Engin. Sci.,, 73-80, 000. [0] M. M. Sigalas and E. N. Economou, Elastic waves in plates with periodically placed inclusions. J. Appl. Phys., 756, ,994. [] S. Venakides, M. A. Haider and V. Papanicolaou, Boundary integral calculations of -D electromagnetic scattering y photonic crystal Fary-Perot structures. SIAM J. Appl. Math., 60, , 000. Constantinos Anestopoulos National Technical University of Athens, Department of Applied Mathematics and Physics, GR-5780 Zografou Campus, Athens, Greece kanesto@mail.ntua.gr Elias Argyropoulos Technological Education Institute, Department of Electrical Engineering, GR-3500 Lamia, Greece protepkste@stellad.pde.sch.gr Drossos Gintides Hellenic Naval Academy. Department of Mathematics, Pireus, Greece dgindi@math.ntua.gr Kiriakie Kiriaki National Technical University of Athens, Department of Applied Mathematics and Physics, GR-5780 Zografou Campus, Athens, Greece kkouli@math.ntua.gr