Pierre Charollois

Robert Sczech

Elliptic Functions according to Eisenstein and Kronecker : An Update. Newly Found Notes of Lectures by Kronecker on the work of Eisenstein

This article introduces a set of recently discovered lecture notes from the last course of Leopold Kronecker, delivered a few weeks before his death in December 1891. The notes, written by F. von Dalwigk, elaborate on the late recognition by Kronecker of the importance of the "Eisenstein summation process", invented by the "companion of his youth", in order to deal with conditionally convergent series known today as Eisenstein series. We take this opportunity to give a brief update of the well known book by André Weil (1976) that brought these results of Eisenstein and Kronecker back to light. We believe that Eisenstein's approach to the theory of elliptic functions was in fact a very important part of Kronecker's planned proof of his visionary "Jugendtraum".

functions depending on two pairs of real variables". We gathered that historical information from recently discovered lecture notes at the library of the University of Saarbrücken. Before discussing the content and the author of these handwritten notes, we wish to recall the circumstances of their discovery. The original discovery is due to Professor Franz Lemmermeyer, who started the task of retyping the text written in old style German cursive handwriting, but did not finish the job. The manuscript then got lost during a library move. Being interested to learn more about Kronecker's work and insight into his so called "liebster Jugendtraum", we decided to enlist the help of Simone Schulze at the library in finding it. After a search effort which lasted a few weeks, the complete set of notes was finally found and the library even produced a high quality digital copy which is now available to the public [Cw]. We thank Ms. Schulze for her help and we also thank Franz-Josef Rosselli who undertook the effort to translate the cursive German handwriting (prevailing in the 19 th century) into modern German typeface [Cw].

The manuscript was written by Friedrich von Dalwigk, a graduate student at that time who was just finishing his dissertation on theta functions of many variables [vD]. Later, von Dalwigk became Professor for applied mathematics at the University of Marburg (1897Marburg ( -1923)). Although the course consisted only of six lectures (due to the premature death of Kronecker), the whole manuscript is more than 120 pages, with many appendices, partly written by von Dalwigk himself relying on published papers of Kronecker as well as on unpublished papers from Kronecker's "Nachlass". Dalwigk explicitly mentions the word "Nachlass" in the manuscript. He is likely to have had access to that specific document from his colleague Hensel in Marburg, who was in possession of all scientific papers of Kronecker at that time. As we learned from Hasse and Edwards [Ed], the personal papers of Kronecker were lost in the chaotic events surrounding WWII. Most probably, they were destroyed by a fire caused by exploding munitions in an old mine near Göttingen. That mine was used to store some of the collections of the Göttingen library in 1945. This dramatic event is only part of the long lasting spell put on the posterity of Eisenstein's ideas, as predicted by André Weil in his essay [START_REF] Weil | Book review : Gotthold Eisenstein, Mathematische Werke[END_REF].

At any rate, the care and the amount of detail included in the manuscript is extraordinary and leads us to think that it was written with the ultimate intention to publish it as a book.

We know from a letter of Kronecker to his friend Georg Cantor who at that time served as the first president of the newly founded German Mathematical Association (DMV) and who invited Kronecker to deliver the opening address at the first annual meeting of the DMV in 1891, that Kronecker intended to talk at that meeting about the "forgotten" work of Eisenstein. In that letter, Kronecker apologizes for not being able to attend the meeting due to the death of his wife Fanni. It is very likely that the lecture notes in question are an expanded version of his intended talk.

Kronecker was indeed a great mathematician with fundamental contributions to algebra and number theory. We mention here only his "Jugendtraum" which, historically, gave rise to class field theory and to Hilbert's 12 th problem (the analytic generation of all abelian extensions of a given number field), one of the great outstanding problems in classical algebraic number theory. The celebrated conjectures of Stark (published in a sequence of 4 papers during the 1970's) offer a partial solution to a problem with ultimate roots in the work of Kronecker and Eisenstein.

In passing, we wish to mention two standard references about the work of Kronecker in number theory. The first chapter of Siegel's Lectures on Advanced Number Theory [Si] is devoted to the so called Kronecker limit formulas with applications (Kronecker's solution of Pell's equation). The second reference is the book by André Weil [START_REF] Weil | Elliptic functions according to Eisenstein and Kronecker. 2nd Edition[END_REF] on elliptic functions according to Eisenstein and Kronecker. The lecture notes under review can be classified roughly as an extension and elaboration of the material discussed by Weil. Besides resurrecting the ideas of Eisenstein from final oblivion, the book of Weil is also a valuable source for many anecdotes about Eisenstein and Kronecker. Our favorite one is the story that Kronecker, in his public PhD defense talk, claimed that mathematics is both science and art. His friend Eisenstein challenged Kronecker publicly by claiming mathematics is art only.

Kronecker and the work of Eisenstein

Except for a letter to Dedekind dated March 15, 1880, there is no comprehensive statement of the Jugendtraum in the papers of Kronecker. In that letter, Kronecker reports on his recent progress towards a proof of his conjecture (the Jugendtraum) that all abelian extensions of an imaginary quadratic field F are generated by division values of suitable elliptic functions admitting complex multiplication by elements of F together with the corresponding singular moduli, that is, the values of the j-invariant of the corresponding elliptic curves. He expresses the hope to complete the proof soon. In closing, he regrets having to postpone the problem of finding the analogue of singular moduli for arbitrary complex number fields (Hilbert's 12 th problem) until the case of imaginary quadratic fields is completely resolved. 10 years later, in his lectures on elliptic functions, he does not mention his work on the Jugendtraum at all. Instead, he concentrates on reviewing and generalizing the work of Eisenstein. In the following, we wish to give a hypothetical explanation of why the approach of Eisenstein may have been an important step in Kronecker's envisioned proof of his Jugendtraum. Namely, we are going to carry out Kronecker's program in the simpler setting of abelian extensions of the field of rational numbers by modifying a basic example given by Eisenstein in his great paper [START_REF] Eisenstein | Genaue Untersuchung der unendlichen Doppelproducte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind, und der mit ihnen zusammenhängenden Doppelreihen[END_REF].

Let u be a complex number which is not an integer. Then the coset Z + u ⊂ C \ Z does not contain the zero element so all the terms of the series φ(u) = m∈Z+u 1 m are well defined, but the series does not converge absolutely. It is therefore necessary to specify an order of summation. Following Eisenstein, we define

φ(u) = lim t→+∞ m∈Z+u, |m|<t 1 m = 1 u + ∞ n=1 1 u + n + 1 u -n = 1 u + ∞ n=1 2u u 2 -n 2 .
The last series on the right converges absolutely. The function φ is odd and 1-periodic, hence φ( 12 ) = 0. Next, we consider the special value φ( 1 4 ) and we obtain

φ 1 4 = m∈Z+ 1 4 1 m = 4 ∈4Z+1 1 = 4 1 - 1 3 + 1 5 - 1 7 + 1 9 + . . . = 4 1 0 dx 1 + x 2 = π.
We are now ready to derive the fundamental property of the φ function, its addition formula. To this end, we let u, v, w be three complex numbers, none of which is a rational integer, such that u + v + w = 0. Then the equation p + q + r = 0

(1) with p ∈ Z + u, q ∈ Z + v, r ∈ Z + w has infinitely many solutions which can be obtained by letting p, q (or q, r or r, p) run independently. Since pqr = 0, Equation (1) becomes

1 pq + 1 qr + 1 rp = 0.
It is the starting point of Eisenstein's method, which he learnt from high school teacher Schellbach, to average this rational identity over all solutions of Equation (1). Since the resulting double series are conditionally convergent, we need again to pay attention to the ordering of the series. To this end, we choose three non-zero fixed real parameters α, β, γ such that α + β + γ = 0. Then αp -βq = γq -αr = βr -γp, which allows us to write

|αp-βq|<t 1 pq + |γq-αr|<t 1 qr + |βr-γp|<t 1 rp = 0.
It is easy to see that each of these series converges absolutely for every t > 0. In order to pass to the limit t → ∞, we need the following lemma :

Lemma 2.1.

lim t→∞   |αp-βq|<t 1 pq   = φ(u)φ(v) + π 2 sign α sign β.
For a generalization and a proof of this lemma, we refer to [START_REF] Sczech | Eisenstein group cocycles for GL n and values of L-functions[END_REF]Th.2]. As a corollary, we obtain

φ(u)φ(v) + φ(v)φ(w) + φ(w)φ(u) = π 2 , (2)
which is the addition formula for the function φ. To prove (2) without using Lemma 2.1, it is enough to show that the left side is independent of u, v because the specialization

u = v = 1 4 , w = -1 2 provides the value π 2 .
That was the approach taken by Eisenstein in his original proof of (2).

To study arithmetic applications, we eliminate the period π by introducing the function

c(u) = φ(u) φ(1/4) = φ(u)/π,
and its Cayley transform

e(u) = c(u) + i c(u) -i .
The addition formula (2) implies immediately

e(u)e(v) = e(u + v).
From here it follows easily that e(u) = e 2πiu and c(u) = cot(πu). Iterating the addition formula for the cotangent function,

c(u + v) = c(u)c(v) -1 c(u) + c(v) ,
we obtain the formula

c(nu) = U n (c(u)) V n (c(u)) , n = 1, 2, 3, 4, . . . (3) with polynomials U n , V n ∈ Z[t] given explicitly by U n (t) = Re (t + i) n , V n (t) = Im (t + i) n .
A formula of type (3) was called a "transformation formula" in the 19 th century. The elliptic analogs of (3) played a prominent role in the work of Kronecker. Taking u ∈ Q with denominator n > 1, we conclude that the numbers c(u) are the roots of V n (t) = 0, that is, c(u) is a real algebraic number whenever u is a non-integral rational number. One can refine this result and prove more precisely :

Proposition 2.2. The number c(u) is an algebraic integer iff n is not the power of an odd prime. If n = p k with an odd prime p and k > 0, then p c(u) is an algebraic integer. Moreover, c(u) is a unit iff neither n nor n/2 is a power of an odd prime.

Theorem 2.3 (Kronecker-Weber). The set of numbers c(Q \ Z) generates the real subfield of the maximal abelian extension of Q.

Kronecker was interested in generalizing these results to the case of elliptic functions with period lattice being an ideal in an imaginary quadratic field. One of the difficulties he was facing is that the addition formula for elliptic functions is in general algebraic and not rational like in the case of the cotangent function. To the best of our knowledge, the proof of the Jugendtraum as envisioned by Kronecker has never been completed 2 . Except for the case of imaginary quadratic fields and the case of the rational number field (both closely related to the work of Kronecker), we do not even know whether Hilbert's 12 th has a solution. This is partly related to a classical theorem of Weierstrass asserting that every meromorphic function with algebraic addition law is either elliptic, circular or rational.

Instead of the series defining φ(u), Kronecker preferred to study the more general series

φ(u, ξ) = ∞ n=-∞ e(-nξ) u + n = 2πi e(ξu) e(u) -1 , 0 < ξ < 1, (4)
again, with a complex number u which is not an integer. The introduction of the second variable ξ is very natural and suggested by Fourier analysis. Note that the limiting case

lim ξ=0 ξ>0 φ(u, ξ) = φ(u) -iπ
relates φ(u, ξ) to the Eisenstein function φ. The right side of Equation ( 4) is essentially the generating function for the Bernoulli polynomials. Expanding the left side into a power series in u, one obtains the Fourier expansion of the Bernoulli polynomials. The above proof of the addition formula for the cotangent function applies to φ(u, ξ) as well and yields addition formulas for the Bernoulli polynomials. Due to the factor e(-nξ) in the numerator, the convergence of this series is slightly better than that of the cotangent series, but is still conditional. A substantial part of the lecture notes is devoted to the study of the elliptic analog of (4), written in Kronecker's notation,

Ser (ξ, η, u, τ ) = lim N →+∞ lim M →+∞ N n=-N M m=-M e(-mξ + nη) u + nτ + m , (5)
where ξ, η is a pair of real variables and τ is a point in the upper half space. The complex variable u must be restricted to the complement of the lattice Z+τ Z in C. Writing u = στ +ρ as a linear combination of τ and 1 with real coefficients σ, ρ, this series can be viewed as a function of two pairs of real variables (ξ, η), (σ, ρ) and these are the two pairs of real variables referred to in the title of the lecture notes.

Various alternative ways to sum the conditionally convergent series (5) are discussed in the manuscript. It is a remarkable fact that in all cases, the value obtained for the sum is independent of the limiting process chosen. Kronecker's main result expresses these series in terms of Jacobi theta series. Let

ϑ(z, τ ) = ϑ 1 (z, τ ) = n∈Z+ 1 2 e n 2 τ 2 + n(z - 1 2 ) = 2 q 1 8 sin(πz) n≥1 (1 -q n )(1 -q n e(z))(1 -q n e(-z)),
with a complex variable z, a point τ in the upper half plane and q = e(τ ).

Theorem 2.4. (Kronecker). Suppose 0 < u < τ, 0 < ξ < 1. Then

Ser (ξ, η, u, τ ) = e(ξu) ϑ (0, τ )ϑ(u + η + ξτ, τ ) ϑ(u, τ )ϑ(η + ξτ, τ ) . ( 6
)
This result is reminiscent of the so called limit formula of Kronecker which is not discussed in the lecture notes but deserves to be stated here : let τ, τ be two complex numbers with τ > 0, τ < 0, and let 0 ≤ ξ, η < 1 be two real numbers, not both equal to zero. Writing u = η -ξτ, v = η -ξτ , the second limit formula is the identity

(τ -τ ) 2πi m,n e(mξ + nη) (mτ + n)(mτ + n) = -log ϑ(u, τ )ϑ(v, -τ ) η(τ )η(-τ ) -πi (u -v) 2 τ -τ , (7)
where η(τ ) refers to the Dedekind eta function, and the term (m, n) = (0, 0) needs to be excluded from the series. Again, convergence is only conditional so a specific order of summation like in (5) needs to be observed.

Historically, the notion of complex multiplication of elliptic functions appeared for the first time in the work of Abel. Pages 64-67 of Kronecker's lecture notes are devoted to the task of expressing the elliptic functions used by Abel in terms of the Kronecker series Ser(ξ, η, u, τ ). This is perhaps one of the highlights of the manuscript and deserves special attention. It is very likely that Kronecker's conception of the Jugendtraum was the result of a close study of the work of Abel.

Recent developments.

It would be wise to leave aside the complete discussion of the legacy of Eisenstein or Kronecker to serious historians. We offer instead a brief survey of several recent developments that feature Eisenstein's summation process and Kronecker's Theorem 2.4 for the series Ser (ξ, η, u, τ ).

Algebraicity and p-adic interpolation of Eisenstein-Kronecker numbers.

It was already observed by Weil in 1976 that the combination of the methods of Eisenstein and Kronecker gives a direct access to the Damerell's classical result (1971) on the algebraicity of values of L-functions attached to a Hecke Grössencharakter of an imaginary quadratic field.

Weil also anticipated that their methods would extend to the investigation of the p-adic properties of these algebraic numbers. In the following 10 years, the works of Manin-Visik, Katz and Yager among others provided the expected p-adic interpolation of this family of special values. To give a taste of the results in question, we wish to introduce a recent work by [START_REF] Bannai | Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers[END_REF], based on Kronecker's series, that enables a similar construction.

Our first task is to connect the series Ser to a series that includes a s-parameter pertaining to the style of Hecke. Let τ be a complex number in the upper half plane, and let A = τ π be the area of the fundamental domain of the lattice Λ = Z + τ Z divided by π. Let ψ be the character ψ(z) := e z-z A . For a ≥ 0 an integer, we introduce the Kronecker-Eisenstein series as

K * a (z, w, s, τ ) := * λ∈Λ (z + λ) a |z + λ| 2s ψ(zw), Re(s) > a/2 + 1,
where the * means that the summation excludes λ = -z if z ∈ Λ. It is a continuous function of the parameters z ∈ C \ Λ, w ∈ C, and it has meromorphic continuation to the whole s-plane, with possible poles only at s = 0 (if a = 0 and z ∈ Λ) and at s = 1 (if a = 0 and w ∈ Λ). Moreover, it satisfies a functional equation relating the value at s to the value at a + 1 -s. Write w = η + ξτ with real variables ξ, η, and abbreviate the central value K * 1 (z, w, 1, τ ) by K(z, w) when there is no ambiguity on the lattice. This specific function is related to Kronecker's series by the identity

K(z, w) = K * 1 (z, w, 1, τ ) = Ser (ξ,
:= ϑ (0, τ )ϑ(z + w, τ ) ϑ(z, τ )ϑ(w, τ ) (8)
denotes the meromorphic function appearing as the ratio in Equation ( 6). It will play a major role in the remainder of this text, so we name it the "Kronecker theta function", in agreement with the terminology in [BK]. Given a pair of integers a ≥ 0, b > 0 and z 0 , w 0 ∈ C, the Eisenstein-Kronecker numbers are defined as e * a,b (z 0 , w 0 , τ ) = K * a+b (z 0 , w 0 , b, τ ). When b > a + 2, these numbers include in particular the values of the absolutely convergent partial Hecke L-series

e * a,b (0, 0, τ ) = λ∈Z+τ Z λ a λ b .
They should be considered as elliptic analogs of the Bernoulli numbers. As such, the Eisenstein-Kronecker numbers can be packaged in a generating series that is the elliptic analog of the cotangent function and its relative φ(u, ξ). To obtain that nice two-variables generating series, it is enough to translate and slightly alter the Kronecker theta function in order to define Θ z 0 ,w 0 (z, w) = e -(z 0 +z)w 0 +wz 0 A Θ(z 0 + z, w 0 + w).

Its Laurent expansion around z = w = 0 displays exactly the collection of Eisenstein-Kronecker numbers.

Proposition 3.1. Fix z 0 , w 0 ∈ C. We have the following Laurent expansion near z = w = 0 :

Θ z 0 ,w 0 (z, w) = ψ(w 0 z 0 ) δ(z 0 ) z + δ(w 0 ) w + a≥0,b>0 (-1) a+b-1 e * a,b (z 0 , w 0 , τ ) a!A a z b-1 w a , (9)
where δ(u) = 1 if u ∈ Λ and 0 otherwise.

If in addition τ is a CM point and z 0 , w 0 are torsion points over the lattice Λ, then these coefficients are algebraic.

Theorem 3.2. Let Λ = Z + τ Z be lattice in C. Assume that the complex torus C/Λ has complex multiplication by the ring of integers of an imaginary quadratic field k, and possess a Weierstrass model E : y 2 = 4x 3 -g 2 x -g 3 defined over a number field F. Fix N > 1 an integer. For z 0 , w 0 two complex numbers such that N z 0 , N w 0 ∈ Λ, the Laurent expansion in (9) has coefficients in F (E[4N 2 ]). In particular, the rescaled Eisenstein-Kronecker numbers e * a,b (z 0 , w 0 , τ )/A a are algebraic.

We refer to [BK], Th. 1 and Cor. 2.11 for the proofs. The above construction enables Bannai and Kobayashi to recover Damerell's result along the way.

Let p be a prime number. Bernoulli numbers and Bernoulli polynomials satisfy a whole collection of congruences modulo powers of p, known as the "Kummer congruences". These congruences are incorporated in the construction of the Kubota-Leopoldt p-adic zeta function ζ p (s), s ∈ Z p , which interpolates the values of the Riemann zeta function at negative integers, as given by the Bernoulli numbers.

Similarly, Eisenstein-Kronecker numbers satisfy a collection of congruences that are the building blocks for the construction of two-variables p-adic L-functions. Since Bannai-Kobayashi also have a generating series at their disposal, they can interpolate the Eisenstein-Kronecker numbers p-adically, not only when p splits in k, like in the work of Katz (ordinary case), but also when p is inert in k (supersingular case).

It seems appropriate to remark in passing that in this p-adic setting, the Kronecker limit formula (7) also has a counterpart. It has been originally obtained by Katz (1976), and proved to be crucial in the study of Euler systems attached to elliptic units.

To complete this p-adic picture and get started with a recurring theme for sections to come, we would like to mention a generalization by Colmez-Schneps [CS] of the above construction of Manin-Visik and Katz. Colmez and Schneps consider the case where the Hecke character is attached to an extension F of degree n over the imaginary quadratic field k (the previous setting thus corresponds to n = 1). The field F might not necessary be a CM field. Building on the techniques of [Co], they can construct a p-adic L-function for F by interpolating the algebraic numbers arising from the Laurent expansion of certain linear combinations of products of n generating series of Kronecker's type, each of them being evaluated at torsion points over the lattice Λ. The juxtaposition of n copies of Proposition 3.1-Theorem 3.2 allows them to bootstrap the case n = 1 to arbitrary n ≥ 1 using their identity [START_REF] Colmez | p-adic interpolation of special values of Hecke L-functions[END_REF]Eq. (31)].

Periods of Hecke eigenforms.

Let (z 0 , w 0 ) be a fixed pair of N -torsion points over the lattice Λ. As functions of the τ variable, the modular forms e * a,b (z 0 , w 0 , τ ) are Eisenstein series for the principal congruence subgroup Γ(N ). In particular, the Laurent expansion (9) for the translated Kronecker theta function at z = w = 0 is a generating series for Eisenstein series of increasing weight and fixed level. Its decomposition under the action of the Hecke operators algebra thus possess only Eisenstein components. From this perspective, interesting new phenomenons start to appear when one considers a product of two Kronecker theta functions.

Such a product encodes all periods polynomials of modular forms of all weights, at least in the level one case3 . This is the content of the main result of Zagier's paper [START_REF] Zagier | Periods of modular forms and Jacobi theta functions[END_REF], which we now describe.

Let M k be the C-vector space of modular forms of weight k ≥ 4 on SL 2 (Z), and let S k ⊂ M k be the subspace of cusp forms, equipped with the Petersson scalar product (f, g) and its basis of normalised Hecke eigenforms B cusp k . The period polynomial attached to f ∈ S k is the polynomial of degree ≤ k -2 defined by

r f (X) = i∞ 0 f (τ )(τ -X) k-2 dτ.
The Eichler-Shimura-Manin theory implies that both maps f → r ev f and f → r od f assigning to f the even and the odd part of r f are both injective. Moreover, if f is a normalized Hecke eigenform, then the two-variables polynomial

R f (X, Y ) := r ev f (X)r od f (Y ) + r od f (X)r ev f (Y ) (2i) k-3 (f, f ) ∈ C[X, Y ] transforms under σ ∈ Gal(C/Q) as R σ(f ) = σ(R f )
, so R f has coefficients in the number field generated by the Fourier coefficients of f. As a consequence, for each integer k > 0, the finite sum

C cusp k (X, Y, τ ) := 1 (k -2)! f ∈B cusp k R f (X, Y )f (τ ), belongs to Q[X, Y ][[q]
]. Zagier starts to complete this cuspidal term by a contribution that arises from the Eisenstein series in M k , using the following convenient recipe. For any even k > 0, let B k be the usual k-th Bernoulli number, and let

E k (τ ) := 1 - 2k B k m≥1 ( d|m d k-1 ) q m
be the normalized weight k Eisenstein series for SL 2 (Z). Its pair of period polynomials in

X -1 Q[X] is defined by r od E k (X) = X k-2 -1, r ev E k (X) = k h=0 B h h! B k-h (k -h)! X h-1 ,
and they make up the contribution of Eisenstein series by the rule

C Eis k (X, Y, τ ) := -r od E k (X)r ev E k (Y ) + r ev E k (X)r od E k (Y ) E k (τ ).
The main identity of Zagier then establishes a remarkable closed formula for the generating series

C τ (X, Y, T ) := (X + Y )(XY -1) X 2 Y 2 T 2 + ∞ k=2 (C cusp k + C Eis k )T k-2 ,
which factorizes as a product of two Kronecker theta functions :

Theorem 3.3. [Za1] In (XY T ) -2 Q[X, Y ][[q, T ]] we have C τ (X, Y, T ) = Θ(XT , Y T ) Θ(T , -XY T )/ω 2 , ( 10
)
where ω = 2πi, T = T /ω.

Equation ( 10) shows that complete information on Hecke eigenforms of any desired weight for SL 2 (Z) and their period polynomial is encoded in the Laurent expansion at T = 0 of the right side, that consists of the product of two building blocks Kronecker theta functions. To further support that claim, Zagier explains in the sequel paper [START_REF] Zagier | Hecke operators and periods of modular forms[END_REF] how to deduce from Equation ( 10) an elementary proof of the Eichler-Selberg formula for traces of Hecke operators on SL 2 (Z).

The period polynomials satisfy a collection of linear relations under the action of SL 2 (Z). These cocycle relations are reflected using Equation (10) by relations satisfied by Kronecker's theta functions, e.g. [START_REF] Zagier | Periods of modular forms and Jacobi theta functions[END_REF]p. 461]. A typical example is

C τ (X, Y, T ) + C τ (1 - 1 X , Y, T X) + C τ ( 1 1 -X , Y, T (1 -X)) = 0, (11)
which is the counterpart of the classical relation for the period polynomial

r f | 1 + U + U 2 = 0, U = 1 -1 1 0 .
In the remainder of this paper, we explain how the method of Eisenstein-Schellbach, when properly modulated, is the adequate tool to produce systematically general (n -1)-cocycle relations for GL n (Z) involving products of n Kronecker theta functions, including Equation (11) as a very special case.

Trigonometric cocycles on GL n .

Let 0 < a < 1, 0 < b < 1 be rational numbers, and x, y ∈ C -Z be complex parameters. We use the shorthand 0 < {t} < 1 for the fractional part of a non-integral real number t. From a direct computation or a mild generalisation of Lemma 2.1, one deduces the following relation. It amounts to the addition formula for the function φ(u, ξ) : As pointed out by the second author in [START_REF] Sczech | Eisenstein group cocycles for GL 2 (Q) and values of L-functions in real quadratic fields[END_REF], this identity can naturally be recasted in terms of the cohomology of the group SL 2 (Z). The building blocks are products of two copies of the trigonometric function φ(u, ξ). Given to pairs u = (u 1 , u 2 ) and ξ = (ξ 1 , ξ 2 ) t , we set Φ(u, ξ) := φ(u 1 , ξ 1 )φ(u 2 , ξ 2 ).

To

any matrix A = a b c d ∈ SL 2 (Z) is associated σ = 1 a 0 c ∈ M 2 (Z). If c = 0 we define Ψ a b c d (u, ξ) = sign(c) µ∈Z 2 /σZ 2 Φ(uσ, σ -1 (µ + ξ)).
If c = 0 then Ψ(A)(u, ξ) = 0 by definition. The next proposition stands for the case n = 2 of a more general (n -1)-cocycle relation for the group GL n (Z) obtained in [START_REF] Sczech | Eisenstein group cocycles for GL n and values of L-functions[END_REF]Cor. p. 598].

Proposition 3.4. Let A, B ∈ SL 2 (Z) be two matrices. For any u in a dense open domain in C 2 and any non zero ξ ∈ Q 2 , the following inhomogenous 1-cocycle relation holds :

Ψ(AB)(u, ξ) -Ψ(A)(u, ξ) -Ψ(B)(uA, A -1 ξ) = 0.
The addition law (12) corresponds to the choice of matrices

A = 0 -1 1 0 , B = 1 1 -1 0 in this proposition.
The coefficients of the Laurent expansion at u = 0 of the general (n -1)-cocycle Ψ carry a great deal of arithmetic information. According to the main result of [START_REF] Sczech | Eisenstein group cocycles for GL n and values of L-functions[END_REF]Th. 1], the pairing of Ψ with a (n -1)-cycle built out of the abelian group generated by the fundamental units of a totally real number field F of degree n produces the values ζ F (k) at non-positive integers k of the Dedekind zeta function of F, and thereby establishes their rationality. This provides a new proof, deeply rooted in the Eisenstein-Schellbach method, of the Klingen-Siegel rationality result.

An integral avatar of the cocycle Ψ has later been introduced by Charollois and Dasgupta to study the integrality properties of the values ζ F (k), enabling them to deduce a new construction the p-adic L-functions of Cassou-Noguès and Deligne-Ribet.

Combining their cohomological construction with the recent work of Spiess, they also show in [CD] that the order of vanishing of these p-adic L-functions at s = 0 is at least equal to the expected one, as conjectured by Gross. This result was already known from Wiles' proof of the Iwasawa Main Conjecture.

On elliptic functions depending on four pairs of real variables.

Our goal in this last section is to propose a q-deformation of the trigonometric cocycle Ψ to construct an elliptic cocycle, where the role of the function φ(u, ξ) is now played by Kronecker's theta function. It will make it clear that the trigonometric relations we have encountered so far are specialization of elliptic ones when q → 0, i.e. τ → i∞.

Kronecker has not been given the opportunity to implement the Eisenstein-Schellbach method in his own elliptic investigations. Let's proceed now by fixing τ in the upper halfplane and first perform in Equation ( 12) the change of variables x ← nτ + x, y ← n τ + y.

We also aim to twist that equation by a pair of roots of unity e(nr)e(n s) with r, s ∈ Q, and then sum over n, n ∈ Z. We write x 0 = aτ + r, y 0 = bτ + s, q = e(τ ), so that the real and imaginary parts of x, x 0 , y, y 0 make up four pair of real variables. The first summand becomes S := e(xa)e(yb) n,n ∈Z e(nr)q na q n b e(n s) (q n e(x) -1)(q n e(y) -1) .

The assumption 0 < a < 1 ensures that 1 > |q a | > |q| and similarly for q b , so that this double series is absolutely convergent. The sum S naturally splits as a product, whose value is deduced from two consecutive uses Kronecker's Theorem 2.4 :

S = e(xa)e(yb) Θ(x, x 0 ) Θ(y, y 0 ).

A similar resummation process can be performed on the second and the third term of Eq. ( 12). After simplification by the common factor e(xa + yb), we obtain the identity Θ(x, x 0 )Θ(y, y 0 ) -Θ(x + y, x 0 )Θ(y, y 0 -x 0 ) -Θ(x + y, y 0 )Θ(x, x 0 -y 0 ) = 0, (13) which can be extended analytically to remove the restrictive assumptions made on the parameters. Equation ( 13) is just another form of the Riemann theta addition relation, also known as Fay trisecant identity. It implies simultaneously Eq. ( 12) when τ → i∞ and Eq. ( 11) under the choice of parameters x = XT , x 0 = Y T , y = -T , y 0 = XY T .

More generally, the resummation of the trigonometric (n -1)-cocycle Ψ gives rise to an elliptic (n -1)-cocycle which we name the Eisenstein-Kronecker cocycle κ on GL n (Z). One recovers Ψ from κ by letting τ → i∞. The coefficients of the Laurent expansion of κ at zero now display modular forms, essentially sums of products of n Eisenstein series of various weights and levels, that are members of a compatible p-adic family.

When paired with a (n -1)-cycle built out of the fundamental units of a totally real number field F of degree n, the elliptic cocycle κ produces a generating series for the pullbacks of Hecke-Eisenstein series over the Hilbert modular group SL 2 (O F ), whose constant terms are the values ζ F (k), k ≤ 0. These classical modular forms already played a prominent role in the original proof by Siegel of the Klingen-Siegel theorem. More details on the construction of the Eisenstein-Kronecker cocycle κ and its properties will be given in [Ch].

  e(xa)e(yb) (e(x) -1)(e(y) -1) -e((x + y)a)e(y{b -a}) (e(x + y) -1)(e(y) -1) -e((x + y)b)e(x{a -b}) (e(x + y) -1)(e(x) -

see the comment linked to interpretation c) in[HH].

A very recent preprint[CPZ] indicates that a similar result, slightly weaker, also holds for arbitrary squarefree level.


Michiel Postema
email: m.postema@erasmusmc.nl

Ayache Bouakaz

Nico De Jong

Noninvasive microbubble-based pressure measurements: a simulation study

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Abstract

This paper describes a noninvasive method to measure local hydrostatic pressures in fluid filled cavities. The method is based on the disappearance time of a gas bubble, as the disappearance time is related to the hydrostatic pressure. When a bubble shrinks, its response to ultrasound changes. From this response, the disappearance time, and with it the hydrostatic pressure, can be determined.

We investigated the applicability of the gases Ar, C 3 F 8 , Kr, N 2 , Ne, and SF 6 , based on their diffusive properties. For pressure measurements with a limited duration, e.g. 150 ms, Kr and Ar bubbles are most suitable, since they are most sensitive to pressure change. If there is also a limitation to bubble size, e.g. a maximum diameter of 6 µm, SF 6 is most suitable.

We present improvements of a method that correlates the duration of the decay of the fundamental ultrasound response to the hydrostatic overpressure. We propose to correlate the duration until subharmonic occurrence in combination with its decay, to hydrostatic overpressure, since the subharmonic decays more rapidly than the fundamental response. For a dissolving Ar gas bubble with an initial diameter of 14 µm, the overpressure can be determined 4 times as precise from the decay of the subharmonic response as from the decay of the fundamental response. Overpressures as small as 11 mmHg may be discriminated with this method.

Introduction

Local pressure measurements in cavities are widely used in medical diagnostics. Local pressure measurements in the heart are generally done by catheterization, causing pain and risk of infection [START_REF]Pulmonalarterienkatheter: Methodik und klinische Anwendung[END_REF]. We propose a renewed noninvasive method to measure pressure in cavities, based on the diffusion of free gas microbubbles and their interaction with ultrasound waves.

Epstein & Plesset [START_REF] Epstein | On the stability of gas bubbles in liquid-gas solutions[END_REF] and De Jong et al. [START_REF] Jong | Principles and recent developments in ultrasound contrast agents[END_REF] demonstrated the relation between the disappearance time of gas bubbles and the hydrostatic pressure applied. Because the sizes of gas bubbles change as a function of the hydrostatic pressure, the acoustic properties of the bubbles are affected. Based on this finding, a relation between bubble disappearance time, its acoustic response, and ambient pressure can be established. Bouakaz et al. gave an overview of pressure measurement methods using this relation, and studied one in vitro [START_REF] Bouakaz | Noninvasive measurement of the hydrostatic pressure in a fluid-filled cavity based on the disappearance time of micrometer-sized free gas bubbles[END_REF]. Instead of free gas bubbles, they inserted ultrasound contrast agent in their setup. Hard-shelled ultrasound contrast agent can act as a vehicle to carry gas to a region of interest. Upon insonification at sufficient acoustic pressure, the gas is released. This process is called sonic cracking [START_REF] Postema | Optically observed microbubble coalescence and collapse[END_REF].

Bouakaz et al. determined overpressures from the decay of the fundamental acoustic response from diffusing released air bubbles. Pressure differences of 50 mmHg could be distinguished theoretically and experimentally. In medical diagnostics a resolution lower than 50 mmHg is desirable. To improve the sensitivity of the measurement approach mentioned above, we investigated the use of different gases. Furthermore, we investigated subharmonics as a marker for half resonant bubble size, since the subharmonic response is more sensitive to bubble size change than the fundamental [START_REF] Palanchon | Subharmonic and ultraharmonic emissions for emboli detection and characterization[END_REF].

Shi et al. had suggested the use of subharmonics for noninvasive pressure measurements [START_REF] Shi | Pressure measurements using contrast microbubbles[END_REF]. They measured the scattering of encapsulated microbubbles over the pressure range 0-186 mmHg, and found a decrease of approximately 10 dB. It had previously been demonstrated theoretically that subharmonic generation from free gas bubbles and from ultrasound contrast agents requires a threshold insonifying pressure, which is minimal when microspheres are insonified at twice their resonance frequency [START_REF] Prosperetti | Application of the subharmonic threshold to the measurement of the damping of oscillating gas bubbles[END_REF][START_REF] Shankar | Subharmonic backscattering from ultrasound contrast agents[END_REF]. Palanchon et al. determined such thresholds with simulations and experiments for free microemboli [START_REF] Palanchon | Subharmonic and ultraharmonic emissions for emboli detection and characterization[END_REF].

Since subharmonics can be generated with the resonant bubble size corresponding to half the transmitted frequency, we propose to correlate the time until subharmonic occurrence, to the hydrostatic overpressure.

In this paper we present improvements of a previously published method that correlates the duration of the decay of the fundamental response to the hydrostatic overpressure [START_REF] Bouakaz | Noninvasive measurement of the hydrostatic pressure in a fluid-filled cavity based on the disappearance time of micrometer-sized free gas bubbles[END_REF]. To improve the method, we simulate the diffusive behavior of six gases, and suggest a qualitative measure for the applicability of a specific gas. We discuss the results with respect to boundary conditions and limitations, which hold for the ultimate in vivo situation. Furthermore, we simulate the scattering behavior of ultrasound-insonified diffusing gas bubbles. We investigate the sensitivity of the subharmonic response in comparison to the fundamental response.

Methods

Theoretical model

The change of gas bubble radius as a function of time, is given by [START_REF] Jong | Principles and recent developments in ultrasound contrast agents[END_REF][START_REF] Bouakaz | Noninvasive measurement of the hydrostatic pressure in a fluid-filled cavity based on the disappearance time of micrometer-sized free gas bubbles[END_REF]:

dR dt " DL ¨Ci C 0 ´1 ´2 σ R p 0 ´pov p 0 1 `4 σ 3 R p 0 ‹ ‹ ' ¨1 R `1 ? π D t ‹ ' , (1)
where C i C 0 is the ratio of the dissolved gas concentration to the saturation concentration (saturation ratio), D is the diffusion constant, L is the Ostwald coefficient, p 0 is the ambient pressure, p ov is the applied overpressure, R is the instantaneous bubble radius, t is the time starting (t " 0) when the bubble surface is exposed to the liquid surface, and σ is the surface tension. Eq. [START_REF]Pulmonalarterienkatheter: Methodik und klinische Anwendung[END_REF] shows that the disappearance of gas bubbles in a liquid medium is highly influenced by gas diffusion parameters and applied overpressure, and that the disappearance time of gas bubbles is shorter when the liquid medium is under pressure. When a gas bubble dissolves into a liquid medium, its acoustic response changes with its radius. The oscillating behavior of a gas bubble in a liquid, subjected to a sound field with a low acoustic pressure, was derived by [START_REF] Morgan | Experimental and theoretical evaluation of microbubble behavior: Effect of transmitted phase and bubble size[END_REF]. The changes in oscillating behavior of the dissolving gas bubble lead to changes in scattering cross sections [START_REF] Hilgenfeldt | Sound scattering and localized heat deposition of pulse-driven microbubbles[END_REF], and thus, the scattering behavior of an insonified, diffusing gas bubble can be calculated, dependent of the applied overpressure.

Simulations

First, we investigate the diffusive behavior of six gases by simulations, and define a qualitative measure for the applicability of a specific gas. Then we explore the sensitivity of the subharmonic acoustic response from diffusing gas bubbles, in comparison to the fundamental response.

The disappearance of free gas bubbles was simulated at hydrostatic overpressures between 0 and 200 mmHg, with gases Ar, C 3 F 8 , Kr, N 2 , Ne, and SF 6 . For our computations we used Matlab R (The MathWorks, Inc., Natick, MA) programs. The parameters used were published in [START_REF] Jong | Higher harmonics of vibrating gasfilled microspheres. Part one: simulations[END_REF]. The diffusion constants and the Ostwald coefficients were used for gas diffusing in water at room temperature [START_REF] Wilhelm | Low-pressure solubility of gases in liquid water[END_REF][START_REF] King | Measurement of the diffusion coefficient of sulfur hexafluoride in water[END_REF][START_REF] Chomas | Mechanisms of contrast agent destruction[END_REF][START_REF] Lide | CRC Handbook of Chemistry and Physics[END_REF][START_REF] Mayer | Myocardial contrast agents: recent advances and future directions[END_REF]. Saturation ratios C i C 0 " 0 were used, indicating that the gases are not present in water. From the diffusion curves, we computed the times it takes for bubbles to diffuse until they reach half their initial diameters. Half-size times were computed for gas bubbles varying in diameters from 5 to 20 µm, at ambient pressure, and at 20 mmHg overpressure. When applying an overpressure, the half-size time of a bubble is shorter than the half-size time at ambient pressure, t1 {2 . This difference in half-size times, ∆t1 {2 , is a qualitative measure for the applicability of a specific gas for the measurement of hydrostatic overpressures: The sensitivity of the bubble to pressure change improves when ∆t1 {2 increases.

Scattering cross-sections were calculated for the diffusing gas bubbles as a function of time [START_REF] Morgan | Experimental and theoretical evaluation of microbubble behavior: Effect of transmitted phase and bubble size[END_REF][START_REF] Hilgenfeldt | Sound scattering and localized heat deposition of pulse-driven microbubbles[END_REF][START_REF] Jong | Higher harmonics of vibrating gasfilled microspheres. Part one: simulations[END_REF], after filtering the fundamental and subharmonic responses from the acoustic bubble responses using a band-pass filter. The acoustic frequencies simulated, ranged from 0.5 MHz to 10 MHz.

Results and discussion

From our simulations it follows that diffusion duration, and with it ∆t1 {2 , increases with the initial bubble diameter I. However, bubble size is a limiting factor in the in vivo situation, as an encapsulated bubble may have to pass through narrow vessels before arriving in the cavity where the gas is released. The measurement duration is another limitation, especially for blood pressure measurements, since pressure changes occur within the cardiac cycle. A measurement of the systolic pressure in the left ventricle is limited to a duration of roughly 150 ms [START_REF]Physiology[END_REF]. Table 1 gives an overview of values ∆t1 {2 , for different limitations. If the half-size time is limited to 150 ms, and there is no limit to the bubble size, Kr and Ar gas bubbles result in the highest ∆t1 {2 . If the bubble diameter is limited to 6 µm, and there is no limit to the measurement duration, C 3 F 8 and SF 6 bubbles result in the highest ∆t1 {2 . If both limitations are combined, SF 6 and N 2 bubbles result in the highest ∆t1 {2 . Hence, the applicability of a specific gas is mainly determined by the limitations that apply. Shi et al. noted, that yet another limitation of our approach may lie in the fact that the disappearance times of the gas bubbles depend also on the gas content of the blood [START_REF] Shi | Pressure measurements using contrast microbubbles[END_REF]. We may overcome this limitation by choosing gases that are not present in the human body, and as such have a saturation ratio C i C 0 " 0. Figure 1 shows the fundamental and subharmonic scattered power of a dissolving m 14 µm Ar bubble, insonified at 1 and 2 MHz, respectively, as a function of time. The solid lines represent the situation at ambient pressure, the dashed at a 50 mmHg hydrostatic overpressure. The maximum of the fundamental is reached gradually, whereas the subharmonic has a rapid rise near double-resonant size. The subharmonic peaks decay with 40 dB in 8.2 ms, whereas the most rapid fundamental decay is only 20 dB in 15.2 ms.

Hence, the subharmonic response is more sensitive to diameter change than the fundamental response, indeed. Since overpressures of 50 mmHg could be distinguished from the decay of the fundamental response [START_REF] Bouakaz | Noninvasive measurement of the hydrostatic pressure in a fluid-filled cavity based on the disappearance time of micrometer-sized free gas bubbles[END_REF], we estimate that the scattering may have a ˘5 dB variation. With this variation, we computed diffusion time differences ∆t d for the decays observed in Figure 1: ∆t d " 7.6 ms for the fundamental response, and ∆t d " 2.0 ms for the subharmonic response. In our simulations, these diffusion time differences correspond to hydrostatic overpressures of 45 and 11 mmHg, respectively. Hence, the overpressure can be determined 4 times as precisely from the decay of the subharmonic response as from the decay of the fundamental response.

As an improvement of noninvasive pressure measurements, we propose to correlate the duration until the subharmonic peak in combination with the subharmonic decay, to hydrostatic overpressure. Evidently, precise knowledge of the initial bubble size is of importance. This might be established by generating subharmonics around the initial bubble size too.

Controlled gas release from a single bubble is currently under investigation in vitro with an ultrafast framing camera system [START_REF] Chin | Optical imaging of ultrasound contrast bubble motions at 25 million frames per second[END_REF].

Conclusions

For pressure measurements with a limited duration, e.g. 150 ms, Kr and Ar bubbles are most suitable, since they are most sensitive to pressure change.

If there is also a limitation to bubble size, e.g. a maximum diameter of 6 µm, SF 6 is most suitable.

When a diffusing gas bubble with known initial diameter is insonified, the duration until subharmonic occurrence in combination with its decay, is an indicator of the hydrostatic overpressure. The subharmonic decays more rapidly than the fundamental response. For a diffusing Ar gas bubble with an initial diameter of 14 µm, the overpressure can be determined 4 times as precisely from the decay of the subharmonic response as from the decay of the fundamental response. Overpressures as small as 11 mmHg may be discriminated with this method. Generating subharmonics may also be useful for verifying the initial bubble size.

Free gas bubbles can be delivered to cavities, and released by means of sonic cracking, which is currently under investigation.

  Noninvasive microbubble-based pressure measurements: a simulation study Michiel Postema a,b, * , Ayache Bouakaz a,b , Nico de Jong a,b,c a Erasmus MC-University Medical Center, Thoraxcentre, Department of Experimental Echocardiography, Dr. Molewaterplein 50, Rm. Ee-2302, 3000 DR Rotterdam, The Netherlands b Interuniversity Cardiology Institute of The Netherlands, Catharijnesingel 52, 3511 GC Utrecht, The Netherlands c University of Twente, Faculty of Science and Technology, Department of Applied Physics, Physics of Fluids, P.O. Box 217, Rm. CTW N-200, 7500 AE Enschede, The Netherlands

Figure 1 :

 1 Figure 1: Fundamental and subharmonic scattering cross-sections of a diffusing m 14 µm Ar bubble, insonified at 1 and 2 MHz, respectively, as a function of time, when applying hydrostatic overpressures of 0 and 50 mmHg.

Table 1 :

 1 Half-size time differences for different limitations.

	Gas	Limitation
						t1 {2 ď150 ms	Îď6
		t1 {2 ď 150 ms	I ď 6 µm		µm
		I	∆t1 {2	t1 {2	∆t1 {2	I	∆t1 {2
		pµmq pmsq	pmsq	pmsq	pµmq pmsq
	Ar	12	3.1	35	0.6	6.0	0.6
	C 3 F 8	1.0	0.8	ą 1000 ą 12	1.0	0.8
	Kr	14	3.2	24	0.4	6.0	0.4
	N 2	7.5	2.8	99	1.6	6.0	1.6
	Ne	8.5	2.7	72	1.2	6.0	1.2
	SF 6	3.5	1.7	453	7.2	3.5	1.7

Acknowledgements

This work has been supported by the Technology Foundation STW (RKG.5104).


M D Titirla¹
email: mtitirla@civil.auth.gr

Finite Element Simulation of the Device CAR1 on Braced Frames

Keywords: Experimental validation, Finite Element verification, Absorption Seismic Energy, Friction, Dynamic Explicit analysis

The developed device, has the codename CAR1, belongs to the passive energy dissipation systems, as it doesn't require external power to generate system control forces. It can be used on new or existing structures and can be easily adapted to the particular demands of structures. It can be installed in a variety of ways such as in single or X diagonal bracing in building frames. Moreover the use of this device may result in improving (i) the increase of stiffness (ii) the absorption of seismic energy, (iii) as well as control of the axial forces that are developed at the diagonal steel braces. The main part of CAR1 device is the groups of superimposed blades, which absorb seismic energy through simultaneous friction and yield. Firstly this paper discusses the experimental and numerical evaluation of the effectiveness of this steel device. Full scale CAR1 device was experimentally investigated under cyclic loading in Laboratory for Strength of Materials and Structures of Aristotle University of Thessaloniki. Finite Element Models of CAR1 device were developed and analyzed using the software ABAQUS, checking the credible documentation of the device. In addition, a numerically robust finite element model of a whole one storey structure is described, for highfidelity simulations of inelastic responses of device CAR1 on braced frame. Aim of this study is to compare the response of one storey structure with and without the existence of device CAR1 on diagonal braces.

Introduction

The safety of construction (existing or new) is one of the major priorities of engineering globally, because structures often subject to large and often devastating, for their viability, loadings. So, great interest is in the study of the innovations of the design and materials of construction that minimize the probability of failure of the structure in any charging. Steel concentrically braced frames have been used widely in high-seismic regions due to their efficiency in meeting lateral-load resisting requirements. Based on extensive research since the 1970's, it is well known that the cyclic loading performance of steel braces depend on their slenderness ratio and on the width-to-thickness ratio of their cross sectional elements, and that adequate detailing of the bracing connection is critical to avoid premature fracture at the end of the brace. Braced frame systems are presently being designed to satisfy performance-based seismic design criteria [START_REF]FEMA 356: Pre-standard and commentary for the seismic rehabilitation of buildings[END_REF][START_REF]AISC 341-10: Seismic provisions for structural steel buildings[END_REF]. In terms of analysis capabilities, researchers have proposed methodologies to predict the occurrence of fracture from cumulative damage and to exhibit significant ductility in life safety and collapse prevention limit states, which are governed by inelastic post-buckling and tensile yielding behaviors of the brace elements. There are essentially two main orientations in order to protect braced elements, first one is to provide the structural system in order to avoid unexpected premature failure modes (mid-length or connections) and the second is to be incorporated in braces a passive energy dissipation devices. Buckling restrained braced frames (BRBFs) for seismic load resistance have been widely used in recent years because it yields under both tension and compression without significant buckling [START_REF] Wada | Passive damping technology for buildings in Japan[END_REF][START_REF] Xie | State of arte of buckling-restrained braces in Asia[END_REF][START_REF] Hoveidae | Overall buckling behavior of all-steel buckling restrained braces[END_REF]. Others researchers create numerical models to approach brace elements. Numerical models can be classified in three categories, the phenomenological models [START_REF] Zayas | Cyclic inelastic buckling of tubular steel braces[END_REF][START_REF] Ikeda | Phenomenological modeling of steel braces under cyclic loading[END_REF][START_REF] Khatib | Seismic behavior of concentrically braced steel frames[END_REF], the beam-column Finite elements models [START_REF] Ikeda | Cyclic response of steel braces[END_REF][START_REF] Jin | Inelastic cyclic model for steel braces[END_REF][START_REF] Lee | Inelastic buckling behavior of steel members under reversed cyclic loading[END_REF] and the 3-D Finite elements models [START_REF] Lotfollahi | Inelastic buckling simulation of steel braces through explicit dynamic analyses[END_REF][START_REF] Lotfollahi | A validated finite element procedure for buckling simulation of diagonally braced moment resisting frames[END_REF][START_REF] Lumpkin | Investigation of the seismic response of three-story special concentrically braced frames[END_REF][START_REF] Nip | Cyclic testing and numerical modelling of carbon steel and stainless steel tubular bracing members[END_REF][START_REF] Yoo | Analytical performance simulation of special concentrically braced frames[END_REF][START_REF] Yoo | Influence of connection design parameters on the seismic performance of braced frames[END_REF]. On the other hand, passive energy dissipation devices such as visco-elastic dampers, metallic dampers and friction dampers have widely been used to reduce the dynamic response of civil engineering structures subjected to seismic loads [START_REF] Aiken | Comparative study of four passive energy dissipation systems[END_REF][START_REF] Soong | Supplemental energy dissipation: state-of-the-art and state-of-thepractice[END_REF][START_REF] Symans | Energy dissipation systems for seismic applications: Current practice and recent developments[END_REF] and can easily replaced or repaired. Their effectiveness for seismic design of building structures is attributed to minimizing structural damages by absorbing the structural vibratory energy and by dissipating it through their inherent hysteresis behavior. So, several of these devices have been selected for seismic strengthening of existing or new buildings in the US, Canada and Japan [START_REF] Pall | Friction-Dampers for seismic control of Concordia University Library Building[END_REF][START_REF] Makris | Spring-Viscous Damper Systems for Combined Seismic and Vibration Isolation[END_REF][START_REF] Μartinez-Romero | Experiences on the Use of Supplemental Energy Dissipators on Building Structures[END_REF]. In order to demonstrate the effectiveness of the devices, many passive energy dissipation systems were studied in experimental research [START_REF] Whittaker | Earthquake Simulator Testing of Steel Plate Added Damping and Stiffness Elements[END_REF][START_REF] Anagnostides | Development and applications of energy absortion devices based on friction[END_REF][START_REF] Aiken | Testing of passive energy dissipation systems[END_REF][START_REF] Papadopoulos | Experimental investigation of a new steel friction device with link element foe seismic strengthening of structures[END_REF] or in numerical research [START_REF] Pall | Seismic response of friction damped braced frames[END_REF][START_REF] Papadopoulos | New nonlinear anti-seismic steel device for the increasing the seismic capacity of multi-storey reinforced concrete frames[END_REF][START_REF] Ramirez | Numerical Simulation and Design of Friction-Damped Steel Frame Structures damped[END_REF]. The Finite Element Method (FEM) has become the most popular method in both research and industrial numerical simulations, as it takes into consideration material laws, contact interface conditions and others parameters, which lead to the exact response of the device. Several algorithms, with different computational costs, are implemented in the finite codes, such as ABAQUS [START_REF]Analysis User's Manual Volume IV[END_REF], which is commonly used software for finite element analysis. Comparison of numerical results with the same experimental one is very useful and necessary as it provides the possibility to researchers to study the behavior of their devices more widely [START_REF] Vasdravellis | Finite element models and cyclic behavior of selfcentering steel post-tensioned connections with web hourglass pins[END_REF][START_REF] Manos | Numerical simulation of the shear behaviour of reinforced concrete rectangular beam specimens with or without FRP-strip shear reinforcement[END_REF]. The calibrated FEM models are used to conduct a series of simulations to study the effect of different parameters. In this way, results come out that are harder to obtain experimentally. In the present paper, a numerically robust finite element model is described, which is based on explicit time-stepping, for high-fidelity simulations of inelastic responses of device CAR1 on braced frame. The effectiveness of the investigated device was recently developed at the Laboratory of Strength of Materials and Structures of Aristotle University of Thessaloniki. Aim of this study is to compare the response of one storey structure with and without the existence of device CAR1 on diagonal braces.

Study of the individual device CAR1

A short description:

The developed device has the codename CAR1 and belongs to the passive energy dissipation system, as it doesn't require external power to generate system control forces. This device proposed by Papadopoulos et al. [START_REF] Papadopoulos | A new seismic energy absorption device through simultaneously yield and friction used for the protection of structures[END_REF] and it consists of 4 main elements, as illustrated in Figure 1. Device CAR1 has the advantage to (i) provide additional stiffness as well as (ii) absorption of seismic energy, through yield and friction, (iii) provision of control of the axial forces that are developed at the diagonal steel rods and last but not least the ability to retain the plastic displacements to a desired level, due to the restrain bolt. Energy dissipation is provided by inelastic bending of superimposed blades. Moreover, device CAR1 can be used on new or existing structures and can easily be adapted to the particular demands of structures. However, it can be installed in a variety of ways which include using them in single diagonal braces or in X braces (Figure 2) and in accordance with the requirements of each construction, it can be used one or more devices.

Experimental set up:

A standard test has been carried out in order to establish the basic material properties of the superimposed blades (Figure 3). These experimentally derived material properties were utilized in the subsequent numerical study.

(α) (β) (γ)
Young Modulus E (Gpa) 203.6

Yield Strength F y (Mpa) 220

Maximum Strength (MPa) 290

(α) (β) (γ)

Finite Element Modeling:

The general purpose FE software ABAQUS was employed to generate FE models to simulate numerically the behavior of the device CAR1. It was selected to use an explicit dynamic solver because this allows the definition of very general contact conditions for complicated contact problems, without generating numerical difficulties. The explicit dynamics analysis procedure is based upon the implementation of an explicit integration rule together with the use of diagonal ("lumped") element mass matrices.

To the comparison with the Standard, the explicit dynamic solver is computationally inefficient for quasi-static problems if real time is used, because the time needed to finish an analysis is proportional to its duration. However, it is often possible to scale the real time to a very small time period if the response of the structure remains basically static. According to classical dynamic theory, when a dynamic system is subjected to a linearly rising load, its response can be approximately treated as static if the duration of the loading stage is large compared to the natural period of the system. For solving this problem, check the ratio of kinetic to internal energy can be used to check if the structure has failed and the analysis is continuing simply as dynamic motion. It is stated in the ABAQUS/Explicit manual [START_REF]Analysis User's Manual Volume IV[END_REF] that the procedure is quasi-static if the ratio of the kinetic energy to the internal energy is less than 2%. Any responses which have an energy ratio larger than this should be treated as dynamic and removed from the results.

The FEM model geometry reproduced the actual geometry of the tests set up of the device CAR1 to characterize the behavior of the device. The geometry of FE model was reproduced in full detail (Figure 5). Several simulations were conducted to identify the best meshing. For the explicit method, blades and interior shaft are meshed using 3D reduced integration solid element C3D8R (eight-node bricks), while exterior tube is meshed using 3D solid element C3D4 (four-node tetrahedron) available in ABAQUS. Normally, a higher mesh density provides for higher accuracy but also increases the computational time without improving substantially the accuracy of the results, therefore, a trade-off between time and accuracy becomes crucial [START_REF] Doudoumis | Finite element modelling and investigation of the behaviour of elastic infilled frames under monotonic loading[END_REF].

The final mesh has 8126 elements and it resulted in a solution that correlated with the experimental results. The uniaxial stress-strain relation of the blades, exterior tube and interior shaft are modeled as elastic with Young's modulus (Es) and Poisson's ratio (v) of which typical values are 200 GPa and 0.3, respectively. Plastic behavior are defined in a tabular form, including yield stress and corresponding plastic strain. The experimentally obtained stress (σ nom )-strain (ε nom ) curves for the blades was converted into the true stress (or Cauchy) (σtrue)-logarithic plastic strain ( ) format according to Eq. 1 and 2 and utilized to define the material response.

(1)

The surface-to-surface contact formulation technique with small sliding between the contacting surfaces was chosen. The contact definition includes the specification of two surfaces, one acting as the ''master'' surface and the other as the ''slave'' surface. The contact algorithm searches whether the nodes of the slave surface are in contact with the nodes of the master surface and enforces contact conditions in an average sense over a region of slave nodes using a Lagrange multiplier formulation [START_REF]Analysis User's Manual Volume IV[END_REF]. A friction coefficient equal to 0.2 [START_REF]Design of steel structures. Part 1-8: design of joints[END_REF] was assumed between the contacting surfaces. A flowchart for carrying out the FEM analysis procedure is presented in Figure 6. Figure 7 plots the force versus relevant displacement from FEM analyses along with the experimental hysteresis. Blue lines illustrate hysteresis loops of experiments, while green lines shows hysteresis loops of Finite Element Models. The predicted values for the load and displacement are in very good agreement with the corresponding experimental ones. The comparisons between the FEM analyses and experiments show that the proposed FEM model is capable of reproducing the inelastic response of the device CAR1. Therefore, it is a reliable tool for the simulation of the hysteretic behavior of the device CAR1 and can be used to contact further studies to investigate the effect of various parameters. In addition, the area within a hysteresis loop is equivalent to the amount of seismic energy that the device is dissipating. Since the shape and consistency of the hysteresis loops, observe the device's ability to absorb CAR1 seismic energy, whereas will not break during the cyclic loading.

-60.00 In addition, the numerical deformed shapes are compared with the corresponding experimental ones for relevant movement Uz=±5mm in Figure 8.

Study of one storey structure with and without the existence of device CAR1on diagonal braces

A one storey reinforced concrete structure (Figure 9) was chosen to be studied. It has a height of 3m and length equal to 4.5m. The horizontal elements are beams with dimensions in plan 25x50cm and the vertical are columns with dimensions in plan 35x35cm. Structure was modeled and analyzed in SAP 2000 ver. 11.0.3 [START_REF]SAP 2000 nonlinear version 11.0.3 User's Reference Manual[END_REF] in order to define floor's displacement drifts for seismic performance "Life Safety" (drift=1.6%) and "Collapse" (drift=2.1%). Columns and beams were modeled by frame elements. As it is drawn, the maximum horizontal displacement was chosen equal to 6cm (drift=2.0%), smaller than the collapse displacement (6.3cm). Also, the 2004 NEHRP provisions [START_REF] Bssc | NEHRP recommended provisions for seismic regulations for new buildings and other structures[END_REF] allow the design of buildings with passive damping systems to experience controlled inelastic deformations associated with typical design drifts limits, e.g. a 2% drift limit. For this horizontal displacement, both braced structures ((i) with diagonal brace and (ii) with diagonal brace and CAR1 device) will compare in software ABAQUS. Figure 12 shows the distribution of horizontal displacement around x-x axis at the end of analysis. In Braced Structure, diagonal brace fracture especially at the middle length of diagonal, while in CAR1-Braced Structure the brace remains un-deformed without plastic hinges. Figures 13 and14 show the peak plastic strains at the end of the analysis. In braced structure, maximum plastic strain observed at the middle length of diagonal brace (fracture point), while in CAR1-Braced Structure at the superimposed blades, which are easily be replaced with minimum cost. As a result, using CAR1 device on diagonal brace, the fracture life of brace is increased. The system exhibited uniform energy absorption with more stability, as strength and maximum deformation of the system increased considerably.

LS C

Details

Fracture of brace

Conclusions

In the present paper, an anti-seismic steel device (with code name CAR1) for seismic strengthening of existing or new buildings, which was recently developed at the Laboratory of Strength of Materials and Structures of Aristotle University of Thessaloniki, is studied experimental. A detailed nonlinear finite element model (FEM) was also developed. This model was calibrated against experimental results and used to explain the response of the device CAR1. In addition, a numerically robust finite element model of a whole one storey structure is described, for high-fidelity simulations of inelastic responses of device CAR1 on braced frame.

Based on the findings of this paper, the following conclusions are drawn:

1. Device CAR1 is a reliable energy dissipated device, which can be used on new or existing structures and minimize the probability of failure of the structure in any charging. 2. The developed nonlinear FEM models can be reliable used to access the behavior of the proposed anti-seismic steel device CAR1 as they are capable to trace the hysteretic behavior and predict the deformed shape of the device with good accuracy. 3. Based on the shape and consistency of the hysteresis loops, it is recommended seismic energy, whereas will not break during repeated cyclic loading.

4. The calibrated FEM model permits a thorough investigation of the stress state in the blades and helps to identify all possible local failures 5. Using CAR1 device on diagonal brace, the fracture life of brace is increased. The system exhibited uniform energy absorption with more stability, as strength and maximum deformation of the system increased considerably.

Figure 1 :

 1 Figure 1: The investigated device CAR1.

Figure 2 :

 2 Figure 2: Possible positions of the device CAR1, incorporated in steel diagonal braces.

Figure 3 :Figure 4 :

 34 Figure 3: A standard test in order to establish the basic properties of the superimposed blades.

Figure 5 :

 5 Figure 5: The FEM model used for the device CAR1 in software ABAQUS.

START

  Establish the basic material properties (experimentally stress-strain curves are converted into the true stress -strain curves) Develop (or re-adjust) the 3D-FEM (Mesh/Element Type/Boundary Conditions/Loading/Contact Surfaces/Friction) Run monotonic Static analyses to validate/verify 3D-Dynamic analyses to calibrate (quasi-static) explicit solution Check the quasi-static response criterion (Kinetic energy <2% input energy)Check the Energy Balance (Ε tot = E input -E output = 0

Figure 6 :

 6 Figure 6: Flowchart in order to develop the Finite Element Model of Device CAR1 in ABAQUS.

Figure 7 :

 7 Figure 7: Comparison of the experimental and the numerical force-displacement hysteresis of the device CAR1

Figure 8 :

 8 Figure 8: Distribution of deformed shapes (values in m)

Figure 9 :

 9 Figure 9: The longitudinal section of structure.

Figure 10 :Figure 11 :

 1011 Figure 10: Deformed shape in SAP2000

Figure 12 :

 12 Figure 12: Deformed model at end of analysis, (i) Braced Structure, (ii) CAR1-Braced Structure (values in m).

Figure 13 :

 13 Figure 13: Deformed shape and maximum plastic strain at the end of analysis.

Figure 14 :

 14 Figure 14: Maximum plastic strain in superimposed blades and diagonal brace at the end of analysis for CAR1-Braced Structure.

Acknowledgments

I would like to sincerely thank my supervisor, Papadopoulo Paniko (Assistant Professor at Aristotle University of Thessaloniki), for his valuable support.