Differential Geometry applied to Acoustics Non Linear Propagation in Reissner Beams : an integrable system?

07/11/2017
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Although acoustics is one of the disciplines of mechanics, its ”geometrization” is still limited to a few areas. As shown in the work on nonlinear propagation in Reissner beams, it seems that an interpretation of the theories of acoustics through the concepts of differential geometry can help to address the non-linear phenomena in their intrinsic qualities. More precisely, the dynamics of the Reissner beam is formulated as a map over spacetime with values in a nonlinear manifold (a Lie group). Fortunatly, this multi-symplectic approach can be related to the study of harmonic maps for which two dimensional cases can be solved exactly. It allows us to identify, among the family of problems, a particular case where the system is completely integrable. Among almost explicit solutions of this fully nonlinear problem, it is tempting to identify solitons, and to test the known numerical methods on these solutions.

Differential Geometry applied to Acoustics Non Linear Propagation in Reissner Beams : an integrable system?

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application/pdf Differential Geometry applied to Acoustics Non Linear Propagation in Reissner Beams : an integrable system? Frédéric Hélein, Joël Bensoam, Pierre Carré
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contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
- Accès libre pour les ayants-droit

Although acoustics is one of the disciplines of mechanics, its ”geometrization” is still limited to a few areas. As shown in the work on nonlinear propagation in Reissner beams, it seems that an interpretation of the theories of acoustics through the concepts of differential geometry can help to address the non-linear phenomena in their intrinsic qualities. More precisely, the dynamics of the Reissner beam is formulated as a map over spacetime with values in a nonlinear manifold (a Lie group). Fortunatly, this multi-symplectic approach can be related to the study of harmonic maps for which two dimensional cases can be solved exactly. It allows us to identify, among the family of problems, a particular case where the system is completely integrable. Among almost explicit solutions of this fully nonlinear problem, it is tempting to identify solitons, and to test the known numerical methods on these solutions.
Differential Geometry applied to Acoustics Non Linear Propagation in Reissner Beams : an integrable system?
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Differential Geometry applied to Acoustics Non Linear Propagation in Reissner Beams : an integrable system? Frédéric Hélein1 , Joël Bensoam2 , and Pierre Carré2 1 IMJ-PRG, Institut de Mathématiques de Jussieu - Paris Rive Gauche (UMR 7586) 2 Ircam, centre G. Pompidou, CNRS UMR 9912, S3AM, 1 place I. Stravinsky 75004 Paris, France Abstract. Although acoustics is one of the disciplines of mechanics, its ”geometrization” is still limited to a few areas. As shown in the work on nonlinear propagation in Reissner beams, it seems that an inter- pretation of the theories of acoustics through the concepts of differential geometry can help to address the non-linear phenomena in their intrinsic qualities. More precisely, the dynamics of the Reissner beam is formulated as a map over spacetime with values in a nonlinear manifold (a Lie group). Fortunatly, this multi-symplectic approach can be related to the study of harmonic maps for which two dimensional cases can be solved exactly. It allows us to identify, among the family of problems, a particular case where the system is completely integrable. Among almost explicit solutions of this fully nonlinear problem, it is tempting to identify solitons, and to test the known numerical methods on these solutions. 1 Introduction The Reissner beam is one of the simplest acoustical system that can be treated in the context of mechanics with symmetry. A Lie group is a mathematical construction that handle the symmetry but it is also a manifold on which a motion can take place. As emphasized by Arnold [1], physical motions of symmetric systems governed by the variational principle of least action correspond to geodesic motions on the corresponding group G. For general problems (wave propagation, field theory), two different geometric approaches are basi- cally available. The first approach, called the ”dynamical” approach, uses, as its main ingredient, an infinite dimensional manifold as configuration space (TQ). The reduction techniques developed in the dynamical framework have been studied thoroughly in the literature (see for example [24] and the references therein cited), but it presents the difficulty to handle geodesic curves in an infinite dimensional function space. As an alternative, the covariant formulation allows to consider a finite dimensional configuration space (the dimension of the symmetry group itself in our case). This can be achieved by increasing the number of independent variables since the validity of the calculus of variations and of the Noether’s theorem is not limited to the previous one-variable setting. Although its roots go back to De Donder [25], Weyl [26], Caratheodory [27], after J. M. Souriau in the seventies [28], the classical field theory has been only well understood in the late 20th century (see for example [29] for an extension from symplectic to multisym- plectic form). It is therefore not surprising that, in this covariant or jet formulation setting, the geometric constructions needed for reduction have been presented even more recently. In this context, the multi-symplectic form is obtained from the differential of the Cartan-Poincaré n- form, and is crucial to give rise to an Hamiltonian framework (Lie-Poisson Schouten-Nijenhuis (SN) brack- ets [30]). The derivation of the conserved quantities from the symmetries of the Lie group is described by a moment map that is no longer a function but must be defined, more generally, as a Noether’s current. This form is the interior product of the Poincaré-Cartan form by the fundamental vector field of the Lie group and leads to the dynamic equations of the problem. To obtain a well-posed problem, a zero-curvature equation (also know as the Maurer-Cartan equation) must be added to the formulation. The multi-symplectic approach is developed in the first section through the Reissner’s beam model. Inspired from [31], it turns out that, under some assumptions, this system is a completely integrable one. This is done, in the next section, by relating the formulation to the study of harmonic maps (also know under the name of chiral fields in Theoretical Physics and Mathematical Physics) for which the two dimensional case can be solved exactly. In Mathematical Physics this was known for the non linear σ-model since the seventies but it was the Russian school in integrable systems who made an exhaustive study of the principal chiral model (chiral fields with values in a Lie group). 2 Nonlinear model for Reissner Beam 2.1 Reissner kinematics A beam of length L, with cross-sectional area A and mass per unit volume ρ is considered. Following the Reissner kinematics, each section of the beam is supposed to be a rigid body. The beam configuration can be described by a position r(s, t) and a rotation R(s, t) of each section. The coordinate s corresponds to the position of the section in the reference configuration Σ0 (see figure 1). Fig. 1: Reference and current configuration of a beam. Each section, located at position s in the reference configuration Σ0, is parametrized by a translation r(s, t) and a rotation R(s, t) ∈ S O3 in the current configuration Σt. 2.2 Lie group configuration space Any material point M of the beam which is located at x(s, 0) = r(s, 0) + w0 = sE1 + w0 in the reference configuration (t = 0) have a new position (at time t) x(s, t) = r(s, t) + R(s, t)w0. In other words, the current configuration of the beam Σt is completely described by a map x(s, t) 1 ! = R(s, t) r(s, t) 0 1 ! | {z } H(s,t) w0 1 ! , R ∈ S O(3), r ∈ R3 , (1) where the matrix H(s, t) is an element of the Lie group S E(3) = S O(3) × R3 , where S O(3) is the group of all 3 × 3 orthogonal matrices with determinant 1 (rotation in R3 ). As a consequence, to any motion of the beam a function H(s, t) of the (scalar) independent variables s and t can be associated. Given some boundary conditions, among all such motions, only a few correspond to physical ones. What are the physical constraints that such motions are subjected to? In order to formulate those constraints the definition of the Lie algebra is helpful. To every Lie group G, we can associate a Lie algebra g, whose underlying vector space is the tangent space of G at the identity element, which completely captures the local structure of the group. Concretely, the tangent vectors, ∂sH and ∂tH, to the group S E(3) at the point H, are lifted to the tangent space at the identity e of the group. The definition in general is somewhat technical3 , but in the case of matrix groups this process is simply a multiplication by the inverse matrix H−1 . This operation gives rise to definition of two left invariant vector fields in g = se(3) ˆ ǫc(s, t) = H−1 (s, t)∂sH(s, t) (2) χ̂c(s, t) = H−1 (s, t)∂tH(s, t), (3) which describe the deformations and the velocities of the beam. Assuming a linear stress-strain relation, those definitions allow to define a reduced Lagrangian by the difference of kinetic and potential energy with Ec(χc) = Z L 0 1 2 χT c Jχcds, (4) Ep(ǫc) = Z L 0 1 2 (ǫc − ǫ0)T C(ǫc − ǫ0)ds, (5) where ǫ̂0 = H−1 (s, 0)∂sH(s, 0) correspond to the deformation of the initial configuration and J and C are matrix of inertia and Hooke tensor respectively, which are expressed by J = Jr 0 0 Jd ! , Jr =           I1 0 0 0 I2 0 0 0 I3           , Jd =           m 0 0 0 m 0 0 0 m           (6) C = Cr 0 0 Cd ! , Cr =           GIρ 0 0 0 EIa 0 0 0 EIa           , Cd =           EA 0 0 0 GA 0 0 0 GA           (7) 3 In the literature, one can find the expression dLg−1 (ġ) where dL stands for the differential of the left translation L by an element of G defined by Lg : G → G h → h ◦ g. where Jr, m, Iρ and Ia are respectively the inertial tensor, the mass, the polar momentum of inertia and the axial moment of inertia of a section, and with E, G and A the Young modulus, shear coefficient and cross-sectional area respectively. The reduced Lagrangian density 2-form yields ℓ = l(χL, ǫL)ω = 1 2  χT L JχL − (ǫL − ǫ0)T C(ǫL − ǫ0)  ds ∧ dt. 2.3 Equations of motion Applying the Hamilton principle to the left invariant Lagrangian l leads to the Euler-Poincaré equation ∂tπc − ad∗ χc πc = ∂s(σc − σ0) − ad∗ ǫc (σc − σ0), (8) where πc = Jχc and σc = Cǫc, (see for example [3], [4] or [5] for details). In order to obtain a well-posed problem, the compatibility condition, obtained by differentiating (2) and (3) ∂sχc − ∂tǫc = adχc ǫc, (9) must be added to the equation of motion. It should be noted that the operators ad and ad∗ in eq. (8) ad∗ (ω,v)(m, p) = (m × ω + p × v, p × ω) (10) ad(ω1,v1)(ω2, v2) = (ω1 × ω2, ω1 × v2 − ω2 × v1), (11) depend only on the group S E(3) and not on the choice of the particular ”metric” L that has been chosen to described the physical problem [6]. Equations (8) and (9) are written in material (or left invariant) form (c subscript). Spatial (or right invari- ant ) form exist also. In this case, spatial variables (s subscript) are introduced by ˆ ǫs(s, t) = ∂sH(s, t)H−1 (s, t) (12) χ̂s(s, t) = ∂tH(s, t)H−1 (s, t) (13) and (8) leads to the conservation law [18] ∂tπs = ∂s(σs − σ0) (14) where πs = Ad∗ H−1 πc and σs = Ad∗ H−1 σc. The Ad∗ map for S E(3) is Ad∗ H−1 (m, p) = (Rm + r × Rp, Rp). (15) Compatibility condition (9) becomes ∂sχs − ∂tǫs = adǫs χs. (16) Equations (8) and (9) (or alternatively ( 14) and (16)) provide the exact non linear Reissner beam model and can be used to handle the behavior of the beam if the large displacements are taking into account. Notations and assumptions vary so much in the literature, it is often difficult to recognize this model (see for example [7] for a formulation using quaternions). However, this generic statement is used to classify publications according to three axes. In the first one, the geometrically exact beam model is the basis for numerical formulations. Starting with the work of Simo [2], special attention is focused on energy and mo- mentum conserving algorithms [8], [9]. Numerical solutions for planar motion are also investigated in [10]. Even, in some special sub-cases (namely where the longitudinal variables do not appear) the non-linear beam model gives rise to linear equations which can be solved by analytical methods [11]. Secondly, much of the literature is also devoted to the so-called Kirchhoff’s rod model. In this case, shear strain is not taken into account along a thin rod (i.e., its cross-section radius is much smaller than its length and its curvature at all points). In this approximation cross-sections are perpendicular to the central axis of the filament and the rotation matrix is given by the Frenet-Serret frame. (see [12] , [14], [15], for example). In that context an interesting geometric correspondence between Kirchhoff rod and Lagrange top can be made [13]. Finally, if only rigid motion is investigated, ( i.e. if the spatial dependence in (8) is canceled: ∂s ≡ 0) the so-called underwater vehicle4 model is obtained. In absence of exterior force and torque, the equation of motion for a rigid body in an ideal fluid become more simply [16], [17] ∂tπc = ad∗ χc πc, that is        ṁ = m × ω + p × v ṗ = n × ω (17) In this simpler form, a geometric interpretation is easier. The solution of the equation of motion mentioned above, if it exists, should be interpreted as a geodesic of the group S E(3) endowed with a non-canonical left invariant metric J. To accomplish the correspondence between the Euler-Poincaré’s equation and geodesic equation the historical definition of the covariant derivative is exposed in the next section. 3 Comparison with integrable systems 3.1 The zero-curvature formulation I turns out that, under some assumptions, the previous system is a completely integrable one. Our dis- cussion is inspired from [31]. In the following we consider for simplicity the previous system on an infinite 2-dimensional space-time R2 . We also make the hypothesis that the tensors J and C are proportional to the identity, so that they can be written JI and CI respectively, the Ad operator and the matrix product commutes, and the relation between πs and χs simply becomes : πs = Ad∗ H−1 (JAdH−1 χs) = Jχs (18) In the same way we have σs = Cǫs. Without loss of generality, J and C are taken equal to one. Making the hypothesis σ0 = 0, the equations (16) and (14) respectively becomes : ( ∂sχs − ∂tǫs − [ǫs, χs] = 0 ∂tχs − ∂sǫs = 0 (19) Consider the 1-form ω = χsdt + ǫsds and denote ⋆ω := χsds + ǫsdt, the previous equations are equivalent to ( dω − ω ∧ ω = 0 d(⋆ω) = 0 (20) We set also ωL := 1 2 (ω+⋆ω) = 1 2 (ǫs +χs)d(s+t) (L for left moving) and ωR := 1 2 (ω−⋆ω) = 1 2 (ǫs −χs)d(s−t) (R for right moving) and remark that ⋆ωL = ωL and ⋆ωR = −ωR. 4 underwater vehicle in the case that the center of buoyancy and the center of gravity are coincident The key in the following is to introduce a so-called spectral parameter λ ∈ (C ∩ {∞}) \ {−1, 1} = CP \ {−1, 1} and the following family of connexion forms ωλ = ωL 1 + λ + ωR 1 − λ = ω − λ ⋆ ω 1 − λ2 . (21) We observe that ωλ=0 = ω and that System (20) is satisfied if and only if dωλ − ωλ ∧ ωλ = 0, ∀λ ∈ CP \ {−1, 1}. (22) This relation is a necessary and sufficient condition for the existence of a family of maps (Hλ)λ∈CP\{±1} from R2 to S E(3)C , the complexification of S E(3), such that dHλ = ωλHλ on R2 . (23) We will assume that Hλ(0, 0) = 1 (the identity element of S E(3)C ). Then the solution of (23) is unique. Since ω and ⋆ω are real, ωλ is also real for any λ ∈ R \ {±1} and hence Hλ is also real, i.e. takes value in S E(3), for these values of λ. In general Hλ satisfies the reality condition Hλ = Hλ, ∀λ ∈ CP \ {±1}. 3.2 The undressing and dressing procedures The family of maps (Hλ)λ∈CP\{±1} is in correspondence with solutions of linear wave equations, through the following transformation, called the undressing procedure. For that purpose fix some small disks D−1 and D1 in C centered respectively at −1 and 1 and denote respectively by Γ−1 and Γ1 their boundaries. We temporarily fix (s, t) ∈ R2 , denote Hλ = Hλ(s, t) and let the variable λ run. By solving a Riemann-Hibert problem (which admits a solution for Hλ close to the identity), we can find two maps [λ 7−→ IL λ ] and [λ 7−→ OL λ] on Γ−1 with value in S E(3)C such that: – Hλ = (IL λ )−1 OL λ, ∀λ ∈ Γ−1; – IL λ can be extended holomorphically inside Γ−1, i.e. in D−1; – OL λ can be extended holomorphically outside Γ−1, i.e. on CP\ D−1, and converges to the unity at infinity. Similarly we can find two maps [λ 7−→ IR λ ] and [λ 7−→ OR λ ] on Γ1 with value in S E(3)C such that: – Hλ = (IR λ )−1 OR λ , ∀λ ∈ Γ1; – IR λ can be extended holomorphically in D1; – OR λ can be extended holomorphically on CP \ D1. Set oL λ := dOL λ ·(OL λ)−1 . We deduce from OL λ = IL λ Hλ that oL λ = dIL λ (IL λ )−1 +IL λ ωλ(IL λ )−1 on Γ−1. From its very definition we deduce that oL λ can be extended holomorphically on CP\D−1. From oL λ = dIL λ (IL λ )−1 +IL λ ωλ(IL λ )−1 and by using (21) we deduce that oL λ can be extended meromorphically inside D−1, with at most one pole, equal to IL λ=−1 ωL 1 + λ (IL λ=−1)−1 . By using Liouville theorem and the fact that oL converges to 0 at infinity we deduce that oL λ actually coincides with the latter expression, i.e. has the form oL λ = v(s, t) 1 + λ d(s + t). (24) But by the definition of oL , we know that doL λ − oL λ ∧ oL λ = 0. Writing this equation gives us then that ∂v ∂s − ∂v ∂t = 0, hence oL λ = v(s+t) 1+λ d(s + t). A similar analysis on Γ1 gives us that oR λ := dOR λ · (OR λ )−1 is of the form oR λ = u(s−t) 1−λ d(s − t). Hence we constructed from Hλ two maps (s, t) 7−→ v(s + t) and (s, t) 7−→ u(s − t) with value in the complexification of the Lie algebra of S E(3) (but which actually satisfy a reality condition) and which are solutions of the linear wave equation (actually v is a left moving solution and u a right moving one). This construction can be reversed through the so-called dressing procedure: starting from the data (s, t) 7−→ (u(s − t), v(s + t)), set oL λ = v(s+t) 1+λ d(s + t) and oR λ = u(s−t) 1−λ d(s − t). We can integrate these forms for λ ∈ Γ−1 and λ ∈ Γ1 respectively. We hence get the maps OL λ and OR λ . We then solve the Riemann–Hilbert problem, consisting of finding a map Hλ on Γ := Γ−1 ∩ Γ1 and two maps IL λ and IR λ , defined respectively on Γ−1 and Γ1, such that – OL λ = IL λ Hλ, ∀λ ∈ Γ−1; – OR λ = IR λ Hλ, ∀λ ∈ Γ1; – IL λ can be extended holomorphically on D−1; – IR λ can be extended holomorphically on D1; – Hλ can be extended holomorphically on CP \ (D−1 ∩ D1) and converges to the unity at infinity. Then Hλ will provides us with a solution of (19). Indeed by setting ωλ := dHλ · H−1 λ , we first deduce that ωλ extends holomorphically on CP \ (D−1 ∩ D1) and converges to 0 at infinity. On the other hand, on Γ−1 we have Hλ = (IL λ )−1 OL λ, from which we deduce that ωλ = −(IL λ )−1 dIL λ + (IL λ )−1 v(s+t) 1+λ IL λ d(s + t). Hence ωλ extends meromorphically inside D−1 with at most a unique pole at −1 with leading term (IL −1)−1 v(s+t) 1+λ IL −1d(s + t). Similarly on Γ1 we have Hλ = (IR λ )−1 OR λ , from which we deduce that ωλ = −(IR λ )−1 dIR λ +(IR λ )−1 u(s−t) 1+λ IR λ d(s− t). Hence ωλ extends meromorphically inside D1 with at most a unique pole at 1 with leading term (IR 1 )−1 u(s−t) 1+λ IR 1 d(s− t). From all these informations on ωλ we deduce by using Liouville theorem that ωλ has the form (21). It also satisfies (22). This leads to a solution to (20) by setting λ = 0 in ωλ. 4 Conclusion A geometrical approach of the dynamic of a Reissner beam has been studied in this article in order to take into account non linear effects due to large displacements. Among the family of problems we identified a particular case where this system is completely integrable. This allows us to find almost explicit solutions of this fully nonlinear problem. Among these solutions, it is tempting to identify solitons, and to test the known numerical methods on these solutions. The existence of such soliton solutions leads also to the question whether soliton solutions exists whenever the tensors J and C are less symmetric. References [1] V. Arnold, ”Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits”, Ann. Inst. Fourier, Grenoble 16, 319-361, (1966) [2] J. Simo, ”A finite strain beam formulation. The three-dimensional dynamic problem. Part I”, Comput. Meth- ods Appl. Mech. Engrg. 49, 55-70, (1985) [3] D. Roze, Simulation d’une corde avec fortes déformations par les séries de Volterra, Master Thesis, Université Pierre et Marie Curie (Paris 6), 2006. [4] J .Bensoam, D. Roze, ”Modelling and numerical simulation of strings based on lie groups and algebras. 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