A symplectic minimum variational principle for dissipative dynamical systems

07/11/2017
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Using the concept of symplectic subdi erential, we propose a modi cation of the Hamiltonian formalism which can be used for dissipative systems. The formalism is rst illustrated through an application of the standard inelasticity in small strains. Some hints concerning possible extensions to non-standard plasticity and nite strains are then given.
Finally, we show also how the dissipative transition between macrostates can be viewed as an optimal transportation problem.

A symplectic minimum variational principle for dissipative dynamical systems

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application/pdf A symplectic minimum variational principle for dissipative dynamical systems Abdelbacet Oueslati, An Danh Nguyen, Géry de Saxcé
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Using the concept of symplectic subdi erential, we propose a modi cation of the Hamiltonian formalism which can be used for dissipative systems. The formalism is rst illustrated through an application of the standard inelasticity in small strains. Some hints concerning possible extensions to non-standard plasticity and nite strains are then given.
Finally, we show also how the dissipative transition between macrostates can be viewed as an optimal transportation problem.
A symplectic minimum variational principle for dissipative dynamical systems
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A symplectic minimum variational principle for dissipative dynamical systems Abdelbacet Oueslati1 , An Danh Nguyen2 , Géry de Saxcé1,∗ 1 Laboratoire de Mécanique de Lille (FRE CNRS 3723), Villeneuve d’Ascq, France {abdelbacet.oueslati, gery.desaxce}@univ-lille1.fr 2 Institute of General Mechanics, RWTH, Aachen, Germany andanh@iam.rwth-aachen.de Abstract. Using the concept of symplectic subdifferential, we propose a modification of the Hamiltonian formalism which can be used for dissipa- tive systems. The formalism is first illustrated through an application of the standard inelasticity in small strains. Some hints concerning possible extensions to non-standard plasticity and finite strains are then given. Finally, we show also how the dissipative transition between macrostates can be viewed as an optimal transportation problem. Keywords: symplectic geometry · convex analysis · non smooth mechanics 1 Introduction Realistic dynamical systems considered by engineers and physicists are subjected to energy loss. It may stem from external actions, in which case we call them non conservative. The behaviour of such systems can be represented by Hamilton’s least action principle. If the cause is internal, resulting from a broad spectrum of phenomena such as collisions, surface friction, viscosity, plasticity, fracture, damage and so on, we name them dissipative. Hamilton’s variational principle failing for such systems, we want to propose another one for them. Classical dynamics is generally addressed through the world of smooth func- tions while the mechanics of dissipative systems deals with the one of non smooth functions. Unfortunately, both worlds widely ignore each other. Our aim is laying strong foundations to link both worlds and their corresponding methods. 2 Non dissipative systems Let us consider a dynamical system, which is described by z = (x, y) ∈ X × Y , where the primal variables x describe the body motion and the dual ones y are the corresponding momenta, both assembled in vectors. X and Y are topological, locally convex, real vector spaces. There is a dual pairing: h·, ·i : X × Y → R which makes continuous the linear forms x 7→ hx, yi and y 7→ hx, yi. The space X × Y has a natural symplectic form ω : (X × Y ) 2 → R defined by: ω(z, z0 ) = hx, y0 i − hx0 , yi For any smooth hamiltonian function (x, t) 7→ H(x, t), we define the symplectic gradient (or Hamiltonian vector field) by: ż = XH ⇔ ∀ δz, ω(ż, δz) = δH In the particular case X = Y , the dual pairing is a scalar product and the space X × Y is dual with itself, with the duality product: hh(x, y), (x0 , y0 )ii = hx, x0 i + hy, y0 i Introducing the linear map J(x, y) = (−y, x) and putting: ω(z, z0 ) = hhJ(z), z0 ii we have J (XH) = DzH that allows to recover the canonical equations governing the motion: ẋ = DyH, ẏ = −DxH (1) Notice that J makes no sense in the general case when X 6= Y . 3 Dissipative systems For such systems, the cornerstone hypothesis is to decompose the velocity in the phase space into reversible and irreversible parts: ż = żR + żI, żR = X H, żI = ż − X H the idea being that for a non dissipative system, the irreversible part vanishes and the motion is governed by the canonical equations. Now, it is a crucial turning-point. We will be confronting the tools of the differential geometry to the ones of the non smooth mechanics. We start with a dissipation potential φ. It is not differentiable everywhere but convex and lower semicontinuous. This weakened properties allow to model set-valued constitutive laws –currently met in mechanics of disipative materials– through the concept of subdifferential, a set of generalized derivatives (a typical example, the plasticity, will be given at the end of Section 5). We introduce a new subdifferential, called symplectic [2]. Mere sleight of hand: all we have to do is to replace the dual pairing by the symplectic form in the classical definition; żI ∈ ∂ω φ(ż) ⇔ ∀ż0 , φ(ż + ż0 ) − φ(ż) ≥ ω(żI, ż0 ) (2) From a mechanical viewpoint, it is the constitutive law of the material. Likewise, we define a symplectic conjugate function, by the same sleight of hand in the definition of the Legendre-Fenchel transform: φ∗ω (żI) = sup ż {ω(żI, ż) − φ(ż)} (3) satisfying a symplectic Fenchel inequality: φ(ż) + φ∗ω (żI) − ω(żI, ż) ≥ 0 (4) The equality is reached in the previous relation if and only if the constitutive law (2) is satisfied. Remarks. Always in the case X = Y where J makes sense, the subdifferen- tial is defined by: żI ∈ ∂φ(ż) ⇔ ∀ż0 , φ(ż + ż0 ) − φ(ż) ≥ hhżI, ż0 ii Comparing to the definition (2) of the symplectic subdifferential, one has: żI ∈ ∂ω φ(ż) ⇔ J(żI) ∈ ∂φ(ż) Recalling the definition of the conjugate function: φ∗ (żI) = sup ż {hhżI, żii − φ(ż)} and comparing to (3), we obtain φ∗ω (ż) = φ∗ (J(ż)). Moreover, an interesting fact is that, taking into account the antisymmetry of ω: hhDzH, żii = hhJ (X H), żii = ω(X H, ż) = ω(ż, ż − X H) = ω(ż, żI) If we suppose that for all couples (ż, żI): φ(ż) + φ∗ω (żI) ≥ 0 the system dissipates for the couples satisfying the constitutive law: hhDzH, żii = −ω(żI, ż) = −(φ(z, ż) + φ∗ω (z, żI)) ≤ 0 4 The symplectic Brezis-Ekeland-Nayroles principle The variational formulation can be obtained by integrating the left hand member of (4) on the system evolution. On this ground, we proposed in [3] a symplectic version of the Brezis-Ekeland-Nayroles variational principle: The natural evolution curve z : [t0, t1] → X × Y minimizes the functional: Π(z) := Z t1 t0 [φ(ż) + φ∗ω (ż − XH) − ω(ż − XH, ż)] dt among all the curves verifying the initial conditions z(t0) = z0 and, remarkably, the minimum is zero. Observing that ω(ż, ż) vanishes and integrating by part, we have also the variant (which is not compulsory): Π(z) = Z t1 t0 [φ(ż) + φ∗ω (ż − XH) − ∂H ∂t (t, z)] dt + H(t1, z(t1)) − H(0, z0) 5 Application to the standard plasticity and viscoplasticity To illustrate the general formalism and to show how it allows to develop power- ful variational principles for dissipative systems within the frame of continuum mechanics, we consider the standard plasticity and viscoplasticity in small de- formations based on the additive decomposition of strains into reversible and irreversible strains: ε = εR + εI where εI is the plastic strain. Let Ω ⊂ Rn be a bounded, open set, with piecewise smooth boundary ∂Ω. As usual, it is divided into two disjoint parts, ∂Ω0 (called support) where the displacements are imposed and ∂Ω1 where the surface forces are imposed. The elements of the space X are fields x = (u, εI) ∈ U × E where εI is the irreversible strain field and u is a displacement field on the body Ω with trace ū on ∂Ω. The elements of the corresponding dual space Y are of the form y = (p, π). Unlike p which is clearly the linear momentum, we do not know at this stage the physical meaning of π. The duality between the spaces X and Y has the form: hx, yi = Z Ω (hu, pi + hεI, πi) where the duality products which appear in the integral are finite dimensional duality products on the image of the fields u, p (for our example this means a scalar product on R3 ) and on the image of the fields ε, π (in this case this is a scalar product on the space of 3 by 3 symmetric matrices). We denote all these standard dualities by the same h·, ·i symbols. The total Hamiltonian of the structure is taken of the integral form: H(t, z) = Z Ω  1 2ρ k p k2 +w(∇u − εI) − f(t) · u  − Z ∂Ω1 ¯ f(t) · u The first term is the kinetic energy, w is the elastic strain energy, f is the volume force and ¯ f is the surface force on the part ∂Ω1 of the boundary, the displacement field being equal to an imposed value ū on the remaining part ∂Ω0. According to (1), its symplectic gradient is: XH = ((DpH, DπH), (−DuH, −DεI H)) where, introducing as usual the stress field given by the elastic law: σ = Dw(∇u − εI) DuH is the gradient in the variational sense (from (1) and the integral form of the duality product): DuH = ∂H ∂u − ∇ ·  ∂H ∂∇u  = −f − ∇ · σ and: DūH = σ · n − ¯ f Thus one has: żI = ż − XH =  u̇ − p ρ , ε̇I  , (ṗ − f − ∇ · σ, π̇ − σ)  We shall use a dissipation potential which has an integral form: Φ(z) = Z Ω φ(p, π) and we shall assume that the symplectic Fenchel transform of Φ expresses as the integral of the symplectic Fenchel transform of the dissipation potential density φ. The symplectic Fenchel transform of the function φ reads: φ∗ω (żI) = sup {hu̇I, ṗ0 i + hε̇I, π̇0 i − hu̇0 , ṗIi − hε̇0 I, π̇Ii − φ(ż0 ) : ż0 ∈ X × Y } To recover the standard plasticity, we suppose that φ is depending explicitly only on π̇: φ(ż) = ϕ(π̇) (5) Denoting by χK the indicator function of a set K (equal to 0 on K and to +∞ otherwise), we obtain: φ∗ω (żI) = χ{0}(u̇I) + χ{0}(ṗI) + χ{0}(π̇I) + ϕ∗ (ε̇I) where ϕ∗ is the usual Fenchel transform. In other words, the quantity φ∗ω (żI) is finite and equal to: φ∗ω (żI) = ϕ∗ (ε̇I) if and only if all of the following are true: (a) p equals the linear momentum p = ρu̇ (6) (b) the balance of linear momentum is satisfied ∇ · σ + f = ṗ = ρü on Ω, σ · n = ¯ f on ∂Ω1 (7) (c) and an equality which reveals the meaning of the variable π: π̇ = σ . (8) Eliminating π̇ by (8), the symplectic Brezis-Ekeland-Nayroles principle ap- plied to standard plasticity states that the evolution curve minimizes: Π(z) = Z t1 t0  ϕ(σ) + ϕ∗ (ε̇I) − ∂H ∂t (t, z)  dt + H(t1, z(t1)) − H(t0, z0) among all curves z : [t0, t1] → X × Y such that z(0) = (x0, y0), the kinematical conditions on ∂Ω0, (6), (7) are satisfied. For instance, in plasticity, the potential ϕ is the indicator function of the closed convex plastic domain K. The constitutive law ε̇I ∈ ∂ϕ(σ) reads for σ ∈ K: ∀σ0 ∈ K, (σ0 − σ) : ε̇I ≤ 0 (9) If σ is an interior point of K, ε̇I vanishes. If σ is a boundary point of K, ε̇I is a called a subnormal to K at σ. An important case of interest is the metal plasticity governed by von Mises model for which K is defined as the section: K = {σ such that f(σ) ≤ 0} where f is differentiable on the boundary of K. In this case, if σ is a boundary point of K, there exists λ > 0 such that: ε̇I = λ Df(σ) that means ε̇I is an exterior normal to K. Otherwise, if σ does not belong to K, there is no solution to this inequation (9). In short, the previous non smooth constitutive law allows to model the following behavior: below a given stress threshold, there is no plastic deformation then no dissipation; at the threshold, plastic yielding and dissipation occur; over the threshold, no stress state may be reached. Remark. The assumption that u, εI and p are ignorable in (5) comes down to introduce into the dynamical formalism a ”statical” constitutive law: ε̇I ∈ ∂ϕ(π̇) = ∂ϕ(σ) Conversely, the symplectic framework suggests to imagine fully ”dynamical” con- stitutive laws of the more general form: (u̇, ε̇I) ∈ ∂ω φ (ṗ, π̇) 6 Extensions to non standard plasticity and finite strains In plasticity and more generally in the mechanics of dissipative materials, some of the constitutive laws are non-associated, i.e. cannot be represented by a dis- sipation potential. A response proposed first in [5] is to introduce a bipotential. The applications to solid Mechanics are various: Coulombs friction law, non- associated Drucker-Prager and Cam-Clay models in Soil Mechanics, cyclic Plas- ticity and Viscoplasticity of metals with the non linear kinematical hardening rule, Lemaitre’s damage law (for more details, see reference [4]). For such constitutive laws, the principle can be easily generalized replacing φ(ż) + φ∗ω (ż0 ) by a symplectic bipotential b(ż, ż0 ): – separatly convex and lower semicontinuous with respect to ż, ż0 , – satisfying a cornerstone inequality: ∀ż, ż0 , b(ż, ż0 ) ≥ ω(ż, ż0 ) extending the symplectic Fenchel inequality (4), leading to generalize the functional of the symplectic BEN principle: Π(z) := Z t1 t0  b(ż, żI) − ∂H ∂t (t, z)  dt + H(t1, z(t1)) − H(t0, z0) For the extension to finite strains, we may modify the original framework by working on the tangent bundle and making for instance φ = φ(z, ż) ([3]). Then the goal is reached in three steps. Firstly, we develop a Lagrangian formalism for the reversible media based on the calculus of variation by jet theory. Next, we propose a corresponding Hamiltonian formalism. Finally, we deduce from it a symplectic minimum principle for dissipative media. This allows, among other things, to get a minimum principle for unstationnary Navier-Stokes models. 7 Dissipative transition between macrostates as an optimal transportation problem We would like to model a dissipative transition between the macrostates at t = tk (k = 0, 1). Now, X and Y are separable metric spaces. X × Y is viewed a the space of microstates zk with Gibbs probability measure µk at t = tk of density: µk = e−(ζk+βkH(zk)) where βk is the reciprocal temperature and ζk is Planck’s (or Massieu’s) poten- tial. Let us consider the set of curves from z0 to z1: Z(z0, z1) = {z : [t0, t1] → X × Y s.t. z(t0) = z0 and z(t1) = z1} We adopt as cost function: c(z0, z1) = inf z∈Z(z0,z1) Π(z) and suppose it is measurable. It is worth to remark that it is not generally zero. For a measurable map T : X × Y → X × Y , T(µ0) denotes the push forward of µ0 such that for all Borel set B: T(µ0)(B) = µ0(B) Inspiring from Monge’s formulation of the optimal transportation problem, we claim that: Among all the transport maps T such that µ1 = T(µ0), the natural one minimizes the functional: Cµ0 (T) := Z X×Y c(z0, T(z0)) dµ0(z0) Following Kantorovich’s formulation, we consider the set Γ(µ0, µ1) of all probability measures γ on X ×Y with marginals µ0 on the first factor X ×Y and µ1 on the second factor X×Y , i.e. for all Borel set B on X×Y , γ(B×(X×Y )) = µ0(B) and γ((X × Y ) × B) = µ1(B). Hence, we claim that: The natural probability measure γ minimizes the functional: C(γ) := Z (X×Y )2 c(z0, z1) dγ(z0, z1) within the set Γ(µ0, µ1). The advantage of the new formulation is that the latter problem is linear with respect γ while the former one is non linear with respect to T. The symplectic Wasserstein distance-like function is then defined as the op- timal value of the cost C: Ws(µ0, µ1) := inf γ∈Γ (µ0,µ1) C(γ) 8 Aknowledgement This work was performed thanks to the international cooperation project Dissi- pative Dynamical Systems by Geometrical and Variational Methods and Appli- cation to Viscoplastic Structures Subjected to Shock Waves (DDGV) supported by the Agence Nationale de la Recherche (ANR) and the Deutsche Forchungs- gemeinschaft (DFG). References 1. Brezis, H., Ekeland I.: Un principe variationnel associé à certaines équations paraboliques. I. Le cas indépendant du temps. C. R. Acad. Sci. Paris Série A- B 282, 971-974 (1976) 2. Buliga M.: Hamiltonian inclusions with convex dissipation with a view towards applications. Mathematics and its Applications 1(2), 228-25 (2009) 3. Buliga, M., de Saxcé, G.: A symplectic Brezis-Ekeland-Nayroles principle. Mathe- matics and Mechanics of Solids, doi: 10.1177/1081286516629532, 1-15 (2016) 4. Dang Van, K., de Saxcé, G., Maier, G., Polizzotto, C., Ponter, A., Siemaszko, A., Weichert, D.: Inelastic Behaviour of Structures under Variable repeated Loads, D. Weichert, G. Maier, Eds., CISM International Centre for Mechanical Sciences, Courses and Lectures, No. 432, Springer, Wien, New York. (2002) 5. de Saxcé, G., Feng, Z.Q.: New inequation and functional for contact with friction: the implicit standard material approach. International Journal of Mechanics of Structures and Machines 19(3), 301-325 (1991) 6. Nayroles B.: Deux théorèmes de minimum pour certains systèmes dissipatifs. C. R. Acad. Sci. Paris Série A-B 282:A1035-A1038 (1976)