David Doyen
email: david.doyen@edf.fr

Alexandre Ern
email: ern@cermics.enpc.fr

Analysis of the modified mass method for the dynamic Signorini problem with Coulomb friction

Keywords: Finite elements, Time-integration scheme, Elastodynamics, Unilateral contact, Coulomb friction, Differential inclusion, Modified mass method Mathematics Subject Classification (2000) 65M12, 65M60, 74H15, 74M10, 74M15, 74S05

The aim of the present work is to analyze the modified mass method for the dynamic Signorini problem with Coulomb friction. We prove that the space semidiscrete problem is equivalent to an upper semi-continuous one-sided Lipschitz differential inclusion and is, therefore, well-posed. We derive an energy balance. Next, considering an implicit time-integration scheme, we prove that, under a certain condition on the discretization parameters, the fully discrete problem is well-posed. For a fixed discretization in space, we prove also that the fully discrete solutions converge to the space semi-discrete solution when the time step tends to zero.

Introduction

The modified mass method is a new approach for solving computationally dynamic problems with unilateral contact. Introduced in [START_REF] Khenous | Mass redistribution method for finite element contact problems in elastodynamics[END_REF] for frictionless contact problems, it is based on a space semi-discrete formulation in which the mass matrix is modified (the entries of the mass matrix associated with the (normal) displacements at the contact boundary are set to zero). This modified semi-discrete problem can then be discretized with various time-integration schemes, either implicit or semi-explicit. In the implicit case, the modified mass method enables to eliminate the large spurious oscillations on the contact pressure, which can appear with a standard mass matrix, while ensuring an exact enforcement of the contact condition. The method does not require extra steps or extra parameters and can easily be implemented. With a suitable scheme such as the Newmark scheme (trapezoidal rule), the modified mass method achieves a tight energy conservation and exhibits a good behavior in long-time. The method can also be used in the context of a semi-explicit time-marching method, with in practice the same stability condition as in the linear case while ensuring an exact enforcement of the contact condition. For a detailed presentation of the modified mass method for frictionless contact and a comparison with other methods, we refer to [START_REF] Doyen | Time-integration schemes for the finite element dynamic Signorini problem[END_REF].

The modified mass method has been formulated for dynamic contact problems with Coulomb friction in [START_REF] Khenous | Problèmes de contact unilatéral avec frottement de Coulomb en élastostatique et élastodynamique[END_REF][START_REF] Hager | A stable energy-conserving approach for frictional contact problems based on quadrature formulas[END_REF][START_REF] Hager | Analysis of a space-time discretization for dynamic elasticity problems based on mass-free surface elements[END_REF] and numerical simulations have been performed in [START_REF] Hager | A stable energy-conserving approach for frictional contact problems based on quadrature formulas[END_REF][START_REF] Hager | Analysis of a space-time discretization for dynamic elasticity problems based on mass-free surface elements[END_REF]. Up to date, no theoretical analysis has been carried out for the modified mass method in the frictional case. In the frictionless case with an elastic material, interesting results have been proven. The space semi-discrete problem is equivalent to a Lipschitz system of ordinary differential equations and is, therefore, well-posed [START_REF] Khenous | Mass redistribution method for finite element contact problems in elastodynamics[END_REF]. The variation of energy is equal to the work of the external forces; the contact forces do not work [START_REF] Khenous | Mass redistribution method for finite element contact problems in elastodynamics[END_REF]. Furthermore, convergence of the semi-discrete solutions to a continuous solution is proven for viscoelastic materials in [START_REF] Doyen | Convergence of a space semi-discrete modified mass method for the dynamic Signorini problem[END_REF].

The aim of the present work is to analyze the modified mass method for the dynamic Signorini problem with Coulomb friction. We prove that the space semi-discrete problem is equivalent to an upper semi-continuous one-sided Lipschitz differential inclusion [START_REF] Deimling | of de Gruyter Series in Nonlinear Analysis and Applications[END_REF][START_REF] Smirnov | Introduction to the theory of differential inclusions[END_REF] and is, therefore, well-posed (Theorem 2). Furthermore, the variation of energy is equal to the work of the external forces and friction forces (Theorem 3). For the time discretization, we propose an implicit scheme. Each time step requires solving a nonlinear problem similar to a static friction problem. It is well-known that such a problem can have several solutions [START_REF] Hild | Local uniqueness and continuation of solutions for the discrete Coulomb friction problem in elastostatics[END_REF]. Here we prove that, under a certain condition on the discretization parameters of CFL-type, the fully discrete problem is well-posed (Theorem 4). For a fixed discretization in space, we prove also that the fully discrete solutions converge to the space semi-discrete solution when the time step tends to zero (Theorem 5).

With a standard mass term, proving the existence of a semi-discrete solution to a dynamic contact problem is quite delicate. It is necessary to add an impact law and to work with BV and measures spaces [START_REF] Brogliato | Nonsmooth impact mechanics[END_REF][START_REF] Stewart | Rigid-body dynamics with friction and impact[END_REF][START_REF] Ballard | Existence and uniqueness for dynamical unilateral contact with Coulomb friction: a model problem[END_REF]. The modification of the mass term greatly simplifies the analysis. Indeed, the unilateral contact condition can be eliminated and replaced by a Lipschitz continuous term in the momentum equation [START_REF] Khenous | Mass redistribution method for finite element contact problems in elastodynamics[END_REF]. Static and quasi-static Coulomb friction problems can have several solutions [START_REF] Hild | Local uniqueness and continuation of solutions for the discrete Coulomb friction problem in elastostatics[END_REF]. Uniqueness is only obtained for small friction coefficients (see [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF]Theorem 11.4] for the static case and [START_REF] Han | Quasistatic contact problems in viscoelasticity and viscoplasticity[END_REF]Theorem 7.2.1] for the quasi-static case). It is worthwhile to notice that in the dynamic case, uniqueness is recovered. We do not examine the convergence of the discrete solutions to a solution of the continuous problem. Nevertheless, it seems possible to extend the convergence result in [START_REF] Doyen | Convergence of a space semi-discrete modified mass method for the dynamic Signorini problem[END_REF] (see above) to the case of a non-local Coulomb friction (the non-local Coulomb friction is a regularization of Coulomb friction [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF][START_REF] Cocou | Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity[END_REF]). This paper is organized as follows. In Section 2, we formulate the continuous problem. Sections 3 and 4 are devoted to the space semi-discrete and fully discrete problems, respectively. In Section 5, we examine the convergence of the fully discrete solutions to the space semi-discrete solution.

Continuous problem

We consider the infinitesimal deformations of a body occupying a reference domain Ω ⊂ R d (d = 2 or d = 3) during a time interval [0, T ]. Let ν be the outward unit normal to Ω. The elasticity tensor is denoted by A and the mass density by ρ. An external load f is applied to the body. Let u : (0,

T ) × Ω → R d , ǫ(u) : (0, T ) × Ω → R d,d
, and σ(u) : (0, T ) × Ω → R d,d be the displacement field, the linearized strain tensor, and the stress tensor, respectively. Denoting time-derivatives by dots, the momentum conservation equation reads

ρü -div σ = f, σ = A : ǫ, ǫ = 1 2 (∇u + T ∇u) in Ω × (0, T ). ( 1
)
The boundary ∂Ω is partitioned into three disjoint open subsets Γ D , Γ N , and Γ c . Dirichlet and Neumann conditions are prescribed on Γ D and Γ N , respectively,

u = u D on Γ D × (0, T ), σ • ν = f N on Γ N × (0, T ). (2)
In what follows, we assume

f ∈ W 1,∞ (0, T ; L 2 (Ω) d ) and f N ∈ W 1,∞ (0, T ; L 2 (Γ N ) d ).
We let uν := u |∂Ω •ν and uτ := u |∂Ω -uν ν the normal and tangential displacements on ∂Ω, respectively. We also let σν (u) := ν •σ(u) |∂Ω •ν and στ (u) := σ(u) |∂Ω •ν -σ(u)νν be the normal and tangential stress on ∂Ω, respectively. Note that uν and σν (u) are scalars while uτ and στ (u) are vectors in R d . Let | • | denote the Euclidean norm in R m , m ≥ 1. On Γ c , a unilateral contact condition, also called Signorini condition, and a Coulomb friction (see Fig. 1) are enforced

uν ≤ g, σν (u) ≤ 0, σν (u)(uν -g) = 0 on Γ c × (0, T ), (3)
|στ (u)| ≤ µ|σν (u)| on Γ c × (0, T ), (4)
στ (u) = -µσν (u) uτ | uτ | if uτ = 0 on Γ c × (0, T ), ( 5
)
where µ is the friction coefficient and g is the initial gap. At the initial time, the displacement and velocity fields are prescribed,

u(0) = u 0 , u(0) = v 0 in Ω. (6)
The mathematical analysis of the above time-dependent problem entails substantial difficulties. The existence of a weak solution is only proven for a viscoelastic material and a non-local Coulomb friction law [START_REF] Cocou | Existence of solutions of a dynamic Signorini's problem with nonlocal friction in viscoelasticity[END_REF]. 3 Space semi-discrete formulation

uτ σ τ (u) ν F σ n (u) -ν F σ n (u)
In this section, we formulate the space semi-discrete problem in space and we prove existence and uniqueness of a solution. We also establish an energy balance. In the frictionless case, the semi-discrete problem is equivalent to a Lipschitz ordinary differential equation, and existence and uniqueness are deduced from the Cauchy-Lipschitz theorem. With friction, the situation is more complicated. We choose to model the friction term as a set-valued map. The semi-discrete problem is then equivalent to a differential inclusion, for which generalizations of the Cauchy-Lipschitz theorem are available [START_REF] Deimling | of de Gruyter Series in Nonlinear Analysis and Applications[END_REF][START_REF] Smirnov | Introduction to the theory of differential inclusions[END_REF].

Preliminaries

To begin with, we introduce some notions needed for the formulation of our problem as a differential inclusion.

-Given a set E, we define P(E) as the set of all subsets of E, and P * (E) := P(E) \ {∅}. -A set-valued map is said to be closed convex if its images are closed convex sets.

-Various notions of continuity can be defined for set-valued maps. One of them is upper semi-continuity1 . A set-valued map F is said to be upper semi-continuous at x if, for every open set V containing F (x), there exists a neighborhood U of x such that F (U ) ⊂ V . Consider, for instance, the following set-valued maps:

F 1 (x) = ( [-1, 1] if x = 0, {0} if x = 0, and F 2 (x) = ( {0} if x = 0, [-1, 1] if x = 0.
It is easy to verify that F 1 is upper semi-continuous for all x ∈ R, whereas F 2 is not upper semi-continuous at x = 0. Here is another example of set-valued-map, closely related to the Coulomb friction term,

F 3 (x, y) = 8 > < > : {-|x|} if y < 0, [-|x|, |x|] if y = 0, {|x|} if y > 0.
It is easy to verify that this map is upper semi-continuous for all (x, y) ∈ R 2 . Finally, we observe that upper semi-continuity applied to single-valued functions is equivalent to continuity. For more details on set-valued maps, we refer to [START_REF] Aubin | Set-valued analysis, volume 2 of Systems & Control: Foundations & Applications[END_REF]. -The existence theorem for differential inclusions we use does not provide continuously differentiable solutions in time. The solutions are only absolutely continuous in time. For brevity, we do not define this concept and refer to [START_REF] Rudin | Real and complex analysis[END_REF]. For our purpose, it suffices to know that an absolutely continuous function y is continuous, differentiable almost everywhere and is equal to the integral of its derivative:

y(t 0 ) = y(0) + Z t0 0 ẏ(t)dt.
Lipschitz continuous functions are absolutely continuous. In what follows, we denote by AC([0, T ]; R m ), the space spanned by absolutely continuous functions from [0, T ] to R m . -The set-valued maps which appear in our space semi-discrete problem are subgradients and for completeness, we define this notion. Let J : R m → R ∪ {+∞} be a convex function and D(J) := {v ∈ R m ; J(v) < +∞} its domain. We define the subgradient of J as the set-valued map ∂J :

D(J) → P * (R m ) such that ∀v ∈ D(J), ∂J(v) := ˘γ ∈ R m ; J(w) -J(v) ≥ (γ, w -v), ∀w ∈ D(J) ¯, (7)
where (•, •) denotes the canonical inner product on R m . It is easy to prove that the subgradient of a convex function is well-defined and is a closed convex set-valued map. For more details on subgradients, we refer to [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF].

We can now state the main result we use for asserting the well-posedness of a problem posed in the form of a differential inclusion.

Theorem 1 Let P : [0, T ] × R m → P * (R m ) be a closed convex set-valued map. Let x 0 ∈ R m and consider the following problem: Find x ∈ AC([0, T ]; R m ) such that ẋ(t) ∈ P (t, x(t)), (8)
x(0) = x 0 . (9)
Assume that 1. the set-valued map

P (t, •) is upper semi-continuous for almost all t ∈ [0, T ]; 2. for any x ∈ R m , there exists a measurable function p(•, x) satisfying p(t, x) ∈ P (t, x) for almost all t ∈ [0, T ]; 3. there exists a function b ∈ L 1 (0, T ; R m ) such that |p(t, x)| ≤ b(t) for almost all t ∈ [0, T ].
Then, there exists a solution to ( 8)- [START_REF] Ern | Theory and Practice of Finite Elements[END_REF]. Furthermore, assume the following one-sided Lipschitz condition: there exists K ∈ R such that, for all t ∈ [0, T ], for all

x 1 , x 2 ∈ R m , (y 1 -y 2 , x 1 -x 2 ) ≤ K x 1 -x 2 2 , ∀y 1 ∈ P (t, x 1 ), ∀y 2 ∈ P (t, x 2 ). ( 10
)
Then, the solution is unique.

Proof For the existence, see [START_REF] Smirnov | Introduction to the theory of differential inclusions[END_REF]Theorem 4.7] or [START_REF] Deimling | of de Gruyter Series in Nonlinear Analysis and Applications[END_REF]Theorem 5.2]. Uniqueness is straightforward owing to the one-sided Lipschitz condition since it implies that two solutions x 1 and x 2 satisfy 1 2

d dt ( x 1 -x 2 2 ) ≤ K x 1 -x 2 2 .
Remark 1 In the single-valued case (P :

[0, T ] × R m → R m ), the hypotheses of Theo- rem 1 become 1. P (t, •) is continuous for almost all t ∈ [0, T ]; 2. for any x ∈ R, P (•, x) is measurable; 3. there exists a function b ∈ L 1 (0, T ; R) such that |P (t, x)| ≤ b(t) for almost all t ∈ [0, T ];
We recover Caratheodory's existence theorem for ordinary differential equations [START_REF] Filippov | Differential equations with discontinuous righthand sides[END_REF]. Furthermore, the one-sided Lipschitz condition means that P (t, •) is Lipschitz continuous for all t ∈ [0, T ] (uniformly).

Remark 2 If P is a monotone operator, i.e., for all t ∈ [0, T ], for all

x 1 , x 2 ∈ R m , (y 1 -y 2 , x 1 -x 2 ) ≥ 0, ∀y 1 ∈ P (t, x 1 ), ∀y 2 ∈ P (t, x 2 ),
then -P satisfies the one-sided Lipschitz condition.

The discrete setting

For simplicity, we suppose that Ω is a polyhedron. Let T be a simplicial mesh of Ω (triangles in 2D and tetrahedra in 3D). Let {x i } i∈N and {φ i } i∈N be the nodes of the mesh and the associated scalar basis functions (continuous and piecewise affine), respectively. We denote by N D the set of indices where a Dirichlet condition is enforced, and we set Ñ := N \ N D . The space of admissible displacements is approximated by the space

V = {v ∈ C 0 (Ω) d ; v |T ∈ (P 1 ) d , ∀T ∈ T , and v(x i ) = 0, ∀i ∈ N D }.
The space V is spanned by {φ i eα} i∈ Ñ ,1≤α≤d , where {eα} 1≤α≤d is the canonical basis of R d . Denote by N c the set of indices of contact nodes (that is, the nodes located on Γ c which is fixed a priori) and by N i := Ñ \ N c the set of indices of the remaining nodes (see Fig. 2). Let {ν i } i∈N c and {τ i,α } i∈N c ,1≤α≤d-1 be the contact normal vectors and tangential vectors, respectively. We set Clearly,

V i = span({φ i eα} i∈N i ,1≤α≤d ), V c = span({φ i ν i } i∈N c ) and V f = span({φ i τ i,α } i∈N c ,1≤α≤d-1 ).
V = V i ⊕ V c ⊕ V f , so that any discrete function v ∈ V can be decomposed as v = v i + vc + v f with v i ∈ V i , vc ∈ V c , v f ∈ V f .
We also introduce the space V * := V i ⊕ V f , so that any discrete function v ∈ V can also be decomposed as

v = v * + vc with v * ∈ V * , vc ∈ V c .
Let (•, •) denote the L 2 inner product on V . Let • denote the norm associated with (•, •). Herein, we always work in finite dimension on a fixed spatial mesh; the specific choice of the norm is therefore not critical. The present choice is made for simplicity. The standard mass term stems from the bilinear form

m : L 2 (Ω) d × L 2 (Ω) d ∋ (v, w) -→ Z Ω ρv • w ∈ R.
The key idea in the modified mass method is to remove the mass associated with the normal components at the contact nodes. We consider an approximate mass term associated with the bilinear form m * such that

m * (φ i ν i , w) = m * (w, φ i ν i ) = 0, ∀i ∈ N c , ∀w ∈ V. (11)
Many choices are possible to build the rest of the mass term. In [START_REF] Hager | A stable energy-conserving approach for frictional contact problems based on quadrature formulas[END_REF][START_REF] Khenous | Mass redistribution method for finite element contact problems in elastodynamics[END_REF], the authors devise various methods to preserve some features of the standard mass term (the total mass, the center of gravity, and the moments of inertia). Here, we focus for simplicity on the choice

m * : V × V ∋ (v, w) -→ m(v * , w * ) ∈ R.
We define the associated operator

M * : V * → V * such that, (M * v * , w * ) = m * (v * , w * ) ∀(v * , w * ) ∈ V * × V * .
We define the bilinear and linear forms

a : H 1 (Ω) d × H 1 (Ω) d ∋ (v, w) -→ Z Ω ǫ(v) : A : ǫ(w), l : [0, T ] × H 1 (Ω) d ∋ (t, v) -→ Z Ω f (t) • v + Z Γ N f N (t) • v.
We define the linear operator A : V → V and the vector L(t) ∈ V such that for all v ∈ V and w ∈ V , and for all t ∈ [0, T ],

(Av, w) = a(v, w), (L(t), w) = l(t, w).

We also need A c : V → V c , and L c (t) ∈ V c such that for all v ∈ V and all wc ∈ V c , and for all t ∈ [0, T ],

(A c v, wc) = a(v, wc), (L c (t), wc) = l(t, wc).
We define the constraint set

K := {v ∈ V ; v(x i ) • ν i ≤ g(x i ), ∀i ∈ N c }.
We define the unilateral contact term

I K : V c → R I K (vc) = ( 0 if vc ∈ K +∞ if vc ∈ K The function I K is non-differentiable, but convex since K is convex. Therefore, it is possible to define its subgradient ∂I K : V c ∩ K → P * (V c ), ∂I K (vc) := ˘γ ∈ V c ; 0 ≥ (γ, wc -vc) ∀wc ∈ V c ∩ K ¯.
Now, we define the friction term j :

V × V f → R such that j(v, w f ) = Z Γ c µ|σν (v)||w f |. ( 12
)
The function j is non-differentiable with respect to its second argument, but convex, and its domain is V f . We can define its subgradient with respect to its second argument such that for all z ∈ V , ∂ 2 j(z, •) :

V f → P * (V f ) with ∂ 2 j(z, v f ) := n γ ∈ V f ; j(z, w f ) -j(z, v f ) ≥ (γ, w f -v f ) ∀w f ∈ V f o . (13)

Formulation of the semi-discrete problem

We can now formulate the semi-discrete problem. Let u 0 * ∈ V * and v 0 * ∈ V * be suitable approximations of the initial displacement and velocity u 0 and v 0 , respectively.

Problem 1 Seek u ∈ C 0 ([0, T ]; K) such that u * ∈ C 1 ([0, T ]; V * ), u * ∈ AC([0, T ]; V * ),
and the following differential inclusion holds true

M * ü * ∈ -Au -∂ 2 j(u, uf ) -∂I K (uc) + L(t) a.e. in [0, T ], (14)
with the initial conditions u * (0) = u 0 * and u * (0) = v 0 * in Ω.

Remark 3

The velocity u * is absolutely continuous. Therefore, it is differentiable almost everywhere, and the acceleration ü * in ( 14) is well-defined. Moreover, uc ∈ K so that ∂I K (uc) is well-defined.

To explicitate the link between the space semi-discrete Problem 1 and the continuous problem formulated in Section 2, we observe that [START_REF] Hild | Local uniqueness and continuation of solutions for the discrete Coulomb friction problem in elastostatics[END_REF] means that, for almost all t ∈ [0, T ], there exist λc ∈ ∂I K (uc) and λ f ∈ ∂ 2 j(u, uf ) such that

M * ü * + Au + λc + λ f = L(t).
Therefore, the vectors λc and λ f are discrete counterparts of the normal and tangential contact stresses. Furthermore, lumping the mass matrices, it is easy to verify that the definitions of ∂I K (uc) and ∂ 2 j(u, uf ) imply that, for all i ∈ N c ,

uc(x i ) • ν i ≤ g(x i ), λc(x i ) ≤ 0, λc(x i )(uc(x i ) • ν i -g(x i )) = 0, ( 15
)
|λ f (x i )| ≤ µ|σν (u)(x i )|, ( 16
)
λ f (x i ) = -µσν (u)(x i ) uf (x i ) | uf (x i )| if uf (x i ) = 0. ( 17
)
Thus, we recover the discrete counterpart of the contact and friction conditions (3)-( 5).

Main results

This section contains our main results concerning the space semi-discrete problem. We define the map q :

[0, T ] × V * → V c ∩ K such that for all t ∈ [0, T ] and for all v * ∈ V * , vc = q(t, v * ) ∈ V c ∩ K solves the following variational inequality a(vc, wc -vc) ≥ l(t, wc -vc) -a(v * , wc -vc) ∀wc ∈ V c ∩ K, ∀t ∈ [0, T ]. (18)
This variational inequality is well-posed since it is equivalent to the minimization of a strictly convex functional over a convex set. We first examine the properties of the map q.

Lemma 1 For all v * ∈ V * , the map t → q(t, v * ) is Lipschitz continuous, and its Lipschitz constant is uniformly bounded in v * . For all t ∈ [0, T ], the map v * → q(t, v * ) is Lipschitz continuous, and its Lipschitz constant is uniformly bounded in t.

Proof Let t 1 , t 2 ∈ [0, T ] and v * , w * ∈ V * . Set vc = q(t 1 , v * ) and wc = q(t 2 , w * ). Owing to [START_REF] Krenk | Energy conservation in Newmark based time integration algorithms[END_REF],

a(vc -wc, vc -wc) ≤ a(v * -w * , wc -vc) + l(t 1 -t 2 , vc -wc). (19)
Since

l(t 1 -t 2 , vc -wc) = Z Ω (f (t 1 ) -f (t 2 )) • (vc -wc) + Z Γ N (f N (t 1 ) -f N (t 2 )) • (vc -wc),
and since f ∈ W 1,∞ (0, T ; L 2 (Ω) d ) and f N ∈ W 1,∞ (0, T ; L 2 (Γ N ) d ), there exists a constant c l such that l(t 1 -t 2 , vc -wc) ≤ c l |t 1 -t 2 | vc -wc .
Moreover, the bilinear form a being continuous (with constant ca) and elliptic (with constant α) for the norm • , a straightforward calculation yields

α vc -wc ≤ ca v * -w * + c l |t 1 -t 2 |,
which proves the desired regularity for q.

We now reformulate the differential inclusion (14) using the map q.

Lemma 2 The differential inclusion ( 14) is equivalent to

M * ü * ∈ -A * (u * + q(t, u * )) -∂ 2 j(u * + q(t, u * ), uf ) + L * (t), a.e. in [0, T ], ( 20
) uc = q(t, u * ), ∀t ∈ [0, T ], (21)
Proof Distinguishing components in V * and V c , the inclusion ( 14) is equivalently split into the following inclusions

M * ü * ∈ -A * u -∂ 2 j(u, uf ) + L * (t) a.e. in [0, T ], ( 22
) 0 ∈ -A c u -∂I K (uc) + L c (t) a.e. in [0, T ]. (23)
Consider [START_REF] Warburton | On the constants in hp-finite element trace inverse inequalities[END_REF]. By continuity, the inclusion ( 23) is valid for all t ∈ [0, T ]. For convenience, we recast it as a variational inequality,

a(u, vc -uc) ≥ l(t, vc -uc) ∀t ∈ [0, T ], ∀vc ∈ V c ∩ K, (24)
or, equivalently,

a(uc, vc -uc) ≥ l(t, vc -uc) -a(u * , vc -uc) ∀t ∈ [0, T ], ∀vc ∈ V c ∩ K. (25)
Hence uc = q(t, u * ) so that the system ( 22)-( 23) is equivalent to the system ( 20)- [START_REF] Stewart | Rigid-body dynamics with friction and impact[END_REF].

We can now state our main existence result for Problem 1.

Theorem 2 There exists a unique solution u to Problem 1. Furthermore, uc ∈ W 1,∞ (0, T ; V c ).

Proof (i) To prove the existence of a solution, we rewrite the second-order differential inclusion ( 14) as a first-order differential inclusion. We define the single-valued map

S : [0, T ] × V * × V * → V * × V * such that, for all t ∈ [0, T ], for all v * , w * ∈ V * , S(t, v * , w * ) = " w * -A * (v * + q(t, v * )) + L * (t) « ,
and the set-valued map P : [0, T ] × V * × V * → {0} × P * (V * ) such that

P (t, v * , w * ) = " 0 -∂ 2 j(v * + q(t, v * ), w f ) « .
We define also the linear single-valued map D : 20) can be recast as

V * × V * → V * × V * such that D(v * , w * ) = " v * M * w * « . Setting X(t) = " u * (t) u * (t) « ∈ V * × V * , the differential inclusion (
D Ẋ(t) ∈ S(t, X(t)) + P (t, X(t)). ( 26
)
We equip the product space V * × V * with the product norm.

(ii) The operator S is a single-valued map. Since q(t, •) is continuous and q(•, x) is Lipschitz continuous, the operator S satisfies the hypotheses of Theorem 1 (see Remark 1).

(iii) We now examine the operator P . This operator is a closed convex set-valued map (owing to the properties of the subgradients of convex functions). Since q(t, •) is continuous and ∂ 2 j(•, •) is upper semi-continuous (see the example given by (3.1)), the map P (t, •) is upper semi-continuous. Hence, Hypothesis 1 of Theorem 1 holds true. Since q(•, x) is Lipschitz continuous, Hypotheses 2 and 3 of this theorem are also satisfied. Next, we check the one-sided Lipschitz condition [START_REF] Filippov | Differential equations with discontinuous righthand sides[END_REF]. Let (u 1 Therefore, P satisfies the one-sided Lipschitz condition. 21), u = u * + uc = u * + q(t, u * ). Therefore, Problem 1 has a unique solution and it is clear that uc = q(t, u * ) ∈ W 1,∞ (0, T ; V c ).

We conclude this section with the energy balance.

Theorem 3 For all t 0 ∈ [0, T ], the following energy balance holds true:

E(u(t 0 )) -E(u(0)) = Z t0 0 n l(t, u(t)) -j(u(t), uf (t)) o dt, ( 27
)
where

E(v) = 1 2 (m( v * , v * ) + a(v, v)). Proof We recast the differential inclusion (20) as a variational inequality, m(ü * , v * -u * ) + a(u, v * -u * ) + j(u, v f ) -j(u, uf ) ≥ l(t, v * -u * ) ∀v * ∈ V * , a.

Fully discrete formulation

In this section, we discretize the space semi-discrete problem with an implicit time scheme. We discretize the elastodynamic part with an implicit Newmark scheme (trapezoidal rule), while the unilateral contact and friction conditions are enforced in an implicit way. This choice of time discretization is very common. It is for instance employed in [START_REF] Hager | A stable energy-conserving approach for frictional contact problems based on quadrature formulas[END_REF]. At each time step, we have thus to solve a nonlinear problem similar to a static friction problem. It is well-known that such a problem may have several solutions. Here we prove that, under a certain condition on the discretization parameters of CFL-type, the fully discrete problem is well-posed. We also derive the energy balance of this time-integration scheme.

For simplicity, the interval [0, T ] is divided into N equal subintervals of length ∆t. We set t n = n∆t and denote by u n , v n , and a n the approximations of u(t n ), u(t n ), and ü(t n ), respectively. We define the convex combination n+α := (1α) n + α n+1 , where stands for u, v, a or t, and α ∈ [0, 1]. In this section, the notation A B means that A ≤ cB with a constant c independent of h and ∆t.

Let T c ⊂ T be the set of simplices such that at least one vertex is a contact node. We set Ω c = int `∪T ∈T c T ´. Let T c ′ ⊂ T be the set of simplices such that at least one vertex belongs to Ω c . We set Ω c ′ = int `∪T ∈T c ′ T ´(see Fig. 2). We define

hc = min T ∈T c diam(T ) and h c ′ = min T ∈T c ′ diam(T ),
where diam(T ) denotes the diameter of the simplex T . Observe that hc and h c ′ are defined using a minimum.

Let us recall some classical discrete trace and inverse inequalities (see, e.g., [START_REF] Warburton | On the constants in hp-finite element trace inverse inequalities[END_REF] and [START_REF] Ern | Theory and Practice of Finite Elements[END_REF]). For all vc ∈ V c ,

vc L 2 (Γ c ) d ≤ 1 √ hc vc L 2 (Ω c ) d , (32)
|vc| H 1 (Ω c ) d = ∇vc L 2 (Ω c ) d×d ≤ 1 hc vc L 2 (Ω c ) d . (33)
The same inequalities hold when Ω c is replaced by Ω c ′ , and hc by h c ′ . We define the operator q n : V * → V c , such that for all 0 ≤ n ≤ N ,

q n (v * ) = q(t n , v * ) ∀v * ∈ V * , (34)
where the map q is defined in Section 3.4.

Lemma 3

The function q n : V * → V c is Lipschitz continuous. More precisely,

|q n (v * ) -q n (w * )| H 1 (Ω c ) d |v * -w * | H 1 (Ω c ′ ) d ∀v * , w * ∈ V * . ( 35
)
Proof Let v * , w * ∈ V * . Set vc = q n (v * ) and wc = q n (w * ). Owing to [START_REF] Rudin | Real and complex analysis[END_REF],

a(vc -wc, vc -wc) ≤ a(v * -w * , wc -vc).
Since vc and wc are zero outside Ω c , a(vcwc, vcwc) |vc -wc| 2 H 1 (Ω c ) d , and

a(v * -w * , wc -vc) = a((v * -w * )1 Ω c ′ , wc -vc) |v * -w * | H 1 (Ω c ′ ) d |vc -wc| H 1 (Ω c ) d ,
whence the assertion.

We can now formulate the fully discrete problem. Next, we define the linear operator à * : V * → V * and the vector Ln+1

Problem 2 Seek u n+1 ∈ V , v n+1 * ∈ V * , and a n+1 * ∈ V * such that M * a n+1 * ∈ -A * u n+1 -∂ 2 j(u n+1 , v n+1 f ) + L * (t n+1 ), (36)
u n+1 c = q n+1 (u n+1 * ), (37)
u n+1 * = u n * + ∆t v n * + ∆t 2 2 a n+ 1 2 * , (38)
v n+1 * = v n * + ∆t a n+ 1 2 * . (39)
∈ V * such that, ∀v * ∈ V * , Ã * v * := A * v * + 1 4∆t 2 M * v * , ( Ln+1 , v * ) := L * (t n+1 ) - 1 4∆t 2 M * ε n+1 * .
Then, using (37), it is straightforward to turn (36) into

0 ∈ Ã * u n+1 * + ∂ 2 j " u n+1 * + q n+1 (u n+1 * ), 2 ∆t (u n+1 * + δ n * ) « -Ln+1 + A * q n+1 (u n+1 *
).

(40) Observe that the last term on the right-hand side of (40) involves the operator A * (and not à * ) owing to [START_REF] Hager | A stable energy-conserving approach for frictional contact problems based on quadrature formulas[END_REF] and the fact that q n+1 (u n+1 * ) ∈ V c .

Theorem 4 Problem 2 has a unique solution under the CFL-condition

∆t h c ′ 1. (41) Proof Define the map Φ n : V * → V * such that for all v * ∈ V * , v * = Φ n (v * ) satisfies 0 ∈ Ã * v * + ∂ 2 j " v, 2 ∆t (v * + δ n * ) « -Ln+1 + A * vc, (42)
where vc := q n+1 (v * ) and v := v * + vc, so that (40) amounts to seeking a fixed-point for Φ n . Setting y * := 2 ∆t (v * + δ n * ), we rewrite the above inclusion as a variational inequality,

ã(v * , z * -y * ) + j(v, z f ) -j(v,
-v * ) + j(v, w f + δ n f ) -j(v, v f + δ n f ) ≥ l n+1 (w * -v * ) -a(vc, w * -v * ). ( 44
)
The variational inequality (44) has one and only one solution. Indeed, it is equivalent to the minimization of a strictly convex functional. The map Φ n is thus well-defined. Now we shall prove that Φ n is a contraction under the CFL condition (41). Let v * ∈ V * and ŵ * ∈ V * . Set v * := Φ n (v * ) and w * := Φ n ( ŵ * ). Using (44), a straightforward calculation yields

ã(v * -w * , v * -w * ) ≤ j(v, w f + δ n f ) -j( ŵ, w f + δ n f ) -j(v, v f + δ n f ) + j( ŵ, v f + δ n f ) -a(vc -ŵc, v * -w * ). (45)
Using the ellipticity of m and a,

ã(v * -w * , v * -w * ) 4 ∆t 2 v * -w * 2 L 2 (Ω) d + |v * -w * | 2 H 1 (Ω) d . (46)
Using a reverse triangle inequality,

j(v, w f + δ n f ) -j( ŵ, w f + δ n f ) -j(v, v f + δ n f ) + j( ŵ, v f + δ n f ) ≤ Z Γ c µ |σν (v) -σν ( ŵ)| ˛|w f + δ n f | -|v f + δ n f | ≤ Z Γ c |µ||σν (v) -σν ( ŵ)||v f -w f | Z Γ c |σν (v) -σν ( ŵ)||v f -w f |.
Using the Cauchy-Schwarz inequality and the trace inequality (32),

j(v, w f + δ n f ) -j( ŵ, w f + δ n f ) -j(v, v f + δ n f ) + j( ŵ, v f + δ n f ) σν (v) -σν ( ŵ) L 2 (Γ c ) v f -w f L 2 (Γ c ) d 1 hc σν (v) -σν ( ŵ) L 2 (Ω c ) v * -w * L 2 (Ω c ) d 1 hc |v -ŵ| H 1 (Ω c ) d v * -w * L 2 (Ω c ) d . (47)
Furthermore, using (35) and the inverse inequality (33),

|v -ŵ| H 1 (Ω c ) d = |vc -ŵc| H 1 (Ω c ) d + |v * -ŵ * | H 1 (Ω c ) d = |q n+1 (v * ) -q n+1 ( ŵ * )| H 1 (Ω c ) d + |v * -ŵ * | H 1 (Ω c ) d |v * -ŵ * | H 1 (Ω c ′ ) d + |v * -ŵ * | H 1 (Ω c ) d |v * -ŵ * | H 1 (Ω c ′ ) d 1 h c ′ v * -w * L 2 (Ω c ′ ) d . (48)
Collecting inequalities (47) and (48), and since

h c ′ ≤ hc, j(v, w f + δ n f ) -j( ŵ, w f + δ n f ) -j(v, v f + δ n f ) + j( ŵ, v f + δ n f ) ≤ 1 h 2 c ′ v * -ŵ * L 2 (Ω) d v * -w * L 2 (Ω c ) d .
Using the inverse inequality (33),

a(vc -ŵc, w * -v * ) 1 h 2 c ′ ŵc -vc L 2 (Ω) d v * -w * L 2 (Ω) d .
Collecting these different estimates,

Φ n (v * ) -Φ n ( ŵ * ) L 2 (Ω) d = v * -w * L 2 (Ω) d " ∆t h c ′ « 2 v * -ŵ * L 2 (Ω) d .
Hence, if the ratio ∆t h c ′ is sufficiently small, the mapping Φ n is a contraction. The Banach fixed-point theorem guarantees that the problem has a unique solution.

Remark 4 In the above proof, the inertial term is essential. By strengthening the coercivity of ã, it enables to prove that Φ n is a contraction (for a time step sufficiently small). In the static case, without the help of the inertial term, this fixed-point proof works only for a certain range of physical parameters, for instance when the Young modulus is large compared with the friction coefficient [START_REF] Kikuchi | Contact problems in elasticity: a study of variational inequalities and finite element methods[END_REF]Theorem 11.4].

To conclude this part, we formulate the energy balance. We define the energy at time t n as

E n := 1 2 (Au n , u n ) + 1 2 (M * v n , v n ). (49)
At each time t n , there exist

λ n c ∈ ∂I K (u n c ) and λ n f ∈ ∂ 2 j(u n , v n f ) such that M * a n * + Au n + λ n c + λ n f = L(t n ). (50)
Proceeding as in [START_REF] Krenk | Energy conservation in Newmark based time integration algorithms[END_REF], it is readily shown that

E n+1 -E n = - 1 2 (λ n c + λ n+1 c , u n+1 -u n ) - 1 2 (λ n f + λ n+1 f , u n+1 -u n ) + 1 2 (L n + L n+1 , u n+1 -u n ). ( 51
)
5 Convergence of the fully discrete solutions

We fix the space discretization and we build the approximate solutions ω ∆t : [0, T ] → V as follows:

ω ∆t (t) := u n + v n * (t -t n ) + 1 2 a n+ 1 2 * (t -t n ) 2 ∀t ∈ [t n , t n+1 ), (52)
ω ∆t (T ) := u N . (53)
It is readily verified that, by construction, ω ∆t ∈ C 0 ([0, T ]; V ) and ω ∆t * ∈ C 1 ([0, T ]; V * ). Furthermore, ω ∆t ∈ W 1,∞ (0, T ; V ). We are now going to prove the convergence of these approximate solutions to the semi-discrete solution u of Problem 1. In this section, the notation A B means that A ≤ cB with a constant c independent of ∆t, but which can depend on h. We assume without loss of generality that ∆t ≤ 1.

Lemma 4 Let (u n , v n , a n ) solve, for all n ∈ {0, . . . , N }, Problem 2. Then, for ∆t small enough,

u n 1, v n * 1, a n * 1. ( 54
)
Proof (i) Let n ∈ {0, . . . , N }. From (50) we deduce A c u n + λ n c = L c (t n ), and then, λ n c u n + L(t n ) . Owing to the inequality [START_REF] Khenous | Mass redistribution method for finite element contact problems in elastodynamics[END_REF], we obtain λ n f u n . Hence, owing to the equilibrium equation (50), a n * u n + L(t n ) . (ii) Using the energy balance (51), it follows that

E n+1 -E n " u n + u n+1 + L(t n ) + L(t n+1 ) " u n+1 -u n .
Observing that

u n+1 -u n ≤ u n+1 * -u n * + q n+1 (u n+1 *
)q n+1 (u n * ) + q n+1 (u n * )q n (u n * )

u n+1 * -u n * + L(t n+1 ) -L(t n ) u n+1 * -u n * + L(t n+1 ) + L(t n ) ,
we infer

E n+1 -E n " u n + u n+1 + L(t n ) + L(t n+1 ) " " u n+1 * -u n * + L(t n+1 ) + L(t n ) " . (55)
Using ( 38),

E n+1 -E n ∆t " u n + u n+1 + L(t n ) + L(t n+1 ) " " v n * + ∆t 2 a n+ 1 2 * + L(t n+1 ) + L(t n ) « . (56)
Thus, using the previous bound on a n * and a n+1 * , and since ∆t ≤ 1,

E n+1 -E n ∆t " u n + u n+1 + L(t n ) + L(t n+1 ) " " v n * + u n + u n+1 + L(t n+1 ) + L(t n ) " . (57)
Now, using Young's inequality and the coercivity of the energy E n ,

E n+1 -E n ≤ C 1 ∆t E n+1 + C 2 ∆t E n + C 3 ∆t " L(t n ) 2 + L(t n+1 ) 2 " ,
where C 1 , C 2 , C 3 are three constants independent of ∆t. Next,

E n+1 -E n ≤ C 1 ∆t (E n+1 -E n )+(C 1 +C 2 )∆t E n +C 3 ∆t " L(t n ) 2 + L(t n+1 ) 2 " ,
For ∆t ≤ 1/(2C 1 ),

1 2 (E n+1 -E n ) ≤ (C 1 + C 2 )∆t E n + C 3 ∆t " L(t n ) 2 + L(t n+1 ) 2 " , so that E n+1 -E n ∆t " E n + L(t n ) 2 + L(t n+1 ) 2
" .

Finally, using a discrete Gronwall lemma,

E n E 0 + n X j=0 ∆t L(t j ) 2 1.
Then, it is straightforward to obtain the estimates (54). (ii) Using standard compactness arguments [START_REF] Temam | Theory and numerical analysis[END_REF], there exists ω ∈ C 0 (0, T ; V ) such that ω * ∈ C 0 (0, T ; V * ), ω * ∈ L ∞ (0, T ; V * ), and, up to a subsequence, By definition of the approximate solutions ω ∆t and ω ∆t , and using relation (38), ∀n ∈ {0, . . . , N }, ∀t ∈ [t n , t n+1 ),

ω

Fig. 1

 1 Fig. 1 Coulomb condition (d = 2).

Fig. 2

 2 Fig. 2 Decomposition of the domain Ω; bullets (resp., circles) indicate nodes indexed by elements of the set N i (resp., N c ). The open sets Ω c and Ω c ′ are defined in Section 3.

(

  iv) Owing to Theorem 1, there exists a unique X ∈ AC([0, T ]; V * × V * ) satisfying (26) with the initial condition X(0) = " u 0 * v 0 * « . Therefore, there exists a unique u * ∈ C 1 (0, T ; V * ) such that u * ∈ AC([0, T ]; V * ) satisfying (20) with the initial conditions u * (0) = u 0 * and u * (0) = v 0 * . Owing to (

  e. in [0, T ]. (28) Taking v * = 0 and then v * = 2 u * in the above inequality, we obtain m(ü * , u * ) + a(u, u * ) + j(u, uf ) = l(t, u * ) a.e. on [0, T ].(29)Recalling that the family {φ i ν i } i∈N c is a basis of V c , we decompose uc on this basis yielding uc =P i∈N c u i φ i ν i . Define C 0 i := {t ∈ [0, T ]; u i = 0} and C - i := {t ∈ [0, T ]; u i < 0}.The sets C 0 i and C - i are respectively closed and open, and they form a partition of [0, T ]. On int(C 0 i ), ui φ i ν i = 0. Owing to (23), a(u, ui φ iν i ) = 0 on C - i . Finally, a(u, ui φ i ν i ) = 0 on int(C 0 i ) ∪ C - i ,and hence almost everywhere (since an open set in R is a countable union of open intervals, so that its boundary has zero measure). Hence, a(u, uc) = l(t, uc) a.e. on [0, T ]. (30) Using (30), we obtain m(ü * , u * ) + a(u, u) + j(u, uf ) = l(t, u) a.e. on [0, T ]. (31) Since u is absolutely continuous in time, by integrating in time (31), we obtain (27).

  To begin with, we reformulate Problem 2 by eliminating v n+1 * and a n+1 * . We set δ n * := -u n * -∆t 2 v n * and ε n * := -u n * -∆tv n * -∆t 2 4 a n * , and we rewrite v n+1 * and a n+1 * as

  vf ) ≥ l n+1 (z *y * )a(vc, z *y * ), ∀z * ∈ V * , (43) where we have set ã(v * , w * ) := ( Ã * v * , w * ) and l n+1 (v * ) := (L(t n+1 ), v * ). Taking z * := 2 ∆t (w * + δ n * ) in (43), then dividing by 2 ∆t , we obtain for all w * ∈ V * , ã(v * , w *

Theorem 5

 5 The following convergence results hold true as ∆t → 0,ω ∆t → u in C 0 ([0, T ]; V ), ω∆t * → u * in C 0 ([0, T ]; V * ), ω∆t * → ü * weakly * in L ∞ (0, T ; V * ),where u solves Problem 1.Proof (i) From the estimates (54), we deduce that

  ∆t → ω in C 0 ([0, T ]; V ), ω∆t * → ω * in C 0 ([0, T ]; V * ), ω∆t * ⇀ ω * weakly * in L ∞ (0, T ; V * ).(iii) Next, we introduce the auxiliary (piecewise constant in time) approximate solutionsω ∆t : [0, T ] → V and ζ ∆t * : [0, T ] → V * such that ζ ∆t * (t) := v n+1 * ∀t ∈ [t n , t n+1 ), ζ ∆t * (T ) := v N * , ω ∆t (t) := u n+1 ∀t ∈ [t n , t n+1), ω ∆t (T ) := u N .

This notion of upper semi-continuity is distinct from the upper semi-continuity for singlevalued functions.


¸ois Laudenbach

A Morse complex on manifolds with boundary

Keywords: 2000 Mathematics Subject Classification, 57R19 , pseudo-gradient, supported

Given a compact smooth manifold M with non-empty boundary and a Morse function, a pseudo-gradient Morse-Smale vector field adapted to the boundary allows one to build a Morse complex whose homology is isomorphic to the (absolute or relative to the boundary) homology of M with integer coefficients. Our approach simplifies other methods which have been discussed in more specific geometric settings.

Introduction

We consider a smooth compact manifold M of dimension n with a non-empty boundary ∂M. A smooth function is said to be Morse when its critical points lie in the interior of M, are not degenerate, and when its resriction to the boundary is Morse. The property of being Morse in this sense, with distinct critical values, is generic among the smooth functions on M. Let f : M → R be a Morse function. There are two types of critical points of f |∂M, called type N and D (we shall see later that N (resp. D) is like Neumann (resp. Dirichlet)): a critical point p ∈ ∂M is of type N (resp. D) when < df (p), n(p) > is negative (resp. positive), where n(p) denotes the outward normal to the boundary at p. Looking at the change of homotopy type of the sub-level set M a := {x ∈ M | f (x) ≤ a} when a is increasing, it is well known that change happens when a crosses a critical value of f |int(M) or of f |∂M only when they are of type N; no change happens when crossing a critical value of type D.

Thirty years ago, I tried to find Morse inequalities in this setting. Of course, f gives rise to a Morse function Df on the double manifold DM := M ∪ ∂M M. But Morse inequalities for Df are not sharp. At that time I did not succeed in finding a geometrical Morse complex in the case of manifolds with boundary. I had even kept in mind the idea that such a complex should not exist. Apparently, the problem seems to be still open, at least in the setting of generic assumptions on the boundary (see below).

In a seminal paper [START_REF] Witten | Supersymmetry and Morse theory[END_REF], E. Witten introduced a deformed Laplacian (now called the Witten Laplacian) on a closed Riemannian manifold equipped with a Morse function and deduced in particular an analytic proof of the Morse inequalities. Recently, Francis Nier explained to me various works concerning the case of manifolds with non-empty boundary. K. Chang and J. Liu [START_REF] Chang | A cohomology complex for manifolds with boundary[END_REF] introduced two Witten complexes of differential forms constrained to satisfy some boundary conditions. Two types of boundary conditions can be distinguished: Dirichlet conditions which cancel tangential components or Neumann conditions which cancel normal components. These authors made the analysis easier by considering only flat metrics near the critical points. These boundary problems with general metrics have been recently studied in [START_REF] Helffer | Quantitative analysis of metastability in reversible processes via a Witten complex approach: the case with boundary[END_REF] (Helffer-Nier) for the Dirichlet problem, in [START_REF] Koldan | Semiclassical asymptotics on manifolds with boundary[END_REF] (Kolan-Prokhorenkov-Shubin) for Dirichlet and Neumann, and in [START_REF] Peutrec | Small eigenvalues of the Neumann realization of the semiclassical Witten Laplacian[END_REF] (Le Peutrec) for the Neumann problem, with additional developments concerned with the asymptotic analysis.

In case of the Dirichlet conditions [START_REF] Chang | A cohomology complex for manifolds with boundary[END_REF] [10], when the deformation parameter h is small enough, the De Rham complex Ω * contains a finite dimensional sub-complex F * (of R-vector spaces) whose cohomology (when M is orientable) is isomorphic to the relative cohomology H * (M, ∂M; R). In that case, a basis of F * is in a bijective correspondence with the critical points of f in int(M) and those of type D on the boundary (up to a shift of their grading). In case of the Neumann boundary conditions [START_REF] Chang | A cohomology complex for manifolds with boundary[END_REF] [13], a similar result holds: F * is generated by the critical points of f in the interior and by those of type N on the boundary; the cohomology of F * is isomorphic to the singular cohomology of M with real coefficients. Hence, Morse inequalities follow. Let us also mention work by M. Braverman and V. Silantyev ( [START_REF] Braverman | Kirwan-Novikov inequalities on a manifold with boundary[END_REF]) which is in the same spirit and deals with the Morse-Novikov theory. But they introduce some extra condition which obliges them to exclude for instance the standard annulus in the plane equipped with the height function.

There are also approaches motivated by Floer homology. In [START_REF] Akaho | Morse homology and manifolds with boundary[END_REF], M. Akaho considers a very specific geometric situation along the boundary. In [START_REF] Kronheimer | Monopoles and three-manifolds[END_REF], P. Kronheimer and T. Mrowka consider functions and their gradients having extensions to the double manifold DM which are invariant by the canonical symmetry. Both settings are non-generic, and the setup and analysis are complicated by the presence of isolated trajectories in the boundary which preserves Morse index.

These results (mainly coming from analysis) beg the existence of a simpler geometric Morse complex associated to any Morse function on a manifold with boundary. In this note, we are going to perform this program, working with Z-coefficients (or orientation-twisted coefficients) rather than with real cohomology as in Witten's work. We use the following notation:

-C k denotes the set of critical points of f : int(M) → R of index k; -N k denotes the set of critical points of f : ∂M → R of type N and index k; -D k denotes the set of critical points of f : ∂M → R of type D and index k -1 (notice that such a point has index k in the double manifold DM). -| • | stands for the cardinality of the mentioned finite set.

Theorem A. Let F N * be the free graded Z-module generated by C * ∪ N * . There exists a differential ∂ :

F N * → F N * -1 , making (F N
As usual, Morse inequalities follow (see J. Milnor [START_REF] Milnor | Morse theory[END_REF], p. 28, or R. Bott [START_REF] Bott | Lectures on Morse theory, old and new[END_REF], p. 338). They are contained in a polynomial identity. Let M N f (T ) be the Morse polynomial of type N, where T is the variable:

M N f (T ) = k |C k ∪ N k | T k .
From this, we deduce another family of Morse inequalities. Let M D f (T ) be the Morse polynomial

M D f (T ) = k |C k ∪ N k | T k .
Let P or (M,∂M ) be the relative Poincaré polynomial

P or (M,∂M ) (T ) = k rank H k (M, ∂M; Z or ) T k ,
which is nothing but the symmetric polynomial of P M (T ):

P or (M,∂M ) (T ) = T n P M (1/T ). Corollary B. We have M D f (T ) -P or (M,∂M ) (T ) = (1 + T )Q D (T )
, where Q D (T ) is a polynomial with non-negative coefficients.

Of course, corollaries A and B, which are a direct consequence of theorems A and B respectively, are due to Chang & Liu [START_REF] Chang | A cohomology complex for manifolds with boundary[END_REF].

I am very indebted to Francis Nier for encouraging me to return to this question and for his careful reading of a first version. I also thank Jonathan Bloom and Claude Viterbo for useful information and comments.

Proof of Theorem A

2.1. An adapted pseudo-gradient. We introduce a pseudo-gradient vector field X for the Morse function f adapted to the boundary in the following sense (in case of closed manifolds it is a reformulation of [START_REF] Meyer | Energy functions for Morse-Smale systems[END_REF] by K. Meyer; a slightly more restrictive, but still generic, definition is given in [START_REF] Grines | Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds[END_REF]). 1) X.f < 0 except at the critical points in int(M) and at the critical points of type N on the boundary; 2) X points inwards along the boundary except near the critical points of type N where it is tangent to ∂M; 3) if p ∈ int(M) is a critical point, X is hyperbolic at p and the quadratic form q := X lin . d 2 p f is negative definite; here X lin denotes the linear part of X at p (that is, X lin = DX(p)) and the second derivative d 2 p f at p is thought of as a quadratic function defined near p which is derived by X lin ; 4) let p ∈ ∂M be a critical point of type N; there are coordinates

x = (y, z) ∈ R n-1 × R of M near p, such that M = {z ≥ 0} and f (x) = f (p)+z+q(y)
, where q is a non-degenerate quadratic form; it is required that X is a vector field tangent to the boundary, vanishing and hyperbolic at p, such that X lin . (q(y) + z 2 ) is negative definite; 5) X is Morse-Smale in the sense that the global unstable manifolds and the local stable manifolds are mutually transverse.

Proposition. Given the Morse function f , there exists an adapted pseudo-gradient X.

Proof. Look first at condition 4). The existence of coordinates where f reads as f (x) = f (p) + z + q(y) mainly follows from the Morse lemma with parameters (or local stability of Morse functions), up to the addition of a function depending on z only. If c(z) denotes the critical value of y → f (y, z), the type N assumption implies dc(z) dz (0) > 0, hence the normal form holds after changing z by applying the inverse function theorem. Moreover, an easy calculation shows that, away from p, X lin . q(y) + z 2 ) < 0 implies X lin . q(y) + z) < 0. Therefore, X is a pseudo-gradient for f , that is X.f < 0 near p (except at p). Now the local existence of X is clear and local pseudo-gradients can be glued together by a partition of unity. By construction, X is positively complete. Let X t denote its flow; it is defined on an open set of

M × R containing M × [0, +∞). The global unstable manifold W u (p) of a critical point p of index k is the image of a non-proper embedding of R k into M.
There is also a local stable manifold W s loc (p) which is diffeomophic to R n-k when p ∈ int(M), or to R n-k ∩ {z ≥ 0} when p is a type N critical point in the boundary (here R n-k is a space on which q(y) + z 2 is positive definite). It is properly embedded when it is truncated to the sub-level set M f (p)+ε . Following Smale [START_REF] Smale | On gradient dynamicalsystems[END_REF], condition 5) is generically fulfilled among the vector fields meeting conditions 1)-4).

If p has index k the frontier of W u (p), that is the set of points in the closure which are not in W u (p), is contained in the union of the unstable manifolds of critical points of index less than k. An orientation of W u (p) is chosen arbitrarily. Then W s loc (p) is co-oriented by the orientation of W u (p).

Given a pair of critical points (p, q) respectively of index k and k -1, there are only finitely many flow lines (up to translation in time) X t (x), t ∈ R, such that X t (x) → p as t → -∞ and X t (x) → q as t → +∞. Each such line has a sign according to the co-orientation of W s loc (q) with respect to the orientation of W u (p). Let m pq be the algebraic sum of these signs, summing up over all connecting orbits from p to q.

2.2. The boundary morphism. We are going to define ∂ : F k → F k-1 à la Witten-Floer (see [START_REF] Witten | Supersymmetry and Morse theory[END_REF][START_REF] Floer | Witten's complex and infinite dimensinal Morse theory[END_REF]). Let p ∈ C k ∪ N k be a generator. The boundary morphism is defined by:

∂ < p >= q m pq < q >, where q runs in C k-1 ∪ N k-1 . Proposition. We have ∂ • ∂ = 0.
Proof. Consider p, a critical point of index k, and q, a critical point of index k -2. We have to prove that < ∂ • ∂(p), q >= 0. In [START_REF] Laudenbach | On the Thom-Smale complex[END_REF] it is given a complete description of the closure of W u (p); only the effect of critical points of index k -1 is useful for yielding the following description. In a level set L := {f (x) = f (q) + ε}, we look at the trace A of W s loc (q), a (nk + 1)-dimensional sphere or proper disk, and the trace B of the closure of W u (p). The intersection A ∩ B lies in the interior of L since W u (p) lies in int(M) except near p when p ∈ ∂M. It is made of simple closed curves and arcs having end points in common and whose interior are mutually disjoint. The closed curves and the open arcs correspond to connecting orbits from p to q; the end points correspond to broken connecting orbits going through some critical points of index k -1. Notice that the sign of m pz m zq does not depend on the chosen orientation of the unstable manifold of z, an index (k -1) point connected to p and to q. Each open arc in A ∩ B is oriented. Then one of his end points is equipped with +, the other with -, and the sum of all these signs is < ∂ • ∂(p), q >. Hence, it is zero.

2.3.

Invariance. (This property is not needed in the proof of theorems A and B, but it is interesting in itself). Of course, the complex (F N , ∂) depends on the Morse function f and on an adapted pseudo-gradient X. Given two pairs (f 0 , X 0 ) and (f 1 , X 1 ) where, for i = 0, 1, f i is a Morse function and X i is an adapted pseudo-gradient, they are connected by a path of pairs (f t , X t ), t ∈ [0, 1], if we allow us to cross "codimension 1 accidents" which arrive at finitely many times t 1 , t 2 , ...t ℓ . When t is distinct of those times, (f t , X t ) is a pair of a Morse function and an adapted pseudo-gradient. The list of the possible accidents is yielded by appropriate transversality theorems. We list below the "codimension 1 accidents" for a function f ; and those of pseudo-gradient X. Here, it is convenient to set When crossing an accident f1), a pair of critical points of consecutive indices is created/cancelled in int(M). When crossing an accident f2), a pair of critical points of consecutive indices and of the same type N or D is created/cancelled for f ∂ in ∂M. Crossing an accident f3) can be modelled as follows: there are local coordinates near p, x = (y, z) ∈ R n-1 × R where M = {z ≥ 0} and f (x) = f (p) + z 2 + q(y) where q is a non-degenerate quadratic form (say of index k). For crossing the accident one leaves this function fixed and translates M by (y, z) → (y, z + t), t ∈ [-ε, +ε]. For t = ε, we have a point of type N and index k on the boundary; for t = -ε, we have a point of type D on the boundary and a new point of index k in int(M). The model for crossing f4) is similar but f (x) = f (p)z 2 + q(y). When moving from t = ε to t = -ε, a point of type D and index k on the boundary is replaced by a point of type N and index k on the boundary and a point of index k + 1 in the interior.

f ∂ := f |∂M. f1) f has a degenerate critical point at p ∈ int(M) and rank d 2 f (p) = n -1; f2) f ∂ has a degenerate critical point at p ∈ ∂M, rank d 2 f ∂ (p) = n
We now list the "codimension 1 accidents" for a pseudo-gradient X; they never happen at the same time as a "codimension 1 accident" of the function whose X is a pseudo-gradient. g1) creating/cancelling a pair of connecting orbits from a point of index k to a point of index k -1 (the considered critical points may lie in int(M) or in ∂M); g2) there is an orbit connecting two points of the same index.

Proposition. Let (F i

* , ∂ i ) be the complex associated to the pair of a Morse function and a pseudo-gradient (f i , X i ), i = 0, 1. Then they are quasi-isomorphic: there is a chain morphism from one to the other inducing an isomorphism in homology.

Proof. According to the preceding discussion, we may assume that there exists a path (f t , X t ) with one accident only. The proposition is proved by examining each accident. When crossing f1), an acyclic complex of rank 2 (that is, 0 → Z ∼ = -→ Z → 0) is added or removed. The same happens when crossing f2), if the considered points are of type N; if they are of type D the complex is unchanged. When crossing f3), the complex remains unchanged. When crossing f4), again an acyclic complex of rank 2 is added or removed. Concerning the accidents of X, there are those which are encountered in the analogous discussion for closed manifolds (see [START_REF] Laudenbach | On the Thom-Smale complex[END_REF]). In each case we get the desired quasi-isomorphism.

The homology of (F

N * , ∂). Proposition. The homology of (F N * , ∂) is isomorphic to H * (M; Z).
Proof. According to the next lemma, it is allowed to assume that there are no critical points of type N. Let us achieve the proof under this assumption. In that case the complex does not "see" the boundary since all connecting orbits lie in int(M). Even, the global unstable manifolds lie in the interior of M. So, we can deal with such a function as on a closed manifold. For instance, an adapted pseudo-gradient being chosen, it is possible to reorder the critical values so that the function becomes self-indexing (the critical value of a critical point is its index). Indeed, we recall the following fact in Morse theory (see Cerf [START_REF] Cerf | La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie[END_REF], II 2.3).

Let (f, X) be a Morse function and an adapted pseudo-gradient. Let p and q two critical points with f (p) > f (q). Assume that the open interval f (q), f (p) contains no critical value and that there are no connecting orbits from p to q. Then there exists a path of Morse functions f t , t ∈ [0, 1], with f 0 = f , f 1 (p) < f 1 (q) = f (q) and X is a pseudogradient for every f t .

Once f is self-indexing, M has the homotopy type of a CW-complex, with one cell of dimension k for each critical point of index k (see Milnor [START_REF] Milnor | Lectures on the h-cobordism theorem[END_REF], theorem 7.4, p. 90). In that case, our Morse chain complex is exactly the so-called cellular chain complex. By the cellular homology theorem (see Milnor [START_REF] Milnor | Characteristic classes[END_REF], theorem A.4 p. 263), its homology is the homology of M. So the proof will be finished after the following lemma.

Lemma. Let (f, X) be a pair of a Morse function on M and an adapted pseudo-gradient. Let p be a critical point of type N in ∂M and U be a neighborhood of p. Then there exist a path of functions f t and an isotopy ϕ t of embeddings M → M, t ∈ [0, 1], with f 0 = f, ϕ 0 = Id, satisfying the following properties :

1) the support of both f t -f and ϕ t is contained in U; 2) f t is Morse when t = 1/2; 3) f 1 has one critical point p 1 in U ∩ int(M); 4) f 1 |∂M has one critical point in U ∩ ∂M and it is of type D; 5) ϕ 1 (W u (p))
is the unstable manifold of p 1 for an adapted pseudo-gradient X 1 of f 1 , whose other invariant manifolds are those of X. In particular, the Morse complexes associated to X and X 1 are the same.

Of course the accident at time t = 1/2 is f3) in list 2.3.

Proof. We start with a model for f in coordinates x = (y, z) ∈ R n-1 × R on a small open neighborhood U of the critical point p: f (x) = f (p) + z + q(y), where q(y) is a non-degenerate quadratic form and M ∩ U ⊂ {z ≥ 0}. We consider f 1/2 defined on U by f 1/2 = f (p) + q(y) + z 2 . Let ρ : [0, δ] → [0, 1] be a smooth bump function: ρ(s) = 1 for s close to 0 and ρ(s) = 0 for s close to δ; here δ is small enough so that the support of (y, z) → ρ( y ), ρ(z) is a compact set in U. Without loss of generality, we may assume that the support of ρ contains in its interior the connected domain of ∂M along which X is tangent to the boundary. For ε > 0 small enough, we look at the restriction of f 1/2 to {z ≥ ερ( y )}. It is conjugate to f |M ∩ U because both functions have no critical points in U and have restrictions to ∂M ∩ U which are Morse with one critical point of the same index. Let M 0 be the closure of M \ {0 ≤ z ≤ ερ( y )} and ψ : M 0 → M be a diffeomorphism such that both functions f 0 := ψ * f and f 1/2 have the same germ along {z = ερ( y )}. For t ∈ [0, 1], introduce the manifold M t and its function f t obtained from (M 0 , f 0 ) by gluing {ε(1 -2t)ρ( y ) ≤ z ≤ ερ( y )} endowed with f 1/2 . Up to a diffeomorphism ψ t : M t → M, this path f t is the desired one.

M 1 = M 0 ∪ the lens z y z = -ερ( y ) p M 0 p y s X-orbits {z = +ερ( y )} ∩ W s loc (p) z s (y s , z s ) stands for coordinates of W s (p)
More precisely, the isotopy ψ t , t ∈ [0, 1], can be chosen so that ψ 1/2 = Id. Recall that X is a pseudo-gradient in U for both functions f and f 1/2 (see 2.1). At t = 1/2, the deformation of functions is stopped and a small isotopy χ t , is applied for pushing W u (p) \ {p} into int(M); it is chosen to be supported in int(M 1 ) and to leave W s loc (p) fixed. As this isotopy is C ∞ small, X 1 := χ 1 * X is still a pseudo-gradient of f 1/2 and χ 1 (W u (p)) is its unstable manifold of p.

Moreover, X 1 points toward int(M) along U ∩ {z = 0} except along W s loc (p). If η > 0 is small, X 1 points inwards along ∂M 1/2+η ∩ U. Therefore, if the path of deformation (M t , f t ) is stopped at time t = 1/2 + η we already get the desired conclusion.

Remark. (C. Viterbo) Instead of deforming (f, X) as in the preceding lemma, one could invoke Conley's theory of isolating blocks [START_REF] Conley | Isolated invariant sets and the Morse index[END_REF] which in our setting states as follows (compare [START_REF] Floer | Witten's complex and infinite dimensinal Morse theory[END_REF] section 3 for a similar situation, but our flow is not negatively complete).

There exists a finite filtration M -1 = ∅ ⊂ M 0 ⊂ . . . ⊂ M k . . . ⊂ M n = M which is positively invariant by the flow X t and such that M k \ M k-1 contains all critical points in C k ∪ N k and every positive orbit in M k which does not go to a critical point enters M k-1 without getting out of.

Our F k can be identified to H k (M k , M k-1 ; Z) and our ∂ is nothing but the connecting morphism Milnor [16] A.4). 2.5. Complement. We have statements similar to theorem A for every local system of coefficients (or flat bundle with discrete fiber). Such a bundle is given by a representation of the fundamental group π := π 1 (M, x 0 ) (where x 0 is a base point) into an Abelian discrete group. For instance, Z or is defined by the first Stiefel-Whitney class of τ M, the tangent bundle of M: w 1 : π → Z/2. The complex F N,or * has the same graded module. But the differential is changed. At the beginning some arc is chosen from the base point to each critical point. Therefore a connecting orbit γ from p to q defines a loop γ and the contribution of γ to m pq is twisted by w 1 (γ). So we have a new differential ∂ or . The proof that ∂ or • ∂ or = 0 is alike. In that case, theorem A states as the following: The homology of (F N,or * , ∂ or ) is isomorphic to H * (M; Z or ). New Morse inequalities follow. A good example is given by the compact Möbius band in R 3 with a height function. In that case the Betti numbers of H * (M; Z or

H k (M k , M k-1 ; Z) → H k-1 (M k-1 , M k-2 ; Z) in the long exact homology sequence of the triple (M k , M k-1 , M k-2 ) (as in
∂ := f |∂M (then it belongs to D k ) is of type N and index n -k for -f ∂ . A critical point in ∂M which is of type N and index k for f ∂ is of type D and index n -k -1 for -f ∂ ; then it belongs to N n-k (-f ).
There is another change to make. Indeed, the vector field -X is not adapted as it points outwards along the boundary except near the critical points of former type N. So we appeal another vector field X -which is adapted to -f . With the data (-f, X -) we can form a Morse complex. But we insist to keep the initial grading, that is: F D k is freely generated by C k ∪ D k , a set of critical points of index nk for -f . The pseudo-gradient X -yields a differential which is of degree +1 in this grading: it is a co-differential and the complex is a cochain complex. According to theorem A its (co-)homology is isomorphic to H * (M, Z), up to the grading which is changed into the complementary one. Therefore, it is the Poincaré dual of the homology, hence the cohomology with integer coefficients twisted by the orientation. Thus theorem B is proved.

As we did in a variant of theorem A in 2.5, we can twist the codifferential d by the orientation. We get a cochain complex (F D,or * , d or ) whose cohomology is isomorphic to H * (M, ∂M; Z). From this we get another Morse inequalities involving the same Morse polynomial:

M D f (T ) -P (M,∂M ) (T ) = (1 + T )Q ′ (T ).
We recall here, that homology and cohomology have the same Poincaré polynomial, as the free factor of H k (.) is the dual of the free factor of H k (.), as long as the same system of local coefficients is used for both. Thus, it is weaker than its summands.

3.3.

Relative homology and its pairing with absolute homology. 1 Here we need the following notation: for p ∈ C k ∪D k , W u (p, X -) will denote the unstable manifold of the pseudogradient vector field X -adapted to the function -f ; it is equipped with an orientation. If we want to think of it as a relative k-cochain, it is necessary to endow p with a local orientation of M. Then, if σ is an oriented k-simplex tranverse to W u (p, X -) it is allowed to count the intersection points in σ ∩W u (p, X -) with signs. It is the way of thinking of the complex (F D * , d) as a cochain complex with orientation-twisted coefficients calculating H * (M, ∂M; Z or ). Now, we consider the dual complex ( F D * , ď) in which the differential is the transpose of d. Then its homology is isomorphic to H * (M, ∂M; Z or ); here we use the universal-coefficient theorem for cohomology ( [START_REF] Spanier | Algebraic topology[END_REF], theorem 5.5.3) and the finite generation of (F D * , d). Geometrically, ď can be described as follows. Let p ∈ C k ∪ D k and q ∈ C k-1 ∪ D k-1 . We have to calculate mpq :=< ď(p), q >. It is the algebraic number of connecting orbits of X -from q to p counted with the orientation of W u (q, X -) and the co-orientation of W s loc (p, X -).

  -2 and df (p) = 0; f3) df (p) vanishes at some point p ∈ ∂M, d 2 f (p) and d 2 f ∂ (p) are both non-degenerate and have the same index; f4) df (p) vanishes at some point p ∈ ∂M, d 2 f (p) and d 2 f ∂ (p) are both non-degenerate and index d 2 f (p)= index d 2 f ∂ (p) + 1.

3 . 1 .

 31 ) are: b 0 = 0, b 1 = 0, b 2 = 0; in relationship to theorem B, we also list the Betti numbers of the relative homology H * (M, ∂M; Z or ): b 0 = 0, b 1 = 1, b 2 = 1. Proof of Theorem B and complements 3.Upside down. We look at the Morse theory of the Morse function -f . The critical points are the same but their indices and types are changed. A critical point in int(M) of index k for f has index nk for -f . A critical point in ∂M which is of type D and index k -1 for f

3. 2 .

 2 Morse inequalities from the double. On the one hand, the set C k (Df ) of critical points of index k for the "double" Morse function Df defined on the double DM of M consists of two copies of C k , one copy of N k and one copy of D k . On the other hand, H k (DM; Z) ∼ = H k (M; Z) ⊕ H k (M, ∂M; Z). Indeed, the inclusion M ֒→ DM gives rise to a long exact sequence. . . → H k (M) → H k (DM) → H k (M, ∂M) → H k-1 (M) → . . . ;but this sequence splits since any relative cycle, glued to its symmetric one, gives a cycle in DM. Thus the Morse inequalities for Df is just obtained by adding both Morse inequalities obtained previously, the one for M N f -P M and the one for M D f -P (M,∂M ) : M Df (T ) -P DM (T ) = (1 + T )Q ′′ (T ).

I thank M. Akaho for asking me about the relative homology.

For making the pairing H k (M, ∂M; Z or )⊗H n-k (M; Z) → R explicit, we need to ask a general position of X -with respect to X: for each pair of integers (k, k ′ ) and for each p

it is easy to see that only compact parts of these invariant manifolds could intersect; therefore, there are finitely many intersection points. The sign is calculated by using three data; the orientation of W u (p ′ , X), the orientation of W u (p, X -) and the local orientation of M at p.

Arguing this way, we avoid to identify F D * to a quotient complex of F N * by a subcomplex isomorphic to the Morse complex of f |∂M. Such an identification should require difficult formulas similar to those in [9], section 2.4.