Alessandro Giust
email: alessandro.giust@dk-compmath.jku.at

Bert Jüttler
email: bert.juettler@jku.at

Angelos Mantzaflaris
email: angelos.mantzaflaris@inria.fr

Local (T)HB-spline projectors via restricted hierarchical spline fitting

Keywords: hierarchical splines, truncation, quasi-interpolation, spline fitting

This paper is devoted to techniques for adaptive spline projection via quasi-interpolation, enabling the efficient approximation of given functions. We employ local least-squares fitting in restricted hierarchical spline spaces to establish novel projection operators for hierarchical splines of degree p. This leads to efficient spline projectors that require O(p d ) floating point operations and O(1) evaluations of the given function per degree of freedom, while providing essentially the same accuracy as global approximation. Our spline projectors are based on a unifying framework for quasi-interpolation in hierarchical spline spaces. We present a detailed comparison with the scheme of Speleers and Manni (2016).

Introduction

Geometric modeling relies on the extensive use of spline technology for the description and manipulation of arbitrary shapes [START_REF] Farin | Curves and surfaces for CAGD: a practical guide[END_REF]. More recently, the introduction of Isogeometric Analysis (IgA) by [START_REF] Hughes | Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement[END_REF] proved the effectiveness of splines also in the context of numerical simulation, where it has been noted that this technology leads to a better accuracy per degree of freedom than classical finite element methods.

The standard approach to multivariate spline functions -both in geometric modeling and isogeometric simulation -relies on the use of the tensor-product construction. As a major drawback, this construction prevents any possibility of employing adaptive (rather than uniform) refinement strategies. This fact has triggered intense research on adaptive spline refinement, resulting in a substantial number of publications. These include the rich literature on T-splines that were introduced by [START_REF] Sederberg | T-splines and T-NURCCs[END_REF] as a generalization of NURBS. The problem of missing adaptivity is addressed by control grids with T-junctions. Properties such as linear independence and suitability for IgA were further analyzed by [START_REF] Li | On linear independence of T-spline blending functions[END_REF]. The related approach of polynomial splines over hierarchical T-meshes (PHT-splines) was introduced by [START_REF] Deng | Polynomial splines over hierarchical T-meshes[END_REF] and has subsequently been employed both in geometric modeling and isogeometric analysis. Another approach was proposed by [START_REF] Dokken | Polynomial splines over locally refined box-partitions[END_REF], resulting in the framework of LR-splines.

We focus our attention on the hierarchical spline refinement originally introduced by Forsey and Bartels (1988). These spline spaces are constructed in terms of a sequence of subdomains representing different refinement levels, and they can be equipped with the basis of hierarchical Bsplines (HB-splines) that was introduced and analyzed by [START_REF] Kraft | Adaptive und linear unabhängige Multilevel B-Splines und ihre Anwendungen[END_REF]. Even though HB-splines do not form a partition of unity (PoU), several important properties such as linear independence and approximation power were established in Kraft's seminal work. In order to restore the PoU Figure 1: Sequence of subdomains and corresponding hierarchical mesh. From left to right: Ω 0 , Ω 1 , Ω 2 and final hierarchical mesh the approximation order of the projector. In Section 6 we describe some experimental tests that illustrate relevant features of our construction. Finally, Section 7 concludes the paper with some remarks and a summary of the achieved results.

Preliminaries

We recall the construction of hierarchical spline bases and the use of the truncation mechanism. We also summarize several known results related to quasi-interpolation methods in this context.

Hierarchical B-splines

In order to define hierarchical B-splines (HB-splines) on a bounded domain Ω 0 ⊂ R d , we consider d-variate tensor product spline spaces V satisfying

V 0 ⊂ V 1 ⊂ • • • ⊂ V N ,
where the upper index = 0, . . . , N is called the level. All spline spaces have the same degree p, and they are defined by d univariate bi-infinite knot sequences 2 -Z at level , one per coordinate direction. The spline spaces are spanned by uniform tensor-product B-splines

B = {β i , i ∈ Z d }
of degree p possessing maximum smoothness C p-1 . The restrictions of the spline functions in V to cells of level

2 -(Z d + [0, 1] d )
are simply tensor-product polynomials of degree p. While the results presented below can be extended to a more general setting (such as splines with non-uniform knot vectors), we try to keep the exposition simple by restricting ourselves to the uniform setting.

In addition, we consider a sequence of subdomains Ω ⊂ R d satisfying

Ω 0 ⊇ Ω 1 ⊇ • • • ⊇ Ω N ,
see Fig. 1. Moreover, it is assumed that Ω is created by forming the union of finitely many square blocks (which are not required to be mutually disjoint) containing p+1 2 cells of level -1 per coordinate direction. Consequently, the subdomain Ω is a union of the (set-theoretic, i.e., without boundaries) supports of a finite number of B-splines β . Note that this includes the assumption that Ω 0 is the union of square blocks of cells of level -1 .

The level cells contained in Ξ = Ω \ Ω +1 are said to be active. These cells cover Ξ , since there are no cells that are only partially refined.

The corresponding hierarchical mesh (cf. Fig. 1) is formed by all the active cells of all levels. Hierarchical B-splines (HB-splines) are based on the selection mechanism devised by [START_REF] Kraft | Adaptive und linear unabhängige Multilevel B-Splines und ihre Anwendungen[END_REF]: A level B-spline is selected, i.e., it is included into the hierarchical basis, if its support is contained in the associated domain but not in the next finer one. We obtain the basis

H = {β i : i ∈ J , = 0, . . . , N } with the index sets J = {i ∈ Z d : supp β i ⊆ Ω ∧ supp β i Ω +1 } (1)
where we set Ω N +1 = ∅ and we consider supports with respect to the domain Ω 0 (on which we approximate the given function). The selected B-splines are denoted as active basis functions.

Any active cell of level belongs to the support of at least one function of that level, since Ω is assumed to be the union of supports of level B-splines.

For future reference we state an assumption that we will need in Theorem 5 (see Section 4.2).

Assumption I (Mesh grading). Only elements of two consecutive levels are present in the support of any selected basis function β i . Consequently, the active cells in the support of any selected basis function

• have level , or

• have either level or level + 1.

Hierarchical B-splines form a non-negative basis of the hierarchical spline space

H = span H = {σ ∈ V N : σ| Ω 0 \Ω +1 ∈ V | Ω 0 \Ω +1 ∀ = 0, . . . , N }, (2)
but they do not form a partition of the unity. This property can be restored by the truncation mechanism of [START_REF] Giannelli | THB-splines: the truncated basis for hierarchical splines[END_REF]. We define the single-level truncation operator trunc :

V → V as trunc (σ) = i∈Z d \J c i β i = σ - i∈J c i β i if σ = i∈Z d c i β i , (3)
and the corresponding multi-level version Trunc :

V → V N Trunc ( i∈Z d c i β i ) = trunc N trunc N -1 . . . trunc ( i∈Z d c i β i ) . . . (4)
Clearly, in order to apply the single level operators to different levels, the output of each operator has to be represented with respect to the basis of the next level. This can be done easily by the spline refinement rules, due to the nestedness of the spaces V . Truncated hierarchical B-splines (THB-splines) are obtained by applying multi-level truncation (4) to the basis functions of H,

T = {τ i = Trunc +1 (β i ) : i ∈ J , = 0, . . . , N } .
They form another basis, which is a non-negative partition of unity, of the hierarchical spline space.

In addition, they possess the preservation of coefficients (PoC) property: By definition, any function σ ∈ H of the hierarchical spline space has local representations

σ| Ω 0 \Ω +1 = i∈Z d d i β i | Ω 0 \Ω +1 , = 1 . . . , N (5)
on each subdomain Ω 0 \ Ω +1 . All coefficients of the representation of σ with respect to the truncated basis are simply inherited from these representations,

σ = N =0 i∈J d i τ i . (6)
In other words, the coefficient of each THB-spline τ i coincides with the coefficient of its mother B-spline β i in the local representation [START_REF] Giannelli | Strongly stable bases for adaptively refined multilevel spline spaces[END_REF].

Quasi-interpolation in hierarchical spline spaces

We consider linear functionals λ i : C(R d ) → R, the construction of which will be discussed below. For any index set J ⊆ Z d and level , these define operators

Q J (f ) = i∈J λ i (f )β i (7)
that transform a continuous function f into a tensor-product spline. In particular, choosing J = Z d gives a scheme that generates an approximation in the full tensor-product spline space V . Each value λ i (f ) depends on the values of f in a certain subdomain, which is called the support of the linear functional λ i . More precisely, we have that

λ i (f ) = 0 if supp λ i ∩ supp f = ∅,
and supp λ i is the smallest set with this property. In order to benefit from the PoC property, linear functionals that satisfy the condition

supp λ i ⊆ supp β i ∩ (Ω \ Ω +1 ) if i ∈ J (8)
have been considered [START_REF] Speleers | Effortless quasi-interpolation in hierarchical spaces[END_REF]. The functionals λ i are said to define tensorproduct spline projectors if Q Z d f = f for all f ∈ V . Given functionals λ i with this property, two particular QI schemes for hierarchical splines have been studied:

• [START_REF] Kraft | Adaptive und linear unabhängige Multilevel B-Splines und ihre Anwendungen[END_REF] proposed to use the telescopic construction

K 0 f = Q 0 J 0 f, K f = K -1 f + Q J (f -K -1 f ), = 1, . . . , N , Kf = K N f .
• [START_REF] Speleers | Effortless quasi-interpolation in hierarchical spaces[END_REF] exploit the PoC property and define their scheme, which we will call the Speleers-Manni scheme (SMS), in terms of THB-splines,

Sf = N =0 i∈J λ i (f )τ i .
These authors also noted that both schemes are equivalent, i.e., they generate the same1 spline function Kf = Sf for any given f , provided that the one level QI schemes (7) are spline projectors with coefficient functionals that fulfill the condition (8). This condition also ensures that both schemes perform spline projection. However, both constructions involve the rather strong restriction (8) on the support of the linear functionals λ i . Fig. 2 (left) shows the support of a cubic (T)HB-spline whose support is refined, except for the top left cell. The associated coefficient is determined by the values of the function in that cell. This cell covers only a small portion of the support of the function, and does not even contain its maximum. Nevertheless, the two spline projectors evaluate the associated coefficient solely by examining the value of the given function within that cell.

We employ local fitting (Section 4) based on restricted hierarchical spline spaces (Section 3) for the construction of spline projectors for the hierarchical space H. These will take the form

P β f = N =0 i∈J µ i (f )β i and P τ f = N =0 i∈J ν i (f )τ i (9)
with linear functionals µ i (f ) and ν i (f ) for HB-and THB-splines, respectively. For future reference we note that

ν i (s) = d i for s ∈ H (10)
where d i is the coefficient appearing in the associated local representation (5), according to PoC (6).

Figure 2: Left: Support of a cubic HB-spline (entire box) and of a THB-spline (grey) of level 0 on a hierarchy with two levels, N = 1. The associated coefficient functional is supported on the larger cell (that belongs to Ω 0 \ Ω 1 ) in the top left corner. Right: Hierarchy of 2 levels Ω 0 and Ω 1 , subdomain ∆ = Ω0 highlighted in red. We have Ω0 = Ω 1 , hence Ĵ0 = ∅ and Ĵ1 J 1 .

Hierarchical splines on restricted domains

We consider the restricted domain, which is a subset ∆ ⊂ Ω 0 formed by active cells, i.e., by cells of the hierarchical mesh. We also associate with ∆ the nested restricted subdomains Ω = ∆ ∩ Ω , in particular Ω0 = ∆. Analogously to (2), the restricted spline spaces and subdomains define the restricted hierarchical spline space Ĥ. Note that this space Ĥ is not simply the restriction of the original one to the restricted domain ∆, see Fig. 2 (right). Still we have the following result:

Lemma 1. The restriction of the hierarchical spline space H to the restricted domain ∆ is a subspace of the restricted hierarchical spline space Ĥ, i.e., H| ∆ ⊆ Ĥ.

Proof. Consider s ∈ H. By definition, it satisfies

s| Ω 0 \Ω +1 ∈ V | Ω 0 \Ω +1 ∀ = 0, . . . , N, see (2). Since Ω0 \ Ω +1 = (∆ ∩ Ω 0 ) \ (∆ ∩ Ω +1 ) = ∆ ∩ (Ω 0 \ Ω +1 ) ⊆ Ω 0 \ Ω +1 , it also satisfies s| Ω0 \ Ω +1 ∈ V | Ω0 \ Ω +1 ∀ = 0, . . . , N,
and thus it is an element of the restricted hierarchical spline space Ĥ.

We denote with β i and τ i the HB-and THB-splines defined by the restricted spline bases with respect to the restricted subdomains Ω . Their indices form the sets

Ĵ = {i ∈ Z d : supp β i ⊆ Ω ∧ supp β i Ω +1 },
which are defined analogously to (1), but based on the hierarchy of nested restricted subdomains Ω and using the supports (which are denoted by supp) with respect to Ω0 .

It should be noted that the index sets J and Ĵ are generally not nested, i.e., J Ĵ in general. Thus, the restricted index sets are not simply the restrictions of their un-restricted counterparts, see Fig. 2 for a counterexample.

Clearly, the HB-splines are identical on ∆, i.e.

β i = β i | ∆ , (11)
if i ∈ J ∩ Ĵ . This, however, is not valid for THB-splines. Nevertheless, the THB-splines satisfy

τ i | Ω \ Ω +1 = τ i | Ω \ Ω +1 = β i | Ω \ Ω +1
We have the following simple result.

Lemma 2. A selected B-spline β i , i ∈ J , is present in the restricted hierarchical basis, i.e., i ∈ Ĵ , if its support is contained in the restricted domain, supp β i ⊆ ∆. This also applies to the associated truncated hierarchical B-spline.

In addition to the spline projectors introduced in ( 9), we will need the restricted spline projectors with respect to restricted domains ∆. The coefficient functionals will be denoted by μ i (f ) and ν i (f ) for HB-and THB-splines, respectively.

Given a THB-spline τ i , i ∈ Ĵ , we identify a simple sufficient condition2 on the restricted domain ∆, which guarantees that the coefficients of any hierarchical spline function s ∈ H are identical.

Proposition 3. Consider the coefficient of a level THB-spline τ j with index j ∈ J . The original and restricted THB-spline coefficients are identical,

ν j (s) = ν j (s| ∆ ) for all s ∈ H,
if the restricted domain ∆ contains the support of the associated HB-spline, i.e., if supp β j ⊆ ∆.

Proof. According to Lemma 2, the index j ∈ J also belongs to Ĵ , since the support of β j is contained in the restricted domain ∆.

Any spline function s ∈ H possesses local representations

s| Ω 0 \Ω +1 = i∈Z d d i β i | Ω 0 \Ω +1 , and s| Ω0 \ Ω +1 = i∈Z d d i β i | Ω0 \ Ω +1 ,
on the "rings" of the original and the restricted hierarchical spline space, respectively. The local linear independence of the B-splines, combined with (11), implies that the coefficients are identical, d i = d i , if the associated basis functions contribute to this representation, i.e., if β i | Ω0 \ Ω +1 = 0 (which implies β i | Ω 0 \Ω +1 = 0). In particular, this condition is satisfied for the considered index j ∈ J , hence

d j = d j . (12)
Indeed, the support of the associated basis function satisfies supp β j ⊆ ∆ = Ω0 according to the assumption, and supp β j ⊆ Ω +1 since j ∈ Ĵ , thus β j | Ω0 \ Ω +1 = 0. Finally we combine (12) with the PoC property, see (10), to conclude

ν j (s) = d j = d j = ν j (s| ∆ ).
We define the extended support of a HB-spline,

supp β j = aabb {supp β k i : supp β k i ⊃ supp β j , i ∈ J k , k = 1, . . . , -1} ,
where aabb denotes the axis-aligned bounding box. For future reference we state an assumption that we will need in Theorem 6 (see Section 4.3).

Assumption II (Strong mesh grading). Only elements of two consecutive levels are present in the extended support of any selected basis function β i . Consequently, the active cells in the extended support of any selected basis function • have level , or

• have either level or level -1, or

• have either level or level + 1. Now we extend the sufficient condition3 stated in the previous proposition to HB-splines.

Proposition 4. Consider a level HB-spline β j with index j ∈ J . The original and restricted HB-spline coefficients are identical,

µ j (s) = μ j (s| ∆ ) for all s ∈ H, (13)
if the restricted domain ∆ contains the extended support of the HB-spline, i.e., if supp β j ⊆ ∆.

Proof. According to Lemma 2, the index j ∈ J also belongs to Ĵ , since the support of β j , which is a subset of the extended support, is contained in the restricted domain ∆. Any given function s ∈ H possesses two representations on ∆,

s| ∆ = N k=0 i∈J k ν k i (s)τ k i | ∆ and s| ∆ = N k=0 i∈ Ĵk νk i (s| ∆ )τ k i , (14)
with respect to the restrictions of the globally defined THB-splines and with respect to the THBsplines defined by the restricted spline bases, respectively. The previous result guarantees the equality ν k i (s) = νk i (s| ∆ ) of the coefficients whenever the associated B-splines satisfy supp β k i ⊆ ∆. Both versions of the THB-splines admit representations in terms of HB-splines, from which they are constructed via truncation, see (3). These representations take the form

τ k i | ∆ = β k i | ∆ - N k =k+1 i ∈J k c k,k i,i β k i | ∆ and τ k i = βk i - N k =k+1 i ∈ Ĵk ĉk,k i,i βk i (15)
with certain truncation coefficients c k,k i,i and ĉk,k i,i The following observations are in order.

(i ) The corresponding truncation coefficients are identical, i.e. c k,k i,i = ĉk,k i,i , if the support of the associated HB-spline satisfies supp β k i ⊆ ∆.

(ii ) The truncation coefficients c k,k i,i and ĉk,k i,i are non-zero only if the supports satisfy supp β k i ⊆ supp β k i and supp βk i ⊆ supp βk i , respectively, Substituting (15) into ( 14) transforms the THB-spline representation into the HB-spline representation. After re-arranging the sums we arrive at

s| ∆ = N k=0 i∈J k µ k i (s)β k i | ∆ and s| ∆ = N k=0 i∈ Ĵk μk i (s| ∆ ) βk i .
In particular, the two coefficients considered in (13) evaluate to

µ j (s) = ν j (s) - -1 k=0 i∈J c k, i,j ν k i (s) ( ) and μ j (s| ∆ ) = ν j (s| ∆ ) - -1 k=0 i∈ Ĵ ĉk, i,j νk i (s| ∆ ) ( )
In both sums, the two coefficients c k, i,j and ĉk, i,j are nonzero only if supp β k i ⊇ supp β j and supp βk i ⊇ supp β j , according to (ii ). Moreover, under these conditions, the coefficients even take the same values, c k, i,j = ĉk, i,j , since supp β k i ⊆ supp β j ⊆ ∆, as noted in (i ). Finally we note that the latter condition also ensures that the associated THB-spline coefficients are identical, ν k i (s) = νk i (s| ∆ ), according to Proposition 3. These three observations imply that the two double sums ( ) take the same value. We complete the proof by using Proposition 3 once more, which confirms ν j (s) = ν j (s| ∆ ).

Defining spline projectors by local fitting

We recall the local fitting procedure and use it for defining THB-and HB-spline projectors.

Local fitting

We consider a linearly independent system of functions Φ = {φ k : k ∈ K} defined on a given restricted domain ∆, where K is a finite index set. We assume that the domain is equipped with nodes (i.e., sampling points) u s ∈ ∆ with associated positive weights w s , s ∈ S, where S is another finite index set.

Discrete least-squares fitting considers the over-constrained linear system Ac = f with

A = ( √ w s φ k (u s )) s∈S,k∈K , c = (c k ) k∈K , and f = ( √ w s f (u s )) s∈S ,
which is solved in the least-squares sense, resulting in c = (A T A) -1 A T f . A unique least-squares solution exists for certain configurations of nodes that are determined by the system Φ. The resulting function σ = s∈S c s φ s minimizes the Euclidean norm of the residual vector ( √ w s f (u s )-σ(u s )) s∈S among all functions in span Φ.

Two specific cases will be considered in the next two sections. Clearly, the result depends on the choice of nodes and weights. If we consider spline functions φ k , then the resulting function is the best approximation with respect to the L 2 norm if the quadrature rule defined by nodes and weights exactly integrates the products φ k φ k and f φ k , and it produces an approximation otherwise.

We focus on two possibilities to define the nodes and weights:

(a) We consider the Gauss points and weights on the restricted mesh. More precisely, we choose (p + 1) d Gauss points per element, and we scale the Gauss weights by the volume of the element. (b) We consider a tensor-product grid of interpolation points, which is unisolvent for the finest tensor product space that is active on ∆. All the weights are set to 1.

Appendix A describes the smart sampling method, which efficiently generates the points needed for the second choice (b). It will be shown that the total number of required function evaluations does not exceed 10 d |H|.

The THB-spline projector

We define the functionals ν i that generate the coefficients of the THB-spline projector. Given a fixed level 0 and an index i 0 ∈ J 0 , we choose the restricted domain ∆ = supp β 0 i0 and we consider the THB-splines τ i defined on it. These functions form the system Φ. Now, given a function f , we perform local fitting as described in the previous section and assign the value of the coefficient associated with τ 0 i0 to ν 0 i0 (f ). For the first choice (a) of nodes, the least-squares solution of the linear system is unique, since the matrix A T A is simply the mass matrix of the system Φ. The uniqueness is also given for the second choice (b), since one may consider the inner product defined by the tensor-product grid of interpolation points instead of the L 2 inner product. This procedure defines a spline projector (9) according to Proposition 3.

We analyze the computational costs per coefficient, focusing on the second choice (b) of nodes: The restricted domain ∆ consists of at most (p + 1) d cells of level 0 . The active cells belong to the levels 0 and 0 + 1 of the subdomain hierarchy, due to Assumption I about the mesh grading. The number of possible cases is thus bounded by 2 (p+1) d , since cells of level 0 + 1 are created by refining blocks of cells of level 0 .

Figure 3 lists the 44 possible configurations for p = d = 2. The upper bound is rather pessimistic. Assumption I also implies that the grid of nodes can be chosen such that it contains no more than C g p d points, where the constant C g does not depend on the polynomial degree p. See Appendix A for more details.

We may assume that all the matrices (A T A) -1 A T that may occur have been pre-computed and stored in a look-up table. In fact, it suffices to store the row vectors that correspond to the coefficients ν 0 i0 (f ). In each specific situation, finding the right entry in the table via binary search requires not more than log 2 (p+1) d = O(p d ) steps.

In addition, we need to evaluate f at a tensor-product grid of nodes containing O(p d ) nodes, and to compute the inner product with the pre-computed row vector. If smart sampling is employed, then the total number of function evaluations (which are assumed to take constant time) is O(|H|).

We summarize the results:

Theorem 5. Consider the functionals ν i defined by local fitting with respect to the restricted domain supp β i , using a unisolvent set of nodes.

(i) The linear operator P τ is a THB-spline projector.

(ii) When using the second choice of nodes, the evaluation of the spline coefficients requires O(p d ) flops and O(1) function evaluations per degree of freedom if Assumption I is satisfied.

In contrast to this, the first choice (a) leads to higher computational costs, since the values of f at O(p 2d ) nodes are used. Even though the first choice should give a better approximation of the (local) L 2 projection, our experiments did not lead us to observe significant differences between the computational results for the two different choices of the nodes.

In addition to this result, we also obtain a bound

|ν i0 (f )| ≤ C τ f ∞,supp β i (16) with the constant C τ = C g p d max all cases (A T A) -1 A T ∞
where . ∞ is the element-wise maximum norm of a matrix. Recall that C g p d is an upper bound on the number of evaluation points.

The HB-spline projector

Now we define the functionals µ i that generate the coefficients of the HB-spline projector.

• Restore the mesh grading by applying a suitable closure operation to the subdomain hierarchy. See [START_REF] Buffa | Complexity of hierarchical refinement for a class of admissible mesh configurations[END_REF] for a detailed discussion of mesh refinement algorithms.

Continue with

Step 2.

The refinement is driven by the maximum error. Given a tolerance ε, we mark an element e if

f -P γ f ∞,e ≥ ε, (18)
where either γ = β or γ = τ . In addition, the refinement depends on the extension parameter, which controls the locality of the refinement process. While the refinement region needs to be large enough to create sufficiently many new degrees of freedom, it should also remain as local as possible in order to keep the data volume small. Moreover, it is desirable to avoid the frequent re-computation of individual coefficients. A more detailed analysis of the choice of the refinement strategies is beyond the scope of the present paper, see also [START_REF] Kiss | Adaptive CAD model (re-)construction with THB-splines[END_REF]. Given a hierarchical mesh, which is defined by a subdomain hierarchy, we define for each element e the index sets that contains indices of active basis functions, Again we get e ⊆ ext(e) ⊆ ext (e). After these preparations we can now state our result concerning the approximation order, which is based on the fact that spline projectors reproduce polynomials: Theorem 7. Given a function f ∈ C p+1 (Ω), the element-wise maximum error can be estimated by

J ,e γ = {i ∈ J : γ i | e = 0}, = 0
f -P γ f ∞,e ≤ Cdiam(e) p+1 max |α|=p+1 D α f ∞,ext [ ] (e) (19)
where either γ = β for HB-splines or γ = τ for THB-splines, where the maximum norm on the right-hand side considers the restriction to the extended support extension ext (e) or to the (standard) support extension ext(e), respectively.

Proof. Denoting by T e f the Taylor polynomial of degree p of f at the center of e, the triangle inequality gives

f -P γ f ∞,e ≤ f -T e f ∞,e + P γ (f -T e f ) ∞,e
since P γ reproduces polynomials of degree p. A bound for the first term is obtained by the standard error estimate for Taylor expansions,

f -T e f ∞,e ≤ C p diam(e) p+1 max |α|=p+1 D α f ∞,e
where the constant is independent of f . We now analyze the second term in the case of THB-splines, obtaining

P τ (f -T e f ) ∞,e = N =0 i∈J ν i (f -T e f )τ i ∞,e ≤ max =0,...,n max i∈J ,e τ |ν i (f -T e f )|
since the THB-splines form a convex partition of unity. Using ( 16) and again the standard estimate for Taylor series gives

|ν i (f -T e f )| ≤ C τ f -T e f ∞,ext(e) ≤ C τ C p diam(ext(e)) p+1 max |α|=p+1 D α f ∞,ext(e)
According to Assumption I (mesh grading), the support extension satisfies diam(ext(e)) ≤ (4p + 2)diam(e).

Combining this observation with the previous four inequalities confirms (19) for THB-splines with

C = C p (1 + (4p + 2) p+1 C τ ).
The case of HB-splines can be dealt with similarly: Analyzing the second term gives

P β (f -T e f ) ∞,e = N =0 i∈J µ i (f -T e f )β i ∞,e ≤ 2 max =0,...,n max i∈J ,e τ |µ i (f -T e f )|
since B-splines of no more than two levels are present in each cell (according to the mesh grading assumptions) and the B-splines within each level form a convex partition of unity. Using ( 17) and the standard result for Taylor series gives

|µ i (f -T e f )| ≤ C β f -T e f ∞,ext (e) ≤ C β C p diam(ext (e)) p+1 max |α|=p+1 D α f ∞,ext (e)
According to the Assumption II (strong mesh grading), the extended support extension satisfies diam(ext (e)) ≤ (4p + 2)diam(e).

Combining this observation with the previous four inequalities confirms (19) with

C = C p (1 + 2(4p + 2) p+1 C β ).

Experimental results

We perform several numerical experiments to validate and underline some relevant features of our approach. The entire implementation has been carried out in C++ with the use of the G+Smo library [START_REF] Mantzaflaris | G+Smo (Geometry plus Simulation modules)[END_REF].

Example 1. We perform adaptive approximation of the function

f (x, y) = 1 -tanh x 2 + y 2 -0.3 0.05 √ 2 , (x, y) ∈ [-1, 1] 2 ,
which was taken from the work of [START_REF] Hennig | Projection and transfer operators in adaptive isogeometric analysis with hierarchical B-splines[END_REF]. The approximation obtained with our scheme is compared with the global fitting strategy described by [START_REF] Kiss | Adaptive CAD model (re-)construction with THB-splines[END_REF], and with uniform refinement. The value of the extension parameter (see Step 3) is set to p. The algorithms terminate if the maximum error is below a certain threshold (in this example ε = 1e-4). Table 1 summarizes the obtained results. The results obtained by global fitting and by local fitting-based quasi-interpolation with HB and THB-splines are quite similar (We even obtain the same meshes and virtually the same maximum error, except for depth 5). In fact, a slightly smaller number of degrees of freedom suffices to reach the desired accuracy via quasi-interpolation. As to be expected, many more degrees of freedom are needed to achieve the same accuracy when using uniformly refined tensor-product (TP) splines. The convergence rates are depicted in Fig. 4. While we got identical meshes for HB-and THB-splines in this experiment, this is not always the case. Table 2 reports the result for biquadratic splines using the extension parameter 1, where the generated hierarchical meshes and the associated numbers of degrees of freedom for HB-and THB-splines are slightly different. Even though the maximum errors are again identical, the distribution of the error (shown in Fig. 5) is not.

Example 2. We approximate the function

f (x, y) = 1 9 (tanh(9y -9x) + 1) + 1 1.5 exp((10x -6) 2 + (10y + 7) 2 ) , (x, y) ∈ [-1, 1] 2
that has been considered by [START_REF] Bracco | Adaptive fitting with THB-splines: Error analysis and industrial applications[END_REF], using adaptive spline refinement. We set again ε = 1e-4. Here we exploit the observation that many of the coefficients have been computed during previous steps of the iterative refinement process. This is in contrast with global approximation strategies that require the solution of one large linear system per iteration. Table 3 shows the error and the number of computed coefficients per iteration. It is worth noting than even though the use of HB-splines leads to a larger restricted domain ∆ associated with each basis function, this does not have much impact on the locality of the approach. (Again, the refinement processes creates the same meshes for HB-and THB-splines, and also the values of the error are virtually identical in almost all cases.) Example 3. We explore the impact of Assumptions I and II, and of the extension parameter. The latter parameter controls the size of the neighborhood of a marked element that is subject to refinement, see Step 3 of the algorithm in Section 5. We perform adaptive approximation of the function f (x, y) = 1 5 exp((10x) 4 + (10y) 4 )

with Ω = [-1, 1] 2 and ε = 1e-4. We use either HB-or THB-splines and vary the degree and the extension parameter. We compare the average size of the linear systems.

Three observations are in order. First, the use of mesh grading leads to a smaller average size of the linear systems. According to the theory, Assumptions I and II ensure that the size varies between (p + 2) d and (2p + 3) d for THB-splines, and between (p + 2) d and (p + 2 3p+3 2 ) d for HB-splines. In practice, the average size is far less than the upper bound. Second, increasing the extension parameter has a similar effect as mesh grading. Third, the linear systems for HB-splines have a larger size. We compare our QI scheme with SMS [START_REF] Speleers | Effortless quasi-interpolation in hierarchical spaces[END_REF], which was constructed in terms of THB-splines. In order to do so, we apply our scheme to two given functions, both defined in [-1, 1] 2 , that were considered in that paper, adopting the same setting. More precisely, • we adopt manually created sequences of refined meshes (see Fig. 3 of [START_REF] Speleers | Effortless quasi-interpolation in hierarchical spaces[END_REF], and

• we evaluate the maximum error on a fixed regular grid of 150 × 150 points.

In addition, we also compare the results created by the automatic mesh refinement algorithm.

Example 4. The first function is f (x, y) = tanh(9y -9x) + 1 9

Figure 6 (left column) compares the approximations obtained by the two schemes. Our method results in a slightly better or comparable accuracy in almost all the cases4 . Note that the finest grid (level 6), when considered on the entire domain, consists of 128 × 128 points. Using only 150 × 150 points for error estimation may not suffice. The higher accuracy of our method is also present for meshes created by automatic refinement. We used ε = 1e -5 for p = 2 and ε = 1e -6 for p ≥ 3.

Example 5. The same setting as in the previous example is applied to the function

f (x, y) = 2 3 exp √ (10x-3) 2 +(10y-3) 2 + 2 3 exp √ (10x+3) 2 +(10y+3) 2 + 2 3 exp √ (10x) 2 +(10y) 2
The obtained results are shown in the right column of Figure 6. Also in this case, our method reveals a slightly better accuracy in almost all cases. For the manually created meshes (top row), the grid of points used to estimate the error might lead to inaccurate results. We used ε = 1e -2 for the automatic mesh refinement algorithm (bottom row).

Example 6. Finally, we present an example that reveals another significant advantage of out method. We fit the function from Example 3 f (x, y) = 1 5 exp((10x) 4 + (10y) 4 ) on Ω = [-1, 1] 2 using automatic refinement with threshold ε = 1e-4. Figure 7 shows the errors and the DoFs obtained with the two schemes (our scheme and SMS) using THB-splines of degree 3. SMS leads to non-symmetric meshes, even though the function is symmetric, see Fig. 8. This is due to the choice of performing quasi-interpolation via interpolation in only one of cells that form the support of the selected basis function, in agreement with (8). Example 7. Finally we present a simple one-dimensional example that illustrates the differences between our scheme for THB-splines and SMS. We use both schemes to approximate y = |x| by quadratic hierarchical splines on [-1,1] defined on two levels with knot spans of length 1/8 and 1/16, where refinement took place on the interval [-1/8, 1/8], see Fig. 9. SMS (blue) preserves the original approximation outside of the refined domain, whereas our scheme recomputes the coefficients of all functions in the support extension of the refined domain. In the first case, the error is more concentrated and larger, while it is more spread out and thus lower for the second method. Remark. Finally we compare the computational costs of our spline projector for THB-splines with SMS:

• First we analyze the number of required sample points. We give a rather conservative (but degree-independent) upper bound (10 d ) for the number of such points per degree of freedom in Proposition 9. In practice, the ratio is closer to the lower bound of 2 d , which is realized when considering uniform meshes (N = 0), see Fig. 10. As noted by Speleers (2017) for d = 2, SMS requires "on average p d " (and not more than (p + 1) d ) samples per degree of freedom.

• Both schemes generate the THB-spline coefficients by evaluating linear combinations of samples values with predefined weights, and the number of floating point operations (flops) amounts to twice the number of sample points. Our method requires 2(4(p + 1) + 1) d flops per coefficient, thus approximately 4 d times as many as SMS, which needs 2(p + 1) d flops per coefficient. Clearly, the evaluation of the THB-spline coefficients is ideally suited for parallel computing. Without using pre-computed inverse matrices, the computational costs would amount to O(p 3d ) flops per degree of freedom for both methods, again with a smaller constant for SMS and with the option of parallelizing the computations.

Conclusion

We proposed a novel QI scheme based on a local least-squares fitting. This defines two new spline projectors, one each for HB-and for THB-splines, which are formulated within a unifying framework. Moreover, we provided a detailed analysis of the computational complexity, which showed that the costs are comparable to [START_REF] Kraft | Adaptive und linear unabhängige Multilevel B-Splines und ihre Anwendungen[END_REF] for HB-splines and [START_REF] Speleers | Effortless quasi-interpolation in hierarchical spaces[END_REF] for THB-splines. We also analyzed the required number of sampled function values, showing that O(1) evaluations are sufficient to obtain spline projectors. Naturally, the computational costs grow with the degree p and with the dimension d, thereby imposing limitations on the applicability of the method, which can, however, be addressed to some extent via parallel computing. The theoretical results ensure the efficiency of the proposed scheme and provides meaningful indications on how to apply it effectively. The approximation properties of the projector were analyzed, and an adaptive refinement strategy was presented. The interdependence between the theoretical results achieved in the paper is summarized in Fig. 11. Numerical experiments were performed, showing that one obtains an accuracy comparable to the one of global approximation techniques.

Furthermore, a comparison with the QI scheme proposed by [START_REF] Speleers | Effortless quasi-interpolation in hierarchical spaces[END_REF] was carried out. Our construction

• is based on a unifying framework for HB-and THB-splines, while SMS is limited to THBsplines,

• guarantees the spline projector property with O(1) evaluations per degree of freedom,

• was shown experimentally to preserve certain symmetries of the given function better than SMS, and

• seems to give slightly lower errors in almost all computational experiments that we performed.

While it was possible to maintain overall the computational complexity, these advantages come at the price of higher computational costs, i.e., of a larger constant in the complexity estimate. Additionally, more computer memory is needed to store the look-up table that is used to generate the spline coefficients. Future work will be devoted to the extension of the proposed approximation scheme to scattered data sets, both concerning the theory and the algorithmic aspects. In fact, local approximation strategies facilitate the use of parallel computing approaches. This might be further addressed to increase the efficiency of the current implementation. Future work may also include the extension to non-uniform meshes, where the use of pre-computed look-up tables is not possible. Finally, the application of the proposed scheme to IgA related problems, such as matrix assembly for hierarchical splines will be addressed, extending the results of [START_REF] Pan | Fast Formation of Isogeometric Galerkin Matrices via Integration by Interpolation and Look-up[END_REF]. Proof. Any restricted domain ∆ consist of O(p d ) cells of the finer level and we sample at most 3 d nodes per cell. The unisolvency is guaranteed since these grids are unisolvent for the full space of tensor-product splines defined on the finer grid, and the restricted hierarchical spline space is a subspace thereof. The unisolvency for tensor-product splines follows from the Schoenberg-Whitney conditions and the fact that the ∆ consists of at least (p + 1) d cells of the finer level.

Proposition 9. The total number of grid points (and hence the number of required function evaluations) does not exceed 10 d |H|.

Proof. First we note that the total number of grid points is bounded by 5 d n ac , where n ac denotes the number of active cells. Second, we consider all pairs of (T)HB-splines and active cells, where the cells are contained in the support of the (T)HB-spline. The number of such pairs is bounded from below by (p + 1) d n ac , since the (T)HB-splines span the full space of tensor-product polynomials on each active cell. It is also bounded from above by 2 d (p + 1) d |H|, according to Assumption I. Consequently, n ac ≤ 2 d |H| and the result follows.

Finally we note that this counting -which is based on set of candidate sample points -leads to an over-estimation of the required number of evaluations. Indeed, it suffices to perform 4-adic subdivision only for active cells that are "close" to the cells of next finer level. More precisely, in order to generate the sample points for THB-splines,

• the active cells within the offset region5 of the subdomain Ω +1 formed by p layers of cells of level need to be subdivided into 4 d cells via 4-adic subdivision,

• it suffices to subdivide all other cells into 2 d cells only, via binary subdivision.

The vertices of the refined mesh form the set of sample points. A similar observation applies to HB-splines.

Figure 3 :

 3 Figure 3: Possible configurations of the restricted domain ∆ for p = 2 in the case of THB-splines (up to symmetries).

  , . . . , N for the cases of HB-splines (γ = β) and THB-splines (γ = τ ). Clearly we have J

Figure 4 :Figure 5 :

 45 Figure 4: Example 1 -Convergence rates for p = 2 (left) and p = 3 (right)

Figure 6 :

 6 Figure 6: Comparison of the maximum error for the Examples 4 (left) and 5 (right) with manually created meshes and errors with respect to a fixed grid (top row) and for meshes created by automatic refinement (bottom row).

Figure 7 :

 7 Figure 7: Maximum error obtained by adaptive refinement using our QI scheme and SMS for p = 3.

Figure 8 :

 8 Figure 8: Meshes obtained at the finest level, see Figure 7, using our QI scheme (left) and SMS (right) for p = 3.

Figure 9 :Figure 10

 910 Figure 9: Approximation (left) and comparison of the error (right)

Figure 11 :

 11 Figure 11: Dependencies between the assumptions and results in this paper

Table 1 :

 1 Example 1 -Maximum errors and degrees of freedom for approximation via quasi interpolation or global fitting with HB-splines, THB-splines and tensor-product (TP) splines.

				Quasi interpolation		Global fitting
	p	depth	DoF	error (HB) error (THB) DoF (THB) error (HB & THB) DoF (TP) error (TP)
		1	324	1.684e-1	1.684e-1	324	1.243e-1	324	1.243e-1
		2	1012	2.213e-2	2.213e-2	1128	1.811e-2	1156	1.811e-2
	2	3	2076	2.016e-3	2.016e-3	2512	1.545e-3	4356	1.545e-3
		4	4648	1.688e-4	1.688e-4	4948	1.485e-4	16900	1.485e-4
		5	7248	9.103e-5	9.103e-5	7420	7.227e-5	66564	1.903e-5
		1	361	1.343e-1	1.343e-1	361	9.676e-2	361	9.676e-2
	3	2 3	1225 2593	1.491e-2 5.780e-4	1.491e-2 5.780e-4	1225 2977	1.142e-3 3.526e-4	1225 4489	1.142e-2 3.527e-4
		4	4753	5.019e-5	2.845e-5	4921	3.913e-5	17161	1.972e-5

Table 3 :

 3 Example 2 -Maximum errors and computed coefficients

				Quasi interpolation	Computed coeffs.
	p	depth	DoF	error (HB) error (THB)	HB	THB
		1	324	1.185e-1	1.185e-1	324	324
		2	1044	2.088e-2	2.088e-2	1016	1016
	2	3	2678	1.421e-3	1.421e-3	2408	2384
		4	3376	1.245e-4	1.245e-4	1064	1030
		5	3785	9.688e-5	9.688e-5	616	616
		1	361	1.660e-1	1.660e-1	361	361
	3	2 3	1150 3040	1.081e-2 2.220e-4	1.081e-2 2.220e-4	1135 2826	1135 2793
		4	3421	4.408e-5	2.781e-5	617	617

Table 4 :

 4 Example 3 -Number and average size of the linear systems during the refinement process

				THB-splines			HB-splines
			without grading	with grading	without grading	with grading
	p	ext. #lin. syst.	avg. size #lin. syst. avg. size #lin. syst. avg. size #lin. syst. avg. size
	2	1 2	5244 6888	27.27 2 26.80 2	5520 6960	26.96 2 26.76 2	5244 6888	45.68 2 38.16 2	5520 6960	39.34 2 37.72 2
		1	3160	52.73 2	3412	51.88 2	3160	103.51 2	3412	82.64 2
	3	2	4024	52.67 2	4228	52.35 2	4024	89.39 2	4228	82.44 2
		3	5320	50.95 2	5344	50.94 2	5320	79.38 2	5344	78.70 2
		1	3976	94.33 2	5412	82.64 2	4672	347.01 2	5864	125.00 2
	4	2 3	4608 6264	89.35 2 83.48 2	5740 6620	82.92 2 80.93 2	4608 6264	192.62 2 144.80 2	5740 6620	128.97 2 116.22 2
		4	7128	82.69 2	7324	81.25 2	7128	130.39 2	7324	115.27 2

although represented with respect to THB-splines for S and HB-splines for K

While the detailed investigation of weaker conditions is beyond the scope of the present paper, we note that it suffices if ∆ contains at least one active cell of level that belongs to supp β j , according to PoC.

Again we note that the investigation of necessary and sufficient conditions is beyond the scope of this paper.

[START_REF] Speleers | Hierarchical spline spaces: quasi-interpolants and local approximation estimates[END_REF] applies his QI schemes (which reproduce polynomials but not splines) to this function also. While the (manually created) sequence of refined meshes is slightly different in that paper, the results still indicate that sacrificing the spline projector property leads to a slight increase of the error, while the rate of convergence remains the same.

This region is swept out the by the level HB-splines with a support intersecting the subdomain Ω +1 .

Acknowledgements

The research was funded by the Austrian Science Fund (FWF): W1214-N15, project DK3. The support is gratefully acknowledged.

Given a fixed level 0 and an index i 0 ∈ J , we choose the restricted domain ∆ as the extended support supp β 0 i0 and we consider the HB-splines β i defined on it. These functions form the system Φ. Similar to the previous case, we have the two main possibilities (a) and (b) for defining interpolation nodes and weights. Now, given a function f , we perform local fitting as described in the previous section and assign the value of the coefficient associated with β 0 i0 to µ 0 i0 (f ). The same arguments as for THB-splines prove the uniqueness of the solution. This procedure defines a spline projector (9) according to Proposition 4.

We analyze the computational costs per coefficient, focusing on the second choice of nodes. Any restricted domain ∆ contains at most two levels, due to Assumption II. We distinguish between two cases: First, if ∆ = supp β 0 i0 , only cells of levels 0 and 0 + 1 are present. The number of possible cases does not exceed 2 (p+1) d , since there are (p + 1) d cells of level 0 . Second, if ∆ = supp β 0 i0 , only cells of levels 0 and 0 -1 are present. In this situation, the number of cells of level 0 -1 does not exceed 3p+3 2 d . Consequently, the number of possible cases does not exceed

Again, Assumption II also implies that the grid of nodes can be chosen such that it contains no more than C g p d points, where the constant C g does not depend on the polynomial degree p. See Appendix A for more details.

Once more, we may assume that all possible matrices (A T A) -1 A T (more precisely, the row vectors that corresponds to the coefficient µ 0 i0 (f )) have been pre-computed and stored in a look-up table . In each specific situation, finding the right entry in the table via binary search requires O(p d ) steps. In addition, we need to evaluate f at a tensor-product grid containing O(p d ) nodes, and to compute the inner product with the pre-computed row vector.

We summarize the results: Theorem 6. Consider the functionals µ i defined by local fitting with respect to the restricted domain supp β i , using a unisolvent set of nodes.

(i) The linear operator P β is a HB-spline projector.

(ii) When using the second choice of nodes, the evaluation of the spline coefficients requires O(p d ) flops and O(1) function evaluations per degree of freedom if Assumption II is satisfied.

Similar to the case of THB-splines, we additionally obtain a bound

with the constant

Approximation using adaptive refinement

Given a function f : Ω 0 → R, we compute a hierarchical spline approximation as follows:

1. The number of levels is initialized by N = 0. 2. Compute the spline approximation P γ f with respect to one of two bases, i.e., for γ = β and γ = τ , see (9). The coefficient functions are defined by local fitting, see Section 4. 3. Mark the elements e ∈ E of the hierarchical mesh associated with P γ that are identified by the refinement indicator and collect the marked elements, and the elements in the neighborhood whose size is specified by the extension parameter, in the set E . 4. Terminate the iteration if E = ∅. Otherwise create the refined subdomain hierarchy, as follows:

• Increase N by 1 and initialize Ω N by the empty set.

• Increase the level of each marked element e ∈ E by one. More precisely, add e to Ω level(e)+1 where level(e) = max{ : e ⊆ Ω }.

Appendix A. Smart sampling

Any restricted domain ∆ contains active cells of at most two levels, and it can be covered by (active and in-active) cells of the finer one among these two levels. We construct the grid on ∆ by splitting each of these covering cells into 2 d cells via dyadic subdivision and placing the grid points at the cells' vertices. Consequently, all the grid points are vertices of the mesh that is created by subdividing each active cell into 4 d cells via 4-adic subdivision. These vertices form the set of candidate sample points, see Fig. 12. Proposition 8. For the restricted domains ∆ considered in Section 4.2 and 4.3, the grid contains O(p d ) nodes and is unisolvent for the restricted hierarchical spline space Ĥ.


Camille Chapelle

Baptiste Quienne

Céline Bonneaud

Ghislain David

Sylvain Caillol

Baptiste Quienne

Diels-Alder-Chitosan based dissociative covalent adaptable networks

Keywords: Chitosan-based films, Diels-Alder crosslinking, thermoresponsive materials, trimaleimide Highlights

come

Introduction

Natural polysaccharides such as gelatin, hyaluronic acid, guar gum or chitosan are currently widely used due to their renewable and biodegradable nature [START_REF] Dragan | Polysaccharides constructed hydrogels as vehicles for proteins and peptides . A review[END_REF] , [START_REF] Hamedi | Chitosan based hydrogels and their applications for drug delivery in wound dressings : A review[END_REF] , [START_REF] Pakdel | Review on recent progress in chitosan-based hydrogels for wastewater treatment application[END_REF]. They are extensively used in biomedical and pharmaceutical applications as their non-toxicity combined with a good biocompatibility make them suitable in these fields. Among them, chitosan shows attractive properties with a low cost and availability. Chitosan comes from the deacetylation of chitin which is extracted from various natural resources like shrimp shells, squids, insects or mushrooms. It is a cationic polysaccharide made of β-D-glucosamine and N-acetyl-β-Dglucosamine units linked by a glyosidic bond (1→4). This polysaccharide is characterized by its source, its acetylation degree (DA) and its molar mass. Chitosan possesses two hydroxyl and one amino groups per glucosamine unit allowing its functionalization. More specifically, the amino group is responsible for the intrinsic properties of the chitosan chains and, thanks to its nucleophilic character, allows the introduction of new functionalities (Gregorio Crini , Pierre-Marie Badot, n.d.).

Chitosan was already described as a film-forming polymer on its own (S. Bhuvaneshwari *, D.Sruthi*, V. Sivasubramanian*, 2014) , [START_REF] Elsabee | Chitosan based edible fi lms and coatings : A review[END_REF], or mixed with other polymers or fibers. [START_REF] Liu | Food Hydrocolloids Preparation and characterization of protocatechuic acid grafted chitosan fi lms with antioxidant activity[END_REF] , (Y.X. Xua, K.M. Kimb, M.A. Hannaa, * , 2005) , [START_REF] Kaya | Antioxidative and antimicrobial edible chitosan films blended with stem, leaf and seed extracts of Pistacia terebinthus for active food packaging[END_REF] Crosslinked chitosan films are generally obtained by reaction with glutaraldehyde, diepoxides, dianhydride, genipin and tripolyphosphate [START_REF] Nataraj | Crosslinked chitosanfilms with controllable properties for comercial applications.pdf[END_REF] , [START_REF] Jin | Novel Chitosan-Based Films Cross-Linked by Genipin with Improved Physical Properties[END_REF] , [START_REF] Singh | Chitosan films crosslinking with EDTA modifies physicochemical.pdf[END_REF] , [START_REF] Tiwary | Cross-linked chitosan films : Effect of cross-linking density on swelling parameters[END_REF] , [START_REF] Kavianinia | Preparation and characterization of chitosan films , crosslinked with symmetric aromatic dianhydrides to achieve enhanced thermal properties[END_REF] and few literature is provided on Diels-Alder based hydrogels for chitosan but not in view of film forming. [START_REF] Matsumoto | A Water-Based Chitosan-Maleimide Precursor for Bioconjugation: An Example of a Rapid Pathway for an In Situ Injectable Adhesive Gel[END_REF] , [START_REF] Montiel-Herrera | N -(furfural) chitosan hydrogels based on Diels-Alder cycloadditions and application as microspheres for controlled drug release[END_REF] , [START_REF] Montiel | Furanchitosan hydrogels based on click chemistry[END_REF] , [START_REF] Elschner | Furfuryl-and Maleimido Polysaccharides: Synthetic Strategies Toward Functional Biomaterials[END_REF] In 2015, Montiel-Herrera et al. [START_REF] Montiel-Herrera | N -(furfural) chitosan hydrogels based on Diels-Alder cycloadditions and application as microspheres for controlled drug release[END_REF] synthesized Diels-Alder chitosan microparticles. They functionalized chitosan with furan chitosan by a two-step reductive amidation (Erreur ! Source du renvoi introuvable. b)). Indeed, furfural was reacted first with the amines to form the corresponding Schiff base followed by its reduction. In their work, microspheres were synthesized by using a bismaleimide having three ethylene glycol units and chitosan with an average molar mass of 130,000 g/mol. The mixture of furan and maleimide compounds was added dropwise into vegetable oil at 65 °C creating the microspheres by ensuring a continuous and efficient stirring. Finally, the microspheres demonstrated low swelling abilities with the different mixtures employed but the release of methylene blue was assessed. Later, Garcia-Astrain et al. (García-astrain et al., n.d.) followed the same procedure but they employed gold nanoparticles covalently bonded to maleimide functionalities. The same team also carried out studies with different maleimide cross-linkers: either a maleimide functionalized chitosan to afford fully chitosan-based hydrogels [START_REF] Guaresti | Synthesis and characterization of a biocompatible chitosan-based hydrogel cross-linked via 'click' chemistry for controlled drug release[END_REF] or a poly(ethylene oxide) and poly(propylene oxide)-based bis maleimide 21, 22 . Finally, supramolecular hydrogels were fabricated by using a bismaleimide containing cyclodextrins in its backbone [START_REF] Zhang | Supramolecular hydrogel formation between chitosan and hydroxypropyl β-cyclodextrin via Diels-Alder reaction and its drug delivery[END_REF]. In any of those cases, all teams used a chitosan substituted via reductive amidation as previously mentioned. Even though, the Diels-Alder reaction was highlighted using UV-vis and IR studies, no deeper investigation has been performed on the reaction mechanisms.

Going further, retro Diels-Alder has never been revealed for chitosan-based compounds even if its use has been recognized for many applications such as self-healing of various polymers [START_REF] Gandini | Macromolecular materials based on the application of the Diels-Alder reaction to natural polymers and plant oils[END_REF] [START_REF] Vauthier | Interfacial Thermoreversible Chemistry on Functional Coatings: A Focus on the Diels-Alder Reaction[END_REF] [START_REF] Yang | Recyclable and self-healing polyurethane composites based on Diels-Alder reaction for efficient solar-tothermal energy storage[END_REF].

Hence, this present study is dedicated for the first time to the synthesis and characterization of dissociative covalent adaptable networks of chitosan by Diels-Alder chemistry. In this work, furan groups were grafted onto chitosan in one step with furfuryl glcidy ether to avoid the use of boric reducing agents. The furan functionalized chitosan furan was then able to react with maleimide by Diels-Alder reaction. This methodology was thoroughly applied to chitosan oligomers (M w =1,800 g/mol) and also to chitosan of higher molar masses i.e. M w =30,000 g/mol and 150 000 g/mol. The work on shorter chains allowed elucidating the Diels-Alder reaction mechanism, using a maleimide model, under different reaction conditions by 1 H NMR spectroscopy. After this first step of optimization, the Diels-Alder and retro Diels-Alder reactions were highlighted for the first time on chitosan systems by infrared spectroscopy (FT-IR), 1 H NMR and differential scanning calorimetry (DSC) on a model reaction between monomaleimide and 30,000 g/mol chitosan. Afterwards, the higher molar masses polysaccharides were tested in the presence of ethylene oxide and propylene oxide-based maleimide cross-linkers, including a trimaleimide, having different degrees of functionality to create films. Both Diels-Alder and retro Diels-Alder were deeply investigated on the films by dynamic mechanical analysis (DMA) in comparison with previous studies where no proof other than FT-IR for DA, of the occurrence for both reactions was brought so far. The influence of the chitosan length as well as the cross-linkers nature was studied, water capacity measurements at different pHs and contact angle determination were performed to evaluate some of the film properties.

Materials & Methods

2.1.

Materials Maleic anhydride, zinc chloride were purchased from Sigma Aldrich. Furfuryl glycidyl ether, Jeffamine EDR-148, chloroform and hexamethyldisilizane were respectively purchased from Acros Organics, Huntsman Corporation, VWR chemicals and ABCR. The monomaleimide (MI) PEO (n=19) PPO (m=3) were provided by Specific Polymers. Chitosan with DA around 10-15% CH30 (Mw 30,000 g/mol), CH150 (Mw 150,000 g/mol) were purchased from Glentham Science. Acetic acid (AcOH), sodium acetate (NaAc), sodium nitrite (NaNO 2 ), lithium bromide (LiBr), sodium azide (NaN 3 ), sodium hydroxide (NaOH), hydrochloric acid (HCl), acetone (>98%) were purchased from Sigma and used without purification, HPLC water (>99%) was purchased from VWR.

Synthesis of chitosan oligomers functionalised with furfuryl glycidyl ether (FGO)

The depolymerization of 30,000 g/mol chitosan was performed by nitrous deamination involving NaNO 2 as previously described [START_REF] Chapelle | Water-Soluble 2,5-Anhydro-d -mannofuranose Chain End Chitosan Oligomers of a Very Low Molecular Weight: Synthesis and Characterization[END_REF]. Briefly, 1g (w/w) of chitosan was dissolved in 90 mL of AcOH (350µL in 90 mL of deionized water) solution overnight (pH 4.5). Once solubilized, a freshly prepared solution (10 mL) containing 0.4 eq (164 mg) of NaNO 2 /NH 2 was added to the solution. The solution was stirred at 50°C for 3 h.

The solutions were then freeze dried for 24 h and the lyophilisates were dried under vacuum for 24 h to obtain chitosan oligomers.

1 g of the obtained oligomers were solubilised in 20 mL of DMSO and 1 eq (818 µL) of furfuryl glycidyl ether was added. After 24 h stirring at 60°C, the product was precipitated in cold acetone, centrifugated and dried to obtain a hydroscopic orange powder.

1 H NMR (400 MHz, D 2 O, 25 °C, δ): δ 8.2-8.5 (s, H aldehydes ), 8-7.1 (H imines ), 7. 4 (s,Hα),6.4 (d,12 Hz,1H Hβ,1H Hγ ),5.0 (d,5 Hz, 1H H 1 gemdiol ), 3.2-4.4 (m, 1H H 2 acetyl , 1H H 3, 1H H 4 , 1H H 5 , 2H H 6 ), 2.5 (m, 1H, H 2 ), 2.0 (s, 3H, H CH3 ), 4.8 (s, HOD).

2.2.

General procedure for the synthesis of furan chitosan via epoxy-amine reaction 1g of CH30 and CH150 were solubilised in 100 mL of deionised water and 1 eq (818 µL) and 1.2 eq (981 µL) respectively of furfuryl glycidyl ether was added in 10 mL of DMSO. After 24h stirring at 60°C, the product was precipitated by addition of a 1M NaOH solution. The product was then centrifugated and washed with acetone and several time with deionised water until neutral pH. The product was finally freeze dried during 12h. Substitution degree (DS) of FGCH30 and FGCH150 were measured by 1 H NMR (400 MHz, D 2 O, 25 °C, δ): 7.6 (s, Hα), 6.5 (d,12 Hz,1H Hβ,1H Hγ ),1H H

2 acetyl , 1H H 3, 1H H 4 , 1H H 5 , 2H H 6 ),
3.2 (m, 1H, H 2 ), 2.0 (s, 3H, H CH3 ), 4.7 (s, HOD).

2.3.

Synthesis of furfuryl chitosan grafted with monomaleimide 1g of FGCH30 was solubilized in 100 mL of 1% acetic acid solution. 1 eq of MI/FG was added in 10 mL of DMSO and the solution was stirred at 65 °C for 24 h. The product was precipitated in cold acetone, centrifuged and washed with acetone to get rid of unreacted MI.

A yellow powder was obtained after drying the product under vacuum for 6 h. 1 H NMR (400 MHz,D 2 O,25 °C,δ) 7.6 (s,Hα),6.5 (d,12 Hz,1H Hβ,1H Hγ ),

(m, 1H Hʊ, 1H Hʎ), 3.5-4.1 (m, 1H H 2 acetyl , 1H H 3, 1H H 4 , 1H H 5 , 2H H 6 ) 3.5 (s, nH H PEO/PEG ), 3.2 (m, 1H, H 2 ), 2.0 (s, 3H, H CH3 ), 1.1 (s, 2H H CH2 ), 4.8 (s, HOD).

2.4.

General procedure for the synthesis of bismaleimide or Synthesis of BMI and TMI Maleic anhydride (2.5 eq) was solubilized in chloroform and Jeffamine EDR-148 or Jeffamine T-403 in chloroform was added dropwise. The reaction mixture was left to stir overnight. Then ZnCl 2 (2.5 eq) was added at R.T. and the mixture was heated at 40 °C.

Finally HMDS (2.5 eq) was added dropwise and the reaction mixture was allowed to heat at 85 °C during 5 h. The reaction was stopped. Silica gel was poured and the reaction mixture was filtrated. A liquid-liquid extraction was performed with D.I. water and the organic phase was washed with D.I. water (*3) and dried with MgSO 4 . The solvent was then evaporated to afford white crystals (yield=62 %).

1 H NMR (400 MHz, CDCl 3 , 25 °C, δ) BMI: 6.70 (s, 4H, H a ), 3.7 (td, 4H, H c , α-azote, 3 J H-H = 5.7 Hz), 3.6 (td, 4H, H d , β-azote, 3 J H-H = 5.7 Hz), 3.55 (s, 4H, H e ) 1 H NMR (400 MHz, CDCl 3 , 25 °C, δ) TMI: 6.63 (s, 6H, H a ), 4.31 (m, 3H, H b , α-azote), 3.0-3.85 (m, 21H, H b , H d , H e ), 1.3-1.32 (dd, 9H, H c , β-azote), 0.95-1.08 (m, 6.5H, H c , H f ), 0.5-0.84 (m, H g )

2.5.

General procedure for the Film formation Films of chitosan were prepared as followed: CH30 and CH150 were dissolved in 2% acetic acid solution at 1wt% and 0.5wt% concentration respectively. 1 eq of maleimide per furan units was added in 250 µL of DMSO and the solution was homogenized using a speed mixer for 3 min at 2500 rpm. 1mL of the prepared solution was deposed on a Teflon sheet and heated in the oven at 65°C for 12h. Control films were realised with the same procedure by adding 250 µL of DMSO to the chitosan solution before going into the oven.

2.6.

Nuclear Magnetic Resonance (NMR) spectroscopy NMR spectra were recorded on a Bruker AVANCE III 400MHz spectrometer instruments using TopSpin 3.5 operating at 400.13 ( 1 H) and100.62 ( 13 C) MHz at room temperature except if specified. D 2 O and CDCl 3 were used as internal references. The letters s, d, t, q, quint, sext and spt stand for singlet, doublet, triplet, quartet, quintuplet, sextet and septuplet respectively.

2.7.

Fourier Transform Infrared Spectroscopy (FTIR) Spectra were recorded on a Nicolet 210 Fourier transform infrared spectroscopy (FTIR) spectrometer. The characteristic IR absorptions mentioned in the text are reported in cm -1 . Materials analyses were recorded using an ATR accessory. Three replicates were made for films analysis to check their homogeneity.

2.8.

Differential Scanning Calorimetry (DSC) Analyses were carried out using a NETZSCH DSC200F3 calorimeter. Constant calibration was performed using indium, n-octadecane and n-octane standards. Nitrogen was used as purge gas. Approximately 10 mg of sample were placed in perforated aluminum pans and the thermal properties were recorded between -35 °C and 190 °C at 10 °C/min to observe the enthalpy of reaction. All the reported temperatures are mean values.

2.9.

Contact angle (CA) Contact angles were measured on a Dataphysics Contact angle System OCA Apparatus coupled with SCA20 software. Briefly, a 3 µl drop of water or hexadecane was placed on the film, picture was captured and the angle was measured. The operation was reproduced 3 times for each solvent and each film.

Swelling ratio (SWR)

Three samples of around 20 mg each were separately put in PBS pH 7.4 0.1 M and acidic water (2 % acetic acid). At selected time intervals after immersion, the samples were removed, the excess of liquid was withdrawn with paper and the samples were weighed before being put in back to their solution. The swelling ratio was calculated using the Equation 2 where m 2 is the mass of the material after swelling in water and m 1 is the initial mass of the material.

-Equation 22.11. Dynamic mechanical analysis Dynamic mechanical analyses (DMA) were carried out on Metravib DMA 25 with Dynatest 6.8 software. Samples were tested according to uniaxial tension mode while heating at a rate of 3 °C/min from -80 °C to + 180 °C, at a frequency of 1 Hz with a fixed strain of 10 -5 m.

These conditions have been chosen to study the elastic behavior of the materials.

Results and Discussion

3.1.

Synthesis of maleimides The synthesis of maleimides was carried out following common procedures. The ring-opening of maleic anhydride was performed using Jeffamine-type amines overnight at room temperature in chloroform. Two starting amines were employed: Jeffamine EDR-148 containing two ethylene oxide units and two primary amines and Jeffamine T-403 containing 5-6 propylene oxide units and three primary amines. By addition of zinc chloride and hexamethyldisilizane, the ring-closure was completed after 5 h at 85 °C. After purification, the final maleimides were obtained in correct yields and called BMI for the bismaleimide from Jeffamine EDR-148 (Figure 1

3.2.

Synthesis of furan-based chitosan compounds: oligosaccharides and polysaccharides Epoxy amine reactions are often used for chitosan functionalisation. Reaction involving furfuryl glycidyl ether (FGE) and carboxymethyl chitosan has been described by [START_REF] Na | Photocurable Ocarboxymethyl chitosan derivatives for biomedical applications: Synthesis, in vitro biocompatibility, and their wound healing effects[END_REF], who confirmed that the reaction occurred on the free amines that are more nucleophiles than alcohol groups of chitosan in the reaction condition (aqueous solution pH 11, 60°C, 48 h). This simple reaction was a one step-procedure as opposed to previous works using furfural and boric reducing agents to perform reductive amination [START_REF] Montiel-Herrera | N -(furfural) chitosan hydrogels based on Diels-Alder cycloadditions and application as microspheres for controlled drug release[END_REF] , [START_REF] Crini | Synthesis of stimuli-responsive chitosan-based hydrogels by Diels-Alder cross-linking `click´ reaction as potential carriers for drug administration[END_REF] (Erreur ! Source du renvoi introuvable. b)).

FGE was covalently linked to free amine groups on chitosan and oligochitosan, as indicated from both FT-IR (SI 1) and 1 H NMR spectra (Figure 3 b)). The 1 H NMR spectrum showed the expected chemical shifts generated by protons bonded to furan rings at 6.3 and 7.4 ppm.

The substitution degrees (DS) were calculated using the signal of the acetylated unit of chitosan at 2.0 ppm (which represents 16% of the macromolecule) compared to furan signals integrating for the three protons of the furan group (α). The DS calculation was done differently in comparison to a previous method [START_REF] Montiel-Herrera | N -(furfural) chitosan hydrogels based on Diels-Alder cycloadditions and application as microspheres for controlled drug release[END_REF] which used the proton in α-position of the free amine (Glc) of chitosan as a reference. However, the chemical shift of the α-proton is dependent on the DS. Thus, it could not be used as a reference as its position changed because of the chemical modification. The acetyl signal (GlAc) at 2.0 ppm was consequently chosen as a reliable reference and the DS was calculated following the equation (1). Three different chitosan-based starting reagents were employed. An oligomer resulting from the depolymerisation of chitosan was firstly obtained with a final molar mass of approximatively 1,800 g/mol. Two molar masses for chitosan products were also used: 30,000 and 150,000 g/mol standing for CH30 and CH150. The three final furan-based products were respectively called FGO, CH30FG and CH150FG from the oligomer, the chitosan of 30 and 150,000 g/mol respectively. It was proved that the higher the molar masses, the lower the DS for the same amount of FGE used. This is why the number of equivalent was increased for longer chains to obtain similar DS independently of the starting chitosan CH30 or CH150, i.e 1.2 eq for CH30FG and 1.5 eq for CH150FG. The DS were respectively 40% for FGO, 25% for both CH30FG and CH150FG. Finally, the furan addition was also confirmed by IR spectroscopy with the signal at 730 cm -1 corresponding to the furan ring (SI 1) In order to determine the working conditions, a model study was carried out. This model reaction was performed between chitosan oligomer FGO and a monomaleimide MI possessing ethylene oxide and propylene oxide units and was monitored by 1 H NMR. Firstly, the reaction was carried out in DMSO at 65 °C, to solubilize both chitosan oligomers and MI.

3.
It was shown that the reaction started in the first minutes under heat. Nevertheless, no furan protons were consumed during the reaction while the double bond of MI fully disappeared after 180 min (Figure 2 a)) indicating that Diels-Alder reaction did not occur. Maleimide double bond was consumed by aza-Michael addition instead (Scheme 1 c)). Indeed, the free amines of chitosan might react with the double bond of maleimide. The same experiment was reproduced in D 2 O with 1 % and 2 % of acetic acid (Figure 2 b)). In this solvent, furan and maleimide were both consumed during the reaction to form the Diels-Alder adduct in acid media (1 and 2%) (Scheme 1 d)). Indeed, free amines of FGO are protonated thus hindering their nucleophilic attack onto MI by Michael addition. By performing the reaction in DMSO with 1% of acetic acid, Michael reaction did not occur, confirming this assumption. Thus, the reaction conditions were set as follows: chitosan was dissolved in 2% acetic acid solution to facilitate the dissolution of the high molar mass chitosan CH150 then the maleimides BMI and TMI were added after dissolution in a minimum amount of DMSO (250 µL).

Reaction between MI and CH30FG

To confirm that the reaction occurs on higher molar masses, it was reproduced with CH30FG and MI. It was shown by 1 H NMR spectroscopy (Figure 3c) that the Diels-Alder adducts (endo and exo forms) were produced with the appearance of peaks between 6.3 and 6.7 ppm as well as 5.2 ppm corresponding to the endo adduct (55%) and at 5 ppm for the exo form (45%). Moreover, the decrease of the furan signal intensities at 6.5 ppm and 7.6 ppm was also observed. Nevertheless, the reaction did not proceed quantitatively, as 10% of unreacted furan groups still remained after 24 h despite the use of 1 eq of MI (Figure 3). However, it was shown by IR spectroscopy that there was no residual unreacted maleimide. This could be explained from the DS calculation based on acetylation degree or from side Michael addition.

3. a) 1 H NMR of CH30 in D 2 O+TFA b) 1 H NMR of CH30FG in D 2 O+TFA c) 1 H NMR of CH30FGMI in D 2 O+TFA
The resulting product from the reaction of MI and CH30FG was analyzed by DSC, a common method to determine if the retro Diels-Alder reaction occurred [START_REF] Dolci | Remendable thermosetting polymers for isocyanate-free adhesives: A preliminary study[END_REF]. The thermogram of CH30FGMI exhibited a complex endothermic peak starting at 110 °C with a first peak at 117 °C and a second one at 128 °C both corresponding to the retro Diels-Alder reaction for endo and exo adducts respectively [START_REF] Froidevaux | Study of the diels-alder and retro-diels-alder reaction between furan derivatives and maleimide for the creation of new materials[END_REF]. Moreover, the enthalpy of the retro Diels-Alder reaction was measured (-15.88 J/g) and similar to the value from literature (-18.09 J/g) with endo peak at 128 °C and exo peak at 157 °C [START_REF] Duval | Biobased and Aromatic Reversible Thermoset Networks from Condensed Tannins via the Diels-Alder Reaction[END_REF] (Figure 4). After DSC analysis, the sample was recovered and characterized by 1 H NMR. It confirmed that the retro Diels-Alder occurred as the peaks at 6.5 ppm and 7.6 ppm corresponding to furan were recovered when peaks between 6.3 and 6.7 ppm corresponding to the Diels-Alder adduct disappeared (see SI 3). Many studies have been performed on cross-linked chitosan systems using the Diels-Alder reaction, but the retro Diels-Alder was never considered in those studies. Hence, to our knowledge our present study is the first one to highlight by structural and thermal analyses the retro Diels-Alder reaction for such polysaccharide systems.

Films making and characterizations

Scheme 2. Synthesis of chitosan-furan based films by Diels-Alder click chemistry with dis and tris-maleimide cross-linkers.

After dissolution of the furan modified-chitosan in 2% acetic acid solution with a concentration of 1 wt% for CH30 and 0.5 wt% for CH150, the maleimides i.e. BMI or TMI dissolved in 250 µL of DMSO, were added. For homogenizing, the solution was then mixed using a speed mixer for 3 min at 2500 rpm. The solution was then deposited on a PTFE-glass films (thickness: 0.23 mm) and allowed to react during 12 h in an oven at 65 °C. The films were then analysed by infrared spectroscopy to demonstrate the reactivity between the maleimide crosslinkers and the furan groups from chitosan to form a network.

Sample

Indeed, a significant signal at 820 cm -1 was attributed to the double bond of maleimide and a signal at 875 cm -1 to the double bond of furan from the starting reagents according to analyses [START_REF] Hou | Mechanical properties, thermal stability, and solubility of sheep bone collagen-chitosan films[END_REF] , [START_REF] Ren | Influence of chitosan concentration on mechanical and barrier properties of corn starch/chitosan films[END_REF]. However, as we previously highlighted the retro Diels-Alder reaction on CH30MI by DSC, dynamic mechanical analyses (DMA) was a promising technique to assess the occurrence of the retro Diels-Alder reaction on our crosslinked chitosan films. Therefore, thermo-dynamical properties of chitosan films have been studied with DMA, storage modulus (E') at room temperature are summarized in Table 1 and a magnification from 45 °C to 190 °C of the analyses is shown in Figure 6. In order to perform DMA analyses, homogeneous films were prepared. No homogeneous films were obtained for the crude CH150. However, once crosslinked with BMI and TMI, homogeneous films were provided and were further analysed. CH30 films were homogeneous either crude or in combination with a maleimide compound. The DMA analysis of CH30 showed a stable E' (50 MPa) over the whole temperature range. CH30BMI, CH30TMI films were much stiffer due to their Diels-Alder cross-linking, with an increase of E' of orders of magnitude of one hundred. Due to the higher functionality of TMI, CH30TMI is stiffer than CH30BMI as TMI may induce a higher cross-link density. Concerning CH150BMI and CH150TMI, the entanglements of 150,000 g/mol chitosan chains was assumed to lower the cross-linker effect.

Thanks to the Diels-Alder cross-linking, E' value of CH30TMI film was rather similar to CH150BMI and CH150TMI. This cross-linking is therefore a good alternative to obtain high mechanical properties from low molar mass chitosan. DMA analyses involved a temperature increase and as we demonstrated previously with CH30FGMI, retro Diels-Alder reaction occurs at 117 °C and 128 °C. As shown in Figure 6, cross-linked films did not show a stable E' compared to CH30. At around 100 °C, E' value of the four cross-linked films started to drop. This fluctuation may be induced by the retro Diels-Alder reaction. Indeed, during this reaction the cross-linking density dropped because cross-linking nodes were destroyed, the network destruction inducing a decrease of E' value. Those dynamical analyses allowed demonstrating the interest of cross-linked chitosan films. Thanks to the cross-linking step, homogeneous films of CH150BMI and CH150TMI were obtained, low molar mass CH30BMI and CH30TMI films showed mechanical properties comparable to high molar mass chitosan films. Moreover, we were the first to our knowledge to characterize chitosan films by DMA and to highlight the retro Diels-Alder reaction at high temperature (117°C and 128°C) on cross-linked chitosan films.

Figure 6. DMA measurements of chitosan films (R 1 : chitosan chain; R 2 : PEG chain; R 3 PPG chain)

3.4.2. Swelling index at different pH The swelling capacity of chitosan films was studied at pH 7.4 and pH 4. Erreur ! Référence non valide pour un signet. shows the swelling index of cross-linked and non-cross-linked films over 24 h. First, swelling index of films was measured in PBS at pH 7.4. When crosslinked with BMI, CH30BMI and CH150BMI showed the lower swelling index, 174% and 220% respectively. Films cross-linked with TMI had slightly higher swelling index with 230% and 280% for CH30TMI and CH150TMI respectively. . It could be assumed that the cross-link density was higher using BMI although its lower functionality indeed, its low size in comparison to TMI chains led to closer cross-linking nodes, therefore BMI films swelled less. For non-cross-linked films, higher molar mass chitosan led to higher swelling index, 335% and 557% for CH30 and CH150 respectively. However, this difference was lowered once films were cross-linked. On the other hand, film swelling capacity was measured at pH 4 where chitosan is fully soluble, especially low molar masses. Indeed, CH30 and CH150 were soluble in these conditions and the swelling index could not be measured, the films were not recovered after their addition in the acidic solution. However, the four others CH30BMI, CH30TMI, CH150BMI and CH150TMI were not soluble, confirming their cross-linking. In the same way as in PBS, CH30TMI and CH150TMI had slightly higher swelling index than their BMI equivalents. Due to electrostatic repulsion of amino groups of chitosan chains, at pH 4 amines are charged and their repulsion leads to higher swelling index. However, once chitosan chains are cross-linked, their electrostatic repulsion is highly reduced, keeping a swelling index almost similar at pH 7.4 and pH 4. 1). For crude CH30, contact angle was in accordance with the literature [START_REF] Luo | Facile fabrication of chitosan active film with xylan via direct immersion[END_REF]. It was shown that cross-linked films have higher hydrophobicity compared to net chitosan films, especially for crude CH150 for which the film was not homogeneous and it was difficult to measure an angle as the drop was directly absorbed (SI 3). Once again, this analysis confirmed that the crosslinking occurred. It was also shown that once cross-linked, the higher are the molar masses, the higher is the contact angle which is in accordance with the chitosan properties. Contact angle between hexadecane and films was also experimented. It was shown that for any film, the drop was immediately absorbed and no angle could be measured. It meant that chitosan films, crosslinked or not, could absorb fatty component as chitosan is well known for being a fat binder [START_REF] Raval | Optimization of Chitosan Nanoparticles Synthesis and Its Applications in Fatty Acid Absorption[END_REF].

All those combined properties could open the way for many application in food packaging for example with the well-known antioxidant properties of chitosan [START_REF] Kabanov | Chitosan Application in Food Technology: a Review of Rescent Advances[END_REF] , [START_REF] Elsabee | Chitosan based edible fi lms and coatings : A review[END_REF] . The proof of concept of the reversible Diels-Alder reaction also opens the way to new types of cross-linkers. Indeed, it could be imagined to use other bis or trimaleimide and even other aromatic compounds, to tune the properties of the film, perhaps making thermoreversible hydrogels as described previously [START_REF] Montiel-Herrera | N -(furfural) chitosan hydrogels based on Diels-Alder cycloadditions and application as microspheres for controlled drug release[END_REF].

Conclusions

In this work, a model study was carried out on Diels-Alder and retro Diels-Alder reaction between maleimide moieties and furan functionalized chitosan. Both reactions were evidenced for the first time by NMR spectroscopy and the retro Diels Alder temperatures were determined by DSC for non-reticulated products. In the meantime, different films were prepared from the chitosan modified with pending furan and maleimide cross-linker. It was shown that efficient crosslinking through Diels Alder chemistry occurred at 65°C. The retro Diels-Alder was also evidenced for the first time in chitosan systems and to occur between 110 and 130 °C. Finally, thanks to the cross-linking, homogeneous films were obtained. It was shown that higher crosslinking density played a more important role than the chitosan molar mass in the resulting mechanical properties, with better properties for TMI based films.

Crosslinked films also showed fat absorption and higher hydrophobicity than chitosan alone, which could be promising materials for bio-based coatings applications.

Associated content

Appendix A supplementary data.

Copies of 13 C spectra, FT-IR, 1 H NMR for the compounds of interest and contact angles pictures of the various films.

  b)) and TMI for the trismaleimide from Jeffamine T-403 (Figure 1 a)).

Figure 1

 1 Figure 1. a) 1 H NMR spectrum of TMI b) 1 H NMR spectrum of BMI

  Scheme 1. Functionalization of chitosan with a) furfural in two steps (Montiel-Herrera et al., 2015) b) FGE in one step and reaction between chitosan and MI in c) DMSO d) H 2 O with 1 or 2 % AcOH

Figure 2

 2 Figure 2. a) Following of 1 H NMR monitoring of FGO+MI at 65 °C in DMSO for 180 min b) Following 1 H NMR of FGO + excess of MI in D 2 O, 1% AcOH for 180 min

Figure

  Figure 3. a) 1 H NMR of CH30 in D 2 O+TFA b) 1 H NMR of CH30FG in D 2 O+TFA c) 1 H NMR of CH30FGMI in D 2 O+TFA

Figure 4 .

 4 Figure 4. DSC measurement of CH30MI

Figure 5 .

 5 Figure 5. Both double bonds were consumed during the reaction as the signals no longer

Figure 7 .

 7 Figure 7. Swelling index of chitosan films over time at pH 4 and pH 7.4 a) films from CH30 b) films from CH150 3.4.3. Contact angle Contact angle between water and films were determined (Table1). For crude CH30, contact

Table 1 .

 1 Contact angle, swelling ratio at different pH and E' of CH30 and CH150 films

				Swelling
		Contact angle (°)	Swelling ratio at pH 4 (%) 24 h at	ratio at 24 h in 7.4 0.1 M PBS pH	E' at 25 °C (GPa)
				(%)
	CH30	90 ± 4	N/A	335	5.10 -2
	CH30BMI	107 ± 10	235	174	1.8
	CH30TMI	99 ± 6	232	230	5.2
	CH150	N/A	N/A	557	N/A
	CH150BMI	120 ± 3	344	220	6.0
	CH150TMI	118 ± 5	450	280	4.8

Address : current address

Notes

The authors declare no competing financial interest.

Acknowledgements

The authors would like to thanks Lucie Grossenbacher for her help in the project, Christine Joly-Duhamel for her advices and Cedric Totee for the NMR analysis.