Moustafa Nakechbandi
email: moustafa.nakechbandi@univ-lehavre.fr

Jean-Yves Colin
email: jean-yves.colin@univ-lehavre.fr

Jean Baptiste

J.-Y Colin

J B Gashumba
email: jb.gashumba@free.fr

An efficient fault-tolerant scheduling algorithm for precedence constrained tasks in heterogeneous distributed systems

Keywords: Scheduling, Heterogeneous Distributed System, Fault-tolerance, Precedence Constraints, Communication Delays, Critical-Path Method

An efficient fault-tolerant scheduling algorithm for precedence constrained tasks in heterogeneous distributed systems

I. INTRODUCTION

Heterogeneous distributed systems have been increasingly used for scientific and commercial applications, some of the current ones include Automated Document Factories (ADF) in banking environments where several hundred thousands documents are produced each day on networks of several multiprocessors servers, or high performance Data Mining (DM) systems [START_REF] Palmerini | On performance of data mining: from algorithms to management systems for data exploration[END_REF] or Grid Computing [START_REF] Ruffner | An Enhanced Application Model for Scheduling in Grid Environments[END_REF][START_REF] Venugopal | A Grid Task Broker for Scheduling Distributed Data-Oriented Applications on Global Grids[END_REF]. However, using efficiently these heterogeneous systems is a hard problem, because the general problem of optimal scheduling of tasks is NP-complete [START_REF] Kwok | Static scheduling algorithms for allocating directed task graphs to multiprocessors[END_REF].

When the application tasks can be represented by Directed Acyclic Graphs (DAGs), many dynamic scheduling algorithms have been devised. For some examples, see [2,3,[START_REF] Chen | Distributed Dynamic Scheduling of Composite Tasks on Grid Computing Systems[END_REF]. Also, several static algorithms for scheduling DAGs in meta-computing systems are described in [1,4,6,[START_REF] Kwok | Static scheduling algorithms for allocating directed task graphs to multiprocessors[END_REF]. Most of them suppose that tasks compete for limited processor resources, and thus these algorithms are mostly heuristics. In [5] is presented an optimal polynomial algorithm that schedules the tasks and communications of an application on a Virtual Distributed System with several clusters' levels, although, in [START_REF] Nakechbandi | Bounding the makespan of best pre-scheduling of task graphs with fixed communication delays and random execution times on a virtual distributed system[END_REF] is studied a static scheduling problem where the tasks execution times are positive independent random variables, and the communication delays between the tasks are perfectly known.

This review shows that a lot of works has been done concerning heterogeneous distributed scheduling.

However, the problems with fault tolerant aspect are less studied. Reliable execution of a set of tasks is usually achieved by task duplication and backup copy [START_REF] Qin | A Novel Fault-tolerant Scheduling Algorithm for Precedence Constrained Tasks in Real-Time Heterogeneous Systems[END_REF][START_REF] Randell | System structure for software faulttolerance[END_REF][START_REF] Chen | N-version programming: a fault tolerant approach to reliability of software operation[END_REF]. Inspired with the works of Xiao Qin, Hong Jiang on Real-Time Heterogenous Systems [START_REF] Qin | A Novel Fault-tolerant Scheduling Algorithm for Precedence Constrained Tasks in Real-Time Heterogeneous Systems[END_REF], we propose in this article a good algorithm to solve the problem of one fault tolerant system using tasks duplication and backup copies technics.

This paper has three main parts: In the first one, we present the problem, in the second part, we present a solution to the problem, and in the third part, we discuss the advantages and disadvantages of the proposed solution.

II. THE CENTRAL PROBLEM

In this part, we present the following problem : Given an application T, we want to build an off-line scheduling of the tasks of this application in an heterogeneous distributed System with some possibilities of limited failures. We suppose that only one permanent failure can occur without any possibility of a temporary failure. This constitute one hypothesis which we call "1-failure". Note that the processing time of the application has to be minimized.

The Distributed Servers System

We call Distributed Servers System (DSS) a virtual set of geographically distributed, multi-users, heterogeneous or not, servers. Therefore, a DSS has the following properties: First, the processing time of a task on a DSS may vary from a server to another. This may be due to the processing power available on each server of the DSS for example. The processing time of each task on each server is supposed known. Second, although it may be possible that some servers of a DSS are potentially able to execute all the tasks of an application, it may also be possible in some applications that some tasks may not be executed by all servers. This could be due to the fact that specific hardware is needed to process these tasks and that this hardware is not available on some servers. Or it could be that some specific data needed to compute these tasks are not available on these servers for some reason. Or it could be that some user input is needed and the user is only located in a geographically specific place. Obviously, in our problem we suppose that the needs of each task of an application are known, and that at least one server of the DSS may process it, else there is no possible solution to the scheduling problem.

Furthermore, an important hypothesis is that the concurrent executions of some tasks of the application on a server have a negligible effect on the processing time of any other task of the application on the same server. Although apparently far-fetched, this hypothesis may hold if the server is a multiprocessors architecture with enough processors to simultaneously execute all the tasks of the application that are to be processed concurrently. Or it may be that the server is a time-shared, multi-user system with a permanent heavy load coming from other applications, and the tasks of an application on this server represent a negligible additional load compared to the rest.

In addition, in the network interconnecting the servers of a DSS, the transmission delay of a result between two tasks varies depending on the tasks and on their respective sites.

Again, we suppose that concurrent communications between tasks of the same application on two servers have a negligible effect on the communication delays between two others tasks located on the same two servers. This hypothesis may hold if the network already has a permanent heavy load due to other applications, and the communications of the application represent a negligible additional load compared to the one already present.

Directed Acyclic Graph

We now describe the application itself in our problem. An application is decomposed into a set of indivisible tasks that have to be processed. A task may need data or results from other tasks to fulfil its function and then send its results to other tasks. The transfers of data between the tasks introduce dependencies between them. The resulting dependencies form a Directed Acyclic Graph. Because the servers are not necessarily identical, the processing time of a given task can vary from one server to the next. Furthermore, the duration of the transfer of a result on the network cannot be ignored. This communication delay is function of the size of the data to be transferred and of the transmission speed that the network can provide between the involved servers. Note that if two dependent tasks are processed themselves on the same server, this communication delay is considered to be 0.

The central scheduling problem P on a Distributed Server System, is represented therefore by the following parameters:

• a set of servers, noted Σ = {σ 1 , ..., σ s }, interconnected by a network, • a set of the tasks of the application, noted I = {1,..., n}, to be executed on Σ. The execution of task i, i ∈ I, on server σ r , σ r ∈ Σ, is noted i/σ r . The subset of the servers able to process task i is noted Σ i , and may be different from Σ,

• the processing times of each task i on a server σ r is a positive value noted r i σ π / . The set of processing times of a given task i on all servers of Σ is noted

Π i (Σ). r i σ
π / = ∞ means that the task i cannot be executed by the server σ r .

• a set of the transmissions between the tasks of the application, noted U. The transmission of a result of an task i, i ∈ I, toward a task j, j ∈ I, is noted (i, j). It is supposed in the following that the tasks are numbered so that if (i, j) ∈ U, then i < j, • the communication delays of the transmission of the result (i, j) for a task i processed by server σ r toward a task j processed by server σ p is a positive value noted . The set of all possible communication delays of the transmission of the result of task i, toward task j is noted ∆ i,j (Σ). Note that a zero in ∆ i,j (Σ) mean that i and j are on the same server, i.e. Π i (Σ) be the set of all processing times of the tasks of P on Σ.

Let ∆ (Σ) = U U j i ∈ ) , (
∆ i,j (Σ) be the set of all communication delays of transmissions (i, j) on Σ.

The central scheduling problem P on a distributed servers system DSS can be modelled by a multi-valued DAG G = {I, U, Π(Σ), ∆(Σ)}. In this case we note P={G, Σ}.

Example: In the following example we consider a problem P of an application with nine tasks that has to be processed by a set of 3 servers on an heterogeneous distributed system. The architecture of distributed system and the task graph of this application are represented by the figure 1 and figure 2. It is supposed that the 3 servers are distributed on two different networks.

Network delay between σi→ σj

Server σ1

Server σ2 Server σ3 In this example there are 9 tasks, the label [x, y, z] on task is the processing cost on the 3 servers'. For example on the task 1 we have the label [START_REF] Colin | A multivalued DAG model and an optimal PERT-like Algorithm for the Distribution of Applications on Heterogeneous, Computing Systems[END_REF][START_REF] Palmerini | On performance of data mining: from algorithms to management systems for data exploration[END_REF]20], that is mean the processing time of task 1 on server σ 1 is 15, on server 2 is 10 on server 3 is 20. The table 2 indicate the complete communication delays for the problem P, this effective communication delay between two tasks is due to the multiplication of cost communication (data in tables 1) and volume communication data between tow tasks. For example, if task 1 is executed on server σ 1 , task 3 is executed on server σ 2 the communication between tasks 1 and 3 noted

Server σ1 0 2 3 Server σ2 2 0 3 Server σ3 3 3 0 Table 1: Cost communication between servers (distance σr → σp )
2 1 / 3 , / 1 σ σ c = 2*2 = 4, because the cost communication (σ 1 →σ 2 )
is 2, the volume of data between task1 and task 3 is 2 also.

(1, 3) σ1 σ2 σ3

(

Definition of a feasible solution

We note PRED(i), the set of the predecessors of task i in G:

{ } ) , ( et / ) PRED( U i k I k k i ∈ ∈ =
And we note SUCC(i), the set of the successors of task i in

G: { } ) , ( et / ) SUCC( U j i I j j i ∈ ∈ =
A feasible solution S for the problem P is a subset of executions { i/σ r , i∈I } with the following properties:

• each task i of the application is executed at least once on at least one server σ r of Σ i , • to each task i of the application executed by a server σ r of Σ i , is associated one positive execution date

r i t σ / ,
• for each execution of a task i on a server σ r , such that PRED(i) ≠ ∅, there is at least an execution of a task k, k ∈PRED(i), on a server σ p , σ p ∈ Σ κ , that can transmit its result to server σ r before the execution date

r i t σ / .
The last condition, also known as the Generalized Precedence Constraint (GPC) [5], can be expressed more formally as:

     + + ≥ Σ ∈ ∃ ∈ ∀ ≥ ∈ ∀ r p p p r r i k k k i k p i r c t t i k t S i σ σ σ σ σ σ π σ σ / , / / / / / / ), PRED( 0 / else ) PRED( if ∅ = i
It means that if a communication must be done between two scheduled tasks, there is at least one execution of the first task on a server with enough delay between the end of this task and the beginning of the second one for the communication to take place. A feasible solution S for the problem P is therefore a set of executions i/σ r of all i tasks, i ∈ I, scheduled at their dates r i t σ / , and verifying the Generalised Precedence Constraints GPC. Note that, in a feasible solution, several servers may simultaneously or not execute the same task. This may be useful to generate less communications.

All the executed tasks in this feasible solution, however, must respect the Generalized Dependence Constraints.

Optimality Condition

Let T be the total processing time of an application (also known as the makespan of the application) in a feasible solution S, with T defined as:

) ( max / / r r r i i S i/ t T σ σ σ π + = ∈ A feasible solution S* of the problem P modelled by a DAG G = {I, U, Π(Σ), ∆(Σ)} is optimal if its total
processing time T* is minimal. That is, it does not exist any feasible solution S with a total processing time T such that T < T*.

III. PROPOSAL SOLUTION

Our algorithm has two phases: the first one is for the scheduling of primary copies where we use the DSS-OPT algorithm [START_REF] Colin | A multivalued DAG model and an optimal PERT-like Algorithm for the Distribution of Applications on Heterogeneous, Computing Systems[END_REF] and the second one is for the scheduling of the backup copies in which a variant of the eFRCD algorithm [START_REF] Qin | A Novel Fault-tolerant Scheduling Algorithm for Precedence Constrained Tasks in Real-Time Heterogeneous Systems[END_REF] is used.

Primary Copies Scheduling (The DSS_OPT Algorithm):

We schedule primary copies of tasks in our algorithm with DSS-OPT algorithm [START_REF] Colin | A multivalued DAG model and an optimal PERT-like Algorithm for the Distribution of Applications on Heterogeneous, Computing Systems[END_REF]. The DSS-OPT algorithm is an extension of PERT algorithms type. In its first phase, it computes the earliest feasible execution date of each task on every server, and in its second phase it builds a feasible solution (without fault) starting from the end of the graph with the help of the earliest dates computed in the first phase.

Let P be a DSS scheduling problem, and let G = {I, U, Π(Σ), ∆(Σ)} be its DAG.

One can first note that there is an optimal trivial solution to this DSS scheduling problem. In this trivial solution, all possible tasks are executed on all possible servers, and their results are then broadcasted to all other tasks that may need them on all others servers. This is an obvious waste of processing power and communication resources, however, and something better and more efficient is usually needed. So, we now present DSS_OPT(P) algorithm's that builds an optimal solution for problem P.

DSS_OPT has two phases:

The first phase, DSS_LWB(P), computes the earliest feasible execution dates r / i b σ for all possible executions i/σ r of each task i of problem P.

The second phase determines, for every task i that does not have any successor in P, the execution i/σ r ending at the earliest possible date b σ , one is chosen, arbitrarily or using other criteria of convenience, and kept in the solution. Then, for each kept execution i/σ r that has at least one predecessor in the application, the subset L i of the executions of its predecessors that satisfy GPC(i/σ r ) is established. This subset of executions of predecessors of i contains at least an execution of each of its predecessors in G. One execution k/σ p of every predecessor task k of task i is chosen in the subset, arbitrarily or using other criteria of convenience, and kept in the solution. It is executed at date

r i k L σ / ← { k/σ q / σ p ∈ Σ κ and r r p p p i i k k k b c b σ σ σ σ σ π / / , / / / ≤ + + } 5: k/σ q ← keepOneFrom( r i k L σ / ) 6: schedule (k/σ q )
end for end if end schedule keepOneFrom(L i ) return an execution i/σ r of task i in the list of the executions L i . end keepOneFrom.

Because the computed execution time of each task on each server is its earliest execution time on this server, and because only the copy with the ealiest ending time, of each task without any successor, is used in the solution calculated by DSS_OPT, and finally because all other copies are used only if they ensure that the final copies receives their data in time else they are not used, it follows that the feasible solution computed by DSS_OPT is optimal in execution tile. Summarizing the above discussion, we have the following theorem.

Theorem : The feasible solution calculated by DSS_OPT algorithm is optimal

Numerical example:

We consider here the problem P definite in figure 1 and2, the DSS-OPT algorithm uses DSS_LWB to compute the earliest possible execution date of all tasks on all possible servers, resulting in the following values b and r (Table 3

= = = Σ ∈ ∀ ∅ = ∀ r i r i i i r T σ σ
In our example, the task 9 does not have any successor. The list L 9 of the executions kept for this task in the solution is reduced therefore to the execution 9/σ 2 . Thus L 9 = {9/σ 2 }.

The execution of task 9 on the server σ 2 is scheduled at date 49. Next, The tasks 6, 7 and 8 are the predecessors in G of task 9. For the task 6, the execution 6/σ 3 may satisfy the Generalised Precedence Constraints relative to 9/σ 2 . Therefore, this execution is kept and is scheduled at date 24 ( 3 / 6 σ b ). For task 7, execution 7/σ 1 is kept and is scheduled at date 31… The table 4 presents the final executions i/σ r kept by the DSS_OPT(P) algorithm, with their date of execution, in an optimal solution S.

1/σ2 2/σ1 2/σ2 3/σ2 4/σ2 5/σ1 5/σ2 6/σ3 7/σ1 8/σ2 9/σ2

Table 4: final executions i/σr kept by the DSS_OPT(P) algorithm

We obtain (figure 3) the following optimal scheduling by DSS_OPT(P) algorithm:

Backup Copies Scheduling (eFRCD Algorithm)

We have just been building a scheduling of the primary copies, with DSS-OPT algorithm. Now, we schedule backup copies of the tasks with the eFRCD algorithm [START_REF] Qin | A Novel Fault-tolerant Scheduling Algorithm for Precedence Constrained Tasks in Real-Time Heterogeneous Systems[END_REF], which allows us to survive failures with the hypothesis [1failure]. Note that the eFRCD Algorithm was originally proposed to schedule tasks with deadlines in a real-time system with 1-fault hypothesis. We slightly modified it, drooping deadlines, and using earliest date of execution of the tasks and the strong copies [START_REF] Colin | A multivalued DAG model and an optimal PERT-like Algorithm for the Distribution of Applications on Heterogeneous, Computing Systems[END_REF] definition in ordering list priority. The modified eFRCD algorithm is detailed below.

Some notices and definitions:

• The OL (Ordered List) used by eFRCD to define priorities of tasks will be determined by earliest start execution task's dates by ascending order on all different servers where primary copies are affected with DSS-OPT algorithm scheduling. • Backup copies are not established for tasks that have already two primary copies with DSS-OPT Algorithm scheduling even it can exist for these tasks the "backup" with good results compared primary copies. • Strong copy definition: Given a task i, i P is a strong primary copy, if and only if only a failure of its executing processor may forbid the execution of this primary copy. Because if its executing processor is not in failure, then for each of its predecessors in the DAG whenever the failure is, there is at least one primary or backup copy able to send it its result on time. For example, on figure 3, all primary copies of tasks 1 and 2 are strong copies, because they have no predecessor. The primary copy of task 3 on server σ 2 is strong too because only a failure of σ 2 may forbid its execution. However, the primary copy of task 6 on server σ 3 will not be executed at the scheduled date if there is a failure on σ 2 , as it must receive the result of task 3, and there is no other copy of task 3 available. Note that as backup copies of its predecessors are added by eFRCD algorithm, it is possible for an initially non strong primary copy to become strong. while SL != φ do for each primary copy i P of a task i in SL do if a backup copy i B of task i can be established, then 3.

Modified eFRCD Algorithm

schedule i B on server σ r where it can be scheduled at low cost end if end for 4.

delete from SL tasks whose backup are scheduled end while while OL != φ do for each primary copy i P of task i in OL do if a backup copy i B of task i can be established, then 5.

schedule i B on server σ r where it can be scheduled at low cost end if delete from OL tasks whose backup are scheduled end for end while End of algorithm

Numerical example:

The execution of our algorithm with the example DAG of figure 2 gives the following results:

1. Ordered List (OL) with precedence constraints by DSS-OPT: {1, 3, 4, 6, 7, 8, 9}; (tasks 2 and 5 are not considered because they have more than one primary copies). Finally, we obtain (figure 4) the following resilient scheduling in the case of one failure : • In the last figure σ 1 use 3 processing units, σ 2 use 3 processing units also, but σ 3 use only 2 processing. • i P is a primary copy of task i, i P1 is the first primary copy of task i, i P2 is the second primary copy of task i. • i B is the backup copy of task i.

IV. ALGORITHM ANALYSIS

The most computationally intensive part of DSS_OPT(P) is the first part DSS_LWB(P). In this part, for each task i, for each server executing i, for each predecessor j of i, for each server executing j, a small computation is done. Thus the complexity of DSS_LWB(P) is Ο(n 2 s 2 ), where n is the number of tasks in P, and s is the number of servers in DSS. Thus, the complexity of the DSS_OPT(P) algorithm is Ο(n 2 s 2 ). eFRCD is polynomial also, for the reason that eFRCD has two loops (line 3 and line 5), each loop need n*s operations, thus the complexity of eFRCD is Ο(n 2 s). Consequently the global complexity of our new algorithm is Ο(n 2 s 2 ).

This algorithm has two advantages:

• When there is no fault on servers, our algorithm is optimal as it uses the optimal solution computed by DSS-OPT. • When there is a fault on one server, our solution is a good one because before the fault the solution by DSS-OPT is better than eFRCD, and after the fault it uses a solution computed by eFRCD, and eFRCD builds a good solution in this case.

Furthermore, note that our solution is guaranteed to finish if one fault occurs, because every tasks has two or more scheduled copies on different servers in the final solution.

If more than one fault occur, the solution may still finish, but there is no guaranty now.

The strong copies notion of eFRCD algorithm has been considered in our algorithm because this notion gives an enhanced order of priority. A strong copy is scheduled with a high priority because it is considered as a task without precedence constraints of starting.

We do not establish backup copies for tasks which have already two primary copies with DSS-OPT Algorithm scheduling even it can exist for these tasks the "backup" with good results compared to primary copies. Related to our hypothesis "1-fault", this option gives scheduling facilities . In our further works, we will study the case of backup copies for tasks, which have more than one primary copy.

The model of failure, as it features at most 1 crash, may seem poor. However, if the probability of any failure is very low, and the probabilities of failures independent, then the probability of two failures will be smaller indeed. Furthermore, the algorithm may be extended to 2 or more failures, by using two or more backup copies.

Finally, the problem solved by this new algorithm suffers from one strong assumption, namely that an unbounded number of tasks may be processed on each server in parallel without any effect on the tasks' processing time.

We mention two cases where this assumption may be satisfied: (a) on systems with enough parallel processors, requiring n processors per server to guarantee schedulability, or (b) constant heavy load on the servers. This limit the usefulness of the direct application of this algorithm to these few cases. However, this algorithm and its model are similar in idea and assumption to the classical Critical-Path Method (or PERT method) [START_REF] Garey | Johnson: Two-Processor Scheduling with Start-Times and Deadlines[END_REF] as that CPM/PERT method do not consider resources constraints in order to get earliest execution dates. By the way these CPM/PERT results are then used in some real-life systems as the priority values of tasks in some list-scheduling algorithms, the result found by our algorithm may be used as the first step of a list scheduling algorithm, in which the earliest execution dates of primary and backup copies are used as priority values to schedule these copies on the servers of a real-life system.

  that either task i cannot be executed by server σ r , or task j cannot be executed by server σ p , or both.Let Π (Σ) = U I i∈

Figure 1 :

 1 Figure 1: Distributed system architecture for the application example. The Matrix cost communication of our example is presented in table 1.

Figure 2 :

 2 Figure 2: Example of a multi-valued DAG

7 :the server σ r 2 :

 72 σ . The examination of the predecessors is pursued in a recursive manner until the studied tasks do not present any predecessors in G. The complete algorithm is the followingInput: G = {I, U, Π(Σ), ∆(Σ)} i such that SUCC(i) = ∅ // second phase 4: i L ← { i/σ r / σ r ∈ Σ ι andT For each task i where PRED(i) = ∅ do 2:for each server σ r such that σ r ∈ Σ i do while there is a non marked task i such that all its predecessors k in G are marked do for each server σ r such that σ r ∈ Σ i do if PRED(i) ≠ ∅ then 3: for each task k such that k ∈ PRED(i) do 4:

Figure 3 :

 3 Figure 3 : DSS_OPT algorithm scheduler

Input:

  Scheduling established by DSS-OPT algorithm Output: Backup copies scheduling. Algorithm 1. order the tasks by their earliest start execution as established by DSS-OPT algorithm. This will generate an Ordered List noted OL; 2. SL ← Strong-copies in OL; // SL: Strong-copies List OL ← (OL -SL)

Figure 4 :

 4 Figure 4 : Modified eFRCD algorithm scheduler Notes on the figure 4: • In the last figure σ 1 use 3 processing units, σ 2 use 3processing units also, but σ 3 use only 2 processing. • i P is a primary copy of task i, i P1 is the first primary copy of task i, i P2 is the second primary copy of task i. • i B is the backup copy of task i.

Table 2 :

 2 Complete communication times for the problem P

					3, 6) σ1 σ2 σ3	(5, 8) σ1 σ2 σ3
	σ1	0	4	6	σ1	0	4	6	σ1	0	4	6
	σ2	4	0	6	σ2	4	0	6	σ2	4	0	6
	σ3	6	6	0	σ3	6	6	0	σ3	6	6	0
	(1, 4) σ1 σ2 σ3	(3, 7) σ1 σ2 σ3	(6, 9) σ1 σ2 σ3
	σ1	0 12 18	σ1	0	8 12	σ1	0 10 15
	σ2	12 0 18	σ2	8	0 12	σ2	10 0 15
	σ3	18 18 0	σ3	12 12 0	σ3	15 15 0
	(2, 4) σ1 σ2 σ3	(4, 8) σ1 σ2 σ3	(7, 9) σ1 σ2 σ3
	σ1	0	6	9	σ1	0	6	9	σ1	0	8 12
	σ2	6	0	9	σ2	6	0	9	σ2	8	0 12
	σ3	9	9	0	σ3	9	9	0	σ3	12 12 0
	(2, 5) σ1 σ2 σ3	(5, 7) σ1 σ2 σ3	(8, 9) σ1 σ2 σ3
	σ1	0	8 12	σ1	0 10 15	σ1	0	6	9
	σ2	8	0 12	σ2	10 0 15	σ2	6	0	9
	σ3	12 12 0	σ3	15 15 0	σ3	9	9	0

  ):

	1 b1 r1	2 b2 r2	3 b3 r3
	σ1 0 15	σ1 0 19	σ1 14 34
	σ2 0 10	σ2 0 12	σ2 10 18
	σ3 0 20	σ3 0 15	σ3 16 26
	4 b4 r4	5 b5 r5	6 b6 r6
	σ1 18 23	σ1 19 31	σ1 22 37
	σ2 12 18	σ2 12 32	σ2 18 40
	σ3 20 29	σ3 15 30	σ3 24 36
	7 b7 r7	8 b8 r8	9 b9 r9
	σ1 31 41	σ1 31 49	σ1 49 74
	σ2 32 52	σ2 32 44	σ2 49 57
	σ3 30 55	σ3 30 62	σ3 53 75

Table 3 :

 3 The earliest possible execution date of all tasks on all possible servers for the problem P

	It then computes the smallest makespan of any solution to
	the P problem :
	/	max ( SUCC	)	min	(	/	)	) 75 , 57 , 74 min(	57

Table 5 :

 5 Table of costs for backup copies on servers

	Tasks number	1		3	4	6	7		8		9
	Candidate_servers	1	3	1	3 1 3 1	2	2	3	1	3	1	3
	for backup copies
	Costs on servers	15 20 20 10 5 9 15 22 20 25 18 32 25 22
	Low cost Server	1		3	1	1	2		1		3
	2. Strong primary copies of the list SL = {1, 3, 4, 8};
	3. Backup copy scheduling of tasks in list SL:
	• Backup copy scheduling of task 1: server σ 1 ,
	bounds 0 to 15;
	• Backup copy scheduling of task 3: server σ 3 , bounds
	22 to 32;

Table 6 :

 6 Start

dates (s) and finish (f) dates table for back-up copies

V. Conclusion

In this paper, we have proposed an efficient scheduling algorithm in which tasks with precedence constraints and communication delays have to be scheduled on an heterogeneous distributed system environment with one fault hypothesis. Our algorithm has combined the DSS-OPT algorithm and the eFRCD algorithm. To provide a fault-tolerant capability, we employed primary and backup copies. But no backup copies were established for tasks which have more than one primary copy.

The result have been a schedule in polynomial time that is optimal when there is no failure, and is a good resilient schedule in the case of one failure. The one fault-tolerant capability was incorporated by using modified eFRCD and the system reliability was further enhanced by using DSS-OPT algorithm where we have optimal scheduling in case of no fault. The execution dates of the primary and backup copies may be used as priority values for list scheduling algorithm in cases of real-life, limited resources, systems.

In our future work, we intend to study the same problem with sub-networks failures. Also, we intend to consider the problem of non permanent failures of servers. Finally, we want to consider the problem of the partial failure of one server, in which one server is not completely down but loses the ability to execute some tasks and keeps the ability to execute at least one other task.


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Development of biomass-based slurry for the manufacture of furan/cellulose composite by cast molding and 3D printing

Keywords: Bio-based, Furan, Cellulose, Composites, Thermomechanical properties

The polycondensation of FA proceeds following two alternative steps, namely head-to-tail [START_REF] Gandini | Furans in polymer chemistry[END_REF] or head-to-head [START_REF] Guigo | Eco-friendly composite resins based on renewable biomass resources: Polyfurfuryl alcohol/lignin thermosets[END_REF] and the latter reaction produces an unstable ether moiety which loses formaldehyde in an acidic medium, thereby giving the same methylene bridge formed in reaction [START_REF] Gandini | Furans in polymer chemistry[END_REF], as shown in Figure 1 [START_REF] Choura | Acid-Catalyzed Polycondensation of Furfuryl Alcohol: Mechanisms of Chromophore Formation and Cross-Linking[END_REF]. This simple growth giving 2,5-furanmethylene units is however perturbed by two reactions involving the polymer, namely the formation of conjugated sequences by loss of hydride ions and protons, leading to a progressive bathochromic shift in color all the way to a black product, accompanied by a polymer-to-polymer Diels-Alder coupling resulting in a crosslinked material [START_REF] Choura | Acid-Catalyzed Polycondensation of Furfuryl Alcohol: Mechanisms of Chromophore Formation and Cross-Linking[END_REF].

This complex structure has puzzled chemists for decades. As analysis techniques evolve, scientists had more data and hence more information about the structure emerged putting together various secondary reactions [START_REF] Tondi | Understanding the Polymerization of Polyfurfuryl Alcohol: Ring Opening and Diels-Alder Reactions[END_REF]. The ensuing FA-based resins (PFA) have found numerous applications, thanks to their excellent flame and chemical resistance and low smoke emission, including metal-casting cores, wood adhesives and binders, coatings, and a precursor to carbonaceous products [START_REF] Choura | Acid-Catalyzed Polycondensation of Furfuryl Alcohol: Mechanisms of Chromophore Formation and Cross-Linking[END_REF].

Recently, they have also been used in the production of composites along with synthetic and natural fibers, e.g. by mold compression of flax and kenaf prepregs [START_REF] Deka | Renewable resource based "all green composites" from kenaf biofiber and poly(furfuryl alcohol) bioresin[END_REF][START_REF] Pohl | Properties of compression moulded new fully biobased thermoset composites with aligned flax fibre textiles[END_REF] and woven glass prepregs [START_REF] Domínguez | Study on the non-isothermal curing kinetics of a polyfurfuryl alcohol bioresin by DSC using different amounts of catalyst[END_REF]. The challenge of the use of this resin was mainly foaming and water condensation which causes high porosity in the composites. The porosity is mainly affected by the moisture content in the composite which should be limited [START_REF] Oishi | Viscosity, pH, and moisture effect in the porosity of poly(furfuryl alcohol)[END_REF] and the condensation of vapor produced during polymerization. The condensation water should be evacuated during the curing process to minimize its effect on porosity [START_REF] Domínguez | Study on the non-isothermal curing kinetics of a polyfurfuryl alcohol bioresin by DSC using different amounts of catalyst[END_REF], which leads to complicated processing of the composite.

PFA has also been coupled with lignocellulosic particles such as humins [START_REF] Pin | Valorization of Biorefinery Side-Stream Products: Combination of Humins with Polyfurfuryl Alcohol for Composite Elaboration[END_REF], cork residues [START_REF] Menager | Green" composites prepared from polyfurfuryl alcohol and cork residues: Thermal and mechanical properties[END_REF], lignin [START_REF] Guigo | Eco-friendly composite resins based on renewable biomass resources: Polyfurfuryl alcohol/lignin thermosets[END_REF], and sisal whiskers [START_REF] Ahmad | Thermal and dynamic mechanical properties of bio-based poly(furfuryl alcohol)/sisal whiskers nanocomposites[END_REF], which displayed good compatibility and improvement in the mechanical performances of the composites. More recently, low concentrations of PFA were added to natural fibers such as micro-fibrillated cellulose [START_REF] Lems | Reinforcing effect of poly(furfuryl alcohol) in cellulosebased porous materials[END_REF] for mechanical reinforcement, and bamboo [START_REF] Wang | Preparation and properties of bamboo/polymer composites enhanced by in situ polymerization of furfuryl alcohol[END_REF] for the improvement of physical-mechanical properties and decay resistance.

An overview of the state of the art of composites developed with PFA highlighting their mechanical and thermal properties is presented in Table 1. According to this data, one can notice that PFA has been mainly coupled either with different classes of natural non-wood fibers such as flax, sisal, and bamboo. These composites were manufactured using different methods but compression is always applied and, if not, PFA was used to produce thin films pobably due to the high porosity which this resin can generate when manufacturing bulk structures. In terms of mechanical properties, it was reported that the neat PFA has a stiffness below 1 GPa (tensile mode) and tensile strength of about 10 MPa [START_REF] Deka | Renewable resource based "all green composites" from kenaf biofiber and poly(furfuryl alcohol) bioresin[END_REF][START_REF] Kherroub | Synthesis of poly(furfuryl alcohol)/montmorillonite nanocomposites by direct in-situ polymerization[END_REF]. After the addition of a continuous reinforcing phase, for instance, aligned flax fiber woven textile, the stiffness of the resulting composite reached a value of 25 GPa and its strength attained 211 MPa. However, when adding discontinuous filler we notice a limited improvement in Young's modulus which remains under 2 GPa and a considerable improvement in the strength of about 400% compared to the neat polymer.

In terms of thermal properties, the results can be quite dispersed depending on the used polymer and filler with a T g ranging from 80°C to over 130°C and a storage modulus over 1 GPa at 160°C in some cases. The addition of some of these fillers (especially nanoparticles) served to confer the polymer with new properties such as electrical conductivity, long wear life, and superhydrophobic surface. In this study, a totally biomass-based composite was developed coupling a commercial oligomeric furfuryl resin (OFR) with cellulose powder by simple casting and curing under atmospheric pressure. The cellulose powder allowed to improve the intrinsic low viscosity of the OFR and favored vapor degassing during thermal curing thus leading to the formulation of a highly viscous composite paste that can be effectively processed both by 3D printing and thermo compression molding. 3D printed and molded/milled objects displayed excellent shape fidelity and porosity/mechanical properties in line with those obtained for casted samples thus paving the way for the use of cellulose/OFR composites in advanced manufacturing processes other than prepreg or sheet compression molding.

MATERIALS AND METHODS

Materials

A commercial OFR and a latent catalyst were kindly supplied by Transfurans Chemicals (Belgium). The commercial resin contains 7.5 wt% water and less than 1% of free FA monomer. According to the supplier's information, the polymer can be stored for a long time at temperatures between 20 °C and 25 °C, and the shelf life of this kind of resin can reach 18 months under ambient conditions [START_REF] Behzadfar | Shelf-life of polyfurfuryl alcohol resin: an accelerated rheokinetics study[END_REF].

Commercial cellulose fibers were kindly supplied by the Rossow Industry. The fibers have a powdery aspect and, according to preliminary SEM analysis, shown in Figure 2, the fibers are stocky particles with an approximated aspect ratio of 5. The water content of the cellulose was measured to be 4.58 ±0.05 wt%. As determined using the image granulometric analysis given in Figure 2, 50% of the cellulosic particle have a size smaller than 9.8 µm.

Composite fabrication

The composite slurry was prepared by first adding the catalyst to the resin and manually mixing the two fluids. The mixture was then put in a planetary mixer and the cellulose powder was gradually added and the whole was blended for at 30 min. The paste was then placed in a refrigerator at 4-5 °C before further processing. In this study, different formulations were tested with different cellulose fractions, as shown in Table 2.

The weight fractions of the cured composite were calculated considering that during the crosslinking 100% of the water was evaporated (added water and condensation water). The density of the cured OFR is estimated to be 1.36 g.cm -3 [START_REF] Domínguez | Study on the non-isothermal curing kinetics of a polyfurfuryl alcohol bioresin by DSC using different amounts of catalyst[END_REF]. The cold slurries were placed in an oven at 50°C for 5 min to decrease their viscosity and facilitate their processing.

The bio-composites were prepared using a simple hand casting in open poly(tetrafluoroethylene) molds. The casting was done manually and without any additional constraints, yet avoiding creating voids because of the sticky nature of the slurries. The molds were then placed in the oven at 90°C for an open-air crosslinking overnight. A post-curing was then performed for 3 hours at 130°C. The processability of the developed paste was tested using a direct ink writing (DIW) 3D printer and by CNC milling composite plated manufactured by hot compression molding (2-4

MPa and 100°C).

Analyses and characterization techniques

To analyze the structure and composition of the commercial OFR, 13C and 1H NMR investigations were done on both liquid and solid polymers. Spectroscopic measurements were recorded on a Bruker AVANCE400 spectrometer (Billerica, MA, USA) equipped with a 5 mm BB/19F-1H/d Z-GRD probe operating at 100.612 MHz for 13C, and 400.130 MHz for 1H. For solid-state RMN the sample was compacted in 4 mm ZrO2 rotors while the liquid NMR measurements were performed on OFR samples dissolved in DMSO-d6 and placed in a 10-mm tube. For further investigation, Fourier Transform Infrared (FT-IR) spectroscopy analyses were also conducted using a PerkinElmer FT-IR spectrometer (Perkin Elmer, USA). For solid samples analysis, each sample was grounded and then mixed with KBr powder with a ratio of 1:125 for pellets preparation. The spectra were acquired in a frequency range of 4000-500 cm -1 with a resolution of 4 cm -1 and then corrected with the environmental spectrum. For liquid resin analysis, the sample was placed between two NaCl plates after removing the water.

The thermal analysis of the thermosetting resin was performed by Differential Scanning Calorimetry (DSC) (TA DSC Q100). An average of 10 mg of each sample was placed in a closed aluminum crucible and heated from 40°C to 200°C at 5°C.min -1 under airflow. The thermogravimetric analysis (TGA) of the composites was performed on TGA-DSC3+ Mettler Toledo. The samples were heated from room temperature to 1000°C with a temperature ramp rate of 10°C.min -1 . The thermo-oxidative degradation of the composites and the neat PFA network were studied under a 40 mL.min -1 oxygen flow, and further tests were performed under a nitrogen flow of 40 mL.min -1 .

To evaluate the shrinkage upon crosslinking of the neat PFA and the bio-composites, the French standard NF 51-401 was followed. 5 specimens of 80 mm × 10 mm × 4 mm were prepared for each sample according to the protocol previously described. The samples were then placed in a normalized atmosphere for 24 hours before measuring their length L' with 0.05 mm precision after curing, using a caliper. For every specimen, the measure was repeated at least 5 times. The shrinkage upon curing R% was calculated according to equation (1):

R (%) = L'/L × 100 (1)
To measure open and close porosity in the composite with different cellulose content, 2.5 cm square specimens with 2 mm thickness were cured. The Buoyancy method was used (ASTM D792) to measure the experimental density (D ex ) with a surfactant aqueous solution as a test liquid. The theoretical density (D th ) was calculated by considering that the cured PFA was perfectly compact using the rule of mixtures. The porosity percentage in the composites was calculated based on the equation (2):

P (%) = ((D th -D ex )×100)/D th (2)
To assess the composite sensitivity to humidity, samples of the cured composite and neat cellulose along with cellulose powder were placed in a Varimass apparatus (TechPap). The relative humidity (RH) was controlled and the mass variation was recorded during the entire test via an integrated precision balance. The humidity program consisted of 3 cycles. In the first cycle, the humidity was fixed at 50% for 2 hours. The second cycle started with two hours at low humidity (20%) followed by two hours at 50%. The third cycle started with two hours at a high humidity level (90%) followed by stabilization at 50% for two extra hours.

The morphology of both fillers and the composite were investigated by scanning electron microscopy (SEM). The specimens were gold coated and examined using a QuantaFEG250 instrument. The accelerating voltage was fixed at 2.50 kV and the magnitude between 60 and 2000.

The dynamic mechanical properties, such as the storage modulus E', were studied with a Dynamic Mechanical Analyzer (DMA). Tests were realized using the 3-points bending mode with a temperature scan ranging from 25 °C

to 300 °C at 2 °C.min -1 at a frequency of 1 Hz. The static force had an average of 25 N and the dynamic deformation was ±15 μm. The samples' dimensions were 40×10×4 mm (length, width, and thickness) and were measured precisely for every specimen before the scan. At least 3 specimens per sample were analyzed.

Before any mechanical test, the composites were conditioned at 23 °C and at 50% relative humidity for at least 16 h.

The mechanical properties were tested with an Instron Universal Testing Machine and the data analyzed with the help of the Blue Hill software. The tensile test was performed according to the ISO 527 standard using the 5A specimen dimension and a crosshead speed of 2 mm.min -1 . The flexural properties were measured according to ISO 178 at a crosshead speed of 2 mm.min -1 . The obtained data were the average of 10 to 15 measurements. The specimens were prepared by open-air casting molding, as described above and therefore shrinkage of up to 5% of the dimensions took place. The surfaces of the samples were polished by sandblasting and the exact dimensions were measured before every test.

RESULTS AND DISCUSSION

Resin investigations

The 1 H NMR spectrum of the commercial OFR in Figure 3 shows a clear peak around 6.2 ppm, which reflects the resonance of the H3 and H4 protons of the furan ring. The peak of the H5 proton around 7.5 ppm is surprisingly very weak because, in the case of the oligomerization of exclusively FA monomers, those protons will be present at least at one of the extremities of the formed oligomers [START_REF] Choura | Acid-Catalyzed Polycondensation of Furfuryl Alcohol: Mechanisms of Chromophore Formation and Cross-Linking[END_REF].

On the other hand, if one compares the signal of the -CH 2 -methylene bridge protons between two furan rings around 3.8 ppm and the signal of CH 2 -O protons, which can be either those derived from the terminal CH 2 OH groups or those in the ether bridge formed after the condensation according to reaction (2), one can notice that the latter is uncommonly more important [START_REF] Choura | Acid-Catalyzed Polycondensation of Furfuryl Alcohol: Mechanisms of Chromophore Formation and Cross-Linking[END_REF][START_REF] González | Furfuryl alcohol polymerisation by iodine in methylene chloride[END_REF][START_REF] Principe | An NMR study of poly(furfuryl alcohol) prepared with ptoluenesulphonic acid[END_REF]. Around 5.3 ppm is the signal of the CH 2 OH hydroxyl protons which is a much more important peak than that of the free H5 protons. The strong peak at 3.3 ppm is related to the water protons, assigned to the remaining moisture in the commercial product.

On the 13 Examining the FT-IR spectra on Figure 5, the first observation relates to the important signal around 3400 cm -1 of the OH groups of the "dry" product which we have verified on the 1 H NMR spectrum, probably partially arising from the residual moisture in the resin. But these hydroxyls are mainly due to the terminal groups of the oligomers, which are the reason behind the partial solubility in water of the commercial resin thanks to the polarity of its molecules. The presence of the furan ring is proven by the two signals around 1000 and 800 cm -1 . One can also distinguish the typical double peak around 2900 cm -1 of the CH 2 groups. The peak around 1200 cm -1 is indicative of ether bonds (CH 2 -O-CH 2 ). Around 1720 cm -1 a peak of the carbonyl groups is due to the furan ring opening during the oligomerization, as discussed above [START_REF] Choura | Acid-Catalyzed Polycondensation of Furfuryl Alcohol: Mechanisms of Chromophore Formation and Cross-Linking[END_REF][START_REF] González | Furfuryl alcohol polymerisation by iodine in methylene chloride[END_REF].

According to the NMR data of the commercial oligomer, the ether bridges are more frequent than the methylene ones, and terminal CH 2 OH groups are much more frequent than unsubstituted H5 end groups, both facts being entirely uncommon for typical FA oligomers, in view of the established mechanism shown in Figure 1. Indeed, in the case of the predominance of reaction (2) without loss of formaldehyde, and hence the important presence of ether bridges, the condensation would be soon blocked for lack of CH 2 OH moieties, i.e. one would only be faced with oligomers ending with 2 unreacted protons at the C5 positions, which is clearly not the present situation.

Conversely, the strong presence of hydroxyl groups on the NMR and FTIR spectra versus a very limited sign of the proton at the C5 position, suggests that these oligomers are ending predominantly with hydroxymethyl groups rather than a furan ring with an unreacted H5 proton. It seems logical to suggest therefore that the synthesis of the commercial OFR required the concomitant use of FA and another furan monomer capable of generating a high frequency of ether bridges and terminal CH 2 -OH groups, e.g. 2,5-hydroxymethylfuran.

This structural investigation showed that the interest of the product developed by Transfurans in that it allows water compatibility while maintaining the potential to polymerize and crosslink following essentially the same mechanisms discussed above for FA alone, using a temperature activation catalyst which allows the process to occur upon heating the system, as detailed below.

Crosslinking thermal analysis

Different PFA/catalyst ratios were analyzed by DSC at a 5°C.min -1 heating rate. The PFA samples were prepared by diluting the commercial product with 15 wt% water and adding the catalyst at 0, 2, 5, and 15 wt%. On the DSC thermograms at different catalyst content, shown in Figure 6 and summarized in Table 3, the variation of the exothermic network-polymerization peak can be observed. The higher the catalyst concentration, the more pronounced the exothermic peak and the lower the exothermic peak temperature. In the same Figure, one can also see that the addition of the catalyst is essential for curing since, in the absence of catalyst, one cannot observe any exothermic peak, but only the endothermic peak of water evaporation at around 100°C. The increase in catalyst concentration up to 15 wt% induced a more exothermic (the maximum heat flow at the peak increases) and rapid reaction (lower onset temperature) which, in the case of PFA, can lead to the quick evaporation of residual water and foaming/high porosity. When adding 5 wt% of catalyst the curing heat has a similar value as when adding 15 wt% catalyst (292-309 J.g -1 ), thus suggesting that 5 wt% of catalyst is enough for a total crosslinking of the polymer in the tested conditions. The given numbers corresponding to different codes of samples detailed in Table 3.

On the other hand, the use of only 2 wt% of the catalyst seems to be insufficient for rapid curing because the reaction curing heat is around 262 J.g -1 , well below the enthalpy of 300 J.g -1 measured when adding 5-15 wt% of catalyst. This reduction is supposed to be due to the fact that this low amount of catalyst is not capable of reaching all the reactive sites in the resin network leading to an incomplete crosslinking reaction [START_REF] Montserrat | Influence of the accelerator concentration on the curing reaction of an epoxy-anhydride system[END_REF][START_REF] Ooi | DSC studies of the curing mechanisms and kinetics of DGEBA using imidazole curing agents[END_REF][START_REF] Domínguez | Study on the non-isothermal curing kinetics of a polyfurfuryl alcohol bioresin by DSC using different amounts of catalyst[END_REF].

It can be concluded that 5 wt% seems to be the best catalyst concentration to minimize the lower water content (the catalyst is water-based) while keeping a rapid curing rate with high conversion and a limited exothermic reaction peak.

Water absorption under controlled relative humidity

The capacity of water absorption and desorption of the neat cured PFA, cellulose powder, and the composite was assessed with the Varimass device. Mass gain and loss during the different relative humidity levels are gathered in Figure 7. Cellulose contains hydroxyl groups that can interact with water molecules which makes this biopolymer highly sensitive to humidity. As one can notice in Figure 7, after mass stabilization at 50% of relative humidity, the cellulose lost up to 1.5% of its weight upon desorption at low humidity levels. At a high humidity level (90%), cellulose gained weight rapidly to reach 5% in two hours. On the other hand, cured PFA is a matrix insensitive to water. Therefore, it is expected that the number of OH groups in FA, available for hydrogen bonding with water molecules, was reduced during PFA polymerization, resulting in a non/slightly-polar polymer. As shown in Figure 7, the weight of the specimen was stable with virtually no mass variation. The composite with up to 27 wt% of cellulose remained insensitive to water its mass variation does not exceed 0.1%, even at high humidity levels. The composite nevertheless is slightly more sensitive to humidity compared with that of the neat matrix. These curves suggest that the ensuing composite is not sensitive to humidity despite the high cellulose fraction, a fact that can be explained by the encapsulation of cellulose particles in the hydrophobic polymer matrix. These findings are in agreement with a previous works [START_REF] Mokhothu | Bio-based coatings for reducing water sorption in natural fibre reinforced composites[END_REF] , [START_REF] Dong | Wood/Polymer Nanocomposites Prepared by Impregnation with Furfuryl Alcohol and Nano-SiO2[END_REF][START_REF] Kumar | Biofilms from soy protein isolate and polyfurfuryl alcohol[END_REF]. Hence, the important cellulose content in the composites developed in the present work should not disrupt the outstanding performance of PFA polymers such as insensitivity to moisture and aging, thermal stability, and water resistance. 3.4 Effect of cellulose on porosity and shrinkage

Effect of cellulose on foaming and porosity

One of PFA major issues is the high water content in the resin and the water generated by the condensation reactions during curing. This issue is particularly noticeable in composite manufacturing because it leads to high porosity and low dimensional stability. Other parameters can trigger pores formation in the furanic resin, but moisture content and viscosity are the dominant factors [START_REF] Oishi | Viscosity, pH, and moisture effect in the porosity of poly(furfuryl alcohol)[END_REF].

The used commercial PFA contains 7.5 wt% of water and generates over 2 wt% during curing [START_REF] Domínguez | Development of new biomass-based furan/glass composites manufactured by the double-vacuum-bag technique[END_REF], added to over 1.75 wt% of water brought by the catalyst. Therefore, at least 11 wt% of the resin's weight should be eliminated during the curing phase, which promotes porosity when the gas release is blocked. This important water content can lead to foaming caused by water boiling during the curing process especially for samples with thick walls which blocks the evacuation of gases through the structure. The foaming is particularly noticeable when the curing kinetic is fast, leading to a low gel time which interferes with gas elimination and hence the bubbles can be trapped inside the rubbery polymer matrix. To further investigate the effect of cellulose on the porosity caused by the water evaporation during curing, the prepared formulations were cured in the oven, but at a higher temperature (around the exothermic peak of the curing reaction) to favor resin foaming and to assess the effect of cellulose on this phenomenon. Some samples fluffed up and the porosity could not be accurately measured. However, the effect of cellulose is clearly shown in Figure 8.

One can notice that when curing at 90°C, below the water boiling temperature, the composite with or without cellulose did not foam, but still displayed micro and macro-porosity. When curing at 130°C, above the water boiling point, and with a lower gel time, the resin foamed because of the internal pressure due to gases and blocked by the external rubbery resin layer, as is illustrated in Figure 9-(a). This foaming was mitigated by the addition of cellulose, particularly in the PFA-Cell25-B composite. This effect can be related to the percolation of highly hydrophilic cellulose fibers which helped steam diffusion through the fibrous network without causing any noticeable deformation in the composite showed in Figure 8 and explained in Figure 9-(b). As shown in Figure 10, and in line with the above explanations, the neat cured PFA displayed almost no internal pores, but one can distinctly see macro-pores with perfect circular shapes all over the surface (Figure 10-(b)),

showing the presence of air bubbles leaving the resin after water evaporation. These pores were more visible on the surface in contact with the mold, which obstructed bubbles degassing, as illustrated in Figure 9-( This porosity might be mainly due to manual molding and mixing and not to vapor bubbles which were probably, as discussed above, drained through the fibrous network. On the micrograph in Figure 10-(d) one can visualize the fibers network. However, pores formation by vapor generation can still be possible and we suppose that they partially coalesced with the pre-existing air bubbles in the composite.

Study of the porosity origin

The porosity content (Vp %) of different composites is displayed in Figure 11. The pure PFA resin cured at 90°C had 8.8% porosity which is in agreement with the porosity found in a previous work for the same commercial PFA resin using the double-vacuum-bag technique [START_REF] Domínguez | Development of new biomass-based furan/glass composites manufactured by the double-vacuum-bag technique[END_REF]. The porosity percentage doubled after the addition of 18 wt% of cellulose. Above 18 wt% of cellulose, the porosity decreased steadily when increasing the cellulose weight fraction. Owing to the particularly viscous and sticky nature of the PFA-Cell16-B slurry, the sudden increase in porosity when adding 18 wt% cellulose might be ascribed to air inclusion during the mixing and casting step and not only to water vapor generation during the thermal curing. In fact, this specific slurry was not as fluid as the pure PFA, which flowed easily in the molds. The stickiness of this particular slurry against the spatula further complicated the molding process leading to porosity embedding. For this reason, when the formulation started to be less sticky, casting was easier and voids inclusion was avoided. However, the porosity percentage of all the composites containing cellulose is higher than the neat PFA sample which is inconsistent with water content 30% lower than that of the bare resin. Therefore, the porosity volume (V p ) in the composites was mainly associated with the mixing and casting process (V p mix ). For a better understanding, the hypothesized internal composite structure is illustrated in Figure 9-(c) where red circles represent the porosity creating during the mixing and molding and those represented with blue circles present the porosity created later during curing. Accordingly, to estimate the porosity created during paste mixing and molding step (V p mix) , few assumptions should be made, namely:

 The porosity volume created in the neat resin in the curing step (V p cure ) is not affected by the cellulose addition (neglecting the draining effect of fibers).



Any porosity in the fibers is neglected.

In this case, the V p cure can be estimated based on the porosity of the neat PFA, which was found to be 8.9% using equation ( 3):

Where V m is the volume fraction without considering any eventual porosity in the composite calculated using equation ( 4).

Where is the added oligomers volume inside the formulation, and the added cellulose powder volume inside the formulation.

The porosity volume caused by the mixing can be therefore estimated as the difference between the total porosity and the porosity created in the resin phase during curing as:

(

The effective volume fractions in the composite considering the pores volume can therefore be recalculated and the results are given in Table 4.

Shrinkage of the biocomposite

PFA, like the majority of thermosetting polymers, shrinks during curing, which, according to the supplier data, can reach 7%. The dimensional stability is important for composite production and 3D printing and it is, therefore, essential to minimize it. As shown in Figure 12, the shrinkage decreased when increasing the volume fraction of cellulose. Shrinkage upon curing of the PFA went from 2.9% for neat PFA to 1.8% when adding at least 24 wt% of cellulose fibers. The shrinkage of the resin is due to the evaporation of water and other volatiles during the curing process and the densification of the network. However, the cured composites have up to 35% of their volume occupied by dimensionally stable components (cellulose and pores), which could explain the shrinkage decrease. A theoretical shrinkage percentage was estimated by considering the shrinkage to be strictly related to the matrix phase shrinkage (which is estimated to be 2.9 %) using equation ( 6) and the resulting values are plotted in Figure 12. Overall, cellulose addition has a major role in improving the composite dimensional stability during the curing process as shown by the fact that, under the tested conditions, cellulose decreased by 35% the shrinkage of the neat PFA down to 1.8% for the formulation PFA-Cell28-B (i.e. 3 times the usual shrinkage of ABS and PLA but less than the shrinkage of PP and PA which is over 2%). Cellulose addition can also prevent the foaming of the resin minimizing the curing porosity. It follows that, when avoiding air bubbles inclusion during the processing of the pasty formulation, the composite porosity can be decreased to less than 6%.

Thermal property

The thermal stability of the crosslinked PFA, cellulose and the composites was assessed by thermogravimetric analysis (TGA). The TGA curves given in Figure 13, show that in an oxidative atmosphere, cellulose loses 3% of its weight between 90 and 200°C, which is considered to be the residual moisture, followed by a classical abrupt degradation starting at 300°C, before a total degradation of the material when the temperature reaches 530°C. In the case of PFA, one notices a three-step degradation process. The first ramp starts at 200°C with only 10% weight loss, which could be explained by a further curing and consequent water evaporation [START_REF] Ma | Pultruded fibre-reinforced furfuryl alcohol resin composites: 1. Process feasibility study[END_REF]. The second step starts when the decomposition rate increases around 400°C to reach a 50% weight loss at 600°C, which is attributed to the decomposition of the furan polymer [START_REF] Guigo | New insights on the thermal degradation pathways of neat poly(furfuryl alcohol) and poly(furfuryl alcohol)/SiO2 hybrid materials[END_REF]. The final degradation step leads to the total burning of the organic polymer at 700°C. The degradation of the composites with 31 wt% cellulose content goes through the same 3 steps as for the neat PFA, with an increase in the cellulose decomposition temperature when the degradation rate increases, to reach a 25% weight loss at 400°C.

Under inert atmosphere (N 2 ), the degradation curves (Figure 13) show the same degradation steps, but the degradation rate decreases considerably starting from 500°C to reach a plateau at 800°C. After the stabilization of the weight loss, the char residue is about 50 wt%, which is in agreement with values given in the literature [START_REF] Gandini | Furans in polymer chemistry[END_REF]. This residue decreases when increasing the thermally unstable cellulose fraction that loses up to 90% of its initial weight.

The char residue of the composite with more than 30 wt% of cellulose is around 40 wt%, which therefore does not exclude the possibility of carbonization of the molded composites.

This thermomechanical stability is competitive with those of PEEK and some other technical polymers used in 3D

printing [START_REF] Diez-Pascual | The influence of a compatibilizer on the thermal and dynamic mechanical properties of PEEK/carbon nanotube composites[END_REF][START_REF] Cao | The surface modifications of multi-walled carbon nanotubes for multiwalled carbon nanotube/poly(ether ether ketone) composites[END_REF][START_REF] Qu | Hybrid poly(ether ether ketone) composites reinforced with a combination of carbon fibres and graphene nanoplatelets[END_REF][35][START_REF] Niu | Mechanical and thermal properties of PEEK composites by incorporating inorganic particles modified phosphates[END_REF]. Unlike the PFA/cellulose composite, these technical thermoplastics can be extremely expensive and also rather complicated to print, apart from the fact that they are fossil-based materials.

Mechanical performances

As can be seen in Figure 15, the composite stiffness improved after the addition of cellulose powder. The tensile modulus increased from 2527 to 2880 MPa and the flexural modulus from 3092 to 4309 MPa by adding 30 wt% of cellulose. This modest improvement suggests a good wetting by the resin of the cellulose particles [START_REF] Ku | A review on the tensile properties of natural fiber reinforced polymer composites[END_REF] and agrees well with the DMA results. On the other hand, the tensile and flexural strength decreased when adding cellulose with an opposite evolution of porosity. This can be explained by the fact that high porosity content in the composite is deteriorating the strength because pores act as stress concentrators and fracture nucleation sites [START_REF] Pfister | Green composites from a conjugated linseed oil-based resin and wheat straw[END_REF]. Therefore, when recalculating the specific modulus by normalizing the measured values to their density, one can notice that cellulose is improving the mechanical properties of the composite by increasing the flexural modulus by 40% and tensile modulus by 10%.

Therefore, voids generated during the paste preparation and manual casting are supposed to be at the origin of the limited enhancement of the mechanical performance of the tested samples.

Biocomposite's applications

Owing to its interesting thermal and mechanical properties, the developed new biocomposite might be used in technical applications such as in automobile, transport fields and for industrial uses in general. However, due to the particular nature of the pasty composite precursor, an adapted manufacturing process should be found. To this end, the use of the novel manufacturing process of additive manufacturing is possible thanks to the rheological properties of the developed slurry. Indeed, these rheological properties, open the possibility of using a special 3D printing technique called DIW (direct ink writing) where a pasty material is pushed through a nozzle and deposited layer per layer without heating followed by a solidification step by crosslinking the thermosetting polymer. As one can see, in Figure 16-(a) a gear made of the formulation PFA-Cell28-B was successfully printed and solidified. This case of application was described and studied in detail in a previous study [START_REF] Bouzidi | 3D printable fully biomass-based composite using poly(furfuryl alcohol) as binder and cellulose as a filler[END_REF]. The pasty precursor can also be molded using compression molding to produce mainly thin pieces to avoid the phenomenon of porosity formation mentioned earlier. The paste is placed in a heated mold and spread into the mold's form thanks to an applied pressure (between 2 and 4 MPa). The mold is then mantained heated to crosslink and solidify the composite to obtain, as we can see in Figure 16-(b) a solid biocomposite which is in this case a 4mm-thick sheet. This sheet can be machined (milling, drilling, shaping, CNC) to manufacture a more specific shape. To implement the use of these manufacturing processes, a demonstrator of a gear pump was manufactured using machined molded biocomposite plates, to shape the casing in which two 3D-printed gears are housed as demonstrated in Figure 16-(c).

CONCLUSION

In the present study, the production of fully bio-based composites by the air casting of a commercial oligomeric furan resin with a curing catalyst and cellulose powder slurries was readily achieved. The experimental results demonstrated that:

 The cellulose addition induced porosity during manual paste lay-out caused by its high viscosity and sticky nature. After adding 18 wt% cellulose the porosity increased by 125% with respect to the pure matrix, but gradually decreased when adding more cellulose until reaching a plateau value of 11%. Despite the increase in the overall porosity, the cellulose fibers limited the resin foaming during the thermal curing at high temperature.

This phenomenon was associated to composite degassing by vapor diffusion through the cellulose percolation network. The porosity content of these composites could be further minimized by applying a pressure to the curing process instead of the simple air curing process chosen in this study.

 The dimensional stability of the resulting composites increased by 35% when adding 30 wt% of cellulose with respect to the cured matrix.

 The thermo-mechanical stability was improved by adding the cellulose fillers. The storage modulus improved by 60% at room temperature and by 125% at 150 °C when adding 30 wt% of cellulose. Moreover, composites were thermally stable up to 200°C, unlike the cured neat matrix. In addition, according to the TGA curves, the composites, like the matrix, showed excellent thermal stability with 40-50% char residue. Nevertheless, the mass loss due to the cellulose degradation (300 °C) was more important for the composites compared to that of the matrix.

 Despite the high cellulose content, the composites were insensitive to humidity with less than 0.05% weight gain when placed in 90% RH for two hours.

 Finally, mechanical performances were enhanced when adding the cellulose filler. The specific tensile and flexural modulus were improved from 2.04 to 2.34 GPa/g.cm -3 and from 2.49 to 3.5 GPa/g.cm -3 , respectively.

Figure 1 :

 1 Figure 1: Stepwise polycondensation reaction of FA according to the two different pathways

Figure 2 :

 2 Figure 2: SEM photograph of the dry cellulose powder and its particles size distribution measured using a granulometer (Cilas particle size analyzer 1190) based on the Fraunhofer method.

  C NMR spectrum of the commercial OFR, shown in Figure4, the signals between 156 ppm and 149 ppm match the C3 and C4 carbons, and those of the C2 and C5 are found between 110 and 113 ppm, which reveals the presence of furan rings. Comparing the methylene bridge and the ether bridge signals, at respectively 27 ppm and 56 ppm, one can see again a stronger presence of the latter. Around 175 ppm, one can also detect a modest signal of carbonyl groups which is due to the furan ring-opening due to its minor acid hydrolysis during the oligomerization[START_REF] Chuang | Carbon-13 NMR study of curing in furfuryl alcohol resins[END_REF].

Figure 3 : 6 Figure 4 :

 364 Figure 3: 1 H NMR spectrum of the commercial OFR in DMSO-d 6

Figure 5 :

 5 Figure 5: FT-IR spectrum of the commercial OFR

Figure 6 :

 6 Figure 6: Curves of DSC scans of PFA containing different catalyst concentrations at a heating rate of 5°C.min -1 .

Figure 7 :

 7 Figure 7: Water absorption of composites, neat PFA, and cellulose under controlled relative humidity.

Figure 8 .

 8 Figure 8. Photographs of different composites aspect cured overnight at 90°C and 130°C.

Figure 9 :

 9 Figure 9: Illustration of the foaming and porosity formation phenomenon under different conditions. (a) Representation of the water evaporation and porosity formation in the mold at 90°C of the neat PFA resin (on the left) and the cellulose-based slurry (on the right). (b) Water evaporation at 130°C through neat PFA with foam formation under the evaporation forces and through cellulose-based slurry. (c) a representation of the different types of forms porosity in molded cured neat PFA and in the biocomposite. Vp cure presents the porosity volume caused by the degassing during the curing, Vp mix the porosity incorporated in the paste during the mixing, Vm is the volume occupied by the matrix, and Vf is that occupied by the fillers.

  a). On the other hand, for the PFA-Cell25-B composite, one can notice on Figure9-(c) numerous internal pores with different sizes and shapes, but mostly oval and recumbent along the casting direction. The surface of the specimen is smooth and presents no bubble marks.

Figure 10 :

 10 Figure 10: SEM images of several cured (90°C) PFA surface: (a) surface of the cured specimen of neat PFA, (b) fractured surface of neat PFA, (c) PFA-Cell25-B composite fractured surface, (d) zoom on the pores in PFA-Cell25-B composite.

Figure 11 :

 11 Figure 11: Porosity percentage in composites with different cellulose contents, cured at 90°C in the oven.

Figure 12 :

 12 Figure 12: Shrinkage percentage of composites with different cellulose contents cured at 90°C in the oven.

Figure 13 :

 13 Figure 13: TGA scans collected at 10°C.min -1 showing the degradation of cured neat PFA, cellulose powder, and biocomposites with different cellulose weight content under N2 (continuous curves) and under an oxygen atmosphere (dashed curves).

Figure 15 :

 15 Figure 15: Tensile and flexural modulus and strengths of neat cured PFA and different biocomposites.

Figure 16 :

 16 Figure 16: Application cases of the developed biocomposite using (a) DIW 3D printing process to manufacture (b) a gear and (c) compression molding to obtain a 4 mm thick composite sheet. The composite sheet was further (d) machined to shape a gear pump casing. (e) The two 3D-printed gears were fixed in the machined pump casing to obtain a static gear pump demonstrator.

Table 1 :

 1 A summary of research work on PFA-based composites and their mechanical and thermal properties.

	Fillers	Filler'

s concentration Manufacturin g methode Curing conditions Mechanical properties Thermal properties Ref

	Fillers	Filler's concentration	Manufacturin g methode	Curing conditions	Mechanical properties	Thermal properties	Ref
	Cellulose whiskers	0.75 wt%	Open-air casting	130°C for 75 min + 210°C for	Young's modulus= 1 GPa Rupture strength=	Char residue at 800°C 50 wt%	[17]
				105 min	32 MPa
					Young's modulus=
	Flax fiber fabric	N.A	Compression molding	140°C	25 GPa Tensile strength=	N.A	[18]
					211 MPa
						Char residue at
	Sisal whisker	2 wt%	Open-air molding	70°C, 100°C and 150°C	Storage modulus= 2.1 GPa at 25°C	800°C 50 wt% Tg= 70°C	[11]
	Kenaf fibers	20 wt%	Compression molding 7 MPa	60°C for 4 h + 80°C for 4 h + 110°C for 1 h	Young's modulus= 1.25 GPa 42 MPa Tensile strength=	Storage modulus= 800°C 45 wt% Char residue at 0.5 GPa at 150°C Tg= 95°C	[5]
	High tenacity rayon filaments	64 wt%	Compression molding 10 MPa	40°C for 12 h + 60°C for 12 h + 80°C for 6 h	Young's modulus= 5.5 GPa 120 MPa Tensile strength=	Tg=122°C Storage modulus= residue at 600°C 35 wt% Char 1.5 GPa at 160°C	[19]
					Compression
	MFC-L gel	50 wt%	Open-air casting	Freeze-drying	modulus= 1 MPa Compression	20 wt% char residue at 800°C	[12]
					strength= 0.21 MPa
	Cork powder	25 wt%	Compression molding 1 MPa	160°C	Young's modulus= 6.5 MPa Tensile strength= 180 MPa	0.8 GPa at 160°C Storage modulus=	[10]
					Modulus of elastic=
	Flax fibers mat Bamboo fibers	50 wt% 30 wt%	Compression molding 2.2 MPa Pressure impregnation	150°C 120°C for 5 h	Young's modulus= 12.2 GPa Rupture strength= 95 MPa rupture= 15 GPa 6 GPa Modulus of	Storage modulus= 1 GPa at 160°C Tg= 105°C 20 wt% char residue at 800°C	[15] [13]
					Young's modulus=
	flax fibers mat	60 wt%	Vacuum bag	150°C for 15 min	8.5 GPa Tensile strength=	N.A	[16]
					64 MPa
	Flax fibers non-woven	57 wt%	Compression molding 2 MPa	150°C	Young's modulus= 48 MPa 9 GPa Rupture strength=	N.A	[6]

Table 2 :

 2 Different tested composite formulation's compositions.

Slurry component by weight (wt%) Slurry component by volume (vol%) Composite by weight (wt%) Formulation PFA Solvent Cellulose PFA Solvent Cellulose PFA (Wm) Cellulose (Wf)

	Neat PFA	91.19	8.81	0	97.73	2.27	0	100	0
	PFA-Cell16-B	76.60	7.40	16.00	81.84	4.95	13.20	82.22	17.78
	PFA-Cell19-B	73.65	7.12	19.23	79.23	4.79	15.98	78.72	21.28
	PFA-Cell22-B	70.93	6.85	22.22	76.78	4.64	18.58	75.51	24.49
	PFA-Cell25-B	68.39	6.61	25.00	74.47	4.51	21.03	72.55	27.45
	PFA-Cell28-B	66.03	6.38	27.59	72.30	4.37	23.33	69.81	30.19
	PFA-Cell29-B	65.14	6.29	28.57	71.46	4.32	24.21	68.77	31.23

Table 3 :

 3 Summary of DSC scans data.

	Sample code	Catalyst content (wt%)	Onset Temperature (°C)	Peak temperature (°C)	Reaction enthalpy (J.g -1 )
	1	2	119	137	262
	2	5	109	124	309
	3	15	100	114	292

Table 4 :

 4 Summary of the weight and volume fractions of different components and their respective shrinkage rates.

			Without porosity		With porosity	Porosity origin	Shrinkage
	Reference	Wm (wt%)	Wf (wt%)	Vf (vol%)	Vm (vol%)	Vm eff (vol%)	Vf eff (vol%)	Vp (vol%)	Vp cure (vol%)	Vp mix (vol%)	Shr. exp	Shr. theo
	Neat PFA	100.0	0.0	0.0	100.0	91.2	0.0	8.8	8.8	0.0	2.9	2.9

Funding: This study was funded by the Region Auvergne Rhone Alpes (Grant: 3D-PapEl, 1701107001-22437-) and the LabCom IDEX Université Grenoble Alpes (Grant: 2018/2019 ThermoBioComp3D).

Thermomechanical behavior

The storage modulus E' of composites as a function of temperature is given in Figure 14. Cellulose addition increased the storage modulus of the composites, which is higher than that of the neat cured PFA over the whole temperature region. This could be ascribed to the tension transfer from the matrix to the high-modulus filler, which enhanced the composite stiffness.

At room temperature, the modulus E' increased when increasing the cellulose fraction, which was maximized for W f = 30 wt%. Cellulose addition also improved the composite thermomechanical stability by increasing the stiffness at high temperatures. For instance, with a 30 wt% weight fraction of cellulose, the modulus remained higher than 4 GPa up to 150°C and higher than 2 GPa up to 200°C, whereas, for the neat cured PFA, it dropped below 2 GPa when reaching 150°C. The thermal stability and the rigidity of both the cured neat PFA and the composites were higher than those reported in the literature when incorporating sisal whiskers and lignin in conventional PFA [START_REF] Guigo | Eco-friendly composite resins based on renewable biomass resources: Polyfurfuryl alcohol/lignin thermosets[END_REF][START_REF] Ahmad | Thermal and dynamic mechanical properties of bio-based poly(furfuryl alcohol)/sisal whiskers nanocomposites[END_REF].

In addition to the typical thermomechanical stability of cellulose fibers, cellulose microparticles are supposed to hinder polymer chains mobility thus further enhancing the composite thermomechanical behavior.

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. All authors contributed equally.

ACKNOWLEDGEMENT

The authors would like to acknowledge Transfurans Chemicals for generously providing the furan resin and the curing agent used in this work.

COMPLIANCE WITH ETHICAL STANDARDS

Conflict of Interest:

The authors declare that they have no conflict of interest.