Marc Boyer

Benoît Dupont De Dinechin

Amaury Graillat

Lionel Havet

Computing Routes and Delay Bounds for the Network-on-Chip of the Kalray MPPA2 Processor

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Introduction

Network-on-Chip [START_REF] Dally | Route packets, not wires: On-chip inteconnection networks[END_REF] (NoC) is the dominant paradigm for on-chip interconnects. While mostly applied to besteffort communication scenarios, NoC interconnects can also be engineered to provide service guarantees by using a Deterministic Network Calculus (DNC) approach [START_REF] Lu | Flow regulation for on-chip communication[END_REF]. In this paper, we present our approach to NoC management for Kalray MPPA2 processor [START_REF] Saidi | The shift to multicores in real-time and safety-critical systems[END_REF], based on DNC. Assuming that application tasks are assigned endpoints on the NoC, we proceed in two steps:

1) route the flows to ensure deadlock-free operations, while fairly allocating maximum rates to the flows; 2) using DNC, compute the maximum queue backlogs and upper-bounds on the end-to-end latencies.

In order to apply DNC to the MPPA2 NoC, we need to determine the network elements traversed by each flow, which implies selecting a route between its endpoints. Moreover, application of DNC require that flows be routed over a feed-forward network, that is, the graph of directed links traversed by the flows has no cycles (circuits) [START_REF] Boudec | Network Calculus: A Theory of Deterministic Queuing Systems for the Internet[END_REF].

The motivation for this work is an anticipation on how avionics certification could be applied to a MPPA manycore processor: each cluster would be considered as a shared memory multicore processor, with certification based on the EASA CRI-MCP [START_REF] Easa | The use of multi-core processors in safety-critical applications[END_REF] guidelines; inter-cluster communication would be carried by the NoC, where the certification principles of AFDX based on DNC analysis would apply.

Organization of the paper is as follows. Section 2 introduces the MPPA2-256 Bostan NoC architecture, discusses previous work, and summarizes basic results of Deterministic Network Calculus (DNC). In Section 3, we show that deadlock-free routing on a wormhole switching network and ensuring feed-forward flows are equivalent. In Section 4, we present a linear formulation based on DNC to compute end-to-end latencies for concurrent flows over the MPPA2 NoC. In Section 5, this linear formulation is compared to a classic formulation originally designed for the AFDX avionics networks and adapted to the MPPA2 NoC.

Background

The MPPA2-256 Bostan NoC

The MPPA2-256 processor [START_REF] Saidi | The shift to multicores in real-time and safety-critical systems[END_REF] integrates 256 processing cores and 32 management cores on a chip, all implementing the same VLIW core architecture. The MPPA2-256 architecture is clustered with 16 compute clusters and 2 I/O clusters, where each cluster is built around a multi-banked local static memory shared by 16+1 (compute cluster) or 4+4 (I/O cluster) processing + management cores. The clusters communicate through a RDMA NoC, with one node per compute cluster and 8 nodes per I/O cluster.

The MPPA2 NoC is a direct network based on a 2Dtorus topology extended with extra links connected to the otherwise unused ports of the NoC nodes on the I/O clusters (see Fig. 1). The MPPA2 NoC implements wormhole switching with source routing and without virtual channels. With wormhole switching, a packet is decomposed into flits (32-bits on the MPPA2 NoC), which travel in a pipelined fashion across the network elements, with buffering and flow control applied at the flit level. The packet follows a route determined by a bit string in the header.

The motivation for implementing wormhole switching with source routing and without virtual channels is the reduction of hardware dedicated to the network elements and interfaces. However, wormhole switching networks are sensitive to deadlocking. Fig. 2 B is using it. Likewise, flow B needs R 1 → R 4 held by flow A. Deadlock requires that router queues be full.

A MPPA2 NoC node is composed of a router (Fig. 3) and a cluster interface. The network elements of the MPPA2 NoC appear in Fig. 3: the links between nodes, the switches that steer packets depending on the routing information, the 'turns' inside nodes, and the link arbiters. Each outgoing link has one such arbiter, which selects whole packets by round-robin across the adjacent queues. There is one queue per incoming direction or 'turn'.

We call a queue active if there is some flow passing through it and there is another queue in the same link arbiter with some flow. A non-active queue either has no flow passing through it, or is the only one in the link arbiter with some flow. If so, there are no effects on the packets beyond a constant delay and the queue can be ignored except for the constant delay that needs to be added to the end-to-end latency bound.

Context and Previous Work

They have been several studies designed to compute upper bounds on the worst case traversal time (WCTT) of a NoC by a data flow. Nevertheless, they mainly consider either a pure wormhole solution, with hop-level flow-control, or systems with hardware dedicated to QoS management such as packet priorization and preemption. Hop-level flow control and packet preemption are not amendable to DNC analysis. In fact, we found only a few approaches based on DNC which are applicable to the MPPA2 NoC architecture. An overview of the state of the art of NoC performance evaluation can be found in [START_REF] Kiasari | Mathematical formalisms for performance evaluation of networks-on-chip[END_REF].

The recursive calculus has been designed to compute bounds on the SpaceWire technology, a wormhole-based technology [START_REF] Ferrandiz | Worst-case end-to-end delays evaluation for spacewire networks[END_REF]. It has been adapted to the MPPA NoC in [START_REF] Ayed | Towards a unified approach for worst-case analysis of tilera-like and kalraylike noc architectures[END_REF] and compared with a DNC-based approach on an example, that will also be considered in this article. In [START_REF] Qian | Analysis of worst-case delay bounds for best-effort communication in wormhole networks on chip[END_REF], authors consider a DNC model for a NoC with wormhole switching, input queuing, and Weighted Round Robin arbitration. However, the MPPA2 NoC only has output queuing.

Both [START_REF] Ayed | Towards a unified approach for worst-case analysis of tilera-like and kalraylike noc architectures[END_REF], [START_REF] Qian | Analysis of worst-case delay bounds for best-effort communication in wormhole networks on chip[END_REF] take into account the back-pressure mechanism associated to the wormhole switching. Back-pressure occurs when the buffer in a switch is full. In classical store and forward systems, when a buffer is full, either incoming or local packets are dropped. In wormhole switching, when a buffer is full, the upstream link stops. Then, its own buffer may fill, up to also being full. This "back-pressure" mechanism may goes back up to the source, and may also block others flows sharing the blocked links. In pathological cases, this may lead to a global network stall, cf. Fig 2 .  Since the back-pressure mechanism has a unbounded impact on the NoC performances, it must not be activated in a real-time application environments. And since this mechanism is activated only when local buffers are full, one can configure the traffic shapers in a way such that NoC buffers are never fully used. From the network calculus point of view, when the back-pressure mechanism is not active, the MPPA2 NoC is simply a network using a Round-Robin arbitration and cut-through forwarding.

A network-calculus model of the Round-Robin policy has been presented in [START_REF] Georges | Network calculus: application to switched real-time networking[END_REF], [START_REF] Bouillard | Worst-Case Analysis of Tandem Queueing Systems Using Network Calculus[END_REF]. It is compared in [START_REF] Long | Analysis and evaluation of perflow delay bound for multiplexing models[END_REF] with the "arbitrary multiplexing" model in the case of a NoC.

A computation of the MPPA NoC delays based on network calculus has been presented in [START_REF] Dupont De Dinechin | Time-critical computing on a single-chip massively parallel processor[END_REF]. The approach has been refined for the MPPA2 NoC in [START_REF] Dupont De Dinechin | Network-on-chip service guarantees on the kalray mppa-256 bostan processor[END_REF], whose results are summarized in Section 4. These authors study the problem of deadlock-free feed-forward routing on the MPPA2 NoC in [START_REF]Feed-forward routing for the wormhole switching networkon-chip of the kalray mppa2 processor[END_REF], whose main results are reported in Section 3.

Note that solving both routing and resource assignment withing the network calculus framework have also been studied in [START_REF] Frangioni | Optimal joint path computation and rate allocation for real-time traffic[END_REF], [START_REF]Delay-constrained routing problems: Accurate scheduling models and admission control[END_REF], [START_REF]Qos routing with worst-case delay constraints: Models, algorithms and performance analysis[END_REF] in a different context: it considers that the flow throughput is an input of the problem and that per flow resource reservation can be done.

Deterministic Network Calculus (DNC)

Deterministic Network Calculus [START_REF] Boudec | Network Calculus: A Theory of Deterministic Queuing Systems for the Internet[END_REF] is a framework for performance analysis of networks based on the representation of flows as cumulative data over time. Given a network element with cumulative arrival and departure functions A and A , the delay for traversing the network element and the backlog of data correspond respectively to the horizontal and the vertical deviations between A and A (Fig. 4).

Network Calculus exploits the properties of the (min, +) algebra, in particular it introduces the following operations:

convolution: (f ⊗ g)(t) inf 0≤s≤t f (t) + g(t -s) deconvolution: (f g)(t) sup s≥0 f (t + s) -g(s)
Let A be a cumulative data function. An arrival curve α is a constraint on A defined by ∀0 ≤ s ≤ t : A(t) -A(s) ≤ α(t -s), which is equivalent to A ≤ A ⊗ α. Fig. 5 illustrates the smoothing constraint of the arrival curve α on the cumulative function A. This particular type of arrival curve α(t) = (ρt + σ)1 t>0 , known as affine or leaky-bucket, is denoted γ ρ,σ with ρ the rate and σ the burstiness.

Let A, A be the cumulative arrival and departure functions of a network element. This element has β as a service curve iff: ∀t ≥ 0 : A (t) ≥ inf 0≤s≤t (A(s)+β(t-s)), which is equivalent to A ≥ A ⊗ β. Fig. 6 illustrates the guarantee offered by the service curve β. This particular type of service curve β(t) = R[t -T ] + , known as rate-latency, is denoted β R,T with R the rate and T the latency.

Key results of Deterministic Network Calculus include:

•

The arrival curve of the aggregate of two flows is the sum of their arrival curves.

• A flow A with arrival curve α that traverses a server with service curve β results in a flow A constrained by the arrival curve α β.

•

The service curve of a tandem of two of servers with service curves β 1 and

β 2 is β 1 ⊗ β 2 .
• If a flow has arrival curve α(t) on a node of service curve β, the backlog bound is max t≥0 (α(t)β(t)) and the delay bound is hDev(α, β) = max t≥0 {inf s ≥ 0 : α(t) ≤ β(t + s)}. These bounds are respectively the maximum vertical deviation and maximum horizontal deviation between α and β.

Max-Min Fair Feed-Forward Routing

The routing problem on the MPPA2 NoC has three objectives. First, routes must be chosen to avoid deadlock, which is always an issue on wormhole switching networks. Second, the resulting routes must compose a feed-forward network, to enable application of the main DNC results. Third, the selection of routes for each flow must optimize use of the global network capacity, while guaranteeing a fair allocation of bandwidth to the flows.

In this section, we include the main contributions of [START_REF]Feed-forward routing for the wormhole switching networkon-chip of the kalray mppa2 processor[END_REF], with the experimental results extended to include the maximum and average of upper bounds on end-to-end delays computed according to the linear formulation of Section 4.

The Feed-Forward Routing Problem

On a wormhole switching network, the objectives of deterministic deadlock-free routing and feed-forward flows are connected:

• In a connection network, deadlock results from circuits of agents and resources connected by a wait-for relation [START_REF] Dally | Deadlock-Free Message Routing in Multiprocessor Interconnection Networks[END_REF]. With wormhole switching, agents are packets, and resources are the router internal turns and the external links between routers.

• Deadlocks can be avoided by eliminating directed cycles in the resource dependence graph. This is accomplished by imposing a partial order on the resources and then insisting that an agent allocates resources in ascending order [START_REF] Dally | Principles and Practices of Interconnection Networks[END_REF].

• A definition of a feed-forward network is that it is possible to find a numbering of its links such that for any flow through the network, the numbering of its traversed links is an increasing sequence [START_REF] Schmitt | The DISCO network calculator: a toolbox for worst case analysis[END_REF].

As the links considered in a feed-forward network form a subset of the resources considered for deadlock, the numbering of these resources so that they are allocated in ascending order by a packet is also a numbering of the links which are traversed in ascending order by the flows. Therefore, application of a deterministic deadlock-free routing algorithm ensures that the flows are feed-forward.

Deadlock-free routing algorithm for wormhole switching have been proposed, mostly for 2D-mesh topologies. The classic ones are X-Y or Dimension-Ordered [START_REF] Dally | Principles and Practices of Interconnection Networks[END_REF] and the Turn Model [START_REF] Glass | The turn model for adaptive routing[END_REF]. The X-Y routing has no path diversity, while the Turn Model variants have irregular diversity. This problem led to the formulation of the Odd-Even [START_REF] Chiu | The odd-even turn model for adaptive routing[END_REF] turn model, which has been improved as the Hamiltonian Odd-Even turn model [START_REF] Bahrebar | The Hamiltonian-based Odd-even Turn Model for Maximally Adaptive Routing in 2D Mesh Networkson-chip[END_REF] to accommodate multicasting.

Independently from the computer architecture community, the communication network community has proposed a series of techniques to ensure that flows are feed-forward. In particular, the Turn Prohibition algorithm [START_REF] Starobinski | Application of network calculus to general topologies using turn-prohibition[END_REF] finds a set of prohibited turns that ensure the feed-forward routing property on an arbitrary network topology of bi-directional links and turns. A turn is a triple of nodes (a, b, c) connected by two links and the algorithm requires that if turn (a, b, c) is prohibited, so must be turn (c, b, a). Turn Prohibition has been improved to yield Simple Cycle Breaking [START_REF] Levitin | Minimal sets of turns for breaking cycles in graphs modeling networks[END_REF], which still requires disabling both directions of a turn at once. In order to ease implementation of routing under Turn Prohibition, [START_REF] Fidler | Routing in turn-prohibition based feedforward networks[END_REF] introduces the Turnnet, which is a graph whose nodes correspond to the external links, and arcs to the internal turns of the network. Application of any cycle-breaking technique to ensure feed-forward flows on the network amounts to removing arcs from the Turnnet so it becomes acyclic. It is then used to compute routes, for instance by applying Dijkstra's shortest path algorithm [START_REF] Fidler | Routing in turn-prohibition based feedforward networks[END_REF].

Observe that in case of wormhole switching, the Turnnet can be expanded into the resource dependence graph by splitting each arc and inserting a node corresponding to the turn queue. As this expansion cannot introduce a (directed) cycle, an acyclic Turnnet implies an acyclic resource dependence graph for wormhole switching. So ensuring feed-forward flows on such networks implies deadlock-free routing when using the corresponding paths.

Routing with Max-Min Fairness

Among the possibly multiple paths proposed by a routing algorithm between each pair of endpoints of a flow, only one can be selected. This constraint of routing a flow under a unique path between its endpoints comes from the MPPA2 NoC architecture, as both packet ordering inside a flow and flow regulation of (rate, burstiness) at ingress can only be ensured along a single path. Global allocation of bandwidth to flows on a network is a fairness problem, and we use max-min fairness [START_REF] Chen | Maxmin Fair Routing in Connection-Oriented Networks[END_REF] whose objective is to maximize the lowest flow rate, then the next lowest flow rate, etc. Precisely, a rate allocation is max-min fair iff an increase of any rate must be at the cost of a decrease of some already smaller rate. Max-min fair allocation is solved by the simple "Water Filling" algorithm [START_REF] Amaldi | On single-path network routing subject to max-min fair flow allocation[END_REF] in case there is a single path available per flow. In case of multiple paths available per flow and splittable flows, the Max-Min Fair with Splittable Paths (MMFSP) problem is still of polynomial time complexity and can be solved as a series of linear programs [START_REF] Nace | A linear programming based approach for computing optimal fair splittable routing[END_REF]. When only a single path among those available can be assigned to the flow, the resulting Max-Min Fairness with Unsplittable Paths (MMFUP) problem is NP-hard [START_REF] Amaldi | On single-path network routing subject to max-min fair flow allocation[END_REF].

Our first approach to routing is to solve the MMFUP problem instances by enumeration. For a given feed-forward routing algorithm, the search tree is deployed by assigning a unique path to each flow, selected among the minimal path diversity obtained by enumerating all the allowed shortest paths between each flow endpoints. On the leaves of this search tree, the problem is solved by applying the Water Filling algorithm of [START_REF] Chen | Maxmin Fair Routing in Connection-Oriented Networks[END_REF]. This gives a vector of flow rates, including the minimum rate. Water Filling is skipped if the problem instance has a link shared by n flows and its capacity divided by n is lower than a previously computed minimum rate. Finally, a solution whose sorted vector of flow rates is lexicographically maximal is selected as in [START_REF] Chen | Maxmin Fair Routing in Connection-Oriented Networks[END_REF].

Our second approach to routing is a heuristic that reduces the search space by not enumerating the paths that carry a split flow of low rate. Each internal node of the search tree corresponds to a MMFSP problem instance, which is solved by a series of linear programs. At a search tree node, iterate over three steps: select a split flow of lowest rate; remove all its path(s) not carrying the maximum amount of sub-flow; solve the resulting MMFSP problem instance. Following this iteration, either all flows follow a unique path, or there exists a flow of minimal rate equally split among its alternate paths. Recursively apply the heuristic on the new problem instances where each of the alternate paths of this flow has become its only allowed path.

Experimental Comparisons

We compare the routing approaches on two representatives of each family of algorithms: X-Y (XY) and Hamiltonian Odd-Even [START_REF] Bahrebar | The Hamiltonian-based Odd-even Turn Model for Maximally Adaptive Routing in 2D Mesh Networkson-chip[END_REF] (HOE) for deadlock-free routing; Turn Prohibition [START_REF] Starobinski | Application of network calculus to general topologies using turn-prohibition[END_REF] (TP) and Simple Cycle Breaking [START_REF] Levitin | Minimal sets of turns for breaking cycles in graphs modeling networks[END_REF] (SCB) for feed-forward flows. The routing problems include the example in [START_REF] Dupont De Dinechin | Network-on-chip service guarantees on the kalray mppa-256 bostan processor[END_REF], an AFDX example from [START_REF] Ayed | Towards a unified approach for worst-case analysis of tilera-like and kalraylike noc architectures[END_REF], and also the classic Bit-Complement, Bit-Reverse, Shuffle and Tornado instances. Only the MPPA2 compute clusters, re-numbered as a 4 × 4 grid, are considered as endpoints (as illustrated in Fig. 8 for Bit-Complement). Except for XY, all routing algorithms allow alternate minimal paths, whose diversity is displayed in Fig. 7. Overall, HOE and SCB provide the highest minimal path diversity.

The results of max-min fair allocation of flow rates appear in Table 3, where we compare the enumeration approach based on Water Filling to the heuristic approach based on MMFSP resolutions. Columns #L display the number of leaves in the respective search trees, however in case of enumeration the exploration is limited to 50K steps in order to limit computation times. The main criterion for comparison is the minimum of flow rates ρ min , although the average of flow rates ρ avg gives an indication on how the global NoC bandwidth is exploited. The maximum and We observe in Table 3 that the heuristic drastically reduces the number of leaves (#L) explored, while providing good solutions. In case of BC-SCB, the heuristic outperforms the enumeration because the latter suffers from the 50K steps limitation. The deadlock-free routing techniques (XY, HOE), although restricted to the 2D-mesh topology subset, appear to perform better than the feedforward techniques (TP, SCB). Surprisingly, XY routing that has no minimal path diversity outperforms HOE on Bit-Complement and Tornado. The explanation is that HOE does not allow some XY path, as illustrated in Fig. 9.

Linear Formulation for the MPPA2 NoC

The MPPA2 NoC is a RDMA-capable network that can be configured for regulating the injection rate ρ i and burstiness σ i at ingress of each flow f i , that is, setting the parameters of a leaky-bucket arrival curve. In this section, we summarize the linear formulation of the DNC equations for the MPPA2 NoC that has been developed in [START_REF] Dupont De Dinechin | Network-on-chip service guarantees on the kalray mppa-256 bostan processor[END_REF], [START_REF]Feed-forward routing for the wormhole switching networkon-chip of the kalray mppa2 processor[END_REF].

The motivation of this formulation is to provide upper bounds on queue backlogs and flow end-to-end delays, while accounting for the shaping implied by the peak rate r of one flit per cycle in the network elements. Replacing the flow rates by the values computed when solving the Feed-Forward Routing problem (Section 3) yields a system of linear inequalities with the flow burstiness as variables. Thanks to the feed-forward flows, the system of linear inequalities is acyclic and solved in one pass.

Effects of Link Shaping

In this linear formulation, each flow f i is associated with a series of arrival curves γ ρi,σ j i , where ρ i is its constant rate and σ j i its burstiness in front of queue q j . Queue q j receives the aggregates of flows F j passing through it, so its arrival curve is of leaky-bucket type γ ρ j ,σ j with ρ j = fi∈F j ρ i and σ j = fi∈F j σ j i , shaped by the turn peak rate r. This yields the arrival curve min(rt, σ j + ρ j t)1 t>0 , which is a special case of the standard T-SPEC arrival curve α(t) = min(M + pt, rt + b)1 t>0 used in IntServ [START_REF] Firoiu | Theories and models for internet quality of service[END_REF].

Assume that a link arbiter offers a rate-latency service curve β R j ,T j to queue q j with R j ≤ r. The delay d j for queue q j is the maximum horizontal deviation between the arrival curve and the service curve. Application of the T-SPEC arrival curve on such service curve yields [START_REF] Boudec | Network Calculus: A Theory of Deterministic Queuing Systems for the Internet[END_REF]:

d j = T j + σ j (r -R j ) R j (r -ρ j ) (1)
Let l max i be the maximum packet size for flow f i . At ingress, whole packets are atomically injected at rate r. Call θ the date when injection ends. We have rθ = l max i and l max i ≤ σ i + ρ i θ, so:

∀f i ∈ F : σ i ≥ σ min i l max i r -ρ i r (2)
We now express the values ρ j i and σ j i for all flows f i ∈ F j for an active queue q j . If q j is the first active queue traversed by the flow, then σ j i = σ i . Else, let q k be predecessor of q j in the sequence of active queues traversed by flow f i , with β R k ,T k its service curve. When flow f i traverses queue q k , its burstiness increases differently whether it is alone or aggregated with other flows in q k . If the flow is alone in queue q k , we apply the classic result of the effects of a rate-latency service curve β R,T on a flow constrained by an affine arrival curve γ ρ,σ . The result is another affine arrival curve γ ρ,σ+ρT [START_REF] Boudec | Network Calculus: A Theory of Deterministic Queuing Systems for the Internet[END_REF], so:

σ j i = σ k i + ρ i T k (3)
Else, we apply Theorem 6.2.2 (Burstiness Increase Due to FIFO Multiplexing, General Case) [START_REF] Boudec | Network Calculus: A Theory of Deterministic Queuing Systems for the Internet[END_REF], where flow 1 is token bucket constrained with rate ρ 1 and burstiness σ 1 , and flow 2 is constrained by a sub-additive arrival curve α 2 . Because of link shaping in q k , α 2 (t) = min(rt, ρ 2 t + σ 2 )1 t>0 . As a result, b 1 = σ 1 + ρ 1 (T + σ2(r+ρ1-R) R(r-ρ2) ) [START_REF] Dupont De Dinechin | Network-on-chip service guarantees on the kalray mppa-256 bostan processor[END_REF].

With ρ 1 = ρ i , σ 1 = σ k i , b 1 = σ j i , ρ 2 = l∈F k ,l =i ρ l
, and σ 2 = l∈F k ,l =i σ k l , this yields:

σ j i = σ k i + ρ i T k + ( l∈F k ,l =i σ k l )(r + ρ i -R k ) R k (r -k ,l =i ρ l ) (4)
Not accounting for the link shaping would yield b 1 = σ 1 + ρ 1 (T + σ2 R ), by application of Corollary 6.2.3 (Burstiness Increase due to FIFO) [START_REF] Boudec | Network Calculus: A Theory of Deterministic Queuing Systems for the Internet[END_REF]. This result is worse than Eq. ( 4) because

ρ 1 + ρ 2 < R ⇒ r + ρ 1 -R < r -ρ 2 .

Link Arbiter Service Curves

On the MPPA2 NoC, the output link arbiters operate in round-robin on turn queues at the packet granularity, while each queue contains flows aggregated in FIFO. As the packets presented to a link arbiter are not processed in FIFO order, previous work (e.g. [START_REF] Bouillard | Worst-Case Analysis of Tandem Queueing Systems Using Network Calculus[END_REF]) would have to assume blind multiplexing between all flows and fail to exploit FIFO aggregation. This is addressed in [START_REF] Dupont De Dinechin | Network-on-chip service guarantees on the kalray mppa-256 bostan processor[END_REF] by exposing the service offered to each queue of a link arbiter: either, the rate and latency ensured by round-robin packet scheduling; or, the residual service guaranteed by blind multiplexing across queues when the round-robin service does not apply. Then, aggregation need only be considered withing the scope of single queues so is FIFO.

The service curve offered by a link arbiter to each of its queues is abstracted as a rate-latency function β j = β R j ,T j . The first approach to derive this curve is to consider the behavior of the round-robin arbiter, assuming that each flow f i has its packet sizes bounded by a minimum l min i and a maximum l max i . Let l min F j min fi∈F j l min i and l max F j max fi∈F j l max i be respectively the minimum and maximum packet sizes for q j . Let A j be the set active queues in the link arbiter of q j , and B j A j -{q j }. The general roundrobin service curve β j = β R j ,T j for q j is:

R j = rl min F j l min F j + k∈B j l max F k and T j = k∈B j l max F k r (5)
The second approach to derive a service curve for queue q j is to consider that the round-robin arbiter serves packets at peak rate r according to a blind multiplexing strategy across the queues. Application of Theorem 6.2.1 (Blind Multiplexing) [START_REF] Boudec | Network Calculus: A Theory of Deterministic Queuing Systems for the Internet[END_REF] yields the blind multiplexing service curve β j = β R j ,T j for q j : R j = r -

k∈B j ρ k and T j = k∈B j σ k r -k∈B j ρ k (6)
The blind multiplexing service curve must be used whenever the sum of flow rates inside q j exceeds R j in Eq. ( 5). Else, we select the formula that evaluates to the lowest T j .

End-to-End Latency Bound

For computing an upper bound on the end-to-end latency of any particular flow f i , we proceed in three steps. First, compute the left-over (or residual) service curve β j i of each active queue q j traversed by f i . Second, find the equivalent service curve β * i offered by the NoC to flow f i through the convolution of the left-over service curves β j i . Last, find the end-to-end latency bound by computing d * i the delay between α i the arrival curve of flow f i and β * i . Adding d * i to the constant delays of flow f i such as the traversal of non-active queues and other logic and wiring pipeline yields the upper bound. This approach is similar in principle to the Separated Flow Analysis (SFA) [START_REF] Bouillard | Worst-Case Analysis of Tandem Queueing Systems Using Network Calculus[END_REF], even though the latter is formulated in the setting of aggregation under blind multiplexing, while we use FIFO multiplexing.

For the first step, we have two cases to consider at each active queue q j . Either f i is the only flow traversing q j , and β j i = β R j ,T j from equations ( 5) or [START_REF] Kiasari | Mathematical formalisms for performance evaluation of networks-on-chip[END_REF]. Or, f i is aggregated in q j with other flows in F j . Packets from the flow aggregate F j are served in FIFO order, so we may apply Corollary 6.2.3 (Burstiness Increase due to FIFO) [START_REF] Boudec | Network Calculus: A Theory of Deterministic Queuing Systems for the Internet[END_REF]. This yields the left-over service curve

β j i = β R j i ,T j i
for an active queue q j traversed by f i :

R j i = R j - l∈F j ,l =i ρ l and T j i = T j + l∈F j ,l =i σ j l R j (7)
For the second step, we compute the convolution β * i of the left-over service curves β j i . Let Q i denote the set of active queues traversed by flow f i . Thanks to the properties of rate-latency curves [START_REF] Boudec | Network Calculus: A Theory of Deterministic Queuing Systems for the Internet[END_REF], β * i is a rate-latency curve whose rate R * i is the minimum of the rates and the latency T * i is the sum of the latencies of the left-over service curves β j i :

R * i = min j∈Qi R j i and T * i = j∈Qi T j i (8)
For the last step, we compute the delay d * i between α i the arrival curve of flow f i at ingress and β * i . This flow is injected at rate ρ i and burstiness σ i , however it is subjected to link shaping at rate r as it enters the network. As a result, α i = min(rt, σ i + ρ i t)1 t>0 and we may apply Eq. ( 1):

d * i = T * i + σ i (r -R * i ) R * i (r -ρ i ) (9)

Comparing DNC Formulations

In this section, we compare the linear DNC formulation of Section 4 to a "local" DNC formulation, which is an adaptation to the MPPA2 NoC of an algorithms designed to analyze AFDX networks [START_REF] Frances | Using network calculus to optimize AFDX network[END_REF], [START_REF] Boyer | An efficient and simple class of functions to model arrival curve of packetised flows[END_REF], [START_REF] Boyer | Experimental assessment of timing verification techniques for AFDX[END_REF]. In these comparisons, we assume that the flow paths are pre-determined.

Local Formulation

An AFDX switch applies a 2-levels non-preemptive static priority scheduling, meaning that in each output port, there are two queues. All flows from the same priority level are set in the same queue. The queue with the low priority is selected only if the queue with the high priority is empty. All flows sharing a queue are served with a FIFO policy.

The network calculus analysis consists in: 1) computing a residual service, β q for each queue q; 2) computing the global arrival curve in this queue α q by summing the arrival curves of all individual flows, and considering the shaping; and 3) computing d q = hDev(α q , β q ) a bound on the delay of the queue, which is also a bound on delay for all flows in the queue due to FIFO aggregation.

The delay bound of a flow f is then computed as the sum of the queue delay bound for all crossed servers. This approach is quite different from the linear formulation presented in Section 4, in several points.

First, it relies on a class more generic than linear curves, which is more accurate in general. For example, consider a periodic (resp. sporadic) flow sending one packet of maximal size L every P time unit (resp. not quicker than every P time unit). Two arrival curves can be computed for this kind of flow: a linear one, α l (t) = L + L P t, or an exact one, α e (t) = tL P , where • denotes the ceiling function (cf. Fig. 10). The AFDX tool, presented in [START_REF] Boyer | Experimental assessment of timing verification techniques for AFDX[END_REF] can handle the Ultimately Pseudo Periodic (UPP) class of functions, that generalizes both kinds of curves.

Second, it does no perform the end-to-end SFA analysis but computes per queue local delay bound.

Third, it does not apply Corollary 6.2.3 (Burstiness Increase due to FIFO) [START_REF] Boudec | Network Calculus: A Theory of Deterministic Queuing Systems for the Internet[END_REF] since it can be applied only in the case of linear arrival curves. [START_REF] Dupont De Dinechin | Network-on-chip service guarantees on the kalray mppa-256 bostan processor[END_REF] Last, AFDX switches use static priorities to share the bandwidth between queues connected to an output port. For the comparisons, the AFDX tool has been adapted to consider round-robin output arbitration policy, using Eq. ( 5), like the linear formulation.

First Example

The first example comes from [START_REF] Dupont De Dinechin | Network-on-chip service guarantees on the kalray mppa-256 bostan processor[END_REF]. Four flows are considered, f 1 , . . . , f 4 , with a maximum packet size of 17 flits. All flows have a long-term rate 1 3 but f 1 that have r 1 = 2 3 . The admissible bursts at network ingress are 34 3 but f 1 that have b 1 = 17 3 . The upper bounds on delays for this example, computed with the linear and the classic formulation, are reported in Table 1. The results are comparable, but the linear formulation gives better (smaller) bounds. This is due to several differences between an AFDX network and the MPPA2 NoC, and also to the fact that the linear formulation fits very well the MPPA2 NoC architecture.

The flow contract is the first difference: in AFDX, a flow (called Virtual Link) is characterized as a sporadic flow, with a maximal frame size and an inter-frame minimal time. As shown in Fig. 10, a linear model cannot capture exactly this behavior, it only does an upper approximation. In the MPPA2 NoC, traffic shaping is done at ingress by a tokenbucket regulator, which enforces a linear constraint. Whereas it had been shown in [START_REF] Boyer | Experimental assessment of timing verification techniques for AFDX[END_REF] than for typical AFDX networks, the linear formulation is 18% pessimistic with respect to a more accurate one, it is sufficient for the MPPA2 NoC.

A second difference is related to the queuing architecture and its impact on bursts. In an AFDX switch, the flows from two different input ports can be merged into the same output port. In a MPPA2 NoC router, for each output port, there is a queue dedicated to each input port (cf. Fig. 3). In an AFDX switch, each queue can receive traffic from several input links, whereas in the MPPA2 NoC, it received only from one input link. This changes the kind of curve required to accurately model a switch. As shown in Fig. 12, the sum of two piece-wise linear functions, each one having two slopes, leads to a piece-wise linear function with three slopes. This implies that a linear model with two slopes is sufficient to capture the MPPA2 NoC behavior, whereas a more complex one is required for AFDX.

f 1 f 2 f 3 f 4 Linear
A third difference stems is the use of Corollary 6.2.3 [START_REF] Boudec | Network Calculus: A Theory of Deterministic Queuing Systems for the Internet[END_REF] by the linear formulation for the left-over service in Eq. [START_REF] Ferrandiz | Worst-case end-to-end delays evaluation for spacewire networks[END_REF]. Considering a set of flows (f i ) i∈ [1,n] sharing a FIFO queue, each flow having a leaky-bucket arrival curve γ ri,bi , the increase of the burst of the flow f i does not depend on its own burst size, but only on the sum of the others ones. In AFDX networks, since a queue is in general shared by a lot of flows, removing its own burst size has a limited positive impact, whereas approximating a sporadic flow as a leakybucket one has a large negative impact. This explains why the the local formulation originally designed for AFDX does not implement this result. These differences explain the numerical results observed in this example. Consider the flow f 2 : its crosses three routers. In the first router, R 2 , it shares the output link with the flow f 1 , with a Round-Robin policy, each flow being alone in its own queue. The situation is the same in the second crossed router, R 10 , with the flow f 3 . In the last router, R 8 , the flows f 2 and f 3 share a common queue, with a FIFO aggregation policy, and both share the output link with the flow f 4 with a Round Robin policy.

In the linear formulation, the Round-Robin link arbitration is used to compute the residual service in routers R 2 and R 10 , using Eq. ( 5), leading in both cases to a latency T 2 = T 10 = 17, since the flow have to wait at most one packet of the competing flow before getting access to the output link, and a rate R 2 = R 10 = 1 2 , since all flows having the same packet sizes, the round-robin arbiter shares the throughput in two equal parts In the last router, R 8 , the flows f 2 and f 3 share a common queue. First, one has to select the residual service offered by the link arbiter. Since the total load of both f 2 and f 3 is 2 3 , it overtakes the service offered by the Round-Robin arbiter, so the Blind Multiplexing result of Eq. ( 6) is used. It leads to a latency T 8 = 17, the latency associated to one packet, and a rate R 8 = 2 3 since the other flow uses only 1 3 of the throughput. But this service is shared by f 2 and f 3 . The left-over service dedicated to flow f 2 is computed using Eq. ( 7), leading to R The interpretation is that, first, the residual throughput is 1 3 , since f 2 shares the output link with two flows, each of throughput 1 3 , second, the latency is decomposed into two terms, the 17 is the frame size of the flow f 4 , in the other queue, and the 17 2/3 is the latency due to the sharing of the queue with flow f 4 , which has a frame size of 17, and this shared queue has a residual service of 2 3 . Once the per router residual services have been computed, one can compute the end-to-end delay, using Pay Burst Only Once principle (PBOO), with Eq.( 8). The latency term is T * 2 = T 2 2 + T 10 2 + T 8 2 = 17 + 17 + 42.5 = 76, 5. The minimal rate is the one observed in the last router, R 8 2 = 1 3 . The application of Eq. ( 9) leads to 110.

In the local formulation, a per queue delay is computed. In the first and the second routers, the Round-Robin residual service is, like for the linear formulation, a service of rate 1 2 and latency 17. Eq. ( 9) can be applied to compute the local delay, leading to a delay of 34: 17 cycles due to the Round-Robin arbitration latency, and 17 cycles related to the service of a packet. In the last router, one can also apply the Blind Multiplexing result of Eq. ( 6), leading also to a latency of 17 cycles and a throughput of 2 3 for the queue that is shared between f 2 and f 3 . The delay associated to this queue is 102 cycles: 17 related to the latency and 85 related to the service of the burst at rate 2 3 . We can now compare the approaches. Both linear and local formulations consider a latency of 17 cycles per router, since at each router, one packet of the other queue can be served before having access to the output link. But the local formulation also has a latency related to the handling of the burst in each queue, whereas the linear formulation pays burst only once by using the SFA principles. Moreover, the use of Corollary 6.2.3 [START_REF] Boudec | Network Calculus: A Theory of Deterministic Queuing Systems for the Internet[END_REF] for the left-over service in Eq. ( 7) allows for a better handling of the FIFO aggregation.

Second Example

The second example comes from [START_REF] Ayed | Towards a unified approach for worst-case analysis of tilera-like and kalraylike noc architectures[END_REF]. It considers 6 flows, f 1 , . . . , f 6 , each one being strictly periodic with a packet size of 50 flits an a period of 1000 cycles. This yields a rate of 5% on each flow. Unlike with the previous example, the network capacity is not fully exploited.

In [START_REF] Ayed | Towards a unified approach for worst-case analysis of tilera-like and kalraylike noc architectures[END_REF], the authors compare on this example the recursive calculus (RC) and a network calculus model given in [START_REF] Dupont De Dinechin | Time-critical computing on a single-chip massively parallel processor[END_REF], presented in here as "Naive" since it was a first DNC model for the NoC of the first-generation MPPA processor. The values of the NoC delay bounds are reported in Table 2 augmented with the results of the local and linear formulations. Note that the results presented here are not exactly the same as in [START_REF] Ayed | Towards a unified approach for worst-case analysis of tilera-like and kalraylike noc architectures[END_REF]. There, the delay includes the arbitration in the node, whereas the methods presented here only consider the NoC delays. Then, the node arbitration delays have been removed for flows f 1 , f 2 and f 3 .

A first observation is that the linear and the local formulations both outperform recursive calculus and an early Figure 13. Case study from [START_REF] Ayed | Towards a unified approach for worst-case analysis of tilera-like and kalraylike noc architectures[END_REF] DNC formulation for the MPPA [START_REF] Dupont De Dinechin | Time-critical computing on a single-chip massively parallel processor[END_REF]. Whereas in the previous experiment, the linear formulation always gave smaller bounds, in this experiment, the local formulation is tighter for half or the flows (f 1 , f 2 , f 6 ).

f 1 f 2 f 3 f 4 f 5 f 6 NC,
Consider the flow f 1 . Its queuing delays in the first and last routers crossed (R 1 and R 3 ) are null, since the output throughput is always less than the input throughput. This is handled by both the local and the linear formulation. The difference come from the handling of the second router. In this router, f 1 shares with f 2 a queue, (of index q) with a FIFO policy, and both share the output link with f 5 with a Round-Robin policy. Both models the residual service offered to the shared queue using Blind Multiplexing, leading to a rate-latency service with latency T q = 50 (one frame from competing flow f 5 ) and a rate R q = 0.95 (since the rate of f 5 is 0.05). Then comes a difference.

The linear formulation computes a residual service for the flow f 1 , using Eq. ( 7), leading to R q 1 = 0.95 and T q 1 = 100 = 50 + 50, since the flow f 2 may have to wait one frame from the flow f 5 (of size 50) and one frame from the flow f 2 (also of size 50). The local formulation considers that f 1 is part of the aggregate flow f 1 + f 2 . This flow has a burst size of 100 flits (two frames), a long rate ρ 1,2 = 0.1 and is shaped by the incoming link at a rate r = 1. It is served at rate R q = 0.95, after latency T q = 50, leading to a local delay of 56, by local application of Eq. ( 9).

The interpretation of this result is that the local formulation better captures the "serialization" between f 1 and f 2 coming from the same input link than the linear formulation.

Conclusions

This paper addresses three related problems of the MPPA2 NoC exploitation. The first is the routing between given endpoints, the second the NoC resource allocation and the third is the end-to-end latency bound computation. For routing, we show that both deterministic deadlockfree wormhole switching routing algorithms from the computer architecture community and feed-forward routing techniques from the communication networks community can be used. On a wormhole switching network, either ensures deadlock-freedom and feed-forward flows, so that deterministic network calculus (DNC) can be applied.

For network resource allocation, we rely on the max-min fairness objective in order to select a single path for each flow among the minimal path diversity allowed by routing. As exhaustive enumeration of minimal path combinations becomes infeasible, we design an LP-based heuristic to reduce the number of combinations to explore. We observe that XY and HOE deterministic deadlock-free routing algorithms perform best in term of higher minimal rates. XY, in spite of having no path diversity, sometimes outperforms HOE, as some XY paths are not included in HOE.

The evaluation of the NoC latency has been done using two DNC-based methods. The first one, called linear formulation, is designed specifically for the MPPA2 NoC architecture. The second one, called local formulation, is adapted from a classic AFDX analysis. Both are compared on two cases and we observe that, depending on the configuration, one or the other gives the best (i.e. smaller) bounds. Some insight is given on the parameters that influence the performance of each method: number of flows sharing a FIFO queue, path length, etc. Nevertheless, both methods achieve comparable results.

From a research point of view, these different results provide the seeds of a cross-fertilization. This comparison has allowed us to highlight the weak points of each method, and to identify some progress perspective. From a production perspective, the lightweight computations involved by XY routing and solving the linear formulations open opportunities for on-line reconfiguration of the MPPA2 NoC.

Figure 1 .Figure 2 .

 12 Figure 1. MPPA2 NoC topology unfolded (I/O nodes are labeled N0..N3, E0..E3, S0..S3, W0..W3).

Figure 3 .

 3 Figure 3. Structure of a MPPA2 NoC router.

Figure 4 .

 4 Figure 4. Flow arrival and departure as cumulative data functions over time.

Figure 5 .

 5 Figure 5. Arrival curve α for cumulative function A.

Figure 6 .

 6 Figure 6. Service curve β for a server A → A .

Figure 7 .Figure 8 .

 78 Figure 7. Minimal path diversity in path count per flow for XY, TP, HOE and SCB (horizontal line is average diversity).

Figure 9 .

 9 Figure 9. HOE path diversity between nodes 3 and 12 on the 4×4 2D-mesh subset of the MPPA2 NoC topology. average end-to-end latencies d * max and d * avg are computed using the linear formulation of Section 4 for 17-flit packets.We observe in Table3that the heuristic drastically reduces the number of leaves (#L) explored, while providing good solutions. In case of BC-SCB, the heuristic outperforms the enumeration because the latter suffers

Figure 10 .Figure 11 .

 1011 Figure 10. Two arrival curves for a periodic/sporadic flow of period P and packet length L

Figure 12 .

 12 Figure 12. Sum of two-sloped functions

TABLE 1 .

 1 BOUNDS ON NOC DELAY PER FLOW IN CYCLES.

	formulation	25	110 100	34
	Local formulation	25	170 136	34

TABLE 2 .

 2 BOUNDS ON NOC DELAY PER FLOW, IN CYCLES.

	Naive form. [13]	4104 4104 5155 3153 5255 3153
	Recursive Calculus [8]	158	158	107	103	156	103
	NC, Local form.	57	57	54	62	153	58
	NC, Linear form.	105	105	52	52	150	100

TABLE 3 .

 3 COMPARISON OF MAX-MIN FAIRNESS ROUTING METHODS. UNDER-PERFORMING SOLUTIONS IN A ROW APPEAR IN BOLD.

			Enumeration	Heuristic
		ρ min ρavg d * max	d * avg	#L ρ min ρavg #L
	E1-XY	0.333 0.500	68 51.3	1 0.333 0.500	1
	E1-HOE 0.333 0.500	94 55.5	9 0.333 0.500	4
	E1-SCB 0.333 0.417	111 68.0	1 0.333 0.417	1
	E1-TP	0.333 0.417	111 68.0	64 0.333 0.417	8
	E2-XY	0.333 0.500	128 56.2	1 0.333 0.500	1
	E2-HOE 0.333 0.500	77 46.0	192 0.333 0.383 13
	E2-SCB 0.333 0.500	85 51.1	1029 0.333 0.433 22
	E2-TP	0.250 0.425	229 89.3	8 0.250 0.425	5
	BC-XY	0.500 0.500	51 51.0	1 0.500 0.500	1
	BC-HOE 0.333 0.427	145 77.6 3.69E06 0.333 0.417 20
	BC-SCB 0.143 0.143	681 282.3 9.41E13 0.200 0.292 11
	BC-TP	0.125 0.125	479 315.7	256 0.125 0.125	9
	BR-XY	0.333 0.563	94 42.8	1 0.333 0.562	1
	BR-HOE 0.500 0.750	34 17.5	256 0.500 0.656	9
	BR-SCB 0.500 0.813	51 19.8	900 0.333 0.729 12
	BR-TP	0.250 0.500	255 105.4	64 0.250 0.500	8
	S-XY	0.500 0.750	34 17.5	1 0.500 0.750	1
	S-HOE	0.500 0.875	34	9.3	36 0.500 0.844	7
	S-SCB	0.500 0.844	51 12.4	200 0.500 0.750 11
	S-TP	0.250 0.500	272 110.5	64 0.250 0.469	7
	T-XY	0.500 0.500	51 51.0	1 0.500 0.500	1
	T-HOE	0.333 0.438	128 76.8 1.19E07 0.250 0.391	4
	T-SCB	0.500 0.594	68 35.3 5.54E06 0.250 0.437 15
	T-TP	0.125 0.125	479 315.7	256 0.125 0.125 10

Acknowledgment

This work was supported by the French DGE and Bpifrance through the "Investissements d'Avenir" program CAPACITES.


Raphaël Cusset

Farida Azzouz

Jacques Besson

Henry Proudhon

Marta Dragon-Louiset

Vincent Jacques

Effect of residual stress and strain on the fatigue crack propagation in Al2024T351 structures

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Manufacturing processes commonly require to bend, stretch and deform material to give its final shape to machined components. Loading and manipulations in-use can change their geometry. All of these events can induce both residual stresses and significantly high plastic strains in localized regions of the component. At the same time, fatigue strength is in many cases, a crucial factor in certification especially for structural elements. Previous studies showed that residual stresses are likely to influence fatigue fracture by load transfer [START_REF] Jin | Mallick Effect of cold work on the tensile and fatigue performance of aluminum alloy 5754[END_REF][START_REF] Kermanidis | Prediction of crack growth following a single overload in aluminum alloy with sheet and plate microstructure[END_REF]. It was observed that pre-strain in a steel structure would reduce fatigue crack growth rate [START_REF] Donato | Effects of plastic prestrain on the fatigue crack growth behavior of ASTM A36 steel[END_REF]. In this work, the case of a structure made of an aluminum alloy with a bend has been chosen. In particular, the effects of residual stresses and strain fields induced by the bending process, on the fatigue crack initiation and propagation are studied. In order to understand interactions between both phenomena, a combination of experimental testing and finite element simulations are used.

Approach

First, an anisotropic elasto-plastic Hill model has been chosen to describe the mechanical behavior of the material. The yield criterion, expressed by f(σ) = σ -R(p), has been chosen to describe both the material anisotropy and non-linear plasticity σ is the equivalent stress [START_REF] Hill | A theory of the yielding and plastic flow of anisotropic metals[END_REF] and R(p) an exponential isotropic hardening function, with p the cumulative plastic strain. A large set of mechanical tests was conducted and used to identify the material parameters Al2024 T351 thick sheets. The model reproduces very well the material anisotropy with respect to the (L,T) directions as well as the properties heterogeneity in the plate thickness direction. A finite elements analysis of the structure geometry and the identified model has been performed to establish the residual stress/strain fields resulting from a bending process (cf.. Figure 1). The conclusion about study structure bending simulation is that the structure presents high residual stresses and strains due to bending, as shown by the simulation. According to the initial position of the sample in the aluminum plate and to the bending load, these stress and strain fields develop in the rolling direction. Compressive plastic strain shows a strong gradient with a maximum as high as 25% localized at the top of the structure. Residual stress

fluctuate from compression to tension but remain in tension where compressive plastic strain is at its highest level.

Next, the fatigue properties of the plate and of the homogeneously pre-deformed material have been characterized. Olygocyclic specimen for crack initiation and CT specimen for crack growth were used in both cases to assess the effect of the pre-strain on the fatigue strength of Al2024T351. As final test, the studied structure will be cyclically loaded and a crack growth will be expected where residual strain/stress are concentrated. All of these experimental data are used to identify a fatigue crack initiation criterion and a LEFM Paris-like propagation law which takes into account crack closure. The final goal of this work is to quantify contribution of residual stress/strain on fatigue crack behavior. A crack growth simulation within the study structure under cyclic loading will be conducted to compare the results with a scale 1 experiment.


J-M Garcia

T Morgeneyer
email: thilo.morgeneyer@mines-paristech.fr

Mechanical characterization of Titanium Alloys Linear Friction Welds

Keywords:

Similar Ti17-Ti17 LFW joints were studied. Microhardness tests were conducted to track the weld zones areas and locations. Digital Image Correlation (DIC) provided the strain field of the specimen surface. Tensile test and DIC results allowed us to identify constitutive law of the joint by fitting an isotropic hardening behavior to the weld zones. Constitutive law coefficients were mapped using the weld zones tracking. A simulation of the tensile test was performed using Z-Set. Tensile test simulations were in good accordance with the experimental results.

Introduction

Linear friction welding is increasingly used in the aerospace industry. The assembly is achieved by fixing a part while the other is rubbed against with an oscillatory motion. Produced heat in the contact surface induces a material softening. A better understanding of the joints properties is required for the safe use in structural applications.

Aim of the Study

Simulation of a tensile test of a Ti17 similar linear friction joint.

Experimental and materials

Timet produced billets of Ti17 from which, blocks of 80mm×70mm×15mm have been machined using a Electrical Discharge Machine (EDM). Blocks have been used to produce LFW joints. Amplitude was 2mm and oscillation frequency was 50Hz. Forging pressure was 90MPa and the axial shortening was 3mm. Mean welding time was 2.9s. Fig. 2 show specimens geometry, which have been produced from the welded blocks at Centre des Matériaux.

Tensile test Tensile test were carried out using a MTS 100kN standard machine. An extensometer was used to measure elongation. Test conditions are exhibited in Fig. 2. Microhardness Microhardness tests have been carried out in a perpendicular line to the weld with a 100µm step size, with a 500g load and a dwell period of 15s.

Digital Image Correlation Specimens were painted using matte white, an airbrush was used to set black ink speckle pattern. Two cameras of 4MP were used for the stereographic imaging. Objects-lens distance was 22cm. Lens-lens distance was 10cm. The angle was 26 • 18 . For correlation values chosen for subset (SS), step size (ST) and filter size (SW) are respectivly 29, 15 and 15.

Results

Tensile test Joints tensile tests are analyzed and compared with the parent materials. Tensile tests done at the Centre des Matériaux appear in blue. Manufacturer tensile tests are plotted in gray. Tensile test curve has been compared with the parent material (PM) curve which has been provided by the manufacturer [START_REF] Millet | Tensile test reports of the titanium alloys: Ti17, ti6242 and ta6v[END_REF]], results are shown in Fig. 3. Localization took place in the surroundings of weld interface and the T-MAZ. Failure started in one of these two zones and then displaced to the other. Further analysis with DIC will be done to get a better understanding of failure localization. Linear friction weldings seems to represent a structural weakness in similar T i17 joints.

Microhardness Result of the microharness level are shown in Fig. 3. The welded zone seems to present the minimal microhardness level. The thermo-mechanically affected zones are highlighted by gradients of microharness. Far away from the welded zone parent material microhardness is found. DIC A discretization of the strain field on the specimen surface was used to analyze weld zones constitutive behavior. A straight line perpendicular to the weld interface was used as reference. A set of points over this line separated of 90µm, allowed us to track every weld zone strain field. Figure 4 shows the strain field of the reference line for increasing macroscopic engineering stress. Material softening is once again highlighted by the increase of ductility of the weld zone. Constitutive behavior identification and Simulation Using the macroscopic force of the tensile test and the strain field of the discretization, an isotropic hardening behavior was fitted to every point in the reference line. A tensile test simulation was launched using a map of isotropic hardening parameters using the weld line as reference. Results are shown in Figure 6. January 24-27 2017

Conclusions

Mechanical characteristics of similar an dissimilar linear friction welding joints depend on the welding parameters such as frequency, amplitude and forging pressure. In this work, tensile test has been performed for a similar Ti17-Ti17 linear friction welding joint. The following observations were made:

(1) Ti17-Ti17 similar joint is weaker than the parent material: its ultimate tensile strength is reduced by 10% and its ductility by about a half.

(2) A 13% microharness decrease is provoked by Linear friction welding, in Ti17-Ti17 similar joints.

(3) Ductility of the weld zone is higher than the parent material.

Figure 1 :

 1 Figure 1: LFW phases and parameters. Micrograph of a weld zone in a Ti6246 LFW sample: (a) a typical LFW microstructure, (b and d) sudden onset of severe plastic deformation on both sides of the weld, (c) recrystallized grains in the weld line.) [Sorina-Müller et al. (2010)]

  Figure 2: Tensile test specimen.

Figure 3 :

 3 Figure3: Tensile test curves comparison of the T i17 parent material and the T i17 -T i17 similar welding. Parent material information has been provided by[START_REF] Millet | Tensile test reports of the titanium alloys: Ti17, ti6242 and ta6v[END_REF]]. Microhardness line perpendicular to the weld line.

FatigueFigure 4 :

 4 Figure 4: Strain field of a line in the specimen surface obtained by Digital Image Correlation (DIC).

Figure 5 :

 5 Figure 5: Comparison of Microhardness profile and Strain Field along a line perpendicular to the weld.

Figure 6 :

 6 Figure 6: Similar Ti17-Ti17 LFW joint tensile test simulation.