François Tessier
email: ftessier@anl.gov

Preeti Malakar
email: pmalakar@anl.gov

Venkatram Vishwanath

Emmanuel Jeannot
email: emmanuel.jeannot@inria.fr

Topology-Aware Data Aggregation for Intensive I/O on Large-Scale Supercomputers

Reading and writing data efficiently from storage systems is critical for high performance data-centric applications. These I/O systems are being increasingly characterized by complex topologies and deeper memory hierarchies. Effective parallel I/O solutions are needed to scale applications on current and future supercomputers. Data aggregation is an efficient approach consisting of electing some processes in charge of aggregating data from a set of neighbors and writing the aggregated data into storage. Thus, the bandwidth use can be optimized while the contention is reduced. In this work, we take into account the network topology for mapping aggregators and we propose an optimized buffering system in order to reduce the aggregation cost. We validate our approach using micro-benchmarks and the I/O kernel of a large-scale cosmology simulation. We show improvements up to 15× faster for I/O operations compared to a standard implementation of MPI I/O.

Introduction

Optimizing data movement is critical for improved performance in high performance computing (HPC). We are witnessing the computational capability of HPC systems growing rapidly and exascale is now within reach. These systems are enabling large-scale simulations with higher fidelity and resolutions, among others, to model more complex phenomena. These simulations are generating and accessing increasing amounts of data. Often, it is more costly to access, move or allocate data than to actually process data. During the data lifetime, efficient access to the storage Input/Output system1 is becoming increasingly critical. The I/O requirements can be extremely important (as depicted by simulations estimates given in Table 1), however, the current I/O middleware and system face several challenges with respect to scalability, contention, latency and diverse application patterns. Also, given the limited scaling of I/O bandwidth in comparison to that of the computational capability of HPC systems and the current expectation that this will be even dire on future HPC systems, scalable I/O mechanisms that fully exploit the platform characteristics will be critical.

Thus, to improve the overall efficiency of a high-end parallel system, novel solutions to cope with efficient and optimized data access to the I/O system are needed. The current large-scale computing infrastructures are often characterized by network interconnects with complex topologies (e.g., multidimensional tori, dragonfly). Additionally, these systems are architected to have a separation of computation and I/O networks to avoid I/O interference and for functional decoupling. In these systems, I/O accesses require data movements along several hops of various networks. Hence, optimizing the data movement requires not only staging the data within these networks, but also to adapt I/O access pattern of the applications to the characteristics of the filesystems and the system topology.

To reduce latency of access and contention to the I/O system while improving its scalability, a common strategy (called two-phase I/O) is to aggregate data to a set of compute nodes (called aggregators) and have only the aggregators communicate with the I/O system. This approach poses several challenges such as: where to map aggregators among the various compute nodes, or, how to optimize communications to and from these aggregators? In this work, we explore the two-phase I/O (and specifically the write access) by carefully placing aggregators taking into account the application's communication needs (I/O access patterns), the topology of the underlying interconnect, and effective pipelining of communications to the aggregators and to the storage. The goal is to balance the aggregation phase cost with the I/O phase cost so as to minimize the overall time the application spends in writing the data to the storage system.

The main contribution of this paper is a novel approach to optimizing I/O data aggregation on large scale HPC infrastructures. First, we present a novel aggregator placement optimization framework and this framework is used to evaluate various approaches for data movement, including our topology-aware method. Next, we discuss a holistic end-to-end approach for I/O that goes beyond aggregator placement to also include pipelined aggregation buffering, file system awareness, and efficient inter-node communication (one-sided) for both the aggregation and I/O. Finally, we evaluate our approaches at scale on supercomputers and demonstrate that our approach significantly outperforms state-of-the-art techniques and represents a promising approach for scalable I/O on HPC systems.

Context and Motivation

In this section we start by describing the I/O subsystems of current and expected large-scale supercomputers. Next, we discuss the two-phase I/O algorithm used in MPI. Finally, we highlight the limitations of the current two-phase approach.

Storage systems on large-scale supercomputers

The current high-performance system architectures have undergone several improvements in order to tackle the I/O challenges. The networks topologies, despite being more complex, tend to reduce the distance between the data and the storage. While the compute nodes and the I/O infrastructures are commonly partitioned to avoid I/O interference, the interconnect networks bring these two entities closer. This partitioning is a characteristic of the IBM BG/Q supercomputers where the I/O nodes are dedicated to the I/O tasks and separated from the 5D-torus topology [START_REF] Chen | The ibm blue gene/q interconnection network and message unit[END_REF]. Similarly, Cray has also chosen a similar strategy for its supercomputers wherein the system has a subset of nodes called LNET nodes to manage I/O. To optimize the I/O bandwidth, a dragonfly network has been implemented reducing the number of hops from a compute node to the LNET node.

However, the amount of data produced by the applications remains extremely high and this sole architectural solution is not sufficient. Writing data out for future analysis suffers an I/O bandwidth limitation whereas storing data in memory for in situ analysis is bounded by the amount of available memory. Similar to solutions to overcome the memory bottleneck by adding more levels of hierarchy with faster but smaller memory close to the computing units, an approach to improve I/O performance is to create new tiers of storage between the main memory of the compute nodes and the storage system. Some supercomputers made the choice of allocatable DRAM by embedding Intel Knights Landing processors. Others chose to add NVRAM (on-node SSD for instance) to have a trade-off between cost, bandwidth and capacity. Burst buffer nodes as the ones used on the Cray Cori infrastructure [START_REF] Bhimji | Accelerating science with the NERSC burst buffer early user program[END_REF] are also a method to achieve high I/O performance. In this case, nodes similar to I/O nodes contain SSDs for data staging.

MPI Two-phase I/O

The MPI-2 standard [START_REF]Message Passing Interface Forum. Mpi-2: Extensions to the message-passing interface[END_REF] introduced the notion of two-phase I/O [START_REF] Miguel Del Rosario | Improved parallel i/o via a two-phase run-time access strategy[END_REF] for collective I/O operations. The goal of this improvement is to optimize the I/O performance by reducing the network contention, increasing the I/O bandwidth and simplifying the data access pattern. In two-phase I/O, a subset of processes called aggregators is responsible for the I/O phase. These aggregators are elected during the MPI collective I/O calls. Each aggregator manages a chunk of contiguous data in file from a subset of processes. For read access, the aggregators load a part of the file and distribute smaller chunks of data to a subset of processes. For write access, an aggregator gathers data from a subset of processes in a contiguous way and writes the aggregated data to the file system (through an I/O node if necessary). A toy example of the two-phase I/O mechanism is shown in Figure 1. In this example, four processes need to write non-contiguous pieces of data to a shared file. During the aggregation phase, two processes elected as aggregators gather these pieces of data into contiguous chunks in memory (X then Y then Z). Once this phase is finished, the aggregators effectively write the data in file (I/O phase).

X

Y Z X Y Z X Y Z X Y Z Processes Data Aggregators X X X X Y Y Y Y Z Z Z Z File X X X X Y Y Y Y Z Z Z Z I/O Phase Aggr. phase 3 2 1 0 0 2

Limitations

The two-phase I/O technique has several limitations. First, this two-phase approach works well for large messages but performs poorly with processes writing small data pieces (e.g., less than 1 MB). Secondly, some MPI I/O implementations of two-phase I/O take into account the topology of the machine to partition and elect aggregators (e.g. ROMIO considers the BG/Q topology to partition the number of elected aggregators). However, they do not use the topology information to propose an efficient aggregator placement policy. Moreover, beyond the characteristics of the underlying topology, the a priori details of the application's I/O patterns also could help to compute the placement, however, this is not currently considered. This is critical for emerging application I/O patterns in multi-physics applications, analysis outputs, etc. Therefore, in this paper we address the problem of mapping aggregators taking into account both the topology and the I/O access pattern of the application while optimizing performance for short messages.

Related Work

Parallel I/O is an active research topic due to the increasing requirements of applications for data movement to memory or storage. From a filesystem perspective, GPFS [START_REF] Schmuck | Gpfs: A shared-disk file system for large computing clusters[END_REF] or Lustre [START_REF]Lustre filesystem website[END_REF] are examples of widespread highly scalable parallel file systems. At a library or application level, parallel I/O libraries such as MPI I/O, part of the MPI-2 [START_REF]Message Passing Interface Forum. Mpi-2: Extensions to the message-passing interface[END_REF] standard, on top of parallel filesystems is commonly deployed. In these, collective I/O allows to achieve improved performance. For this, Chaarawi et al. [START_REF] Chaarawi | Performance Evaluation of Collective Write Algorithms in MPI I/O[END_REF] evaluate various collective I/O write algorithms. Other approaches to optimizing collective I/O have also been undertaken using techniques such as process placement based on the I/O pattern [START_REF] Venkatesan | Optimized process placement for collective i/o operations[END_REF] or collective I/O autotuning with machine learning [START_REF] Isaila | Collective i/o tuning using analytical and machine-learning models[END_REF]. One of the de facto collective I/O algorithm is called two-phase I/O [START_REF] Miguel Del Rosario | Improved parallel i/o via a two-phase run-time access strategy[END_REF]. This method adds a level of hierarchy in collective I/O phases by aggregating data on a subset of processes before writing it onto the storage system (more details are given in section 2.2). ROMIO [START_REF] Thakur | A case for using MPIs derived datatypes to improve I/O performance[END_REF] is a popular implementation of MPI I/O using two-phase I/O included in the widely-used MPICH library [START_REF] Gropp | Mpich2: A new start for mpi implementations[END_REF]. There have been a number of approaches to improve this library and the two-phase I/O algorithm [START_REF] Thakur | Data sieving and collective i/o in romio[END_REF][START_REF] Thakur | On implementing mpi-io portably and with high performance[END_REF][START_REF] Thakur | Optimizing noncontiguous accesses in mpi i/o[END_REF]. Approaches based on multi-threading to overlap aggregation and I/O phases using double buffering have been studied in [START_REF] Tsujita | Improving collective i/o performance using pipelined two-phase i/o[END_REF][START_REF] Tsujita | Multithreaded two-phase i/o: Improving collective mpi-io performance on a lustre file system[END_REF]. The number of aggregators or the buffer size needed in collective I/O remains still an open topic [START_REF] Chaarawi | Automatically selecting the number of aggregators for collective I/O operations[END_REF]. Finally, the placement of aggregators is a well-known problem. Certain approaches focus on data locality and a polynomial time assignment algorithm (the Hungarian algorithm) to reduce the communication between compute nodes and aggregators [START_REF] Filgueira | Data locality aware strategy for two-phase collective I/O[END_REF]. Others concentrated their efforts on the specific problem of sparse data patterns on BG/Q by offering an algorithm to take paths on the network topology into account [START_REF] Bui | Improving data movement performance for sparse data patterns on the blue gene/q supercomputer[END_REF]. A more general method designed to increase the I/O bandwidth of collective I/O for the previous version of IBM supercomputers BG/P has been proposed in [START_REF] Vishwanath | Topology-aware data movement and staging for i/o acceleration on blue gene/p supercomputing systems[END_REF]. Our approach differs from the above solutions by attempting to combine both an optimized buffering system and a topology-aware quantitative aggregators mapping strategy targeting any kind of architecture and being extensible to address new tiers of storage. It does so while also accounting for the application's I/O pattern.

Our Approach

Our approach consists of optimizing the data aggregation by taking into account critical parameters including the topology of the underlying architecture, the filesystem block size, and the double buffering with pipelined data movement achieved via one-sided communications. Thus, we developed a new data movement optimization library implemented in C++. This provides two simple functions to the user of parallel I/O while hiding all complexities of the underlying system. In fact, from the application developer point of view, using this aggregation mechanism comes down to describing the upcoming I/O operations (data sizes and offset in file) through an initialization function and commit the data instead of directly calling MPI File write in the application. This section first describes the parameters we identified to optimize the two-phase I/O method, and, next, discusses implementation details. Finally, we describe the challenges we address for scalable performance.

Aggregator Placement

The various implementations of the MPI-2 standard propose a couple of algorithms for two-phase I/O and particularly for aggregators mapping. In MPICH a strategy consists in choosing, for n aggregators, the first one on the bridge node (the compute node connected to the I/O node) and select the n -1 remaining ones on different nodes based only on their rank. For instance, let's consider 4 aggregators. The first one will be assigned on the bridge node. For the next three, the ranks of all processes are sorted and the aggregators are picked in this order such that no two aggregators are on the same node. Depending on the process mapping strategy, this placement can easily choose aggregators on neighboring nodes and thus create contention, or these could be selected far from the storage system leading to a large additional latency cost. Our strategy consists in electing the aggregators whose number is based on a fixed ratio (i.e. n aggregators per k nodes) in order to find a compromise between the cost of aggregating data and the cost of sending data to the storage system (we can extend this to other types of memory: burst buffers, NVRAM, ...).

To achieve this, we create a level of partitioning where a partition, which is subset of processes, contains processes that write a contiguous piece of data. Inside each partition, a process is elected as aggregator according to the topology and the amount of data to manage. Thus, the number of partitions corresponds to the number of aggregators. It has to be noted that a process can be involved in more than one partition. In this case we consider only the amount of data related to a given partition.

Figure 2 shows a simple example of partitioning and aggregator election on a grid according to our approach. In this figure, two more strategies are depicted. Indeed, to evaluate our topologyaware placement, we implemented three other methods. These simpler strategies can be described as follow:

• Shortest path: a rank hosted on the node with the smallest distance to the storage system is elected as an aggregator;

• Longest path: same as the previous strategy except that the longest distance to the storage system is considered.

• Greedy: the lowest rank in partition is the aggregator is selected; As explained before, our topology-aware strategy aims at taking into account the network topology and the amount of data exchanged between ranks and their aggregator. More precisely, we defined an objective function to minimize the time to perform the I/O and find an efficient aggregator placement.

Given, for each partition:

• V C : The set of compute nodes performing aggregation in the partition.

• A ∈ V C : An aggregator chosen among compute nodes

• ω(u, v): The data size exchanged between nodes u and v

• d(u, v):
The number of hops between nodes u and v

• IO: The storage system (I/O node) of the partition.

• l: The interconnect latency

• B i→j : The bandwidth from node i to node j.

For the first step of our strategy, each process involved in an aggregation partition computes the cost of aggregating data from all other ranks if it were chosen as the aggregator. This can be done in a distributed way once all the processes know the amount of data produced by each of them. We then keep the maximum cost as all the others will be bounded by this one. Formally, each process A computes the cost C 1 :

C 1 = max l × d(i, A) + ω(i, A) B i→A , i ∈ V C , i = A
The second step consists in computing the cost of sending the aggregated data to the storage system. The first version of our model took into account the total amount of aggregated data to compute this cost. However, while cost C 1 considers a small amount of data (the data sent by the rank with the maximal cost), C 2 considers the sum of the aggregated data. This creates an imbalance between these two cost and can make C 1 negligible compared to C 2 . To avoid this, in C 2 , we normalize the aggregated data with the number of processes involved in the aggregation phase to have C 1 and C 2 of the same order of magnitude. For each process A, we define the cost C 2 as:

C 2 = l × d(A, IO) + ω(A, IO) |V C | × B A→IO
Our topology-aware strategy minimizes the objective function defined as the sum of the costs C 1 and C 2 . More generally, this placement policy can be formulated as the solution for each partition of this objective function:

T opoAware(A) = min (C 1 + C 2 )
This sum is computed by each process independently in O(n), n = |V C |. Indeed the topology characteristics d, l, B can be precomputed at start time and the data exchange between processes depends on the data distribution and is accessible in constant time after a preliminary call to MPI Allgather. Hence, finding the process having the minimum cost is done through a MPI Allreduce with the MPI MINLOC operation. All these MPI calls involve meta-data of very small size compared to the actual data and hence have a negligible cost compared to sending the application data.

File System Block Size

The block size of a filesystem corresponds to an indivisible block of memory on disk requested for each read or write operation, no matter the size of data involved. Thus, writing a piece of data smaller than the block size implies a lock on a piece of memory of size blocksize. In the context of parallel I/O, the multiplication of locks on disks creates an important bottleneck. To evaluate the impact of the block size, we wrote a simple benchmark and ran it on a BG/Q supercomputer and its GPFS high-performance filesystem. This benchmark works as follow: one process per node writes the same amount of data to a single shared file at the corresponding offset. The results are depicted in Figure 3. The purple curve shows the bandwidth achieved while writing a chunk of data which is a multiple of the filesystem block size (8 MB in this case). The green curve shows this bandwidth with the same chunk of data plus 1 kB. We see that writing a multiple of the block size can perform up to 3x better than the non-aware case. In order to reduce this penalty, our algorithm has been designed to aggregate and write on an aggregator an amount of data which is a multiple of the filesystem block size. In other words, the buffers used by the aggregators to stage data during the I/O phase are allocated as a multiple of the filesystem block size. The default buffer size used in MPI two-phase I/O implementations has this property as well.

Pipelined Aggregation Buffers

In order to optimize both the aggregation phase and the I/O phase, each aggregator manages two buffers and overlaps the communications. In fact, as the aggregation phase is performed with RMA operations (one-sided communication), no synchronization is needed between the processes sending data to the aggregators and the aggregators themselves. Moreover, the aggregators perform nonblocking independent writes to the storage system making themselves available for other operations. In this way the aggregators are able to flush a full buffer while receiving data in the second one. This loop is performed as many times as necessary to process the data. Each instance of buffer filling and flushing is called a round. A global round is equivalent to the same buffer of all the aggregators filled (if applicable) and flushed.

Algorithm

The Algorithm 1 describes the core function of our aggregation process; that is, the Commit function called by the processes to write a piece of data. This function is recursive in order to split a piece of data among different buffers and/or aggregators. As we initialize our aggregation step with the upcoming writes, each process knows for each piece of data the target tuple {round, aggregator, buffer}. The lines 2 to 5 return these values. The chunk size corresponds to the amount of data to be written. If the piece of data fits in one buffer, this amount is equal to the data size given as parameter. In case of data splitting, this amount is smaller than the data size and an extra round is necessary. The while loop starting from line 8 blocks the processes whose current round is different from the global round in a fence (barrier in the context of MPI one-sided communication). Only the processes with the matching round can lift the barrier. If a process passing this fence is an aggregator, it flushes the appropriate buffer into the file. Line 16 just puts the data into the target buffer. If the process has written all its data, it enters a portion of code similar to the one starting from line 8. Else, we recursively call this Commit function again while updating the parameters.

Challenges at scale

When bringing these optimizations at scale, we needed to address two challenges in our model. Firstly, the partitioning phase can appear costly for certain cases. In particular, if one needs to partition a large MPI communicator, the fact that a process can be involved with several aggregators limits the parallelization of this part of the algorithm. In MPI, a communicator splitting cannot produce sub-communicators with an intersection. Put it in another way a process cannot belong to more than one sub-communicator resulting from a MPI Comm split. It implies that we have to perform the aggregator mapping partly sequentially. Moreover, some MPI Comm split implementations can be very unoptimized (e..g., on BG/Q) increasing the computation time. However, even if in most cases this limitation is absent, it has to be noted that we compute the aggregators mapping once for the whole application lifetime reducing its impact. Secondly, it is clear that to avoid contention our implementation needs to keep at most one aggregator per node. As a consequence, we have to update a list of nodes already selected as aggregators. This list is shared among at least a subset of processes. Because of this, some synchronization steps are necessary.

Evaluation

To validate our approach, we carried out experiments with both micro-benchmarks as well as with a real application called HACC from which we extracted the I/O kernel. We conducted our experiments on the Mira supercomputer, an IBM BG/Q supercomputer located at Argonne National Laboratory.

Targeted supercomputer

Mira is an IBM BG/Q supercomputer hosted at Argonne National Laboratory with 49,152 compute nodes each with 16 hyper-threaded PowerPC A2 cores (1600 MHz). Each node has 16 GB of main memory. These nodes are interconnected through a 5D torus high-speed network giving a theoretical bandwidth of 1.8 GBps per link. The BG/Q architecture splits the nodes into Psets. A Pset is a subset of nodes sharing the same I/O node. On Mira, this subset contains 128 nodes including two bridge nodes. A bridge node is a regular compute node connected to an I/O node (so each I/O node is connected to two bridge nodes). The Figure 4 shows this architecture. Note that for all the experiments using our aggregation strategy, we set the bandwidth between two nodes to 1.8 GBps (theoretical bandwidth) and the latency to 20 ms (measured). We compiled the test applications and our library with GCC v4.4.7 and used the default MPI installation on Mira, which is based on MPICH2 v1.5.

The MPI I/O part is based on ROMIO, an open-source high-performance implementation. MPI I/O has a highly configurable set of parameters; the default number of aggregators on Mira is set to 16 per Pset and the size of the buffer employed to aggregate the data is set to 16 MB. The default aggregator mapping strategy used by ROMIO on BG/Q is the one described in Section 4.1. The version of MPI-2 hosted on Mira has been optimized for the BG/Q platform.

Evaluation of the placement strategies

We evaluate our topology-aware strategy with a micro-benchmark and compare this approach to the other approaches described in section 4. micro-benchmark works as follow: Each rank produces an amount of data distributed randomly between 0 and 2 MB. In order to reduce the file system noise, the data were sent to the null device of the I/O nodes instead of a real file. This effectively measures the performance of just aggregating the data and moving this out to the storage system. It eliminates any I/O contention, and thus, performance degradation at the storage system. The experiments were carried out on 512 Mira-nodes with 16 ranks per node (8192 ranks). We set the number of aggregators to 16 for the MPI COMM WORLD communicator (i.e. 4 aggregators per Pset) to intensify the impact of the placement policy. The aggregator's buffer size was set to 16 MB. The results are shown in Table 2 and are calculated from 20 runs. These results illustrate the I/O bandwidth achieved and the time needed per round for the twophase I/O step. We notice that the topology-aware strategy gives the best bandwidth compared to the other methods. The shortest-path strategy offers the second best performance and greedy is the worst.

Impact of the number of aggregators and the buffer size

Finding the tradeoff between the number of aggregators and the buffer size is an open problem [START_REF] Chaarawi | Automatically selecting the number of aggregators for collective I/O operations[END_REF]. Even though it is not the main purpose of this paper, it is important to understand the behavior of our topology-aware aggregation algorithm with these parameters. For this, we carried out experiments with a micro-benchmark where we vary the number of aggregators and the buffer size. We ran these experiments on 1024 Mira-nodes, with 16 ranks per node, and with 1 MB produced by each process. The data is again written to the null device of the I/O nodes to understand the performance of the aggregation and data movement. Table 3 presents these results. In these experiments, the third column and row corresponds to the default parameters set in ROMIO on Mira: 8 aggregators per Pset and 16 MB of buffer size. With our algorithm, these settings produce a good I/O bandwidth. Moreover, the best I/O bandwidth is reached with larger buffers sizes. These observations are useful for future improvements and tuning.

HACC-IO

HACC-IO is the I/O kernel of HACC (Hardware Accelerated Cosmology Code). This large-scale cosmological application requires the massive compute power of supercomputers to simulate the mass evolution of the universe with particle-mesh techniques. In terms of I/O, every process of a HACC simulation manages a number of particles. Each particle is defined by nine variables: XX, Y Y , ZZ, V X, V Y , V Z, phi, pid and mask, corresponding to the coordinates, the velocity vector and relevant physics properties. The size of a particle is 38 bytes. A useful base value is: 25,000 particles require approximately 1 MB.

Impact of data layout

During our experiments, we observed that ROMIO performs very poorly in case of an array of structure (AoS) data layout. This layout is the default one used by HACC. To have a good overview of the performance with the different standard strategies, we implemented an alternative data layout based on structures of array (SoA). Figure 5 describes the differences between these two layouts in the context of HACC. For all the next experiments with this application, results for both these layouts are shown (except for POSIX I/O which performs independent I/O). The difference between the two types of organizations in terms of I/O can be noted. In the case of HACC, where the particles are written by variables, an AoS design needs more time to fill an aggregator buffer as it is bounded by the time necessary to write all the variables from one process. With a SoA layout, all the processes write the same variables at the same time. The time required to fill a buffer gets closer to the time needed to write one variable from one process. Except in few cases, we observed this impact in our results. It also explains the performance of ROMIO.

X Y Z X Y Z X Y Z X Y Z Processes Data X Y Z X Y Z X Y Z X Y Z Data layouts in file X X X X Y Y Y Y Z Z Z Z

Results

The Figures 6 and7 show respectively the results when writing to a single file shared by all the processes and to one file per Pset. These experiments were run on 1024 Mira-nodes and 16 ranks per node. For each strategy, we vary the number of particles per rank. To conduct fair experiments, we set the number of aggregators to 16 per Pset and the buffer size to 16 MB, as the default settings of ROMIO on Mira.

Writing to a single shared file (Figure 6) results in poor performance in general. On 1024 nodes, the effective peak I/O bandwidth is estimated to be 22.4 GBps (while the theoretical bandwidth is 28.8 GBps). The best performance we can achieve in this case does not exceed 5 GBps. Nevertheless, only our topology-aware strategy is able to achieve this. We can also notice that our approach outperforms both POSIX I/O and MPI I/O regardless of the data size or the data layout. This difference is particularly substantial on small messages and tends to decrease while the data size per rank increases. When 5000 particles are written by a process (∼200 KB), in the case of an array of structures data layout, our method provides an I/O bandwidth 15× higher than MPI I/O. This factor is 4× larger with a different data layout (SoA). As explained before, we can clearly see the poor performance obtained with MPI I/O on a specific data layout. Dividing the output data into several files, also known as sub-filing, appears to be an attractive solution. The results presented in Figure 7 are a relevant illustration with I/O bandwidth close to the peak value. Like the previous results, we observe that our strategy based on topologyaware placement and optimized buffers (block size awareness and pipelining) achieves much better performance than the standard approaches. Again, this gap decreases slightly as the data size increases. However, the I/O bandwidth remains higher with our method, even with large messages (2 MB per process).

Figure 8 presents results from the same experiments as we perform weak scaling to 4096 nodes (64K MPI processes). These results show very similar behavior at scale. As on 1024 nodes, we are able to significantly improve the I/O bandwidth, irrespective of the data layout in file and particularly on messages smaller than 2 MB.

Conclusions and Future Work

In this paper, we have demonstrated the importance of the data movement optimizations for intensive I/O operations. In particular, we developed an I/O library on top of MPI I/O based on the two-phase I/O scheme and this takes into account the topology of the infrastructure, an efficient buffering system and the access patterns of applications. This model achieves very good performance at scale and outperforms standard approaches. For the I/O kernel of a cosmological application, our solution was able to achieve a 15 × improvement over default parallel I/O implementation. Additionally, we demonstrated the needs to design an algorithm capable of dealing with different data layout by overlapping communications. We demonstrate the necessity to fully exploit the architecture characteristics in order to achieve performance at scale and meet the expectations of large-scale simulations. As part of our future work, we plan to consider routing strategies in our algorithm. This information could be extremely useful to reduce performance degradation due to network contention. Another research track is to extend this aggregation method to a larger varieties of data patterns (2D or 3D arrays for example). Finally, we will need to adapt our model to the various expected tiers of storage and memory.

Figure 1 :

 1 Figure 1: Two-phase I/O mechanism

Figure 2 :

 2 Figure 2: Data aggregation for I/O: simple partitioning and aggregator election on a grid with one different strategy per partition.

  rank (in MB) Impact of the file system block size on I/O 2048 Mire-nodes -1 rank/node -Indep. MPI I/O Multiple of BS Multiple of BS + 1 KB

Figure 3 :

 3 Figure 3: Benchmark measuring the impact of the filesystem block size for write operations.

Algorithm 1 :

 1 Data Aggregation 1 Function Commit (data, size, of f set) 2 round ← GetRound(); 3 aggr ← GetAggregatorRank(); 4 chunkSize ← GetRoundSize(round); 5 buf f erId ← globalRound % 2;

7 7 8 while round = globalRound do 9 Fence (); 10 if I am an aggregator then 11 Write

 891011 Buffer (buf f erId); 12 globalRound ← globalRound + 1; 13 buf f erId ← globalRound % 2;

19 if chunkSize = size then 20 while round = m round do 21 Fence

 192021 , chunkSize, of f set, aggr, buf f erId);18 18 + roundSize, size -chunkSize, of f set + chunkSize);

Figure 4 :

 4 Figure 4: BG/Q architecture

Figure 5 :

 5 Figure 5: Data layouts implemented in HACC

Figure 6 :

 6 Figure 6: Single shared file from 1024 nodes (16 ranks/node)

TopologyFigure 7 :

 7 Figure 7: One file per Pset from 1024 nodes (16 ranks/node)

TopologyFigure 8 :

 8 Figure 8: One file per Pset from 4096 nodes (16 ranks/node)

Table 1 :

 1 I/O requirements of diverse large-scale applications

	Scientific domain	Simulation	Data size
	Cosmology	Q Continuum 2 PB / simulation
	High-Energy Physics Higgs Boson	10 PB / year
	Climate / Weather	Hurricane	240 TB / simulation

  1, namely, greedy, shortest path and longest path. The

	Pset	5D Torus network
	128 nodes	2 GBps per link	2 GBps per link	4 GBps per link
					QDR Infiniband switch	Storage GPFS filesystem
	Compute nodes	Bridge nodes	I/O nodes
	PowerPC A2, 16 cores		2 per I/O node	IO forwarding daemon
	16 GB of DDR3			GPFS client

Table 2 :

 2 Impact of aggregators placement strategy

	Strategy	I/O Bandwidth (MBps) Aggr. Time/round (ms)
	Topology-Aware	2638.40	310.46
	Shortest path	2484.39	327.08
	Longest path	2202.91	370.40
	Greedy	1927.45	421.33

Table 3 :

 3 I/O Bandwidth (in MBps) achieved by a simple benchmark with a topology-aware aggregator placement while varying the number of aggregators and the buffer size.

	#Aggr/Pset	8 MB	Buffer size 16 MB	32 MB
	8	7652.49	8848.28	9050.71
	16	7318.15	8774.58	9331.84
	32	6329.95	7797.12	8134.41

Called I/O for the remaining of the paper. Even if an application performs other kind of I/O (to local disk or the network), here we consider only I/O to the storage/parallel file system

Acknowledgment

This research has been funded in part and used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under contract no. DE-AC02-06CH11357. This work was supported in part by the DOE Office of Science, ASCR, under award numbers 57L32, 57L11, 57K50, and 5080500. This research is partially supported by the NCSA-Inria-ANL-BSC-JSC-Riken Joint-Laboratory


Eric Ragneau

Wei Liu

Dominique Guines
email: dominique.guines@insa-rennes.fr

Lionel Leotoing

Identification of strain rate-dependent mechanical behaviour of DP600 under in-plane biaxial loadings

Keywords: Biaxial test, Cruciform specimen, Hardening model, Rate-dependent behaviour, Metallic sheet, DP600

come

Identification of strain rate-dependent mechanical behaviour of DP600 under in-plane biaxial loadings

Introduction

In many industrial fields (automotive, aeronautical, shipbuilding, railway, …), sheet metal forming processes are widely adopted to produce more and more complex parts. The lightening, the enhancement of safe design, the cost decrease of both the design stage and the manufacturing process are challenging issues in these industries. These objectives lead to the increasing adoption of the simulation tool, usually based on Finite Element (FE) numerical simulations. For material behaviour modelling, accurate constitutive models are then required and have to be implemented in FE codes.

Today, phenomenological approach remains predominant to model material behaviours. These models have to be characterized and calibrated by means of experimental techniques carefully designed to carry out tests at conditions met both in the considered sheet metal forming processes and in service. Depending on the forming operation and service conditions, multi-axial loadings, large plastic strains, thermal and rate-dependencies of the material usually have to be investigated.

The most popular experimental test used to characterize sheet metal behaviour is the uniaxial tensile test. Nevertheless, only uniaxial stress and strain states and relatively low strain levels, compared to the ones obtained in forming operations, can be reached. In order to obtain information concerning multiaxial loading conditions encountered in sheet forming processes, several tests have been developed such as, tension tests on wide specimens [START_REF] An | A novel and simple method for the measurement of the plane strain work hardening[END_REF][START_REF] Flores | Accurate stress computation in plane strain tensile tests for the sheet metal using experimental data[END_REF], combined plane strain-simple shear test on only one specimen [START_REF] Pijlman | Sheet material characterisation by multi-axial experiments[END_REF][START_REF] Mohr | A new experimental technique for the multi-axial testing of advanced High strength steel sheets[END_REF], hydraulic bulge tests with circular or elliptical die openings [START_REF] Rees | Plastic flow in the elliptical bulge test[END_REF], combined uniaxial tension-internal pressure tests [START_REF] Kuwabara | Advanced in experiments on metal sheets and tubes in support of constitutive modeling and forming simulations[END_REF], Marciniak [START_REF] Naka | The effects of temperature and forming speed on the forming limit diagram for type 5083 aluminium-magnesium alloy sheet[END_REF] or Nakazima stamping tests [START_REF] Kim | Forming limit diagram of auto-body steel sheets for highspeed sheet metal forming[END_REF] and biaxial tensile tests on flat cruciform specimen [START_REF] Léotoing | Cruciform shape benefits for experimental and numerical evaluation of sheet metal formability[END_REF][START_REF] Liu | Identification of sheet metal hardening for large strains with an in-plane biaxial tensile test and a dedicated cross specimen[END_REF]. As shown in these last works, the biaxial tensile test on a flat cruciform specimen presents several advantages: it is a frictionless test, it allows to reach large plastic strains and linear and non-linear strain paths can be easily achieved. Nevertheless, strain-rate dependency of metallic sheets has not yet been studied with such experimental device.

Various dynamic tests have been performed to identify strain rate dependent hardening law in the intermediate strain rate range. For dynamic tests, depending on the initial impact velocity at the beginning of the test, an oscillation phenomenon of the force signal measured by the load cell, called as loading ringing [START_REF] Roebuck | Load cell ringing in high rate compression tests[END_REF], can be observed. Those variations of the experimental force can be induced by the propagation of an elastic stress or strain wave or by the vibratory response of mechanical parts of the experimental device [START_REF] Borsuzki | Recommendations for dynamic tensile testing of sheet steels[END_REF]. Huh et al. [START_REF] Huh | High speed tensile tests of steel sheets for the stress-strain curve at the intermediate strain rate[END_REF] have developed a high speed uniaxial testing machine to investigate dynamic tensile behaviour of steel sheets in the strain rate range from s . The grip attached to the load cell has been shorted to increase the natural frequency. This designed grip has been validated to reduce the load ringing. For testing materials in the intermediate strain rate range from s . The force on the bar/specimen interface is firstly determined by inverse analysis based on the elastic FE model of tools. The plastic hardening model of material is then identified by applying the preliminarily identified force to the testing specimen. Fitoussi et al. [START_REF] Fitoussi | Experimental methodology for high strain-rates tensile behaviour analysis of polymer matrix composites[END_REF] have conducted high speed uniaxial tensile tests on fibre reinforced polymer composites by a servo-hydraulic machine. A damping joint of a low impedance material is placed between the sliding bar and the hydraulic jack to reduce the wave effects caused by the dynamic shock. Boyce et al. [START_REF] Boyce | The dynamic tensile behaviour of tough, ultrahigh-strength steels at strain-rates from 1 0002 . 0  s to 1 200  s[END_REF] have performed uniaxial tensile tests on ultrahigh strength steels at the strain rates from s . A rubber damper placed in the slack adapter is validated to reduce the initial oscillations by more than % 50 . Xiao [START_REF] Xiao | Dynamic tensile testing of plastic materials[END_REF] has also adopted a damping layer to reduce load ringing for dynamic uniaxial tensile tests on plastic materials. As one can see, many works were interested in strain rate dependency of metallic alloy behaviour under uniaxial stress state.

Concerning stain rate sensitivity under biaxial loadings, very few studies have been published.

Recently, Grolleau et al. [START_REF] Grolleau | Biaxial testing of sheet materials at high strain rates using viscoelastic bars[END_REF] have proposed a dynamic bulge testing device to investigate the material behaviour under biaxial loadings conditions at intermediate strain rates. The hardening law of aluminium alloy A1611-T4 sheet at high strain rate of up to 1 500  s is identified. A movable bulge cell is designed and installed in a split Hopkinson pressure bar (SHPB) system. Ramezani et al. [START_REF] Ramezani | Combined experimental and numerical analysis of bulge test at high strain rates using split Hopkinson pressure bar apparatus[END_REF] have also introduced a similar high strain rate bulge test technique on the basis of SHPB system. The analytical and finite element methods are compared with experiments to verify the developed method.

In order to investigate the material behaviour under dynamic and different biaxial stress states, Shimamoto et al. [START_REF] Shimamoto | The development of servo dynamic biaxial loading device[END_REF] have developed a servo-hydraulic dynamic biaxial loading device. The inplane dynamic biaxial test on cruciform specimen is carried out to reveal the crack extension behaviour of aluminium alloy AA7075-T6 at 1m/s. Nevertheless, no calibration model is proposed in this work. Recently, a specific dynamic biaxial tensile device with four independent servo-hydraulic actuators (each one with a loading capacity of 50kN) has been developed in the Laboratory of Civil and Mechanical Engineering (LGCGM) of National Institute of Applied Sciences (INSA) at Rennes [START_REF] Zidane | Development of an in-plane biaxial test for forming limit curve (FLC) characterization of metallic sheets[END_REF]. This biaxial tensile machine has been used to lead in-plane biaxial tensile tests on cruciform specimens at quasi-static strain rate for different applications: (i) calibration of yield criteria [START_REF] Zhang | Potential of the cross biaxial test for anisotropy characterization based on heterogeneous strain field[END_REF], (ii) identification of hardening laws at large strains [START_REF] Liu | Identification of sheet metal hardening for large strains with an in-plane biaxial tensile test and a dedicated cross specimen[END_REF] and (iii) determination of forming limit curves [START_REF] Léotoing | Cruciform shape benefits for experimental and numerical evaluation of sheet metal formability[END_REF] of metallic sheets. Depending on applications, several in-plane specimen shapes have been proposed and validated and linear or non-linear strain paths can be imposed. Based on an instantaneous release of a great quantity of high pressure flow accumulated, this machine is also dedicated to perform dynamic tensile tests (a maximum velocity of 2m/s can be reached on each actuator) on flat cross specimens.

In the last few years, optical measurement techniques and image correlation techniques have been intensively used to calibrate material models. In most analyses reported in the literature, the identification is performed by minimizing a least-square functional based on the difference between experimental and numerical fields or discrete values of strains or displacement. Within the framework of quasi-static characterization, Meuwissen [START_REF] Meuwissen | Determination of the elasto-plastic properties of aluminium using a mixed numerical-experimental method[END_REF] designed sheet specimens of nonstandard geometry and optically measured the displacement of markers placed on the surface specimen. Belhabib [START_REF] Belhabib | Heterogeneous tensile test on elastoplastic metallic sheets: comparison between FEM simulations and full-field strain measurements[END_REF] identify material parameters by fitting simulated and optical heterogeneous strain fields obtained from tensile tests on elastoplastic metallic sheet. On flat cruciform specimen, Zhang [START_REF] Zhang | Potential of the cross biaxial test for anisotropy characterization based on heterogeneous strain field[END_REF] used discrete values of principal strains along a diagonal direction of the central zone of a cruciform specimen. With a more complex cross specimen shape, Liu [START_REF] Liu | Identification of sheet metal hardening for large strains with an in-plane biaxial tensile test and a dedicated cross specimen[END_REF] used local strain measures in the specimen centre to identify hardening law of a 5086 aluminium alloy up to large strains. Due to the increasing acquisition frame rate of cameras, these non-contact techniques are now well adapted to dynamic characterizations and may be used for homogeneous or non-homogeneous tests.

Kajberg [START_REF] Kajberg | Characterization of materials subjected to large strains by inverse modelling based on in-plane displacement fields[END_REF] used specimens with non-uniform widths tested in tension. The state of deformation is quantified by an optical method, digital speckle photography (DSP), providing field information for both in-plane displacements and strains. The strain fields were then analysed for the numerical identification of the stress-strain relationship up to large strains.

In the present work, a dual phase advanced high strength steel (DP600), which combines high strength and good formability, is considered. It has been shown in previous works [START_REF] Peixinho | Study of Viscoplasticity Models for the Impact Behavior of High-Strenght Steels[END_REF] that such steel grades exhibited rate-dependent behaviour. This strain rate dependency is first evaluated experimentally from uniaxial tensile tests at room temperature in a moderate strain rate range (from quasi-static conditions up to 20s -1 ). Based on these observations, two phenomenological ratedependent hardening models are selected and calibrated. Then, static and dynamic equi-biaxial tests are performed on a dedicated cross specimen shape, previously developed and validated for quasi-static applications [START_REF] Liu | Identification of sheet metal hardening for large strains with an in-plane biaxial tensile test and a dedicated cross specimen[END_REF], up to an equivalent plastic strain of 30%. Parameters of the two ratedependent hardening formulations are identified from biaxial experiments by means of an optimization procedure minimizing the difference between local simulated and experimental strains in the specimen centre. The identified biaxial flow stress curves are then compared with uniaxial ones for the quasi-static and intermediate strain rates to show the benefits of the proposed methodology.

Uniaxial tensile tests

Initially, uniaxial characterizations of DP600 thin plate specimens are performed to establish the strain rate dependency of the elastoplastic behaviour of such alloys in an intermediate strain rate range. Based on these experimental observations, two phenomenological strain rate-dependent constitutive relationships are selected and calibrated up to the maximum strain level reached in uniaxial tension.

Uniaxial flow stress curves

Uniaxial tensile tests on DP600 were performed by dynamic servo-hydraulic testing machine at room temperature. The specimens have a constant cross section (width×thickness: 10mm×2mm) over an initial gauge length of 50mm. For all specimens, the tensile direction corresponds to the It can be seen that DP600 exhibits a positive strain rate sensitivity in the tested strain rate range at room temperature. The maximum stress values at and are respectively 760MPa and 820MPa, corresponding to an increase of about 8%. In the observed strain range, the strain rate dependency seems to affect both the initial yield stress and the strain hardening. These trends (moderate rate sensitivity) have been already observed in previous works on the same material [START_REF] Peixinho | Study of Viscoplasticity Models for the Impact Behavior of High-Strenght Steels[END_REF][START_REF] Rahmaan | effect of strain rate on flow stress and anisotropy of DP600, TRIP780, and AA5182-O sheet metal alloys[END_REF].

Strain rate dependent hardening law

Many integrated constitutive equations relating equivalent flow stress with equivalent plastic strain ̅ , equivalent total strain rate ̇ and temperature have been proposed to describe the mechanical behaviour of metallic alloys. In the current work, the temperature is not considered so the different formulations used to describe the influence of this parameter on the flow stress are not discussed. Concerning the strain hardening function ( ̅ ), two main types of formulations are usually encountered. The first one includes "saturation" models which present a saturation stress at large strains while the second one concerns "power-law" models which are unbounded at large strain.

A typical formulation of these models is given by Ludwick's (Eq. 1) and Voce's (Eq. 2.) equations which present respectively the saturated and unsaturated behaviour:

( ̅ ) ̅ (1)
( ̅ ) ( ( ̅ )) (2)
Where is the initial yield stress, ̅ is the equivalent plastic strain and and are material constants.

The strain-rate influence on the plastic flow curve, described by the function ( ̇), can be introduced in a multiplicative or additive way on the general strain hardening law. According to the coupling effects, four types of rate-dependent hardening model have been identified by Paul [32].

These models are summed up below (Eq. 3 to 6):

Type 1 : ( ̅ ) ( ̇) (3)
( ̇) ( ̅ ) (4)
( ( ̅ )) ( ̇) (5)
( ̇) ( ̅ ) ( ̇) (6)
In those models strain and strain-rate hardening parameters can be linear or non-linear functions of temperature. In this work, deformation-induced thermal effects are not considered, even if these phenomena could be significant at tested strain rates which are closed to the adiabatic limit for typical steel [START_REF] Sung | A plastic constitutive equation incorporating strain, strainrate and temperature[END_REF]. The purpose of the current work being to validate a new dynamic experimental device in order to investigate the rate-dependent behaviour of metallic alloys under biaxial loadings, the flow stress temperature sensitivity is considered as a secondary material factor in this study.

Classically, behaviour of many metallic alloys is described by Type 1 models assuming the independency of the initial yield stress to the strain rate.

Concerning the DP600 steel grade, Yu [33] have proposed a hardening model on the basis of Khan-Huang (KH) model [34] by neglecting the influence of strain rate on the strain hardening term (Type 2) as follows:

̅ [ ( ̅ ̇ ̅ ̇ ) ] ̅ ( ̅ ) (7)
The strain-rate dependency of the modified model is applied only on the initial yield stress though it was applied on both initial yield stress and strain hardening in the initial KH model (Type 3). This modified model describes well the hardening behaviour for DP600 steel at the strain rates from to . In this model, , , , and are the material parameters and the reference strain rate is ̅ ̇ .

In [34], Paul proposed a hardening model with different strain rate dependencies on the initial yield stress and the strain hardening term as follows (Type 4):

̅ [ ( ̅ ̇ ̅ ̇ ) √ ̅ ̇ ̅ ̇ ] [ ̅ ( ̅ )] [ ( ̅ ̇ ̅ ̇ ) √ ̅ ̇ ̅ ̇ ] (8)
, , , , , and are the material parameters. This model has been validated to describe the strain rate influence on both the initial yield stress and strain hardening term for DP600 steel sheets in the strain rate range from to .

Based on observations of experimental uniaxial true-stress versus plastic strain curves (Fig. 1)

and recent published works on this grade (cited above), a general formulation of the constitutive law assuming both a strain-rate dependency on the initial yield stress and the strain hardening seems to be well adapted to describe the hardening behavior of DP600 steel sheets. Here, a constitutive hardening law (Type 4) with two different strain rate dependency terms, ( ̇) and ( ̇) , respectively on the initial yield stress and the strain hardening is chosen. For these two terms, a power model is adopted to describe the strain rate sensitivity:

( ̇) ̇ and ( ̇) ̇ (9)
Where and are the strain rate sensitivity parameters.

Calibration of strain-rate dependent hardening laws from uniaxial tensile tests

Parameters of the strain rate dependent hardening laws are obtained by minimizing the error function  expressed below (Eq. 10). The calculated stresses are determined from the analytical model of the homogeneous uniaxial tensile test.

      2 exp, , exp exp, , exp exp, , 1 1 1 1
( 10
) cal i j i j p p q pp j ij j j i p q                       
Where p is the number of true stress-plastic strain curves (or number of tests in the experimental database), q is the point number on each curve, ̅ and ̅ are respectively the analytical and experimental equivalent true-stress.

The identified parameters on the basis of Ludwick's law are presented in Table 1. As it can be seen, both initial yield stress and hardening behaviour are strain rate dependent. As shown in the literature, it seems that initial yield stress is more sensitive than strain hardening to the strain rate ( ), nevertheless, as observed in Fig. 2, the best set of parameters for the Ludwick's formulation leads to an over-estimation of the hardening level at low strains.

Tab. 1 Identified material parameters of rate-dependent Ludwick's law from uniaxial tests The identified flow stress curves on the basis of Ludwick's and Voce's laws are compared with the experimental ones at each strain rate, as shown in Fig. 2.

1 2 0 m m n p K        Error  (%) 0  (MPa) K (MPa
    1 2 0 1 exp m m p K n          Error  (%) 0  (MPa) K (MPa
It can be seen that all identified flow stress curves are in relatively good agreement with experimental ones. Nevertheless, the Voce's formulation is closer to the experimental results than the Ludwick's one. This last formulation leads to a slight over-estimation for small strains (under 5%) and for large strains (up to 13%).

Fig. 2 Comparison of experimental and identified flow curves from uniaxial tensile tests

For the two analytical models (Ludwick's and Voce's) calibrated from experimental uniaxial tensile tests in an equivalent plastic strain ranging up to 17%, the equivalent stress versus equivalent plastic strain curves are plotted for an equivalent plastic strain range extrapolated up to 0.3 (Fig. 3).

These two formulations are compared at three strain rates , and . As shown in this figure, these two models give very different trends. The identified material parameters for the Voce's formulation lead to a complete saturation of the strain hardening for deformation between 17% and 30%. As contrary, Ludwick's model, identified on the same experimental data, shows a constant increase of the flow stress for large strains (between 15 and 30%). As one can see it, it is difficult to define the best formulation for large strains on the only base of uniaxial tensile tests.

Biaxial tensile tests

In this section, the quasi-static and dynamic biaxial tensile tests on cruciform specimens are performed to obtain large strains at the central zone.

Testing specimen and experiments

Recently, for static characterizations, Kuwabara [37] has proposed a standard cruciform shape to determine elastic constants or 2D initial yield surfaces of metallic sheet under in-plane biaxial loadings. This shape is only suitable for low strain levels and no standard specimen exists for large strains. To allow characterizations for large strains, Zidane et al. [START_REF] Zidane | Development of an in-plane biaxial test for forming limit curve (FLC) characterization of metallic sheets[END_REF] have proposed a specific cruciform shape to determine forming limit curves (FLC) of metallic sheet. In this specimen, a central thickness reduction is defined by a curved profile to obtain the onset of necking at the central point. In the present study, a cruciform shape with a flat central zone of reduced thickness to localize the deformation, four slits in each arm to reduce the transverse rigidity, notches at the intersections of arms to reduce the localisation is optimized as shown in Fig. 4. This shape has been validated for large strains and static applications by present authors in previous works [START_REF] Liu | Identification of sheet metal hardening for large strains with an in-plane biaxial tensile test and a dedicated cross specimen[END_REF] for the determination of hardening law parameters. The sample thickness outside the thinned zone is 2mm.

The final thickness in the central zone of the specimen is 0.625mm. The specimen is manufactured in DP600 steel sheet by milling the central zone and by wire cutting the slits.

As explained above, the biaxial machine (Fig. 5) is equipped with four independent servohydraulic actuators. Each end of the cruciform specimen is connected to an actuator rod through a mechanical part assembly as shown (for only one specimen end) in Fig. 6. For dynamic tests, the impactor is accelerated along the sliding bar to obtain the specific velocity before the specimen is loaded. Due to the initial impact, oscillations appear on force signal measured by the load sensor (see Fig. 6). To reduce these oscillations a 3mm thick elastomer damping layer is placed at the interface between the sliding bar and impactor. The thickness of the damping layer has been determined by mean of numerical simulations. These simulations are based on a FE model of the equi-biaxial tensile test including the mechanical part assembly presented Fig. 6, under dynamic conditions. After the impact, the displacement of the sliding bar is transmitted to each specimen end through the load sensor and the bi-articulated link. During the test, the velocity is maintained almost constant due to the inertia effect of impactor. The experimental forces along the two perpendicular directions are measured by deformation gauge load sensors. where p and q are the numbers of points to be averaged in each direction and t is the time. Here, the average strains are calculated for p = q = 3, corresponding to a square area defined by . Maximum value of 50% of the equivalent plastic strain is reached for the tensile test velocity of 0.02mm/s (Fig. 10a) and this value is about 40% for the velocities of 1mm/s and 250mm/s (Fig. 10b and Fig. 10c). Due to a little desynchronization between the two directions and material anisotropy, major principal strain curves does not exactly coincide with the minor principal strain curves. The major and minor principal strains are close to 20% for the two first tested velocities, with a maximum variation of 5% between the major and minor strain just before rupture. Due to a higher desynchronization, this variation is about 10% for the last tested configuration at 250mm/s.

          2 22 exp exp 1 1 1 1 11 2 p q q sim k i j i j i j k i j Q Q t t pp                  ( 12
)
where q is the total number of time points and p is the number of tests at different velocities, sim  are the experimental principal strains in the same zone of the specimen (Fig. 10(a), (b), (c)).

The error function Q is evaluated through an optimization loop built in the software platform ModFrontier. The Simplex algorithm has been chosen to optimize the material parameters. In order to make sure that the global minimum is found, different sets of initial values have been tested.

           N M H G F ij          (13)
where F , G , H , L , M and N are the material parameters. These parameters, given in Table 3, have been determined from the three anisotropic coefficients ( ) The simulated and experimental principal strains at the central point of the surface specimen have been compared as shown in Fig. 13. Although the simulated principal strains are lower than the experimental ones at the beginning of tests, the observed trends are in good agreement for principal strains above 4% whatever the hardening formulation. The equivalent stress versus equivalent plastic strain curves identified on the basis of Ludwick's and Voce's models from the equi-biaxial tensile tests are compared in Fig. 14. For low strains, due to the difference observed on the values of the two initial yield stresses identified, significant differences can be raised between the two formulations. For larger strains (more than 5%), the two formulations give approximately the same description of the strain-rate dependent hardening behaviour of DP600, even if some discrepancies appear for equivalent plastic strain close to 30%. Comparison between From these biaxial dynamic experiments, benefits of the proposed methodology are clearly shown by comparing the two identified hardening laws, the "saturation" model and the "power-law" model, which present very different evolutions when identified on uniaxial tensile tests and predict very similar hardening behaviour when calibrated on equi-biaxial tensile tests up to 30% of equivalent plastic strain. Finally, with such biaxial loadings, the work hardening behaviour can be identified for strain paths corresponding to those encountered in forming processes.

proposed

  et al.[START_REF] Gilat | A new compression intermediate strain rate testing apparatus[END_REF] have introduced a new compression apparatus consisting in a large hydraulic actuator and a long transmitter bar of m 40 length. The experiment is completed before the reflected wave in the transmitter bar returns back to the strain gauge so that the measured force is not disturbed. Othman et al.[START_REF] Othman | A modified servo-hydraulic machine for testing at intermediate strain rates[END_REF] have developed a modified servo-hydraulic machine to assess the mechanical properties of aluminium alloy 2017-T4 at the strain rate of approximately 1 2 10  s . A long duration Hopkinson bar technique and a wave separation method are adopted for less oscillating force measurement. Wood et al.[START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF] have developed uniaxial tensile tests at the strain rates from quasi-static to 1 500  s . The force measured by the local force transducer on the specimen is less oscillating than the one measured by the dynamic load cell located between the grip and the machine frame. Diot et al.[START_REF] Diot | Two-step procedure for identification of metal behaviour from dynamic compression tests[END_REF] have carried out uniaxial compressive tests to identify the constitutive model of a steel up to large strains of %

Fig. 1

 1 Fig. 1 Equivalent stress-plastic strain curves of DP600 by uniaxial tensile tests

Fig. 3

 3 Fig. 3 Strain rate dependent hardening law identified by uniaxial tests

Fig. 4

 4 Fig. 4 Cruciform specimen of DP600

Filmed zoneFig. 5 3 . 2

 532 Fig. 5 Biaxial testing machine

Fig. 7 Fig. 9

 79 Fig. 7 Displacement vectors at 250mm/s Fig. 8 Initial cracks at 250mm/s

Fig. 10 Fig. 11 Fig. 12 .

 101112 Fig. 10 Evolution of equivalent and principal strains at central point

Fig. 12 Experimental forces 3 . 3

 33 Fig. 12 Experimental forces

1  and sim 2  1  and exp 2

 1212 are the simulated major and minor principal strains at the central zone ofFE model, exp

  Identified material parameters correspond to the parameter set which minimizes the error function Q.As explained in[START_REF] Liu | Identification of sheet metal hardening for large strains with an in-plane biaxial tensile test and a dedicated cross specimen[END_REF], FE simulations with solid elements (C3D4) are very time-consuming.So to keep simulation times compatible with the inverse procedure which requires many simulations, a FE model with 2D linear shell element mesh (S4R) is proposed. The shell model has been divided into three zones with different thicknesses obtained from successive simulations in order to obtain the major and minor principal strain curves at the centre of the specimen for the two models.For the isotropic elasticity, a Young's modulus of GPa Plastic behaviour of DP600 steel sheets is modelled assuming the associated normal flow rule and Hill 48 anisotropic yield criterion defined by :

Fig. 13

 13 Fig. 13 Comparison of experimental and simulated strain curves

Fig. 3 andFig. 14

 314 Fig.3and Fig.14, shows clearly the interest of the biaxial tensile test which permits a better strain and strain-rate hardening identification for large strains whatever the hardening formulation. If maximum values of the equivalent stress at 30% of equivalent plastic strain are compared for the Voce's formulation, one can notice that the strain rate sensitivity identified by mean of classical

  Dynamic experiments presented in this study validate the capabilities of the servo-hydraulic device used in this study to investigate the rate-dependent behaviour of metallic sheets under biaxial loadings for an intermediate strain rate range. Concerning the geometry of the specimen, it is shown that the cross specimen shape, already validated under quasi-static conditions, can also be used when dynamic biaxial tensile tests are performed. The two work hardening formulations identified in this study give similar results in terms of error function with a strain rate index more important on the strain hardening term than on the initial yield stress.

  ) nThe identified parameters on the basis of Voce's law are presented in Table2. As observed on Fig.2, a better fit is obtained with Voce's formulation than with Ludwick's one which is confirmed by error function values  which are respectively 6.53% and 3.85% for Ludwick's and Voce's formulations. Both the initial yield stress and the strain hardening term are observed to be strain-rate dependent. Nevertheless, Voce's formulation gives close values of sensitivity indexes and , with a higher sensitivity observed on the yield stress than on the strain hardening.

				1 m	m	2	6.53
	351.6	1079.4	0.47	0.0223	0.0044

Tab. 2 Identified material parameters of rate-dependent Voce's law from uniaxial tests

  ) n

				1 m	m	2	3.85
	392.4	432.4	14.45	0.0137	0.0123

  by Ozturk et al.[36]. The parameters of the yield function are supposed to be independent to the strain rate and strain level.The general strain rate dependent hardening laws on the basis of Ludwick and Voce formulations, already calibrated from uniaxial tensile tests in section 2.3, are now identified from experimental biaxial tensile tests. The identified parameters are presented in Table4and Table5for respectively the Ludwick and Voce formulations. Contrary to the results observed from uniaxial tensile test calibration, the influence of strain rate on the strain hardening term is more important than the one on the initial yield stress (

	Tab. 3 Parameters of Hill 48 yield function
	F	G		H					M	N
	4204 . 0	5291 . 0	4709 . 0				5 . 1	5 . 1	2819 . 1
				2 m  ) for the two identified models. 1 m
	Tab. 4 Identified material parameters of rate-dependent Ludwick law from biaxial tests
				0    	1 m		K	p   n	2 m
	0  (MPa)	K (MPa)	n					1 m	m	2
	339.2	839.7	0.3864			0.0052	0.0158
		0.02mm/s	1mm/s			250mm/s
									Average error 
	Error (%)	  1 	  2 	  1 	  2 	  1 	  2 	(%)
		7.	8.4	13.4		7.7		9.2	6.4	8.7

Table 5

 5 Identified material parameters of rate-dependent Voce law from biaxial tests

			0    	1 m	 1 exp  	  p n    	2 m
	0  (MPa)		K (MPa)			n		1 m	m	2
	437.9		446.2			8.24	0.0032	0.0245
		0.02mm/s			1mm/s	250mm/s
									Average error 
	Error (%)	  1 	  2 	  1 	  2 	  1 	  2 	(%)
		8.	9.8		10.4	8.9	11.2	9.5	9.6

Acknowledgments

We express our gratitude to ArcelorMittal company for supplying dual phase (DP600) steel sheets characterized in this study.