Manuel Selva

Abdoulaye Gamatié

David Novo

Gilles Sassatelli

Speed and Accuracy Dilemma in NoC Simulation: What about Memory Impact?

Network on Chip (NoC) communication infrastructures are increasingly being used in modern manycore architectures. Many industrial and research NoC simulators have been proposed in the last years in order to facilitate the design of such communication infrastructures. As any simulator, all of them have to trade off speed and accuracy. Simulation time directly depends on the simulation accuracy. It also directly depends on the complexity of the system to be simulated, e.g., the number of cores and their unit complexity. In this work, we show that the memory footprint of NoC simulators can be a serious factor limiting the simulation of manycore architectures with a large number of cores. We first quantitatively compare the memory footprint of a transactional level modeling NoC simulator and its cycle-accurate counterpart to show that memory footprint is a concern. Then, we show that memory footprint is also largely impacted by the choice of the programming abstraction by comparing two cycle-accurate simulators written using different application programming interfaces, i.e., plain C++ and SystemC.

I. INTRODUCTION

Simulation software is a key component in system design process. During this process, it helps handling the increasing complexity of modern processors at different levels. Simulation tools first allow to perform large design space explorations purely in software. They also help reducing the time to market by allowing software development to start before the first version of the hardware is available.

Processors with 256 cores [START_REF] De Dinechin | Guaranteed Services of the NoC of a Manycore Processor[END_REF] are already on the market. According to the ITRS road-map, this core count in a single chip will continue to increase in the near future. To help designing these future processors, simulation tools must be able to handle this huge core count. One main concern when simulating such large systems is simulation time, which heavily depends on simulation accuracy and on the number of components to be simulated.

Among the existing studies on simulator design, most of the addressed issues concern the speed and accuracy dilemma. In other words, how to make the simulation faster while providing a relevant accuracy in the results. Nevertheless, none of the existing studies focus directly on the memory footprint, which can be a strong limitation to the practical simulation of large systems. As shown intuitively in Figure 1, for a given level of accuracy, simulation time becomes very long when the simulated manycore exceeds a particular number of cores. The point where performances drastically decrease is reached when the memory requirement of a simulation exceeds the physical memory size of the simulation host. In such a situation, the host system needs to use swapping mechanisms in order to handle the simulation. This drastically increases simulation time.

Even if the memory requirements of the simulation are below the capacity of the host, the swapping limit can be reached depending on the load of the system induced by other running applications. This is particularly true for desktop computers being more and more used to run simulations. Memory footprint is thus a concern that must be taken into account when choosing or building a simulation tool.

Swapping required

Number of cores in simulated manycore Simulation time Fig. 1. Simulation time as a function of the number of cores in the simulated manycore. Because of swapping, simulation time increases drastically when the memory requirements of a simulation exceed the physical memory capacity of the host machine.

In this work, we focus on the simulation of Network on Chip (NoC) communication infrastructures that are increasingly being used in manycore architectures. We measure the memory footprint of different NoC simulators to evaluate when the swapping limit is reached in practice. We compare simulators classified according to two criteria:

• Simulation abstraction defines the level of details taken into account by the simulation. It can be either Transactional Level Modeling (TLM) simulation or cycleaccurate simulation; • Programming abstraction concerns the level of abstraction (and thus the programming language) used to develop a simulator. Our first contribution concerns simulators of NoCs for realtime embedded systems. We empirically compare the memory requirements of two simulators for a real-time NoC using [START_REF] Jiang | A detailed and flexible cycle-accurate network-on-chip simulator[END_REF], a cycle-accurate NoC simulator written in plain C++. In this paper we focus on the comparison of memory footprint depending on simulation and programming abstractions for NoC simulation. However, we believe that the conclusion drawn from the current study apply more generally to full system simulation.

The rest of the paper is organized as follows. Section II first introduces NoCs and simulation and programming abstractions commonly used to build NoC simulators. It then presents related work. Section III compares the memory footprint of McSim-CA and McSim-TLM both simulating the same NoC architecture but using different simulation abstractions. To show the impact of the programming abstraction on memory footprint, we compare in Section IV McSim-CA based on SystemC with BookSim, which is written in C++ only. Finally, Section V concludes and presents perspectives.

II. BACKGROUND AND RELATED WORK

This section first introduces NoCs and the different ways of designing and implementing a simulator for such communication medium. Then, related work is reviewed.

A. Network on Chip (NoC)

NoCs alleviate the bottleneck of usual buses and the prohibitive cost of crossbars in the context of a large number of cores by allowing parallel communications. Figure 2 shows an example of a NoC-based manycore architectures with distributed memory. It is made of tiles communicating with each other by exchanging packets through the NoC. Each tile comprises a router dedicated to receive and forward packets over the NoC. Each tile also comprises a core and a private memory. Among existing NoC topologies, the 2Dmesh illustrated in Figure 2 is one of the most used.

As for distributed systems, routing, control flow and resource arbitration are required in NoCs. Routing consists in deciding which links of the NoC must be used to carry a message between different tiles. Control flow, also known as packet switching, decides the packet allocation to the internal router resources and to the NoC links. Resource arbitration decides in case of conflict between packets, which packet should be given priority. For performance concerns, this control flow and arbitration are often done at the flit level. The flits are the basic elements composing a packet. Many routing algorithms and control flow mechanisms have been proposed in literature [START_REF] Dally | Principles and Practices of Interconnection Networks[END_REF].

Figure 3 shows the router architecture supporting the packet switching and arbitration policies considered in this work. For routing, we consider the XY strategy that first routes packets according to the X axis and then according to Y axis. Each router comprises five ports (some ports may not exist for routers at the boundaries of the mesh). Four ports are used to communicate with neighboring tiles. The extra port, called local port, allows communication with the core of the tile. Each port contains several buffers, called virtual channels, which temporarily store the flits of exchanged packets. These buffers provide a better utilization of the NoC links by storing blocked flits and still making usable the resources of the router and the links of the NoC. The arbitration block handles resource allocation in case of conflicts between packets. Different arbitration policies such as oldest-first or priority-based exist. In this work, we focus on priority-based arbitration [START_REF] Bolotin | QNoC: QoS architecture and design process for network on chip[END_REF]. With such a policy, each virtual channel is associated with a priority level, and the arbiter always gives priority to virtual channels with the highest priority. The priority of a flit is assigned according to the priority of the packet containing the flit. This packet priority is assigned by packet producers (cores) before sending them into the NoC. A credit-based mechanism, not shown in Figure 3 for the sake of simplicity, is used to ensure that routers do not forward flits to neighbors without enough space in their virtual channels.

B. Simulation Abstraction

In order to simulate a NoC in an accurate way regarding timing, many micro-architectural details should be taken into account. Indeed, most of them have an impact on the timing behavior of the NoC. As a consequence, a cycle-accurate simulator, i.e., accurate at the clock cycle granularity, needs to simulate all these details impacting timing. The simulation of all the details affecting accuracy comes at the price of simulation time; the more detailed the system, the longer the simulation. Cycle-accurate simulation also has a cost on the development of the simulator, again because all design details need to be taken into account. To speed up the simulation and to reduce the complexity of building a simulator, the TLM abstraction has been proposed. In TLM, communications between components are represented as high level transactions abstracting away some details impacting timing. In other words, the accuracy of the results is decreased to speedup the simulation. TLM simulators are also often simpler to implement than their cycle-accurate counterpart because less details are taken into account.

The choice of the simulation abstraction depends on the expected result accuracy. In practice, TLM simulators are often built in first stages of hardware design to perform large design space explorations. They are also widely used to start software development before the hardware is available. Later on, cycleaccurate simulators are used in the design process for a more accurate validation of design choices.

C. Programming Abstraction

From the perspective of a simulator developer, beyond the simulation abstraction described above, the programming abstraction should be considered carefully. When building a simulator, at least, three levels of abstraction can be considered as shown in Figure 4.

1) Low level programming languages: At a low level, the developer of a simulator can decide to use a programming language such as C. In this case, all the simulator logic devoted to the management of time and concurrent processes, should be implemented from scratch. This task can result in high simulation speed if it is carefully handled, but it is time consuming and tedious.

2) Object-oriented languages: The next level consists in using higher level programming languages such as object oriented ones. As for low level programming languages, the developer has to manually implement the logic related to time and concurrent processes. Nevertheless, the high level concepts of the language such as classes and inheritance can make this task easier compared to using a language such as C.

3) Simulation frameworks: A higher programming abstraction is provided by simulation frameworks such as Ptolemy or SystemC. The latter, considered in this work, is a C++ based framework widely adopted in both industry and academia. It allows to implement simulators very easily. Indeed, all the management of timing and parallel activities is handled by the framework. To that end, the developer uses a high level Application Programming Interface (API) providing access to concepts such as hardware modules and hardware communication mechanisms such as signals and first-in-first-out buffers.

In addition to the above programming abstractions, further alternatives are not shown in Figure 4. Among them, hardware description languages provide a high accuracy, but they often induce low productivity and simulation speed. On the opposite, analytic models are very fast to simulate but provide low accuracy.

As in any other programming domains not related to simulation, abstractions have a cost regarding performances. In this work we evaluate this cost regarding memory footprint, impacting directly simulation time when swapping is required.

D. Related Work

Among all the cycle-accurate NoC simulators proposed in the past, BookSim [START_REF] Jiang | A detailed and flexible cycle-accurate network-on-chip simulator[END_REF] is a state-of-the-art simulator written in C++ only. It is able to simulate NoCs under different usage conditions. It provides support for configuring the NoC topology, the routing algorithm and the router architecture. NoCTweak [START_REF] Tran | NoCTweak: A highly parameterizable simulator for early exploration of performance and energy of networks on-chip[END_REF] is a SystemC-based NoC simulator focusing on 2D-mesh topologies. It provides results for common performance metrics such as throughput and latency. Noxim [START_REF] Catania | Noxim: An open, extensible and cycle-accurate network on chip simulator[END_REF] is also written in SystemC and allows to configure different parameters regarding the internal micro-architecture of the routers to be simulated. TOPAZ [START_REF] Abad | Topaz: An open-source interconnection network simulator for chip multiprocessors and supercomputers[END_REF] is another simulator that enables the modeling of a wide variety of message routers. It was originally conceived to obtain results very close to those obtained by using hardware description languages.

To speedup simulations of NoC-based systems, TLM concepts can be applied. In this work we restrict the NoCs to be simulated to NoCs with priority preemptive virtual channels. It allows to simulate the NoC 1000 times faster with more than 90% of accuracy [START_REF] Indrusiak | Fast and accurate transactionlevel model of a wormhole network-on-chip with priority preemptive virtual channel arbitration[END_REF], [START_REF] Indrusiak | Fast simulation of networks-on-chip with priority-preemptive arbitration[END_REF]. Horsinka et al. [START_REF] Horsinka | On RTL to TLM abstraction to benefit simulation performance and modeling productivity in NoC design exploration[END_REF] also proposed a TLM NoC simulator integrated in a multi-level simulation framework.

Most of the simulators described above, have been subject to analysis regarding simulation speed. Standalone analyses regarding the simulation speed and accuracy dilemma have also been proposed [START_REF] Schirner | Quantitative analysis of the speed/accuracy trade-off in transaction level modeling[END_REF], [START_REF] Lehtonen | Analysis of modeling styles on network-on-chip simulation[END_REF]. To the best of our knowledge, none of these studies focused on the memory footprint criteria.

III. IMPACT OF SIMULATION ABSTRACTION ON MEMORY FOOTPRINT

We compare the memory footprint of two simulators built in the context of the DreamCloud 2 and3. From the cores perspective, both simulators simulate application models described at a high level. This application model is used only as a packet injector for simulating the NoC traffic and has been described in a previous work [START_REF] Latif | Design space exploration for complex automotive applications: An engine control system case study[END_REF]. McSim-TLM and McSim-CA simulators only differ in the accuracy of the NoC simulator as described in the next two subsections.

A. McSim-TLM

The McSim-TLM simulator is based on existing work [START_REF] Indrusiak | Fast simulation of networks-on-chip with priority-preemptive arbitration[END_REF] and has been implemented in SystemC. To speedup simulations of such NoC architectures we rely on the fact that a packet with high priority will always preempt a packet with lowest priority. Based on this preemption property and considering static routing only, McSim-TLM simulates only the instants at which a packet enters and at which a packet leaves the NoC. Internally, McSim-TLM maintains a map containing the expected leaving time for all the packets into the NoC. Each time a packet enters the NoC, the expected leaving time of all the packets impacted by the new packet is delayed. Impacted packets are the ones with a lower priority and having a route interfering with the one of the new packet. Compared to the simulation of all the events happening inside all the routers of the NoC, McSim-TLM drastically reduces the number of simulated events.

Abstracting the NoC in this way has nevertheless some impact on the accuracy of results. McSim-TLM may consider conflicts that do not occur in practice. These false conflicts come from the fact that packets may not use all their route all the time in practice. The number of these false conflicts mainly depends on the NoC size and the packet length. In the opposite way, the assumption about using the entire route all the time can also lead to scenarios where the simulator ignores some conflicts occurring in practice [START_REF] Indrusiak | Fast simulation of networks-on-chip with priority-preemptive arbitration[END_REF].

B. McSim-CA

For McSim-CA, because of the complexity of writing a cycle-accurate NoC simulator from scratch, we reuse an existing simulator. We opted for NoCTweak [START_REF] Tran | NoCTweak: A highly parameterizable simulator for early exploration of performance and energy of networks on-chip[END_REF] for two main reasons. First, because the NoCTweak simulator is based on SystemC, the integration of the cores simulation engine already developed for McSim-TLM was straightforward. Second, thanks to its modular design, NoCTweak was easily extended to support arbitration based on virtual channels. In McSim-CA, all the internal components of each router, e.g., virtual channels, arbitration logic, are simulated. McSim-CA extends NoCTweak with a new virtual channel allocator and a new switch allocator both based on priority.

C. Experimental Setup

All the experiments described in the following have been performed on a system with an Intel i5-4670 processor, 4 gigabytes of main memory and running Linux. We use the last 2.3.1 version of the Accellera SystemC implementation 4 . In order to monitor memory footprint during each simulation, we periodically read the /proc/PID/smaps virtual file reflecting kernel memory data structures for the process identified by PID. We simulate the same application model using different NoC sizes for both McSim-TLM and McSim-CA using 8 virtual channels with 16 flit entries. The application model has a uniform randomly distributed injection rate of 0.0069. It is worth mentioning that the simulated application and thus the injected NoC traffic, do not impact the memory footprint. Indeed, a SystemC simulation is made of two phases. The first one, usually called elaboration, instantiates all the objects and their connections, i.e., modules and signals in SystemC terminology. The second phase, called execution, consists in effectively running the simulation. Most of the memory allocations take place in the elaboration phase. Thus, considered as static, even if they are dynamic memory allocation from the operating system point of view. As a consequence, the memory footprint measured at the end of the elaboration phase is present until the end of the execution phase. To support this claim, Figure 6 reports the evolution of the memory footprint for the cycle-accurate simulator simulating a 16x16 manycore architecture. From this plot, we see that the memory footprint is growing during elaboration phase and then stays constant up to the end of the simulation. Because memory footprint is almost constant from the end of the elaboration phase to the end of the simulation, we report only the average memory footprint in the following sections. With McSim-TLM, we are able to simulate a platform made of 128x128 cores using one gigabyte of memory while the swapping limit of four gigabytes of our host machine is reached for a manycore with 20x20 cores with McSim-CA. Above this NoC size limit, the simulation time grows exponentially.

D. Results and Analysis

In McSim-TLM, a single object is created to simulate the NoC. The increase of the total memory footprint mainly comes from the creation of additional core objects. On the other hand, in McSim-CA, all internal components of the NoC have an associated object in memory. The simulator creates objects for each router, each virtual channel and each internal component of the routing and packet switching block. So, the total number of these objects and their corresponding memory footprint, is proportional to the size of the 2D-mesh.

From this experiment we see that simulating a 20x20 NoC based manycore with McSim-CA consumes all the four gigabytes of memory of our simulation host. Simulating bigger manycores will require swapping and thus will lead to impractically long simulations. More generally, this study gives additional information that can help the choice of going from TLM to cycle-accurate simulation. In addition to the wellknown simulation speed degradation because of the number of events simulated, cycle-accurate simulation can lead to impractical simulation times if the swapping limit is reached.

IV. IMPACT OF PROGRAMMING ABSTRACTION ON MEMORY FOOTPRINT

In this section, we compare McSim-CA with BookSim [START_REF] Jiang | A detailed and flexible cycle-accurate network-on-chip simulator[END_REF], a state-of-the-art cycle accurate NoC simulator written purely in C++, i.e., not using SystemC.

A. BookSim Configuration

In order to be able to compare McSim-CA with BookSim, we calibrate the latter to simulate as close as possible the same NoC architecture as McSim-CA. We use the following configuration parameters:

• 2D-mesh topology,

• 8 virtual channels with 16 flit entries,

• default virtual channel and switch allocators,

• uniform injection rate. Regarding virtual channel and switch allocators, BookSim does not support the notion of per channel priority compared to McSim-CA. We decided to use the default switch and virtual channel allocator provided in BookSim. Even if this difference could have a strong impact on the accuracy of the results, it does not influence significantly the memory footprint because in both McSim-CA and BookSim we use the same 2D-mesh topology, the same number of virtual channels and the same number of entries in each virtual channel.

Regarding the injection of packets, we can also fairly compare BookSim with McSim-CA even if the latter uses an application model to inject packets into the NoC while the former uses synthetic injection patterns. The memory footprint contributed by the objects corresponding to the NoC architecture is very high compared to the footprint of the objects corresponding to packets. To assess this claim, we performed memory evaluation of BookSim for a 20x20 and a 128x128 manycore architectures under different injection rates. Figure 8 shows the average memory usage of BookSim for injection rates varying from 0.0001 to 1 (for example, an injection rate of 0.20 means each core injects one packet every 5 clock cycles). From this figure, we see that most of the memory used by BookSim is not related to the number of injected packets. Over 1 gigabyte of memory is required run the simulation of the 128x128 architecture with an injection rate as small as 0.0001. The memory footprint then increases slightly up to an injection rate of 1. This slight increase is due to the memory required to store packets into temporary queues at cores level before they are injected into the network. These queues grow infinitely for injection rates above the saturation point of the NoC. Nevertheless, the graph shows that the memory footprint induced by these packets is much lower than that of the objects corresponding to the NoC architecture. Moreover, in the following we report results for an injection rate of 0.0069 because it is the injection rate of the application used with McSim-CA. As shown in Figure 8, the additional memory footprint caused by packet objects is negligible for such a low injection rate.

We report average memory footprint for each simulation. Figure 9 shows that BookSim also performs most of its memory allocations before starting the effective simulation, i.e., most of the memory is allocated at the end of the initialization, which takes about two seconds. Then the memory footprint slightly increases, but not significantly, during the simulation phase up to the end of the simulation (time nine). Because both BookSim and McSim-CA simulators create an internal memory representation for each object in the NoC, and because they simulate the same NoC architecture, the main explanation for this large difference is the individual size of each object type among router, virtual channel, buffer, link, and all the other objects required for the cycle-accurate simulation. We performed a memory analysis of McSim-CA by instrumenting the code to measure the size of all the allocated objects. No memory killer objects were identified even if we were able to save few megabytes by removing some object fields that were not strictly required. Further investigations on BookSim implementation are required to have a better understanding of this big gap, nevertheless our study leads to the conclusion that this big memory footprint is the result of a sub-optimized usage of SystemC. Indeed, to easily write a cycle-accurate NoC simulator, one is tempted to use extensively all the abstractions provided by the API, such as signals and buffers.

B. Results and Analysis

The main conclusion we draw from this comparison is that the programming abstraction has a strong impact on the memory requirement for a cycle-accurate simulator. Large memory footprint has an impact on the simulation when swapping is required. Depending on the size of the simulated architecture, memory footprint must be an important criterion for the choice of the programming abstraction. Moreover, it is worth mentioning that the memory limitation presented for NoC simulator in this work will be even more important for a full system simulator including cores. The swapping limit reached for a 20x20 manycore architecture with McSim-CA on our simulation host machine will be reached for smaller architectures in the case where McSim-CA also simulates cores accurately. The 4 gigabytes footprint of 20x20 simulations is also a problem when they run on server machines. On such machines, the memory available per core is few gigabytes by default. Running one simulation on each core of the system at the same time may also lead to the swapping limit.

V. CONCLUSION AND PERSPECTIVES

This work presents an evaluation of the memory footprint of three NoC simulators that differ from the accuracy point of view and from the programming abstraction used to implement them. The study provides quantitative results to support the natural intuition that TLM NoC simulators also save memory in addition to time. By comparing the memory footprint of two cycle-accurate simulator based on different programming abstractions, we also evaluated the cost of using programming abstraction provided by framework such as SystemC. This study concludes that depending on the size of the NoC to be simulated, great care must be taken when building or choosing a NoC cycle-accurate simulator. Even if this study focuses on NoC simulators, we believe that the tendencies of the results presented would be similar for the simulation of other hardware components. In the case of a full system cycle-accurate simulator, the swapping limit would be reach earlier because of all the memory used to simulate the cores behavior accurately.

From this work, we think that it will be very interesting to study how existing SystemC model could benefit or not from lazy memory allocations. For large systems, instantiating a lot of components that are not used from the beginning of the simulation, such lazy allocation mechanisms could help for saving memory.

Fig. 2 .

 2 Fig.2. Example of a 2D-mesh NoC-based manycore architecture with distributed memory. The architecture is made of tiles, each comprising a core, a private memory and a router allowing communication with other tiles.

Fig. 4 .

 4 Fig. 4. Programming abstractions for hardware simulation. As in any domain, abstraction has a cost regarding performance.

Fig. 5 .

 5 Fig. 5. Accuracy comparison between McSim-TLM and McSim-CA. McSim-TLM can under and over estimate the real contention occurring in the NoC.

  Figure 5 illustrates these inaccuracies by showing the comparison of the execution time of the same application simulated with McSim-TLM and McSim-CA. In all the different NoC sizes except for 2x2, the McSim-TLM simulator considers conflicts that do not occur in practice leading to a higher execution time than the one reported by McSim-CA. In the 2x2 configuration, we face the opposite situation where the TLM abstractions lead to ignoring conflicts occurring in practice.

Fig. 6 .

 6 Fig. 6. Memory footprint of McSim-CA over time for during the simulation of a 16x16 manycore architecture. All the memory allocations are performed during the elaboration phase and never freed until the end of the simulation.

Figure 7

 7 Figure 7 reports average memory footprint for both simulators according to the NoC size. The vertical axis reports average memory footprint in megabytes on a logarithmic scale while the horizontal axis shows different NoC sizes. As expected, this plot shows that McSim-TLM uses far less memory than McSim-CA. Nevertheless, the memory footprint of McSim-CA is surprisingly high. It uses around 3.8 gigabytes of memory to simulate a 20x20 platform while McSim-TLM only uses 31 megabytes which is 121 times lower.With McSim-TLM, we are able to simulate a platform made of 128x128 cores using one gigabyte of memory while the swapping limit of four gigabytes of our host machine is reached for a manycore with 20x20 cores with McSim-CA. Above this NoC size limit, the simulation time grows exponentially.In McSim-TLM, a single object is created to simulate the NoC. The increase of the total memory footprint mainly comes from the creation of additional core objects. On the other hand,

Fig. 7 .

 7 Fig. 7. Average memory footprint of McSim-TLM and McSim-CA. The memory footprint is expressed in megabytes on a logarithmic scale. Memory footprint of McSim-TLM is really small compared to the one of McSim-CA which is almost 4 gigabytes for a 20x20 NoC.

Figure 10 Fig. 8 .Fig. 9 .

 1089 Figure 10 shows the results of the average memory footprint of BookSim compared to McSim-CA. For McSim-CA the results are the same as the ones presented in Section III. From this plot it is clear that BookSim memory footprint is much lower than the one of McSim-CA. BookSim can simulate manycore architecture with 128x128 (i.e., 16384) cores using

Fig. 10 .

 10 Fig. 10. Average memory footprint of McSim-CA and BookSim. Footprint is expressed in megabytes on a logarithmic scale. BookSim, written in plain C++, uses far less memory than McSim-CA which is based on SystemC.

  978-1-5090-2520-6/16/$31.00 c 2016 IEEE different simulation abstractions. The first simulator, named McSim-CA, is cycle-accurate while the second one, named McSim-TLM is based on TLM concepts. The second contribution of this work compares the memory footprint of two cycle-accurate NoC simulators built upon different programming abstractions. We compare McSim-CA based on SystemC with BookSim

  1 European project. Both simulators are included in the Manycore platform Simulation (Mc-Sim) tool suite. They simulate the same NoC-based manycore architecture, but they use different simulation abstractions. The fastest simulator, called McSim-TLM 2 , represents the NoC at a transactional level. The second one, called McSim-CA 3 , is cycle-accurate. It handles all the internals of the NoC architecture impacting timing. Both McSim-TLM and McSim-CA simulate the 2D-mesh NoC architecture depicted by Figures

http://www.accellera.org/downloads/standards/systemc

ACKNOWLEDGMENT

The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under the DreamCloud Project: http: //www.dreamcloud-project.org, grant agreement no. 611411.


Penny S Loosier

Michael Windle

Eun Young Mun

M Hadj-Bachir
email: hadjbachir@celia.u-bordeaux1.fr

R Capdessus

E D'hummieres

V T Tikhonchuk

Youth, Family, and Contextual Characteristics Predicting Violence Exposure: Disruptive Behavior Disorder Symptoms as a Moderator

Introduction Modeling shocks in astrophysical laboratories

Distribution functions of the electrons.

Asymetric case 10 keV / 100 eV

Colliding plasmas in universe.

PIC simulations of sub-relativistic plasmas

We present here an analysis of a very large PIC simulations of two inter-penetrating plasmas at sub-relativistic velocities. At the initial time moment in the center-of-mass reference frame two homogenous plasmas of the density are facing each other in the simulation plane (x,y). All lengths are measured in terms of electron inertia length, , the time is in inverse electron plasma frequencies, , the energy is measured in units of electron rest energy, ,the electric and magnetic fields is are in relativistic plasma units, , and , respectively.

An experiment for studying the collisionnelss mechanism of fast ion energy in laboratory plasma is proposed. We have performed two dimensional PIC simulations of electron-ion sub-relativistic counter-streaming plasma beams propagating without an external magnetic field. Electron heating is an important stage of shock formation. This is probably due to stochastic processes that occurs due to the strong charge separation. Radiation emission losses is due to electron synchrotron emission [START_REF] Medvedev | Radiation of in Weibel-generted fields: a general case[END_REF]. Simulations of similar physical processes at the GRB scale are ongoing.

Evolution of filaments

Conclusion

Ions distribution functions

The ion streams are mixing creating and modifing the electric fields in the interaction zone.

The filament merges and are drawn in transverse direction.

The lifetime of a filament is

  Scheme of laboratory experiment on the collisionless dissipation of fast ion beams [1].

  Very high energy of electrons in the filament. Strong electrostatic field due to the charge separation. Collisionless shocks are frequent events in the universe. They transform energy of star explosion into hot plasma, high energy particules and radiation. It has been shown that the Weibel instability is a very good candidate to explain GRBs [7]. We demonstrate with particule-in-cell simulations that a part of directed kinetic energy of proton beams is converted into energy of electron, ions, electric and magnetic fields [1]. We study the shock structure and energetics of flow and we demonstrate that electron heating is an important catalyst to shock formation via filamentation instability [3], [4]. We use the 2D version of PICLS. For these kind of numerical simulation we use at least :160 cores, 14500 *1600 cells of plasma , 32 of macro-particules per cell. the time step is .Time computation is typically 2 weeks.


Michaël Peigney
email: michael.peigney@enpc.fr

ON SHAKEDOWN OF ELASTIC-PLASTIC BODIES WITH TEMPERATURE DEPENDENT PROPERTIES

Keywords: Plasticity, Shakedown, Temperature-dependent properties, Melan's theorem

For elastic-perfectly plastic structures under prescribed loading histories, the wellknown Melan's theorem gives a sufficient condition for shakedown to occur, i.e. for the evolution to become elastic in the large-time limit. The original Melan's theorem rests on the assumption that the material properties remain constant in time, independently on the applied loading. This communication addresses the long standing issue of extending Melan's theorem to temperaturedependent (or time-dependent) elastic moduli. The main motivation is to extend the range of applications of Melan's theorem to thermomechanical loading histories with large temperature fluctuations: In such case, the variation of the elastic properties with the temperature cannot be neglected. Some recent results obtained in perfect plasticity and viscoplasticity are presented and discussed.

INTRODUCTION

For elastic-perfectly plastic structures under prescribed loading histories, the well-known Melan's theorem [START_REF] Melan | Theorie statisch unbestimmter systeme aus ideal-plastischen baustoff[END_REF][START_REF] Symonds | Shakedown in continuous media[END_REF][START_REF] Koiter | General problems for elastic solids[END_REF] gives a sufficient condition for shakedown to occur, i.e. for the evolution to become elastic in the large-time limit. Intuitively, shakedown means that the plastic strain tends to a limit as time tends to infinity. The Melan's theorem has the distinctive property of being path-independent, i.e. independent on any residual stress that may exist initially in the structure. The shakedown theory has been the object of numerous developments, regarding both extensions of the original theorem to various nonlinear behaviors [START_REF] Pham | On shakedown theory for elastic-plastic materials and extensions[END_REF][START_REF] Nguyen | On shakedown analysis in hardening plasticity[END_REF][START_REF] Ahn | Shakedown of coupled two-dimensional discrete frictional systems[END_REF][START_REF] Peigney | Shakedown theorems and asymptotic behaviour of solids in non-smooth mechanics[END_REF][START_REF] Peigney | On shakedown of shape memory alloys structures[END_REF][START_REF] Weichert | A historical view on shakedown theory[END_REF] and numerical methods for assessing the shakedown limits in the case of parametrized loading histories [START_REF] Zarka | A new approach to inelastic analysis of structures[END_REF][START_REF] Maitournam | Détermination de la réponse asymptotique d'une structure anélastique sous chargement cyclique[END_REF][START_REF] Carvelli | Shakedown analysis of defective pressure vessels by a kinematic approach[END_REF][START_REF] Peigney | Approche par contrôle optimal des structures élastoviscoplastiques sous chargement cyclique[END_REF][START_REF] Peigney | An optimal control approach to the analysis of inelastic structures under cyclic loading[END_REF][START_REF] Simon | Shakedown analysis of engineering structures with limited kinematical hardening[END_REF][START_REF] Spiliopoulos | A direct method to predict cyclic steady states of elastoplastic structures[END_REF].

This communication addresses the issue of extending Melan's theorem to temperature-dependent (or time-dependent) elastic moduli. The main motivation is to extend the range of applications of Melan's theorem to thermomechanical loading histories with large temperature fluctuations: In such case, the variation of the material properties with the temperature cannot be neglected. Whereas the case of temperature-dependent yield limits is well understood [START_REF] Borino | Consistent shakedown theorems for materials with temperature dependent yield functions[END_REF], the case of temperature-dependent elastic moduli remains a long standing issue and has been the object of several conjectures [START_REF] König | A shakedown theorem for temperature dependent elastic moduli[END_REF][START_REF] Halphen | Evolution des structures élastoplastiques dont les coefficients d'élasticité dépendent de la température, 17ème congrès franc ¸ais de mécanique[END_REF][START_REF] Hasbroucq | Inelastic responses of a two-bar system with temperature-dependent elastic modulus under cyclic thermomechanical loadings[END_REF][START_REF] Hasbroucq | Analytical study of the asymptotic behavior of a thin plate with temperature-dependent elastic modulus under cyclic thermomechanical loadings[END_REF]. The main difficulty is that the proof used in the original Melan's theorem -as well as in most of its knows extensions -crucially relies on some contraction properties that are lost when the elastic moduli are allowed to vary in time. A shakedown theorem has recently been proposed for elastic-perfectly plastic materials with time-periodic elastic properties [START_REF] Peigney | Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli[END_REF]. The statement and proof of that theorem differ significantly from the case of constant material properties. A salient result is that time fluctuations of the elastic moduli need to remain small (in a certain sense) for shakedown to occur in a path-independent fashion.

In this communication, the theorem of [START_REF] Peigney | Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli[END_REF] is presented and improved upon. The extension to elastic-viscoplastic materials is discussed.

EVOLUTION EQUATION FOR THE STRESS FIELD

Consider an elastic-plastic material occupying a domain Ω in the reference configuration. Under the small strains assumption, the strain ε, stress σ and plastic strain ε p at position x and time t are related by the constitutive equations

ε(x,t) = L(x,t) : σ (x,t) + ε θ (x,t) + ε p (x,t), (1)
ε p (x,t) = φ (σ (x,t), x,t), ( 2
)
where φ is the dissipation potential, assumed to be positive, convex, and null at the origin. In (1), L is the (symmetric positive definite) elastic moduli tensor and ε θ is the thermal strain tensor. Note the time dependence of L, ε θ and φ , which may result from imposed variations of the temperature. The elastic domain of the material is denoted by C (x,t), i.e. C (x,t) = {σ : φ (σ , x,t) = 0}. Assuming quasi-static evolutions, the stress field σ satisfies the equilibrium equations

div σ + f = 0 in Ω, σ • n = T on ∂ Ω T , (3)
where f (x,t) are body forces and T (x,t) are tractions prescribed on a part ∂ Ω T of the boundary ∂ Ω. The strain field ε and displacement field u satisfy

ε = 1 2 (∇u + ∇ T u) in Ω, u = v on ∂ Ω u = ∂ Ω -∂ Ω T (4)
where v(x,t) is a given function.

As usual in shakedown analysis, it is convenient to introduce the fictitious elastic response (u E , σ E ), i.e. the response of the system if it were purely elastic, defined by

ε E = L : σ E + ε θ , div σ E + f = 0 in Ω, σ E • n = T on ∂ Ω T , ε E = 1 2 (∇u E + ∇ T u E ) in Ω, u E = v on ∂ Ω u . (5)
The boundary value problem defined by Eqs. (1-4) governs the evolution of the fields (u, ε, σ ). An evolution equation involving only the stress field can be formulated from Eqs.

(1-4), as is now detailed. Let E denote the space of stress fields, chosen as a subspace of symmetric second-order tensor fields with square-integrable components. The space E is equipped with the scalar product σ , σ = Ω σ (x) : σ (x)dω. The associated norm is denoted by , i.e. σ = σ , σ . For simplicity we assume in the following that E is of finite dimension. Any stress field σ satisfying (3) can be written as σ = σ E + ρ where ρ belongs to the vectorial space H ⊂ E of self-equilibrated stress fields, defined by

H = {ρ ∈ E : div ρ = 0 in Ω, ρ • n = 0 on ∂ Ω T }. (6)
For an arbitrary ρ ∈ H, it follows from (1) that

Ω (ρ -ρ) : d(ε -ε E ) dt dω = Ω (ρ -ρ) : d dt (L : ρ) dω + Ω (ρ -ρ) : φ (σ ) dω. ( 7
)
The principle of virtual power shows that the left-hand side of ( 7) is equal to 0, hence

- Ω (ρ -ρ) : d dt (L(x,t) : ρ) dω = Ω (ρ -ρ) : φ (σ , x,t) dω. (8)
Let Φ(ρ,t) : H → R be the convex, positive function defined by

Φ(ρ,t) = Ω φ (σ E (x,t) + ρ(x), x,t) dω. Note that Φ(•,t) vanishes on the set K(t) = {ρ ∈ H : ρ(x) + σ E (x,t) ∈ C (x,t) ∀x ∈ Ω}. For later reference, note that K(t) = K 0 (t) ∩ H where K 0 (t) = {ρ ∈ E : ρ(x) + σ E (x,t) ∈ C (x,t) ∀x ∈ Ω}.
We have (at least formally)

Ω (ρ -ρ) : φ (σ ) dω = ρ -ρ, Φ (ρ,t) .
Further introducing the linear symmetric operator L(t) : H → H defined by

ρ, L(t)ρ = Ω ρ(x) : L(x,t) : ρ (x,t) dω ∀ρ, ρ ∈ H,
Eq. ( 8) can be rewritten as

- d dt (L(t)ρ) = Φ (ρ,t). (9)
Since the elastic moduli tensor L(x,t) is symmetric positive definite, it can easily be verified that L(t) is self-adjoint and positive definite. Its inverse is denoted by M(t). Starting from a given initial state ρ(t = 0), the evolution of the stress field in H is governed by the ordinary differential equation [START_REF] Weichert | A historical view on shakedown theory[END_REF]. The uniqueness of the stress rate ρ has been proved in [START_REF] Halphen | Elastic perfectly plastic structures with temperature dependent elastic coefficients[END_REF].

In the case of time-independent elastic moduli, a change of scalar product allows Eq. ( 9) to be put in the form

- dρ dt = Φ (ρ,t). (10)
Properties of Eq. ( 10) are well understood. In particular, the distance between two solutions (as measured by the energy norm) is decreasing with time: For two solutions ρ(t) and ρ (t) of ( 10), we have indeed

1 2 d dt ρ(t) -ρ (t) 2 = ρ -ρ , ρ -ρ = -Φ (ρ,t) -Φ (ρ ,t), ρ -ρ ≤ 0 ( 11
)
where the convexity of Φ(•,t) has been used. The contraction property (11) plays a crucial role when studying the asymptotic behaviour of solutions as t → +∞. If L(t) is a radial function of time, i.e. of the form

L(t) = f (t)L 0 (12)
then a change of variables allows Eq. ( 9) to be cast in the form (10) [START_REF] Peigney | Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli[END_REF]. In the general case when L(t) is not radial, then the properties of Eq. ( 9) differ significantly from those of Eq. ( 10) -In particular, the contraction property is lost.

In the following it will be useful to measure 'how much' L(t) (or M(t)) differs from a radial function. To that purpose, set

µ(t) = 1/ tr M(t) , M 0 (t) = µ(t)M(t),
and define

γ(a, b) = b a ||| Ṁ0 (t)||| dt (13)
where

|||•||| is the norm operator in H, i.e. |||M(t)||| = sup ρ ∈H, ρ =1 M(t)ρ . Observe that γ(a, v) vanishes if L(t) is of the form (12) on the time interval [a, b],
and is non negative otherwise.

Remark: For simplicity, we have assumed so far that the dissipation potential φ is differentiable, which is the case for viscoplasticity. For perfect plasticity, the dissipation potential is only subdifferentiable: Eqs. ( 9) and [START_REF] Zarka | A new approach to inelastic analysis of structures[END_REF] have to be replaced with

- d dt (L(t)ρ) = ∂ I K(t) (ρ,t) (14)
and

- dρ dt = ∂ I K(t) (ρ,t), (15)
respectively. In Eqs (14-15), I K(t) is the indicator function of the set K(t) and ∂ denotes the subdifferential operator. Eq. ( 15) corresponds to the so-called 'sweeping process' [START_REF] Moreau | Evolution problem associated with a moving convex set in a hilbert space[END_REF].

SHAKEDOWN THEOREM

In the following we study the asymptotic behavior of solutions to (9) as t → ∞. We consider the case where L(t), K(t) are periodic in time (with the same period T ) and assume that K 0 (t) and L(t) are bounded by some constants denoted respectively by M and C, i.e.

ρ ≤ M , |||L(t)||| ≤ C (16)
for all t and ρ ∈ K 0 (t). As mentioned in the introduction, shakedown corresponds to the situation where the evolution becomes elastic in the large time limit. Elastic evolutions are characterized by the fact that φ (σ (x,t), x,t) = 0. Concerning the evolution equation ( 9) for the stress field, it follows that the distinctive properties of an elastic solution ρ * (t) are

d dt (L(t)ρ * (t)) = 0, ρ * (t) ∈ K(t) ∀t ∈ [0, T ]. (17)
Shakedown corresponds to the situation where the residual stress converges towards an elastic solution as t → ∞. Our main objective is to examine conditions under which every solution of (9) converges towards an elastic solution in the large time limit. More precisely, the main result we discuss is the following Shakedown theorem: If (i) There exists an elastic solution ρ * to Eq. ( 9) and some r > 0 such that

∀ρ ∈ H, ∀t ∈ [0, T ], ρ -ρ * (t) ≤ r =⇒ ρ ∈ K(t);
(ii) the elastic moduli are such that

γ(0, T ) inf µ < r 2CM ;
then shakedown occurs, whatever the initial state is.

That theorem holds for elastic-perfectly plastic materials as well as for Duvaut-Lions elastoviscoplastic materials (see Sect. 3.3). Condition (i) means that there exists an elastic solution that remains in the interior of the elasticity domain K(t): It is a reformulation of the Melan's condition found in the classical Melan's theorem. Condition (ii) sets restriction on the time variations of the elastic moduli. Setting such a restriction is necessary to obtain global convergence results: One can indeed find counterexamples in which condition (i) is fulfilled and some solutions of ( 9) do not become elastic in the large time limit [START_REF] Peigney | Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli[END_REF].

Variation of the elastic energy

Let ρ * (t) be an elastic solution satisfying (i) and let ρ(t) denotes an arbitrary solution of (9). We consider the positive function f defined as

f (t) = 1 2 τ(t), L 0 (t)τ(t) . ( 18
)
where τ(t) = µ(t)(ρ(t)ρ * (t)). The function f is referred to as the elastic energy. The shakedown theorem stated above rests critically on the property

f (a + T ) -f (a) ≤ 2Nγ(0, T ) -r inf µ a+T a η(t) dt, (19)
where

N = sup a≤t≤a+T { L(t)ρ(t) , L(t)ρ * (t) }.
Eq. ( 19) bounds the variation of the elastic energy over a cycle. That bound plays the role of the contraction property that holds only in the case of time-independent elastic moduli. Observe in particular from ( 19) that f (a + T )f (a) ≤ 0 if γ(0, T ) is sufficiently small. The rest of this Section is devoted to a proof of Eq. ( 19). The inequality ( 19) is obtained by performing minor modifications to a similar result obtained in [START_REF] Peigney | Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli[END_REF] for perfect plasticity, hence we only sketch the main steps of the proof and refer to [START_REF] Peigney | Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli[END_REF] for details. Recalling that d(L(t)ρ * (t))/dt = 0 and setting

H(t) = -1 2 τ(t), L0 (t)τ(t) , some straightforward manipula- tion shows that ḟ (t) = τ(t), η(t) + H(t)
where η(t) = L(t)τ(t). Integrating on the time interval [a, a + T ] gives

f (a + T ) -f (a) = a+T a τ(t), η(t) dt + a+T a H(t) dt. ( 20
)
In the following, the two integrals in the right-hand side of Eq. ( 20) are bounded separately. Let us first consider the integral a+T a τ(t), η(t) dt. Noting from ( 9) that η(t) = -Φ (ρ(t),t) and recalling that Φ is convex, positive and vanishes in K(t), we have

0 ≤ Φ(ρ(t),t) ≤ η(t), ρ (t) -ρ(t) for all ρ ∈ K(t). (21)
Condition (i) implies that ρ * (t) -r η/ η is in K(t). Hence (21) gives 0 ≤ η(t), ρ * (t) - r η η -ρ(t) i.e. η(t), ρ(t) -ρ * (t) ≤ -r η(t) . Therefore we obtain a+T a τ(t), η(t) dt ≤ -r(inf µ) a+T a η dt. ( 22
)
Now consider the integral a+T a H(t) dt. We have τ(t) = L -1 0 (t)(η(t)η(a) + s) where s = L(a)(ρ(a)ρ * (a)) is independent on t. Some manipulations (see [START_REF] Peigney | Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli[END_REF]) show that

a+T a H(t)dt ≤ 1 2 a+T a η(t) -η(a) •||| Ṁ0 (t)||| • η(t) -η(a) + 2s dt. (23)
Since

η(t) -η(a) + 2s = L(t)τ(t) + L(a)τ(a) -2L(a)ρ * (a), we have η(t) -η(a) + 2s ≤ 4N. ( 24
)
We also have, for t ∈ [a, a + T ],

η(t) -η(a) = t a η(t )dt ≤ t a η(t ) dt ≤ a+T a η(t ) dt . (25)
Substituting [START_REF] Moreau | Evolution problem associated with a moving convex set in a hilbert space[END_REF][START_REF] Peigney | On Melan's theorem in temperature-dependent viscoplasticity[END_REF] in [START_REF] Halphen | Elastic perfectly plastic structures with temperature dependent elastic coefficients[END_REF] and using the definition (13) of γ(a, b), we obtain

a+T a H(t)dt ≤ 2Nγ(a, a + T ) a+T a η(t) dt. (26)
Replacing [START_REF] Peigney | Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli[END_REF][START_REF] Halphen | Elastic perfectly plastic structures with temperature dependent elastic coefficients[END_REF][START_REF] Moreau | Evolution problem associated with a moving convex set in a hilbert space[END_REF][START_REF] Peigney | On Melan's theorem in temperature-dependent viscoplasticity[END_REF](26) in [START_REF] Halphen | Evolution des structures élastoplastiques dont les coefficients d'élasticité dépendent de la température, 17ème congrès franc ¸ais de mécanique[END_REF] and observing by periodicity that γ(a, a + T ) = γ(0, T ), Eq. ( 19) is obtained.

Case of elastic-perfectly plastic materials

In perfect plasticity, the stress is constrained to remain in the elasticity domain C (x,t), which by [START_REF] Spiliopoulos | A direct method to predict cyclic steady states of elastoplastic structures[END_REF] implies that ρ(t) ≤ M and ρ * (t) ≤ M for all t. By ( 16) we thus have L(t)ρ(t) ≤ CM and L(t)ρ * (t) ≤ CM. In [START_REF] Halphen | Evolution des structures élastoplastiques dont les coefficients d'élasticité dépendent de la température, 17ème congrès franc ¸ais de mécanique[END_REF], the constant N thus satisfies N ≤ CM. Applying Eq. ( 19) on the time interval [iT, (i + 1)T ] gives

f ((i + 1)T ) -f (iT ) ≤ -m (i+1)T iT η(t) dt where m = r(inf µ) -2CMγ(0, T ) is non-negative by (ii). Summing over i = 0, • • • , I gives IT 0T η(t) dt ≤ 1 m f (0).
Taking the limit I → ∞ shows that the integral S 0 η(t) dt converges as S -→ +∞. It follows that η(t) also converges towards a limit η ∞ as t -→ +∞. Setting ρ ∞ (t) = M(t)η ∞ , it can be checked that ρ(t)ρ ∞ (t) -→ 0 as t → ∞, and that ρ ∞ (t) is an elastic solution.

Remark: For perfect plasticity, the theorem stated above slightly improves on the result obtained in [START_REF] Peigney | Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli[END_REF]. In [START_REF] Peigney | Shakedown of elastic-perfectly plastic materials with temperature-dependent elastic moduli[END_REF], a condition analog to (ii) was formulated, with the factor 2 in (ii) replaced by a factor 3.

Case of elastic-viscoplastic materials

Compared to perfect plasticity, an additional difficulty is that the stress is not restricted to remain in the elasticity domain: There is no direct bound on ρ(t), hence no direct on the constant N in [START_REF] Halphen | Evolution des structures élastoplastiques dont les coefficients d'élasticité dépendent de la température, 17ème congrès franc ¸ais de mécanique[END_REF] . That difficulty, however, can be overcome in the case of a Duvaut-Lions viscoplastic behavior, i.e. for a dissipation potential of the form

φ (σ , x,t) = α 2 |σ -P C (x,t) σ | 2 (27)
where α > 0 is a viscosity parameter (assumed to be independent on (x,t) for simplicity) and P C (x,t) denotes the projection on the elasticity domain C (x,t). Full details will be published elsewhere [START_REF] Peigney | On Melan's theorem in temperature-dependent viscoplasticity[END_REF], but the main point is that η(t) = L(t)ρ(t) is bounded by a constant N for large time. Moreover, that constant N can be chosen as arbitrary close to CM. Using [START_REF] Halphen | Evolution des structures élastoplastiques dont les coefficients d'élasticité dépendent de la température, 17ème congrès franc ¸ais de mécanique[END_REF] with a value of N such that CM < N < (r inf µ)/2γ(0, T ), one can show that η(t) converges as t → ∞ and consequently that ρ(t) converges towards an elastic solution (the reasoning is similar to that presented in Sect. 3.2).

EXAMPLE

As an illustrative example, consider the 3-bar truss represented in Fig. 1. We denote by σ i , ε i , ε p i the uniaxial stress, total strain and plastic strain in bar i. The constitutive relation in each bar reads as

ε i = σ i E i (t) + ε p i + ε θ i , ε p i = φ (σ i ).
The truss is stress-free, i.e. σ 1 + σ 2 + σ 3 = 0. The loading consists of a thermal strain ε θ 2 (t) = a sin(2πt/T ) in bar 2. We assume that E 1 = E 3 remains independent on time and that E 2 (t) vary as E 2 (t) = E/(1 + δ sin 2 (πt/T )). The elasticity domain is the same for all bars and taken as [-σ y , σ y ] where the yield limit σ y is independent on temperature. For the problem at hand, the vectorial space E of stress fields identifies with R 3 and is equipped with the scalar product σ , σ = ∑ 3 i=1 σ i σ i . Writing the stress state of the system as σ = (σ 1 , σ 2 , σ 3 ), the fictitious elastic response of the system is

σ E (t) = E ε θ 2 (t) 3 + 2δ sin 2 (πt/T ) (1, -2 , 1).
It follows that σ E (t) ≤ √ 6σ θ /3 where σ θ = E sup t |ε θ 2 (t)|. Any σ in K 0 verifies σ +σ E ≤ √ 3σ y . Hence the constant M in ( 16) can be taken as

M = √ 3σ y + √ 6σ θ /3.
In the present case, the space H of self-equilibrated stress fields is of dimension 2. Calculating the operator L(t) shows that

C = 3 + 2δ 3E , γ(0, T ) = δ 3 + δ , inf µ = 1 2E .
Note that (0, 0, 0) is an elastic solution. As long as σ θ /σ y ≤ 3/2, it can be verified that (0, 0, 0) satisfies condition (i) with a value or r given by

r = ( σ y 2 - σ θ 3 ) √ 6.
In Fig. 2 are represented the values (σ θ , δ ) satisfying conditions (i)-(ii): Shakedown occurs for any (σ θ , δ ) below the solid line shown in Fig. 2. For δ = 0, we obtain that shakedown occurs provided that σ θ ≤ 3/2σ y , in agreement with the classical Melan's theorem for timeindependent moduli. As can be observed on Fig. 2, the time fluctuations of the elastic moduli result in a decrease of the shakedown limit on σ θ . Those results apply both to elastic-perfectly plastic materials and to elasto-viscoplastic material of the Duvaut-Lions type. In Fig. 3 are plotted the incremental evolution of the plastic strain and residual stress in bar 2, starting from an initial residual stress ρ(0) = (0.46, -0.53, 0.07)E. The plots on the left correspond to a elastic-perfectly plastic behavior, while the plots on the right correspond to a elastic-viscoplastic behavior. We set δ = 0.1 and ε θ (t) = 0.1(σ y /E) sin(2πt/T ): Such values are below the shakedown limit shown in Fig. 2. As expected from the shakedown theorem, we observe the convergence of the plastic strain towards a stabilized value. In the case of perfect plasticity, that convergence is reached after the first loading cycle. In contrast,the convergence is slow in the case of viscoplasticity. Observe that the residual stress does not converge towards a stabilized value: This is a direct consequence of the time dependence of the elastic moduli.

Figure 1 :

 1 Figure 1: Three-bar truss with large thermal loading.

Figure 2 :

 2 Figure 2: Shakedown limit for the three-bar truss.

Figure 3 :

 3 Figure 3: Residual stress and plastic strain in bar 2: (left) elastic-perfectly plastic material; (right) elasticviscoplastic material