Prototype P4.26, Report R4.25 Parallelization with increased performance based on model partitioning with TLM-couplings December, 2012 Peter Fritzson, Mahder Gebremedhin, Martin Sjölund (LIU) ••••••••••••••••••••••••••••••••••••••••••••• This document is public. Deliverable OPENPROD (ITEA 2) Page 2 of 2 Summary This deliverable includes two prototypes using two different approaches to parallelism, described in the following papers. Both prototypes are implemented within OpenModelica. Paper 1, “A Data-Parallel Algorithmic Modelica Extension for Efficient Execution on Multi-Core Platforms” describes a parallel extension to the algorithmic part of the Modelica. This extension is fully integrated in the OpenModelica, and was used in a parallel programming tutorial at the Modelica conference in Munich, December 2012. Code written in this language extension is compiled to the OpenCL parallel programming C-style language, which is portable both to multi-core GPUs and CPUs. Speedup up to 300 for large problems has been achieved for some applications. Paper 2, “TLM and Parallelization”, describes a way of using transmission line modeling to partition equation-based model, thus enabling the parts to be simulated partly in parallel. Transmission line modeling (TLM) is a technique where the wave propagation of a signal in a medium over time can be modeled. The propagation of this signal is limited by the time it takes for the signal to travel across the medium. By utilizing this information it is possible to partition the system of equations in such a way that the equations can be partitioned into independent blocks that may be simulated in parallel. This leads to improved efficiency of simulations since it enables taking advantage of most of the full performance of multi-core CPUs. An early prototype implementation has been developed in OpenModelica where the Modelica delay() built-in function is used to introduce TLM-style decoupling between model parts, which is then detected by the compiler for parallelization purposes. Publications and Reports included in this Document 1. Mahder Gebremedhin, Afshin Hemmati Moghadam, Peter Fritzson, Kristian Stavåker. A Data-Parallel Algorithmic Modelica Extension for Efficient Execution on Multi-Core Platforms. In Proceedings of the 9th International Modelica Conference (Modelica'2012), Munich, Germany, Sept.3-5, 2012 2. Martin Sjölund. TLM and Parallelization. Internal Research Report, Linköping University, December 2012. OPENPROD A Data-Parallel Algorithmic Modelica Extension for Efficient Execution on Multi-Core Platforms Mahder Gebremedhin, Afshin Hemmati Moghadam, Peter Fritzson, Kristian Stavåker Department of Computer and Information Science Linköping University, SE-581 83 Linköping, Sweden {mahder.gebremedin, peter.fritzson, Kristian.stavaker}@liu.se, [email protected] Abstract New multi-core CPU and GPU architectures promise high computational power at a low cost if suitable computational algorithms can be developed. However, parallel programming for such architectures is usually non-portable, low-level and error-prone. To make the computational power of new multi-core architectures more easily available to Modelica modelers, we have developed the ParModelica algorithmic language extension to the high-level Modelica modeling language, together with a prototype implementation in the OpenModelica framework. This enables the Modelica modeler to express parallel algorithms directly at the Modelica language level. The generated code is portable between several multi-core architectures since it is based on the OpenCL programming model. The implementation has been evaluated on a benchmark suite containing models with matrix multiplication, Eigen value computation, and stationary heat conduction. Good speedups were obtained for large problem sizes on both multi-core CPUs and GPUs. To our knowledge, this is the first high-performing portable explicit parallel programming extension to Modelica. Keywords: Parallel, Simulation, Benchmarking, Modelica, Compiler, GPU, OpenCL, Multi-Core 1 Introduction Models of large industrial systems are becoming increasingly complex, causing long computation time for simulation. This makes is attractive to investigate methods to use modern multi-core architectures to speedup computations. Efficient parallel execution of Modelica models has been a research goal of our group for a long time [4], [5], [6], [7], involving improvements both in the compilation process and in the run-time system for parallel execution. Our previous work on compilation of dataparallel models, [7] and [8], has primarily addressed compilation of purely equation-based Modelica models for simulation on NVIDIA Graphic Processing Units (GPUs). Several parallel architectures have been targeted, such as standard Intel multi-core CPUs, IBM Cell B.E, and NVIDIA GPUs. All the implementation work has been done in the OpenModelica compiler framework [2], which is an open-source implementation of a Modelica compiler, simulator, and development environment. Related research on parallel numeric solvers can for example be found in [9]. The work presented in this paper presents an algorithmic Modelica language extension called ParModelica for efficient portable explicit parallel Modelica programming. Portability is achieved based on the OpenCL [14] standard which is available on several multi-core architectures. ParModelica is evaluated using a benchmark test suite called Modelica PARallel benchmark suite (MPAR) which makes use of these language extensions and includes models which represent heavy computations. This paper is organized as follows. Section 2 gives a general introduction to Modelica simulation on parallel architectures. Section 3 gives an overview of GPUs, CUDA and OpenCL, whereas the new parallel Modelica language extensions are presented in Section 4. Section 5 briefly describes measurements using the parallel benchmark test suite. Finally, Section 6 gives programming guidelines to use ParModelica, and Section 7 presents conclusions and future work. 2 Parallel Simulation of Modelica Models on Multi-Core Computers The process of compiling and simulating Modelica models to sequential code is described e.g. in [3] and [12]. The handling of equations is rather complex and involves symbolic index reduction, topological sorting according to the causal dependencies between the equations, conversion into assignment statement form, etc. Simulation corresponds to "solving" the compiled equation system with respect to time using a numerical integration method. Compiling Modelica models for efficient parallel simulation on multi-core architectures requires additional methods compared to the typical approaches described in [3] and [12]. The parallel methods can be roughly divided into the following three groups: In Section 3.1 the NVIDIA GPU with its CUDA programming model is presented as an influential example of GPU architecture, followed by the portable OpenCL parallel programming model in Section 3.2. Automatic parallelization of Modelica models. Several approaches have been investigated: centralized solver approach, distributed solver approach and compilation of unexpanded array equations. With the first approach the solver is run on one core and in each time-step the computation of the equation system is done in parallel over several cores [4]. In the second approach the solver and the equation system are distributed across several cores [5]. With the third approach Modelica models with array equations are compiled unexpanded and simulated on multi-core architectures. Coarse-grained explicit parallelization using components. Components of the model are simulated in parallel partly de-coupled using time delays between the different components, see [11] for a summary. A different solver, with different time step, etc., can be used for each component. A related approach has been used in the xMOD tool [26]. Explicit parallel programming language constructs. This approach is explored in the NestStepModelica prototype [10] and in this paper with the ParModelica language extension. Parallel extensions have been developed for other languages, e.g. parfor loop and gpu arrays in Matlab, Visual C++ parallel_for, Mathematica parallelDo, etc. An important concept in NVIDIA CUDA (Computer Unified Device Architecture) for GPU programming is the distinction between host and device. The host is what executes normal programs, and the device works as a coprocessor to the host which runs CUDA threads by instruction from the host. This typically means that a CPU is the host and a GPU is the device, but it is also possible to debug CUDA programs by using the CPU as both host and device. The host and the device are assumed to have their own separate address spaces, the host memory and the device memory. The host can use the CUDA runtime API to control the device, for example to allocate memory on the device and to transfer memory to and from the device. 3 GPU Architectures, CUDA, and OpenCL Graphics Processing Units (GPUs) have recently become increasingly programmable and applicable to general purpose numeric computing. The theoretical processing power of GPUs has in recent years far surpassed that of CPUs due to the highly parallel computing approach of GPUs. However, to get good performance, GPU architectures should be used for simulation of models of a regular structure with large numbers of similar data objects. The computations related to each data object can then be executed in parallel, one or more data objects on each core, so-called data-parallel computing. It is also very important to use the GPU memory hierarchy effectively in order to get good performance. 3.1 NVIDIA GPU CUDA – Compute Unified Device Architecture Figure 1. Simplified schematic of NVIDIA GPU architecture, consisting of a set of Streaming Multiprocessors (SM), each containing a number of Scalar Processors (SP) with fast private memory and on-ship local shared memory. The GPU also has off-chip DRAM. The building block of the NVIDIA CUDA hardware architecture is the Streaming Multiprocessor (SM). In the NVIDIA Fermi-Tesla M2050 GPU, each SM contains 32 Scalar Processors (SPs). The entire GPU has 14 such SMs totaling to 448 SPs, as well as some offchip DRAM memory, see Figure 1. This gives a scalable architecture where the performance of the GPU can be varied by having more or fewer SMs. To be able to take advantage of this architecture a program meant to run on the GPU, known as a kernel, needs to be massively multi-threaded. A kernel is just a C-function meant to execute on the GPU. When a kernel is executed on the GPU it is divided into thread blocks, where each thread block contains an equal number of threads. These thread blocks are automatically distributed among the SMs, so a programmer need not consider the number of SMs a certain GPU has. All threads execute one common instruction at a time. If any threads take divergent execution paths, then each of these paths will be executed separately, and the threads will then converge again when all paths have been executed. This means that some SPs will be idle if the thread executions diverge. It is thus important that all threads agree on an execution path for optimal performance. This architecture is similar to the Single Instruction, Multiple Data (SIMD) architecture that vector processors use, and that most modern general-purpose CPUs have limited capabilities for too. NVIDIA call this architecture Single Instruction, Multiple Thread (SIMT) instead, the difference being that each thread can execute independently, although at the cost of reduced performance. It is also possible to regard each SM as a separate processor, which enables Multiple Instructions, Multiple Data (MIMD) parallelism. Using only MIMD parallelism will not make it possible to take full advantage of a GPU’s power, since each SM is a SIMD processor. To summarize: Streaming Multiprocessors (SM) can work with different code, performing different operations with entirely different data (MIMD execution, Multiple Instruction Multiple Data). All Scalar processors (SP) in one streaming multiprocessor execute the same instruction at the same time but work on different data (SIMT/SIMD execution, Single Instruction Multiple Data). 3.1.1 NVIDIA GPU Memory Hierarchy As can be seen in Figure 1 there are several different types of memory in the CUDA hardware architecture. At the lowest level each SP has a set of registers, the number depending on the GPU’s capabilities. These registers are shared between all threads allocated to a SM, so the number of thread blocks that a SM can have active at the same time is limited by the register usage of each thread. Accessing a register typically requires no extra clock cycles per instruction, except for some special cases where delays may occur. Besides the registers there is also the shared (local) memory, which is shared by all SPs in a SM. The shared memory is implemented as fast on-chip memory, and accessing the shared memory is generally as fast as accessing a register. Since the shared memory is accessible to all threads in a block it allows the threads to cooperate efficiently by giving them fast access to the same data. Most of the GPU memory is off-chip Dynamic Random Access Memory (DRAM). The amount of off- chip memory on modern graphics cards range from several hundred megabytes to few gigabytes. The DRAM memory is much slower than the on-chip memories, and is also the only memory that is accessible to the host CPU, e.g. through DMA transfers. To summarize: Each scalar processor (SP) has a set of fast registers. (private memory) Each streaming multiprocessor (SM) has a small local shared memory (48KB on Tesla M2050 ) with relatively fast access. Each GPU device has a slower off-chip DRAM (2GB on Tesla M2050) which is accessible from all streaming multiprocessors and externally e.g. from the CPU with DMA transfers. 3.2 OpenCL – the Open Computing Language OpenCL [14] is the first open, free parallel computing standard for cross-platform parallel programming of modern processors including GPUs. The OpenCL programming language is based on C99 with some extensions for parallel execution management. By using OpenCL it is possible to write parallel algorithms that can be easily ported between multiple devices with minimal or no changes to the source code. The OpenCL framework consists of the OpenCL programming language, API, libraries, and a runtime system to support software development. The framework can be divided into a hierarchy of models: Platform Model, Memory model, Execution model, and Programming model. Figure 2. OpenCL platform architecture. The OpenCL platform architecture in Figure 2 is similar to the NVIDIA CUDA architecture in Figure 1: Compute device – Graphics Processing Unit (GPU) Compute unit – Streaming Multiprocessor (SM) Processing element – Scalar Processor (SP) Work-item – thread Work-group – thread block The memory hierarchy (Figure 3) is also very similar: Global memory – GPU off-chip DRAM memory Constant memory – read-only cache of off-chip memory Local memory – on-chip shared memory that can be accessed by threads in the same SM Private memory – on-chip registers in the same Figure 4. OpenCL execution model, work-groups depicted as groups of squares corresponding to workitems. Each work-group can be referred to by a unique ID, and each work-item by a unique local ID. Figure 3. Memory hierarchy in the OpenCL memory model, closely related to typical GPU architectures such as NVIDIA. The memory regions can be accessed in the following way: Memory Regions Constant Memory Local Memory Private Memory Global Memory 3.2.1 Access to Memory All work-items in all work-groups All work-items in a work-group Private to a work-item All work-items in all work-groups OpenCL Execution Model The execution of an OpenCL program consists of two parts, the host program which executes on the host and the parallel OpenCL program, i.e., a collection of kernels (also called kernel functions), which execute on the OpenCL device. The host program manages the execution of the OpenCL program. Kernels are executed simultaneously by all threads specified for the kernel execution. The number and mapping of threads to Computing Units of the OpenCL device is handled by the host program. Each thread executing an instance of a kernel is called a work-item. Each thread or work item has unique id to help identify it. Work items can have additional id fields depending on the arrangement specified by the host program. Work-items can be arranged into work-groups. Each work-group has a unique ID. Work-items are assigned a unique local ID within a work-group so that a single work-item can be uniquely identified by its global ID or by a combination of its local ID and work-group ID. The work-items in a given work-group execute concurrently on the processing elements of a single compute unit as depicted in Figure 4. Several programming models can be mapped onto this execution model. OpenCL explicitly supports two of these models: primarily the data parallel programming model, but also the task parallel programming model 4 ParModelica: Extending Modelica for Explicit Algorithmic Parallel Programming As mentioned in the introduction, the focus of the current work is an extension (ParModelica) of the algorithmic subset of Modelica for efficient explicit parallel programming on highly data-parallel SPMD (Single Program Multiple Data) architectures. The current ParModelica implementation generates OpenCL [14] code for parallel algorithms. OpenCL was selected instead of CUDA [15] because of its portability between several multi-core platforms. Generating OpenCL code ensures that simulations can be run with parallel support on OpenCL enabled Graphics and Central Processor Units (GPUs and CPUs). This includes many multicore CPUs from [19] and Advanced Micro Devices (AMD) [18] as well as a range of GPUs from NVIDIA [17] and AMD [18]. As mentioned earlier most previous work regarding parallel execution support in the OpenModelica compiler has been focused on automatic parallelization where the burden of finding and analyzing parallelism has been put on the compiler. In this work, however, we have decided to leave this responsibility to the end user programmer. The compiler provides additional high level language constructs needed for explicitly stating parallelism in the algorithmic part of the modeling language. These, among others, include parallel variables, parallel functions, kernel functions and paral- lel for loops indicated by the parfor keyword. There are also some target language specific constructs and functions (in this case related to OpenCL). 4.1 Parallel Variables OpenCL code can be executed on a host CPU as well as on GPUs whereas CUDA code executes only on GPUs. Since the OpenCL and CUDA enabled GPUs use their own local (different from CPU) memory for execution, all necessary data should be copied to the specific device's memory. Parallel variables are allocated on the specific device memory instead of the host CPU. An example is shown below: function parvar protected Integer m = 1000; Integer A[m,m]; Integer B[m,m]; // global and local parglobal Integer parglobal Integer parglobal Integer parlocal Integer parlocal Integer end parvar; // Host Scalar // Host Matrix // Host Matrix device memories pm; // Global Scalar pA[m,m];// Glob Matrix pB[m,m];// Glob Matrix pn; // Local Scalar pS[m]; // Local Array The first two matrices A and B are allocated in normal host memory. The next two matrices pA and pB are allocated on the global memory space of the OpenCL device to be used for execution. These global variables can be initialized from normal or host variables. The last array pS is allocated in the local memory space of each processor on the OpenCL device. These variables are shared between threads in a single work-group and cannot be initialized from hast variables. Copying of data between the host memory and the device memory used for parallel execution is as simple as assigning the variables to each other. The compiler and the runtime system handle the details of the operation. The assignments below are all valid in the function given above Normal assignment - A := B Copy from host memory to parallel execution device memory - pA := A Copy from parallel execution device memory to host memory - B := pB Copy from device memory to other device memory – pA := pB Modelica parallel arrays are passed to functions only by reference. This is done to reduce the rather expensive copy operations. 4.2 Parallel Functions ParModelica parallel functions correspond to OpenCL functions defined in kernel files or to CUDA device functions. These are functions available for distributed (parallel) independent execution in each thread executing on the parallel device. For example, if a parallel array has been distributed with one element in each thread, a parallel function may operate locally in parallel on each element. However, unlike kernel functions, parallel functions cannot be called from serial code in normal Modelica functions on the host computer just as parallel OpenCL functions are not allowed to be called from serial C code on the host. Parallel functions have the following constraints, primarily since they are assumed to be called within a parallel context in workitems: Parallel function bodies may not contain parforloops. The reason is that the kernel containing the parallel functions is already distributed on each thread. Explicitly declared parallel variables are not allowed since execution is already taking place on the parallel device. All memory allocation will be on the parallel device's memory. Nested parallelism as in NestStepModelica [10] is not supported by this implementation. Called functions must be parallel functions or supported built-in functions since execution is on the parallel device. Parallel functions can only be called from the body of a parfor-loop, from parallel functions, or from kernel functions. Parallel functions in ParModelica are defined in the same way as normal Modelica functions, except that they are preceded by the parallel keyword as in the multiply function below: parallel function multiply input parglobal Integer a; input parlocal Integer b; output parprivate Integer c; output Integer c; algorithm c := a * b; end multiply; 4.3 // same as Kernel Functions ParModelica kernel functions correspond to OpenCL kernel functions [14] or CUDA global functions [16]. They are simply functions compiled to execute on an OpenCL parallel device, typically a GPU. ParModelica kernel functions are allowed to have several return- or output variables unlike their OpenCL or CUDA counterparts. They can also allocate memory in the global address space. Kernel functions can be called from serial host code, and are executed by each thread in the launch of the kernel. Kernels functions share the first three constraints stated above for parallel functions. However, unlike parallel functions, kernel functions cannot be called from the body of a parfor-loop or from other kernel functions. Kernel functions in ParModelica are defined in the same way as normal Modelica functions, except that they are preceded by the kernel keyword. An example usage of kernel functions is shown by the kernel function arrayElemtWiseMult. The thread id function oclGetGlobalId() (see Section 4.5) returns the integer id of a work-item in the first dimension of a work group. kernel function arrayElemWiseMultiply input Integer m; input Integer A[m]; input Integer B[m]; output Integer C[m]; protected Integer id; algorithm id := oclGetGlobalId(1); // calling the parallel function multiply is OK from kernel functions C[id] := multiply(A[id],B[id]); // multiply can be replaced by A[id]*B[id] end arrayElemWiseMultiply; 4.4 ParModelica parallel for loops, compared to normal Modelica for loops, have some additional constraints: All variable references in the loop body must be to parallel variables. Iterations should not be dependent on other iterations i.e. no loop-carried dependencies. All function calls in the body should be to parallel functions or supported built-in functions only. 4.5 // Matrix multiplication using parfor loop parfor i in 1:m loop for j in 1:pm loop ptemp := 0; for h in 1:pm loop // calling the // parallel function multiply is OK // from parfor-loops ptemp := multiply(pA[i,h], pB[h,j]) + ptemp; end for; pC[i,j] := ptemp; end for; end parfor; OpenCL Code There are also some additional ParModelica features available for directly compiling and executing userwritten OpenCL code: oclbuild(String) takes a name of an OpenCL source file and builds it. It returns an OpenCL program object which can be used later. oclkernel(oclprogram, String) takes a previously built OpenCL program and create the kernel specified by the second argument. It returns an OpenCL kernel object which can be used later. oclsetargs(oclkernel,...) takes a previously created kernel object variable and a variable number of arguments and sets each argument to its corresponding one in the kernel definition. oclexecute(oclkernel) executes the specified kernel. Parallel For Loop: parfor The iterations of a ParModelica parfor-loop are executed without any specific order in parallel and independently by multiple threads. The iterations of a parfor-loop are equally distributed among available processing units. If the range of the iteration is smaller than or equal to the number of threads the parallel device supports, each iteration will be done by a separate thread. If the number of iterations is larger than the number of threads available, some threads might perform more than one iteration. In future enhancements parfor will be given the extra feature of specifying the desired number of threads explicitly instead of automatically launching threads as described above. An example of using the parfor-loop is shown below: Executing User-written from ParModelica. All of the above operations are synchronous in the OpenCL jargon. They will return only when the specified operation is completed. Further functionality is planned to be added to these functions to provide better control over execution. 4.6 Synchronization and Thread Management All OpenCL work-item functions [20] are available in ParModelica. They perform the same operations and have the “same” types and number of arguments. However, there are two main differences: Thread/work-item index ids start from 1 in ParModelica, whereas the OpenCL C implementation counts from 0. Array dimensions start from 1 in Modelica and from 0 in OpenCL and C. For example oclGetGlobalId(1) call in the above arrayElemWiseMultiply will return the integer ID of a work-item or thread in the first dimension of a work group. The first thread gets an ID of 1. The OpenCL C call for the same operation would be ocl_get_global_id(0) with the first thread obtaining an ID of 0. 5 Benchmarking and Evaluation To be able to evaluate the relative performance and behavior of the new language extensions described in Section 4, performing systematic benchmarking on a set of appropriate Modelica models is required. For this purpose we have constructed a benchmark test suite containing some models that represent heavy and highperformance computation, relevant for simulation on parallel architectures. 5.1 The MPAR Benchmark Suite The MPAR benchmark test suite contains seven different algorithms from well-known benchmark applications such as the LINear equations software PACKage (LINPACK) [21], and Heat Conduction [23]. These benchmarks have been collected and implemented as algorithmic time-independent Modelica models. The algorithms implemented in this suite involve rather large computations and impose well defined workloads on the OpenModelica compiler and the run-time system. Moreover, they include different kinds of forloops and function calls which provide parallelism for domain and task decomposition. For space reasons we have provided results for only three models here. Time measurements have been performed of both sequential and parallel implementations of three models: Matrix Multiplication, Eigen value computation, and Stationary Heat Conduction, on both CPU and GPU architectures. For executing sequential codes generated by the standard sequential OpenModelica compiler we have used the Intel Xeon E5520 CPU [24] which has 16 cores, each with 2.27 GHz clock frequency. For executing generated code by our new OpenCL based parallel code generator, we have used the same CPU as well as the NVIDIA Fermi-Tesla M2050 GPU [25]. 5.2 Measurements In this section we present the result of measurements for simulating three models from the implemented benchmark suite. On each hardware configuration all simulations are performed five times with start time 0.0, stop time of 0.2 seconds and 0.2 seconds time step, measuring the average simulation time using the clock_gettime() function from the C standard li- brary. This function is called once when the simulation loop starts and once when the simulation loop finishes. The difference between the returned values gives the simulation time. All benchmarks have been simulated on both the Intel Xeon E5520 CPU (16 cores) and the NVIDIA Fermi-Tesla M2050 GPU (448 cores). 5.3 Simulation Results The Matrix Multiplication model (Appendix A) produces an M×K matrix C from multiplying an M×N matrix A by an N×K matrix B. This model presents a very large level of data-parallelism for which a considerable speedup has been achieved as a result of parallel simulation of this model on parallel platforms. The simulation results are illustrated in Figure 5 and Figure 6. The obtained speedup of matrix multiplication using kernel functions is as follows compared to the sequential algorithm on Intel Xeon E5520 CPU: Intel 16-core CPU – speedup 26 NVIDIA 448-core GPU – speedup 115 Speedup 114,67 CPU E5520 GPU M2050 35,95 26,34 24,76 4,36 0,61 13,41 64 4,61 128 256 512 Parameter M (Matrix sizes MxM) Figure 5. Speedup for matrix multiplication, Intel 16-core CPU and Nvidia 448 core GPU. The measured matrix multiplication model simulation times can be found in Figure 6. Simulation Time (second) In addition to the above features, special built-in functions for building user written OpenCL code directly from source code, creating a kernel, setting arguments to kernel and execution of kernels are also available. In addition parallel versions of some built-in algorithm functions are also available. 512 256 128 64 32 16 8 4 2 1 0,5 0,25 0,125 0,0625 32 64 128 256 512 CPU E5520 (Serial) 0,093 0,741 5,875 58,426 465,234 CPU E5520 (Parallel) 0,137 0,17 0,438 2,36 17,66 GPU M2050 (Parallel) 1,215 1,217 1,274 1,625 4,057 Figure 6. Simulation time for matrix multiplication, Intel 1-core, 16-core CPU, NVidia 448 core GPU. The second benchmark model performs Eigen-value computation, with the following speedups: Intel 16-core CPU – speedup 3 Simulation Time (second) NVIDIA 448-core GPU – speedup 48 Speedup 47,71 CPU E5520 GPU M2050 33,25 16,95 1,020,71 256 6,68 2,24 1,992,27 512 1024 2,75 2,51 2,32 2048 4096 Figure 7. Speedup for Eigen value computation as a function of model array size, for Intel 16-core CPU and NVIDIA 448 core GPU, compared to the sequential algorithm on Intel Xeon E5520 CPU. The measured simulation times for the Eigen-value model are shown in Figure 8. 1024 512 Simulation Time (second) 128 256 512 1,543 5,116 16,7 52,462 147,411 363,114 574,057 CPU E5520 (Parallel) 3,049 5,034 8,385 23,413 63,419 144,747 208,789 GPU M2050 (Parallel) 7,188 7,176 7,373 7,853 CPU E5520 (Serial) 1024 2048 4096 8,695 8192 10,922 12,032 Figure 8. Simulation time for Eigen-value computation as a function of model array size, for Intel 1-core CPU, 16core CPU, and NVIDIA 448 core GPU. The third benchmark model computes stationary heat conduction, with the following speedups: Intel 16-core CPU – speedup 7 NVIDIA 448-core GPU – speedup 22 Speedup 22,46 CPU E5520 GPU M2050 10,1 2,04 5,85 4,21 0,22 128 0,87 256 6,23 6,41 3,32 512 1024 128 256 512 102 4 204 8 CPU E5520 (Serial) 1,958 7,903 32,104 122,754 487,342 CPU E5520 (Parallel) 0,959 1,875 5,488 19,711 76,077 GPU M2050 (Parallel) 8,704 9,048 9,67 12,153 21,694 8192 Figure 10. Simulation time (seconds) for heat conduction model as a function of model size parameter M, for 1-core CPU, 16-core CPU, and 448 core GPU. Array size 256 128 64 32 16 8 4 2 1 512 256 128 64 32 16 8 4 2 1 0,5 2048 Parameter M (Matrix size MxM) Figure 9. Speedup for the heat conduction model as a function of model size parameter M, Intel 16-core CPU and Nvidia 448 core GPU, compared to sequential algorithm on Intel Xeon E5520 CPU. The measured simulation times for the stationary heat conduction model are shown in Figure 10. According to the results of our measurements illustrated in Figure 5, Figure 7, and Figure 9, absolute speedups of 114, 48, and 22 respectively were achieved when running generated ParModelica OpenCL code on the Fermi-Tesla M2050 GPU compared to serial code on the Intel Xeon E5520 CPU with the largest data sizes. It should be noted that when the problem size is not very large the sequential execution has better performance than the parallel execution. This is not surprising since for executing even a simple code on OpenCL devices it is required to create an OpenCL context within those devices, allocate OpenCL memory objects, transfer input data from host to those memory objects, perform computations, and finally transfer back the result to the host. Consequently, performing all these operations normally takes more time compared to the sequential execution when the problem size is small. It can also be seen that, as the sizes of the models increase, the simulations get better relative performance on the GPU compared to multi-core CPU. Thus, to fully utilize the power of parallelism using GPUs it is required to have large regular data structures which can be operated on simultaneously by being decomposed to all blocks and threads available on GPU. Otherwise, executing parallel codes on a multi-core CPU would be a better choice than a GPU to achieve more efficiency and speedup. 6 Guidelines for Using the New Parallel Language Constructs The most important task in all approaches regarding parallel code generation is to provide an appropriate way for analyzing and finding parallelism in sequential codes. In automatic parallelization approaches, the whole burden of this task is on the compiler and tool developer. However, in explicit parallelization approaches as in this paper, it is the responsibility of the modeler to analyze the source code and define which parts of the code are more appropriate to be explicitly parallelized. This requires a good understanding of the concepts of parallelism to avoid inefficient and incorrect generated code. In addition, it is necessary to know the constraints and limitations involved with using explicit parallel language constructs to avoid compile time errors. Therefore we give some advice on how to use the ParModelica language extensions to parallelize Modelica models efficiently: Try to declare parallel variables as well as copy assignments among normal and parallel variables as less as possible since the costs of data transfers from host to devices and vice versa are very expensive. In order to minimize the number of parallel variables as well as data transfers between host and devices, it is better not to convert forloops with few iterations over simple operations to parallel for-loops (parfor-loops). It is not always useful to have parallel variables and parfor-loops in the body of a normal for-loop which has many iterations. Especially in cases where there are many copy assignments among normal and parallel variables. Although it is possible to declare parallel variables and also parfor-loops in a function, there are no advantages when there are many calls to the function (especially in the body of a big for-loop). This will increase the number of memory allocations for parallel variables as well as the number of expensive copies required to transfer data between host and devices. Do not directly convert a for-loop to a parfor-loop when the result of each iteration depends on other iterations. In this case, although the compiler will correctly generate parallel code for the loop, the result of the computation may be incorrect. Use a parfor-loop in situations where the loop has many independent iterations and each iteration takes a long time to be completed. Try to parallelize models using kernel functions as much as possible rather than using parfor-loops. This will enable you to explicitly specify the desired number of threads and work-groups to get the best performance. If the global work size (total number of threads to be run in parallel) and the local work size (total number of threads in each work-group) need to be specified explicitly, then the following points should be considered. First, the work-group size (local size) should not be zero, and also it should not exceed the maximum work-group size supported by the parallel device. Second, the local size should be less or equal than the global-size. Third, the global size should be evenly divisible by the local size. The current implementation of OpenCL does not support recursive functions; therefore it is not possible to declare a recursive function as a parallel function. 7 Conclusions New multi-core CPU and GPU architectures promise high computational power at a low cost if suitable computational algorithms can be developed. The OpenCL C-based parallel programming model provides a way of writing portable parallel algorithms that perform well on a number of multi-core architectures. However, the OpenCL programming model is rather low-level and error-prone to use and intended for parallel programming specialists. This paper presents the ParModelica algorithmic language extension to the high-level Modelica modeling language together with a prototype implementation in the OpenModelica compiler. This makes it possible for the Modelica modeler to directly write efficient parallel algorithms in Modelica which are automatically compiled to efficient low-level OpenCL code. A benchmark suite called MPAR has been developed to evaluate the prototype. Good speedups have been obtained for large problem sizes of matrix multiplication, Eigen value computation, and stationary heat condition. Future work includes integration of the ParModelica explicit parallel programming approach with automatic and semi-automatic approaches for compilation of equation-based Modelica models to parallel code. Autotuning could be applied to further increase the performance and automatically adapt it to varying problem configurations. Some of the ParModelica code needed to specify kernel functions could be automatically generated. 8 Acknowledgements This work has been supported by Serc, by Elliit, by the Swedish Strategic Research Foundation in the EDOp and HIPo projects and by Vinnova in the RTSIM and ITEA2 OPENPROD projects. The Open Source Modelica Consortium supports the OpenModelica work. Thanks to Per Östlund for contributions to Section 3.1. References [1] Modelica Association. The Modelica Language Specification Version 3.2, March 24th 2010. http://www.modelica.org. Modelica Association. Modelica Standard Library 3.1. Aug. 2009. http://www.modelica.org./ [2] Open Source Modelica Consortium. OpenModelica System Documentation Version 1.8.1, April 2012. http://www.openmodelica.org/ [3] Peter Fritzson. Principles of Object-Oriented Modeling and Simulation with Modelica 2.1. Wiley-IEEE Press, 2004. [4] Peter Aronsson. Automatic Parallelization of Equation-Based Simulation Programs, PhD thesis, Dissertation No. 1022, Linköping University, 2006. [5] Håkan Lundvall. Automatic Parallelization using Pipelining for Equation-Based Simulation Languages, Licentiate thesis No. 1381, Linköping University, 2008. [6] Håkan Lundvall, Kristian Stavåker, Peter Fritzson, Christoph Kessler: Automatic Parallelization of Simulation Code for Equation-based Models with Software Pipelining and Measurements on Three Platforms. MCC'08 Workshop, Ronneby, Sweden, November 27-28, 2008. [7] Per Östlund. Simulation of Modelica Models on the CUDA Architecture. Master Thesis. LIUIDA/LITH-EX-A{09/062{SE. Linköping University, 2009. [8] Kristian Stavåker, Peter Fritzson. Generation of Simulation Code from Equation-Based Models for Execution on CUDA-Enabled GPUs. MCC'10 Workshop, Gothenburg, Sweden, November 1819, 2010. [9] Matthias Korch and Thomas Rauber. Scalable parallel rk solvers for odes derived by the method of lines. In Harald Kosch, Laszlo Böszörményi, and Hermann Hellwagner, editors, Euro-Par, volume 2790 of Lecture Notes in Computer Science, pages 830-839. Springer, 2003. [10] Christoph Kessler and Peter Fritzson. NestStepModelica – Mathematical Modeling and BulkSynchronous Parallel Simulation. In Proc. of PARA'06, Umeå, June 19-20, 2006. In Lecture Notes of Computer Science (LNCS) Vol 4699, pp 1006-1015, Springer Verlag, 2006. [11] Martin Sjölund, Robert Braun, Peter Fritzson and Petter Krus. Towards Efficient Distributed Simulation in Modelica using Transmission Line Modeling. In Proceedings of the 3rd International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools, (EOOLT'2010), Published by Linköping University Electronic Press, www.ep.liu.se, In conjunction with MODELS’2010, Oslo, Norway, Oct 3, 2010. [12] Francois Cellier and Ernesto Kofman. Continuous System Simulation. Springer, 2006. [13] Khronos Group, Open Standards for Media Authoring and Acceleration, OpenCL 1.1, accessed Sept 15, 2011. http://www.khronos.org/opencl/ [14] The OpenCL Specication, Version: 1.1, Document Revision: 44, accessed June 30 2011. http://www.khronos.org/registry/cl/specs/opencl1.1.pdf [15] NVIDIA CUDA, accessed September 15 2011. http://www.nvidia.com/object/cuda home new.html [16] NVIDIA CUDA programming guide, accessed 30 June 2011. http://developer.download.nvidia.com/ compute/cuda/4 0 rc2/toolkit/docs/CUDA C Programming Guide.pdf [17] OpenCL Programming Guide for the CUDA Architecture, Appendix A, accessed June 30 2011. http://developer.download.nvidia.com/compute/D evZone/docs/html/OpenCL/doc/OpenCL Programming Guide.pdf [18] AMD OpenCL, System Requirements & Driver Compatibility, accessed June 30 2011. http://developer.amd.com/sdks/AMDAPPSDK/pa ges/DriverCompatibility.aspx [19] INTEL OpenCL, Technical Requirements, accessed June 30 2011. http://software.intel.com/enus/articles/openclrelease-notes/ [20] OpenCL Work-Item Built-In Functions, accessed June 30 2011. http://www.khronos.org/registry/cl/sdk/1.0/docs/ man/xhtml/workItemFunctions.html [21] Jack J. Dongarra, J. Bunch, Cleve Moler, and G. W. Stewart. LINPACK User's Guide. SIAM, Philadelphia, PA, 1979. [22] Ian N. Sneddon. Fourier Transforms. Dover Publications, 2010. ISBN-13: 978-0486685229. [23] John H. Lienhard IV and John H. Lienhard V. A Heat Transfer Textbook. Phlogiston Press Cambridge, Massachusetts, U.S.A, 4th edition, 2011. [24] Intel Xeon E5520 CPU Specifications, accessed October 28 2011. http://ark.intel.com/products/40200/Intel-XeonProcessor-E5520-(8M-Cache-2 26-GHz-5 86GTs-Intel-QPI) [25] NVIDIA Tesla M2050 GPU Specifications, accessed June 30 2011. http://www.nvidia.com/docs/IO/43395/BD05238-001 v03.pdf [26] Cyril Faure. Real-time simulation of physical models toward hardware-in-the-loop validation. PhD Thesis. University of Paris East, October 2011. Appendix A. Serial Matrix Multiply model MatrixMultiplication parameter Integer m=256 ,n=256 ,k =256; Real result ; algorithm result := mainF (m,n,k); end MatrixMultiplication ; function mainF input Integer m; input Integer n; input Integer k; output Real result ; protected Real A[m,n]; Real B[n,k]; Real C[m,k]; algorithm // initialize matrix A, and B (A,B) := initialize (m,n,k); // multiply matrices A and B C := matrixMultiply (m,n,k,A,B); // only one item is returned to speed up // computation result := C[m,k]; end mainF; function initialize input Integer m; input Integer n; input Integer k; output Real A[m,n]; output Real B[n,k]; algorithm for i in 1:m loop for j in 1:n loop A[i,j] := j; end for; end for; for j in 1:n loop for h in 1:k loop B[j,h] := h; end for; end for; end initialize ; function matrixMultiply input Integer m; input Integer p; input Integer n; input Real A[m,p]; input Real B[p,n]; output Real C[m,n]; Real localtmp ; algorithm for i in 1:m loop for j in 1:n loop localtmp := 0; for k in 1:p loop localtmp := localtmp +(A[i,k]* B[k,j]); end for; C[i,j] := localtmp ; end for; end for; end matrixMultiply; Appendix B. Parallel Matrix-Matrix Multiplication with parfor and Kernel functions model MatrixMultiplicationP parameter Integer m=32,n=32,k=32; Real result; algorithm result := mainF(m,n,k); end MatrixMultiplicationP ; function mainF input Integer m; input Integer n; input Integer k; output Real result ; protected Real C[m,k]; parglobal Real pA[m,n]; parglobal Real pB[n,k]; parglobal Real pC[m,k]; parglobal Integer pm; parglobal Integer pn; parglobal Integer pk; // the total number of global threads // executing in parallel in the kernel Integer globalSize [2] = {m,k}; // the total number of local threads // in parallel in each workgroup Integer localSize [2] = {16 ,16}; algorithm // copy from host to device pm := m; pn := n; pk := k; (pA ,pB) := initialize(m,n,k,pn ,pk); // specify the number of threads and // workgroups // to be used for a kernel function // execution oclSetNumThreads(globalSize, localSize); pC := matrixMultiply(pn ,pA ,pB ); // copy matrix from device to host // and resturn result C := pC; result := C[m,k]; // set the number of threads to // the available number // supported by device oclSetNumThreads(0); end mainF ; function initialize input Integer m; input Integer n; input Integer k; input parglobal Integer pn; input parglobal Integer pk; output parglobal Real pA[m,n]; output parglobal Real pB[n,k]; algorithm parfor i in 1:m loop for j in 1: pn loop pA[i,j] := j; end for; end parfor; parfor j in 1:n loop for h in 1: pk loop pB[j,h] := h; end for; end parfor ; end initialize ; parkernel function matrixmultiply input parglobal Integer pn; input parglobal Real pA [: ,:]; input parglobal Real pB [: ,:]; output parglobal Real pC[size(pA,1), size(pB,2)]; protected Real plocaltmp ; Integer i,j; algorithm // Returns unique global thread Id value // for first and second dimension i := oclGetGlobalId (1); j := oclGetGlobalId (2); plocaltmp := 0; for h in 1: pn loop plocaltmp := plocaltmp + (pA[i,h] * pB[h,j]); end for; pC[i,j] := plocaltmp; end matrixmultiply; TLM and Parallelization Martin Sjölund December 3, 2012 Abstract Transmission line modeling (TLM) is a technique where the wave propagation of a signal in a medium over time can be modelled. The propagation of this signal is limited by the time it takes for the signal to travel across the medium. By utilizing this information it is possible to partition the system of equations in such a way that the equations can be partitioned into independent blocks that may be simulated in parallel. This leads to improved efficiency of simulations since it enables full performance of multi-core CPUs. 1 Background and Related Work An increasingly important way of creating efficient computations is to use parallel computing, i.e., dividing the computational work onto multiple processors that are available in multi-core systems. Such systems may use either a CPU [10] or a GPU using GPGPU techniques [13, 28]. Since multi-core processors are becoming more common than single-core processors, it is becoming important to utilize this resource. This requires support in compilers and development tools. However, while parallelization of models expressed in equation-based objectoriented (EOO) languages is not an easily solved task, the increased performance if successful is important. A hardware-in-the-loop real-time simulator using detailed computationally intensive models certainly needs the performance to keep short real-time deadlines, as do large models that take days or weeks to simulate. There are a few common approaches to parallelism in programming: • No parallelism in the programming language, but accessible via library calls. You can divide the work by executing several processes or jobs at once, each utilizing one CPU core. • Explicit parallelism in the language. You introduce language constructs so that the programmer can express parallel computations using several CPU cores. • Automatic parallelization. The compiler itself analyzes the program or model, partitions the work, and automatically produces parallel code. 1 Automatic parallelization is the preferred way because the users do not need to learn how to do parallel programming, which is often error-prone and timeconsuming. This is even more true in the world of equation-based languages because the ”programmer/modeler” can be a systems designer or modeler with no real knowledge of programming or algorithms. However, it is not so easy to do automatic parallelization of models in equation-based languages. Not only is it needed to decide which processor to perform a particular operation on; it is also needed to determine in which order to schedule computations needed to solve the equation system. This scheduling problem can become quite difficult and computationally expensive for large equation systems. It might also be hard to split the sequence of operations into two separate threads due to dependencies between the equations [1]. There are methods that can make automatic parallelization easier by introducing parallelism over time, e.g. distributing solver work over time [24]. However, parallelism over time gives very limited speedup for typical ODE systems of equations. A single centralized solver is the normal approach to simulation in most of today’s simulation tools. Although great advances have been made in the development of algorithms and software, this approach suffers from inherent poor scaling. That is, execution time grows more than linearly with system size. By contrast, distributed modeling, where solvers can be associated with or embedded in subsystems, and even component models, has almost linear scaling properties. Special considerations are needed, however, to connect the subsystems to each other in a way that maintains stability properties without introducing unwanted numerical effects. Technologies based on bilateral delay lines [2], also called transmission line modeling, TLM, have been developed for a long time at Linköping University. It has been successfully implemented in the Hopsan simulation package, which is currently almost the only simulation package that utilizes the technology, within mechanical engineering and fluid power. It has also been demonstrated in [15] and subsequently in [4]. Although the method has its roots already in the sixties, it has never been widely adopted, probably because its advantages are not evident for small applications, and that wave-propagation is regarded as a marginal phenomenon in most areas, and thus not well understood. In this paper we focus on introducing distributed simulation based on TLM technology in Modelica, and combining this with solver inlining which further contributes to avoiding the centralized solver bottleneck. In a future paper we plan to demonstrate these techniques for parallel simulation. Summarizing the main contents of the paper. • We propose using a structured way of modeling with model partitioning using transmission lines in Modelica that is compatible with existing Modelica tools (Section 6). 2 • We investigate two different methods to model transmission lines in Modelica and compare them to each other (Section 6). • We show that such a system uses a distributed solver and may contain subsystems with different time steps, which may improve simulation performance dramatically (Section 7). • We demonstrate that solver inlining and distributed simulation using TLM can be combined, and that the resulting simulation results are essentially identical to those obtained using the Hopsan simulation package. We use the Modelica language [7, 21] and the OpenModelica Compiler [8, 9] to implement our prototype, but the ideas should be valid for any similar language. 1.1 Related Work Several people have performed work on parallelization of Modelica models [1, 18, 19, 20, 23, 30], but there are still many unsolved problems to address. The work closest to ours is [23], where Nyström uses transmission lines to perform model partitioning for parallelization of Modelica simulations using computer clusters. The problem with clusters is the communication overhead, which is huge if communication is performed over a network. Real-time scheduling is also a bit hard to reason about if you connect your cluster nodes through TCP/IP. Today, there is an increasing need to parallelize simulations on a single computer because most CPUs are multi-core. One major benefit is communication costs; we will be able to use shared memory with virtually no delay in interprocessor communication. Another thing that is different between the two implementations is the way TLM is modeled. We use regular Modelica models without function calls for communication between model elements. Nyström used an external function interface to do server-client communication. His method is a more explicit way of parallelization, since he looks for the submodels that the user created and creates a kind of co-simulation. Inlining solvers have also been used in the past to introduce parallelism in simulations [18]. Parallelization of Modelica-based simulation on GPUs has been explored by Stavåker [27] and Östlund [30]. 2 Transmission Line Element Method A computer simulation model is basically a representation of a system of equations that model some physical phenomena. The goal of simulation software is to solve this system of equations in an efficient, accurate and robust way. To achieve this, the by far most common approach is to use a centralized solver algorithm which puts all equations together into a differential algebraic equation 3 system (DAE) or an ordinary differential equation system (ODE). The system is then solved using matrix operations and numeric integration methods. One disadvantage of this approach is that it often introduces data dependencies between the central solver and the equation system, making it difficult to parallelize the equations for simulation on multi-core platforms. Another problem is that the stability of the numerical solver often will depend on the simulation time step. An alternative approach is to let each component in the simulation model solve its own equations, i.e. a distributed solver approach. This allows each component to have its own fixed time step in its solvers. A special case where this is especially suitable is the transmission line element method. Such a simulator has numerically highly robust properties, and a high potential for taking advantage of multi-core platforms [14]. Despite these advantages, distributed solvers have never been widely adopted and centralized solvers have remained the de facto strategy on the simulation software market. One reason for this can perhaps be the rapid increase in processor speed, which for many years has made multi-core systems unnecessary and reduced the priority of increasing simulation performance. Modeling for multi-core-based simulation also requires applications of significant size for the advantages to become significant. With the recent development towards an increase in the number of processor cores rather than an increase in speed of each core, distributed solvers are likely to play a more important role. The fundamental idea behind the TLM method is to model a system in a way such that components can be somewhat numerically isolated from each other. This allows each component to solve its own equations independently of the rest of the system. This is achieved by replacing capacitive components (for example volumes in hydraulic systems) with transmission line elements of a length for which the physical propagation time corresponds to one simulation time step. In this way a time delay is introduced between the resistive components (for example orifices in hydraulic systems). The result is a physically accurate description of wave propagation in the system [14]. The transmission line element method (also called TLM method) originates from the method of characteristics used in Hytran [16], and from Transmission Line Modeling [12], both developed back in the nineteen sixties [2]. Today it is used in the Hopsan simulation package for fluid power and mechanical systems, see Section 3, and in the SKF TLM-based co-simulation package [25]. Mathematically, a transmission line can be described in the frequency domain by the four pole equation [29]. Assuming that friction can be neglected and transforming these equations to the time domain, they can be described according to equation 1 and 2. p1 (t) = p2 (t − T ) + Zc q1 (t) + Zc q2 (t − T ) (1) p2 (t) = p1 (t − T ) + Zc q2 (t) + Zc q1 (t − T ) (2) Here p equals the pressure before and after the transmission line, q equals the volume flow and Zc represents the characteristic impedance. The main property of these equations is the time delay they introduce, representing the 4 x Figure 1: Transmission line components calculate wave propagation through a line using a physically correct separation in time. communication delay between the ends of the transmission line, see Figure 1. In order to solve these equations explicitly, two auxiliary variables are introduced, see equations 3 and 4. c1 (t) = p2 (t − T ) + Zc q2 (t − T ) (3) c2 (t) = p1 (t − T ) + Zc q1 (t − T ) (4) These variables are called wave variables or wave characteristics, and they represent the delayed communication between the end nodes. Putting equations 1 to 4 together will yield the final relationships between flow and pressure in equations 5 and 6. p1 (t) = c1 + Zc q1 (t) (5) p2 (t) = c2 + Zc q2 (t) (6) These equations can now be solved using boundary conditions. These are provided by adjacent (resistive) components. In the same way, the resistive components get their boundary conditions from the transmission line (capacitive) components. One noteworthy property with this method is that the time delay represents a physically correct separation in time between components of the model. Since the wave propagation speed (speed of sound) in a certain liquid can be calculated, the conclusion is that the physical length of the line is directly proportional to the time step used to simulate the component, see equation 7. Note that this time step is a parameter in the component, and can very well differ from the time step used by the simulation engine. Keeping the delay in the 5 transmission line larger than the simulation time step is important, to avoid extrapolation of delayed values. This means that a minimum time delay of the same size as the time step is required, introducing a modeling error for very short transmission lines. s β l = ha = (7) ρ Here, h represents the time delay and a the wave propagation speed, while β and ρ are the bulk modulus and the density of the liquid. With typical values for the latter two, the wave propagation speed will be approximately 1000 m/s, which means that a time delay of 1 ms will represent a length of 1 m. [15] 3 Hopsan Hopsan is a simulation software for simulation and optimization of fluid power and mechanical systems. This software was first developed at Linköping University in the late 1970’s [6]. The simulation engine is based on the transmission line element method described in Section 2, with transmission lines (called C-type components) and restrictive components (called Q-type) [17]. In the current version, the solver algorithms are distributed so that each component uses its own local solvers, although many common algorithms are placed in centralized libraries. In the new version of Hopsan, which is currently under development, all equation solvers will be completely distributed as a result of an object-oriented programming approach [3]. Numerical algorithms in Hopsan are always discrete. Derivatives are implemented by first or second order filters, i.e. a loworder rational polynomial expression as approximation, and using bilinear transforms, i.e. the trapetzoid rule, for numerical integration. Support for built-in compatibility between Hopsan and Modelica is also being investigated. 4 Example Model with Pressure Relief Valve The example model used for comparing TLM implementations in this paper is a simple hydraulic system consisting of a volume with a pressure relief valve, as can be seen in Figure 2. A pressure relief valve is a safety component, with a spring at one end of the spool and the upstream pressure, i.e., the pressure at the side of the component where the flow is into the component, acting on the other end, see Figure 3. The preload of the spring will make sure that the valve is closed until the upstream pressure reaches a certain level, when the force from the pressure exceeds that of the spring. The valve then opens, reducing the pressure to protect the system. In this system the boundary conditions are given by a constant prescribed flow source into the volume, and a constant pressure source at the other end of the pressure relief valve representing the tank. As oil flows into the volume the 6 Figure 2: The example system consists of a volume and a pressure relief valve. Boundary conditions is represented by a constant flow source and a constant pressure source. Figure 3: A pressure relief valve is designed to protect a hydraulic system by opening at a specified maximum pressure. pressure will increase at a constant rate until the reference pressure of the relief valve is reached. The valve then opens, and after some oscillations a steady state pressure level will appear. A pressure relief valve is a very suitable example model when comparing simulation tools. The reason for this is that it is based on dynamic equations and also includes several non-linearities, making it an interesting component to study. It also includes multiple physical domains, namely hydraulics and mechanics. The opening of a relief valve can be represented as a step or ramp response, which can be analyzed by frequency analysis techniques, for example using bode plots or Fourier transforms. It also includes several physical phenomena useful for comparisons, such as wave propagations, damping and self oscillations. If the complete set of equations is used, it will also produce non-linear phenomena such as cavitation and hysteresis, although these are not included in this paper. The volume is modeled as a transmission line, in Hopsan known as a Ctype component. In practice this means that it will receive values for pressure and flow from its neighboring components (flow source and pressure relief valve), and return characteristic variables and impedance. The impedance is calculated from bulk modulus, volume and time step, and is in turn used to calculate the characteristic variables together with pressures and flows. There is also a low- 7 pass damping coefficient called α, which is set to zero and thereby not used in this example. mZc = mBulkmodulus/mVolume ∗ mTimestep ; c10 = p2 + mZc ∗ q2 ; c20 = p1 + mZc ∗ q1 ; c1 = mAlpha∗ c1 + (1.0 −mAlpha ) ∗ c10 ; c2 = mAlpha∗ c2 + (1.0 −mAlpha ) ∗ c20 ; The pressure relief valve is a restrictive component, known as Q-type. This means that it receives characteristic variables and impedance from its neighboring components, and returns flow and pressure. Advanced models of pressure relief valves are normally performance oriented. This means that parameters that users normally have little or no knowledge about, such as the inertia of the spool or the stiffness of the spring are not needed as input parameters but are instead implicitly included in the code. This is however complicated and not very intuitive. For this reason a simpler model was created for this example. It is basically a first-order force equilibrium equation with a mass, a spring and a force from the pressure. Hysteresis and cavitation phenomena are also excluded from the model. The first three equations below calculate the total force acting on the spool. By using a second-order filter, the x position can be received from Newton’s second law. The position is used to retrieve the flow coefficient of the valve, which in turn is used to calculate the flow using a turbulent flow algorithm. Pressure can then be calculated from impedance and characteristic variables according to transmission line modeling. mFs = mPilotArea ∗ mPref ; p1 = c1 + q1 ∗ Zc1 ; Ftot = p1∗ mPilotArea − mFs ; x0 = m F i l t e r . v a l u e ( Ftot ) ; mTurb . s e t F l o w C o e f f i c i e n t (mCq∗mW∗ x0 ) ; q2 = mTurb . getFlow ( c1 , c2 , Zc1 , Zc2 ) ; q1 = −q2 ; p1 = c1 + Zc1 ∗ q1 ; p2 = c2 + Zc2 ∗ q2 ; 5 OpenModelica and Modelica OpenModelica [8, 9] is an open-source Modelica-based modeling and simulation environment, whereas Modelica [21] is an equation-based, object-oriented modeling/programming language. The Modelica Standard Library [22] contains almost a thousand model components from many different application domains. Modelica supports event handling as well as delayed expressions in equations. We will use those properties later in our implementation of a distributed TLMstyle solver. It is worth mentioning that Hopsan may access the value of a state 8 variable, e.g. x, from the previous time step. This value may then be used to calculate derivatives or do filtering since the length of time steps is fixed. In standard Modelica, it is possible to access the previous value before an event using the pre() operator, but impossible to access solver time-step related values, since a Modelica model is independent of the choice of solver. This is where sampling and delaying expressions comes into play. Note that while delay(x,0) will return a delayed value, if the solver takes a time step > 0, it will extrapolate information. Thus, it needs to take an infinite number of steps to simulate the system, which means a delay time > 0 needs to be used. 6 Transmission Lines in an Equation-based Language There are some issues when trying to use TLM in an equation-based language. TLM has been proven to work well using fixed time steps. In Modelica however, events can happen at any time. When an event is triggered due to an event-inducing expression changing sign, the continuous-time solver is temporarily stopped and a root-finding solution process is started in order to find the point in time where the event occurs. If the event occurs e.g. in the middle of a fixed time step, the solver will need to take a smaller (e.g. half) time step when restarted, i.e. some solvers may take extra time steps if the specified tolerance is not reached. However, this occurs only for hybrid models. For pure continuous-time models which do not induce events, fixed steps will be kept when using a fixed step solver. The delay in the transmission line can be implemented in several ways. If you have a system with fixed time steps, you get a sampled system. Sampling works fine in Modelica, but requires an efficient Modelica tool since you typically need to sample the system quite frequently. An example usage of the Modelica sample() built-in function is shown below. Variables defined within whenequations in Modelica (as below) will have discrete-time variability. when sample(−T, T) then l e f t . c = p r e ( r i g h t . c ) + 2 ∗ Zc ∗ p r e ( r i g h t . q ) ; r i g h t . c = p r e ( l e f t . c ) + 2 ∗ Zc ∗ p r e ( l e f t . q ) ; end when ; Modelica tools also offer the possibility to use delays instead of sampling. If you use delays, you end up with continuous-time variables instead of discretetime ones. The methods are numerically very similar, but because the variables are continuous when you use delay, the curve will look smoother. l e f t . c = d e l a y ( r i g h t . c + 2 ∗ Zc ∗ r i g h t . q , T) ; r i g h t . c = d e l a y ( l e f t . c + 2 ∗ Zc ∗ l e f t . q , T) ; Finally, it is possible to explicitly specify a derivative rather than obtaining it implicitly by difference computations relating to previous values (delays or 9 Figure 4: Pressure increases until the reference pressure of 10 MPa is reached, where the relief valve opens. Figure 5: Comparison of spool position using different TLM implementations. sampling). This then becomes a transmission line without delay, which is a good reference system. d e r ( l e f t . p ) = ( l e f t . q+r i g h t . q ) /C ; der ( r i g h t . p) = der ( l e f t . p) ; Figure 4 contains the results of simulating our example system, i.e., the pressure relief valve from section 4. Figures 5 and 6 are magnified versions that show the difference between our different TLM implementations. The models used to create the Figures, are part of the Modelica package DerBuiltin in Appendix A. If you decrease the delay in the transmission even closer to zero (it is now 10−4 ), the signals are basically the same (as would be expected). It does however come at a significant increase in simulation times and decreased numerical stability. This is not acceptable if stable real-time performance is desired. We use the same step size as the delay of the transmission line since that is the maximum allowed time step using this method, and better shows numerical issues than a tiny step size. Due to the nature of integrating solvers, we calculate the value der(x), and use reinit() when der(x) changes sign. The OpenModelica dassl solver cannot be used in all of these models due to an incompatibility with the delay() operator (the solver does not limit its step size as it should). Dassl is used together with sampling since the solver does limit its step size if a zero crossing occurs; in the other simulations the Euler solver is used. Because of these reasons we tried to use another method of solving the equation system, see package DerInline in Appendix A. We simply inlined a derivative approximation (x−delay(x,T))/T instead of der(x), which is much closer to the discrete-time approximation used in the Hopsan model. This is quite slow in practice because of the overhead delay adds, but it does implicitly inline the solver, which is a good property for in parallelization. If you look at Figures 7 and 8, you can see that all simulations now have the same basic shape. In fact, the OpenModelica ones have almost the same values. The time step is still 10−4 , which means you get the required behavior even without sacrificing simulation times. Even in this small example, the implementation using delays has 1 state variable, while the ideal, zero-delay, implementation has 3 state variables. This Figure 6: Comparison of system pressure using different TLM implementations. 10 Figure 7: Comparison of spool position with inlined explicit euler. Figure 8: Comparison of system pressure with inlined explicit euler. makes it easier to automatically parallelize larger models since the centralized solver handles fewer calculations. When inlining the der() operator, we end up with 0 continuous-time state variables. Table 1 contains some performance numbers on the models used. At this early stage in the investiagtion the numbers are not that informative for several reasons. We only made single-core simulations so far. Models that have better parallelism will get better speedups when we start doing multi-core simulations. The current OpenModelica simulation runtime system implementation does not have special efficient implementations of time events or delayed expressions. The inlined solver uses delay explicitly instead of being an actual inlined solver. This means it needs to search an array for the correct value rather than accessing it directly, resulting in an overhead that will not exist once inline solvers are fully implemented in OpenModelica. We used the -noemit flag in OpenModelica to disable generation of result files. Generating them takes between 20% and 90% of total simulation runtime depending on solver and if many events are generated. Do not compare the current Dymola [5] performance numbers to OpenModelica. We run Dymola inside a Windows virtual machine, while we run OpenModelica on Linux. The one thing that the performance numbers really tells you is not to use sampling in OpenModelica until performance is improved, and that the overhead of inlining the derivative using delay is a lot lower in Dymola than it is in OpenModelica. Table 1: Performance comparison between different models in the DerBuiltin and DerInline packages. Method OpenModelica Delay OpenModelica Ideal OpenModelica Sample Dymola Ideal Dymola Sample Builtin (sec) 0.13 0.04 3.65 0.64 1.06 11 Inlined (sec) 0.40 0.27 63.63 0.75 1.15 7 Distributed Solver The implementation using an inlined solver in Section 6 is essentially a distributed solver. It may use different time steps in different submodels, which means a system can be simulated using a very small time step only for certain components. The advantage of such a distributed system becomes apparent in [15]. In the current OpenModelica implementation this is not yet taken advantage of, i.e., the states are solved in each time step regardless. 8 Further Work To progress further we need to introduce replace the use of the delay() operator in the delay lines with an algorithm section. This would make the initial equations easier to solve, the system would simulate faster, and it would retain the property that connected subsystems don’t depend on each other. Once we have partitioned a Modelica model into a distributed system model, we will be able to start simulating the submodels in parallel, as described in [11, 15]. Some of the problems inherent in parallelization of models expressed in EOO languages are solved by doing this partitioning. By partitioning the model, you essentially create many smaller systems, which are trivial to schedule on multicore systems. To progress this work further a larger more computationally intensive model is also needed. Once we have a good model and inlined solvers, we will work on making sure that compilation and simulation scales well both with the target number of processors and the size of the problem. 9 Conclusions We conclude that all implementations work fine in Modelica. The delay line implementation using delays is not considerably slower than the one using the der() operator, but can be improved by using for example algorithm sections here instead. Sampling also works fine, but is far too slow for real-time applications. The delay implementation should be preferred over using der(), since the delay will partition the whole system into subsystems, which are easy to parallelize. Approximating integration by inline euler using the delay operator is not necessary to ensure stability although it produces results that are closer to the results of the same simulation in Hopsan. When you view the simulation as a whole, you can’t see any difference (Figure 4). 12 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (a) Regular system ∗ ∗ ∗ ∗ ∗ ∗ ∗ (c) Partitioned systems (b) TLM system Figure 9: Adjacency matrices in lower triangular form. 10 Partitioning The reason for using TLM is that you implicitly gain a course-grained parallelization of the system. We present a general approach to partitioning a system of equations that utilizes the time-delay introduced by TLM. Each partition of the equation system will be independent from any other within the current time step. This means they can be parallelized by synchronizing between time steps. To illustrate what our algorithm does, consider the standard approach to causalize (i.e. deciding what order and which assignments should be performed) an equation system. The equation system is put in block lower triangle (BLT) form, where each BLT block corresponds to either a single equation of the original system or a strong component (several equations). The examples below are lower triangular matrices (the strong components have been replaced with a single one to make them easier to read). Since the matrix has no variables above the diagonal (see Figure 9a), no block depends on a subsequent block and the whole system can solved sequentially. What happens when TLM is used to model the system is that some entries in the adjacency matrix disappear since delay expressions are allowed to decouple the system if they only access data in former time steps. In Figure 9b, one such entry has been removed from Figure 9a. The new matrix can be cut into two separate ones and solved sequentially in parallel since the two blocks are now totally independent from each other (Figure 9c). The basic data structure needed to perform this analysis is the incidence matrix (which may be represented by an adjacency list or matrix depending on the sparsity of the system). The benefit of only looking at the adjacency matrix is that we may partition the equation system before we perform optimizations, some of which are costly to perform on large systems since they do not have linear time complexity. 10.1 Testcase The testcase (see Appendix A) is a scalable version of the model in [26], where instead of a volume, we can scale it to a sequence of volumes and orifices. This means the automatic partitioning will be able to create a lot of independent subsystems, which can then be run in parallel (synchronized only between 13 timesteps). 10.2 Algorithm ... 11 Restrictions Step size of the solver has to be enforced to be lower than or equal to the delay of the shortest delay line in the system. 12 Runtime The runtime uses OpenMP to run the independent equation systems in parallel. The main changes to the runtime is that it needs to be thread-safe. References [1] Peter Aronsson. Automatic Parallelization of Equation-Based Simulation Programs. Doctoral thesis No 1022, Linköping University, Department of Computer and Information Science, June 2006. [2] D. M. Auslander. Distributed System Simulation with Bilateral Delay-Line Models. Journal of Basic Engineering, Trans. ASME:195–200, 1968. [3] Mikael Axin, Robert Braun, Petter Krus, Alessandro dell’Amico, Björn Eriksson, Peter Nordin, Karl Pettersson, and Ingo Staack. Next Generation Simulation Software using Transmission Line Elements. In Proceedings of the Bath/ASME Symposium on Fluid Power and Motion Control (FPMC), Sep 2010. [4] JD Burton, KA Edge, and CR Burrows. Partitioned Simulation of Hydraulic Systems Using Transmission-Line Modelling. In ASME WAM, 1993. [5] Dassault Systèmes. Dymola 7.3, 2009. [6] Björn Eriksson, Peter Nordin, and Petter Krus. Hopsan, A C++ Implementation Utilising TLM Simulation Technique. In Proceedings of the 51st Conference on Simulation and Modelling (SIMS), October 2010. [7] Peter Fritzson. Principles of Object-Oriented Modeling and Simulation with Modelica 2.1. Wiley-IEEE Press, February 2004. [8] Peter Fritzson, Peter Aronsson, Håkan Lundvall, Kaj Nyström, Adrian Pop, Levon Saldamli, and David Broman. The OpenModelica Modeling, Simulation, and Software Development Environment. Simulation News Europe, 44/45, December 2005. 14 [9] Peter Fritzson et al. Openmodelica 1.5.0 system documentation, June 2010. [10] Jason Howard et al. A 48-core IA-32 message-passing processor with DVFS in 45nm CMOS. In Solid-State Circuits Conference Digest of Technical Papers (ISSCC), 2010 IEEE International, pages 108 –109, 7-11 2010. [11] Arne Jansson, Petter Krus, and Jan-Ove Palmberg. Real Time Simulation Using Parallel Processing. In The 2nd Tampere International Conference on Fluid Power, 1991. [12] P. B. Johns and M. A. O’Brien. Use of the transmission line modelling (t.l.m) method to solve nonlinear lumped networks. The Radio and Electronic Engineer, 50(1/2):59–70, 1980. [13] David B. Kirk and Wen-Mei W. Hwu. Programming Massively Parallel Processors: A Hands-On Approach. Morgan Kaufmann Publishers, 2010. [14] Petter Krus. Robust System Modelling Using Bi-lateral Delay Lines. In Proceedings of the 2nd Conference on Modeling and Simulation for Safety and Security (SimSafe), Linköping, Sweden, 2005. [15] Petter Krus, Arne Jansson, Jan-Ove Palmberg, and Kenneth Weddfelt. Distributed Simulation of Hydromechanical Systems. In The Third Bath International Fluid Power Workshop, 1990. [16] Air Force Aero Propulsion Laboratory. Aircraft hydraulic system dynamic analysis. Technical report, Air Force Aero Propulsion Laboratory, AFAPLTR-76-43, Ohio, USA, 1977. [17] Linköping University, Linköping University. The hopsan Simulation Program, User’s Manual, 1985. LiTH-IKP-R-387. [18] Håkan Lundvall. Automatic Parallelization using Pipelining for EquationBased Simulation Languages. Licentiate thesis No 1381, Linköping University, Department of Computer and Information Science, 2008. [19] Håkan Lundvall, Kristian Stavåker, Peter Fritzson, and Christoph Kessler. Automatic Parallelization of Simulation Code for Equation-based Models with Software Pipelining and Measurements on Three Platforms. Computer Architecture News. Special Issue MCC08 – Multi-Core Computing, 36(5), December 2008. [20] Martina Maggio, Kristian Stavåker, Filippo Donida, Francesco Casella, and Peter Fritzson. Parallel Simulation of Equation-based Object-Oriented Models with Quantized State Systems on a GPU. In Proceedings of the 7th International Modelica Conference, September 2009. [21] Modelica Association. Modelica: A unified object-oriented language for physical systems modeling, language specification version 3.2, 2010. 15 [22] Modelica Association. Modelica Standard Library version 3.1, 2010. [23] Kaj Nyström and Peter Fritzson. Parallel Simulation with Transmission Lines in Modelica. In Christian Kral and Anton Haumer, editors, Proceedings of the 5th International Modelica Conference, volume 1, pages 325–331. Modelica Association, September 2006. [24] Thomas Rauber and Gudula Rünger. Parallel execution of embedded and iterated runge-kutta methods. Concurrency - Practice and Experience, 11(7):367–385, 1999. [25] Alexander Siemers, Dag Fritzson, and Peter Fritzson. Meta-Modeling for Multi-Physics Co-Simulations applied for OpenModelica. In Proceedings of International Congress on Methodologies for Emerging Technologies in Automation (ANIPLA), November 2006. [26] Martin Sjölund, Robert Braun, Peter Fritzson, and Petter Krus. Towards efficient distributed simulation in Modelica using transmission line modeling. In Peter Fritzon, Edward Lee, François Cellier, and David Broman, editors, Proceedings of the 3rd International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools, pages 71–80. Linköping University Electronic Press, October 2010. [27] Kristian Stavåker. Contributions to Parallel Simulation of Equation-Based Models on Graphics Processing Units. Licentiate thesis No 1507, Linköping University, Department of Computer and Information Science, 2011. [28] Ryoji Tsuchiyama, Takashi Nakamura, Takuro Iizuka, Akihiro Asahara, and Satoshi Miki. The OpenCL Programming Book. Fixstars Corporation, 2010. [29] T. J. Viersma. Analysis, Synthesis and Design of Hydraulic Servosystems and Pipelines. Elsevier Scientific Publishing Company, Amsterdam, The Netherlands, 1980. [30] Per Östlund. Simulation of Modelica Models on the CUDA Architecture. Master’s thesis, Linköping University, Department of Computer and Information Science, November 2009. A ParallelPRV package TLM connector Connector Q output Real p ; output Real q ; input Real c ; 16 input Real Zc ; end Connector Q ; connector Connector C input Real p ; input Real q ; output Real c ; output Real Zc ; end Connector C ; model FlowSource Connector Q s o u r c e ; parameter Real f l o w V a l ; equation so ur ce .q = flowVal ; source.p = source.c + source.q ∗ source.Zc ; end FlowSource ; model P r e s s u r e S o u r c e Connector C p r e s s u r e ; parameter Real P ; equation pressure.c = P; pressure.Zc = 0; end P r e s s u r e S o u r c e ; model H y d r a u l i c A l t e r na t i v e P R V Connector Q l e f t ; Connector Q r i g h t ; parameter Real P r e f = 20000000 ” R e f e r e n c e Opening Pressure ” ; parameter Real cq = 0 . 6 7 ” Flow C o e f f i c i e n t ” ; parameter Real s p o o l d i a m e t e r = 0 . 0 1 ” S p o o l Diameter ” ; parameter Real f r a c = 1 . 0 ” F r a c t i o n o f S p o o l C i r c u m f e r e n c e t h a t i s Opening ” ; parameter Real W = s p o o l d i a m e t e r ∗ f r a c ; parameter Real p i l o t a r e a = 0 . 0 0 1 ” Working Area o f P i l o t Pressure ” ; parameter Real k = 1 e6 ” S t e a d y S t a t e C h a r a c t e r i s t i c s o f Spring ” ; parameter Real c = 1000 ” S t e a d y S t a t e Damping Coefficient” ; parameter Real m = 0 . 0 1 ”Mass” ; parameter Real x h y s t = 0 . 0 ” H y s t e r e s i s o f S p o o l Position ” ; 17 constant Real xmax = 0 . 0 0 1 ”Maximum S p o o l P o s i t i o n ” ; constant Real xmin = 0 ”Minimum S p o o l P o s i t i o n ” ; parameter Real T ; parameter Real Fs = p i l o t a r e a ∗ P r e f ; Real Ftot = l e f t . p ∗ p i l o t a r e a − Fs ; Real Ks = cq ∗W∗x ; Real x ( s t a r t = xmin, min = xmin, max = xmax ) ; parameter I n t e g e r one = 1 ; Real x f r a c = x∗ P r e f /xmax ; Real v = der ( xtmp ) ; Real a = der ( v ) ; Real v2 = c ∗v ; Real x2 = k∗x ; Real xtmp ; equation left.p = left.c + left.Zc∗ left.q ; right.p = right.c + right.Zc ∗ right.q ; l e f t . q = −r i g h t . q ; r i g h t . q = s i g n ( l e f t . c − r i g h t . c ) ∗ Ks ∗ ( noEvent ( s q r t ( abs ( l e f t . c − r i g h t . c ) +(( l e f t . Z c+r i g h t . Z c ) ∗Ks ) ˆ 2 / 4 ) ) − Ks ∗ ( l e f t . Z c+r i g h t . Z c ) / 2 ) ; xtmp = ( Ftot − c ∗v − m∗ a ) /k ; x = i f noEvent ( xtmp < xmin ) then xmin e l s e i f noEvent ( xtmp > xmax ) then xmax e l s e xtmp ; end H y d r a u l i c A l t e r n a t i v e P R V ; model Volume parameter Real V; parameter Real Be ; f i n a l parameter Real Zc = Be∗T/V; parameter Real T ; Connector C l e f t ; Connector C r i g h t ; equation l e f t . Z c = Zc ; r i g h t . Z c = Zc ; l e f t . c = d e l a y ( r i g h t . c +2∗Zc∗ r i g h t . q , T ) ; r i g h t . c = d e l a y ( l e f t . c +2∗Zc∗ l e f t . q , T ) ; end Volume ; 18 model O r i f i c e parameter Real K; Connector Q l e f t ; Connector Q r i g h t ; equation l e f t . q = ( r i g h t . p − l e f t . p ) ∗K; r i g h t . q = −l e f t . q ; left.p = left.c + left.Zc∗ left.q ; right.p = right.c + right.Zc ∗ right.q ; end O r i f i c e ; end TLM; model PRVSystem import TLM.{ Volume,Orifice,FlowSource,PressureSource,HydraulicAlternativePRV }; constant Boolean b = f a l s e ; parameter Real T = 1e−4 ; Volume volume1 ( f i n a l V=1 e − 3 , f i n a l Be=1 e 9 , f i n a l T=T) ; O r i f i c e o r i f i c e 1 ( f i n a l K=1) i f b ; Volume volume2 ( f i n a l V=1 e − 3 , f i n a l Be=1 e 9 , f i n a l T=T) i f b; FlowSource f l o w S o u r c e ( f l o w V a l = 1e−5 ) ; P r e s s u r e S o u r c e p r e s s u r e S o u r c e (P = 1 e5 ) ; H y d r a u l i c A l t e r n a t i v e P R V hydr ( P r e f=1 e 7 , c q = 0 . 6 7 , s p o o l d i a m e t e r = 0 . 0 0 2 5 , f r a c = 1 . 0 , p i l o t a r e a =5e−5,xmax = 0 . 0 1 5 ,m= 0 . 1 2 , c =400, k =150000,T=T) ; equation connect ( f l o w S o u r c e . s o u r c e , v o l u m e 1 . l e f t ) ; i f b then connect ( v o l u m e 1 . r i g h t , o r i f i c e 1 . l e f t ) ; connect ( o r i f i c e 1 . r i g h t , v o l u m e 2 . l e f t ) ; connect ( v o l u m e 2 . r i g h t , h y d r . l e f t ) ; else connect ( v o l u m e 1 . r i g h t , h y d r . l e f t ) ; end i f ; connect ( h y d r . r i g h t , p r e s s u r e S o u r c e . p r e s s u r e ) ; end PRVSystem ; 19

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