Platform Based Physical Response Modeling

Platform based design uses high speed embedded processors, FPGAs, and ASICs as targets for implementation of a system model. Application of this design approach to physical response modeling is described and results are given. The approach shows promise for providing a PC based simulation acceleration environment. A new modeling approach using a cascading cellular automata methodology is discussed. The method utilizes a computationally explicit yet numerically stable set of rules for physical processes which are necessary for the FPGA implementations. Introduction Most approaches to the acceleration of the solution of physical equations rely on large processor arrays in a super computer environment. Here we describe a different approach that provides an environment where by a scientist can formulate model equations using Mathematica or Matlab on a PC, and have the solution accelerated locally on a low cost attached platform. Platform Based Design When designing Systems On A Chip, an approach frequently used is Platform Based Design. In this approach, the nature of the major components is known apriori: 1) processor architecture and 2) bus protocol, 3) Field Programmable Gate Arrays (FPGAs) or Application Specific Integrated Circuits (ASICs). High speed specialized computations are mapped onto ASICs and FPGAs. Processors perform control functions. Figure 1 shows the ARM ASIC Development platform that we are using for our research. Components include: a) 2 ARM 720 RISC general purpose processors, b) 1 966 ARM Digital Signal Processing Processor, and c) Two Xilinx XCV 2000E FPGA boards, each with a capacity of 2.5 million equivalent gates, all interconnected by an ARM bus. ARM Ltd. sells processor cores for embedded systems. Platforms such as this allow customers to check out their designs. In our application we use the platform as a hardware accelerator. Physical Response Modeling The intent of this approach to modeling is to give the scientist a good picture of the time response of the system to stimuli. The gold standard model is a floating point arithmetic model formulated in Matlab or Mathematica by the scientist. FPGA models are compared with the gold standard. In our FPGA implementation of the model, fixed point arithmetic is used to achieve speed up, but we select word size to achieve a maximum of 1% error of the fixed point solution vs. the gold standard floating point solution. We feel that this is reasonable since in many cases the values of input parameters are known only within 5% to 20% accuracy. FPGA Execution Scheme In our approach to execution we use explicit methods because with this approach there is a remarkable potential speedup using FPGAs. The reason is as follows. In explicit methods, cells exchange information from the previous iteration with their nearest neighbors. Thus the solution rate is the cell clock rate. A conservative value for an FPGA cell clock rate is 10 MHz [Xilinx, 2003]. Many FPGA designs today run at 100 MHz. The clock period varies inversely with cell complexity. Recent advances in integrated circuit technology have resulted in the fabrication of 27 GHz transistors [Lapedus, 2001 and Taur, 1999]. Thus, 20 GHz IC clock rates for future integrated circuits are likely. Explicit methods have the disadvantage that they may be unstable under certain conditions. However, in many cases this instability can be eliminated by choosing a smaller algorithm time step. In terms of FPGA implementation, this requires running longer, but if the FPGA clock speed is high, the execution time can still be low. However, it is best to have an explicit algorithm that is stable for all conditions. Below, we show how to achieve this. The Test Case As a test case involving several physical mechanisms in a heterogeneous material, we extend the thermal model for heterogeneous materials of Vick and Scott [1998] to include the effects of chemical reactions associated with a curing process. Consider the situation of heat transfer in a material consisting of a matrix with embedded, separated particles, as shown schematically in Fig. 2. The particles are separated and are small enough that the lumped capacity approximation is valid within each particle. The microstructure is characterized by the number density of the particles (number of particles per volume), N, and an average particle diameter, d. In addition a contact conductance, h, characterizes the contact between the particles and the matrix. The volume fractions of particles and matrix are N*Vp and (1-N)*Vp respectively, where Vp = π d/6 is the average volume of a typical particle. The physical mechanism of heat transfer consists of a two step process where energy is first absorbed at the surface by the matrix material. The energy is then conducted through the matrix and transferred to the particles in thermal contact with the matrix. In addition, the matrix is in the process of curing, causing an effective volumetric release of energy. With this physical model in mind, the energy equations for the matrix and particles as well as the rate equation for the curing reaction, assuming onedimensional heat flow can be expressed per total cross-sectional area as: