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In addition to their increased accuracy and robustness, highorder methods can lead to better utilization of modern multicore and heterogeneous computer architectures. We demonstrate this by combining several layers of parallelism in the software implementation of our numerical algorithms:
 MPIbased parallel finite elements in MFEM (domaindecomposed between CPUs)
 CUDAbased parallel kernels in BLAST (zonedecomposed on GPUs)
 OpenMPbased parallel kernels in BLAST (datadecomposed between CPU cores)
In particular, our hybrid MPI+CUDA+OpenMP highorder algorithms can take full advantage of parallel clusters with multicore processors and GPU accelerators on each node.
Parallel MPI Weak Scalability of a 3D Sedov Blast Simulation with Q2Q1 Finite Elements
MPI simulation up to 512 processors with fixed problem size per processor (approximately 512 Q2Q1 zones). Unstructured parallel domain decomposition and weak scalability are shown below.
Parallel MPI Strong Scalability of a 3D Noh Implosion Simulation with Q2Q1 Finite Elements
MPI simulation up to 512 processors with fixed total problem size. Unstructured parallel domain decomposition and strong scalability are shown below. Note that we get good performance on 512 processors with only 64 Q2Q1 zones per processor.
Comparing Hardware Utilization with HighOrder Finite Elements for 2Drz Sedov Problem
Parallel simulation on 48 processors (Intel Xeon EP X5660) with fixed number of unknowns, using Q1Q0, Q2Q1 and Q4Q3 finite elements on unstructured 2D mesh. Highorder finite element methods have much greater FLOP/byte ratios in 2D leading to improved hardware utilization.
Q1Q0  Q2Q1  Q4Q3  

no. cycles  4,799  5,017  4,660 
run time(s)  10.88  11.39  15.62 
no. qpts/zone  4  16  64 
no. floating point operations  1.00e11  1.54e11  4.75e11 
% floating point ops/total instructions  13.56%  16.4%  22.2% 
GFlops  9.2  13.5  30.4 
Comparing Hardware Utilization with HighOrder Finite Elements for 3D Sedov Problem
Parallel simulation on 256 processors (Intel Xeon E52670) with fixed number of unknowns, using Q1Q0, Q2Q1 and Q4Q3 finite elements on 3D mesh. Highorder finite element methods have much greater FLOP/byte ratios in 3D leading to improved hardware utilization.
Q1Q0

Q2Q1

Q4Q3



no. cycles  3,273  1,017  983 
run time (s)  34.90  14.43  76.95 
no. qpts/zone  8  64  512 
no. floating point operations  3.82e12  2.53e12  3.34e13 
% floating point ops/total instructions  19.42%  22.85%  26.84% 
GFlops  109.5  175.3  434.1 
Extreme Parallel MPI Strong Scalability of a 2D Sedov Simulation with Very HighOrder Finite Elements
Comparing extreme strong scaling results on 131,072 processors of LLNL’s Vulcan computer for the 2D Sedov test problem using two different approaches for highorder parallel finite element computations: our default approach where highorder matrices are assembled and applied globally (left) and a new “actionbased” approach where finite element matrices are computed and applied on the fly (right). The “action based” approach favors the very highorder methods (Q4 and beyond) by trading memory usage / bandwidth for inline floating point operations, resulting in a reduced memory foot print and improved run times for very highorder methods. For each case, the mesh size is fixed, leading to increased spatial resolution as the order of the finite elements is increased. Highorder finite element methods have much greater FLOP/byte ratios than traditional low order staggered grid hydro (SGH) methods, leading to better strong parallel scaling, including the ability to strongscale all the way down to a single computational zone per core.
Onn [Inline Matrix Computation / Multiplication] ode MPI/CUDA/OpenMP Scalability
Results from 2D Q3Q2 triplepoint shock interaction and 3D Q2Q1 Sedov blast simulations on a node of a parallel cluster with two Intel 6 Core Xeon CPUs and two NVIDIA M2050 GPU cards, as shown on the left below. The critical corner force routine in BLAST is parallelized by different methods, compared on the right below. In these settings, one GPU performs similarly to a sixcore CPU, and the hybrid MPI+CUDA+OpenMP approach is 2.5x faster compared to full MPI (12 tasks) on the node.
The HyperQ technology on the Kepler architecture allows multiple MPI tasks to simultaneously use the same GPU. Results from 3D Q2Q1 Sedov blast problem run on nodes of SNL’s Shannon testbed with a Sandy bridge CPU and 2 Tesla K20X GPUs show significant improvements (up to 4X) in the corner force kernel computation, as well as the overall BLAST simulation time.
Strong scaling results on up to 30 CPUs and 60 GPUs are also promising.