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Numerical integration on GPUs for higher order finite elements
Banaś K., Płaszewski P., Macioł P. Computers & Mathematics with Applications67 (6):1319-1344,2014.Type:Article
Date Reviewed: Oct 22 2014

Finite element methods (FEMs) are numerical methods for finding approximate solutions to partial differential equations. When a problem is presented, a mesh and a finite element space are required to be defined. Regular choices for meshes are tetrahedral meshes and hexahedral meshes. Recently, adaptive techniques are widely adopted. The finite element space is defined by the mesh types and the shape functions. The solution of the classic FEMs consists of three steps, including numerical integration on each element, assembly of the global linear system, and finding the solution to the linear system. The numerical integration and the solution of the global linear system occupy most of the computation time. Usually, Gaussian integration techniques are employed for numerical integration. When the order of the quadrature formula is low, the numerical integration is fast and the cost is low. However, when the order of the formula is high, such as formulas for hp-adaptive FEMs, the numerical integration could dominate the whole computation.

This paper describes work on developing fast integration techniques using graphics processing units (GPUs). The GPUs are designed for display, and since each pixel on the screen is independent from each other and they can be processed simultaneously, the GPUs have highly parallel architectures, which makes them proper devices for computation. Now GPU computing is becoming more and more popular.

In this paper, the formulation problem is studied on the reference element, in which unified formulas and algorithms are presented. The programming models and GPU architectures are then presented, as well as basic performance optimization techniques. For GPU computing, CUDA and OpenCL are the two most popular programming models. CUDA is developed by NVIDIA, which supports GPUs from NVIDIA only. The advantages of CUDA are its efficiency and its simplicity. OpenCL is supported by more companies, such as Apple, Intel, NVIDIA, and AMD. OpenCL targets a broader hardware, including central processing units (CPUs) and GPUs from AMD and NVIDIA. It also exposes more details of the hardware and offers opportunities to implement aggressive optimization techniques. In the end, high-performance computation kernels for high order numerical integration are designed. The numerical experiments show that GPU performance is lower than CPU when the order of integration is low. When the order is greater than five, the GPUs are much faster than the CPUs. The results indicate that the techniques developed by this paper should be applied to high order numerical integrations only. The results also show the time for transfer of output data and the time for preprocessing are comparable with the calculation time, which indicate the whole finite element computation should be performed within the GPUs and communications should be avoided. Special techniques should be developed to avoid preprocessing.

The paper studies fast integration techniques for high order numerical integration on GPUs. The performance of the GPU codes is high and is much faster than CPUs. The algorithms are helpful to researchers who study higher order FEMs and for problems when high accuracy integrations are required.

Reviewer:  Hui Liu Review #: CR142851 (1501-0076)
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Finite Element Methods (G.1.8 ... )
 
 
Graphics Processors (I.3.1 ... )
 
 
Parallel And Vector Implementations (G.4 ... )
 
 
General (C.1.0 )
 
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