For FEM solution of complex hydraulic structures, parallel method is usually needed to deal with large amount of numerical computations. In this paper, the authors present an EBE-PCG solver which adopt Jacobi-conditioned conjugate gradient algorithm. For data exchange, a working scheme in which only necessary data is needed to gather and scatter is submitted to make EBE method available for distributedmemory parallel computers. This method could dramatically reduce data exchange and consequently improve efficiency of parallel computing. At the same time, it is very easy to reach load balance for solving three-dimensional FEM. For any complicated three-dimensional structures, the algorithm can be convenient for automatically allocate the task in parallel computing. Based on this algorithm, the authors developed PFEM code using MPICH and C/C++ language. Numerical example of an Arch dam-foundation system demonstrates that PFEM is applicable for complicated three-dimensional underground structures.


In hydraulic engineering problems, large scale numerical analysis of geological structures and other relevant details increases the demand for high-speed and large scale computing. To meet this demand, parallel computing methods are proposed to increase the calculational scale and shorten computing time. Parallel methods of solving linear systems of equations and domain decomposition method are both popular approaches to finite element parallel computations. When the size of structural analysis problems increases, the requirement for the capacity of memory storage will increase greatly because of the storage and assembly of global stiffness matrix. If the analysis model is regular in geometric shape, it is suitable to adopt domain decomposition method. However, in case of three-dimensional model with complicated geometrical shapes, the process of domain decomposition itself will become the bottleneck in computation.

The finite element EBE method can solve the above problems very well. The basic idea is to convert the calculation of the vector product of an whole matrix into that of the vector product of a group of element stiffness matrices. During computation in EBE method, the global stiffness matrix and global force vector won't be assembled. All the computation occurs at element level. As a result, the entire process of finite element computation can be paralleled. The finite element EBE idea was initially proposed by Hughes T. J. R. in 1983 (Hughes 1983a), and applied to the problems of heat-conduction analysis. Before long, Hughes T. J. R. applied it to the Problems of Structural and Solid Mechanics (Hughes 1983b). Law K. H. (Law 1986) developed the ESE conjugate gradient solution procedures that can be used on distributed-memory parallel computers. Carey G. F. and Barragy E systematically studied the EBE-PCG (Carey 1988), discussed the parallelism of EBE-PCG method. Bova & Carey (1998) and Gullerud & Dodds (2001) suggested the approaches of combining ESE method and domain decomposition method together, and achieved good efficiency of parallel computation when implementing them on distributed-memory parallel computers.

In this paper, a new type of parallel computation algorithm of finite element ESE method is introduced based on Jacobi preprocessing conjugate gradient method.

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