The Parallel Implementation Of PCG Method For Structural Static Reanalysis On Cluster

In structural design or optimization, the procedures are generally iterative and require repeated analysis as the structure is progressively modified. For each trial design, the analysis equations must be solved and the corresponding analyses usually involve much computational effort. Reanalysis methods are intended to analyze efficiently structures that are modified due to changes in design. The object of reanalysis is to evaluate the structural response for successive modifications in the design without solving the set of the modified implicit equations so that the computational cost is significantly reduced.With the development of science and technology, many engineering problems are of large-scale and require high-precision calculations. But traditional calculation methods are often powerless, we need research on parallel algorithms to solve these problems. In this paper, the author presents a parallel algorithm of PCG method for structural static reanalysis on cluster.This paper is consisted of four parts. In Chapter 1, we introduce the engineering back-ground of reanalysis problems and describe status all over the world in this research field. Parallel computation based on MPI is introduced in Chapter 2, and then we construct a cluster computing system based on MPI by using MPICH. The major research work is composed of Chapter 3 which presents a parallel algorithm of PCG method of structural static reanalysis. In Chapter 4, we analyze the algorithm by utilizing the numerical experimental results, and suggest some improvements.The main ideas of the algorithm are as follow.Step 1. Considering the load balancing,at first, we calculate the number m of matrix rows which are stored in each processor, and then the matrix is divided into small pieces which are stored in each processor.Step 2. Solve K₀ z=r. We take equation U₀^T x =r ( U₀^TU₀ = K₀)as an example. First, x₀ , x₁ ,…, x_n-1 are calculated by the processors, respectively. Once x_i calculated, it is immediately broadcast to all processors. Multiplying the corresponding coefficients and summing up yield the solution. Then,ρ_k-1=r^Tz can be calculated.Step 3. All processors (my_rank=0,…, p - 1) implement the following algorithm at the same time(i=0,…, p - 1): if k =1 p = z elseβ_k =ρ_k-1/ρ_k-2 p = z+β_kpStep 4. Compute w = Kp. To reduce transmission time, we increase use of the memory, therefore p is stored in each processor, but K is allocated to different processors. In such a way, much time in the transmission can be reduced.Step 5. At first,α_k =ρ_k-1/p^Tw is calculated. And then, the following algorithm at the same time(i=0,…, p - 1)is implemented: for j=0 to m d = d+α_kp r = r-α_kw end forStep 6, Print the results and calculation time.When the number p of processors increases, efficiency and speedup of the algorithm decrease. The main reason is that communication time increases. When p keeps unchanged, speedup and efficiency of the algorithm increase with the increase of the size of the problem. Because the proportion of communication and synchronization in total compution decreases. It also demonstrates the advantages of parallel compution.The parallel algorithm of PCG. in this paper has the merits of less communication, 1ower time complexity. In addition, parallel implementation is also very easy under the Cluster environment.Experimental results show that good parallel results have been obtained using the algorithm.However, the algorithm we designed can be further improved. When the size of problem is very big , system memory needs more time to read the data even memory overflows. It shows that the algorithm depends on the allocation of resources. The main reason is that we have adopted a shared memory model, all of data should be read into memory. If using the distributed memory, speedup and efficiency of parallel algorithm will yield better results.