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Optimization of Finite-Element Software Systems on Supercomputers
Published in Hojjat Adeli, Supercomputing in Engineering Analysis, 2020
The nested dissection algorithm uses the concept of the separator. A separator is a set of nodes in the graph that, if removed, disconnects the graph into disjoint parts. The algorithm is executed recursively as follows: Find a separator with the minimum number of nodes in it.Number the nodes of the separator in any order.Remove the separator from the structure.Repeat the steps on all substructures.
Numerical Methods for Large-Scale Electronic State Calculation on Supercomputer
Published in Klaus D. Sattler, st Century Nanoscience – A Handbook, 2019
Takeo Hoshi, Yusaku Yamamoto, Tomohiro Sogabe, Kohei Shimamura, Fuyuki Shimojo, Aiichiro Nakano, Rajiv Kalia, Priya Vashishta
To parallelize the sparse-direct solver, we can use the nested dissection ordering. To be concrete, let us consider the Poisson equation discretized on a two-dimensional grid using 5-point finite difference formula. Assume that the grid is partitioned into two regions A and B and a border S (called separator) in such a way that the grid points in A and B are not directly connected. Further, assume that the grid points in region A are numbered first, those in B are numbered next, and those in S are numbered last (Figure 15.5a). Then, it is easy to see that the resulting coefficient matrix has the bordered block diagonal form as shown in Figure 15.5b. Now, apply Gaussian elimination (which is equivalent to the Cholesky decomposition) to the first block row of A, which corresponds to the grid points in region A. Then, it is observed that the second block row, corresponding to the grid points in B, is not modified due to this elimination (Figure 15.5c). This means that we do not need to wait until the first block row has been eliminated to start the elimination of the second block row. In other words, the elimination of the first block row and the second block row can be done in parallel. This is the source of parallelism of the nested dissection ordering. Note that the last block row corresponding to the separator S can be eliminated only after both the first and the second block rows have been eliminated. Thus, this part is sequential in nature.
Topology optimization of repetitive near-regular shell structures using preconditioned conjugate gradients method
Published in Mechanics Based Design of Structures and Machines, 2022
A. Kaveh, M. Pishghadam, A. Jafarvand
The focus here is on the effective solution of the linear system of equations Ku = f arising from a finite element discretization of the state problem. The stiffness matrix is large, sparse and positive definite. The solution u can be obtained using either direct methods (e.g., Davis 2006) or iterative methods (Saad 2003). For full matrices, the computational cost of direct Cholesky decomposition is of while for sparse matrices direct methods such as nested dissection require operations, where is the size of the stiffness matrix (Davis 2006). For small problems, particularly in 2D problems, direct solvers are often preferable due to their robustness. However, solution time and memory requirements become prohibitive for 3D problems as well as for large-scale 2D problems. For these cases, the sparsity of the stiffness matrix results in an inexpensive matrix vector product operation, which can effectively be utilized by iterative solution algorithms.