Direct methods for sparse matrices
Direct methods for sparse matrices
Ordering methods for preconditioned conjugate gradient methods applied to unstructured grid problems
SIAM Journal on Matrix Analysis and Applications
Orderings for Parallel Conjugate Gradient Preconditioners
SIAM Journal on Scientific Computing
Orderings for Incomplete Factorization Preconditioning of Nonsymmetric Problems
SIAM Journal on Scientific Computing
Incomplete Cholesky Factorizations with Limited Memory
SIAM Journal on Scientific Computing
Orderings for Factorized Sparse Approximate Inverse Preconditioners
SIAM Journal on Scientific Computing
Numerical Linear Algebra for High Performance Computers
Numerical Linear Algebra for High Performance Computers
Computer Solution of Large Sparse Positive Definite
Computer Solution of Large Sparse Positive Definite
The effect of orderings on sparse approximate inverse preconditioners for non-symmetric problems
Advances in Engineering Software - Engineering computational technology
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Crout Versions of ILU for General Sparse Matrices
SIAM Journal on Scientific Computing
A Block FSAI-ILU Parallel Preconditioner for Symmetric Positive Definite Linear Systems
SIAM Journal on Scientific Computing
Parallel inexact constraint preconditioners for saddle point problems
Euro-Par'11 Proceedings of the 17th international conference on Parallel processing - Volume Part II
Adaptive Pattern Research for Block FSAI Preconditioning
SIAM Journal on Scientific Computing
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The Finite Element Method (FEM) is widely used in civil and mechanical engineering to simulate the behavior of complex structures and, more specifically, to predict stress and deformation fields of structural parts or mechanical bodies. In the former case, the coupling between different types of elements, such as beams, trusses, and shells, is often required, while in the latter fully 3D discretizations are typically used. For both, FEM leads to symmetric positive definite (SPD) matrices that, depending on the type of discretization and especially on the topology of the nodal connections, may be efficiently solved by either the Preconditioned Conjugate Gradient (PCG) or a direct solver such as the routine MA57 of the Harwell Software Library. Numerical experiments are shown and discussed where the effect of spatial discretization, different solution techniques, and a possible nodal reordering, is explored. The PCG preconditioner used is a variant of the incomplete Cholesky factorization with variable fill-in. It is shown that for structures with 1D or 2D connections, such as for example a bridge, MA57 performs usually better than PCG. In this case it is noted that some reorderings specifically designed and implemented for direct elimination methods can be very helpful for PCG as well as they yield a cheaper preconditioner and lead to a much faster PCG convergence. The main disadvantage is the need for an appropriate degree of fill-in for the preconditioner which turns out to be problem dependent and must be found empirically. However, in fully 3D problems, arising for example from the FE discretization of structural components or geomechanical structures, PCG outperforms MA57 while also requiring much less memory, and thus allowing for the use of much refined grids, if needed. With the aid of a large geomechanical problem it is shown that direct solvers may not be (even) used on serial computers due to their prohibitive computational cost with PCG the only viable alternative solver.