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Particle-spring systems
Published in Sigrid Adriaenssens, Philippe Block, Diederik Veenendaal, Chris Williams, Shell Structures for Architecture, 2014
Shajay Bhooshan, Diederik Veenendaal, Philippe Block
For small systems, a direct solving method such as Gaussian elimination or Cholesky decomposition suffices. For larger systems, typical implementations as those mentioned in Kilian and Ochsendorf (2005) use iterative methods such as the Conjugate Gradient Method (CG) or the Biconjugate Gradient Stabilized Method (BiCGSTAB). Given that the matrix A is symmetric, positive definite, CG is the most appropriate (Barrett et al., 1994) and employed by Baraff and Witkin (1998). BiCGSTAB, however, is mostly used for non-symmetric matrices, and would therefore constitute unnecessary overhead in our case.
A Quasi–Monte Carlo Method With Krylov Linear Solvers for Multigroup Neutron Transport Simulations
Published in Nuclear Science and Engineering, 2023
Sam Pasmann, Ilham Variansyah, C. T. Kelley, Ryan McClarren
For deterministic solutions, Source Iteration (SI) is the simplest and most common deterministic solution technique for solving the discrete ordinates method.3 SI is equivalent to a fixed-point Picard iteration and as problems become collision dominated, the convergence rate of the SI can become arbitrarily slow.4,5 More advanced iteration techniques such as Krylov subspace methods, including the Generalized Minimal RESidual method (GMRES) and the BiConjugate Gradient STABilized method (BiCGSTAB), have been shown to outperform standard SI, particularly when there are highly scattering materials.4 Nonetheless, as the dimensionality and fidelity of the problem increase, the deterministic quadrature techniques used to evaluate the system of equations become intractable.6,7
Implicit discrete ordinates discontinuous Galerkin method for radiation problems on shared-memory multicore CPU/many-core GPU computation architecture
Published in Numerical Heat Transfer, Part B: Fundamentals, 2021
As what would be demonstrated in Section 2, the implicit SNDG implementation fully transforms RTE approximation to the solution of a set of linear systems, each in general matrix form is The number of discrete ordinates determines the number of to solve for each iterative step. Meanwhile, spatial aspects of the problem, including dimension, geometry, mesh topology for spatial discretization, and so on, are all reflected in dimension (number of rows and columns), sparsity pattern, and algebraic condition of A. For most problems of realistic meaning, the associated are sparse and ill conditioned. Depending on the problem size, As could be very large as well. To solve with A of such characteristics, iterative algorithms such as the Krylov subspace based generalized minimal residual (GMRES) [10, 11] and biconjugate gradient stabilized method (BiCGSTAB) [10, 12, 13] are more economic and efficient than direct algorithms such as Gaussian elimination, in terms of intermediate data storage, computation time and parallel perspective [10, 11, 14]. This work has adopted the restarted GMRES. In the following text, the SNDG implementation that relies on GMRES for solution would be referred as SNDG/GMRES.