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A Synopsis of the Strategies and efficient Resolution Techniques Used for Modelling and Numerically Simulating the Drying Process
Published in Ian Turner, Arun S. Mujumdar, Mathematical Modeling and Numerical Techniques in Drying Technology, 1996
Furthermore, if the same problem is discretised using the CV-FE method or the CV-UM with hexagonal or octagonal control volumes the band structure of the resulting matrix system becomes more complex [Turner and Ferguson, 1995b, 1995c ] and the ADLSOR or SOR solution techniques become either no longer appropriate, or extremely inefficient. In these cases the more generalised solution method concerns the family of iterative schemes which are associated with the Conjugate Gradient Methods. Two of the more robust schemes such as the Generalised Minimal Residual Method (GMRES) [Saad, 1981; Saad and Schultz, 1985 ; Brown and Hindmarsh, 1989] or the Bi-Conjugate Gradient Method Stabilised (Bi-CGSTAB) [van der Vorst, 1992] are discussed in great depth within the literature. In this work the performance of the Bi-CGSTAB scheme with appropriate preconditioners based on Diagonal, Gauss Seidel, Successive Overrelaxation, Symmetric Successive Over-relaxation [Ajmani et al, 1994] or Incomplete LU decompositions [Behie and Forsyth, 1984; D'Azevedo, Forsyth and Tang, 1992] will be analysed. Note that for these classes of iterative methods the constraint of diagonal dominance is not essential for rapid convergence and this important feature renders these schemes to be attractive alternatives for the resolution of the systems obtained by the Newton outer iteration technique.
Complexities of the Molecular Conductance Problem
Published in Sergey Edward Lyshevski, Nano and Molecular Electronics Handbook, 2018
Gil Speyer, Richard Akis, David K. Ferry
For this calculation, the density from the transmission calculation must be projected onto the wave-functions for each atomic orbital centered at the accurate positions in three-dimensional space. The solver employed is the symmetric successive over-relaxation preconditioned bi-conjugate gradient stabilized algorithm (SSOR-BiCGSTAB) originally developed by van der Vorst [75]. To comply with the non-orthogonal mesh given by the unit cell from the energy spectrum code, the solver works over a 15-diagonal matrix. The applied bias is implemented as Dirichlet boundary conditions. It should be added that due to the low applied bias used in the experiments which were analyzed in this work, the self-consistent potential introduced fairly small corrections.
Measuring stiffness of soils in situ
Published in Fusao Oka, Akira Murakami, Ryosuke Uzuoka, Sayuri Kimoto, Computer Methods and Recent Advances in Geomechanics, 2014
Fusao Oka, Akira Murakami, Ryosuke Uzuoka, Sayuri Kimoto
Geotechnical finite element computations may often involve complex constitutive relations, and hence, nonlinear iterative methods such as the Newtontype iteration may be utilized to linearize the resultant nonlinear finite element equations (Sloan et al. 2000, Smith & Griffiths 2004). Traditionally, to solve the elasto-plastic finite element linear equations, the global elasto-plastic stiffness matrices were assembled and preconditioned when a Krylov subspace iterative method was employed (Borja 1991, Ferronato 2012). Recently, Gambolati et al. (2011) suggested that the elasto-plastic stiffness matrix be partitioned into 2×2 block according to the unknowns associated with linear FE nodes and those associated with nonlinear FE nodes (Chen et al. 2013). However, other studies indicate that separating elasto-plastic stiffness matrices into the elastic and the plastic part and then treating them separately may be more suitable for some applications. For instance, Chen & Cheng (2011) proposed an accelerated symmetric stiffness method based on the separated elastic and plastic stiffness matrices. Furthermore, Chen & Phoon (2012) examined the modified symmetric successive over-relaxation (MSSOR) preconditioners proposed by Chen et al. (2006) for the finite element analyses of consolidation problems involving associated and non-associated soil plasticity, respectively. With the non-symmetric IDR(s) solver developed recently, Tran et al. (2013) applied the zero-level fill-in incomplete Cholesky factorization (i.e. IC0) to the global elastic stiffness matrix based on the separated elastic and plastic stiffness matrices in FEM analysis of shallow foundation. Numerical results show that the proposed scheme is efficient for those geotechnical applications with non-associated soil plasticity and low yielding percentage.
Aerodynamic optimisation of a parametrised engine pylon on a mission path using the adjoint method
Published in International Journal of Computational Fluid Dynamics, 2019
Damien Guénot, François Gallard, Joël Brézillon, Yann Mérillac
The spatial convective fluxes of the mean flow are discretized with the upwind Roe scheme (Roe 1981) with Harten's entropic correction. A MUSCL scheme (Monotone Upstream-centred Schemes for Conservation Laws) (van Leer 1979) associated with a Van Albada limiter (Van Albada, van Leer, and Roberts 1982) provides a second-order accurate scheme. The spatial convective fluxes of the turbulent flow are discretized with the first-order upwind Roe scheme. Spatial diffusive fluxes are approximated with a second-order central scheme. The turbulent equations are solved separately from the mean flow equations at each time step with the same time-marching method. The backward Euler implicit scheme drives the time integration. The resulting linear systems are solved with the scalar Lower–Upper Symmetric Successive Over-Relaxation (LU-SSOR) method (Yoon and Jameson 1988). A standard nonlinear multigrid algorithm (Jameson 1982) combined with local time stepping accelerates the convergence to steady-state solutions.
The DTEQ Code for Toroidal MHD Equilibria with Diamagnetic Current Modeling Using the deal.II Finite Element Library
Published in Fusion Science and Technology, 2019
K. S. Han, B. H. Park, A. Y. Aydemir, J. Seol
where the matrix is and the right side is . In this problem, comes from a Laplace-like equation with spatial variable coefficients that lead to symmetric positive definite matrix. We choose conjugate gradients solver with symmetric successive over relaxation preconditionor to solve Eq. (5).
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
According to Saad (2003), for the Block Jacobi, Block Gauss-Seidel, Block SOR (Successive Over Relaxation), and Block SSOR (Symmetric Successive Over Relaxation) iterations, the preconditioning matrices are, respectively as follows: