Multigridding techniques

The optimal choice of preconditioner will ultimately depend on the computer architecture, in as much as some are more readily vectorizable or parallelizable. For example, the initial incomplete Cholesky decomposition methods work well on serial machines, but the forward and backward substitutions are not vectorizable. Simpler decompositions, such as diagonal scaling, run faster on machines like the Cray YMP. More complicated, vectorizable variations of the incomplete Cholesky decompositions have been developed (see, e.g., ref. 24) and are currently under investigation for their applicability to problems in biomolecular electrostatics. Studies of multigridding techniques are also very exciting. 

The multigrid technique can powerfully save considerable CPU time in the direct SCF procedure. 

The general numerical method of solution is described in Sec. V. In the Debye-Hiickel approximation, all the equations are linear. First, the equilibrium potential is solved second, the ionic potential and the velocity field are alternately obtained in an iteration loop. These three systems are solved by means of classical iteration schemes such as the conjugate gradient for the potentials and a multigrid technique for the velocity. General estimates of the expected precision are given. 

The use of multigrid techniques is advantageous because the computational work per time step is reduced and because large volumes in coarse meshes can result a quick overall balance of the solution [3]. 

BRANDT A. (1984), "Multigrid Techniques 1984 guide with applications to fluid dynamics". Monograph. Available as G.H.D.-studle No. 85, from G.M.D.-FIT, postfach 1240, D-5205, St. Augustin 1 W.-Germany. 

The author has found several basic sources to be instructive, especially in the early stages of learning about multigrid techniques. This is strictly a personal bias, but the book by Briggs, Henson, and MeCormick and the early review studies by Brandt have been particularly helpful and are recommended. They lay out both the theory and applications in clear language and are less mathematically oriented than some other multigrid sources. 

A. Brandt, Multigrid Techniques 1984 Guide with Applications to Fluid Dynamics,... 

Ai, X and Cheng, H S, 1993, "A transient EHL analysis for line contacts with measured surface roughness using multigrid technique", ASME Journal of Tribology, Paper No 93-Trib-56. 

Detailed understanding of the performance of elastohydrodynamically lubricated contacts can be gained through theoretical studies. The first theoretical studies concerning elastohydrodynamic lubrication took place in the 1940s [1-2]. More recently, multigrid techniques have successfully been introduced as a fast and efficient numerical method to simulate EHL line and point contact... 

These thermal effects have been extensively studied for many years since the initial theoretical work by Crook [2] in 1961, with numerical methods for the thermal solutions in line contacts developed in the 1960 s by Sternlicht [3], Cheng and Sternlicht [4], Cheng [5] and by Dowson and Whitaker [6]. Recent papers, for example, those by Lee ei al [7], [8], by Kim et al [9,10] and by Kazamaet al [11] have utilised the multigrid techniques applied to EHL problems by Venner and... 

In this paper the transient EHL model developed in the CPDE Unit at Leeds, [13,14] using the multigrid techniques applied to EHL problems by Venner and Lubrecht [12] is extended to include thermal effects [16]. This involves extensions to the non-Newtonian fluid model [16] governing the density and viscosity and introducing three additional equations to the code, one for mean temperature and one for each of the two contact surfaces. Inclusion of these into the equation set for the line contact case will be discussed along with the solution method. An extension to the point contact case is also described. 

The equations, excluding the thermal equations, are solved using multigrid techniques in in the standard manner as described in [21,20,22] and utilising the multilevel multi-integration algorithm of Brandt and Lubrecht [23]. 

Multigrid methods have proven to be powerful algorithms for the solution of linear algebraic equations. They are to be considered as a combination of different techniques allowing specific weaknesses of iterative solvers to be overcome. For this reason, most state-of-the-art commercial CFD solvers offer the multigrid capability. 

Fio. 4. Types of multiscale modeling and solution strategies. Hybrid models (one model at each scale) apply well when there is separation of scales (onion or nested-type models). When there is lack of separation of scales, mesoscale models need to be developed where the same technique (e.g., MD or MC) is accelerated. Alternatively, multigrid (heterogeneous) hybrid models can be employed where the unresolved degrees of freedom are determined from a finer scale model and passed to a coarser scale model. 

The PB equation may be solved numerically for macromolecules (for reviews, see References 36-38. The finite difference, finite element, and multigrid methods are used most commonly to solve the PB equation. Usually, this technique is performed by mapping the molecules onto a three-dimensional cubic grid. To solve the PB equation, a suitable interior relative dielectric constant and definition of the dielectric boundary should be assigned (39, 40). 

Linear-scaling techniques for solving Poisson s equation include multigrid approaches [58-61] and fast Poisson solvers [62]. 

Once we have stated or derived the mathematical equations which define the physics of the system, we must figure out how to solve these equations for the particular domain we are interested in. Most numerical methods for solving boundary value problems require that the continuous domain be broken up into discrete elements, the so-called mesh or grid, which one can use to approximate the governing equation (s) using the particular numerical technique (finite element, boundary element, finite difference, or multigrid) best suited to the problem. 

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