Moving mesh method

Hyman, J. M., Moving Mesh Methods for Partial Differential Equations, Mathematics Applied to Science, Academic Press, Inc. (1988), 129-153... 

The moving mesh method implies that the entire domain is translated as the train moves. This means that all results are presented for locations with constant relative distances from the train, but in a moving frame of reference. The results are presented below. 

There are two main categories of interface solvers. The first category of methods involve mesh motion and deformation while the second group are based on fixed grids. In the moving mesh methods, the grid points move according... 

Several different methods may be used to generate and adaptive mesh. Saltzman (1) categorized them as follows i) fixed meshes, ii) selectively refined meshes, and iii) moving meshes. 

An alternative and complementary method is to follow the deformation of the material by following the boundaries of a fluid domain, by either the front capturing or front tracking technique. In the former, marker particles are distributed over the fluid domain volume and tracked over time, with the surface of the domain being restored by interpolation techniques in the latter, a separate moving mesh is used to describe the interface between the subdomain and the rest of the volume (86-89). 

Murthy et al [63] were the first who used the sliding grid method for the simulation of unsteady flows in mixing vessels. The model formulation used with moving meshes is of the arbitrary-Lagrangian-Eulerian (ALE) t rpe [70]. In this particular approach the flow domain is divided into two cylindrical, non-overlapping sub-domains, each gridded as a separate block. The outer... 

In the macroscale, the particles can also be considered as moving mesh nodes, as it is so conceived in SPH [86,96] and moving particles semi-implicit method (MPS) [96]. Both of the algorithms belong to the wider class of approximation methods defined in Ref. [33]. The particle system is defined by mass m, distribution in space, where fluid density in r is given, by the approximation formula... 

Z. Tukovic and H Jasak. Simulation of fiee-rising bubble with soluble surfactant using moving mesh finite volume/aiea method. In 6th International Conference on CFD in Oil dc Gas, Metallurgical and Process Industries. SINTEF/NTNU, June 2008. 

Lagrangian-Eulerian (ALE) method. In the ALE technique the finite element mesh used in the simulation is moved, in each time step, according to a predetermined pattern. In this procedure the element and node numbers and nodal connectivity remain constant but the shape and/or position of the elements change from one time step to the next. Therefore the solution mesh appears to move with a velocity which is different from the flow velocity. Components of the mesh velocity are time derivatives of nodal coordinate displacements expressed in a two-dimensional Cartesian system as... 

Another implementation of homotopy-continuation methods is the use of problem-dependent homotopies that exploit some physical aspect of the problem. Vickeiy and Taylor [AIChE J., 32, 547 (1986)] utilized thermodynamic homotopies for K values and enthalpies to gradually move these properties from ideal to ac tual values so as to solve the MESH equations when veiy nonideal hquid solutions were involved. Taylor, Wayburn, and Vickeiy [I. Chem. E. Symp. Sen No. 104, B305 (1987)] used a pseudo-Murphree efficiency homotopy to move the solution of the MESH equations from a low efficiency, where httle separation occurs, to a higher and more reasonable efficiency. 

Remapping methods must also be able to move material into neighbor eells that share only a node. This is ealled eorner eoupling and is required to aeeurately move material diagonally aeross the mesh. For example, the flow shown in Fig. 9.15 is at a 30° angle and material from element zero will flow into elements one, two, and three. Elements one and two share a faee with element zero. But element three shares only a node with element zero. The multistep algorithm moves material from element zero to element one, then from element one to element three when the direetion is switehed. 

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