Grid technique

A finite difference formula is used to estimate the second derivatives of the coordinate vector with respect to time and S is now a function of all the intermediate coordinate sets. An optimal value of S can be found by a direct minimization, by multi-grid techniques, or by an annealing protocol [7]. We employed in the optimization analytical derivatives of S with respect to all the Xj-s. 

Pyrolysis experiments were performed using a heated wire grid technique. Reactor details have been reported previously [7, 8]. The technique has been successfully applied in both pyrolysis and gasification studies [8, 10, 11]. In brief, the mesh or grid is housed in a stainless steel chamber known as the grid reactor (Fig. 8.1). The reactor is 227 mm long and has an inner diameter of 15 mm. The grid (9x4 mm) is constructed of interwoven wires (platinum/rhodium 10%)... 

Location defined in Table 1 based on critical sampling site determination by grid technique. 

To identify sampling points for the monitoring particulate count (nonviable) by generating data for nonviable particulate matter in class 100 filling and lyophilization room by grid technique. 

Attarakih M, Bart HJ, Faqir NM. An approximate optimal moving-grid technique for the solution of discretized population balances in batch systems. Proceedings of ESCAPE 12 European Symposium on Computer-Aided Process Engineering, The Hague, 2002. 

Typically, the numerical solutions techniques used are very specific to the problem. Particularly challenging problems include moving front problems where concentration profiles, for example, may vary widely over a short distance but may not change much at other spatial locations. The spatial discretization must be small close to the front for accuracy and numerical stability, but must be larger at other locations to reduce computation time. Various adaptive grid techniques to change the spatial step sizes have been developed for these problems. One of the more common codes to solve fluid-flow-related problems is FLUENT. 

Voller, V.R., Swaminathan, C.R. and Thomas, B.G. (1990). Fixed grid techniques for phase-change problems - a review. International Journal for Numerical Methods in Engineering, 30(4) 875-898. 

In the H30 /D30 experiments in CRYRING/ also used to exemplify the grid technique, the results were ... 

For multimode problems, it is sometimes advantageous to use a dual grid technique in order to minimize the computational expense associated with storing and evaluating view factors. A course mesh can be used for the radiation heat transfer, while finer meshes can be used for the conduction and/or convection heat transfer. This technique is discussed in detail, and associated computational error (which is small) is reported in Zhao [178]. 

The direct-shadow method has been employed in the study of free convection by many authors following Benard, and unfortunately has often been mislabelled as a schlieren method. Levengood (L2) in a study of evaporative convection in pools of methyl alcohol, and Hickman (H2) in a study of surface behavior of boiling liquids, used modifications of Benard s deformed grid technique. 

Note that similar methods can be applied also to the solution of systems arising from the time-dependent heat conduction - convection problems. There are also other space decomposition methods. Let us mention the displacement decomposition technique for solving the elasticity problems and composite grid technique, for solving problems, which need a local resolution. More details can be found in Blaheta (2002) and Blaheta et al. (2002b). 

Interpolation or gridding of sparse dataset may help to recover missing data points and use conventional FT processing. This, however, may lead to significant disturbances if the simplest, pol3momial interpolation is used [77]. More advanced gridding techniques are helpful here [78]. 

Recently Yao and Chu have developed complex-scaling grid techniques which can be used to discretize the time-independent Floquet Hamiltonian... 

L. K. Bieniasz and C. Bureau. Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations Part 7. Testing of the finite-difference patch-adaptive strategy on example kinetic models with moving reaction fronts, in one-dimensional space geometry, J. Electroanal. Chem. 481, 152-167 (2000). 

As discussed throughout the previous chapters, for the calculation of the concentration profiles at a given timestep, fc, we need the values of the concentrations at the same spatial nodes at the previous timestep A — 1. However, because the grids for k and k— are in general different in adaptive grid techniques, the values at the nodes of the new grid need to be... 

Gridless discrete-particle methods have a few significant advantages over grid techniques. These advantages can be enumerated as follows ... 

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